Batch digester blowdown operations exhibit highly nonlinear hydraulic transport dynamics due to evolving slurry consistency and rheological uncertainty. This work presents a nonlinear dynamic model and a robust Sliding Mode Control (SMC) strategy for discharge-flow regulation in industrial batch digester systems. The proposed framework incorporates non-Newtonian slurry rheology, consistency-dependent hydraulic resistance, drainability effects, channeling phenomena, and nonlinear hydraulic transport behavior.

We introduce a geometric framework for constructing superintegrable systems from Poisson centralizers (commutants) in the Lie-Poisson algebra $S(\mathfrak{g})$ of a complex semisimple Lie algebra. Starting from a chain of reductive subgroups, we study the corresponding invariant Poisson subalgebras and their Poisson centers, and formulate superintegrability in terms of a \emph{Poisson projection chain} of affine Poisson varieties. For a maximal torus $T\subset G$, we prove that the inclusions $S(\mathfrak{g})^G\subset S(\mathfrak{g})^T\subset S(\mathfrak{g})$ determine a superintegrable chain and identify the associated quotient maps $\mathfrak{g}\xrightarrow{\chi_T}\mathfrak{g}//T\xrightarrow{\rho}\mathfrak{g}//G$. The rank (transcendence degree) computations yield the expected dimension split between commuting Hamiltonians and first integrals, and we describe the corresponding symplectic leaves in the intermediate space. Several examples illustrate how the centralizer generators organize into explicit superintegrable Poisson chains.

We study the weakly nonlinear saturation of the flutter instability of a planar Cosserat rod in a viscous fluid driven by a terminal follower force. This instability, established in our preceding work as a Hopf bifurcation of a non-self-adjoint linear operator, produces stable limit-cycle oscillations in the fully nonlinear overdamped dynamics. Here we derive an analytical description of the emergence of this limit cycle near threshold. Working close to the critical follower force, we perform a multiple-scale expansion about the compressed straight base state and systematically remove secular growth at higher orders. Solvability at cubic order, enforced using the adjoint eigenmode of the non-Hermitian operator, yields a Stuart-Landau amplitude equation for the critical oscillatory mode. The Landau coefficients are expressed as explicit inner products involving the critical eigenmode, its adjoint, and quadratic corrections. The resulting reduced theory predicts a supercritical Hopf bifurcation with a steady-state tip oscillation amplitude scaling as the square root of the distance from threshold. These predictions rationalize the near-threshold scaling observed in nonlinear simulations and provide an analytical normal form for the onset of self-sustained beating in pressure-driven soft robotic arms at low Reynolds number.

Conservation laws of a class of time-dependent damped nonlinear multidimensional wave equations are derived by Noether's theorem. For arbitrary nonzero damping coefficient and nonlinear interaction term, its infinitesimal variational symmetries span a Euclidean algebra $\euclid(n)$ of space translations and rotations. They produce conservation of linear and angular momentums. For some specific forms of these two terms symmetry algebra is enlarged to a subalgebra of the conformal algebra $\conf(1,n)$ and in this case more interesting conservation laws are found.

In this paper, we discuss the spherically symmetric gravitational collapse of matter fields in the de Sitter universe. The energy-momentum tensor of the matter field is assumed to admit a wide variety including dust, perfect fluids with equations of state, fluids with tangential and radial pressure, and with bulk and shear viscosity. Under different initial conditions imposed on the velocity and the density profiles, and by combining the results from exact analytical methods with those obtained from numerical techniques, we track the formation and evolution of spherical marginally trapped spheres as the matter suffers continual gravitational collapse. We show that the quasilocal formalism of trapped surfaces provides an ideal framework to study the evolution of horizons. More precisely, black hole and cosmological horizons may be viewed as the time development of marginally trapped surfaces.

For a local quantum field theory in anti-de Sitter space with conformal boundary conditions but without dynamical gravity, the boundary theory is generically a non-local conformal field theory. Such theories can support conformal defects, but the standard local-CFT arguments based on a boundary stress tensor and conserved currents do not apply. We argue that, under general assumptions, displacement and tilt operators nevertheless exist and have protected quantum numbers. The mechanism is a Goldstone-type phenomenon in AdS: defect-induced symmetry breaking on the boundary is spontaneous from the viewpoint of the local bulk theory, whose Ward identities enforce the corresponding protected defect operators. We illustrate the mechanism in weakly coupled defect RG flows, long-range Landau--Ginzburg models, 4D Maxwell theory, and Yang--Mills theory in AdS.

Understanding the mutual influence between social behavior and physical health is crucial for designing effective epidemic mitigation strategies. Individual interactions drive the evolution of opinions, which in turn shape how infectious diseases are perceived and consequently how they spread within a population, for instance through the adoption or rejection of preventive measures. At the same time, the distribution and dynamics of physical contacts play a fundamental role in determining transmission patterns. To this end, we develop a mathematical framework to analyze the coupled dynamics of opinion formation, disease transmission, and physical contacts by employing graphon-based networks, which capture heterogeneous and large-scale connectivity patterns typical of realistic social structures. The epidemic compartmental model further incorporates a kinetic description of microscopic level physical contacts, allowing for a consistent multiscale representation of interaction patterns. Starting from a microscopic description governed by interpersonal compromise and intrinsic self-thinking processes, we derive a kinetic compartmental epidemic model on graphons via a mean-field limit. This formulation allows us to investigate the joint evolution of the disease state and the opinion distribution, with a particular focus on the role of social networks and physical contacts. Numerical experiments demonstrate that the graphon-kinetic approach provides a comprehensive representation of the coupled opinion-epidemic dynamics, revealing new possibilities for controlling disease spread by shaping population opinion patterns.

The robustness of XRD methods for the determination of the lattice parameters of crystals is well established. These methods have been extended to helical atomic structures using twisted x-rays \cite{friesecke_twisted_2016}. Building on an integral form used in \cite{friesecke_twisted_2016}, we construct integral representations of a broad class of time-harmonic solutions to Maxwell's equations in a vacuum or, more generally, in a homogeneous medium without source terms. The representation includes assignable generalized functions (distributions) that can be tailored to specific boundary or far-field conditions. When the assignable functions satisfy mild periodicity and smoothness conditions, the solutions can be approximated using multi-dimensional trapezoidal rules with exponentially fast convergence. This approximation can be physically interpreted as utilizing finite sources of plane waves to approximate the broad class of time-harmonic solutions to Maxwell's equations. Using these solutions, we show that radiation from suitably placed and oriented sources can serve as incoming radiation for structures with icosahedral symmetry to achieve constructive interference after interacting with the icosahedral structure. The finite source approximations are sufficiently general to satisfy the general Dirichlet conditions at an arbitrarily large number of assigned locations in a source-free domain. The integral representation also extends to a broad class of physical phenomena governed by Helmholtz-type equations. Examples include the scalar wave equation for acoustic waves and elastic wave propagation in linear isotropic solids, which involve both scalar and vector wave equations.

Using the multi-index approach to regularity structures due to F. Otto et al., we construct a regularity structure and a model for it associated to the stochastic Langevin equation for the 3D Euclidean Yang-Mills functional. For the model we also obtain global stochastic and global pointwise weighted Besov type estimates which hold almost surely. The model is defined as a limit of a sequence of smooth models introduced with the help of a mollified noise. When the mollification is removed the sequence converges in a certain topology defined with the help of the stochastic estimates. To obtain these results we develop the multi-index approach for systems of equations with vector-valued white noises. This project is motivated by the problem for constructing 3D Euclidean Yang-Mills measure and by the earlier results of the author on the related problem of canonical quantization of the Yang-Mills field on the Minkowski space.

These notes have been written for a series of lectures to be given at the 44th Finnish Summer School on Probability and Statistics in Lammi, Finland, from 25th to 29th May, 2026. They contain an introduction to Wiener chaos decomposition in finite dimension, a construction of Gaussian fields on the torus, including white noise and the Gaussian free field, and applications to the $\Phi^4$ model. They do not cover other important aspects of the topic, such as stochastic integration, stochastic PDEs and Malliavin calculus.

We study scattering for continuous-time quantum walks on finite graphs with two attached leads. We derive explicit formulae for the two-terminal scattering matrix in terms of characteristic polynomials of the finite graph and its vertex-deleted subgraphs. For real-weighted two-terminal graphs, we then introduce three real quantities, $\mu_1$, $\mu_2$, and $\nu$, which are each additive under parallel composition of graphs. In these variables, perfect transmission at fixed momentum is characterized by the condition $\mu_1=\mu_2$ together with a hyperbola in the corresponding $(\mu,\nu)$-plane, whose points determine the transmission phase. This turns the search for graphs with prescribed transmission properties into a geometric vector-sum problem for smaller building blocks.

We show that Glauber dynamics for $ p$-spin glass mixes exponentially slowly at inverse temperatures larger than a constant times $ \ln (p)/p $ for large enough $ p $. This is done by analyzing the energy landscape using Gaussian decompositions and establishing a bottleneck bound.

We study a generic quantum Markovian master equation for a linearly displaced or driven harmonic oscillator. It was known that the displacement dynamics of Gaussian mixed states depends on the unitary part of the Liouvillian, the decay rate of the system but not on the bath temperature. Here we further show that the fast-rotating modes do not affect the system's displacement dynamics under linear driving forces. Analytical solutions of the quantum master equation are obtained for displaced Gaussian mixed states. Because the non-driven and driven Liouvillians are related by a unitary displacement operator, they are expected to share the same exceptional points structure. At the exceptional points, the displacement of critically damped oscillator displays a characteristics polynomial-in-time prefactor multiplied by an exponential decay. We discuss how external time-dependent forces affect the displacement dynamics using impulsive force and harmonic force as examples. The results obtained for constant driving remain valid in the presence of time-dependent driving.

Motivated by the central role of the Zel'dovich approximation in the description of cosmic structure formation through gravitational collapse, we investigate Burgers-type dynamics in a spherically symmetric gravitational field. In the Newtonian setting, we derive perturbatively the catastrophe time for radial motion by imposing the loss of invertibility of the Lagrangian map. We show that the perturbative expansion is controlled by the dimensionless parameter $ \alpha=\mu/{r_0^3 v_0(r_0)'^2}, $ rather than by the local gravitational acceleration alone. Hence, the expansion remain valid even when gravity is strong. We then extend the analysis to radial geodesic motion in Schwarzschild spacetime.

Independent and identically distributed (i.i.d.) states are ubiquitous in quantum information theory. However, in a practical setting, the i.i.d. assumption is too stringent, and possibly not realistic. A physically more compelling class of 'almost i.i.d.' sources was recently proposed by [Mazzola/Sutter/Renner, arXiv:2603.15792]. In this paper, we introduce two alternative definitions of almost i.i.d. states, based on the normalised quantum Wasserstein distance and on the idea of looking at the average $k$-body marginal. We explore some basic properties of these notions and prove a strict hierarchical relation among them, with Mazzola et al.'s notion being the strictest, the one based on $k$-body marginals the loosest, and the one based on the quantum Wasserstein distance in between. Strict separation is established by means of explicit examples.

The performance of quantum resource manipulation protocols, including key examples such as distillation of quantum entanglement, is measured in terms of the rate at which desired target states can be produced from a given noisy state. However, to achieve optimal rates, known protocols require precise tailoring to the quantum state in question, demanding a perfect knowledge of the input and allowing no errors in its preparation. Here we show that distillation of quantum resources in the framework of resource non-generating operations can be performed universally: optimal rates of distillation can be achieved with no knowledge of the input state whatsoever, certifying the robustness of quantum resource distillation. The findings apply in particular to the purification of quantum entanglement under non-entangling maps, where the optimal rates are governed by the regularised relative entropy of entanglement. Our result relies on an extension of the generalised quantum Stein's lemma in quantum hypothesis testing to a composite setting where the null hypothesis is no longer a fixed quantum state, but is rather composed of i.i.d. copies of an unknown state. The solution of this asymptotic problem is made possible through new developments in one-shot quantum information and a refinement of the blurring technique from [Lami, arXiv:2408.06410].

We show by a counting argument that even though translation symmetry admits symmetric short-range entangled (SRE) eigenstates, there are not enough such SRE eigenstates to span the zero momentum sector. This means that the fixed point strong-to-weak spontaneous symmetry breaking state of translation symmetry is long-range entangled: it cannot be written as a mixture of SRE states. This is a subtle form of long-range entanglement in mixed states that cannot be detected by long-range connected correlation functions.

We investigate the extreme first-passage statistics of $N$ non-interacting random walkers on discrete, hierarchical networks. {By distinguishing between transport limited by escape from localized initial states (injection-limited) and transport limited by the extended network (bulk-limited), we identify a class of extreme value statistics that arises in geometries dominated by source traps (e.g., the Comet graph).} In this regime, the distribution of the minimum arrival time does not converge to any of the classical generalized extreme value distributions. Instead, it follows a discrete distribution with a {strict lower time bound} determined by the properties of the hierarchical network. We analytically derive the asymptotic behavior of this class and validate our predictions against Monte Carlo simulations. Crucially, we identify the mechanism of ``entropic collapse" that destroys this scaling in bulk-dominated geometries like the Bethe lattice, where the phase space of delayed paths diverges with distance. This work establishes a geometry-encoding function that acts as a diagnostic tool for ascertaining whether or not a given graph is hierarchical.

At the beginning of the study of the bispectral problem, see [18], the ad-conditions played a crucial role in finding non-classical instances. The connection with the ad-conditions has reappeared in several different incarnations of the bispectral problem. Here we show that, properly adapted versions of these conditions, see [44], can play an important role in areas including, for instance, the study of exceptional orthogonal polynomials. This is also true in the non-commutative case. Even at this more advanced stage of the field one may hope that finding explicit solutions of these ad-conditions will provide an additional route to new examples.

Duality transformations reveal unexpected equivalences between seemingly distinct models. We introduce an out-of-equilibrium generalisation of matrix product operators to implement duality transformations in one-dimensional boundary-driven Markov processes on lattices. In contrast to local dualities associated with generalised symmetries, here the duality operator intertwines two Markov processes via generalised exchange relations and realises the out-of-equilibrium duality globally. We construct these operators exactly for the symmetric simple exclusion process with distinct out-of-equilibrium boundaries. In this case, out-of-equilibrium boundaries are dual to equilibrium boundaries satisfying Liggett's condition, implying that the Gibbs-Boltzmann measure captures out-of-equilibrium physics when leveraging the duality operator. We illustrate this principle through physical applications.

We develop the discrete set of Dyson-Schwinger equations for scalar fields and solve them for some cases. We show that their solutions are Gaussian in the continuum limit as expected from the theorems of Aizenman and of Aizenman and Duminil-Copin for $d\ge 4$. Extension to lower dimensionality fails, as it should, by observing that the triviality theorems used in our proof are not applicable in such cases.

The Vlasov-Poisson system for ions is a kinetic equation for dilute, unmagnetised plasma. It describes the evolution of the ions in a plasma under the assumption that the electrons are thermalized. Consequently, the Poisson coupling for the electrostatic potential contains an additional exponential nonlinearity not present in the electron Vlasov-Poisson system. The system can be formally derived through a mean-field limit from a microscopic system of ions interacting with a thermalized electron distribution. However, it is an open problem to justify this limit rigorously for ions modelled as point charges. Existing results on the derivation of the three-dimensional ionic Vlasov-Poisson system, obtained by the author and Iacobelli [J. Math. Pures Appl. 135 (2020), pp. 199-255], require a truncation of the singularity in the Coulomb interaction at spatial scales of order $N^{-\beta}$ with $\beta<1/15$, which is more restrictive than the available results for the electron Vlasov-Poisson system. In this article, we prove that the Vlasov-Poisson system for ions can be derived from a microscopic system of ions and thermalized electrons with interaction truncated at scale $N^{-\beta}$ with $\beta<1/3$. We develop a generalisation of the probabilistic approach to mean-field limits developed in the works of Boers and Pickl [J. Stat. Phys. 164(1) (2016), pp. 1-16] and Lazarovici and Pickl [Arch. Ration. Mech. Anal. 225(3) (2017), pp. 1201-1231] that is applicable to interaction forces defined through a nonlinear coupling with the particle density. The proof is based on a quantitative uniform law of large numbers for convolutions between empirical measures of independent, identically distributed random variables and locally Lipschitz functions.

A given topological manifold can sometimes be endowed with inequivalent differential structures. Physically this means that what is meant by a differentiable function (smooth) is simply different for observers using inequivalent differential structures. {The 7-sphere, $\bS^7$, was the first topological manifold where the possibility of inequivalent differential structures was discovered \cite{Milnor}.} In this paper, we examine the import of inequivalent differential structures on the physics of fields obeying the Dirac equation on $\bS^7$. { $\bS^7$ is a fibre bundle of the 3-sphere as a fibre on the 4-sphere as a base. We consider the Kaluza-Klein limit of such a fibre bundle which reduces to a SO(4) Yang-Mills gauge theory over $\bS^4$. We find, for certain specific symmetric set of gauge potentials, that the spectrum of the Dirac operator can be computed explicitly, for each choice of the differential structure. Hence identical topological manifolds have different physical laws. We find this the most important conclusion of our analysis.

On a Hamiltonian $G$-manifold $X$, we define the notion of $G$-invariance of coisotropic A-branes $B$. Under neat assumptions, we give a Marsden-Weinstein-Meyer type construction of a coisotropic A-brane $B_{\operatorname{red}}$ on $X // G$ from $B$, recovering the usual construction when $B$ is Lagrangian. For a canonical coisotropic A-brane $B_{\operatorname{cc}}$ on a holomorphic Hamiltonian $G_\mathbb{C}$-manifold $X$, there is a fibration of $(B_{\operatorname{cc}})_{\operatorname{red}}$ over $X // G_\mathbb{C}$. We also show that `intersections of A-branes commute with reduction'. When $X = T^*M$ for $M$ being compact Kähler with a Hamiltonian $G$-action, Guillemin-Sternberg `quantization commutes with reduction' theorem can be interpreted as $\operatorname{Hom}_{X // G}(B_{\operatorname{red}}, (B_{\operatorname{cc}})_{\operatorname{red}}) \cong \operatorname{Hom}_X(B, B_{\operatorname{cc}})^G$ with $B = M$.

We study the dissipation measure arising in the inviscid limit of two-dimensional incompressible fluids. It is proved that the dissipation is Lebesgue in time and, for almost every time, it is absolutely continuous with respect to the defect measure of strong compactness of the solutions. When the initial vorticity is a measure, the dissipation is proved to be absolutely continuous with respect to a ''quadratic'' space-time vorticity measure. This results into the trivial measure if the initial vorticity has singular part of distinguished sign, or a spatially purely atomic measure if wild oscillations in time are ruled out. In fact, the dynamics at the Batchelor-Kraichnan dissipative scale is the only relevant one, in turn offering new criteria for anomalous dissipation. We provide kinematic examples highlighting the strengths and the limitations of our approach. Quantitative rates, dissipation life-span and steady fluids are also investigated.

Tensor neural networks (TNNs) have demonstrated their superiority in solving high-dimensional problems. However, similar to conventional neural networks, TNNs are also influenced by the Frequency Principle, which limits their ability to accurately capture high-frequency features of the solution. In this work, we analyze the training dynamics of TNNs by Fourier analysis and enhance their expressivity for high-dimensional multi-scale problems by incorporating random Fourier features. Leveraging the inherent tensor structure of TNNs, we further propose a novel approach to extract frequency features of high-dimensional functions by performing the Discrete Fourier Transform to one-dimensional component functions. This strategy effectively mitigates the curse of dimensionality. Building on this idea, we propose a frequency-adaptive TNNs algorithm, which significantly improves the ability of TNNs in solving complex multi-scale problems. Extensive numerical experiments are performed to validate the effectiveness and robustness of the proposed frequency-adaptive TNNs algorithm.

The tidal deformation of compact objects, characterised by their Love numbers, provides insights into the internal structure of neutron stars and black holes. While static bosonic tidal Love numbers vanish for black holes in general relativity, it has been recently revealed that static fermionic tidal perturbations can induce non-zero Love numbers for Kerr black holes. In this paper, we investigate the response of the Reissner-Nordström black hole to the fermionic Weyl field. As a result, we find that the corresponding fermionic tidal Love numbers are also non-vanishing for the Reissner-Nordström black holes except for the extremal ones, which highlights the universal distinct behavior of the static fermionic tidal Love numbers compared to the bosonic counterparts.

In this paper I study properties of the generators $\triangle_\gamma$ of non-local Dirichlet forms $\mathcal{E}^\mu_\gamma$ on ultrametric spaces which are the path space of simple stationary Bratteli diagrams. The measures used to define the Dirichlet forms are taken to be the Gibbs measures $\mu_\psi$ associated to Hölder continuous potentials $\psi$ for one-sided shifts. I also define a cohomology $H_{lc}(X_B)$ for $X_B$ which can be seen as dual to the homology of Bowen and Franks. Besides studying spectral properties of $\triangle_\gamma$, I show that for $\gamma$ large enough (with sharp bounds depending on the diagram and the measure theoretic entropy $h_{\mu_\psi}$ of $\mu_\psi$) there is a unique $\mathcal{E}^\mu_\gamma$-minimizing representative of any class $c\in H_{lc}(X_B)$.

In this paper, we establish a connection between the formalism of $\mathcal{R}$-transforms for non-Hermitian random matrices and the framework of spherical integrals, using the replica method. This connection was previously proved in the Hermitian setting and in the case of bi-invariant random matrices. We show that the $\mathcal{R}$-transforms used in the non-Hermitian context in fact originate from a single scalar function of two variables. This provides a new and transparent way to compute $\mathcal{R}$-transforms, which until now had been known only in restricted cases such as bi-invariant, Hermitian, or elliptic ensembles.

We study finite-rank normal deformations of rotationally invariant non-Hermitian random matrices. Extending the classical Baik-Ben Arous-Péché (BBP) framework, we characterize the emergence and fluctuations of outlier eigenvalues in models of the form $\mathbf{A} + \mathbf{T}$, where $\mathbf{A}$ is a large rotationally invariant non-Hermitian random matrix and $\mathbf{T}$ is a finite-rank normal perturbation. We also describe the corresponding eigenvector behavior. Our results provide a unified framework encompassing both Hermitian and non-Hermitian settings, thereby generalizing several known cases.

One of the great miracles of random matrix theory is that, in the $N \to \infty$ limit, many otherwise intractable matrix problems with horrendously complicated finite-$N$ expressions admit remarkably simple and elegant asymptotic solutions. In this paper, we illustrate this phenomenon in the context of spectral boundaries (or spectral edges) for deformed random matrices. Specifically, we consider matrices of the form $\mathbf{A} + \mathbf{B}$, where $\mathbf{A}$ is a deterministic $N\times N$ matrix (not necessarily Hermitian) and $\mathbf{B}$ is a rotationally invariant random matrix. In the large-$N$ limit, we show that the complex eigenvalue distribution of $\mathbf{A} + \mathbf{B}$ satisfies remarkably simple boundary equations that depend on the $\mathcal{R}_1$ and $\mathcal{R}_2$ transforms of $\mathbf{B}$. We illustrate our results on several explicit random matrix ensembles and support them with numerical simulations.

We define the Hasse-Witt invariant of Calabi-Yau varieties in two different ways. The first method is through Cartier operator and the second method is through the theory of Calabi-Yau modular forms developed by the third author. We conjecture that these two definitions are equivalent and provide many examples of Calabi-Yau varieties in support of this conjecture.

Positive and negative flows of the Chen-Lee-Liu model and its various reductions, including Burgers hierarchy, are formulated within the framework of Riemann-Hilbert-Birkhoff decomposition with the constant grade two generator. Two classes of vacua, namely zero vacuum and constant non-zero vacuum can be realized within a centerless Heisenberg algebra. The tau functions for soliton solutions are obtained by a dressing method and vertex operators are constructed for both types of vacua. We are able to select and classify the soliton solutions in terms of the type of vertices involved. A judicious choice of vertices yields in a closed form a particular set of multi soliton solutions for the Burgers hierarchy. We develop and analyze a class of gauge-Bäcklund transformations that generate further multi soliton solutions from those obtained by dressing method by letting them interact with various integrable defects.

We study the interconversion of families of quantum states ("statistical experiments") via positive, trace-preserving (PTP) maps and clarify its mathematical structure in terms of minimal sufficient Jordan algebras, which can be seen to generalize the Koashi-Imoto decomposition to the PTP setting. In particular, we show that Neyman-Pearson tests generate the minimal sufficient Jordan algebra, and hence also the minimal sufficient *-algebra corresponding to the Koashi-Imoto decomposition. As applications, we show that a) equality in the data-processing inequality for the relative entropy or the $\alpha$-$z$ quantum Rényi divergence implies the existence of a recovery map also in the PTP case and b) that two dichotomies can be interconverted by PTP maps if and only if they can be interconverted by decomposable, trace-preserving maps. We thoroughly review the necessary mathematical background on Jordan algebras. As a step beyond the finite-dimensional case, we prove Frenkel's formula for approximately finite-dimensional von Neumann algebras.

Establishing the fusion rules of anyonic quasiparticles in fractional quantum Hall fluids is essential for understanding their underlying topological order. Building on the conjecture that key topological properties are encoded in the "DNA" of candidate many-body wave functions - that is, the pattern of dominant orbital occupations restricted to a finite number of lowest Landau levels - we propose a combinatorial framework that derives these fusion rules directly from microscopic data. By extending Schrieffer's counting argument and introducing classes of topological excitations, our framework provides a unified route to the fusion rules for both Abelian and non-Abelian excitations. This approach elucidates the emergence of topological features from first principles in both fermionic and bosonic systems.