This paper presents closed-form evaluations of two new Apéry-like series of weight $4$ that involve harmonic numbers of the form $H_{2k}$. Several key results are derived and subsequently used to establish connections to the main series.

In this paper, we construct a bisymmetric, strictly increasing, symmetric binary operation $F$ on an interval which is not continuous. This answers a natural question in the study of bisymmetric and mean-type operations by showing that continuity may fail even for non-reflexive operations of the form \[ F(x,y)=f^{-1}\bigl(\alpha f(x)+\beta f(y)\bigr), \] where $f$ is a bijection between an interval and a perfect, nowhere dense fractal-type set, and $\alpha,\beta>0$ with $\alpha+\beta\neq 1$. As a consequence, we also obtain a noncontinuous, associative, strictly increasing, symmetric operation on an interval. We generalize these constructions to the multivariate case. We also prove a complementary result: if a symmetric, bisymmetric, strictly increasing operation is reflexive at the endpoints of an interval, then it must be continuous and coincide with a quasi-arithmetic mean on that interval.

PRIMES STEP is a mathematical outreach program established at MIT in 2015. STEP students study advanced topics beyond the school curriculum and conduct group research projects, often leading to publication. This article discusses the program's history, admissions process, lesson organization, interactive teaching style, and teamwork, and provides advice on how to choose research projects, encourage students, and keep them engaged. This paper would be useful for math teachers and instructors in after-school math programs.

We construct a generalisation of what we call Bureau-Guillot systems, i.e. systems of first order equations with coefficient functions being Painlevé transcendents. The same Painlevé equation is related to the system and it appears as regularising condition in the regularisation process. The systems considered are birationally equivalent to the Okamoto polynomial Hamiltonian systems with rational coefficients for Painlevé equations, hence they possess the Painlevé property. This work extends the results of Bureau-Guillot in a two-fold way. On one side, we consider polynomial systems with degree larger than 2 that are free of movable critical points. These systems contain not only transcendents $\text{P}_{\text{I}}$ and $\text{P}_{\text{II}}$ in the coefficients, but also transcendents $\text{P}_{\text{III}}$, $\text{P}_{\text{IV}}$, $\text{P}_{\text{V}}$ and $\text{P}_{\text{VI}}$ (and/or their derivatives). On the other side, we explore examples of rational systems with the Painlevé transcendents in the coefficients birationally equivalent to the Okamoto polynomial systems. Lastly, we present a simpler version of the change of variables to obtain the analogues of the Bureau-Guillot systems. In this framework we discuss generalisations including the mixed case: systems related to the equation $(\text{P}_{\text{J}})$, with ${\text{J}=\text{I}, \dots, \text{VI}}$, containing coefficient functions that are solutions to $(\text{P}_{\text{K}})$ with $\text{K}\neq \text{J}$. In the latter, the equation $(\text{P}_{\text{K}})$ appears as a regularising condition during the regularisation process. Although we are primarily interested in systems possessing the Painlevé property, we also briefly discuss an analogous construction for systems including coefficient functions solving quasi-Painlevé equation.

It is known that every real orthogonal matrix can be brought into the domain of the Cayley transform by multiplication with a suitable diagonal signature matrix. In this paper we provide a constructive and numerically efficient algorithm that, given a real orthogonal matrix $U$, computes a diagonal matrix $D$ with entries in $\{\pm1\}$ such that the Cayley transform of $DU$ is well defined. This yields a representation of $U$ in the form \[ U = D(I-S)(I+S)^{-1}, \] where $S$ is a skew-symmetric matrix. The proposed algorithm requires $O(n^{3})$ arithmetic operations and produces an explicit quantitative bound on the associated skew-symmetric generator. As an application, we show how this construction can be used to control singularities in Cayley-transform-based optimization methods on the orthogonal group.

We establish anisotropic uncertainty principles (UPs) for general metaplectic operators acting on $L^2(\mathbb{R}^d)$, including degenerate cases associated with symplectic matrices whose $B$-block has nontrivial kernel. In this setting, uncertainty phenomena are shown to be intrinsically directional and confined to an effective phase-space dimension given by $\mathrm{rank}(B)$. First, we prove sharp Heisenberg-Pauli-Weyl type inequalities involving only the directions corresponding to $\ker(B)^\perp$, with explicit lower bounds expressed in terms of geometric quantities associated with the underlying symplectic transformation. We also provide a complete characterization of all extremizers, which turn out to be partially Gaussian functions with free behavior along the null directions of $B$. Building on this framework, we extend the Beurling-Hörmander theorem to the metaplectic setting, obtaining a precise polynomial-Gaussian structure for functions satisfying suitable exponential integrability conditions involving both $f$ and its metaplectic transform. Finally, we prove a Morgan-type (or Gel'fand--Shilov type) uncertainty principle for metaplectic operators, identifying a sharp threshold separating triviality from density of admissible functions and showing that this threshold is invariant under metaplectic transformations. Our results recover the classical Fourier case and free metaplectic transformations as special instances, and reveal the geometric and anisotropic nature of uncertainty principles in the presence of symplectic degeneracies.

In this paper we prove that the first Dirichlet eigenvalue $\lambda_1^N$ of an $N$-sided regular polygon of fixed area is a monotonically decreasing function of $N$ for all $N \geq 3$, as well as the monotonicity of the quotients $\displaystyle \frac{\lambda_1^{N}}{\lambda_1^{N+1}}$. This settles a conjecture of Antunes-Freitas from 2006 [P. Antunes, P. Freitas, Experiment. Math., 15(3):333-342, 2006].

The maximization of reach-avoid probabilities for stochastic systems is a central topic in the control literature. Yet, the available methods are either restricted to low-dimensional systems or suffer from conservative approximations. To address these limitations, we propose control architectures that combine the flexibility of Markov Decision Processes with the scalability of Model Predictive Controllers. The Model Predictive Controller tracks reference signals while remaining agnostic to the stochasticity and reach-avoid objective. Instead, the reach-avoid probability is maximized by optimally updating the controller's reference online. To achieve this, the closed-loop system, consisting of the system and Model Predictive Controller, is abstracted as a Markov Decision Process in which a new reference can be chosen at every time-step. A feedback policy generating optimal references is then computed via Dynamic Programming. If the state space of the system is continuous, the Dynamic Programming algorithm must be executed on a finite system approximation. Modifications to the Model Predictive Controller enable a computationally efficient robustification of the Dynamic Programming algorithm to approximation errors, preserving bounds on the achieved reach-avoid probability. The approach is validated on a perturbed 12D quadcopter model in cluttered reach-avoid environments proving its flexibility and scalability.

We consider a random walk $(Y_N)_{N\geq 0}$ on $\mathbb{R}^2$ generated by successively applying independent random isometries, drawn from a fixed measure $\mu$, to the point $0$. When the support of $\mu$ is finite and includes an irrational rotation satisfying a Diophantine condition, we establish a local central limit theorem (LCLT) for $Y_N$ down to super-polynomially small scales. When $\mu$ includes rotations satisfying a further algebraic condition, we prove that a LCLT holds down to the scale $\exp(-cN^{1/3}/(\log N)^2)$. Due to group-theoretic obstructions, this is sharp for symmetric $\mu$, up to the $\log$ factor. Lastly for a special class of asymmetric $\mu$, we obtain an LCLT down to the much finer scale $\exp(-cN^{1/2})$. The proofs relate the fine-scale distribution of $Y_N$ to a question about the values of integer polynomials on the unit circle.

Orthogonal polynomials appear naturally in the study of compositions of Möbius transformations. In this paper, we consider several classes of orthogonal polynomials associated to non-autonomous perturbations of a parabolic Möbius map. Our results can be viewed as instances of non-autonomous parabolic implosion, including a random perturbative regime in which convergence holds almost surely.

We extend bounds, proved by R.C. Thompson in 1966, on the sum of the $j$-th largest eigenvalues of the $(n-1) \times (n-1)$ principal matrices of an $n \times n$ Hermitian matrix. Our bounds are stronger than just summing up Thompson's bounds. We achieve the extensions as a corollary of a more general result giving bounds on the zeros of the generalized derivatives of polynomials with real roots. We use the extended bounds to obtain majorization relationships between the eigenvalues of all $m \times m$ principal matrices of an $n \times n$ Hermitian matrix. These majorization relationships imply both a well-known majorization result by Schur and the well-known Szasz's inequalities.

This paper establishes a novel combinatorial framework at the intersection of Lie theory and algebraic combinatorics, based on a generalization of the Kostant game. We begin by reviewing the foundations of root systems, the classification of Dynkin diagrams, and the structure of Weyl groups. Subsequently, we analyze the original Kostant game as a tool for generating positive roots, demonstrating its unique termination on simply-laced diagrams and its role in an alternative classification thereof. The main contribution of this work -- which, to our knowledge, has not been studied before -- is a multi-vertex generalization of the game that allows for the simultaneous modification of multiple vertices of a Dynkin diagram. We prove that the resulting configurations of this new game establish a natural bijection with the elements of the quotient W/W_J of Weyl groups by parabolic subgroups. This formalism is applied to problems in algebraic geometry, specifically addressing cases of the Mukai conjecture via Hilbert polynomials, and is accompanied by a computational implementation in Java. These results offer new combinatorial perspectives for studying root counting problems, the regularity of reduced word languages, and the construction of Young Tableaux.

We develop Algebraic Phase Theory (APT), an axiomatic framework for extracting intrinsic algebraic structure from phase based analytic data. From minimal admissible phase input we prove a general phase extraction theorem that yields algebraic Phases equipped with functorial defect invariants and a uniquely determined canonical filtration. Finite termination of this filtration forces a structural boundary: any extension compatible with defect control creates new complexity strata. These mechanisms are verified in the minimal nontrivial setting of quadratic phase multiplication operators over finite rings with nontrivial Jacobson radical. In this case nilpotent interactions produce a finite filtration of quadratic depth, and no higher degree extension is compatible with the axioms. This identifies the radical quadratic Phase as the minimal example in which defect, filtration, and boundary phenomena occur intrinsically.

Motivated by an ongoing collaboration with clinical oncologists and pathologists, we develop a hybrid partial differential equation--ordinary differential equation (PDE--ODE) framework that captures (i) competition between susceptible and resistant phenotypes, (ii) stromal state switching, and (iii) a clinically realistic open-loop, single-dose therapeutic agent $I$ with diffusion and clearance. Clinical management of solid tumors is increasingly limited by spatial heterogeneity and therapy-induced resistance niches that are difficult to predict from well-mixed models. We establish a rigorous mathematical backbone with forward invariance of the nonnegative cone and global-in-time well-posedness. Exploiting the decoupled drug equation $\partial_t I=d_I\Delta I-\gamma_I I$, we prove a long-time reduction during washout and show that the damped base dynamics admit no diffusion-driven (Turing-type) instability. We then formulate a directionality--damping principle: unidirectional (open-loop) sensing yields at most transient focusing, whereas bidirectional (closed-loop) feedback reshapes the effective mobility and produces explicit thresholds separating stable homogeneity, finite-band patterning (resistance niche formation), and aggregation when strong parabolicity is violated. Reproducible simulations corroborate this classification and highlight when flux regularization is required for physical realism.

We investigate when the Rees algebra of an integrally closed $\mathfrak{m}$-primary ideal in a regular local ring is a Cohen-Macaulay normal domain. While this property always holds in dimension two, it fails in general in higher dimensions, prompting a search for sufficient conditions on the ideal. We show that if an integrally closed ideal contains a part of regular system of parameters of length $d-2$, where $d$ is the dimension of the regular local ring, then its Rees algebra is Cohen-Macaulay and normal. We also extend results of Goto and Ciupercă by proving the same conclusion when the minimal number of generators of an ideal is at most $d+2$. Furthermore, we treat the case of integrally closed zero-dimensional ideals generated by $d+3$ homogeneous polynomials. Finally, using generic Bourbaki ideals, we generalize these results to integrally closed torsionfree modules of finite colength.

We develop the representation theory intrinsic to Algebraic Phase Theory (APT) in regimes where defect and canonical filtration admit faithful algebraic realisation. This extends the framework introduced in earlier work by incorporating a representation-theoretic layer that is compatible with defect and filtration. In this setting, algebraic phases act naturally on filtered module categories rather than on isolated objects, and classical irreducibility must be replaced by a filtration-compatible notion of indecomposability forced by defect. As a central application, we analyse the Frobenius Heisenberg algebraic phase, which occupies a rigid boundary regime within the broader APT landscape, and show that it satisfies the axioms of APT in a strongly admissible form. We study the representations realising this phase and show that their non-semisimplicity and rigidity properties are consequences of the underlying algebraic structure rather than analytic or semisimple hypotheses. In particular, we establish a Stone von Neumann type rigidity theorem for Heisenberg groups associated with finite Frobenius rings. For each such ring $R$, we construct a canonical Schrödinger representation of the Frobenius Heisenberg group $H_R$, and show that, for a fixed nontrivial central character, every centrally faithful representation of $H_R$ is equivalent to this model. The proof is entirely algebraic and uses no topology, unitarity, Fourier analysis, or semisimplicity. Instead, rigidity emerges as a boundary phenomenon governed by defect and canonical filtration. The Frobenius hypothesis is shown to be sharp: it precisely delineates the structural boundary within APT at which Heisenberg rigidity persists, and outside the Frobenius class this rigidity necessarily fails.

Shift graphs, introduced by Erdős and Hajnal in 1964, form one of the simplest known non-recursive constructions of triangle-free graphs with arbitrarily large chromatic number. In this note, we identify a suprising property: for each integer $k \geq 1$, the smallest $k$-chromatic shift graph contains a unique $k$-vertex-critical subgraph. We give an explicit description of this subgraph and prove its uniqueness. This provides a new and remarkably simple family of triangle-free vertex-critical graphs of arbitrarily large chromatic number.

We introduce a generalized Fourier ratio, the \(\ell^1/\ell^2\) norm ratio of coefficients in an \emph{arbitrary} orthonormal system, as a single, basis-invariant measure of \emph{effective dimension} that governs fundamental limits across signal recovery, localization, and learning. First, we prove that functions with small Fourier ratio can be stably recovered from random missing samples via \(\ell^1\) minimization, extending and clarifying compressed sensing guarantees for general bounded orthonormal systems. Second, we establish a sharp \emph{localization obstruction}: any attempt to localize recovery to subslices of a product space necessarily inflates the Fourier ratio by a factor scaling with the square root of the slice count, demonstrating that global complexity cannot be distributed locally. Finally, we show that the same parameter controls key complexity-theoretic measures: it provides explicit upper bounds on Kolmogorov rate-distortion description length and on the statistical query (SQ) dimension of the associated function class. These results unify analytic, algorithmic, and learning-theoretic constraints under a single complexity parameter, revealing the Fourier ratio as a fundamental invariant in information-theoretic signal processing.

We develop the quantum component of Algebraic Phase Theory by showing that quantum phase, Weyl noncommutativity, and stabiliser codes arise as unavoidable algebraic consequences of Frobenius duality. Working over finite commutative Frobenius rings, we extract nondegenerate phase pairings, Weyl operator algebras, and quantum stabiliser codes directly from admissible phase data, without assuming Hilbert spaces, analytic inner products, or an externally imposed symplectic structure. Within this framework, quantum state spaces appear as minimal carriers of faithful phase action, and stabiliser codes are identified canonically with self-orthogonal submodules under the Frobenius phase pairing. CSS-type constructions arise only as a special splitting case, while general Frobenius rings admit intrinsically non-CSS stabilisers. Nilpotent and torsion structure in the base ring give rise to algebraically protected quantum layers that are invisible to admissible Weyl-type errors. These results place quantum stabiliser theory within Algebraic Phase Theory: quantisation emerges as algebraic phase induction rather than analytic completion, and quantum structure is information-complete at the level of algebraic phase relations alone. Throughout, we work over finite Frobenius rings, which are precisely the base rings for which admissible phase data become strongly admissible, and in this regime the full quantum formalism is forced by Frobenius duality.

Let $f \in S_{k}(\Gamma_{0}(N))$ be a normalized Hecke eigenform. We study the Fourier coefficients $\lambda_{f \otimes \cdots \otimes_{\ell} f}(n)$ of the $\ell$-fold product $L$-function for odd $\ell \ge 3$. Our focus is the distribution of this sequence over the sparse set of integers represented by a primitive, positive-definite binary quadratic form $Q$ of a fixed discriminant $D$. We establish an explicit upper bound for the summatory function of these coefficients, with dependencies on the weight, level, and discriminant. As a key application, we provide a bound for the first sign change of the sequence in this setting. We also generalize this result to find the first sign change among integers represented by any of the $h(D)$ forms of discriminant $D$, showing the bound improves as the class number increases.

In \cite{[CZ]}, Cohen and Zemel showed that for a partition $\lambda \vdash k$, the dimension of the irreducible representation of $S_{n}$ corresponding to the partition $(n-k,\lambda) \vdash n$ is a polynomial of degree $k$ in $n$, whose coefficients in the binomial basis count standard Young tableaux of shape $\lambda$ with special restrictions. In this paper, we generalize their results on the representation's dimension to character values on arbitrary cycles.

Directional notions in topology and analysis naturally lead to nonsymmetric structures such as quasi-metrics, quasi-uniformities, and modular spaces. In these settings, classical notions of connectedness and completion based on symmetric uniformities are often inadequate. In this paper, we study \emph{antisymmetric connectedness} and \emph{local antisymmetric connectedness} within the setting of quasi-uniform and quasi-modular pseudometric spaces. We associate to each quasi-modular pseudometric family compatible forward and backward modular topologies and quasi-uniformities, yielding a canonical bitopological structure. Using this setting, we establish characterization and stability results for local antisymmetric connectedness, including invariance under subspaces, uniformly continuous mappings, and bicompletion. We further relate these notions to Smyth completeness and Yoneda-type completions and show how precompactness combined with asymmetric completeness yields compactness in the join topology. Applications to asymmetric normed and modular spaces illustrate the theory.

A well-known and old result of Hazewinkel and Koszul states that the cohomology of a finite-dimensional Lie algebra is isomorphic, up to a suitable shift, to its twisted homology, a Lie-theoretical version of Poincare duality. This paper establishes an analogue of this result for graded (or super) Lie algebras and, more generally, differential graded Lie algebras. Closely related to this result is a calculation of the cohomology of a graded Lie algebra g with coefficients in its universal enveloping algebra U(g) as a one-sided module. This cohomology turns out to be one-dimensional and serves as a dualizing module for the cohomology of g. Moreover, it is shown that for a unimodular graded Lie algebra g, the derived category of U(g) has a Calabi-Yau structure. As a consequence, the category of rational infinity local systems on a simply connected topological space with totally finite-dimensional rational homotopy groups, has a Calabi-Yau structure, a generalization of Poincare duality for elliptic spaces.

We carry out the asymptotic analysis of repulsive ensembles of N particles which are discrete analogues of continuous 1d log-gases or beta-ensembles of random matrix theory. The ensembles that we study have several groups of particles which can have different intensities of repulsion. They appear naturally in models of random domino and lozenge tilings, random partitions, supersymmetric gauge theory, asymptotic representation theory, discrete orthogonal polynomial ensembles, etc. We allow filling fractions to be either fixed, or free, or to vary while respecting affine constraints. We are interested in the macroscopic behavior of the distribution of particles, captured by linear statistics, partition functions, and their finite-size corrections as N is large. We prove the law of large numbers and large deviations for the empirical measure around the equilibrium measure. To reach finite-size correction we assume off-criticality. For fixed filling fractions, we prove an asymptotic expansion for the partition function and for the cumulants of linear statistics, in particular establishing a central limit theorem. For varying filling fractions, we prove that the central limit theorem is perturbed by an additional discrete Gaussian component oscillating with N. We apply our general results to the study of uniformly random lozenge tilings on a large class of domains -- not necessarily planar, simply-connected, nor orientable. When the analogues of filling fractions are fixed and this domain is orientable, we show that the Gaussian fluctuations on the vertical extend to the whole liquid region and are governed there by the Gaussian free field, as predicted by the Kenyon-Okounkov conjecture. We also establish a modification of the Kenyon-Okounkov conjecture in the non-orientable case. Complementarily, we prove discrete Gaussian fluctuations for filling fractions, when they are not fixed.

Graph theory on surfaces extends classical graph structures to topological surfaces, providing a theoretical foundation for characterizing the embedding properties of complex networks in constrained spaces. The study of bounding the spectral radius $\rho(G)$ of graphs on surfaces has a rich history that dates back to the 1990s. In this paper, we establish tight bounds for graphs of order $n$ that are embeddable on a surface with Euler genus $\gamma$. Specifically, if graph $G$ achieves the maximum spectral radius, then \begin{equation*} \begin{array}{ll} \frac32\!+\!\sqrt{2n\!-\!\frac{15}4}\!+\!\frac{3\gamma\!-\!1}{n}<\rho(G)<\frac32\!+\!\sqrt{2n\!-\!\frac{15}4}\!+\!\frac{3\gamma\!-\!0.95}{n}, \end{array} \end{equation*} which improves upon the earlier bound $\rho(G)\leq2+\sqrt{2n+8\gamma-6}$ by Ellingham and Zha [JCTB, 2000]. Furthermore, we prove that any extremal graph is obtained from $K_2 \nabla P_{n-2}$ by adding exactly $3\gamma$ edges, where `$\nabla$' means the join product. As a corollary, for $\gamma = 0$ and $n \geq 4.5 \times 10^6$, the graph $K_2 \nabla P_{n-2}$ is the unique planar extremal graph, thereby confirming a long-standing conjecture resolved by Tait and Tobin [JCTB, 2017]. Let $K_r^n$ be the graph of order $n$ obtained by attaching two paths of nearly equal length to two distinct vertices of $K_r$. Integrating spectral techniques with considerable structural analysis on surface graphs, we further derive the following sharp bounds: $\rho(G) \leq \rho(K_2 \nabla K_4^{n-2})$ for projective-planar graphs, and $\rho(G) \leq \rho(K_2 \nabla K_5^{n-2})$ for toroidal graphs. Our study presents a novel framework for exploring the eigenvalue-extremal problem on surface graphs with high Euler genus.

While the filtered-x normalized least mean square (FxNLMS) algorithm is widely applied due to its simple structure and easy implementation for active noise control system, it faces two critical limitations: the fixed step-size causes a trade-off between convergence rate and steady-state residual error, and its performance deteriorates significantly in impulsive noise environments. To address the step-size constraint issue, we propose the switched \mbox{step-size} FxNLMS (SSS-FxNLMS) algorithm. Specifically, we derive the \mbox{mean-square} deviation (MSD) trend of the FxNLMS algorithm, and then by comparing the MSD trends corresponding to different \mbox{step-sizes}, the optimal step-size for each iteration is selected. Furthermore, to enhance the algorithm's robustness in impulsive noise scenarios, we integrate a robust strategy into the SSS-FxNLMS algorithm, resulting in a robust variant of it. The effectiveness and superiority of the proposed algorithms has been confirmed through computer simulations in different noise scenarios.

Introduced in [BB], simplicially stable spaces are alternative compactifications of $\mathcal{M}_{g,n}$ generalizing Hassett's moduli spaces of weighted stable curves. We give presentations of the Chow rings of these spaces in genus $0$ using techniques developed by the author in [New25]. When considering the special case of $\overline{\mathcal{M}}_{0,n}$, this gives a new proof of Keel's presentation of $\operatorname{CH}(\overline{\mathcal{M}}_{0,n})$.

Let $M=V\setminus D$ be a smooth quasi-projective variety for some smooth projective variety $V$ and a divisor $D$ with normal crossings. Assume that $M$ is diffeomorphic to a non-compact nilmanifold $\Gamma\backslash N\times\mathbb{R}^m$. We show that $M$ is diffeomorphic to a trivial bundle $T^n\times \mathbb{R}^m$ over a torus $T^n$ if the first cohomology $H^1(V)$ of $V$ vanishes. Moreover, in general, we show that $M$ is diffeomorphic to a trivial bundle $T^{b_1(M)}\times \mathbb{R}^m$ over a $b_1(M)$-dimensional torus $T^{b_1(M)}$, or a trivial bundle $E\times \mathbb{R}^m$ such that $E$ is a torus bundle $E\rightarrow T^{b_1(M)}$ over a torus $T^{b_1(M)}$. Conversely, we consider whether non-compact nilmanifolds are diffeomorphic to a smooth quasi-projective variety. We determine the Lie groups of dimension up to $8$ such that corresponding non-compact nilmanifolds may be diffeomorphic to smooth quasi-projective varieties.

We study equivariant operations on the periodic cyclic homology of a dg algebra that arise from the chain level action of the two-colored Kontsevich-Soibelman operad. Using classical computations of Cohen [Coh], we explicitly compute a set of generators for these operations under composition, and show that they agree with the p-fold equivariant cap products previously studied by the author [Che2] in relation to equivariant Gromov-Witten theory with mod p coefficients. The main technical novelty is a re-formulation of the Kontsevich-Soibelman operad in terms of a two-colored version of the cacti operad, and a proof that it is equivariantly quasi-equivalent to the two-colored operad of little disks on a disk/cylinder. We give applications of the main results to symplectic topology, and more specifically, arithmetic aspects of Fukaya category and classical obstructions to realizing a middle cohomology class of a symplectic manifold by Lagrangian submanifold

Twisted generalized Reed-Solomon (TGRS) codes, as a flexible extension of classical generalized Reed-Solomon (GRS) codes, have attracted significant attention in recent years. In this paper, we construct two classes of LCD codes from the $(\mathcal{L},\mathcal{P})$-TGRS code $\mathcal{C}_h$ of length $n$ and dimension $k$, where $\mathcal{L}=\{0,1,\ldots,l\}$ for $l\leq n-k-1$ and $\mathcal{P}=\{h\}$ for $1\leq h\leq k-1$. First, we derive the parity check matrix of $\mathcal{C}_h$ and provide a necessary and sufficient condition for $\mathcal{C}_h$ to be an AMDS code. Then, we construct two classes of LCD codes from $\mathcal{C}_h$ by suitably choosing the evaluation points together with certain restrictions on the coefficient of $x^{h-1}$ in the polynomial associated with the twisting term. From the constructed LCD codes we further obtain two classes of LCD MDS codes. Finally, several examples are presented.

We construct a quadratic Morse-Bott function on the real Grassmannian of a symplectic vector space from a compatible linear complex structure. We show that its critical loci consist of linear subspaces that split into isotropic and complex parts and that its stable manifolds coincide with the orbits of the linear symplectomorphism group. These orbits generalize the Lagrangian, symplectic, isotropic, and coisotropic Grassmannians to include the Grassmannians of linear subspaces that are neither isotropic, coisotropic, nor symplectic. The negative gradient flow deformation retracts these spaces onto compact homogeneous spaces for the unitary group.

Integrated sensing and communication (ISAC) have been widely recognized as a key enabler for future wireless networks, where the Cramér-Rao bound (CRB) plays a central role in quantifying sensing this http URL this paper, we present the first study on CRB minimization in flexible intelligent metasurface (FIM)-enabled ISAC this http URL, we first derive an average CRB expression that explicitly depends on FIM surface shape and demonstrate that array reconfigurability can substantially reduce the CRB, thereby significantly enhancing sensing this http URL, to tackle the challenging CRB minimization problem, we adopt average Fisher information maximization as a surrogate objective and use the Gauss-Hermite quadrature method to obtain an explicit approximation of the objective this http URL resulting problem is then decoupled into three subproblem, i.e., beamforming optimization and transmit/receive FIM surface shape this http URL beamforming optimization, we employ the Schur complement and penalty-based semi-definite relaxation (SDR) technique to solve this http URL, we propose a fixed-point equation method and a projected gradient algorithm to optimize the surface shapes of the receive and transmit FIMs, this http URL results demonstrate that, compared to rigid arrays, surface shaping of both transmit and receive FIMs can significantly reduce the average sensing CRB while maintaining communication quality, and remains effective even in multi-target scenarios.

We study lossy source coding under a distortion measure defined by the negative log-likelihood induced by a prescribed conditional distribution $P_{X|U}$. This \emph{log-likelihood distortion} models compression settings in which the reconstruction is a semantic representation from which the source can be probabilistically generated, rather than a pointwise approximation. We formulate the corresponding rate-distortion problem and characterize fundamental properties of the resulting rate-distortion function, including its connections to lossy compression under log-loss, classical rate-distortion problems with arbitrary distortion measures, and rate-distortion with perfect perception.

We analyze the inviscid Langmuir layer--Stokesian subfluid (ILLSS) model for two-phase Langmuir monolayers coupled to a Stokes flow in the underlying subfluid. Eliminating the bulk variables, we reformulate the coupled three-dimensional system as an evolution on the film involving the Dirichlet-to-Neumann (DtN) operator. We identify the Fourier symbol of the DtN operator and show it coincides with that of the fractional Laplacian, which yields an explicit Fourier-multiplier representation and allows construction of the corresponding fundamental solution. Using this representation we express the surface velocity as a convolution of the fundamental solution with the interfacial curvature forcing and analyze its normal limit to derive a boundary integral equation for the moving curve. Independently, exploiting the DtN representation we establish a curve-shortening identity: the interfacial perimeter decreases monotonically and its time derivative is controlled by $\dot{H}^{1/2}(\mathbb{R}^2)$-norm of the surface velocity. Building on the boundary integral equation, we prove local well-posedness via maximal $L^2$-regularity for quasilinear parabolic systems, employing a DeTurck-type reparametrization, and show equivalence with the original ILLSS system. Finally, we introduce a linearly implicit parametric finite-element scheme which captures experimentally observed relaxation dynamics.

Let $\nu$ be a positive measure on $[0,1]$. A Shimorin-type operator $T_\nu$ is an integral operator on the unit disk given by \[ T_\nu f(z) = \int_{\mathbb{D}} \frac{1}{1 - z\overline{\lambda}} \left( \int_0^1 \frac{d\nu(r)}{1 - r z \overline{\lambda}} \right) f(\lambda) \, dA(\lambda), \] which originates from Shimorin's work on Bergman-type kernel representations for logarithmically subharmonic weighted Bergman spaces. In this paper, we study $L^p$--$L^q$ estimates for $T_\nu$. Unlike classical Bergman-type operators, the critical line on the $(1/p,1/q)$-plane that separates the boundedness and unboundedness regions of $T_\nu$ is not immediately evident. Moreover, even along this line, new phenomena arise. In the present work, by introducing a quantity $c_\nu$, \begin{itemize} \item we first determine the critical boundary in the $(1/p,1/q)$-plane for bounded $T_\nu$; \item furthermore, on this critical line, we establish necessary and sufficient conditions for $T_\nu$ which have standard Bergman-type $L^p$--$L^q$ estimates, meaning that it is bounded in the interior of the region and admits weak-type and BMO-type estimates at endpoints. \end{itemize}

This paper presents an energy-efficient downlink cell-free massive multiple-input multiple-output (CF-mMIMO) integrated sensing and communication (ISAC) network that serves ultra-reliable low-latency communication (URLLC) users while simultaneously detecting a target. We propose a load-balancing algorithm that minimizes the total network power consumption; including transmit power, fixed static power, and traffic-dependent fronthaul power at the access points (APs) without degrading system performance. To this end, we formulate a mixed-integer non-convex optimization problem and introduce an iterative joint power allocation and AP load balancing (JPALB) algorithm. The algorithm aims to reduce total power usage while meeting both the communication quality-of-service (QoS) requirements of URLLC users and the sensing QoS needed for target detection. Proposed JPALB algorithm for ISAC systems was simulated with maximum-ratio transmission (MRT) and regularized zero-forcing (RZF) precoders. Simulation results show approximately 33% reduction in power consumption, using JPALB algorithm compared to a baseline with no load balancing, without compromising communication and sensing QoS requirements.

We study fibers with isolated singularities of Landau-Ginzburg models for Fano threefolds of Picard rank one. We compare the data we get with maximal known lengths of exceptional collections in derived categories of coherent sheaves on the Fano threefolds, verify some predictions of Homological Mirror Symmetry, and present some expectations about exceptional collections for Fano threefolds.

We present an algorithm to compute the torsion component $\mathrm{Pic}^\tau X$ of the Picard scheme of a smooth projective variety $X$ over a field $k$. Specifically, we describe $\mathrm{Pic}^\tau X$ as a closed subscheme of a projective space defined by explicit homogeneous polynomials. Furthermore, we compute the group scheme structure on $\mathrm{Pic}^\tau X$. As applications, we provide algorithms to compute various homological invariants. Among these, we compute the abelianization of the geometric étale fundamental group $\pi^{\mathrm{{e}t}}_1(X_{\bar{k}}, x)^{\mathrm{ab}}$. Moreover, we determine the Galois module structure of the first étale cohomology groups $H^1_{\mathrm{{e}t}}(X_{\bar{k}}, \mathbb{Z}/n\mathbb{Z})$ without requiring $n$ to be prime to the characteristic of $k$.

Encoding digital information into DNA sequences offers an attractive potential solution for storing rapidly growing data under the information age and the rise of artificial intelligence. However, practical implementations of DNA storage are constrained by errors introduced during synthesis, preservation, and sequencing processes, and traditional error-correcting codes remain vulnerable to noise levels that exceed predefined thresholds. Here, we developed a Partitioning-mapping with Jump-rotating (PJ) encoding scheme, which exhibits exceptional noise resilience. PJ removes cross-strand information dependencies so that strand loss manifests as localized gaps rather than catastrophic file failure. It prioritizes file decodability under arbitrary noise conditions and leverages AI-based inference to enable controllable recovery of digital information. For the intra-strand encoding, we develop a jump-rotating strategy that relaxes sequence constraints relative to conventional rotating codes and provides tunable information density via an adjustable jump length. Based on this encoding architecture, the original file information can always be decoded and recovered under any strand loss ratio, with fidelity degrading smoothly as damage increases. We demonstrate that original files can be effectively recovered even with 10% strand loss, and machine learning datasets stored under these conditions retain their classification performance. Experiments further confirmed that PJ successfully decodes image files after extreme environmental disturbance using accelerated aging and high-intensity X-ray irradiation. By eliminating reliance on prior error probabilities, PJ establishes a general framework for robust, archival DNA storage capable of withstanding the rigorous conditions of real-world preservation.

We introduce the \emph{parameter-geometrization} to the Hitchin system, a paradigm embedding deformation parameters into geometry via the coupled Hitchin-He equations on a surface with boundary. A boundary term couples a second Higgs field $\psi$, recovering the classical system at $\alpha=0$. We prove a unique, smooth solution branch exists near $\alpha=0$ (Theorem A). The system is integrable, admitting a Lax pair (Theorem B). Crucially, the moduli space $\mathcal{M}_\alpha$ is analytically isomorphic to $\mathcal{M}_0$ for small $|\alpha|$, preserving the Hitchin fibration -- revealing a deep rigidity where all moduli are controlled by the primary Higgs field (Theorem C). Using the \emph{nonlinear embedding} technique that casts the deformed system into the form of a classical Higgs bundle system, whose integrability and geometry are well-understood, we extends the framework to compact Kähler manifolds (Theorem D).

Phase-field simulations are a practical but also expensive tool to calculate microstructural evolution. This work aims to compare explicit time integrators for a broad class of phase-field models involving coupling between the phase-field and concentration. Particular integrators are adapted to constraints on the phase-field as well as storage scheme implications. Reproducible benchmarks are defined with a focus on having exact sharp interface solutions, allowing for identification of dominant error terms. Speedups of 4 to 114 over the classic forward Euler integrator are achievable while still using a fully explicit scheme without appreciable accuracy loss. Application examples include final stage sintering with pores slowing down grain growth as they move and merge over time.

The main goal of this paper is to address the following problem: given a positive integer $n$, find the largest value $S(n)$ such that a square of edge length $S(n)$ in the Euclidean plane can be covered by $n$ unit squares. We investigate also the variant in which the goal is to cover only the boundary of a square. We show that these two problems are equivalent for $n \leq 4$, but not for $n=5$. For both problems, we also present the solutions for $n=5$.

Asymptotic expressions for an integral appearing in the solution of a d-bar problem are presented. The integral is a solid Cauchy transform of a function with a rapidly oscillating phase with a small parameter $h$, $0<h\ll 1$. Whereas standard steepest descent approaches can be applied to the case where the stationary points of the phase $\omega_{k}$, $k=1,\ldots, N$ are far from the singularity $\zeta$ of the integrand, a polarization approach is proposed for the case that $|\zeta-\omega_{k}|<\mathcal{O}(\sqrt{h})$ for some $k$. In this case the problem is studied in $\mathbb{C}^{2}$ ($\tilde{\omega}:=\bar{\omega}$ is treated as an independent variable) on steepest descent contours. An application of Stokes' theorem allows for a decomposition of the integral into three terms for which asymptotics expressions in terms of special functions are given.

We study the boundary regularity of local weak solutions to nonlinear parabolic systems of the form \begin{equation*} \partial_t u^i - \mathrm{div} \big( a(|Du|) Du^i \big)= f^i, \qquad i=1,\dots,N, \end{equation*} in a space-time cylinder $\Omega_T = \Omega \times (0,T)$, where $\Omega \subset \mathbb{R}^n$ ($n \geq 2$) is a bounded, convex $C^2$-domain and $T>0$. The inhomogeneity $f=(f^1,\dots,f^N)$ belongs to $L^{n+2+\sigma}(\Omega_T,\mathbb{R}^N)$ for some $\sigma>0$. The coefficients $a\colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ are of Uhlenbeck-type and satisfy a nonstandard $(p,q)$-growth condition with \[ 2 \leq p \leq q < p + \frac{4}{n+2}. \] Our main result establishes a local Lipschitz estimate up to the lateral boundary for any local weak solution that vanishes on the lateral boundary of the cylinder.

We consider the discontinuous semi-classical Laguerre weight function with a jump $w(x;t,s)=\mathrm{e}^{-x^2+tx}(A+B\theta(x-s))$, where $x\in\mathbf{R}$, $t,s\ge0$, $A\ge0$, $A+B\ge0$, where $\theta(x)$ is 1 for $x > 0$ and 0 otherwise. Based on the ladder operator approach, we obtain some important difference and differential equations about the auxiliary quantities and the recurrence coefficients. By proper tranformation, It is shown that $R_{n}(t,s)$ is related to Painlevé IV equations and $r_{n}(t,s)$ satisfies the Chazy II equations. With the aid of Dyson's Coulomb fluid approach, we derive the asymptotic expansions for $\alpha_{n}$ and $\beta_{n}$ as $n\rightarrow\infty$. Furthermore, This enables us to obtain the lagre $n$ behavior of the orthogonal polynomials and derive that they satisfy the biconfluent Heun equation. We also consider the Hankel determinant $D_{n}(t,s)$ generated by the discountinuous semi-classical Laguerre weight. We find that the quantity $\sigma_{n}(t,s)$, allied to the logarithmic derivative of $D_{n}(t,s)$, satisfies the Jimbo-Miwa-Okamoto $\sigma$-form of Painlevé IV.

We investigate clones in the interval between the group polynomials and the ring polynomials of ${\mathbb Z}_8$. This is the simplest open case of the problem, as the answer is known for ${\mathbb Z}_{p^2}$ (with $p$ prime) and, in general, ${\mathbb Z}_n$ reduces to the case when $n$ is a prime power. The investigated structure proves to be very complicated, so we provide only a partial description. We restrict our attention to polynomials whose nonlinear monomials have even coefficients.

Distribution of the sum of independent identically distributed symmetric lattice vectors is approximated by the accompanying compound Poisson law and the second-order Hipp-type signed compound Poisson measure. Bergström -type asymptotic expansion is constructed. The accuracy of approximation is estimated in the total variation metric and, in many cases, is of the order $O(n^{-1})$.

Hodge Theory of $p$-adic analytic varieties was initiated by Tate in his 1967 paper on $p$-divisible groups, where he conjectured the existence of a Hodge-like decomposition for the $p$-adic étale cohomology of proper analytic varieties. Tate's conjecture was refined by Fontaine who gave the theory its definite shape. A lot of work has been done for algebraic varieties and a number of proofs of Fontaine's conjectures have been obtained between years 1985 and 2011. But the study of Hodge Theory of $p$-adic analytic varieties started really only in 2011 with Scholze's proof of Tate's conjecture using perfectoid methods. Methods that opened the way to an avalanche of results. In this paper, we survey our results and conjectures (comparison theorems and their geometrization, dualities, etc.), focusing on the case of nonproper analytic varieties, where a number of new phenomena occur. We also describe the new objects that appeared along the way.

Stick-breaking has a long history and is one of the most popular procedures for constructing random discrete distributions in Statistics and Machine Learning. In particular, due to their intuitive construction and computational tractability they are ubiquitous in modern Bayesian nonparametric inference. Most widely used models, such as the Dirichlet and the Pitman-Yor processes, rely on iid or independent length variables. Here we pursue a completely unexplored research direction by considering Markov length variables and investigate the corresponding general class of stick-breaking processes, which we term Markov stick-breaking processes. We establish conditions under which the associated species sampling process is proper and the distribution of a Markov stick-breaking process has full topological support, two fundamental desiderata for Bayesian nonparametric models. We also analyze the stochastic ordering of the weights and provide a new characterization of the Pitman-Yor process as the only stick-breaking process invariant under size-biased permutations, under mild conditions. Moreover, we identify two notable subclasses of Markov stick-breaking processes that enjoy appealing properties and include Dirichlet, Pitman-Yor and Geometric priors as special cases. Our findings include distributional results enabling posterior inference algorithms and methodological insights.

The inverse of the star discrepancy, $N(\epsilon, s)$, defined as the minimum number of points required to achieve a star discrepancy of at most $\epsilon$ in dimension $s$, is known to depend linearly on $s$. However, explicit constructions achieving this optimal linear dependence remain elusive. Recently, Dick and Pillichshammer (2025) made significant progress by showing that a multiset union of randomly digitally shifted Korobov polynomial lattice point sets almost achieve the optimal dimension dependence with high probability. In this paper, we investigate the analog of this result in the setting of classical integer arithmetic using Fourier analysis. We analyze point sets constructed as multiset unions of Korobov rank-1 lattice point sets modulo a prime $N$. We provide a comprehensive analysis covering four distinct construction scenarios, combining either random or fixed integer generators with either continuous torus shifts or discrete grid shifts. We prove that in all four cases, the star discrepancy is bounded by a term of order $O(s \log(N_{tot}) / \sqrt{N_{tot}})$ with high probability, where $N_{tot}$ is the total number of points. This implies that the inverse of the star discrepancy for these structured sets depends quadratically on the dimension $s$. While the proofs are probabilistic, our results significantly reduce the search space for optimal point sets from a continuum to a finite set of candidates parameterized by integer generators and random shifts.

We consider a class of singular Riemannian metrics on a compact Riemannian manifold with boundary and the eigenfunctions of the corresponding Laplace-Beltrami operator. In our setting, the average density of eigenfunctions with eigenvalue less than $\lambda$ converges weakly to the uniform normalised measure on the boundary as $\lambda\to\infty$. In this work, we show a quantitative estimate on the speed of this convergence in the Wasserstein-sense in the transverse coordinate to the boundary.

In this paper, we develop optimization methods for a new model of multifacility location problems defined by a Minkowski gauge with Laplace-type regularization terms. The model is analyzed from both theoretical and numerical perspectives. In particular, we establish the existence of optimal solutions and study qualitative properties of global minimizers. By combining Nesterov's smoothing technique with recent advances in difference-of-convex optimization, following the pioneering work of P. D. Tao and L. T. H. An and others, we propose efficient numerical algorithms for minimizing the objective function of this model. As an application, our approach provides an effective method for determining the number of centers in gauge-based multifacility location and clustering problems. Our results extend and complement recent developments.

Penney's Ante exhibits non-transitivity when two target strings race to appear in a shared stream of coin tosses. We study instead independent string races, where each player observes their own independent and identically distributed (i.i.d.) coin/die stream (possibly biased), and the winner is the player whose target appears first (under an explicit tie convention). We derive compact generating-function formulas for waiting times and a Hadamard-generating-function calculus for head-to-head odds. Our main theorem shows that for a fair -sided die, stochastic dominance induces a total pre-order on all strings, ordered by expected waiting time. For binary coins, we also prove a converse: total comparability under stochastic dominance characterises the fair coin (), and any bias yields patterns whose waiting times are incomparable under stochastic dominance. In contrast, bias allows both (i) reversals between mean waiting time and win probability and (ii) non-transitive cycles; we give explicit examples and certified computational classifications for short patterns.

Integrated learning and optimization (ILO) is a framework in contextual optimization which aims to train a predictive model for the probability distribution of the underlying problem data uncertainty, with the goal of enhancing the quality of downstream decisions. This framework represents a new class of stochastic bilevel programs, which are extensively utilized in the literature of operations research and management science, yet remain underexplored from the perspective of optimization theory. In this paper, we fill the gap. Specifically, we derive the first-order necessary optimality conditions in terms of Mordukhovich limiting subdifferentials. To this end, we formulate the bilevel program as a two-stage stochastic program with variational inequality constraints when the lower-level decision-making problem is convex, and establish an optimality condition via sensitivity analysis of the second-stage value function. In the case where the lower level optimization problem is nonconvex, we adopt the value function approach in the literature of bilevel programs and derive the first-order necessary conditions under stochastic partial calmness conditions. The derived optimality conditions are applied to several existing ILO problems in the literature. These conditions may be used for the design of gradient-based algorithms for solving ILO problems.

We prove that for knots, the evaluation of the Jones polynomial at the sixth root of unity, as well as the evaluation of the $Q$-polynomial at the reciprocal of the golden ratio, are uniquely determined by the oriented homeomorphism type of the double branched covering. We provide explicit formulae for these evaluations in terms of the linking pairing. The proof proceeds via so-called singular determinants, from which we also extract new lower bounds for the unknotting numbers of knots and links.

While exact and approximate Riemann solvers are widely used, they exhibit two fundamental limitations: 1) Fail to represent continuous entropy transport processes, resulting in thermodynamic incompatibility that limits their applicability to compressible flows. 2) Consider only the effects of normal components at interfaces while neglecting the effects of tangential flux and source term, making them unsuitable for multidimensional problems and cases involving source terms. These limitations persist in Riemann problem-based ghost fluid methods. To address these challenges, we developed a novel spatiotemporal coupling high-resolution ghost fluid method featuring two key advancements: 1) Integration of nonlinear geometrical optics to properly account for thermodynamic entropy evolution. 2) Implementation of the Lax-Wendroff/Cauchy-Kowalevski approach to incorporate tangential fluxes and source term effects. These enhancements have been systematically applied to Riemann problem-based ghost fluid methods. Comprehensive numerical experiments demonstrate significant improvements in simulation accuracy and robustness compared to conventional approaches.

The Chabauty--Kim method was developed with the aim of approaching effective Faltings', the problem of explicitly determining the finite set of rational points on a hyperbolic curve. This method has seen success with the more particular Quadratic Chabauty method, but this method still applies only to certain curves. Previous applications of Chabauty--Kim beyond the quadratic level, as pursued by the authors, by S. Wewers, and by others, use mixed Tate motives and the $p$-adic period map of Chatzistamatiou-Ünver to approach the particular hyperbolic curve $\mathbb{P}^1\setminus\{0,1,\infty\}$. The main purpose of this article is to lay foundations for extending the above approach to more general hyperbolic curves, in particular by defining an analogous $p$-adic period map for more general categories of motives and their non-conjectural cousins such as systems of realizations and $p$-adic Galois representations. We use this to describe a general setup for non-abelian Chabauty for an arbitrary hyperbolic curve. Our period map also connects the study of $p$-adic iterated integrals with Goncharov's theory of motivic iterated integrals, and allows us to investigate Goncharov's conjectures from a $p$-adic point of view. In particular, it suggests the possibility of evaluating syntomic regulators by writing elements of $K$-theory in terms of motivic iterated integrals. Lastly, it forms the basis for a certain generalization of the $p$-adic period conjecture of Yamashita for mixed Tate motives well-suited to applications in Chabauty--Kim theory.

We derive a few extended versions of the Kraft inequality for information lossless finite-state encoders. The main basic contribution is in defining a notion of a Kraft matrix and in establishing the fact that a necessary condition for information losslessness of a finite-state encoder is that none of the eigenvalues of this matrix have modulus larger than unity, or equivalently, the generalized Kraft inequality asserts that the spectral radius of the Kraft matrix cannot exceed one. For the important special case where the FS encoder is irreducible, we derive several equivalent forms of this inequality, which are based on well known formulas for spectral radius. It also turns out that in the irreducible case, Kraft sums are bounded by a constant, independent of the block length, and thus cannot grow even in any subexponential rate. Finally, two extensions are outlined - one concerns the case of side information available to both encoder and decoder, and the other is for lossy compression.

The deep neural network multigrid solver (DNN-MG) combines a coarse-grid finite element simulation with a deep neural network that corrects the solution on finer grid levels, thereby improving the computational efficiency. In this work, we discuss various design choices for the DNN-MG method and demonstrate significant improvements in accuracy and generalizability when applied to the solution of the instationary Navier-Stokes equations. We investigate the stability of the hybrid simulation and show how the neural networks can be made more robust with the help of replay buffers. By retraining on data derived from the hybrid simulation, the error caused by the neural network over multiple time-steps can be minimized without the need for a differentiable numerical solver. Furthermore, we compare multiple neural network architectures, including recurrent neural networks and Transformers, and study their ability to utilize more information from an increased temporal and spatial receptive field. Transformers allow us to make use of information from cells outside the predicted patch even with unstructured meshes while maintaining the locality of our approach. This can further improve the accuracy of DNN-MG without a significant impact on performance.

This work is motivated by the long-standing open problem of designing asymptotically order-optimal aperiodic polyphase sequence sets with respect to the celebrated Welch bound. Attempts were made by Mow over 30 years ago, but a comprehensive understanding to this problem is lacking. Our first key contribution is an explicit upper bound of generalized quadratic Gauss sums which is obtained by recursively applying Paris' asymptotic expansion and then bounding it by leveraging the fast convergence property of the Fibonacci zeta function. Building upon this major finding, our second key contribution includes four systematic constructions of order-optimal sequence sets with low aperiodic correlation and/or ambiguity properties via carefully selected Chu sequences and Alltop sequences. For the first time in the literature, we reveal that the full Alltop sequence set is asymptotically optimal for its low aperiodic correlation sidelobes. Besides, we introduce a novel subset of Alltop sequences possessing both order-optimal aperiodic correlation and ambiguity properties for the entire time-shift window.

In this paper, we study the ground state solutions of the following coupled nonlinear Schrödinger system (P) $-\Delta u_1-\tau_1 u_1 =\mu_1u_1^3+\beta u_1u_2^2$, $ -\Delta u_2-\tau_2 u_2 =\mu_2u_2^3+\beta u_1^2u_2$ in $\Omega$, $u_1=u_2=0$ on $\partial\Omega$, where $\mu_1, \mu_2>0$, $\beta>0$ and $\Omega\subset \mathbb{R}^N (N\le3)$ is a bounded domain with smooth boundary. We are concerned with the indefinite case, i.e., $\tau_1, \tau_2$ are greater than or equal to the principal eigenvalue of $-\Delta$ with the Dirichlet boundary datum. By delicate variational arguments, we obtain the existence of ground state solution to $(P)$, and also provide information on critical energy levels for coupling parameter $\beta$ in some ranges.

We consider a card guessing game with complete feedback. An ordered deck of $n$ cards labeled $1$ up to $n$ is riffle-shuffled exactly one time. Given a value $p\in(0{,}1)\setminus\{\frac12\}$, the riffle shuffle is assumed to be unbalanced, such that the cut is expected to happen at position $p\cdot n$. The goal of the game is to maximize the number of correct guesses of the cards: one after another a single card is drawn from the top, and shown to the guesser until no cards remain. We provide a detailed analysis of the optimal guessing strategy and study the distribution of the number of correct guesses.

We consider the output-feedback stabilization of a one-dimensional cascade coupling a reaction-diffusion equation and a wave equation through an internal term, with Neumann boundary control acting at the wave endpoint. Two measurements are available: the wave velocity at the controlled boundary and a temperature-type observation of the reaction-diffusion component, either distributed or pointwise. Under explicit, necessary and sufficient conditions on the coupling and observation profiles, we show that the generator of the open-loop system is a Riesz-spectral operator. Exploiting this structure, we design a finite-dimensional dynamic output-feedback law, based on a finite number of parabolic modes, which achieves arbitrary exponential decay in both the natural energy space and a stronger parabolic norm. The construction relies on a spectral reduction and a Lyapunov argument in Riesz bases. We also extend the design to pointwise temperature or heat-flux measurements.

Term Coding asks: given a finite system of term identities $\Gamma$ in $v$ variables, how large can its solution set be on an $n$--element alphabet, when we are free to choose the interpretations of the function symbols? This turns familiar existence problems for quasigroups, designs, and related objects into quantitative extremal questions. We prove a guessing-number sandwich theorem that connects term coding to graph guessing numbers (graph entropy). After explicit normalisation and diversification reductions, every instance yields a canonical directed dependency structure with guessing number $\alpha$ such that the maximum code size satisfies $\log_n \Sn(\Gamma)=\alpha+o(1)$ (equivalently, $\Sn(\Gamma)=n^{\alpha+o(1)}$), and $\alpha$ can be bounded or computed using entropy and polymatroid methods. We illustrate the framework with examples from extremal combinatorics (Steiner-type identities, self-orthogonal Latin squares) and from information-flow / network-coding style constraints (including a five-cycle instance with fractional exponent and small storage/relay maps).

We consider individuals of two species distributed over m patches, each with a hosting capacity \(d_i N\) , where \(d_i \in (0, 1]\). We assume that all the patches are linked by the dispersal of individuals. This work examines how the metacommunity evolves in these patches. The model incorporates Wright-Fisher intra-patch reproduction and a general exchange function representing dispersal. Under minimal assumptions, we demonstrate that as \(N\) approaches infinity, the processes converge to a diffusion process for which we establish the Feller property. We prove that the limiting process almost surely reaches the absorbing states in finite time.

In this paper, for a given variety $\Var$, we present a universal algorithm that constructs a subvariety of $\Var$-dialgebras from which one can recover an algebra belonging to $\Var$. We call such a variety an initial $\Var$-dialgebra. In addition, we construct the basis of the free initial Lie and associative dialgebras.

Using a discrete Bakry-{É}mery method based on the JKO scheme, relying on the dissipation of entropy and Fisher information along a discrete flow, we establish new generalized logarithmic Sobolev inequality for log-concave measures of the form $e^{-V} under strict convexity assumptions on $V$ . We then show how this method recovers some well-known inequalities. This approach can be viewed as interpolating between the Bakry-{É}mery method and optimal transport techniques based on geodesic convexity.

This paper studies a continuous-time joint sampling-and-preemption problem, incorporating sampling and preemption penalties under general service-time distributions. We formulate the system as an impulse-controlled piecewise-deterministic Markov process (PDMP) and derive coupled integral average-cost optimality equations via the dynamic programming principle, thereby avoiding the smoothness assumptions typically required for an average-cost Hamilton-Jacobi-Bellman quasi-variational inequality (HJB-QVI) characterization. A key invariance in the busy phase collapses the dynamics onto a one-dimensional busy-start boundary, reducing preemption control to an optimal stopping problem. Building on this structure, we develop an efficient policy iteration algorithm with heavy-tail acceleration, employing a hybrid (uniform/log-spaced) action grid and a far-field linear closure. Simulations under Pareto and log-normal service times demonstrate substantial improvements over AoI-optimal non-preemptive sampling and zero-wait baselines, achieving up to a 30x reduction in average cost in heavy-tailed regimes. Finally, simulations uncover a counterintuitive insight: under preemption, delay variance, despite typically being a liability, can become a strategic advantage for information freshness.

We determine, for any n $\ge$ 1, the generalized eigenvalues of an n x n MAX matrix to the corresponding MIN matrix. We also show that a similar result holds for the generalized eigenvalues of an nxn LCM matrix to the corresponding GCD matrix when n $\le$ 4, but breaks down for n > 4. In addition, we prove Cauchy's interlacing theorem for generalized eigenvalues, and we conjecture an unexpected connection between the OEIS sequence A004754 and the appearance of -1 as a generalized eigenvalue in the LCM-GCD setting.

We investigate the applications of ovals in projective planes to distributed storage, with a focus on the Service Rate Region problem. Leveraging the incidence relations between lines and ovals, we describe a class of non-systematic MDS matrices with a large number of small and disjoint recovery sets. For certain parameter choices, the service-rate region of these matrices contains the region of a systematic generator matrix for the same code, yielding better service performance. We further apply our construction to analyze the PIR properties of the considered MDS matrices and present a one-step majority-logic decoding algorithm with strong error-correcting capability. These results highlight how ovals, a classical object in finite geometry, re-emerge as a useful tool in modern coding theory.

In this paper, we derive a practical, general framework for creating adaptive iterative (linearization or splitting) algorithms to solve multi-physics problems. This means that, given an iterative method, we derive \textit{a posteriori} estimators to predict the success or failure of the method. Based on these estimators, we propose adaptive algorithms, including adaptively switching between methods, adaptive time-stepping methods, and the adaptive tuning of stabilization parameters. We apply this framework to two-phase flow in porous media, surfactant transport in porous media, and quasi-static poroelasticity.

Famously, every finite simple group $G$ can be generated by a pair of elements. Moreover, Liebeck and Shalev (1995) proved that the probability that a pair of elements generate $G$ tends to $1$ as $|G| \to \infty$. More generally, work of Lucchini and Menegazzo (1997) implies that $G$ can be generated by a pair of elements whenever $G$ has a unique chief series. In this paper, we generalize the theorem of Liebeck and Shalev by proving that if $G$ has a unique chief series and the unique simple quotient of $G$ is $S$, then the probability that a pair of elements generate $G$ tends to $1$ as $|S| \to \infty$. As a consequence of our main theorem, for any profinite group $G$ where the open normal subgroups form a chain, the probability that a pair of elements topologically generate $G$ is positive. Along the way, we establish results on the maximal subgroup zeta function of groups with a unique minimal normal subgroup.

Solving Stefan problems via neural networks is inherently challenged by the nonlinear coupling between the solutions and the free boundary, which results in a non-convex optimization problem. To address this, this work proposes an Operator Splitting Method (OSM) based on Extreme Learning Machines (ELM) to decouple the geometric interface evolution from the physical field reconstruction. Within a predictor-corrector framework, the method splits the coupled system into an alternating sequence of two linear and convex subproblems: solving the diffusion equation on fixed subdomains and updating the interface geometry based on the Stefan condition. A key contribution is the formulation of both steps as linear least-squares problems; this transforms the computational strategy from a non-convex gradient-based optimization into a stable fixed-point iteration composed of alternating convex solvers. From a theoretical perspective, the relaxed iterative operator is shown to be locally contractive, and its fixed points are consistent with stationary points of the coupled residual functional. Benchmarks across 1D to 3D domains demonstrate the stability and high accuracy of the method, confirming that the proposed framework provides a highly accurate and efficient numerical solution for free boundary problems.

Erdős, Sárközy, and Sós studied the asymptotics of the maximum size of a subset of $\{1,2,\ldots, N\}$ such that it does not contain $k$ distinct elements whose product is a perfect square. More generally, Verstraëte proposed a conjecture regarding the asymptotic behavior of the same quantity with the set of perfect squares replaced by the value set of a polynomial in $\mathbb{Z}[x]$. In this paper, we study a finite field analogue of Verstraëte's conjecture.

In this paper we obtain uniformly locally $L^{\infty}$-estimate of solutions to non-autonomous quasilinear system involving operators in divergence form and a family of nonlinearities that are allowed to grow also critically.

We propose an algorithm that approximates a given matrix polynomial of degree $d$ by another skew-symmetric matrix polynomial of a specified rank and degree at most $d$. The algorithm is built on recent advances in the theory of generic eigenstructures and factorizations for skew-symmetric matrix polynomials of bounded rank and degree. Taking into account that the rank of a skew-symmetric matrix polynomial is even, the algorithm works for any prescribed even rank greater than or equal to $2$ and produces a skew-symmetric matrix polynomial of that exact rank. We also adapt the algorithm for matrix pencils to achieve a better performance. Lastly, we present numerical experiments for testing our algorithms and for comparison to the previously known ones.

A source encoder is stable if a small change in the source sequence (e.g., changing a few symbols) results in a small (or bounded) change in the output codeword. By this definition, the common technique of random binning is unstable; because the mapping is random, two nearly identical source sequences can be assigned to completely unrelated bin indices. We study compression rates of stable lossless source codes. Using combinatorial arguments, we derive information-theoretic limits on the achievable rate as a function of the stability parameters.

We focus on the optimization problem with smooth, possibly nonconvex objectives and a convex constraint set for which the Euclidean projection operation is practically available. Focusing on this setting, we carry out a general convergence and complexity analysis for algorithmic frameworks. Consequently, we discuss theoretically sound strategies to integrate momentum information within classical projected gradient type algorithms. One of these approaches is then developed in detail, up to the definition of a tailored algorithm with both theoretical guarantees and reasonable per-iteration cost. The proposed method is finally shown to outperform the standard (spectral) projected gradient method in two different experimental benchmarks, indicating that the addition of momentum terms is as beneficial in the constrained setting as it is in the unconstrained scenario.

We propose a new asymptotic test for the separability of a covariance matrix. The null distribution is valid in wide matrix elliptical model that includes, in particular, both matrix Gaussian and matrix $t$-distribution. The test is fast to compute and makes no assumptions about the component covariance matrices. An alternative, Wald-type version of the test is also proposed. Our simulations reveal that both versions of the test have good power even for heavier-tailed distributions and can compete with the Gaussian likelihood ratio test in the case of normal data.

We study the recurrence properties of certain skew products over symmetric interval exchange transformations, including rotations, with cocycles of the form $f(x)=-\frac{1}{x^a}+\frac{1}{(1-x)^a}$, where $a>1$. We prove that typically, such systems are dissipative. However, at the same time they are \emph{topologically recurrent}, i.e. for every open rectangle $A\subset[0,1)\times \R$, there exists an infinite sequence $(q_n)_{n=1}^{\infty}$ such that $T^{q_n}_f(A)\cap A\neq\emptyset$.

We formulate and solve two resource allocation problems motivated by a preparedness question of emergency response services. First, we consider the assignment of vehicles to stations, and, in a second step, assign crews to vehicles. In both cases, we work in a minimax framework and define the objective function for a spatial catchment area as the total risk in this area per resource unit allocated to it. The solutions are explicit and can be calculated in practice by a greedy algorithm that successively allocates a resource unit to an area having maximal relative risk, with suitable tie breaker rules. The approach is illustrated on a data set of incidents reported to the Twente Fire Brigade.

In this paper, we investigate the optimal transport problem when the source is a non-convex polygonal domain in $\mathbb{R}^2$. We show a global $W^{2,1+\epsilon}$ estimate for potentials of optimal transport. Our method applies to a more general class of domains.

This work presents the numerical analysis of a barotropic-baroclinic splitting in a nonlinear multilayer framework with exchanges between the layers in terrain-following coordinates. The splitting is formulated as an exact operator splitting. The barotropic step handles free surface evolution and depth-averaged velocity via a well-balanced one-layer model, while the baroclinic step manages vertical exchanges between layers and adjusts velocities to their mean values. We show that the barotropic-baroclinic splitting preserves total energy conservation and meets both a discrete maximum principle and a discrete entropy inequality. Several numerical experiments are presented showing the gain in computational cost, particularly in low Froude simulations, with no loss of accuracy. The benefits of using a well-balancing strategy in the barotropic step to preserve the geostrophic equilibrium are inherited in the overall scheme.

Spatial Poisson point processes on finite-dimensional Euclidean space provide fundamental mathematical tools for modeling random spatial point patterns. In this paper, we introduce and analyze several Poisson-type spatial point processes. In particular, we propose and study a point process, namely, the generalized Poisson random field (GPRF), in which more than one point can be observed with positive probability, within a rectangular region having infinitesimal Lebesgue measure. A thinning of the GPRF into independent GPRFs with reduced rate parameters is discussed. Furthermore, we consider these processes indexed by the positive quadrant of the plane and analyze their fractional variants. Various distributional properties of these processes and related governing differential equations are obtained. Later, we define and analyze a spatial Skellam-type point process via GPRF. Moreover, a fractional variant of it in the two parameter case is studied in detail.

This work presents a Discrete de Rham (DDR) numerical scheme for solving linear elasticity problems on general polyhedral meshes, with a focus on preventing volumetric locking in the quasi-incompressible regime. The method is formulated as a nodal-based approach using the lowest-order gradient space of the DDR complex, enriched with scalar face bubble degrees of freedom that effectively capture the normal flux across element faces. This face-bubble enrichment is crucial for ensuring sufficient approximation flexibility of the divergence field, thereby eliminating the {volumetric locking} phenomenon that typically occurs as the Lamé parameter $\lambda$ approaches infinity. We establish $H^1$-error estimates that are independent of $\lambda\ge 0$, and depend only on the lower bound of $\mu$, guaranteeing robustness across the entire range from compressible to nearly incompressible regimes. We also show how to adapt our scheme to the frictionless contact mechanics model, maintaining a locking-free estimate for the primal variable (displacement). Numerical experiments confirm that the proposed {locking-free} method delivers accurate and stable approximations on general polytopal discretizations, even when the material behaves as an incompressible medium. The flexibility and robustness of this approach make it a practical alternative to mixed formulations for engineering applications involving nearly incompressible elastic materials.

For a Serre subcategory $\mathscr L$ and a resolving subcategory $\mathscr A$ of an abelian category, we show that the derived equivalence $D^b(\overline{\mathscr A} \cap \mathscr L) \simeq D^b_{\mathscr L}(\mathscr A)$ holds under certain conditions. We apply this to obtain derived equivalences in the contexts of chain complexes of graded modules or coherent sheaves, with finite $\mathscr A$-dimension, supported on closed sets having eventually finite $\mathscr A$-dimension.

We give a counterexample to a conjecture made by Cigler, Jerman and Wojciechowski stating that all posets are conclusive. We also provide combinatorial characterizations for conclusiveness of finite posets and the existence of outer derivations.

We study the space of all bimodule homomorphisms $R_x\otimes_R R(\underline{t})\otimes_R R_y\to R_z\otimes_R R(\underline{t}')\otimes_R R_w$ as a one-sided module, where $R_x,R_y,R_z,R_w$ are standard twisted bimodules and $R(\underline{t})$ and $R(\underline{t}')$ are the Bott-Samelson bimodules corresponding to sequences of reflections $\underline{t}$ and $\underline{t}'$ respectively. We prove that this module is always reflexive under some reasonable restrictions on the representation of the underlying Coxeter group. However, unlike the case where $\underline{t}$ and $\underline{t}'$ contain only simple reflections, this module does not need any longer to be free. We provide a series of counterexamples already for the symmetric groups $S_n$, where $n\ge4$. The projective dimension of the modules dual to them is $n-3$ and thus serves to measure the deviation from the free modules. When placed within a geometric framework, these examples show how to find fibers of points fixed by the compact torus in the Bott-Samelson resolutions (as in the original definition by Raoul Bott and Hans Samelson) with non-vanishing odd cohomology.

In this paper we consider a model of the Dirac vacuum in classical electromagnetic fields at positive temperature. We adopt the Pauli-Villars regularisation technique in order to properly define the free energy of the vacuum, extending the previous work by the second named author on the purely magnetic case. This work is intended as a first step in understanding polarisation effects in the vacuum at positive temperature, in presence of both electrostatic and magnetic potentials.

In this paper, we present a new SDC scheme for solving semi-explicit DAEs with the ability to be parallelized in which only the differential equations are numerically integrated is presented. In Shu et al. (2007) it was shown that SDC for ODEs achieves one order per iteration. We show that this carries over to the new SDC scheme. The method is derived from the approach of spectral deferred corrections and the idea of enforcing the algebraic constraints without numerical integration as in the approach of $\varepsilon$-embedding in Hairer and Wanner (1996). It enforces the algebraic constraints to be satisfied in each iteration and allows an efficient solve of semi-explicit DAEs with high-accuracy. The proposed scheme is compared with other DAE methods. We demonstrate that the proposed SDC scheme is competitive with Runge-Kutta methods for DAEs in terms of accuracy and its parallelized versions are very efficient in comparison to other SDC methods.

We formulate a general version of the Peierls-Onsager substitution for a finite family of Bloch eigenvalues under a local spectral gap hypothesis, via strongly localized tight-frames and magnetic matrices. This extends the existing results to long-range magnetic fields without any slow-variation hypothesis and without any triviality assumption for the associated Bloch sub-bundle. Moreover, our results cover a large class of periodic, elliptic pseudo-differential operators. We also prove the existence of an approximate time evolution for initial states supported inside the range of the isolated Bloch family, with a precise error control.

In this paper, we study an optimal control problem for a cascade of hydroelectric power plants with reversible turbines and uncontrolled spillways. The system dynamics are governed by a linear control model subject to path constraints. The aim is to maximize the power production profit while respecting operational restrictions on reservoir water levels. The challenge is the presence of uncontrollable spillways: their discontinuous nature and the fact that they are activated at the state boundary prevent the application of known necessary conditions of optimality. To overcome this, we derive necessary conditions by approximating the original problem through a sequence of standard optimal control problems using exponential penalty functions. The applicability of resulting conditions are illustrated by an example.

In this paper, we are concerned with the nonlinear Helmholtz system of Hamiltonian type \begin{equation*} \left\{\begin{array}{l} -\Delta u-k^2 u=P(x)|v|^{p-2}v,\quad \text{in}\ \mathbb{R}^N, \\ -\Delta v-k^2v=Q(x)|u|^{q-2}u,\quad \text{in}\ \mathbb{R}^N, \end{array} \right. \end{equation*} where $N\geq3$, $P,Q: \mathbb{R}^N\rightarrow \mathbb{R}$ are two positive continuous functions, the exponents $p,q>2$ satisfy $\frac{1}{p}+\frac{1}{q}>\frac{N-2}{N}$. First, we obtained the existence of a ground state solution via a dual variational method. Moreover, the concentration behavior of such dual ground state solutions is established as $k\rightarrow\infty$, where a rescaling technique and the generalized Birman-Schwinger operator are involved. In addition, we also investigated the relation between the number of solutions and the topology of the set of the global maxima of the functions $P$ and $Q$.

Naciri proved that for any integer $k\geq2$, the Brocard--Ramanujan equation $n!+1=x^2$ has only finitely many integer solutions, assuming $x\pm1$ is a $k$-free integer or a prime power. In the present paper we prove similar statements for equations of the form $P(x)=\prod_{i=1}^rH_{n_i}$, where $P(x)$ is a polynomial and $H_{n_i}$ are divisible sequences.

We study the existence and the schematic structure of elementary components of the nested Hilbert scheme on a smooth quasi-projective variety. Precisely, we find a new lower bound for the existence of non-smoothable nestings of fat points on a smooth $n$-fold, for $n\geqslant 4$. Moreover, we implement a systematic method to build generically non-reduced elementary components. We also investigate the problem of irreducibility of the Hilbert scheme of points on a singular hypersurface of $\mathbb A^3$. Explicitly, we show that the Hilbert scheme of points on a hypersurface of $\mathbb{A}^3$ having a singularity of multiplicity at least 5 admits elementary components.

We compute the sum and the alternating sum of the reciprocals of triangular numbers using two methods: a telescoping series approach and a power series approach. We then extend these results to generalized (higher-order) triangular numbers and obtain closed-form expressions for both the non-alternating and alternating series in all orders.

Under the assumption that data lie on a compact (unknown) manifold without boundary, we derive finite sample bounds for kernel smoothing and its (first and second) derivatives, and we establish asymptotic normality through Berry-Esseen type bounds. Special cases include kernel density estimation, kernel regression and the heat kernel signature. Connections to the graph Laplacian are also discussed.

This paper presents a complete classification of minimal graph surfaces that admit graphical transformations into other minimal surfaces. These transformations are functions that map the height function of a minimal graph surface to another height function, which also describes a minimal graph surface. While trivial maps such as translations and reflections exist, we formulate and solve the Non-Trivial Minimal Graph Transformation Problem, governed by a coupled system of partial differential equations. A central result establishes the rigorous equivalence of this original system to a modified problem for a harmonic function. Through a complex variable approach and a weakening technique, the analysis is reduced to solving a fundamental ordinary differential equation parameterized by a real constant k. The explicit integration of this ordinary differential equation involves various elliptic integrals and identities of elliptic functions. Solving the ordinary differential equation for the three cases: when the constant k equals zero, when k is greater than zero, and when k is less than zero yields the full classification of all admissible surfaces and their associated transformations. This process yields several classes of minimal surfaces that, to the best of the author's knowledge, constitute new families of minimal surfaces.

An adaptive method connected with 3-point Gauss quadrature and 4-point Lobatto quadrature is introduced and investigated for 5-convex functions.

The 6G communication systems use mmWave and MIMO technologies to achieve wide bandwidth and high throughout, leading to indispensable need for beam alignment to overcome severe signal attenuation. Traditional sector-search-based beam alignment algorithms rely on sequential sampling to identify the best sector, resulting in a significant latency burden on 6G communication systems. Recently proposed adaptive beam alignment algorithms based on the active learning framework address the problem, aiming to identify the optimal sector with the fewest possible samples under an identical sector partition. Nevertheless, these algorithms either lack feasibility (Chiu, Ronquillo and Javidi, JSAC 2019) due to ideal assumptions or lack interpretability (Sohrabi, Chen and Yu, JSAC 2021) due to the use of end-to-end black-box neural networks. To avoid ideal assumptions and maintain interpretability, we address all above problems by proposing an adaptive beam alignment algorithm using the framework of noisy twenty questions estimation with a trained questioner. Specifically, we use two methods for training the questioner to eliminate reliance on ideal assumptions. The first method maps queries of twenty questions estimation to beamforming vectors via weighted summation of steering vectors, as an initial attempt to address the feasibility problem encountered in prior pioneering study by Chiu, Ronquillo and Javidi (JSAC 2019). The second method uses multi-layer fully connected neural networks to achieve improved performance while only employing them to train the questioner, which can effectively mitigate the interpretability issues in prior study by Sohrabi, Chen and Yu (JSAC 2021). Furthermore, we provide numerical simulations to illustrate the effectiveness of our proposed adaptive beam alignment algorithms and demonstrate that our algorithms outperform all benchmark algorithms.

We define a notion of marked length spectrum for $S^1$-symmetric Riemannian metrics on the two-sphere having only one equator. We prove that isospectral metrics in this class have conjugate geodesic flows. Under a further $\mathbb{Z}_2$-symmetry assumption, we show that the marked length spectrum determines the metric. Finally, we show that every isospectral class of metrics contains a unique $\mathbb{Z}_2$-symmetric metric and give an explicit description of this isospectral class as an infinite dimensional convex set, generalizing the known description of $S^1$-symmetric Zoll metrics. This paper contains also two appendices, in which we provide an elementary proof of the fact that a $C^2$ real valued function on an interval is determined by the set of tangent lines to its graph, and we classify a class of $S^1$-invariant contact forms on three-manifolds.

The present contribution aims at developing a non-overlapping Domain Decomposition (DD) approach to the solution of acoustic wave propagation boundary value problems based on the Helmholtz equation, on both bounded and unbounded domains. This DD solver, called Generalized Optimized Schwarz Method (GOSM), is a substructuring method, that is, the unknowns of an iteration are associated with the subdomains interfaces. We extend the analysis presented in a previous paper of one of the author to a fully discrete setting. We do not consider only a specific set of boundary conditions, but a whole class including, e.g., Dirichlet, Neumann, and Robin conditions. Our analysis will also cover interface conditions corresponding to a Finite Element Method - Boundary Element Method (FEM-BEM) coupling. In particular, we shall focus on three classical FEM-BEM couplings, namely the Costabel, Johnson-Nédélec and Bielak-MacCamy couplings. As a remarkable outcome, the present contribution yields well-posed substructured formulations of these classical FEM-BEM couplings for wavenumbers different from classical spurious resonances. We also establish an explicit relation between the dimensions of the kernels of the initial variational formulation, the local problems and the substructured formulation. That relation especially holds for any wavenumber for the substructured formulation of Costabel FEM-BEM coupling, which allows us to prove that the latter formulation is well-posed even at spurious resonances. Besides, we introduce a systematically geometrically convergent iterative method for the Costabel FEM-BEM coupling, with estimates on the convergence speed.

This paper studies the existence of multiple non-trivial stationary solutions of a partial differential equation (PDE) model introduced in [3], motivated by collective ant behavior. Previous work suggested the presence of two types of non-trivial stationary solutions for this PDE system: spot and lane solutions. In this paper, we establish the existence of these families of solutions along a bifurcation sequence as the interaction strength grows, with progressively increasing numbers of clusters and parallel lanes, respectively. Finally, we show that, for small values of the anticipation parameter, the first bifurcating spot solutions are locally dynamically stable, while the lane solutions are unstable.

We revisit noisy twenty questions estimation and study the privacy-resolution tradeoff for adaptive query procedures. Specifically, in twenty questions estimation, there are two players: an oracle and a questioner. The questioner aims to estimate target variables by posing queries to the oracle that knows the variables and using noisy responses to form reliable estimates. Typically, there are adaptive and non-adaptive query procedures. In adaptive querying, one designs the current query using previous queries and their noisy responses while in non-adaptive querying, all queries are posed simultaneously. Generally speaking, adaptive query procedures yield better performance. However, adaptive querying leads to privacy concerns, which were first studied by Tsitsiklis, Xu and Xu (COLT 2018) and by Xu, Xu and Yang (AISTATS 2021) for the noiseless case, where the oracle always provides correct answers to queries. In this paper, we generalize the above results to the more practical noisy case, by proposing a two-stage private query procedure, analyzing its non-asymptotic and second-order asymptotic achievable performance and discussing the impact of privacy concerns. Furthermore, when specialized to the noiseless case, our private query procedure achieves better performance than above-mentioned query procedures (COLT 2018, AISTATS 2021).

We find explicit solutions and singularities of the Ricci-harmonic flow of $\mathrm{G_2}$-structures, the Ricci-like flows of $\mathrm{G_2}$-structures studied by Gianniotis-Zacharopoulos in arXiv:2505.06872 (J. Geom. Anal. 36.2 (2026)) and of the negative gradient flow of an energy functional of $\mathrm{G_2}$-structures, on $7$-dimensional contact Calabi-Yau manifolds and the $7$-dimensional Heisenberg group. We prove that the natural co-closed $\mathrm{G_2}$-structure on a contact Calabi-Yau manifold as the initial condition leads to an ancient solution of the Ricci-harmonic flow with a finite time Type I singularity, and it gives an immortal solution to the Ricci-like flows with an infinite time singularity which are Type III if the transversal Calabi-Yau distribution is flat, and Type IIb otherwise. The same ansatz gives ancient solution to the negative gradient flow of $\mathrm{G_2}$-structures. These are the first examples of Type I singularities of the Ricci-harmonic flow and Type IIb and Type III singularities of the Ricci-like flows. We also obtain similar solutions for all the three flows on the $7$-dimensional Heisenberg group.

In this paper, we consider the problem of parametric empirical Bayes estimation of an i.i.d. prior in high-dimensional Bayesian linear regression, with random design. We obtain the asymptotic distribution of the variational Empirical Bayes (vEB) estimator, which approximately maximizes a variational lower bound of the intractable marginal likelihood. We characterize a sharp phase transition behavior for the vEB estimator -- namely that it is information theoretically optimal (in terms of limiting variance) up to $p=o(n^{2/3})$ while it suffers from a sub-optimal convergence rate in higher dimensions. In the first regime, i.e., when $p=o(n^{2/3})$, we show how the estimated prior can be calibrated to enable valid coordinate-wise and delocalized inference, both under the \emph{empirical Bayes posterior} and the oracle posterior. In the second regime, we propose a debiasing technique as a way to improve the performance of the vEB estimator beyond $p=o(n^{2/3})$. Extensive numerical experiments corroborate our theoretical findings.

The distinguishability quantified by information measures after being processed by a private mechanism has been a useful tool in studying various statistical and operational tasks while ensuring privacy. To this end, standard data-processing inequalities and strong data-processing inequalities (SDPI) are employed. Most of the previously known and even tight characterizations of contraction of information measures, including total variation distance, hockey-stick divergences, and $f$-divergences, are applicable for $(\varepsilon,0)$-local differential private (LDP) mechanisms. In this work, we derive both linear and non-linear strong data-processing inequalities for hockey-stick divergence and $f$-divergences that are valid for all $(\varepsilon,\delta)$-LDP mechanisms even when $\delta \neq 0$. Our results either generalize or improve the previously known bounds on the contraction of these distinguishability measures.

In this paper we prove that the smallest eigenvalue $\lambda_1$ of the eigenvalue problem for a quasilinear elliptic systems introduced by de Thélin in \cite{DT}, is not only simple (in a suitable sense), but also isolated. Moreover, we characterize variationally a sequence $\{\lambda_k\}_k$ of eigenvalues, taking into account a suitable deformation lemma for $C^1$ submanifolds proved in \cite{BON}. Furthermore we prove the existence of a weak solution for a quasilinear elliptic systems in resonance around $\lambda_1$, under new sufficient Landesman-Lazer type conditions, extending the results by Arcoya and Orsina \cite{AO}.

This work presents a stochastic analysis of fifth-order KdV soliton momentum distribution in a damping regime. An explicit representation of the soliton momentum associated with amplitude variation is derived in terms of a random time function in the presence of dissipation. Statistical interpretations of soliton propagation modes, amplitude fluctuations, and amplification are analyzed within a $\delta$-correlated Gaussian random framework. Graphical results obtained using Python illustrate the physical insight into amplitude fluctuation and energy flow. Finally, under a dominant approximation, the nonlinear momentum evolution equation is shown to reduce to the Painlevé second equation, a well-known integrable model appearing in diverse physical systems.

We consider the problem of sharing correlated data under a perfect information-theoretic privacy constraint. We focus on redaction (erasure) mechanisms, in which data are either withheld or released unchanged, and measure utility by the average cardinality of the released set, equivalently, the expected Hamming distortion. Assuming the data are generated by a finite time-homogeneous Markov chain, we study the protection of the initial state while maximizing the amount of shared data. We establish a connection between perfect privacy and window-based redaction schemes, showing that erasing data up to a strong stationary time preserves privacy under suitable conditions. We further study an optimal sequential redaction mechanism and prove that it admits an equivalent window interpretation. Interestingly, we show that both mechanisms achieve the optimal distortion while redacting only a constant average number of data points, independent of the data length~$N$.

For a finite metric graph $X=(V,E,\ell)$, where $V$ is endowed with the shortest path metric, we consider the transportation cost problem associated with the distance $d$ on $V$. Namely, for $f$ a function with total sum 0 on $V$, write $f=\sum_{a,b\in V}P(a,b)(\delta_a-\delta_b)$ where the transportation plan $P$ satisfies $P(a,b)\geq 0$ for $(a,b)\in V\times V$. The cost of $P$ is $W(P):=\sum_{a,b\in V}P(a,b)d(a,b)$ and the transportation norm of $f$ is $\|f\|_{TC}=\min_P W(P)$ where $P$ runs over all transportation plans for $f$. In this semi-survey paper, we give short proofs for the following statements: 1)There always exists an optimal transportation plan supported in $V_+\times V_-$ where $V_+=\{x\in V: f(x)>0\}$ and $V_-=\{x\in V: f(x)<0\}$. If $X$ is a metric tree, we may moreover assume that this plan involves at most $|Supp(f)|-1$ transports. 2) There always exists an optimal transportation plan supported in the set of edges of $X$. 3) Better, there always exists an optimal transportation plan supported in some spanning tree of $X$. We use this to reprove known formulae for the transportation norm when $X$ is either a tree or a cycle.

We recall the Lounesto classification of 1/2-spin spinor fields, based on the vanishing of spinorial bilinear quantities: the classes are the regular spinor fields (i.e. the Dirac field), as well as singular spinor fields, also known as flag-dipole spinor fields, admitting two limiting sub-classes, given by the dipole spinors (i.e. the Weyl spinor) and the flagpole spinors (i.e. the Majorana spinor). We discuss each class in terms of its representatives, with particular emphasis upon the flag-dipole spinor fields.

We consider a three-dimensional domain occupied by a homogeneous, incompressible, non-Newtonian, heat-conducting fluid with prescribed nonuniform temperature on the boundary and no-slip boundary conditions for the velocity. No external body forces are assumed. The constitutive relation for the Cauchy stress tensor is assumed in a general form that includes, in particular, the power-law and Ladyzhenskaya models with the power-law exponent in the range where neither regularity, uniqueness, nor the validity of the energy equality is known to hold. Nevertheless, we introduce a novel concept of solution suitable for this setting, which enables us to establish the existence of global-in-time solutions for arbitrary physically relevant initial data. A remarkable feature of this formulation is that the steady-state solution is nonlinearly stable: every such solution converges, in a suitable sense, to the steady state as time tends to infinity. This provides the first result that combines existence with long-time stability in this physically relevant yet mathematically challenging regime.

We study virtual retracts in groups acting on rooted trees. We show that finitely generated branch groups do not have the local retraction (LR) property. Furthermore, we specialize to iterated monodromy groups of post-critically finite quadratic complex polynomials and show that the (LR) property characterizes, among post-critically finite quadratic complex polynomials, those with a euclidean orbifold, i.e. the powering map and the Chebyshev polynomial. Lastly, we show that periodic quadratic complex polynomials provide new examples of pro-$2$ groups with complete finitely generated Hausdorff spectrum.

We generalize a result of Popa-Schnell and show that the isogeny class of the Picard variety is twisted derived invariant. Using this, we prove that any twisted Fourier-Mukai partner of an abelian variety is an abelian variety. We then provide a necessary and sufficient isogeny-based condition for two abelian varieties to be twisted derived equivalent.

We construct an explicit isomorphism between the HOMFLY-PT homology of $(n,n+1)$ torus knots and the direct sum of hook isotypic components of the space of diagonal coinvariants. As a consequence, we compute the action of tautological classes in HOMFLY-PT homology of $(n,n+1)$ torus knots and prove that it extends to an action of the Lie algebra of Hamiltonian vector fields on the plane. We also compute the action of differentials $d_N$ in Rasmussen spectral sequences from HOMLFY-PT to $\mathfrak{gl}(N)$ homology of $(n,n+1)$ torus knots.

Many applications, such as systems of interacting particles in physics, require the simulation of diffusion processes with singular coefficients. Standard Euler schemes are then not convergent, and theoretical guarantees in this situation are scarce. In this work we introduce a Lyapunov-tamed Euler scheme, for drift coefficients for which the weak derivative is dominated by a function that obeys a certain generic Lyapunov-type condition. This allows for a range of coefficients that explode to infinity on a bounded set. We establish that, in terms of Lp-strong error, the Lyapunov-tamed scheme is consistent and moreover achieves the same order of convergence as the standard Euler scheme for Lipschitz coefficients. The general result is applied to systems of mean-field particles with singular repulsive interaction in 1D, yielding an error bound with polynomial dependency in the number of particles.

A new notion of thickness for subsets of $B[0,1]\subset \mathbb{R}^n$ called affine thickness is defined; this notion of thickness is a generalisation of Falconer-Yavicoli thickness and is adapted to be used in the study of certain sets with affine cut outs. Thick sets are proven to be winning for the matrix potential game introduced in (arXiv:2508.11577) and as an application we can prove that for a thick set, there exists $M\in\mathbb{N}$ depending on the thickness of the set, such that the set contains a homothetic copy of every finite set with at most $M$ elements. Additionally, the author provides a counter-example to the gap lemma in $\mathbb{R}^n$ ($n\geq 2$) for Falconer-Yavicoli thickness, stated in (Math. Z., 2022) proving this result does not hold in the generality stated. We go on to provide a gap lemma for affine thickness in $\mathbb{R}^n$ (for $n\geq 2$) under additional conditions to the classical Newhouse gap lemma.

We study canonical basis elements in higher-level Fock spaces associated with the quantum group $U_q(\mathfrak{gl}_\infty)$, which are conjecturally related to Calogero-Moser theory for complex reflection groups. We generalize the Leclerc-Miyachi formula to arbitrary levels by introducing new explicit constructions based on symbols, including a column removal theorem and closed formulas in several cases. These results provide explicit descriptions of canonical basis elements with applications to Calogero-Moser cellular characters and to the decomposition matrices of Ariki-Koike algebras.

Representations of the operator system determined by the canonical generators of the free product of two cyclic groups of order $2$ and $k$, or $d$ cyclic groups of order $2$, are studied for the purpose of shedding light on the noncommutative geometry of noncommutative $d$-cubes and $k$-prisms. By way of the duality of the categories NCConv and OpSys of noncommutative convex sets and operator systems, respectively, an analysis of noncommutative extreme points, exactness, the lifting property, automatic complete positivity, controlled completely positive extensions, tensor products, and operator system duality is undertaken. Of note is the pairing of two classical dilation theorems of Halmos and Mirman to give a complete description of the noncommutative triangular prism in terms of joint unitary dilations.

We propose a new kind of localized shock capturing for continuous (CG) and discontinuous Galerkin (DG) discretizations of hyperbolic conservation laws. The underlying framework of dissipation-based weighted essentially nonoscillatory (WENO) stabilization for high-order CG and DG approximations was introduced in our previous work. In this general framework, Hermite WENO (HWENO) reconstructions are used to calculate local smoothness sensors that determine the appropriate amount of artificial viscosity for each cell. In the original version, candidate polynomials for WENO averaging are constructed using the derivative data from von Neumann neighbors. We upgrade this standard `cell-cell' reconstruction procedure by using WENO polynomials associated with mesh vertices as candidate polynomials for cell-based WENO averaging. The Hermite data of individual cells is sent to vertices of those cells, after which vertex-averaged HWENO data is sent back to cells containing the vertices. The new `cell-vertex' averaging procedure includes the data of vertex neighbors without explicitly adding them to the reconstruction stencils. It mitigates mesh imprinting and can also be used in classical HWENO limiters for DG methods. The second main novelty of the proposed approach is a quadrature-driven distribution of artificial viscosity within high-order finite elements. Replacing the linear quadrature weights by their nonlinear WENO-type counterparts, we concentrate shock-capturing dissipation near discontinuities while minimizing it in smooth portions of troubled cells. This redistribution of WENO stabilization preserves the total dissipation rate within each cell and improves local shock resolution without relying on subcell decomposition techniques. Numerical experiments in one and two dimensions demonstrate substantial improvements in accuracy and robustness for high-order elements.

The Fourier transform of a bounded measurable function, $f$, on the real line is shown to be the second distributional derivative of a Hölder continuous function. The Fourier transform is written as the difference of $\int_{-1}^1 e^{-ist}f(t)\,dt$ and the second distributional derivative of the integral $\int_{\lvert{t}\rvert>1}e^{-ist}f(t)\,dt/t^2$. The space of such Fourier transforms is isometrically isomorphic to $L^\infty(\mathbb{R})$. There is an exchange theorem, inversion and convolution results. The Fourier transform of the functions $x\mapsto\cos^m(a/x)$ for each natural number $m$ are computed. Also for $x\mapsto x\sin(a/x)$ and $x\mapsto\arctan(x/a)$.

A cap set in projective or affine geometry over a finite field is a set of points no three of which are collinear. In this paper, we propose a new construction for complete cap sets that yields a cap set of size 124928 in the affine geometry AG(15,3). It should be noted that the constructed cap set in AG(15,3) is more powerful and exceeds at least by 4096 points than those that can be obtained from the previously known ones using the product or doubling constructions.

We construct characteristic-zero lifts of partial Hasse invariants for genus zero non-compact curves in Hilbert modular varieties. The construction is based on recent results on the associated Picard-Fuchs differential equations. As an application, we relate the size of the non-ordinary locus of the modulo $p$ reduction of these curves to the dimension of spaces of (twisted) modular forms. We compute it explicitly for several Teichmüller curves, obtaining Deuring-like formulae. Moreover, we study the modulo $p$ reduction of (twisted) modular forms on not necessarily arithmetic genus-zero Fuchsian groups with modular embedding.

We prove the (graded) Jordan-Hölder multiplicities of (mixed) tilting sheaves on flag varieties admit a geometric interpretation as the hypercohomology of certain sheaves on Richardson varieties in the Langlands dual flag variety. These sheaves are a motivic variant of geometric extensions, and provide a replacement for parity sheaves on the Richardson variety. We also provide an explicit formula for these multiplicities in terms of $\ell$-Kazhdan-Lusztig polynomials.

We study controlability issues for the group of volume-preserving diffeomorphisms of the torus $\mathbb T^d$ for system $\dot x=f(x)+u(t)$, where $f$ is a fixed divergence free vector field on $\mathbb T^d$ and $u(t)$ are constant vector fields which generate translations of the torus. Main results concern $d$ equals two or three.

Generalizing the work of Atkin and Kaneko-Zagier in the elliptic case, we describe the non-ordinary locus of a genus-zero non-compact curve $Y$ in a Hilbert modular variety in terms of the zeros of generalized Atkin's orthogonal polynomials. The argument relies on the recent construction of lifts of partial Hasse invariants for $Y$. We further describe these orthogonal polynomials as denominators of Padé approximants to the logarithmic derivatives of solutions of the Picard-Fuchs differential equations associated with $Y$. This provides a new link between Padé approximation and the geometry of the non-ordinary locus, extending a classical observation of Igusa for the Legendre family and applying, in particular, to situations where the Picard-Fuchs equations do not admit modular solutions. As applications, we determine the three-term recurrence relations for Atkin polynomials attached to triangle curves via hypergeometric identities, and compute the supersingular locus of a double cover of the Teichmüller curve $W_{17}$. In the latter case, we conjecture that the associated supersingular polynomial is self-reciprocal, implying that supersingular points occur in pairs.

We study Bayesian model selection in colored Gaussian graphical models (CGGMs), which combine sparsity of conditional independencies with symmetry constraints encoded by vertex- and edge-colored graphs. A computational bottleneck in Bayesian inference for CGGMs is the evaluation of Diaconis-Ylvisaker normalizing constants, given by gamma-type integrals over cones of precision matrices with prescribed zeros and equality constraints. While explicit formulas are known for standard Gaussian graphical models only in special cases (e.g. decomposable graphs) and for a limited class of RCOP models, no general tractable framework has been available for broader families of CGGMs. We introduce a new subclass of RCON models for which these normalizing constants admit closed-form expressions. On the algebraic side, we identify conditions on spaces of colored precision matrices that guarantee tractability of the associated integrals, leading to Block-Cholesky spaces (BC-spaces) and Diagonally Commutative Block-Cholesky spaces (DCBC-spaces). On the combinatorial side, we characterize the colored graphs inducing such spaces via a color perfect elimination ordering and a 2-path regularity condition, and define the resulting Color Elimination-Regular (CER) graphs and their symmetric variants. This class strictly extends decomposable graphs in the uncolored setting and contains all RCOP models associated with decomposable graphs. In the one-color case, our framework reveals a close connection between DCBC-spaces and Bose-Mesner algebras. For models defined on BC- and DCBC-spaces, we derive explicit closed-form formulas for the normalizing constants in terms of a finite collection of structure constants and propose an efficient method for computing them in the commutative case. Our results broaden the range of CGGMs amenable to principled Bayesian structure learning in high-dimensional applications.

The Isometry Theorem of Chazal et al. and Lesnick is a fundamental result in persistence theory, which states that the interleaving distance between two one-parameter persistence modules is equal to the bottleneck distance between their barcodes. Significant effort has been devoted to extending this result to modules defined over more general posets. As these modules do not generally admit nice decompositions, one must restrict attention to the class of interval-decomposable modules in order to define an appropriate notion of bottleneck distance. Even with this assumption, it is known that bottleneck distance may not be equivalent to interleaving distance, but that it is Lipschitz stable under certain, fairly restrictive, assumptions. In this paper, we consider the more basic question of stability of the Hausdorff distance with respect to interleaving distance for interval-decomposable modules. Our main theorem is a Lipschitz stability result, which holds in a fairly general setting of interval-decomposable modules over arbitrary posets, where intervals are assumed to be taken from any family satisfying certain closure conditions. Along the way, we develop some new tools and results for interval-decomposable modules over arbitrary posets, in the form of geometrically-flavored characterizations of the existence of morphisms and interleavings between interval modules.

Given a graph $G$, the family of all independent sets of size $k$ containing a fixed vertex $v$ is called a star with centre $v$, and is denoted by $\mathcal{I}_G^k(v)$. Motivated by a generalisation of the Erdős-Ko-Rado Theorem to the setting of independent sets in graphs, Hurlbert and Kamat conjectured that for every tree $T$ and every $k$, the maximum of $|\mathcal{I}_T^k(v)|$ can always be attained by a leaf of $T$. While this conjecture turns out to be false in general, it is known to hold for specific families of trees like spiders and caterpillars. In this paper, we prove that this conjecture holds for a new family of trees, the perfect $r$-ary trees, by constructing injections from stars centred at arbitrary vertices to stars centred at leaves. We also show that the analogous property holds for every forest $\mathcal{T}$ that is the disjoint union of perfect trees with possibly varying sizes and arities, and determine the leaf that maximises $|\mathcal{I}_{\mathcal{T}}^k(v)|$.

We obtain strict stability inequalities for homogeneous solutions of the one-phase Bernoulli problem. We prove that in dimension $7$ and above, cohomogeneity one solutions with bi-orthogonal symmetry are strictly stable. As a consequence, we obtain a bound on the first eigenvalue and the decay rates of Jacobi fields, with applications to the generic regularity of the one-phase problem.

The evaluation of heuristic optimizers on test problems, better known as \emph{benchmarking}, is a cornerstone of research in multi-objective optimization. However, most test problems used in benchmarking numerical multi-objective black-box optimizers come from one of two flawed approaches: On the one hand, problems are constructed manually, which result in problems with well-understood optimal solutions, but unrealistic properties and biases. On the other hand, more realistic and complex single-objective problems are composited into multi-objective problems, but with a lack of control and understanding of problem properties. This paper proposes an extensive problem generation approach for bi-objective numerical optimization problems consisting of the combination of theoretically well-understood convex-quadratic functions into unimodal and multimodal landscapes with and without global structure. It supports configuration of test problem properties, such as the number of decision variables, local optima, Pareto front shape, plateaus in the objective space, or degree of conditioning, while maintaining theoretical tractability: The optimal front can be approximated to an arbitrary degree of precision regarding Pareto-compliant performance indicators such as the hypervolume or the exact R2 indicator. To demonstrate the generator's capabilities, a test suite of 20 problem categories, called \emph{BONO-Bench}, is created and subsequently used as a basis of an illustrative benchmark study. Finally, the general approach underlying our proposed generator, together with the associated test suite, is publicly released in the Python package \texttt{bonobench} to facilitate reproducible benchmarking.

Confirming a conjecture by Erd\H os and Pomerance, we prove that there exist intervals of length $\frac{cn\log n}{\log \log n}$ that do not contain distinct multiples of $1, 2, \ldots, n$.

We describe a method to show a plane quartic over a number field has no rational points. The method can be adapted to show that a curve does not have divisors of degree 1 or 2 and can be generalized to arbitrary smooth projective curves. Our approach significantly improves on the applicability over previous 2-cover descent methods by not requiring the computation of the full $S$-unit group of the étale algebras involved. We illustrate the practicality with several examples, including examples where we determine plane quartics to be of index 2 or 4 when the maximum local index is strictly smaller.

We introduce the resilience rate as a measure of financial resilience. It captures the expected rate at which a dynamic risk measure recovers, i.e., bounces back, when the risk-acceptance set is breached. We develop the corresponding stochastic calculus by establishing representation theorems for expected time-derivatives of solutions to backward stochastic differential equations (BSDEs) with jumps, evaluated at stopping times. These results reveal that the resilience rate can be represented as a suitable expectation of the generator of a BSDE. We analyze the main properties of the resilience rate and the formal connection of these properties to the BSDE generator. We also introduce resilience-acceptance sets and study their properties in relation to both the resilience rate and the dynamic risk measure. We illustrate our results in several canonical financial examples and highlight their implications via the notion of resilience neutrality.

We introduce a quantitative method to compare arbitrary pairs of graph centrality measures, based on the ordering of vertices induced by them. The proposed method is conceptually simple, mathematically elegant, and allows for a quantitative restatement of many conjectures that were previously cumbersome to formalize. Moreover, it produces an approximation scheme useful for network scientists. We explore some of these uses and formulate new conjectures that are of independent interest.

Topological transitivity is a fundamental notion in topological dynamics and is widely regarded as a basic indicator of global dynamical complexity. For general cellular automata, topological transitivity is known to be undecidable. By contrast, positive decidability results have been established for one-dimensional group cellular automata over abelian groups, while the extension to higher dimensions and to non-abelian groups has remained an open problem. In this work, we settle this problem by proving that topological transitivity is decidable for the class of $d$-dimensional ($d\geq 1$) group cellular automata over arbitrary finite groups. Our approach combines a decomposition technique for group cellular automata, reducing the problem to the analysis of simpler components, with an extension of several results from the existing literature in the one-dimensional setting. As a consequence of our results, and exploiting known equivalences among dynamical properties for group cellular automata, we also obtain the decidability of several related notions, including total transitivity, topological mixing and weak mixing, weak and strong ergodic mixing, and ergodicity.

We study a general framework of distributional computational graphs: computational graphs whose inputs are probability distributions rather than point values. We analyze the discretization error that arises when these graphs are evaluated using finite approximations of continuous probability distributions. Such an approximation might be the result of representing a continuous real-valued distribution using a discrete representation or from constructing an empirical distribution from samples (or might be the output of another distributional computational graph). We establish non-asymptotic error bounds in terms of the Wasserstein-1 distance, without imposing structural assumptions on the computational graph.

When parameters of a dynamical system change sufficiently fast, critical transitions can take place even in the absence of bifurcations. This phenomenon is known as rate-induced tipping and has been reported in a variety of systems, from simple ordinary differential equations and maps to mathematical models in climate sciences and ecology. In most examples, the transition happens at a critical rate of parameter change, a rate-induced tipping point, and is associated with a simple unstable orbit (edge state). In this work, we show how this simple picture changes when non-attracting fractal sets exist in the autonomous system, a ubiquitous situation in non-linear dynamics. We show that these fractals in phase space induce fractals in parameter space, which control the rates and parameter changes that result in tipping. We explain how such rate-induced fractals appear and how the fractal dimensions of the different sets are related to each other. We illustrate our general theory in three paradigmatic systems: a piecewise linear one-dimensional map, the two-dimensional Hénon map, and a forced pendulum.

Spherically embedded spatial data are spatially indexed observations whose values naturally reside on or can be equivalently mapped to the unit sphere. Such data are increasingly ubiquitous in fields ranging from geochemistry to demography. However, analysing such data presents unique difficulties due to the intrinsic non-Euclidean nature of the sphere, and rigorous methodologies for statistical modelling, inference, and uncertainty quantification remain limited. This paper introduces a unified framework to address these three limitations for spherically embedded spatial data. We first propose a novel spherical spatial autoregressive model that leverages optimal transport geometry and then extend it to accommodate exogenous covariates. Second, for either scenario with or without covariates, we establish the asymptotic properties of the estimators and derive a distribution-free Wald test for spatial dependence, complemented by a bootstrap procedure to enhance finite-sample performance. Third, we contribute a novel approach to uncertainty quantification by developing a conformal prediction procedure specifically tailored to spherically embedded spatial data. The practical utility of these methodological advances is illustrated through extensive simulations and applications to Spanish geochemical compositions and Japanese age-at-death mortality distributions.

We study a structured bi-level optimization problem where the upper-level objective is a smooth function and the lower-level problem is policy optimization in a Markov decision process (MDP). The upper-level decision variable parameterizes the reward of the lower-level MDP, and the upper-level objective depends on the optimal induced policy. Existing methods for bi-level optimization and RL often require second-order information, impose strong regularization at the lower level, or inefficiently use samples through nested-loop procedures. In this work, we propose a single-loop, first-order actor-critic algorithm that optimizes the bi-level objective via a penalty-based reformulation. We introduce into the lower-level RL objective an attenuating entropy regularization, which enables asymptotically unbiased upper-level hyper-gradient estimation without solving the unregularized RL problem exactly. We establish the finite-time and finite-sample convergence of the proposed algorithm to a stationary point of the original, unregularized bi-level optimization problem through a novel lower-level residual analysis under a special type of Polyak-Lojasiewicz condition. We validate the performance of our method through experiments on a GridWorld goal position problem and on happy tweet generation through reinforcement learning from human feedback (RLHF).

Vapnik--Chervonenkis' theorem is a seminal result in machine learning. It establishes sufficient conditions for empirical probabilities to converge to theoretical probabilities, uniformly over families of events. It also provides an estimate for the rate of such uniform convergence. We revisit the probabilistic component of the classical argument. Instead of applying Hoeffding's inequality at the final step, we use a normal approximation with explicit Berry--Esseen error control. This yields a moderate-deviation sharpening of the usual VC estimate, with an additional factor of order $(\varepsilon\sqrt{n})^{-1}$ in the leading exponential term when $\varepsilon\sqrt{n}$ is large.

This note introduces a family of circulant quantum channels -- a subclass of the mixed-permutation channels -- and investigates its key structural and operational properties. We show that the image of the circulant quantum channel is precisely the set of circulant matrices. This characterization facilitates the analysis of arbitrary $n$-th order Bargmann invariants. Furthermore, we prove that the channel is entanglement-breaking, implying a substantially reduced resource cost for erasing quantum correlations compared to a general mixed-permutation channel. Applications of this channel are also discussed, including the derivation of tighter lower bounds for $\ell_p$-norm coherence and a characterization of its action in bipartite systems.

We reexamine the energy-momentum tensor in classical electrodynamics from the perspective of spacetime-dependent translations, i.e., diffeomorphism invariance in flat spacetime. When energy-momentum is identified through local translations rather than constant ones, a unique, symmetric, and gauge-invariant energy-momentum tensor emerges that satisfies a genuine off shell Noether identity without invoking the equations of motion. For the free electromagnetic field, this tensor coincides with the familiar Belinfante-Rosenfeld and Bessel-Hagen expressions, but arises here directly from spacetime-dependent translation symmetry rather than from improvement procedures or compensating gauge transformations. In interacting classical electrodynamics, comprising a point charge coupled to the electromagnetic field, diffeomorphism invariance yields well-defined energy-momentum tensors for the field and the particle, while the interaction term itself generates no independent local energy-momentum tensor. Its role is instead entirely encoded in the coupled equations of motion governing energy-momentum exchange, thereby resolving ambiguities in energy-momentum localization present in canonical and improvement-based approaches.

Solving massive-scale optimization problems requires scalable first-order methods with low per-iteration cost. This tutorial highlights a shift in optimization: using differentiable programming not only to execute algorithms but to learn how to design them. Modern frameworks such as PyTorch, TensorFlow, and JAX enable this paradigm through efficient automatic differentiation. Embedding first-order methods within these systems allows end-to-end training that improves convergence and solution quality. Guided by Fenchel-Rockafellar duality, the tutorial demonstrates how duality-informed iterative schemes such as ADMM and PDHG can be learned and adapted. Case studies across LP, OPF, Laplacian regularization, and neural network verification illustrate these gains.

Understanding why Transformers perform so well remains challenging due to their non-convex optimization landscape. In this work, we analyze a shallow Transformer with $m$ independent heads trained by projected gradient descent in the kernel regime. Our analysis reveals two main findings: (i) the width required for nonasymptotic guarantees scales only logarithmically with the sample size $n$, and (ii) the optimization error is independent of the sequence length $T$. This contrasts sharply with recurrent architectures, where the optimization error can grow exponentially with $T$. The trade-off is memory: to keep the full context, the Transformer's memory requirement grows with the sequence length. We validate our theoretical results numerically in a teacher-student setting and confirm the predicted scaling laws for Transformers.

We demonstrate how resonances in a quantum graph consisting of a compact core and semi-infinite leads can be identified from the eigenvalue behavior of the cut-off system.

Nonlinear dimensionality reduction techniques, particularly UMAP, are widely used for visualizing high-dimensional data. However, UMAP's local Euclidean distance assumption often fails to capture intrinsic manifold geometry, leading to topological tearing and structural collapse. We identify UMAP's sensitivity to the k-nearest neighbor graph as a key cause. To address this, we introduce Ollivier-Ricci curvature as a geometric prior, reinforcing edges at geometric bottlenecks and reducing redundant links. Since curvature estimation is noise-sensitive, we also incorporate a topological prior using Jaccard similarity to ensure neighborhood consistency. The resulting method, JORC-UMAP, better distinguishes true manifold structure from spurious connections. Experiments on synthetic and real-world datasets show that JORC-UMAP reduces tearing and collapse more effectively than standard UMAP and other DR methods, as measured by SVM accuracy and triplet preservation scores, while maintaining computational efficiency. This work offers a geometry-aware enhancement to UMAP for more faithful data visualization.

Low-complexity multiple-input multiple-output (MIMO) detection remains a key challenge in modern wireless systems, particularly for 5G reduced capability (RedCap) and internet-of-things (IoT) devices. In this context, the growing interest in deploying machine learning on edge devices must be balanced against stringent constraints on computational complexity and memory while supporting high-order modulation. Beyond accurate hard detection, reliable soft information is equally critical, as modern receivers rely on soft-input channel decoding, imposing additional requirements on the detector design. In this work, we propose recurSIC, a lightweight learning-based MIMO detection framework that is structurally inspired by successive interference cancellation (SIC) and incorporates learned processing stages. It generates reliable soft information via multi-path hypothesis tracking with a tunable complexity parameter while requiring only a single forward pass and a minimal parameter count. Numerical results in realistic wireless scenarios show that recurSIC achieves strong hard- and soft-detection performance at very low complexity, making it well suited for edge-constrained MIMO receivers.

Learning a stationary diffusion amounts to estimating the parameters of a stochastic differential equation whose stationary distribution matches a target distribution. We build on the recently introduced kernel deviation from stationarity (KDS), which enforces stationarity by evaluating expectations of the diffusion's generator in a reproducing kernel Hilbert space. Leveraging the connection between KDS and Stein discrepancies, we introduce the Stein-type KDS (SKDS) as an alternative formulation. We prove that a vanishing SKDS guarantees alignment of the learned diffusion's stationary distribution with the target. Furthermore, under broad parametrizations, SKDS is convex with an empirical version that is $\epsilon$-quasiconvex with high probability. Empirically, learning with SKDS attains comparable accuracy to KDS while substantially reducing computational cost and yields improvements over the majority of competitive baselines.

We develop and numerically analyze an algebraic-geometric framework for spin-adapted coupled-cluster (CC) theory. Since the electronic Hamiltonian is SU(2)-invariant, physically relevant quantum states lie in the spin singlet sector. We give an explicit description of the SU(2)-invariant (spin singlet) many-body space by identifying it with an Artinian commutative ring, called the excitation ring, whose dimension is governed by a Narayana number. We define spin-adapted truncation varieties via embeddings of graded subspaces of this ring, and we identify the CCS truncation variety with the Veronese square of the Grassmannian. Compared to the spin-generalized formulation, this approach yields a substantial reduction in dimension and degree, with direct computational consequences. In particular, the CC degree of the truncation variety -- governing the number of homotopy paths required to compute all CC solutions -- is reduced by orders of magnitude. We present scaling studies demonstrating asymptotic improvements and we exploit this reduction to compute the full solution landscape of spin-adapted CC equations for water and lithium hydride.

Quantum neural networks (QNNs) provide expressive probabilistic models by leveraging quantum superposition and entanglement, yet their practical training remains challenging due to highly oscillatory loss landscapes and noise inherent to near-term quantum devices. Existing training approaches largely rely on gradient-based procedural optimization, which often suffers from slow convergence, sensitivity to hyperparameters, and instability near sharp minima. In this work, we propose an alternative inverse-probability algebraic learning framework for QNNs. Instead of updating parameters through incremental gradient descent, our method treats learning as a local inverse problem in probability space, directly mapping discrepancies between predicted and target Born-rule probabilities to parameter corrections via a pseudo-inverse of the Jacobian. This algebraic update is covariant, does not require learning-rate tuning, and enables rapid movement toward the vicinity of a loss minimum in a single step. We systematically compare the proposed method with gradient descent and Adam optimization in both regression and classification tasks using a teacher-student QNN benchmark. Our results show that algebraic learning converges significantly faster, escapes loss plateaus, and achieves lower final errors. Under finite-shot sampling, the method exhibits near-optimal error scaling, while remaining robust against intrinsic hardware noise such as dephasing. These findings suggest that inverse-probability algebraic learning offers a principled and practical alternative to procedural optimization for QNN training, particularly in resource-constrained near-term quantum devices.

The properties of the solution to the adjoint two-dimensional boundary layer equations on a flat plate are investigated from the viewpoint of Libby-Fox theory that describes the algebraic perturbations to the Blasius boundary layer. The adjoint solution is obtained from the Green's function of the perturbation equation as a sum over the infinite perturbation modes of the Blasius solution. The analysis of the solution allows us to obtain constraints on the eigenvalues and eigenfunctions. The extension of the analysis to the case with non-zero pressure gradient, corresponding to the Falkner-Skan solution, is also briefly discussed.

We introduce SeeMPS, a Python library dedicated to implementing tensor network algorithms based on the well-known Matrix Product States (MPS) and Quantized Tensor Train (QTT) formalisms. SeeMPS is implemented as a complete finite precision linear algebra package where exponentially large vector spaces are compressed using the MPS/TT formalism. It enables both low-level operations, such as vector addition, linear transformations, and Hadamard products, as well as high-level algorithms, including the approximation of linear equations, eigenvalue computations, and exponentially efficient Fourier transforms. This library can be used for traditional quantum many-body physics applications and also for quantum-inspired numerical analysis problems, such as solving PDEs, interpolating and integrating multidimensional functions, sampling multivariate probability distributions, etc.

Variational Quantum Algorithms (VQAs) are a class of hybrid quantum-classical algorithms that leverage on classical optimization tools to find the optimal parameters for a parameterized quantum circuit. One relevant application of VQAs is the Variational Quantum Eigensolver (VQE), which aims at steering the output of the quantum circuit to the ground state of a certain Hamiltonian. Recent works have provided global convergence guarantees for VQEs under suitable local surjectivity and smoothness hypotheses, but little has been done in characterizing convergence of these algorithms when the underlying quantum circuit is affected by noise. In this work, we characterize the effect of different coherent and incoherent noise processes on the optimal parameters and the optimal cost of the VQE, and we study their influence on the convergence guarantees of the algorithm. Our work provides novel theoretical insight into the behavior of parameterized quantum circuits. Furthermore, we accompany our results with numerical simulations implemented via Pennylane.

Recent years have witnessed a resurgence in using ReLU neural networks (NNs) to represent model predictive control (MPC) policies. However, determining the required network complexity to ensure closed-loop performance remains a fundamental open problem. This involves a critical precision-complexity trade-off: undersized networks may fail to capture the MPC policy, while oversized ones may outweigh the benefits of ReLU network approximation. In this work, we propose a projection-based method to enforce hard constraints and establish a state-dependent Lipschitz continuity property for the optimal MPC cost function, which enables sharp convergence analysis of the closed-loop system. For the first time, we derive explicit bounds on ReLU network width and depth for approximating MPC policies with guaranteed closed-loop performance. To further reduce network complexity and enhance closed-loop performance, we propose a non-uniform error framework with a state-aware scaling function to adaptively adjust both the input and output of the ReLU network. Our contributions provide a foundational step toward certifiable ReLU NN-based MPC.

The Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) construction encodes all the data of a topological sigma-model in the finite-dimensional symplectic $Q$-manifold. Relaxing the nondegeneracy condition i.e. considering a presymplectic form instead, extends the construction to non-topological models. The gauge-invariant action functional of (presymplectic) AKSZ sigma model is written in terms of space-time differential forms and can be seen as a covariant multidimensional analogue of the usual 1st order Hamiltonian action. In this work, we show that the AKSZ construction has a natural generalisation where the target space $Q$-manifold is equipped with a form of arbitrary degree $\Omega$ (possibly inhomogeneous) which is $(\mathrm{d}+L_Q)$-closed. This data defines a higher-derivative generalisation of the AKSZ action which is still invariant under the natural gauge transformations determined by $Q$ and which is efficiently formulated in terms of a version of Chern-Weil map introduced by Kotov and Strobl. It turns out that a variety of interesting gauge theories, including higher-dimensional Chern-Simons theory, MacDowell-Mansouri-Stelle-West action and self-dual gravity as well as its higher spin extension, can be concisely reformulated as such multisymplectic AKSZ models. We also present a version of the construction in the setup of PDE geometry and demonstrate that the counterpart of the multisymplectic AKSZ action is precisely the standard multisymplectic formulation, where the Chern-Weil map corresponds to the usual pullback map.

Accurate and fast calculations of localized surface plasmon resonances (LSPR) in metallic nanoparticles is essential for applications in sensing, nano-optics, and energy harvesting. Although full-wave numerical techniques such as the boundary element method (BEM) or the discrete dipole approximation (DDA) provide high accuracy, their computational cost often hinders rapid parametric studies. Here it is presented an ultrafast method that avoids solving large eigenproblems. Instead, only the dipolar component of the induced surface charge density \((\sigma_{dipolar})\) is retained through a expansion into Cartesion dipole basis, yielding a compact $3\times3$ geometric formulation that avoids full boundary-integral solves. The spectral response is obtained in a similar way, by projecting the Neumann--Poincaré surface operator onto the dipole subspace and evaluating a Rayleigh quotient, giving geometry-only eigenvalues again without an $N\times N$ eigenproblem. A major advantage of this method is that all geometry-dependent quantities are computed once per nanoparticle, while material dispersion and environmental changes enter only through simple algebraic expressions for the polarizability, enabling rapid evaluation across wavelengths. Retardation effects are incorporated through the modified long-wavelength approximation (MLWA), extending accuracy into the weakly retarded regime. The resulting framework provides a valuable tool for fast modelling and optimization of plasmonic nanoparticles at a significant lesser computational cost than BEM, DDA, and other standard tools.

In many learning tasks, certain requirements on the processing of individual data samples should arguably be formalized as strict constraints in the underlying optimization problem, rather than by means of arbitrary penalties. We show that, in these scenarios, learning can be carried out exploiting a sequential penalty method that allows to properly deal with constraints. The proposed algorithm is shown to possess convergence guarantees under assumptions that are reasonable in deep learning scenarios. Moreover, the results of experiments on image processing tasks show that the method is indeed viable to be used in practice.

We give analytical results for propagation of uncertainty through trained multi-layer perceptrons (MLPs) with a single hidden layer and ReLU activation functions. More precisely, we give expressions for the mean and variance of the output when the input is multivariate Gaussian. In contrast to previous results, we obtain exact expressions without resort to a series expansion.

Edge intelligence enables AI inference at the network edge, co-located with or near the radio access network, rather than in centralized clouds or on mobile devices. It targets low-latency, resource-constrained applications with large data volumes, requiring tight integration of wireless access and on-site computing. Yet system performance and cost-efficiency hinge on joint pre-deployment dimensioning of radio and computational resources, especially under spatial and temporal uncertainty. Prior work largely emphasizes run-time allocation or relies on simplified models that decouple radio and computing, missing end-to-end correlations in large-scale deployments. This paper introduces a unified stochastic framework to dimension multi-cell edge-intelligent systems. We model network topology with Poisson point processes, capturing random user and base-station locations, inter-cell interference, distance-based fractional power control, and peak-power constraints. By combining this with queueing theory and empirical AI inference workload profiling, we derive tractable expressions for end-to-end offloading delay. These enable a non-convex joint optimization that minimizes deployment cost under statistical QoS guarantees, expressed through strict tail-latency and inference-accuracy constraints. We prove the problem decomposes into convex subproblems, yielding global optimality. Numerical results in noise- and interference-limited regimes identify cost-efficient design regions and configurations that cause under-utilization or user unfairness. Smaller cells reduce transmission delay but raise per-request computing cost due to weaker server multiplexing, whereas larger cells show the opposite trend. Densification reduces computational costs only when frequency reuse scales with base-station density; otherwise, sparser deployments improve fairness and efficiency in interference-limited settings.

We study instrumental-variable designs where policy reforms strongly shift the distribution of an endogenous variable but only weakly move its mean. We formalize this by introducing distributional relevance: instruments may be purely distributional. Within a triangular model, distributional relevance suffices for nonparametric identification of average structural effects via a control function. We then propose Quantile Least Squares (Q-LS), which aggregates conditional quantiles of X given Z into an optimal mean-square predictor and uses this projection as an instrument in a linear IV estimator. We establish consistency, asymptotic normality, and the validity of standard 2SLS variance formulas, and we discuss regularization across quantiles. Monte Carlo designs show that Q-LS delivers well-centered estimates and near-correct size when mean-based 2SLS suffers from weak instruments. In Health and Retirement Study data, Q-LS exploits Medicare Part D-induced distributional shifts in out-of-pocket risk to sharpen estimates of its effects on depression.

The minimal norm weight perturbations of DNNs required to achieve a specified change in output are derived and the factors determining its size are discussed. These single-layer exact formulae are contrasted with more generic multi-layer Lipschitz constant based robustness guarantees; both are observed to be of the same order which indicates similar efficacy in their guarantees. These results are applied to precision-modification-activated backdoor attacks, establishing provable compression thresholds below which such attacks cannot succeed, and show empirically that low-rank compression can reliably activate latent backdoors while preserving full-precision accuracy. These expressions reveal how back-propagated margins govern layer-wise sensitivity and provide certifiable guarantees on the smallest parameter updates consistent with a desired output shift.

We study multigrade deep learning (MGDL) as a principled framework for structured error refinement in deep neural networks. While the approximation power of neural networks is now relatively well understood, training very deep architectures remains challenging due to highly non-convex and often ill-conditioned optimization landscapes. In contrast, for relatively shallow networks, most notably one-hidden-layer $\texttt{ReLU}$ models, training admits convex reformulations with global guarantees, motivating learning paradigms that improve stability while scaling to depth. MGDL builds upon this insight by training deep networks grade by grade: previously learned grades are frozen, and each new residual block is trained solely to reduce the remaining approximation error, yielding an interpretable and stable hierarchical refinement process. We develop an operator-theoretic foundation for MGDL and prove that, for any continuous target function, there exists a fixed-width multigrade $\texttt{ReLU}$ scheme whose residuals decrease strictly across grades and converge uniformly to zero. To the best of our knowledge, this work provides the first rigorous theoretical guarantee that grade-wise training yields provable vanishing approximation error in deep networks. Numerical experiments further illustrate the theoretical results.

We introduce FedSGM, a unified framework for federated constrained optimization that addresses four major challenges in federated learning (FL): functional constraints, communication bottlenecks, local updates, and partial client participation. Building on the switching gradient method, FedSGM provides projection-free, primal-only updates, avoiding expensive dual-variable tuning or inner solvers. To handle communication limits, FedSGM incorporates bi-directional error feedback, correcting the bias introduced by compression while explicitly understanding the interaction between compression noise and multi-step local updates. We derive convergence guarantees showing that the averaged iterate achieves the canonical $\boldsymbol{\mathcal{O}}(1/\sqrt{T})$ rate, with additional high-probability bounds that decouple optimization progress from sampling noise due to partial participation. Additionally, we introduce a soft switching version of FedSGM to stabilize updates near the feasibility boundary. To our knowledge, FedSGM is the first framework to unify functional constraints, compression, multiple local updates, and partial client participation, establishing a theoretically grounded foundation for constrained federated learning. Finally, we validate the theoretical guarantees of FedSGM via experimentation on Neyman-Pearson classification and constrained Markov decision process (CMDP) tasks.

We introduce a definition of thickness in $\mathbb{R}^d$ and obtain a lower bound for the Hausdorff dimension of the intersection of finitely or countably many thick compact sets using a variant of Schmidt's game. As an application we prove that given any compact set in $\mathbb{R}^d$ with thickness $\tau$, there is a number $N(\tau)$ such that the set contains a translate of all sufficiently small similar copies of every set in $\mathbb{R}^d$ with at most $N(\tau)$ elements; indeed the set of such translations has positive Hausdorff dimension. We also prove a gap lemma and bounds relating Hausdorff dimension and thickness.

Using the concept of prime submodule introduced by Raggi this http URL. we extend the notion of reduced rank to the module-theoretic context of $\sigma[M]$. We study the quotient category of $\sigma[M]$ modulo the hereditary torsion theory cogenerated by the $M$-injective hull of $M$, when $M$ is a semiprime Goldie module. We prove that this quotient category is spectral. We then consider the hereditary torsion theory in $\sigma[M]$ cogenerated by the $M$-injective hull of $M/\mathfrak{L}(M)$, where $\mathfrak{L}(M)$ is the prime radical of $M$, and we determine when the module of quotients of $M$, with respect to this torsion theory, has finite length in the quotient category. Finally, we give conditions on a module $M$ with endomorphism ring $S$ under which $S$ is an order in an Artinian ring, extending Small's Theorem.

Let $R \to S$ be a cyclically pure map of Noetherian $\mathbb{Q}$-algebras. In this paper, we show that if $S$ has Du Bois singularities, then $R$ has Du Bois singularities. Our result is new even when $R \to S$ is faithfully flat. Our proof also yields interesting results in prime characteristic and in mixed characteristic. As a consequence, we show that if $R \to S$ is a cyclically pure map of rings essentially of finite type over the complex numbers $\mathbb{C}$, $S$ has log canonical type singularities, and $K_R$ is Cartier, then $R$ has log canonical singularities. Along the way, we prove a version of the key injectivity theorem of Kovács and Schwede for Noetherian schemes of equal characteristic zero that have isolated non-Du Bois points. Throughout the paper, we use the characterization of the complex $\underline{\Omega}^0_X$ and of Du Bois singularities in terms of sheafification with respect to Grothendieck topologies.

We establish an algebra isomorphism between the center of the category $\mathcal{O}$ for a hybrid quantum group at a root of unity $\zeta$ and the cohomology of $\zeta$-fixed locus on affine Grassmannian. A deformed version of this isomorphism was established in the previous paper of the author. For the Steinberg block of $\mathcal{O}$, we construct an abelian equivalence to the category of equivariant sheaves on the Springer resolution.

In this paper, we introduce generalized Gorenstein local (GGL) rings. The notion of GGL rings is a natural generalization of the notion of almost Gorenstein rings, which can thus be treated as part of the theory of GGL rings. For a Cohen-Macaulay local ring $R$, we explore the endomorphism algebra of the maximal ideal, the trace ideal of the canonical module, Ulrich ideals, and Rees algebras of parameter ideals in connection with the GGL property. We also give numerous examples of numerical semigroup rings, idealizations, and determinantal rings of certain matrices.

We provide an alternative proof of the recent result by Park and Pham (2022) on the expected suprema of positive selector and empirical processes. We extend it to positive infinitely divisible processes.

The NEPv approach has been increasingly used lately for optimization on the Stiefel manifold arising from machine learning. General speaking, the approach first turns the first order optimality condition, also known as the KKT condition, into a nonlinear eigenvalue problem with eigenvector dependency (NEPv) or a nonlinear polar decomposition with orthogonal factor dependency (NPDo) and then solve the nonlinear problem via some variations of the self-consistent-field (SCF) iteration. The difficulty, however, lies in designing a proper SCF iteration so that a maximizer is found at the end. Currently, each use of the approach is very much individualized, especially in its convergence analysis to show that the approach does work or otherwise. In this paper, a unifying framework is established. The framework is built upon some basic assumptions. If the basic assumptions are satisfied, globally convergence is guaranteed to a stationary point and during the SCF iterative process that leads to the stationary point, the objective function increases monotonically. Also a notion of atomic functions is proposed, which include commonly used matrix traces of linear and quadratic forms as special ones. It is shown that the basic assumptions are satisfied by atomic functions and by convex compositions of atomic functions. Together they provide a large collection of objectives for which the NEPv/NPDo approach is guaranteed to work.

Physics-informed neural networks (PINNs) are a promising approach that combines the power of neural networks with the interpretability of physical modeling. PINNs have shown good practical performance in solving partial differential equations (PDEs) and in hybrid modeling scenarios, where physical models enhance data-driven approaches. However, it is essential to establish their theoretical properties in order to fully understand their capabilities and limitations. In this study, we highlight that classical training of PINNs can suffer from systematic overfitting. This problem can be addressed by adding a ridge regularization to the empirical risk, which ensures that the resulting estimator is risk-consistent for both linear and nonlinear PDE systems. However, the strong convergence of PINNs to a solution satisfying the physical constraints requires a more involved analysis using tools from functional analysis and calculus of variations. In particular, for linear PDE systems, an implementable Sobolev-type regularization allows to reconstruct a solution that not only achieves statistical accuracy but also maintains consistency with the underlying physics.

We study the derived tensor product of the representation rings of subgroups of a given compact Lie group G. That is, given two such subgroups H_1 and H_2, we study the tensor product of the associated representation rings R(H_1) and R(H_2) over the representation ring RG, and prove a vanishing result for the associated higher Tor-groups. This result can be viewed as a natural generalization of the Theorem of Steinberg that asserts that the representation rings of maximal rank subgroups of G are free over RG. It my also be viewed as an analogue of a result of Singhof on the cohomology of classifying spaces. We include an immediate application to the complex K-theory of biquotient manifolds.

We analyze stabilization with respect to ${\mathbb P}^1$ in the Morel--Voevodsky unstable motivic homotopy theory. We introduce a refined notion of cellularity (a.k.a., biconnectivity) in various motivic homotopy categories taking into account both the simplicial and Tate circles. Under suitable cellularity hypotheses, we refine the Whitehead theorem by showing that a map of nilpotent motivic spaces can be seen to be an equivalence if it so after taking (Voevodsky) motives. We then establish a version of the Freudenthal suspension theorem for ${\mathbb P}^1$-suspension, again under suitable cellularity hypotheses. As applications, we resolve Murthy's conjecture on splitting of corank $1$ vector bundles on smooth affine algebras over algebraically closed fields having characteristic $0$ and compute new unstable motivic homotopy of motivic spheres.

This paper studies planar algebras of Jones' style associated with the Young graph. We first see that, given a positive real valued function on the Young graph, we may obtain a planar algebra whose structure is defined in terms of a state sum over the ways of filling planar tangles with Young diagrams. We delve into the case that the function is harmonic and related to the Plancherel measures on Young diagrams. Along with an element that is depicted as a cross of two strings, we see that the defining relations among morphisms for the Khovanov Heisenberg category are recovered in the planar algebra. We also identify certain elements in the planar algebra with particular functions of Young diagrams that include the moments, Boolean cumulants and normalized characters. This paper thereby bridges diagramatical categorification and asymptotic representation theory. In fact, the Khovanov Heisenberg category is one of the most fundamental examples of diagramatical categorification whereas the harmonic functions on the Young graph have been a central object in the asymptotic representation theory of symmetric groups.

We extend Moonen's definition of Ekedahl-Oort types of smooth curves in terms of Hasse-Witt triples to all stable curves and show that it matches Ekedahl and van der Geer's definition of Ekedahl-Oort types of their generalized Jacobians as semi-abelian varieties. Using this intrinsic insight, we can compute the dimensions of certain Ekedahl-Oort loci of curves and generalize some previously known results about the dimensions of the $p$-rank and $a$-number loci of curves.

We study the low lying zeros of $GL(2) \times GL(2)$ Rankin-Selberg $L$-functions. Assuming the generalized Riemann hypothesis, we compute the $1$-level density of the low-lying zeroes of $L(s, f \otimes g)$ averaged over families of Rankin-Selberg convolutions, where $f, g$ are cuspidal newforms with even weights $k_1, k_2$ and prime levels $N_1, N_2$, respectively. The Katz-Sarnak density conjecture predicts that in the limit, the $1$-level density of suitable families of $L$-functions is the same as the distribution of eigenvalues of corresponding families of random matrices. The 1-level density relies on a smooth test function $\phi$ whose Fourier transform $\widehat\phi$ has compact support. In general, we show the Katz-Sarnak density conjecture holds for test functions $\phi$ with $\operatorname{supp} \widehat\phi \subset (-\frac{1}{2}, \frac{1}{2})$. When $N_1 = N_2$, we prove the density conjecture for $\operatorname{supp} \widehat\phi \subset (-\frac{5}{4}, \frac{5}{4})$ when $k_1 \ne k_2$, and $\operatorname{supp} \widehat\phi \subset (-\frac{29}{28}, \frac{29}{28})$ when $k_1 = k_2$. A lower order term emerges when the support of $\widehat\phi$ exceeds $(-1, 1)$, which makes these results particularly interesting. The main idea which allows us to extend the support of $\widehat\phi$ beyond $(-1, 1)$ is an analysis of the products of Kloosterman sums arising from the Petersson formula. We also carefully treat the contributions from poles in the case where $k_1 = k_2$. Our work provides conditional lower bounds for the proportion of Rankin-Selberg $L$-functions which are non-vanishing at the central point and for a related conjecture of Keating and Snaith on central $L$-values.

In the present paper, we formulate two versions of Frank--Wolfe algorithm or conditional gradient method to solve the DC optimization problem with an adaptive step size. The DC objective function consists of two components; the first is thought to be differentiable with a continuous Lipschitz gradient, while the second is only thought to be convex. The second version is based on the first and employs finite differences to approximate the gradient of the first component of the objective function. In contrast to past formulations that used the curvature/Lipschitz-type constant of the objective function, the step size computed does not require any constant associated with the components. For the first version, we established that the algorithm is well-defined of the algorithm and that every limit point of the generated sequence is a stationary point of the problem. We also introduce the class of weak-star-convex functions and show that, despite the fact that these functions are non-convex in general, the rate of convergence of the first version of the algorithm to minimize these functions is ${\cal O}(1/k)$. The finite difference used to approximate the gradient in the second version of the Frank-Wolfe algorithm is computed with the step-size adaptively updated using two previous iterations. Unlike previous applications of finite difference in the Frank-Wolfe algorithm, which provided approximate gradients with absolute error, the one used here provides us with a relative error, simplifying the algorithm analysis. In this case, we show that all limit points of the generated sequence for the second version of the Frank-Wolfe algorithm are stationary points for the problem under consideration, and we establish that the rate of convergence for the duality gap is ${\cal O}(1/\sqrt{k})$.

We establish finite-dimensional central limit theorems for local, additive, interaction functions of temporally evolving point processes. The dynamics are those of a spatial Poisson process on the flat torus with points subject to a birth-death mechanism, and which move according to Brownian motion while alive. The results reveal the existence of a phase diagram describing at least three distinct structures for the limiting processes, depending on the extent of the local interactions and the speed of the Brownian motions. The proofs, which identify three different limits, rely heavily on Malliavin-Stein type CLTs for $U$-statistics on a representation of the dynamic point process via a distributionally equivalent marked point process.

The trust-region (TR) method is renowned historically for its robustness in nonconvex problems and extraordinary numerical performance, but the study of its performance in convex optimization is somehow limited. This paper complements the existing literature by presenting a universal trust-region method that simultaneously incorporates the quadratic regularization and ball constraint. In particular, we introduce a novel descent property tailored for trust-region-type algorithms, enabling us to unify and streamline the analysis for both convex and nonconvex optimization. Our method exhibits an iteration complexity of $\tilde O(\epsilon^{-3/2})$ to find an $\epsilon$-approximate second-order stationary point for nonconvex optimization. Meanwhile, the analysis reveals that the universal method attains an $O(\epsilon^{-1/2})$ complexity bound for convex optimization. Finally, we develop an adaptive universal method to address practical implementations. The numerical results show the effectiveness of our method in both nonconvex and convex problems.

This paper shows that quantization of $\pi$-finite spaces, as a functor out of a higher category of spans, is equivariant in two ways: Symmetries of a given polarization/Lagrangian always induce coherent symmetries of the quantization. On the other hand, symmetries of the entire phase space a priori only induce projective symmetries, with an invertible once-categorified theory, the anomaly theory, encoding the projectivity. We give projective symmetries of three-dimensional finite gauge theories a concrete description via a twice-categorified analogue of Blattner-Kostant-Sternberg kernels and the associated integral transforms, such as the Fourier transform. This establishes an analogy between certain instances of the $\pi$-finite quantization procedure considered herein and the geometric quantization of a symplectic vector space.

We study the complexity of horizontality in the twistor space $\hat{E}$ associated with an oriented vector bundle $E$ of rank $4$ with a positive-definite metric over a torus. If the horizontality has finite complexity of degree $d>2$ for an element of a fiber of $\hat{E}$, then the complexity is expressed in terms of a finite subgroup of $SO(3)$ ([3]). In the present paper, we observe that if the horizontality has infinite complexity derived from one of the cases studied in [3], then the complexity is expressed by a dense subset of $S^2$.

We study a new class of functions that arise naturally in quaternionic analysis, we call them "quasi regular functions". Like the well-known quaternionic regular functions, these functions provide representations of the quaternionic conformal group. However, unlike the regular functions, the quasi regular ones do not admit an invariant unitary structure but rather a pseudounitary equivalent. The reproducing kernels of these functions have an especially simple form: (Z-W)^{-1}. We describe the K-type bases of quasi regular functions and derive the reproducing kernel expansions. We also show that the restrictions of the irreducible representations formed from the quasi regular functions to the Poincare group have three irreducible components. Our interest in the quasi regular functions arises from an application to the study of conformal-invariant algebras of quaternionic functions. We also introduce a factorization of certain intertwining operators between tensor products of spaces of quaternionic functions. This factorization is obtained using fermionic Fock spaces constructed from the quasi regular functions.

Given a class of (undirected) graphs $\mathcal{C}$, we say that a Feedback Arc Set (FAS for short) $F$ is a $\mathcal{C}$-FAS if the graph induced by the edges of $F$ (forgetting their orientations) belongs to $\mathcal{C}$. We show that deciding if a tournament has a $\mathcal{C}$-FAS is NP-complete when $\mathcal{C}$ is the class of all forests. We are motivated by connections between $\mathcal{C}$-FAS and structural parameters of tournaments, such as the dichromatic number, the clique number of tournaments, and the strong Erdős-Hajnal property.

We define two notions. The first one is a $rank\ compression\ system$ $\xi$ for a finite poset $\mathbf{P}$ that assigns each interval subposet $I$ to an order-preserving map $\xi_I \colon I^{\xi} \to \mathbf{P}$ satisfying some conditions, where $I^{\xi}$ is a connected finite poset. An example is given by the $total$ compression system that assigns each $I$ to the inclusion of $I$ into $\mathbf{P}$. The second one is an $I$-$rank$ of a persistence module $M$ under $\xi$, the family of which is called the $interval\ rank\ invariant$ of $M$ under $\xi$. A compression system $\xi$ makes it possible to define the $interval\ replacement$ (also called the interval-decomposable approximation) not only for 2D persistence modules but also for any persistence modules over any finite poset. We will show that the forming of the interval replacement preserves the interval rank invariant, which is a stronger property than the preservation of the usual rank invariant. Moreover, to know what is preserved by the replacement explicitly, we will give a formula of the $I$-rank of $M$ under $\xi$ in terms of the structure linear maps of $M$ for any compression system $\xi$. The formula leads us to a concept of essential cover, which gives us a sufficient condition for the $I$-rank of $M$ under $\xi$ to coincide with that under another compression system $\zeta$. This is applied to the case where $\xi = \mathrm{tot}$, the value of $I$-rank under which is equal to the generalized rank invariant introduced by Kim--Mémoli, to give an alternative proof of the Dey--Kim--Mémoli theorem computing the generalized rank invariant by using a zigzag path.

We investigate full-duplex (FD) multi-user multiple input single-output systems with coarse quantization, aiming to characterize the impact of employing low-resolution analog-to-digital converters (ADCs) on self-interference (SI) and to develop a quantization- and SI-aware beamforming method that alleviates quantization-induced performance degradation in the FD systems. We first present an analysis on the perantenna signal-to-quantization noise ratio for conventional linear beamformers to provide the desired range of the number of analog-to-digital converter (ADC) bits, providing system insights for reliable FD operation in regard to the ADC resolution and beamforming strategy. Motivated by the insights, we then propose an SI-aware beamforming method that mitigates residual SI and quantization distortion. The resulting spectral efficiency (SE) maximization problem is decomposed into two tractable subproblems solved via alternating optimization: precoder and combiner design. The precoder optimization is formulated as a generalized eigenvalue problem, where the dominant eigenvector yields the best stationary solution through power iteration, while the combiner is derived as a quantization-aware minimum meansquared error (MMSE) filter. Numerical studies show that the number of required ADC bits with the proposed beamforming falls within the derived theoretical range while achieving the highest SE compared to benchmarks.

This paper collects polynomial Diophantine equations that are amazingly simple to write down but are apparently difficult to solve.

We consider the problem of maximizing the expected average reward obtained over an infinite time horizon by $n$ weakly coupled Markov decision processes. Our setup is a substantial generalization of the multi-armed restless bandit problem that allows for multiple actions and constraints. We establish a connection with a deterministic and continuous-variable control problem where the objective is to maximize the average reward derived from an occupancy measure that represents the empirical distribution of the processes when $n \to \infty$. We show that a solution of this fluid problem can be used to construct policies for the weakly coupled processes that achieve the maximum expected average reward as $n \to \infty$, and we give sufficient conditions for the existence of solutions. Under certain assumptions on the constraints, we prove that these conditions are automatically satisfied if the unconstrained single-process problem admits a suitable unichain and aperiodic policy. In particular, the assumptions include multi-armed restless bandits and a broad class of problems with multiple actions and inequality constraints. Also, the policies can be constructed in an explicit way in these cases. Our theoretical results are complemented by several concrete examples and numerical experiments, which include multichain setups that are covered by the theoretical results.

Sard's theorem asserts that the set of critical values of a smooth map from one Euclidean space to another one has measure zero. A version of this result for infinite-dimensional Banach manifolds was proven by Smale for maps with Fredholm differential. It is well-known, however, that when the domain is infinite dimensional and the range is finite dimensional, the result is not true -- even under the assumption that the map is ``polynomial'' -- and a general theory is still lacking. Addressing this issue, in this paper, we provide sharp quantitative criteria for the validity of Sard's theorem in this setting. Our motivation comes from sub-Riemannian geometry and, as an application of our results, we prove the sub-Riemannian Sard conjecture for the restriction of the Endpoint map of Carnot groups to the set of piece-wise real-analytic controls with large enough radius of convergence, and the strong Sard conjecture for the restriction to the set of piece-wise entire controls.

Let $p$ be a prime number, $K$ a finite extension of $\mathbb{Q}_p$ and $n$ an integer $\geq 2$. We completely and explicitly describe the global sections $\Omega^\bullet$ of the de Rham complex of the Drinfeld space over $K$ in dimension $n-1$ as a complex of (duals of) locally $K$-analytic representations of $\mathrm{GL}_n(K)$. Using this description, we construct an explicit section in the derived category of (duals of) finite length admissible locally $K$-analytic representations of $\mathrm{GL}_n(K)$ to the canonical morphism of complexes $\Omega^\bullet \twoheadrightarrow H^{n-1}(\Omega^\bullet)[-(n-1)]$.

Based on [1], we study the complexity of horizontality in each twistor space $\hat{E}_{\varepsilon}$ associated with an oriented vector bundle $E$ of rank $4$ with a positive-definite metric over the $2$-torus $T^2$, and obtain classification of the topological holonomy groups in $SO(3)$. We observe that there exist many topological holonomy groups in $SO(3)$ generated by two finite order elements and equipped with noncommutative pairs which consist of infinite order elements. We find topological holonomy groups which are dense in $SO(4)$.

This paper investigates aliasing effects emerging from the reconstruction from discrete samples of spin spherical random fields defined on the two-dimensional sphere. We determine the location in the frequency domain and the intensity of the aliases of the harmonic coefficients in the Fourier decomposition of the spin random field and evaluate the consequences of aliasing errors in the angular power spectrum when the samples of the random field are obtained by using some very popular sampling procedures on the sphere, the equiangular and the Gauss-Jacobi sampling schemes. Finally, we demonstrate that band-limited spin random fields are free from aliases, provided that a sufficiently large number of nodes is used in the selected quadrature rule.

The critical length phenomenon of the Korteweg-de Vries equation is well known; however, in higher dimensions, it is unknown. This work explores this property in the context of the Kadomtsev-Petviashvili equation, a two-dimensional generalization of the Korteweg-de Vries equation. Specifically, we demonstrate observability inequalities for this equation, which allow us to deduce the exact boundary controllability and boundary exponential stabilization of the linear system, provided that the spatial domain length avoids certain specific values, a direct consequence of the Paley-Wiener theorem. To the best of our knowledge, our work introduces new results by identifying a set of critical lengths for the two-dimensional Kadomtsev-Petviashvili equation.

We show that on a $\sigma$-finite measure preserving system $X = (X,\nu, T)$, the non-conventional ergodic averages $$ \mathbb{E}_{n \in [N]} \Lambda(n) f(T^n x) g(T^{P(n)} x)$$ converge pointwise almost everywhere for $f \in L^{p_1}(X)$, $g \in L^{p_2}(X)$, and $1/p_1 + 1/p_2 \leq 1$, where $P$ is a polynomial with integer coefficients of degree at least $2$. This had previously been established with the von Mangoldt weight $\Lambda$ replaced by the constant weight $1$ by the first and third authors with Mirek, and by the Möbius weight $\mu$ by the fourth author. The proof is based on combining tools from both of these papers, together with several Gowers norm and polynomial averaging operator estimates on approximants to the von Mangoldt function of ''Cramér'' and ''Heath-Brown'' type.

MacWilliams identities relate the weight enumerators of a code with those of its dual and are classically formulated with respect to the Hamming weight. For other metrics, however, these identities often fail when considering the weight partition of the ambient space. It is known that MacWilliams identities hold for enumerators associated with Fourier-reflexive partitions, and that orbits of subgroups of the linear isometry group always yield such partitions. This raises the question whether, for metrics beyond the Hamming metric, there exist meaningful partitions that lie strictly between the weight partition and the orbit partition: finer than the latter, yet still coarse enough to retain useful MacWilliams-type identities. In this work, we study this question for finite chain rings endowed with additive metrics. For the Lee metric, we show that the partition induced by the action of the full group of linear isometries is already the coarsest Fourier-reflexive partition refining the weight partition. In particular, no intermediate partition exists that is both finer than the Lee weight partition and Fourier-reflexive. We refer to this partition as the Lee partition and show that it allows the recovery of all additive weight enumerators over the ring. In contrast, for the homogeneous metric and for the subfield metric, we identify new, significantly coarser symmetrized partitions that remain Fourier-reflexive and still allow the recovery of the corresponding weight enumerators. We prove that these partitions are the coarsest such symmetrized partitions for which MacWilliams-type identities hold. As an application, we derive linear programming bounds based on the resulting MacWilliams identities.

We investigate the formal principle for holomorphic line bundles on neighborhoods of an analytic subset of a complex manifold mainly in the case where it can be realized as an open subset of a compact Kähler manifold. Our approach identifies the obstruction as a global analytic class supported on a neighborhood of $Y$, and relates its vanishing to the solvability of a $\partial\overline{\partial}$-problem on neighborhoods of $Y$. As a consequence we obtain cohomological criteria ensuring the formal principle. We also construct a holomorphic family of compact Kähler surfaces containing a curve with topologically trivial normal bundle in which the formal principle holds for almost every fiber but fails for uncountably many fibers, exhibiting an instability phenomenon in families.

Solving large scale entropic optimal transport problems with the Sinkhorn algorithm remains challenging, and domain decomposition has been shown to be an efficient strategy for problems on large grids. Unbalanced optimal transport is a versatile variant of the balanced transport problem and its entropic regularization can be solved with an adapted Sinkhorn algorithm. However, it is a priori unclear how to apply domain decomposition to unbalanced problems since the independence of the cell problems is lost. In this article we show how this difficulty can be overcome at a theoretical and practical level and demonstrate with experiments that domain decomposition is also viable and efficient on large unbalanced entropic transport problems.

We derive new Hanson-Wright-type inequalities tailored to the quadratic forms of random vectors with sparse independent components. Specifically, we consider cases where the components of the random vector are sparse $\alpha$-subexponential random variables with $\alpha>0$. When $\alpha=\infty$, these inequalities can be seen as quadratic generalizations of the classical Bernstein and Bennett inequalities for sparse bounded random vectors. To establish this quadratic generalization, we also develop new Bernstein-type and Bennett-type inequalities for linear forms of sparse $\alpha$-subexponential random variables that go beyond the bounded case $(\alpha=\infty)$. Our proof relies on a novel combinatorial method for estimating the moments of both random linear forms and quadratic forms. We present two key applications of these new sparse Hanson-Wright inequalities: (1) A local law and complete eigenvector delocalization for sparse $\alpha$-subexponential Hermitian random matrices, generalizing the result of He et al. (2019) beyond sparse Bernoulli random matrices. To the best of our knowledge, this is the first local law and complete delocalization result for sparse $\alpha$-subexponential random matrices down to the near-optimal sparsity $p\geq \frac{\mathrm{polylog}(n)}{n}$ when $\alpha\in (0,2)$ as well as for unbounded sparse sub-gaussian random matrices down to the optimal sparsity $p\gtrsim \frac{\log n}{n}.$ (2) Concentration of the Euclidean norm for the linear transformation of a sparse $\alpha$-subexponential random vector, improving on the results of G{ö}tze et al. (2021) for sparse sub-exponential random vectors.

The surface charges associated with $p$-form gauge fields in the Bondi patch of $D$-dimensional Minkowski spacetime are computed. We show that, under the Hodge duality between the field strengths of the dual formulations, electric-like charges for $p$-forms are mapped to magnetic-like charges for the dual $q$-forms, with $q=D-p-2$. We observe that the complex combination of electric-like and magnetic-like charges transforms under duality according to a specific Möbius transformation. This leads to a possible construction of CCFT in $D=4$ as a Möbius-principal equivariant bundle, together with its associated bundles, in order to recover celestial operators. We prove an existence and uniqueness theorem for the duality map relating the asymptotic electric-like charges of the dual descriptions, and we provide an algebraic-topological interpretation of this map. As a result, the duality map has a topological nature and ensures that the charge of one formulation contains information about the dual formulation, leading to a deeper understanding of gauge theories, the non-trivial charges associated with them, and the duality of their observables. Moreover, we propose a link between higher-form symmetry charges, naturally associated with a $p$-form gauge theory, and their asymptotic charges. The higher-form charges are reproduced by choosing the gauge parameter to be constant and supported only on an appropriate codimension submanifold. This could partially answer an open question in the celestial holography program.

We study higher uniformity properties of the von Mangoldt function $\Lambda$, the Möbius function $\mu$, and the divisor functions $d_k$ on short intervals $(x,x+H]$ for almost all $x \in [X, 2X]$. Let $\Lambda^\sharp$ and $d_k^\sharp$ be suitable approximants of $\Lambda$ and $d_k$, $G/\Gamma$ a filtered nilmanifold, and $F\colon G/\Gamma \to \mathbb{C}$ a Lipschitz function. Then our results imply for instance that when $X^{1/3+\varepsilon} \leq H \leq X$ we have, for almost all $x \in [X, 2X]$, \[ \sup_{g \in \text{Poly}(\mathbb{Z} \to G)} \left| \sum_{x < n \leq x+H} (\Lambda(n)-\Lambda^\sharp(n)) \overline{F}(g(n)\Gamma) \right| \ll H\log^{-A} X \] for any fixed $A>0$, and that when $X^{\varepsilon} \leq H \leq X$ we have, for almost all $x \in [X, 2X]$, \[ \sup_{g \in \text{Poly}(\mathbb{Z} \to G)} \left| \sum_{x < n \leq x+H} (d_k(n)-d_k^\sharp(n)) \overline{F}(g(n)\Gamma) \right| = o(H \log^{k-1} X). \] As a consequence, we show that the short interval Gowers norms $\|\Lambda-\Lambda^\sharp\|_{U^s(X,X+H]}$ and $\|d_k-d_k^\sharp\|_{U^s(X,X+H]}$ are also asymptotically small for any fixed $s$ in the same ranges of $H$. This in turn allows us to establish the Hardy-Littlewood conjecture and the divisor correlation conjecture with a short average over one variable. Our main new ingredients are type $II$ estimates obtained by developing a "contagion lemma" for nilsequences and then using this to "scale up" an approximate functional equation for the nilsequence to a larger scale. This extends an approach developed by Walsh for Fourier uniformity.

We solve a long-standing open problem posed by Goodman \& Pollack in 1988 by establishing a necessary and sufficient condition for a family of convex sets in $\mathbb{R}^d$ to admit a $k$-transversal for any $0 \le k \le d-1$. This result is a common generalization of Helly's theorem ($k=0$) and the Goodman-Pollack-Wenger theorem ($k=d-1$). Additionally, we obtain an analogue in the complex setting by characterizing the existence of a complex $k$-transversal to a family of convex sets in $\mathbb{C}^d$, extending the work of McGinnis ($k=d-1$). Our approach is topological and employs a Borsuk-Ulam-type theorem on Stiefel manifolds. Finally, we demonstrate how our results imply the central transversal theorems of Živaljević-Vrećica and Dol'nikov in the real case and of Sadovek-Soberón in the complex case.

We study necessary conditions for local maximizers of the isotropic constant that are related to notions of decomposability. Our main result asserts that the polar body of a local maximizer of the isotropic constant can only have few Minkowski summands; more precisely, its dimension of decomposability is at most $\frac12(n^2+3n)$. Using a similar proof strategy, a result by Campi, Colesanti and Gronchi concerning RS-decomposability is extended to a larger class of shadow systems. We discuss the polytopal case, which turns out to have connections to (affine) rigidity theory, and investigate how the bound on the maximal number of irredundant summands can be improved if we restrict our attention to convex bodies with certain symmetries.

Let R be a commutative ring with identity. The concept of second submodule of an R-module (as a dual notion of prime submodules) was introduced and studied by this http URL in 2001. This notion has obtained a great attention by many authors and now there is a considerable amount of research concerning this class of modules. The main purpose of this paper is to collect these results and provide a useful source for those who are interested in research in this field.

The paper is concerned with a scalar conservation law with discontinuous gradient-dependent flux. Namely, the flux is described by two different functions $f(u)$ or $g(u)$, when the gradient $u_x$ of the solution is positive or negative, respectively. We study here the unstable case where $f(u)>g(u)$ for all $u\in {\mathbb R}$. Assuming that both $f$ and $g$ are strictly convex, solutions to the Riemann problem are constructed. Even for a smooth initial data, examples show that the Cauchy problem can have infinitely many solutions. For an initial data which is piecewise monotone, i.e., increasing or decreasing on a finite number of intervals, a solution can be constructed globally in time. It is proved that such solution is unique under the additional requirement that the number of interfaces, where the flux switches between $f$ and $g$, remains as small as possible.

Let $N_a(x)$ denote the number of primes up to $x$ for which the integer $a$ is a primitive root. We show that $N_a(x)$ satisfies the asymptotic predicted by Artin's conjecture for almost all $1\le a\le \exp((\log \log x)^2)$. This improves on a result of Stephens (1969). A key ingredient in the proof is a new short character sum estimate over the integers, improving on the range of a result of Garaev (2006).

We prove Libgober's divisibility relations for Oka and Alexander polynomials of symplectic curves in the complex projective plane. Along the way, we give new proofs of the divisibility relations for the Alexander polynomials of complex algebraic curves with respect to a generic line at infinity.

This paper provides a blueprint for the construction of a symplectic $(A_\infty,2)$-category, $\mathsf{Symp}$. We develop two ways of encoding the information in $\mathsf{Symp}$ -- one topological, one algebraic. The topological encoding is as an $(A_\infty,2)$-flow category, which we define here. The algebraic encoding is as a linear $(A_\infty,2)$-category, which we extract from the topological encoding. In upcoming work, we plan to use the adiabatic Fredholm theory developed by us to construct $\mathsf{Symp}$ as an $(A_\infty,2)$-flow category, which thus induces a linear $(A_\infty,2)$-category. The notion of a linear $(A_\infty,2)$-category developed here goes beyond the proposal of Bottman and Carmeli. The recursive structure of the 2-associahedra identifies faces with fiber products of 2-associahedra over associahedra, which led Bottman and Carmeli to associate operations to singular chains on 2-associahedra. The innovation in our new definition of linear $(A_\infty,2)$-category is to extend the family of 2-associahedra to include all fiber products of 2-associahedra over associahedra. This allows us to associate operations to cellular chains, which in particular enables us to produce a definition that involves only one operation in each arity, governed by a collection of $(A_\infty,2)$-equations.

Let $E/\mathbb{Q}$ be an elliptic curve and $p > 2$ be a prime of good ordinary reduction for $E$. Assume that the residue representation associated with $(E, p)$ is irreducible. In this paper, we prove more cases on several Iwasawa main conjectures for $E$. As applications, we prove more general cases of $p$-converse theorem and $p$-part BSD formula when the rank is less than or equal to $1$.

In this work, we establish higher-order div-curl type estimates in the sense of Coifman, Lions, Meyer & Semmes, in a local setting for elliptic homogeneous linear differential operators with smooth coefficients acting on localizable Hardy spaces. Our results imply and extend previously known estimates for first-order operators associated with elliptic systems and complexes of vector fields. As tools of independent interest, we develop a new smooth atomic decomposition for localizable Hardy-Sobolev spaces and prove a Poincaré-type inequality in this framework.

We address the question how a given connection structure (directed graph) can be realised as a heteroclinic network that is complete in the sense that it contains all unstable manifolds of its equilibria. For a directed graph consisting of two cycles we provide a constructive method to achieve this: (i) enlarge the graph by adding some edges and (ii) apply the simplex method to obtain a network in phase space. Depending on the length of the cycles we derive the minimal number of required new edges. In the resulting network each added edge leads to a positive transverse eigenvalue at the respective equilibrium. We discuss the total number of such positive eigenvalues in an individual cycle and some implications for the stability of this cycle.

This paper proposes a unified framework for designing robustness in optimization under uncertainty using gauge sets, convex sets that generalize distance and capture how distributions may deviate from a nominal reference. Representing robustness through a gauge set reweighting formulation brings many classical robustness paradigms under a single convex-analytic perspective. The corresponding dual problem, the upper approximator regularization model, reveals a direct connection between distributional perturbations and objective regularization via polar gauge sets. This framework decouples the design of the nominal distribution, distance metric, and reformulation method, components often entangled in classical approaches, thus enabling modular and composable robustness modeling. We further provide a gauge set algebra toolkit that supports intersection, summation, convex combination, and composition, enabling complex ambiguity structures to be assembled from simpler components. For computational tractability under continuously supported uncertainty, we introduce two general finite-dimensional reformulation methods. The functional parameterization approach guarantees any prescribed gauge-based robustness through flexible selection of function bases, while the envelope representation approach yields exact reformulations under empirical nominal distributions and is asymptotically exact for arbitrary nominal choices. A detailed case study demonstrates how the framework accommodates diverse robustness requirements while admitting multiple tractable reformulations.

Second-order optimization methods are computationally expensive for large-scale problems. Recently, Doikov, Chayti, and Jaggi (ICML 2023) proposed the LazyCRN method that reduces computation by studying the gradient complexity of second-order methods. Their method can achieve a gradient complexity of $\mathcal{O}( d + d^{1/2} \epsilon^{-3/2})$ and $\mathcal{O}( d + d^{1/2} \epsilon^{-1/2})$ for nonconvex and convex optimization, respectively, where $d$ is the effective dimension and $\epsilon$ is the target precision. Very recently, Adil, Bullins, Sidford, and Zhang (NeurIPS 2025) improved the gradient complexity to $\mathcal{O}( d + d^{1/3} \epsilon^{-3/2} \ln^{18} \epsilon^{-1})$ for nonconvex optimization. However, the tightness of these methods remains open. In this work, we propose new methods that achieve an improved complexity of $\mathcal{O}( d + d^{1/3} \epsilon^{-3/2})$ and $\mathcal{O}( (d + d^{13/21} \epsilon^{-2/7}) \ln d)$ for nonconvex and convex optimization, respectively, improving best-known results for both setups.

Dynamical systems can be analyzed via their Frobenius-Perron transfer operator and its estimation from data is an active field of research. Recently entropic transfer operators have been introduced to estimate the operator of deterministic systems. The approach is based on the regularizing properties of entropic optimal transport plans. In this article we generalize the method to stochastic and non-stationary systems and give a quantitative convergence analysis of the empirical operator as the available samples increase. We introduce a way to extend the operator's eigenfunctions to previously unseen samples, such that they can be efficiently included into a spectral embedding. The practicality and numerical scalability of the method are demonstrated on a real-world fluid dynamics experiment.

The depth of a subgroup $H$ of a finite group $G$ is a positive integer defined with respect to the inclusion of the corresponding complex group algebras $\mathbb{C}H \subseteq \mathbb{C}G$. This notion was originally introduced by Boltje, Danz and Külshammer in 2011, and it has been the subject of numerous papers in recent years. In this paper, we study the depth of core-free subgroups, which allows us to apply powerful computational and probabilistic techniques that were originally designed for studying bases for permutation groups. We use these methods to prove a wide range of new results on the depth of subgroups of almost simple groups, significantly extending the scope of earlier work in this direction. For example, we establish best possible bounds on the depth of irreducible subgroups of classical groups and primitive subgroups of symmetric groups. And with the exception of a handful of open cases involving the Baby Monster, we calculate the exact depth of every subgroup of every almost simple sporadic group. We also present a number of open problems and conjectures.

We characterize the function $\varphi$ of minimal $L^1$ norm among all functions $f$ of exponential type at most $\pi$ for which $f(0)=1$. This function, studied by Hörmander and Bernhardsson in 1993, has only real zeros $\pm \tau_n$, $n=1,2, \ldots$. Starting from the fact that $n+\frac12-\tau_n$ is an $\ell^2$ sequence, established in an earlier paper of ours, we identify $\varphi$ in the following way. We factor $\varphi(z)$ as $\Phi(z)\Phi(-z)$, where $\Phi(z)= \prod_{n=1}^\infty(1+(-1)^n\frac{z}{\tau_n})$ and show that $\Phi$ satisfies a certain second order linear differential equation along with a functional equation either of which characterizes $\Phi$. We use these facts to establish an odd power series expansion of $n+\frac12-\tau_n$ in terms of $(n+\frac12)^{-1}$ and a power series expansion of the Fourier transform of $\varphi$, as suggested by the numerical work of Hörmander and Bernhardsson. The dual characterization of $\Phi$ arises from a commutation relation that holds more generally for a two-parameter family of differential operators, a fact that is used to perform high precision numerical computations.

Chemotaxis models describe the movement of organisms in response to chemical gradients. In this paper, we present a stochastic interacting particle-field algorithm with a random batch approximation (SIPF-$r$) for the three-dimensional (3D) parabolic-parabolic Keller-Segel (KS) system, also referred to as the fully parabolic KS system. The SIPF-$r$ method approximates the KS system by coupling particle-based representations of the density with a smooth field variable computed using spectral methods. By incorporating the random batch method (RBM), we bypass the mean-field limit and significantly reduce computational complexity. Under mild assumptions on the regularity of the original KS system and the boundedness of numerical approximations, we prove that the empirical measure of the SIPF-$r$ particle system converges, with high probability, to the exact measure of the limiting McKean-Vlasov process in the $1$-Wasserstein distance. Finally, we present numerical experiments to validate the theoretical convergence rates, and demonstrate the performance and robustness of the SIPF-$r$ method as a diagnostic tool for intense focusing and potential finite-time singularity in 3D, subject to critical initial mass thresholds in the system.

Graph pebbling is a game played on graphs with pebbles on their vertices. A pebbling move removes two pebbles from one vertex and places one pebble on an adjacent vertex. A configuration $C$ is a supply of pebbles at various vertices of a graph $G$, and a distribution $D$ is a demand of pebbles at various vertices of $G$. The $D$-pebbling number, $\pi(G, D)$, of a graph $G$ is defined to be the minimum number $m$ such that every configuration of $m$ pebbles can satisfy the demand $D$ via pebbling moves. The special case in which $t$ pebbles are demanded on vertex $v$ is denoted $D=v^t$, and the $t$-fold pebbling number, $\pi_{t}(G)$, equals $\max_{v\in G}\pi(G,v^t)$. It was conjectured by Alcón, Gutierrez, and Hurlbert that the pebbling numbers of chordal graphs forbidding the pyramid graph can be calculated in polynomial time. Trees, of course, are the most prominent of such graphs. In 1989, Chung determined $\pi_t(T)$ for all trees $T$. In this paper, we provide a polynomial-time algorithm to compute the pebbling numbers $\pi(T,D)$ for all distributions $D$ on any tree $T$, and characterize maximum-size configurations that do not satisfy $D$.

A graph is perfectly divisible if for each of its induced subgraph $H$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $\omega(H[B]) < \omega(H)$. A graph $G$ is perfectly weight divisible if for every positive integral weight function on $V(G)$ and each of its induced subgraph $H$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and the maximum weight of a clique in $H[B]$ is smaller than the maximum weight of a clique in $H$. In this paper, we prove that the perfect divisibility of a graph is equivalent to its perfect weighted divisibility.

Given a selfadjoint magnetic Schrödinger operator \begin{equation*} H = ( i \partial + A(x) )^2 + V(x) \end{equation*} on $L^{2}(\mathbb{R}^n)$, with $V(x)$ strictly subquadratic and $A(x)$ strictly sublinear, we prove that the flow $u(t)=e^{-itH}u(0)$ satisfies an Amrein--Berthier type inequality \begin{equation*} \|u(t)\|_{L^{2}}\lesssim_{E,F,T,A,V} \|u(0)\|_{L^{2}(E^{c})} + \|u(T)\|_{L^{2}(F^{c})}, \qquad 0\le t\le T \end{equation*} for all compact sets $E,F \subset \mathbb{R}^{n}$. In particular, if both $u(0)$ and $u(T)$ are compactly supported, then $u$ vanishes identically. Under different assumptions on the operator, which allow for time--dependent coefficients, the result extends to sets $E,F$ of finite measure. We also consider a few variants for Schrödinger operators with singular coefficients, metaplectic operators, and we include applications to control theory.

Onsager reciprocal relations model physical irreversible processes from complex systems. Recently, it has been shown that Onsager principles for master equations on finite states introduce a class of Riemannian metrics on the probability simplex, leading to probability manifolds or finite-state Wasserstein--2 spaces. In this paper, we study geometric calculations on probability manifolds, deriving the Levi-Civita connection, gradient, Hessian operators of energies, parallel transport, and calculating both the Riemannian and sectional curvatures. We present two examples of geometric quantities in probability manifolds. One example is the Levi-Civita connection from the chemical monomolecular triangle reaction. The other example is the sectional, Ricci, and scalar curvatures in Wasserstein space on a three-point lattice graph.

We study the problem of estimating a distribution over a finite alphabet from an i.i.d. sample, with accuracy measured in relative entropy (Kullback-Leibler divergence). While optimal bounds on the expected risk are known, high-probability guarantees remain less well-understood. First, we analyze the classical Laplace (add-one) estimator, obtaining matching upper and lower bounds on its performance and establishing its optimality among confidence-independent estimators. We then characterize the minimax-optimal high-probability risk and show that it is achieved by a simple confidence-dependent smoothing technique. Notably, the optimal non-asymptotic risk incurs an additional logarithmic factor compared to the ideal asymptotic rate. Next, motivated by regimes in which the alphabet size exceeds the sample size, we investigate methods that adapt to the sparsity of the underlying distribution. We introduce an estimator using data-dependent smoothing, for which we establish a high-probability risk bound depending on two effective sparsity parameters. As part of our analysis, we also derive a sharp high-probability upper bound on the missing mass.

In this paper, we survey physically related applications of a class of weighted quasi-Monte Carlo methods from a theoretical, deterministic perspective, and establish quantitative universal rapid convergence results via various regularity assumptions. Specifically, we introduce weighting with compact support to the Birkhoff ergodic averages of quasi-periodic, almost periodic, and periodic systems, thereby achieving universal rapid convergence, including both arbitrary polynomial and exponential types. This is in stark contrast to the typically slow convergence in classical ergodic theory. As new contributions, we not only discuss more general weighting functions but also provide quantitative improvements to existing results; the explicit regularity settings facilitate the application of these methods to specific problems. We also revisit the physically related problems and, for the first time, establish universal exponential convergence results for the weighted computation of Fourier coefficients, in both finite-dimensional and infinite-dimensional cases. In addition to the above, we explore results from a probabilistic perspective, including the weighted strong law of large numbers and the weighted central limit theorem, by building upon the historical results.

We compare eight versions of finite-dimensional categorifications of the colored Jones polynomial and show that they yield isomorphic results over a field of characteristic zero. As an application, we verify a physics-motivated conjectural formula for colored superpolynomials based on Poincaré polynomials of the Khovanov homology of cables. We also obtain a conjectural closed formula for the Poincaré series of the skein lasagna module of $\overline{\mathbb{CP}^2}$. Accompanying this note is an online database of colored superpolynomials.

We study a class of nonlinear Burgers-type stochastic partial differential equations driven by additive space-time white noise in one spatial dimension. Building on the rough path framework initiated by Hairer, which provides a pathwise solution theory under spatial regularity $\alpha \in(\frac{1}{3}, \frac{1}{2})$, we extend this approach to the full subcritical regime $\alpha \in(0, \frac{1}{2})$. Our main contribution is the establishment of pathwise existence and uniqueness of mild (equivalently, weak) solutions when the spatial regularity of the solution lies strictly below the classical rough path threshold. This is achieved through refined estimates for controlled rough paths, including a new upper bound for compositions with smooth functions and a scaling analysis for rough integrals against heat kernels. In particular, we extend and sharpen key analytic estimates originating from Hairer's work, incorporating refined scaling arguments that are effective in the low-regularity regime. As a result, our framework significantly enlarges the class of Burgers-type SPDEs that can be treated pathwise using rough path techniques.

High temperatures and structural deformations can compromise the functionality and reliability of new components for mechatronic systems. Therefore, high-fidelity simulations (HFS) are employed during the design process, as they enable a detailed analysis of the thermal and structural behavior of the system. However, such simulations are both computationally expensive and tedious, particularly during iterative optimization procedures. Establishing a parametric reduced order model (pROM) can accelerate the design's optimization if the model can accurately predict the behavior over a wide range of material and geometric properties. However, many existing methods exhibit limitations when applied to wide design ranges. In this work, we introduce the parametric Box Reduction (pBR) method, a matrix interpolation technique that minimizes the non-physical influence of training points due to the large parameter ranges. For this purpose, we define a new interpolation function that computes a local weight for each design variable and integrates them into the global function. Furthermore, we develop an intuitive clustering technique to select the training points for the model, avoiding numerical artifacts from distant points. Additionally, these two strategies do not require normalizing the parameter space and handle every property equally. The effectiveness of the pBR method is validated through two physical applications: structural deformation of a cantilever Timoshenko beam and heat transfer of a power module of a power converter. The results demonstrate that the pBR approach can accurately capture the behavior of mechatronic components across large parameter ranges without sacrificing computational efficiency.

Information theory is a powerful framework for quantifying complexity, uncertainty, and dynamical structure in time-series data, with widespread applicability across disciplines such as physics, finance, and neuroscience. However, the literature on these measures remains fragmented, with domain-specific terminologies, inconsistent mathematical notation, and disparate visualization conventions that hinder interdisciplinary integration. This work addresses these challenges by unifying key information-theoretic time-series measures through shared semantic definitions, standardized mathematical notation, and cohesive visual representations. We compare these measures in terms of their theoretical foundations, computational formulations, and practical interpretability -- mapping them onto a common conceptual space through an illustrative case study with functional magnetic resonance imaging time series in the brain. This case study exemplifies the complementary insights these measures offer in characterizing the dynamics of complex neural systems, such as signal complexity and information flow. By providing a structured synthesis, our work aims to enhance interdisciplinary dialogue and methodological adoption, which is particularly critical for reproducibility and interoperability in computational neuroscience. More broadly, our framework serves as a resource for researchers seeking to navigate and apply information-theoretic time-series measures to diverse complex systems.

We investigate properties of boundary orbits (separatrices) of canonical regions (basins/neighbourhoods of equilibria) in holomorphic flows with real-valued time. We establish the continuity of transit times along these boundary orbits and classify possible path components of the boundary of flow-invariant domains. Thus, we provide central tools for topological and geometric constructions aimed at examining the role of blow-up scenarios in separatrix configurations of basins of simple equilibria and global elliptic sectors: First, we prove that the separatrices of basins of centers is entirely composed of double-sided separatrices with a blow-up in finite positive and finite negative time. Second, we show that the separatrices of node and focus basins (sinks and sources) exhibit a finite-time blow-up in the same time direction in which the orbits within the basin tend towards the equilibrium. Additionally, we propose a counterexample to the claim in Theorem 4.3 (3) in ["The structure of sectors of zeros of entire flows", K. Broughan (2003)], demonstrating that a blow-up does not necessarily have to occur in both time directions. Third, we describe the boundary structure of global elliptic sectors. It consists of the multiple equilibrium, one incoming and one outgoing separatrix attached to it, and at most countably many double-sided separatrices.

When using the finite element method (FEM) in inverse problems, its discretization error can produce parameter estimates that are inaccurate and overconfident. The Bayesian finite element method (BFEM) provides a probabilistic model for the epistemic uncertainty due to discretization error. In this work, we apply BFEM to various inverse problems, and compare its performance to the random mesh finite element method (RM-FEM) and the statistical finite element method (statFEM), which serve as a frequentist and inference-based counterpart to BFEM. We find that by propagating this uncertainty to the posterior, BFEM can produce more accurate parameter estimates and prevent overconfidence, compared to FEM. Because the BFEM covariance operator is designed to leave uncertainty only in the appropriate space, orthogonal to the FEM basis, BFEM is able to outperform RM-FEM, which does not have such a structure to its covariance. Although inferring the discretization error via a model misspecification component is possible as well, as is done in statFEM, the feasibility of such an approach is contingent on the availability of sufficient data. We find that the BFEM is the most robust way to consistently propagate uncertainty due to discretization error to the posterior of a Bayesian inverse problem.

Time-fractional semilinear and quasilinear parabolic equations with a Caputo time derivative of order $\alpha\in(0,1)$ are considered, solutions of which exhibit a singular behaviour at an initial time of type $t^\sigma$ for any fixed $\sigma \in (0,1) \cup (1,2)$. The L1 scheme in time is combined with a general class of discretizations for the semilinear term. For such discretizations, we obtain sharp pointwise-in-time error bounds on graded temporal meshes with arbitrary degree of grading. Both semi-discretizations in time and full discretizations using finite differences and finite elements in space are addressed. The theoretcal findings are illustrated by numerical experiments.

Markov chains are fundamental models for stochastic dynamics, with applications in a wide range of areas such as population dynamics, queueing systems, reinforcement learning, and Monte Carlo methods. Estimating the transition matrix and stationary distribution from observed sample paths is a core statistical challenge, particularly when multiple independent trajectories are available. While classical theory typically assumes identical chains with known stationary distributions, real-world data often arise from heterogeneous chains whose transition kernels and stationary measures might differ from a common target. We analyse empirical estimators for such parallel Markov processes and establish sharp concentration inequalities that generalise Bernstein-type bounds from standard time averages to ensemble-time averages. Our results provide nonasymptotic error bounds and consistency guarantees in high-dimensional regimes, accommodating sparse or weakly mixing chains, model mismatch, nonstationary initialisations, and partially corrupted data. These findings offer rigorous foundations for statistical inference in heterogeneous Markov chain settings common in modern computational applications.

D-finite power series appear ubiquitously in combinatorics, number theory, and mathematical physics. They satisfy systems of linear partial differential equations whose solution spaces are finite-dimensional, which makes them enjoy a lot of nice properties. After attempts by others in the 1980s, Lipshitz was the first to prove that the class they form in the multivariate case is closed under the operation of diagonal. In particular, an earlier work by Gessel had addressed the D-finiteness of the diagonals of multivariate rational power series. In this paper, we give another proof of Gessel's result that fixes a gap in his original proof, while extending it to the full class of D-finite power series. We also provide a single exponential bound on the degree and order of the defining differential equation satisfied by the diagonal of a D-finite power series in terms of the degree and order of the input differential system.

The second eigenfunction of the Neumann Laplacian on convex, planar domains is considered. Inspired by the famous hot spots conjecture and a related result of Steinerberger, we show that potential critical points of this eigenfunction (and, in particular, interior ''hot spots'') cannot be located ''near the center'' of the domain. The region in which critical points are excluded is described explicitly.

We construct a family of smooth supersingular curves of genus $5$ in characteristic $2$ with several notable features: its dimension matches the expected dimension of any component of the supersingular locus in genus $5$, its members are non-hyperelliptic curves with non-trivial automorphism groups, and each curve in the family admits a double cover structure over both an elliptic curve and a genus-$2$ curve. We also provide an explicit parametrization of this family.

Let $\mathcal{PMD}_{n}$ be the semigroup consisting of all monotone and order-decreasing partial transformations, and let $\mathcal{IMD}_{n}$ be the subsemigroup of $\mathcal{PMD}_{n}$ consisting of all injective monotone and order-decreasing transformations on the finite chain $X_{n}=\{ 1<\cdots<n \}$. For $2\leq r\leq n$, let $\mathcal{PMD}(n,r) =\{ \alpha\in \mathcal{PMD}_{n} : |\textrm{im}(\alpha)| \leq r\}$ and $\mathcal{IMD}(n,r)=\{ \alpha \in \mathcal{IMD}_{n} :|\textrm{im}(\alpha)| \leq r\}$. In this paper, we determine the cardinalities, maximal subsemigroups and ranks of $\mathcal{PMD}(n,r)$ and $\mathcal{IMD}(n,r)$, and moreover, we verify that the semigroups $\mathcal{PMD}(n,r)$ and $\mathcal{IMD}(n,r)$ are non-regular but abundant for any $2\leq r\leq n$.

In this paper, we obtain classification of the topological holonomy groups in $SO(3)$. Such a group is given by one of the following: a finite group (such groups are classified by Klein); a commutative infinite group which is generated by one or two elements, and dense in a subgroup of $SO(3)$ isomorphic to $SO(2)$; a non-commutative infinite group generated by two elements of order $2$, $\infty$ such that these rotation axes are orthogonal; a non-commutative infinite group which is dense in $SO(3)$.

In this paper, we introduce compact group actions with the weak tracial Rokhlin property. This concept simultaneously generalizes finite group actions with the weak tracial Rokhlin property and compact group actions with the tracial Rokhlin property (in the sense of the Elliott program). Under this framework, we prove that simplicity, pure infiniteness, tracial $\mathcal{Z}$-stability and the combination of nuclearity and $\mathcal{Z}$-stability can be transferred from the original algebra to the crossed product. We also show that the radius of comparison of the fixed point algebra does not exceed that of the original algebra. Furthermore, we discuss the relationship between our definition and natural generalization of the finite group case in non-Elliott program settings. Finally, we provide a nontrivial example of a compact group action with the weak tracial Rokhlin property with comparison: an action of $(S_2)^\mathbb{N}$ on the Jiang-Su algebra $\mathcal{Z}$. Since $\mathcal{Z}$ contains no nontrivial projections, this action does not possess the tracial Rokhlin property.

For $n\leq 4$, we compute the indecomposible higher Chow groups $\overline{\operatorname{CH}}(\mathcal{M}_{1,n},1)$ with integer coefficients. As an application, we give new proofs of presentations of the integral Chow rings $\operatorname{CH}(\overline{\mathcal{M}}_{1,n})$ for $n\leq 4$ and determine formulas for the classes of boundary strata in these rings.

We provide a criterion for non-vanishing of period integrals on automorphic representations of a general linear group over a division algebra. We consider three different periods: linear periods, twisted-linear periods and Galois periods. Our criterion is a local-global principle, which is stated in terms of local distinction, a further local obstruction, and poles of certain global L-functions associated to the underlying involution via the Jacquet-Langlands correspondence. Our local-global principle follows from a new method, relying on the Maass-Selberg relations and a careful analysis of singularities of local and global intertwining periods. Our results generalize to inner forms, known results for split general linear groups. Moreover, our result for twisted linear periods is new even in the split situation. As a consequence of our local-global principle, we complete the proof of one direction of the Guo-Jacquet conjecture.

Whether two distinct APN functions can have a Hamming distance of $1$ remains an open problem. In 2020, L. Budaghyan et al. introduced a new CCZ-invariant $\Pi_F$ which can be used to provide lower bounds on the Hamming distance between a given APN function $F \colon \mathbb{F}_2^n \to \mathbb{F}_2^n$ and other APN functions. Lower bounds on the distance from an APN function $F$ to any other are known for almost bent (AB) functions and when $F$ is a 3-to-1 quadratic function with $n$ even. In this paper, we reinterpret $\Pi_F$ in terms of the exclude multiplicities of the graph $\mathcal{G}_F=\{(x, F(x)) : x \in \mathbb{F}_2^n\}$ of $F$ as a Sidon set. We establish lower bounds on the distance for even $n$ when $F$ is plateaued APN, generalize the known lower bounds for quadratic $3$-to-$1$ function to all 3-to-1 plateaued functions (e.g. Kasami functions), and derive new lower bounds for when $F$ is the APN inverse function over $\mathbb{F}_{2^n}$ for $n$ odd. We also study how the exclude multiplicities of $\mathcal{G}_F$ are directly connected to the existence of linear structures of $\gamma_F$ when $F$ is plateaued and APN and the ortho-derivative when $F$ is a quadratic APN function. We also use the CCZ-invariance of exclude multiplicities to prove that the Brinkmann-Leander-Edel-Pott function is not CCZ-equivalent to a plateaued function.

In \cite{Ca2016} and \cite{Ca2018}, we introduced a notion of effective reducibility between set-theoretical $\Pi_{2}$-statements; in \cite{Ca2025}, this was extended to statements of arbitrary (potentially even infinite) quantifier complexity. We also considered a corresponding notion of Weihrauch reducibility, which allows only one call to the effectivizer of $\psi$ in a reduction of $\phi$ to $\psi$. In this paper, we obtain a considerably refined analysis through interpolating between these two notions: Namely, we ask how many calls to an effectivizer for $\psi$ are required for effectivizing $\phi$. This allows us to make formally precise questions such as ``how many ordinals does one need to check for being cardinals in order to compute the cardinality of a given ordinal?'' and (partially) answer many of them. Many of these anwers turn out to be independent of ZFC.

In this paper, we address the Skolem problem for the $k$-generalized Pell sequence $(P_n^{(k)})_{n\geq2-k}$ extended to negative indices. We focus on identifying and bounding the indices $n<0$ for which $P_n^{(k)}=0.$ In particular, we establish that the zero multiplicity of $P_n^{(k)}$ is $ \chi_k = \lfloor k^2/4\rfloor$ for all $k \in [4, 500].$

Resolvent compositions were recently introduced as monotonicity-preserving operations that combine a set-valued monotone operator and a bounded linear operator. They generalize in particular the notion of a resolvent average. We analyze the resolvent compositions when the monotone operator is a positive linear operator. We establish several new properties, including Löwner partial order relations, concavity, and asymptotic behavior. In addition, we show that the resolvent composition operations are nonexpansive with respect to the Thompson metric. We also introduce a new form of geometric interpolation and explore its connections to resolvent compositions. Finally, we study two nonlinear equations based on resolvent compositions.

We continue the work of Belliard, Pimenta and Slavnov (2024) studying the modified rational six vertex model. We find another formula of the partition function for the inhomogeneous model, in terms of a determinant that mix the modified Izergin one and a Vandermonde one. This expression enables us to compute the partition function in the homogeneous limit for the rectangular lattice, and then to study the thermodynamic limit. It leads to a new result, we obtain the first order of free energy with boundary effects in the thermodynamic limit.

An analysis of the boundary representations and C$^*$-envelopes of some finite-dimensional operator systems $\mathcal R$ is undertaken by considering relationships between operator-theoretic properties of a $d$-tuple $\mathfrak x=(x_1,\dots, x_d)$ of elements in a unital C$^*$-algebra $\mathcal A$ and operator-system properties of the linear span $\mathcal O_{\mathfrak x}$ of $\{1_\mathcal A,x_1,x_1^*,\dots,x_d,x_d^*\}$. This approach lends itself well to the study of certain phenomena in single- and several-variable operator theory, such as the Smith-Ward property. The matrix range of a $d$-tuple of elements $\mathfrak x$ is matrix-affinely homeomorphic to the matrix state space of the operator system determined by $\mathfrak x$, and many of our methods connect the geometry of these compact matrix convex sets to operator system properties, including various forms of nuclearity and lifting properties.

Let $p$ be a prime, and let $\mathrm{X}$ be a smooth $p$-adic formal scheme over $\mathrm{Spf} \mathcal{O}_K$ where $K/\mathbf{Q}_p$ is a finite extension. We show that reflexive sheaves on the stack $\mathrm{X}^{\mathrm{Syn}}$ are equivalent to $\mathbf{Z}_p$-lattices in crystalline local systems on the rigid generic fiber $\mathrm{X}_\eta$, and then use this to study the essential image of the étale realization functor on the isogeny category of perfect complexes on $\mathrm{X}^{\mathrm{Syn}}$. We also show that when $\mathrm{X}/\mathrm{Spf} \mathcal{O}_K$ is smooth and proper that $\mathsf{Perf}(\mathrm{X}^{\mathrm{Syn}})[1/p]$ is equivalent to a category of admissible filtered $F$-isocrystals in perfect complexes.

We establish some new bilateral double-sum Rogers-Ramanujan identities involving parameters. As applications, these identities yield several new multi-sum Rogers-Ramanujan type identities. Our proofs utilize the theory of basic hypergeometric series in conjunction with the integral method.

In this note we propose a two-layer viscous mechanism for preventing finite time singularity formation in the Moffatt-Kimura model of two counter-rotating vortex rings colliding at a nontrivial angle. In the first layer the scenario is recast within the framework of the study of turbulent dissipation based on a suitably defined `scale of sparseness' of the regions of intense fluid activity. Here it is found that the problem is (at worst) critical, i.e., the upper bound on the scale of sparseness of the vorticity super-level sets is comparable to the lower bound on the radius of spatial analyticity. In the second layer, an additional more subtle mechanism is identified, potentially capable of driving the scale of sparseness into the dissipation range and preventing the formation of a singularity. The mechanism originates in certain analytic cancellation properties of the vortex-stretching term in the sense of compensated compactness in Hardy spaces which then convert information on local mean oscillations of the vorticity direction (boundedness in certain log-composite weighted local bmo spaces) into log-composite faster decay of the vorticity super-level sets.

Fix a commutative monoid $(T,+,0)$, a commutative monoid $(\Gamma,+,0_\Gamma)$, and a map \[ (a,\alpha,b,\beta,c)\longmapsto a\,\alpha\,b\,\beta\,c\in T \] which is additive in each variable and associative in the ternary sense. A left additive ternary $\Gamma$-module is an abelian group $M$ equipped with an action $(a,\alpha,b,\beta,m)\mapsto a\,\alpha\,b\,\beta\,m$ satisfying the same associativity constraints. Two scalars are intrinsic, so the action is genuinely polyadic; in particular, we do not assume a canonical unary action $T\times M\to M$. The first part constructs the \emph{operator ring} $\mathcal O_{T,\Gamma}$ generated by the left translations $(a,\alpha,b,\beta)$. It is shown that $T\!\!\;\!-\!\Gamma\mathrm{Mod}$ is equivalent to the ordinary module category $\mathcal O_{T,\Gamma}\mathrm{Mod}$. The category is therefore abelian. Under the unital operator hypothesis \emph{(U)} (i.e. when $\mathcal O_{T,\Gamma}$ is unital), it has enough projectives and enough injectives. The second part defines a tensor product over $T$ by a \emph{bi-balanced} universal property forced by the ternary action, proves right exactness of $-\otimes_T-$, and establishes a Tensor--Hom adjunction for bimodules. Assuming \emph{(U)}, derived functors $\mathrm{Ext}$ and $\mathrm{Tor}$ are developed inside the additive track. A finite example with $T=\ZZ/4\ZZ$ gives explicit computations \[\mathrm{Ext}^1_T(\ZZ/2\ZZ,\ZZ/2\ZZ)\cong \ZZ/2\ZZ,\qquad \mathrm{Tor}^T_1(\ZZ/2\ZZ,\ZZ/2\ZZ)\cong \ZZ/2\ZZ,\] with a concrete nonsplit extension and an explicit failure of tensor exactness. Counterexamples isolate the precise points where naive binary-module arguments break.

We investigate the values of the Riemann zeta function at odd integers and the Dirichlet beta function at even integers, by collecting several distinct analytic frameworks converging to these values, thus providing a unifying perspective. Beyond analytic interest, these formulas motivate linear independence conjectures which, if established, would imply the irrationality of the quantities $\zeta(2n + 1)/\pi^{2n+1}$ and $\beta(2n)/\pi^{2n}$

We establish strong connections between two fundamental nonlinear 0/1 optimization problems coming from the area of experimental design, namely maximum entropy sampling and 0/1 D-Optimality. The connections are based on maps between instances, and we analyze the behavior of these maps. Using these maps, we transport basic upper-bounding methods between these two problems, and we are able to establish new domination results and other inequalities relating various basic upper bounds. Further, we establish results relating how different branch-and-bound schemes based on these maps compare. Additionally, we observe some surprising numerical results, where bounding methods that did not seem promising in their direct application to real-data MESP instances, are now useful for MESP instances that come from 0/1 D-Optimality.

Let $E(X)$ denote the number of even integers below $X$ which are not a sum of two primes. We prove the bound $E(X)=O(X^{\frac{7}{10}})$, where the implicit constant is ineffective. The method applied here also leads to $P(q)=O(q^5)$, where $P(q)$ denotes the least prime, if it exists, in any arithmetic progression modulo $q$.

We study the topos $\mathcal{E}=\mathsf{Sh}(H\ltimes 2^{\mathbb{N}})$ arising from a nontrivial finite group $H$ acting freely on Cantor space. Using a local embedding property for the relevant epimorphisms together with effective descent for monomorphisms, we show that the \emph{internal} set universe $V$ obtained from algebraic set theory (AST) inside $\mathcal{E}$ satisfies the Partition Principle. On the other hand, the quotient $q:X\to X/H$ is a small epimorphism in $\mathcal{E}$ with no section, and this yields (via the display interpretation) an internal surjection in $V$ with no internal section; hence $V\models\neg\mathsf{AC}$. In summary, $\mathcal{E}$ contains an internal model of $\mathsf{IZF}+\mathsf{PP}+\neg\mathsf{AC}$ (and if $\mathcal{E}$ is Boolean, equivalently after $\neg\neg$-sheafification, this upgrades to $\mathsf{ZF}+\mathsf{PP}+\neg\mathsf{AC}$).

We develop a unified categorical framework for gauging both continuous and finite symmetries in arbitrary spacetime dimensions. Our construction applies to geometric categories i.e. categories internal to stacks. This generalizes the familiar setting of fusion categories, which describe finite group symmetries, to the case of Lie group symmetries. Within this framework, we obtain a functorial Symmetry Topological Field Theory together with its natural boundaries, allowing us to compute associated endomorphism categories and Drinfeld centers in a uniform way. For a given symmetry group $G$, our framework recovers the electric and magnetic higher-form symmetries expected in $G$-gauge theory. Moreover, it naturally encodes electric breaking symmetry in the presence of charged matter, reproducing known physical phenomena in a categorical setting.

Goal. We analyze when the Partition Principle ($\mathsf{PP}$) holds without $\mathsf{AC}$ in models arising from a free finite $H$-action on Cantor space, and reconcile two standard routes to such models. Approach. Route I proceeds via a Boolean-valued presentation $\mathrm{Sh}(\mathbb{B})$ and symmetric names; Route II uses direct forcing with $\mathrm{Fn}(\mathbb{N}\times H,2)$ and finite-support automorphisms. We prove a unification theorem identifying the resulting symmetric submodels and develop a Local-to-Global Embedding Principle (LEP) for hereditarily symmetric names. Results. We prove external $\mathsf{PP}$ in the symmetric model $N$ built via Route II. From LEP we obtain that $\mathsf{PP}$ holds in the symmetric model, hence $N\models \mathsf{ZF}+\mathsf{PP}+\neg\mathsf{AC}$. Along the way, we unify the Route I/Route II presentations functorially. Limitations. Our proof exploits the countable-support stratification of $\mathrm{Fn}(\mathbb{N}\times H,2)$; extending the LEP/gluing to uncountable presentations (e.g. $\mathrm{Fn}(\kappa\times H,2)$ for $\kappa>\omega$) or, more generally, to $\kappa$-directed families of finite supports remains open.

In this manuscript, we introduce a family of parametrized nonhomogeneous linear complex differential equations on $[1,\infty)$, depending on a complex parameter lying in the critical strip. We identify a "suitable" rigidity hypothesis under which a phenomenon of structural asymmetry is established: if two solutions with the same initial condition equal to $1$, corresponding respectively to the parameters $s$ and $1-\bar{s}$, are both bounded on $[1,+\infty)$, then $\Re(s)=\frac{1}{2}$.

We consider the problem of joint estimation of the parameters of $m$ linear dynamical systems, given access to single realizations of their respective trajectories, each of length $T$. The linear systems are assumed to reside on the nodes of an undirected and connected graph $G = ([m], \mathcal{E})$, and the system matrices are assumed to either vary smoothly or exhibit small number of ``jumps'' across the edges. We consider a total variation penalized least-squares estimator and derive non-asymptotic bounds on the mean squared error (MSE) which hold with high probability. In particular, the bounds imply for certain choices of well connected $G$ that the MSE goes to zero as $m$ increases, even when $T$ is constant. The theoretical results are supported by extensive experiments on synthetic and real data.

The theory of algebras with polynomial identities has developed significantly, with special attention devoted to the classification of varieties according to the asymptotic behavior of their codimension sequences. This sequence is a fundamental numerical invariant, as it captures the growth rate of the polynomial identities of a given algebra. Special partial classification results have been obtained, with particular interest devoted to algebras equipped with additional structure. In this paper, we consider associative G-graded algebras endowed with a graded involution. We provide a complete classification, up to equivalence, of unitary algebras with quadratic codimension growth. Our approach establishes a direct correspondence between the algebras generating minimal varieties and the nonzero multiplicities appearing in the decomposition of the proper cocharacters. As a consequence, we establish that every variety with at most quadratic growth is generated by an algebra that decomposes as a direct sum of algebras generating minimal varieties of at most quadratic growth.

Generative models have shown immense potential for wireless communication by learning complex channel data distributions. However, the iterative denoising process associated with these models imposes a significant challenge in latency-sensitive wireless communication scenarios, particularly in channel estimation. To address this challenge, we propose a novel solution for one-step generative channel estimation. Our approach bypasses the time-consuming iterative steps of conventional models by directly learning the average velocity field. Through extensive simulations, we validate the effectiveness of our proposed method over existing state-of-the-art diffusion-based approach. Specifically, our scheme achieves a normalized mean squared error up to 2.65 dB lower than the diffusion method and reduces latency by around 90%, demonstrating the potential of our method to enhance channel estimation performance.

We develop a dynamical method for proving the sharp Berezin--Li--Yau inequality. The approach is based on the volume-preserving mean curvature flow and a new monotonicity principle for the Riesz mean $R_\Lambda(\Omega_t)$. For convex domains we show that $R_\Lambda$ is monotone non-decreasing along the flow. The key input is a geometric correlation inequality between the boundary spectral density $Q_\Lambda$ and the mean curvature $H$, established in all dimensions: in $d=2$ via circular symmetrization, and in $d\ge 3$ via the boundary Weyl expansion together with the Laugesen--Morpurgo trace minimization principle. Since the flow converges smoothly to the ball, the monotonicity implies the sharp Berezin--Li--Yau bound for every smooth convex domain. As an application, we obtain a sharp dynamical Cesàro--Pólya inequality for eigenvalue averages.

For any sufficiently large $\ell$, suppose that $\ell$ can be expressed as $$ \ell=p_1^4+p_2^4+p_3^4+ \cdots +p_8^4,$$ where $p_1, p_2,p_3,\cdots, p_8$ are this http URL such $\ell$, in this paper we will use circle method and sieves to prove that the proportion of $\ell$ in positive integers is at least $\frac{1}{414.465}$ .

We compute the $RO(C_2)$-graded real Brown--Peterson homology of the representation-loop space $\Omega^\rho S^{\rho + 1}$, where $\rho$ is the regular representation of the cyclic group of order two. This calculation gives a $C_2$-equivariant analogue of the classical computation of Brown--Peterson homology of the double loop space $\Omega^2 S^3$ due to Ravenel. Along the way, we develop comodule Nishida relations for $\rho$-loop spaces.

Halin characterised the chordal locally finite graphs as those that admit a tree-decomposition into cliques. We show that these tree-decompositions can be chosen to be canonical, that is, so that they are invariant under all the graph's automorphisms. As an application, we show that a locally finite, connected graph $G$ is $r$-locally chordal (that is, its $r/2$-balls are chordal) if and only if the unique canonical graph-decomposition $\mathcal{H}_r(G)$ of $G$ which displays its $r$-global structure is into cliques. Our results also serve as tools for further characterisations of $r$-locally chordal graphs.

High-mobility wireless communication systems suffer from severe Doppler spread and multi-path delay, which degrade the reliability and spectral efficiency of conventional modulation schemes. Orthogonal time frequency space (OTFS) modulation offers strong robustness in such environments by representing symbols in the delay-Doppler (DD) domain, while faster-than-Nyquist (FTN) signaling can further enhance spectral efficiency through intentional symbol packing. Meanwhile, reconfigurable intelligent surfaces (RIS) provide a promising means to improve link quality via passive beamforming. Motivated by these advantages, we propose a novel RIS-empowered OTFS modulation with FTN signaling (RIS-OTFS-FTN) scheme. First, we establish a unified DD-domain input-output relationship that jointly accounts for RIS passive beamforming, FTN-induced inter-symbol interference, and DD-domain channel characteristics. Based on this model, we provide comprehensive analytical performance for the frame error rate, spectral efficiency, and peak-to-average power ratio (PAPR), etc. Furthermore, a practical RIS phase adjustment strategy with quantized phase selection is designed to maximize the effective channel gain. Extensive Monte Carlo simulations under a standardized extended vehicular A (EVA) channel model validate the theoretical results and provide key insights into the trade-offs among spectral efficiency, PAPR, input back-off (IBO), and error performance, with some interesting this http URL proposed RIS-OTFS-FTN scheme demonstrates notable performance gains in both reliability and spectral efficiency, offering a viable solution for future high-mobility and spectrum-constrained wireless systems.

In Part I, we established the construction of the Prequantum Groupoid for simply connected spaces. This second part extends the theory to arbitrary connected parasymplectic diffeological spaces $(\mathrm{X}, \omega)$. We identify the obstruction to the existence of the Prequantum Groupoid as the non-additivity of the integration of the prequantum form $\mathbf{K}\omega$ on the space of loops. By defining a Total Group of Periods $\mathrm{P}_\omega$ directly on the space of paths, which absorbs the periods arising from the algebraic relations of the fundamental group, we construct a Prequantum Groupoid $\mathbf{T}_\omega$ with connected isotropy isomorphic to the torus of periods $\mathrm{T}_\omega = \mathbf{R}/\mathrm{P}_\omega$. Furthermore, we propose that this groupoid $\mathbf{T}_\omega$ constitutes the Quantum System itself. The classical space $\mathrm{X}$ is embedded as the Skeleton of units, surrounded by a Quantum Fog of non-identity morphisms. We prove that the group of automorphisms of the Quantum System is isomorphic to the group of symmetries of the Dynamical System, $\mathrm{Diff}(\mathrm{X}, \omega)$.

We use numerics to construct solitary waves $\phi_\omega(x) e^{-\mathrm{i}\omega t}$ in Dirac--Klein--Gordon (in one and three spatial dimensions) and study the dependence of energy and charge of $\omega$. To construct solitary waves, we use two different procedures: the iterative method and the nested shooting method. We also consider the case of massless scalar field where we show that the standard shooting method becomes available. We use the virial identities to control the error of simulations. We discuss possible implications for the stability of solitary waves.

We give an algebraic proof of the unitarity of the vertex operator algebra $L(21/22, 0)\oplus L(21/22, 8)$ and of all its irreducible ordinary modules, using a coset realization arising from the $3C$-algebra. Motivated by the structure of the resulting module decomposition, we establish a general result on fusion rules for commutant vertex operator subalgebras within the framework of modular tensor categories. As an application of this general result, we explicitly determine the fusion rules of all irreducible $L(21/22, 0)\oplus L(21/22, 8)$-modules.

In this article, we study Chow polynomials of weakly ranked posets and prove the existence of Gorenstein algebras with the Kähler package such that their Hilbert--Poincaré series agrees with the Chow polynomial. Our statement provides evidence in support of a conjecture by Ferroni, Matherne and the second author about the existence of an algebra for every weakly ranked poset that generalizes the Feichtner--Yuzvinsky Chow ring for matroids. This allows us to prove strong inequalities for the coefficients of Chow polynomials; we prove log-concavity for all posets of weak rank at most six and provide counterexamples to log-concavity for any higher rank. For ranked posets we recover an even stronger condition, showing that the differences between consecutive coefficients constitute a pure O-sequence.

We study the complexity of the $KK$-equivalence relation on unital $C^*$-algebras, in the sense of descriptive set theory. We prove that $KK$-equivalence is analytic, which in turn shows that the set of separable $C^*$-algebras satisfying the UCT is analytic. This allows us to prove a game-theoretic refinement of the unital classification theorem: there is a transfer of strategies between Ehrenfeucht-Fraïssé games (of various lengths) on classifiable $C^*$-algebras and their invariants.

We construct, from a ground model of $ZFC$, a transitive symmetric model $M$ satisfying $ZF + DC + PP + AC_{wo} + \neg AC$. The construction starts with a Cohen symmetric seed model $N$ over $Add(\omega,\omega_1)$ and performs an Ord-length countable-support symmetric iteration. For fixed parameters $S:=A^\omega$ and $T:=PowerSet(S)$ (as computed in $N$), successor stages add orbit-symmetrized packages which force the localized splitting principle $PP^{\mathrm{split}}\!\restriction T$ (hence $PP\restriction T$) and the choice principle $AC_{wo}$, while preserving $DC$ and keeping $A$ non-well-orderable. A diagonal-lift/diagonal-cancellation scheme produces $\omega_1$-complete normal limit filters. A persistence argument yields $SVC^+(T)$ in M, and Ryan--Smith localization then upgrades $PP\restriction T$ and $AC_{wo}$ to $PP$.

We study data-driven construction of spatial discretizations for the one-dimensional Maxwell system. Using high-fidelity training data from a spectral discretization, we learn a \emph{linear convolution stencil} that approximates the spatial derivative operator in Maxwell's equations. We formulate a convex quadratic program for the stencil coefficients with linear constraints that enforce skew-adjointness of the discrete derivative; these constraints guarantee a semi-discrete electromagnetic energy identity and yield a CFL condition expressed directly in terms of the stencil's Fourier symbol. We compare several convex solvers for the resulting quadratic program -- projected gradient, Nesterov-accelerated gradient, ADMM, and an interior-point reference implemented in CVXPY -- and evaluate the learned operators in time-dependent Maxwell simulations using a Crank--Nicolson (CN) discretization. Numerical experiments, including cases with nonstandard target operators and noisy training data, show that (i) energy-constrained learned stencils achieve accuracy comparable to standard central differences while exactly preserving the discrete electromagnetic energy under CN time-stepping, and (ii) ADMM and interior-point solvers produce nearly identical operators, with ADMM offering a favorable tradeoff between accuracy, constraint satisfaction, and runtime.

The celebrated Mason's conjecture states that the sequence of independent set numbers of any matroid is log-concave, and even ultra log-concave. The strong form of Mason's conjecture was independently solved by Anari, Liu, Oveis Gharan and Vinzant, and by Brändén and Huh. The weak form of Mason's conjecture was also generalized to a polynomial version by Dowling in 1980 by considering certain polynomial analogue of independent set numbers. In this paper we completely solve Dowling's polynomial conjecture by using the theory of Lorentzian polynomials.

We extend a property of Mazzola's theory of cadential sets in relation to the modulation between minor and major tonalities from triadic to tetradic harmony, using the PLRQ group of Cannas et al. (2017) as the analogue of the classical PLR group. While the PLR group connects triadic cadential sets via the relative morphism $R$, the tetradic case reveals a richer structure: two pairs of cadential sets connected by distinct morphisms forming a "prism" in the slice category over the tonic seventh chord, and a single pair for those that allow quantized modulations. We demonstrate this structure through analysis of Charlie Parker's "Blues for Alice" (1951) and Ray Noble's "Cherokee" (1938), showing how the prism morphism, PLRQ transformations and quantized modulations organize harmonic navigation in bebop. The categorical framework captures what syntactic approaches miss: the transformational "vécu" that musicians actually experience when navigating between cadential regions.

In this paper we compute an explicit closed formula for the total Tjurina number $\tau(C)$ of a reduced projective plane curve $C$ in terms of the graded Betti numbers of the corresponding Jacobian algebra. This formula allows a completely new view point on the classical upper bounds for the total Tjurina number $\tau(C)$ of a plane curve $C$ given by A. du Plessis and C. T. C. Wall. This approach yields in particular a new necessary condition for a set of positive integers to be the graded Betti numbers of the Jacobian algebra of a reduced plane curve.

The Addition Theorem for the algebraic entropy of group endomorphisms of torsion abelian groups was proved by Dikranjan, Goldsmith, Salce and Zanardo. It was later extended by Shlossberg to torsion nilpotent groups of class 2. As our main result, we prove the Addition Theorem for endomorphisms of torsion nilpotent groups of arbitrary nilpotency class. As an application, we show that if $G$ is a torsion nilpotent group, then for every $\phi\in\operatorname{End}(G)$ either the entropy is infinite or $h(\phi)=\log(\alpha)$ for a positive integer $\alpha$. We further obtain, for automorphisms of locally finite groups, the Addition Theorem with respect to every term of the upper central series; in particular, it holds for automorphisms of $\omega$-hypercentral groups. Finally, we establish a reduction principle: if $\mathfrak X$ is a class of locally finite groups closed under taking subgroups and quotients, then the Addition Theorem for endomorphisms holds in $\mathfrak X$ if and only if it holds for locally finite groups generated by bounded sets.

We show that any positive, continuous, and bounded function can be realised as the diffusion coefficient of an evolution equation associated with a gradient interacting particle system. The proof relies on the construction of an appropriate model and on the entropy method.

Causal discovery based on Independent Component Analysis (ICA) has achieved remarkable success through the LiNGAM framework, which exploits non-Gaussianity and independence of noise variables to identify causal order. However, classical LiNGAM methods rely on the strong assumption that there exists an ordering under which the noise terms are exactly independent, an assumption that is often violated in the presence of confounding. In this paper, we propose a general information-theoretic framework for causal order estimation that remains applicable under arbitrary confounding. Rather than imposing independence as a hard constraint, we quantify the degree of confounding by the multivariate mutual information among the noise variables. This quantity is decomposed into a sum of mutual information terms along a causal order and is estimated using Bayesian marginal likelihoods. The resulting criterion can be interpreted as Bayesian ICA for causal discovery, where causal order selection is formulated as a model selection problem over permutations. Under standard regularity conditions, we show that the proposed Bayesian mutual information estimator is consistent, with redundancy of order $O(\log n)$. To avoid non-identifiability caused by Gaussian noise, we employ non-Gaussian predictive models, including multivariate $t$ distributions, whose marginal likelihoods can be evaluated via MCMC. The proposed method recovers classical LiNGAM and DirectLiNGAM as limiting cases in the absence of confounding, while providing a principled ranking of causal orders when confounding is present. This establishes a unified, confounding-aware, and information-theoretically grounded extension of ICA-based causal discovery.

We prove some results on the structure of ind-pro completions of Noetherian rings along flags of prime ideals. In particular, we compute the Krull dimension and deduce the criterion on semilocality in the case of essentially of finite type algebras over a field. We also show that ind-pro completion inherits properties of the base ring such as normality, regularity, local equidimensionality, etc.

We study a one-parameter family of binomial-convolution operators acting on sequences. These operators form an additive semigroup with an explicit inverse, and they subsume iterated classical binomial transforms as a special case. We describe the action in terms of ordinary and exponential generating functions, interpret the transform in the Riordan-array framework, and prove a general root-shift principle for constant-coefficient linear recurrences: applying the transform shifts the characteristic roots by a fixed amount. Several classical families (Fibonacci, Lucas, Pell, Jacobsthal, Mersenne) are treated uniformly as illustrative examples.

Motivated by the Onsager statistical mechanics description of turbulent Euler flows with point singularities, we make a first step in the generalization of the mean field theory in [Caglioti, Lions, Marchioro, Pulvirenti; Comm. Math. Phys. (1995)]. On one side we prove the equivalence of statistical ensembles, on the other side we are bound to the analysis of a new blow up phenomenon, which we call "blow up and concentration without quantization", where the mass associated with the concentration is allowed to take values in a full interval of real numbers. This singular behavior may be regarded as lying between the classical blow up-concentration-quantization and the blow up without concentration phenomenon first proposed in [Lin, Tarantello; C.R. Math. Acad. Sci. Paris (2016)]. A careful analysis is needed to generalize known pointwise estimates in this non standard context, resulting in a complete description of the allowed asymptotic profiles.

Volume polynomials form a distinguished class of log-concave polynomials with remarkable analytic and combinatorial properties. I will survey realization problems related to them, review fundamental inequalities they satisfy, and discuss applications to the combinatorics of algebraic matroids. These notes are based on lectures given at the 2025 Summer Research Institute in Algebraic Geometry at Colorado State University.

We extend the results of (Liu and Firoozi, 2025), which develops the theory of linear-quadratic (LQ) mean field games (MFGs) in Hilbert spaces, by incorporating a common noise. This common noise is modeled as an infinite-dimensional Wiener process affecting the dynamics of all agents. In the presence of common noise, the mean-field consistency condition is characterized by a system of coupled forward-backward stochastic evolution equations (FBSEEs) in Hilbert spaces, whereas, in its absence it is represented by coupled forward-backward deterministic evolution equations. We establish the existence and uniqueness of solutions to the coupled linear FBSEEs associated with the LQ MFG framework for small time horizons and prove the $\epsilon$-Nash property of the resulting equilibrium strategy. Furthermore, we establish the well-posedness of these coupled linear FBSEEs for arbitrary finite time horizons. Beyond the specific context of MFGs, our analysis also yields a broader contribution by providing, to the best of our knowledge, the first well-posedness result for a class of infinite-dimensional linear FBSEEs, for which only mild solutions exist, over arbitrary finite time horizons.

This paper develops a systematic approach to infinitesimal variations of Hodge structure for singular and equisingular families by means of logarithmic geometry and residue theory. The central idea is that logarithmic vector fields encode precisely those deformation directions that preserve singularities and act trivially on Hodge structures, while the effective variation is entirely governed by residue calculus. This viewpoint provides a conceptual reinterpretation of classical results of Griffiths, Green, and Voisin, and extends them to settings involving singular varieties and equisingular deformations. The resulting framework yields a geometric explanation for the appearance of Jacobian rings in infinitesimal Hodge theory and clarifies the structure of deformation spaces underlying Severi varieties and related moduli problems.

Channel estimation is a fundamental challenge in massive multiple-input multiple-output systems, where estimation accuracy governs the spectral efficiency and link reliability. In this work, we introduce Recursive Flow (RC-Flow), a novel solver that leverages pre-trained flow matching priors to robustly recover channel state information from noisy, under-determined measurements. Different from conventional open-loop generative models, our approach establishes a closed-loop refinement framework via a serial restart mechanism and anchored trajectory rectification. By synergizing flow-consistent prior directions with data-fidelity proximal projections, the proposed RC-Flow achieves robust channel reconstruction and delivers state-of-the-art performance across diverse noise levels, particularly in noise-dominated scenarios. The framework is further augmented by an adaptive dual-scheduling strategy, offering flexible management of the trade-off between convergence speed and reconstruction accuracy. Theoretically, we analyze the Jacobian spectral radius of the recursive operator to prove its global asymptotic stability. Numerical results demonstrate that RC-Flow reduces inference latency by two orders of magnitude while achieving a 2.7 dB performance gain in low signal-to-noise ratio regimes compared to the score-based baseline.

The visibility of lattice points from the origin along a polynomial family of curves constitutes a significant generalization of visibility along straight lines. Following the classical notion, where the density equals 1/2, and its generalization to monomial curves of the form y = a x^b, where the density equals 1/(b+1), we study a family of polynomial curves defined by y = q(a_n x^n + ... + a_1 x), where q is a positive rational number. We introduce a new criterion based on a polynomial greatest common divisor condition that provides a lower bound on the number of visible lattice points in N^2. Conversely, we derive conditions under which a given lattice point becomes the next visible point along such a polynomial curve. Using the principle of inclusion-exclusion, we also obtain an exact double-sum formula for the number of pairs (a, b) less than or equal to N that are visible with respect to this polynomial family. Finally, we extend the framework to related problems and pose several open questions concerning gap distributions and quantitative bounds for non-visible points. This work provides a broader theoretical foundation for lattice point visibility beyond linear and monomial settings.

Let $p$ be a prime. For $p=2$, the fields of values of the complex irreducible characters of finite groups whose degrees are not divisible by $p$ have been classified; for odd primes $p$, a conjectural classification has been proposed. In this work, we extend this conjecture to characters whose degrees are divisible by arbitrary powers of $p$, and we provide some evidence supporting its validity.

Time-optimal control for triple integrator under full box constraints is a fundamental problem in the field of optimal control, which has been widely applied in the industry. However, scenarios involving asymmetric constraints, non-stationary boundary conditions, and active position constraints pose significant challenges. This paper provides a complete characterization of time-optimal switching surfaces for the problem, leading to novel insights into the geometric structure of the optimal control. The active condition of position constraints is derived, which is absent from the literature. An efficient algorithm is proposed, capable of planning time-optimal trajectories under asymmetric full constraints and arbitrary boundary states, with a 100% success rate. Computational time for each trajectory is within approximately 10$\mu$s, achieving a 5-order-of-magnitude reduction compared to optimization-based baselines.

We prove that the Ginzburg-Landau equation correctly predicts the dynamics of quasilinear pattern-forming reaction-diffusion-advection systems, close to the first instability. We present a simple theorem which is easily applicable for such systems and relies on key maximal regularity results. The theorem is applied to the Gray-Scott-Klausmeier vegetation-water interaction model and its application to general reaction-diffusion-advection systems is discussed.

This paper addresses the problem of robust option pricing within the framework of Vectorial Martingale Optimal Transport (VMOT). We investigate the geometry of VMOT solutions for $N$-period market models and demonstrate that, when the number of underlying assets is $d=2$ and the payoff is sub- or supermodular, the extremal model reduces to a single-factor structure in the first period. This structural result allows for a significant dimension reduction, transforming the problem into a more tractable format. We prove that this reduction is specific to the two-asset case and provide counterexamples showing it generally fails for $d \geq 3$. Finally, we exploit this monotonicity to develop a reduced-dimension Sinkhorn algorithm. Numerical experiments demonstrate that this structure-preserving approach reduces computational time by approximately 99\% compared to standard methods while improving accuracy.

The complexity of molecular and cellular processes forces experimental studies to focus on subsystems. To study the functioning of biological systems across levels of structural and functional organisation, we require tools to compose and organise networks with different levels of detail and abstraction. Systems Biology Graphical Notation (SBGN) is a standardised notational system that visualises biochemical processes as networks. Despite their widespread adoption, SBGN languages remain purely visual and lack an underlying mathematical framework, limiting their compositional analysis, abstraction, and integration with formal modelling approaches. SBGN comprises three complementary visual languages-Process Description (SBGN-PD), Activity Flow (SBGN-AF), and Entity Relationship (SBGN-ER)-each operating at a different level of abstraction. In this manuscript, we introduce a category-theoretic formalism for SBGN-PD, a visual language to describe biochemical processes as biochemical reaction networks. Using the theory of structured cospans, we construct a symmetric monoidal double category whose horizontal 1-morphisms correspond to SBGN-PD diagrams. We also analyse how a designated subnetwork influences the surrounding network and how external entities, in turn, affect the internal reactions of the subnetwork. Our work addresses a key gap between biological visualisation and mathematical structure. It provides precise organising principles for SBGN-PD, including compositionality, enabling the construction of large biochemical reaction networks from smaller ones, and zooming out, allowing the abstraction of detailed biochemical mechanisms while preserving their functional interfaces. Throughout the paper, the proposed framework is illustrated using standard SBGN-PD examples, demonstrating its applicability to large-scale biochemical reaction networks.

Efficient and scalable decoding of quantum codes is essential for high-performance quantum error correction. In this work, we introduce Reliable Subset Reduction (RSR), a reliability-driven preprocessing framework that leverages belief propagation (BP) statistics to identify and remove highly reliable qubits, substantially reducing the effective problem size. Additionally, we identify a degeneracy condition that allows high-order OSD to be simplified to order-0 OSD. By integrating these techniques, we present an ADOSD algorithm that significantly improves OSD efficiency. Our BP+RSR+ADOSD framework extends naturally to circuit-level noise and can handle large-scale codes with more than $10^4$ error variables. Through extensive simulations, we demonstrate improved performance over MWPM and Localized Statistics Decoding for a variety of CSS and non-CSS codes under the code-capacity noise model, and for rotated surface codes under realistic circuit-level noise. At low physical error rates, RSR reduces the effective problem size to less than 5\%, enabling higher-order OSD with accelerated runtime. These results highlight the practical efficiency and broad applicability of the BP+ADOSD framework for both theoretical and realistic quantum error correction scenarios.

When the phase shift of X-shaped solutions before and after interaction is finite but approaches infinity, the vertices of the two V-shaped structures become separated due to the phase shift and are connected by a localized structure, which is referred to as the stem structure. This special type of elastic collision is known as a quasi-resonant collision. This study investigates quasi-resonant solutions and the associated localized stem structures in the context of the KPII and KPI equations. For the KPII equation, we classify quasi-resonant 2-solitons into weakly and strongly types, depending on whether the phase approaches \(-\infty\) or \(+\infty\). We analyze their asymptotic forms to detail the trajectories, amplitudes, velocities, and lengths of their stem structures. These results of quasi-resonant 2-solitons are used to to provide analytical descriptions of interesting patterns of the water waves observed on shallow water surface. Similarly, for the KPI equation, we construct quasi-resonant breather-soliton solutions and classify them into weakly and strongly types, based on the behavior of their internal parameters. We compare the similarities and differences between the stem structures in the quasi-resonant soliton and the quasi-resonant breather-soliton. Additionally, we provide a comprehensive and rigorous analysis of their asymptotic forms and stem structures. Our results indicate that the resonant solution, i.e. resonant breather-soliton of the KPI and soliton for the KPII, represents the limiting case of the quasi-resonant solution as phase approaches \(\infty\).

Temporal graphs arise when modeling interactions that evolve over time. They usually come in several flavors, depending on the number of parameters used to describe the temporal aspects of the interactions: time of appearance, duration, delay of transmission. In the point model, edges appear at specific points in time, whereas in the more general interval model, edges can be present over specific time intervals. In both models, the delay for traversing an edge can change with each edge appearance. When time is discrete, the two models are equivalent in the sense that the presence of an edge during an interval is equivalent to a sequence of point-in-time occurrences of the edge. However, this transformation can drastically change the size of the input and has implications for complexity. Indeed, we show a gap between the two models with respect to the complexity of the classical problem of computing a fastest temporal path from a source vertex to a target vertex, i.e. a path where edges can be traversed one after another in time and such that the total duration from source to target is minimized. It can be solved in near-linear time in the point model, while we show that the interval model requires quadratic time under classical assumptions of fine-grained complexity. With respect to linear time, our lower bound implies a factor of the number of vertices, while the best known algorithm has a factor of the number of underlying edges. We also show a similar complexity gap for computing a shortest temporal path, i.e. a temporal path with a minimum number of edges. Here our lower bound matches known upper bounds up to a logarithmic factor. Interestingly, we show that near-linear time for fastest temporal path computation is possible in the interval model when it is restricted to uniform delay zero, i.e., when traversing an edge is instantaneous. However, this special case is not exempt from our lower bound for shortest temporal path computation. These two results should be contrasted with the computation of a foremost temporal path, i.e., a temporal path that arrives as early as possible. It is well known that this computation can be solved in near-linear time in both models. We also show that there is no gap in testing the all-to-all temporal connectivity of a temporal graph. We demonstrate a quadratic lower bound that applies to both the interval and point models and aligns with the existing upper bounds.

We consider a general model for high-dimensional empirical risk minimization whereby the data $\mathbf{x}_i$ are $d$-dimensional Gaussian vectors, the model is parametrized by $\mathbf{\Theta}\in\mathbb{R}^{d\times k}$, and the loss depends on the data via the projection $\mathbf{\Theta}^\mathsf{T}\mathbf{x}_i$. This setting covers as special cases classical statistics methods (e.g. multinomial regression and other generalized linear models), but also two-layer fully connected neural networks with $k$ hidden neurons. We use the Kac-Rice formula from Gaussian process theory to derive a bound on the expected number of local minima of this empirical risk, under the proportional asymptotics in which $n,d\to\infty$, with $n\asymp d$. Via Markov's inequality, this bound allows to determine the positions of these minimizers (with exponential deviation bounds) and hence derive sharp asymptotics on the estimation and prediction error. As a special case, we apply our characterization to convex losses. We show that our approach is tight and allows to prove previously conjectured results. In addition, we characterize the spectrum of the Hessian at the minimizer. A companion paper applies our general result to non-convex examples.

We study an online dynamic pricing problem where the potential demand at each time period $t=1,2,\ldots, T$ is stochastic and dependent on the price. However, a perishable inventory is imposed at the beginning of each time $t$, censoring the potential demand if it exceeds the inventory level. To address this problem, we introduce a pricing algorithm based on the optimistic estimates of derivatives. We show that our algorithm achieves $\tilde{O}(\sqrt{T})$ optimal regret even with adversarial inventory series. Our findings advance the state-of-the-art in online decision-making problems with censored feedback, offering a theoretically optimal solution against adversarial observations.

In nonperturbative formulation of Euclidean signature quantum field theory (QFT), the vacuum state is characterized by the Wilsonian renormalization group (RG) flow of Feynman measures. Such an RG flow is a family of Feynman measures on the space of ultraviolet (UV) regularized fields, linked by the Wilsonian renormalization group equation. In this paper we show that under mild conditions, a Wilsonian RG flow of Feynman measures extending to arbitrary regularization strengths has a factorization property: there exists an ultimate Feynman measure (UV limit) on the distribution sense fields, such that the regularized instances in the flow are obtained from this UV limit via taking the marginal measure against the regulator. Existence theorems on the flow and UV limit of the corresponding action functional are also discussed.

Tipping behavior can occur when an equilibrium of a dynamical system loses stability in response to a slowly varying parameter crossing a bifurcation threshold, or where noise drives a system from one attractor to another, or some combination of these effects. Similar behavior can be expected when a multistable system is forced by a chaotic deterministic system rather than by noise. In this context, the chaotic tipping window was recently introduced and investigated for discrete-time dynamics. In this paper, we find tipping windows for continuous-time nonlinear systems forced by chaos. We characterize the tipping window in terms of forcing by unstable periodic orbits of the chaos, and we show how the location and structure of this window depend on the relative timescales between the forcing and the responding system. We illustrate this by finding tipping windows for two examples of coupled bistable ODEs forced with chaos. Additionally, we describe the dynamic tipping window in the setting of a changing system parameter.

The Skolem Problem asks to determine whether a given linear recurrence sequence (LRS) has a zero term. Showing decidability of this problem is equivalent to giving an effective proof of the Skolem-Mahler-Lech Theorem, which asserts that a non-degenerate LRS has finitely many zeros. The latter result was proven over 90 years ago via an ineffective method showing that such an LRS has only finitely many $p$-adic zeros. In this paper we consider the problem of determining whether a given LRS has a $p$-adic zero, as well as the corresponding function problem of computing exact representations of all $p$-adic zeros. We present algorithms for both problems and report on their implementation. The output of the algorithms is unconditionally correct, and termination is guaranteed subject to the $p$-adic Schanuel Conjecture (a standard number-theoretic hypothesis concerning the $p$-adic exponential function). While these algorithms do not solve the Skolem Problem, they can be exploited to find natural-number and rational zeros under additional hypotheses. To illustrate this, we apply our results to show decidability of the Simultaneous Skolem Problem (determine whether two coprime linear recurrences have a common natural-number zero), again subject to the $p$-adic Schanuel Conjecture.

The real part of the focusing modified Korteweg-de Vries (MKdV) equation defined over the complex field $\mathbb{C}$ is reduced to the focusing gauged MKdV (FGMKdV) equation. In this paper, we construct the real hyperelliptic solutions of FGMKdV equation in terms of data of the hyperelliptic curves of genus $g$ and demonstrate the closed hyperelliptic plane curves of genus $g=5$ whose curvature obeys the FGMKdV equation by extending the previous results of genus three (Matsutani, {\it{J. Geom. Phys}} {\bf{215}} (2025) 105540). These are a generalization of Euler's elasticae.

We prove that kernel covariance embeddings lead to information-theoretically perfect separation of distinct continuous probability distributions. In statistical terms, we establish that testing for the \emph{equality} of two non-atomic (Borel) probability measures on a locally compact Polish space is \emph{equivalent} to testing for the \emph{singularity} between two centered Gaussian measures on a reproducing kernel Hilbert space. The corresponding Gaussians are defined via the notion of kernel covariance embedding of a probability measure, and the Hilbert space is that generated by the embedding kernel. Distinguishing singular Gaussians is structurally simpler from an information-theoretic perspective than non-parametric two-sample testing, particularly in complex or high-dimensional domains. This is because singular Gaussians are supported on essentially separate and affine subspaces. Our proof leverages the classical Feldman-Hájek dichotomy, and shows that even a small perturbation of a continuous distribution will be maximally magnified through its Gaussian embedding. This ``separation of measure phenomenon'' appears to be a blessing of infinite dimensionality, by means of embedding, with the potential to inform the design of efficient inference tools in considerable generality. The elicitation of this phenomenon also appears to crystallize, in a precise and simple mathematical statement, a core mechanism underpinning the empirical effectiveness of kernel methods.

This paper develops an interpretable, non-intrusive reduced-order modeling technique using regularized kernel interpolation. Existing non-intrusive approaches approximate the dynamics of a reduced-order model (ROM) by solving a data-driven least-squares regression problem for low-dimensional matrix operators. Our approach instead leverages regularized kernel interpolation, which yields an optimal approximation of the ROM dynamics from a user-defined reproducing kernel Hilbert space. We show that our kernel-based approach can produce interpretable ROMs whose structure mirrors full-order model structure by embedding judiciously chosen feature maps into the kernel. The approach is flexible and allows a combination of informed structure through feature maps and closure terms via more general nonlinear terms in the kernel. We also derive a computable a posteriori error bound that combines standard error estimates for intrusive projection-based ROMs and kernel interpolants. The approach is demonstrated in several numerical experiments that include comparisons to operator inference using both proper orthogonal decomposition and quadratic manifold dimension reduction.

We study the complexity of computing the mixed Schatten $\|\Phi\|_{q\to p}$ norms of linear maps $\Phi$ between matrix spaces. When $\Phi$ is completely positive, we show that $\| \Phi \|_{q \to p}$ can be computed efficiently when $q \geq p$. The regime $q \geq p$ is known as the non-hypercontractive regime and is also known to be easy for the mixed vector norms $\ell_{q} \to \ell_{p}$ [Boyd, 1974]. However, even for entanglement-breaking completely-positive trace-preserving maps $\Phi$, we show that computing $\| \Phi \|_{1 \to p}$ is $\mathsf{NP}$-complete when $p>1$. Moving beyond the completely-positive case and considering $\Phi$ to be difference of entanglement breaking completely-positive trace-preserving maps, we prove that computing $\| \Phi \|^+_{1 \to 1}$ is $\mathsf{NP}$-complete. In contrast, for the completely-bounded (cb) case, we describe a polynomial-time algorithm to compute $\|\Phi\|_{cb,1\to p}$ and $\|\Phi\|^+_{cb,1\to p}$ for any linear map $\Phi$ and $p\geq1$.

In multi-source learning with discrete labels, distributional heterogeneity across domains poses a central challenge to developing predictive models that transfer reliably to unseen domains. We study multi-source unsupervised domain adaptation, where labeled data are available from multiple source domains and only unlabeled data are observed from the target domain. To address potential distribution shifts, we propose a novel Conditional Group Distributionally Robust Optimization (CG-DRO) framework that learns a classifier by minimizing the worst-case cross-entropy loss over the convex combinations of the conditional outcome distributions from sources domains. We develop an efficient Mirror Prox algorithm for solving the minimax problem and employ a double machine learning procedure to estimate the risk function, ensuring that errors in nuisance estimation contribute only at higher-order rates. We establish fast statistical convergence rates for the empirical CG-DRO estimator by constructing two surrogate minimax optimization problems that serve as theoretical bridges. A distinguishing challenge for CG-DRO is the emergence of nonstandard asymptotics: the empirical CG-DRO estimator may fail to converge to a standard limiting distribution due to boundary effects and system instability. To address this, we introduce a perturbation-based inference procedure that enables uniformly valid inference, including confidence interval construction and hypothesis testing.

Time-dependent partial differential equations are ubiquitous in physics-based modeling, but they remain computationally intensive in many-query scenarios, such as real-time forecasting, optimal control, and uncertainty quantification. Reduced-order modeling (ROM) addresses these challenges by constructing a low-dimensional surrogate model but relies on a fixed discretization, which limits flexibility across varying meshes during evaluation. Operator learning approaches, such as neural operators, offer an alternative by parameterizing mappings between infinite-dimensional function spaces, enabling adaptation to data across different resolutions. Whereas ROM provides rigorous numerical error estimates, neural operator learning largely focuses on discretization convergence and invariance without quantifying the error between the infinite-dimensional and the discretized operators. This work introduces the reduced-order neural operator modeling (RONOM) framework, which bridges concepts from ROM and operator learning. We establish a discretization error bound analogous to those in ROM, and get insights into RONOM's discretization convergence and discretization robustness. Moreover, three numerical examples are presented that compare RONOM to existing neural operators for solving partial differential equations. The results demonstrate that RONOM using standard vector-to-vector neural networks can achieve comparable performance in input generalization and achieves superior performance in both spatial super-resolution and discretization robustness, while also offering novel insights into temporal super-resolution scenarios and ROM-based approaches for learning on time-dependent data.

We calculate the large-$N$ expansion of the sphere free energy $F=-\log Z_{S^d}$ of the O(N) $\phi^4$ and the Gross-Neveu $(\bar{\psi} \psi)^2$ CFTs to order $1/N$. Analytic regularization of these theories requires consistently shifting the UV scaling dimension of the auxiliary field: this can only be done by modifying its kinetic term. This modification combines with the counterterms to give the result that matches the $\epsilon$-expansion, resolving a puzzle raised by Tarnopolsky in arXiv:1609.09113. These $F$s can be written compactly in terms of the anomalous dimensions, for both the short-range and the long-range versions of these CFTs. We also provide various technical results including a computation of the counterterms on the sphere and a neat derivation of the sphere free energy of a free conformal field. Finally, we observe that the long-range CFT becomes the short-range CFT at exactly the point where its $\tilde{F} =-\sin \tfrac{\pi d}{2} F$ is maximized as a function of the vector's scaling dimension.

We propose the Soft Graph Transformer (SGT), a soft-input-soft-output neural architecture designed for MIMO detection. While Maximum Likelihood (ML) detection achieves optimal accuracy, its exponential complexity makes it infeasible in large systems, and conventional message-passing algorithms rely on asymptotic assumptions that often fail in finite dimensions. Recent Transformer-based detectors show strong performance but typically overlook the MIMO factor graph structure and cannot exploit prior soft information. SGT addresses these limitations by combining self-attention, which encodes contextual dependencies within symbol and constraint subgraphs, with graph-aware cross-attention, which performs structured message passing across subgraphs. Its soft-input interface allows the integration of auxiliary priors, producing effective soft outputs while maintaining computational efficiency. Experiments demonstrate that SGT achieves near-ML performance and offers a flexible and interpretable framework for receiver systems that leverage soft priors.

We apply Lie symmetry analysis of partial differential equations (PDEs) to the Euler-Lagrange equations of the two-Higgs-doublet model (2HDM), to determine its scalar Lie point symmetries. A Lie point symmetry is a structure-preserving transformation of the spacetime variables and the fields of the model, which is also continuous and connected to the identity. Symmetries of PDEs may, in general, be divided into strict variational, divergence and non-variational symmetries, where the first two are collectively referred to as variational symmetries. Variational symmetries are usually preserved under quantization, and variational Lie symmetries yield conservation laws. We demonstrate that there are no scalar Lie point divergence symmetries or non-variational Lie point symmetries in the 2HDM, and re-derive its well-known strict variational Lie point symmetries, thus confirming the consistency of our implementation of Lie's method. Moreover, we prove three general results that may simplify Lie symmetry calculations for a wide class of particle physics models. Lie symmetry analysis of PDEs is broadly applicable for determining Lie symmetries. As demonstrated in this work, the method can be applied to models with many variables, parameters, and reparametrization freedom, while any missing discrete symmetries can be identified through the automorphism groups of the resulting Lie symmetry algebras.

We propose a kernel-based nonparametric framework for mean-variance optimization that enables inference on economically motivated shape constraints in finance, including positivity, monotonicity, and convexity. Many central hypotheses in financial econometrics are naturally expressed as shape relations on latent functions (e.g., term premia, CAPM relations, and the pricing kernel), yet enforcing such constraints during estimation can mask economically meaningful violations; our approach therefore separates learning from validation by first estimating an unconstrained solution and then testing shape properties. We establish statistical properties of the regularized sample estimator and derive rigorous guarantees, including asymptotic consistency, a functional central limit theorem, and a finite-sample deviation bound achieving the Monte Carlo rate up to a regularization term. Building on these results, we construct a joint Wald-type statistic to test shape constraints on finite grids. An efficient algorithm based on a pivoted Cholesky factorization yields scalability to large datasets. Numerical studies, including an options-based asset-pricing application, illustrate the usefulness of the proposed method for evaluating monotonicity and convexity restrictions.

Most of the work on checking spherical symmetry assumptions on the distribution of the $p$-dimensional random vector $Y$ has its focus on statistical tests for the null hypothesis of exact spherical symmetry. In this paper, we take a different point of view and propose a measure for the deviation from spherical symmetry, which is based on the minimum distance between the distribution of the vector $\big (\|Y\|, Y/ \|Y\| )^\top $ and its best approximation by a distribution of a vector $\big (\|Y_s\|, Y_s/ \|Y_s \| )^\top $ corresponding to a random vector $Y_s$ with a spherical distribution. We develop estimators for the minimum distance with corresponding statistical guarantees (provided by asymptotic theory) and demonstrate the applicability of our approach by means of a simulation study and a real data example.

We propose a practical hybrid decoding scheme for the parity-encoding architecture. This architecture was first introduced by N. Sourlas as a computational technique for tackling hard optimization problems, especially those modeled by spin systems such as the Ising model and spin glasses, and reinvented by W. Lechner, P. Hauke, and P. Zoller to develop quantum annealing devices. We study the specific model, called the SLHZ model, aiming to achieve a near-term quantum annealing device implemented solely through geometrically local spin interactions. Taking account of the close connection between the SLHZ model and a classical low-density-parity-check code, two approaches can be chosen for the decoding: (1) finding the ground state of a spin Hamiltonian derived from the SLHZ model, which can be achieved via stochastic decoders such as a quantum annealer or a classical Monte Carlo sampler; (2) using deterministic decoding techniques for the classical LDPC code, such as belief propagation and bit-flip decoder. The proposed hybrid approach combines the two approaches by applying bit-flip decoding to the readout of the stochastic decoder based on the SLHZ model. We present simulations demonstrating that this approach can reveal the latent potential of the SLHZ model, realizing soft-annealing concept proposed by Sourlas.

Population heterogeneity is a key factor in epidemic dynamics, influencing both transmission and final epidemic size. While heterogeneity is often modelled through age structure, spatial location, or contact patterns, differences in host susceptibility have recently gained attention, particularly during the COVID-19 pandemic. Building on the framework of Diekmann and Inaba (Journal of Mathematical Biology, 2023), we focus on the special case of SEIR epidemic models, assuming that at the epidemic start there is no pre-existing immunity. Under two distinct assumptions linking susceptibility and infectiousness, one obtains a closed system of 3 ODEs, which can be easily simulated and for which some analytical results are obtained. In particular, we proved that heterogeneity in susceptibility reduces the epidemic final size compared to homogeneous models with the same basic reproduction number $R_0$. We specialised in the case where susceptibility is distributed according to a gamma or extended Beta distribution, showing how the epidemic final size depends on the variance of the distribution. In the case of a gamma-distributed susceptibility, the resulting model consists of a system of ODEs with just one parameter more than the classical SEIR model; this makes it practical for fitting epidemic data. We illustrate its use by fitting data on seasonal influenza in Italy, and comparing the results to those obtained with simple SEIR models with pre-existing immunity.

This paper presents a minimal operator-only term rewriting system with seven constructors and eight reduction rules. The main contribution is a mechanically-verified proof of strong normalization for a guarded fragment using a novel triple-lexicographic measure combining a phase bit, multiset ordering (Dershowitz-Manna), and ordinal ranking. From strong normalization, a certified normalizer with proven totality and soundness is derived. Assuming local confluence (verified through critical pair analysis), Newman's Lemma yields confluence and therefore unique normal forms for the safe fragment. Impossibility results showing that simpler measures, such as additive counters, polynomial interpretations, and single-bit flags, provably fail for rules with term duplication, are established. The work demonstrates fundamental limitations in termination proving for self-referential systems. A conjecture is stated: no relational operator-only TRS can have its full-system termination proved by internally definable methods. Here "relational" is equivalent to "capable of ordered computation": systems with a recursor enabling iteration over successors, comparison, or sequential counting. Such recursors necessarily redistribute step arguments across recursive calls, defeating all additive termination measures. This structural limitation applies to any TRS expressive enough to encode ordered data. All theorems have been formally verified in a proof assistant. The Lean formalization is available at this https URL.

Typical stabilizer codes aim to solve the general problem of fault-tolerance without regard for the structure of a specific system. By incorporating a broader representation-theoretic perspective, we provide a generalized framework that allows the code designer to take this structure into account. For any representation of a finite group, we produce a quantum code with a code space invariant under the group action, providing passive error mitigation against errors belonging to the image of the representation. Furthermore, errors outside this scope are detected and diagnosed by performing a projective measurement onto the isotypic components corresponding to irreducible representations of the chosen group, effectively generalizing syndrome extraction to symmetry-resolved quantum measurements. We show that all stabilizer codes are a special case of this construction, including qudit stabilizer codes, and show that there is a natural one logical qubit code associated to the dihedral group. Thus we provide a unifying framework for existing codes while simultaneously facilitating symmetry-aware codes tailored to specific systems.

We construct automata with input(s) in base $k$ recognizing some basic relations and study their number of states. We also consider some basic operations on $k$-automatic sequences and discuss their state complexity. We find a relationship between subword complexity of the interior sequence $(h'(i))_{i \geq 0}$ and state complexity of the linear subsequence $(h(ni+c))_{i \geq 0}$. We resolve a recent question of Zantema and Bosma about linear subsequences of $k$-automatic sequences with input in most-significant-digit-first format. We also discuss the state complexity and runtime complexity of using a reasonable interpretation of Büchi arithmetic to actually construct some of the studied automata recognizing relations or carrying out operations on automatic sequences.

Recent advances in quantum simulators allow direct experimental access to ensembles of pure states generated by measuring part of an isolated quantum many-body system. These projected ensembles encode fine-grained information beyond thermal expectation values and provide a new window into quantum thermalization. In chaotic dynamics, projected ensembles exhibit universal statistics governed by maximum-entropy principles, known as deep thermalization. At infinite temperature this universality is characterized by Haar-random ensembles. More generally, physical constraints such as finite temperature or conservation laws lead to Scrooge ensembles, which are maximally entropic distributions of pure states consistent with these constraints. Here we introduce Scrooge $k$-designs, which approximate Scrooge ensembles, and use this framework to sharpen the conditions under which Scrooge-like behavior emerges. We first show that global Scrooge designs arise from long-time chaotic unitary dynamics alone, without measurements. Second, we show that measuring a complementary subsystem of a scrambled global state drawn from a global Scrooge $2k$-design induces a local Scrooge $k$-design. Third, we show that a local Scrooge $k$-design arises from an arbitrary entangled state when the complementary system is measured in a scrambled basis induced by a unitary drawn from a Haar $2k$-design. These results show that the resources required to generate approximate Scrooge ensembles scale only with the desired degree of approximation, enabling efficient implementations. Complementing our analytical results, numerical simulations identify coherence, entanglement, non-stabilizerness, and information scrambling as essential ingredients for the emergence of Scrooge-like behavior. Together, our findings advance theoretical explanations for maximally entropic, information-stingy randomness in quantum many-body systems.

Search processes are a fundamental part of natural and artificial systems. In such settings, the number of searchers is rarely constant: new agents may be recruited while others can abandon the search. Despite the ubiquity of these dynamics, their combined influence on search efficiency remains unexplored. Here we present a general framework for stochastic search in which independent agents progressively join and leave the process, a mechanism we term dynamic redundancy and mortality (DRM). Under minimal assumptions on the underlying search dynamics, this framework yields exact first-passage time statistics. It further reveals surprising connections to stochastic resetting, including a regime in which the resetting mean first-passage time emerges as a universal lower bound for DRM, as well as regimes in which DRM search is faster. We illustrate our results through a detailed analysis of one-dimensional Brownian DRM search. Altogether, this work provides a rigorous foundation for studying first-passage processes with a fluctuating number of searchers, with direct relevance across physical, biological, and algorithmic systems.

In this paper, we introduce monoidal rewriting systems (MRS), an abstraction of string rewriting in which reductions are defined over an arbitrary ambient monoid rather than a free monoid of words. This shift is partly motivated by logic: the class of free monoids is not first-order axiomatizable, so "working in the free setting" cannot be treated internally when applying first-order methods to rewriting presentations. To analyze these systems categorically, we define $\mathbf{NCRS_2}$ as the 2-category of Noetherian Confluent MRS. We then prove the existence of a canonical biadjunction between $\mathbf{NCRS_2}$ and $\mathbf{Mon}$. Finally, we classify all Noetherian Confluent MRS that present a given fixed monoid. For this, we introduce Generalized Elementary Tietze Transformations (GETTs) and prove that any two presentations of a monoid are connected by a (possibly infinite) sequence of these transformations, yielding a complete characterization of generating systems up to GETT-equivalence.

We study nonasymptotic (finite-sample) confidence intervals for treatment effects in randomized experiments. In the existing literature, the effective sample sizes of nonasymptotic confidence intervals tend to be looser than the corresponding central-limit-theorem-based confidence intervals by a factor depending on the square root of the propensity score. We show that this performance gap can be closed, designing nonasymptotic confidence intervals that have the same effective sample size as their asymptotic counterparts. Our approach involves systematic exploitation of negative dependence or variance adaptivity (or both). We also show that the nonasymptotic rates that we achieve are unimprovable in an information-theoretic sense.

We derive exact general solutions (as opposed to attractor particular solutions) for the evolution of a scalar field $\phi$ in a universe dominated by a background fluid with equation of state parameter $w_B = -1$, extending earlier work on exact solutions with $w_B > -1$. Straightfoward exact solutions exist when the evolution is described by a linear differential equation, corresponding to constant, linear, and quadratic potentials. In the nonlinear case, exact solutions are derived for $V = V_0\ln \phi$, $V = V_0 \phi^{1/2}$ and $V = V_0/\phi$, and the logarithmic potential also yields an exact first integral. These complicated parametric solutions are considerably less useful than those derived previously for a universe dominated by a barotropic fluid such as matter or radiation with $w_B > -1$. However, we generalize the slow-roll approximation and show that it applies to all sufficiently flat potentials in the case of a vacuum-dominated expansion, while it never applies when the universe is dominated by a background fluid with $w_B > -1$.