Papers

Let $\mathbb D$ be a division ring, and let $\mathcal{R}\subseteq M_n(\mathbb{D})$ be a structural matrix ring over $\mathbb{D}$, that is, the subring of $M_n(\mathbb{D})$ supported on the ordered pairs of a preorder on $\{1,\ldots,n\}$. We study injective Jordan semi-triple maps $\phi:\mathcal{R}\to M_n(\mathbb{D})$, namely injective maps satisfying \[ \phi(XYX)=\phi(X)\phi(Y)\phi(X), \qquad \text{for all } X,Y\in\mathcal{R}. \] Assuming that the centre of $\mathbb{D}$ has more than two elements, we give a criterion for automatic additivity and show that there are exactly two obstructions. The first one is scalar: it occurs precisely when $\mathcal{R}$ has a direct ring summand isomorphic to $\mathbb{D}$ and $\mathbb{D}$ is isomorphic to neither $\mathbb{F}_3$ nor $\mathbb{F}_4$. The second one is order-theoretic: it occurs when a nonsymmetric comparable pair $i\preceq j$, $j\not\preceq i$, admits no third index $k\notin\{i,j\}$ comparable with both $i$ and $j$. If neither obstruction occurs, all injective Jordan semi-triple maps are additive. The centre-size hypothesis is sharp: for $n\ge3$, the upper-triangular ring $T_n(\mathbb{F}_2)$ has neither obstruction but nevertheless admits nonadditive injective Jordan semi-triple maps. Finally, in the additive case, we describe the maps componentwise, in terms of endomorphisms, anti-endomorphisms, and transitive multipliers.

In this paper, we present a complete characterization of solvable symplectic Lie algebras via a symplectic double extension process. We demonstrate that any such algebra is either symplectically irreducible or can be constructed through a finite sequence of symplectic double extensions by a line or a plane, starting from symplectically irreducible Lie algebras. Furthermore, we show that if a symplectic Lie algebra has a nondegenerate derived ideal, then it is necessarily unimodular and, in particular, solvable. Finally, we present a novel algebraic proof of a classical structural theorem on symplectically irreducible symplectic Lie algebras and classify all Lie algebras of dimension up to $6$ that admit such structures.

We employ the recently developed Kac-Rice formalism for non-Hermitian random matrices to derive the joint distribution of an eigenvalue and its associated normalised right eigenvector in a Gaussian ensemble that interpolates between complex Ginibre (Class A) and complex symmetric matrices (Class AI$^\dagger$). This distribution is valid at finite matrix size, $N$, for any value of the interpolation parameter $\sigma \in [0,1]$, with $0$ and $1$ corresponding to classes A and AI$^\dagger$ respectively. The marginal distribution for the density of the eigenvalues is derived at finite $N$ and then considered asymptotically as $N \to \infty$. When considering bulk eigenvalues, we recover the standard circular law for all $\sigma$. Furthermore, for edge eigenvalues we find that for fixed $\sigma$, the eigenvalues follow the edge density associated with matrices in Class A. However, a transitional regime is discovered for the interpolation parameter being scaled as $\sigma = 1 - \kappa N^{-1/2}$, where new edge behaviour is observed for the density of eigenvalues - smoothly interpolating two previously known results. This transitional regime and the associated density of eigenvalues is conjectured to be universal for non-Gaussian matrices and we provide numerical evidence in support of this.

A primary method for constructing orthogonal polynomials on the unit ball consists of combining a Jacobi-type radial component with a spherical harmonic angular part. Building upon this framework and using Jacobi-Piñeiro multiple orthogonal polynomials, this paper introduces Type I and Type II multiple orthogonal polynomials on the multidimensional ball. To demonstrate the practical utility of these definitions, we establish multivariate extensions of several fundamental results from univariate multiple orthogonality. Finally, we extend the construction to more general domains by introducing multiple orthogonality with respect to radial weights.

The paper consists of a brief survey of the use of Stein's method in network analysis. Topics covered include normal and Poisson approximation of graph–based statistics, approximating an exponential random graph by a Bernoulli model, and comparison of different random geometric graph models.

Building on the recently discovered origami-amplituhedron correspondence, we prove that the BCFW (Britto-Cachazo-Feng-Witten) cells triangulate the $m=4$ amplituhedron in full generality at all loop orders, both in momentum and momentum-twistor space. Along the way, we develop two natural "$L$-punctured" extensions of the positive Grassmannian and relate them via T-duality.

A long-standing question is whether chromatic symmetric functions can distinguish non-isomorphic trees. Campbell introduced chromatic noncommutative symmetric functions for digraphs, which lift chromatic symmetric functions to NSym, and asked to what extent they can distinguish non-isomorphic oriented trees. In this article, we prove that chromatic noncommutative symmetric functions can reconstruct oriented stars, oriented double stars, some oriented caterpillars, and some oriented paths.

In this paper we give estimate of $|S'_{f}(0)|$, where $S_{f}(z)= \left(\frac{f''(z)}{f'(z)}\right)'-\frac{1}{2}\left(\frac{f''(z)}{f'(z)}\right)^{2}$ is the Schwartian derivative, and $f$ belongs to different classes of functions univalent in the open unit disc $\ID$.

Modeling high-dimensional data is challenging, yet essential to understanding many complex systems. Maximum entropy models such as Ising and Potts models have been used extensively to capture pairwise interactions from correlation patterns in data, allowing to infer graphical representations of complex systems from observations (e.g., from protein sequences or neural population activity). Recently, there has been growing interest in modeling higher-order correlation patterns involving simultaneously three or more variables. While progress has been made in binary data with high-order Ising models, we extend this framework to the more general case of discrete data. We introduce q-state spin models, a complete family of maximum entropy models that generalize the vector Potts model to include long-range and arbitrary high-order interactions. In the pairwise case, our models allow for more diverse interaction types compared to the standard vector Potts model. We discuss their statistical interpretation with examples and relate them to discrete Fourier analysis. Using a loop expansion of the partition function, we show that the statistical properties of spin models are fully captured by the algebraic structure of their interactions. We define gauge transformations under which this structure, and thus the partition function, remains invariant. Models equivalent under gauge transformations can be seen as different representations of the same abstract statistical model, despite generally having interactions of different orders, extending results from the binary case. For practical application to data analysis, we focus on a subset of models known in the binary case as Minimally Complex Models, generalizing them to discrete data. We obtain a closed-form expression for the marginal likelihood of these models, enabling fast model selection. We illustrate their use with simple real-world examples.

In this article, we analyze how (projective and injective) complexity, curvature, and complete intersection dimension behave under linkage of modules and ideals. Let $R$ be a Gorenstein local ring. Consider a Gorenstein perfect ideal $\mathfrak{a}$ (e.g., $\mathfrak{a}$ is generated by an $R$-regular sequence). Let $M$ and $N$ be two Cohen-Macaulay $R$-modules linked by $\mathfrak{a}$. We prove that $\mathrm{cx}_R(M)= \mathrm{inj\,cx}_R(N)$ and $\mathrm{curv}_R(M)= \mathrm{inj\,curv}_R(N)$. In particular, when $R$ is complete intersection, $\mathrm{cx}_R(M)= \mathrm{cx}_R(N)$ and $\mathrm{curv}_R(M)= \mathrm{curv}_R(N)$. Furthermore, we show that $\mathrm{pd}_R(M)= \mathrm{pd}_R(N)$ and $\operatorname{CI-dim}_R(M)= \operatorname{CI-dim}_R(N)$. If any of these dimensions is finite, it is equal to $\mathrm{ht}(\mathfrak{a})$. Similar results are obtained for linkage of ideals. All these results highly extend a classical result of Peskine and Szpiro in many directions. We construct several examples that complement our results. These also show how properties like `integrally closed', `$\mathfrak{m}$-full' and `Burch' behave under linkage of ideals.

Bondal's conjecture in Poisson geometry gives lower bounds on the degeneracy loci of Poisson Fano manifolds, where the rank of the Poisson structure drops. By work of several authors, it was previously known to hold for Fano manifolds of dimension at most four. We give the first proof of this conjecture for Fano manifolds of dimension five, and partial results for Fano manifolds of all odd dimensions. The proof uses: (i) an algebraic integrability criterion for codimension-one foliations on weak Fano manifolds, extending a previous result of the first author; (ii) the "modular residues" of Poisson structures introduced by Gualtieri and the third author; and (iii) a cohomological constraint on invariant subvarieties for Pfaff fields, extending earlier results of Esteves–Kleiman to the case in which the Pfaff distribution on the subvariety admits a closed strongly directed positive current.

In this paper, we settle the problem of optimal hypercontractivity for free group von Neumann algebras. Namely, for $n\ge 2$ and the free group $\mathbb{F}_n$ on $n$ generators, we prove that for any $1<p\le q<\infty$, the Poisson semigroup $P_t$ associated with the word-length function satisfies $$ \|P_t:L_p(\widehat{\mathbb{F}_n})\to L_q(\widehat{\mathbb{F}_n})\|\le 1 \qquad\text{ if and only if }\qquad t\ge \frac{1}{2}\log \frac{q-1}{p-1}. $$ The main idea is to apply a refined cubic majorant estimate from a recent work of Frank and Ivanisvili \cite{FrankIvanisvili2026} to the equivalent logarithmic Sobolev inequality, and use the Haagerup-type cancellation estimate \cite{Haagerup1979}. Similar ideas and techniques extend to free products \[ G=\left(*_{\alpha\in A}\mathbb Z\right)*\left(*_{\beta\in B}\mathbb Z_2\right) \] and the free Gaussian von Neumann algebras. In the former setting, partial sharp estimates were previously obtained by Junge–Palazuelos–Parcet–Perrin–Ricard \cite{JungePalazuelosParcetPerrinRicard2015}; in the latter, our approach recovers Biane's free hypercontractivity theorem \cite{Biane1997}.

The 2026 disproof of Erdős's unit-distance conjecture and Sawin's quantitative refinement show that the maximum number $u(n)$ of unit distances among $n$ planar points can exceed $n^{1+\varepsilon}$ for a fixed positive $\varepsilon$. Sawin's explicit bound gives more than $n^{1.014}$ unit distances for arbitrarily large $n$ and exposes integer parameters whose choice is not fully optimized. This report treats Sawin's parameter selection as a nonlinear integer optimization problem and develops an open-source Python optimization and verification pipeline for certificates involving prime sets $T$ and $S_Q$, integer multiplicities $k(p)$, and a rationally encoded real parameter $R$. After reproducing Sawin's certificate with $\delta=0.014114\ldots$, the pipeline yields improved certificates with the same $T$. We develop a tailored integer evolution strategy achieving a certificate with $\delta=0.015263\ldots$ and supporting the cautious statement $u(n)>n^{1.0152}$ for arbitrarily large $n$. For extended ramified prime ranges, the Emmerich–Cordella certificate obtained with the same framework reports $u(n)>n^{1.031}$ for $\#T=67$, illustrating the importance of enlarging $T$. Very recent MathOverflow discussions, brought to the author's attention as of version~4, report further improvements, including certificates above $\delta>0.035$ and beyond $\delta>0.036$. Some of these improvements may rely not only on larger prime ranges but also on modified constraint systems and additional degrees of freedom that deviate from Sawin's original formulation. Beyond this application, the work illustrates how randomized optimization heuristics can improve, verify, and refine explicit certificates for combinatorial geometry through nonlinear integer optimization.

We introduce the formalism of non-wellfounded and cyclic linear nested sequent calculi, developing concrete systems for linear temporal logic (LTL). The paper addresses two central problems, which we call "cycle recognition" and "unraveling." Cycle recognition concerns identifying cycles in non-wellfounded proofs in order to extract corresponding cyclic proofs, while unraveling studies the converse transformation, from cyclic proofs to non-wellfounded ones. Although these processes are well understood for Gentzen sequents, they have received little attention for more expressive sequent formalisms and become more challenging in the linear nested sequent setting. To address cycle recognition, we show the completeness of non-wellfounded proofs relative to a particular normal form exhibiting a property we call "saturation recurrence," which enables the systematic extraction of cyclic proofs. To address unraveling, we introduce a specialized procedure that shifts rule applications forward along linear nested sequents, allowing non-wellfounded proofs to be reconstructed from cyclic ones. Overall, our work provides new proof-theoretic techniques for cycle recognition and unraveling in expressive multisequent formalisms.

We give detailed analysis and circuit design of structure-preserving quantum algorithms for second-order linear evolutionary PDEs, including parabolic equations and hyperbolic equations with mixed Dirichlet, Neumann, and periodic boundary conditions and source terms. While prior quantum algorithms usually neglect the stability problem from the PDE-to-ODE reduction, our method-of-lines approach investigates the boundary lifting via Coons interpolation and boundary-aware discretization, so that the resulting semi-discrete systems are stable and compatible with efficient quantum ODE primitives. For the parabolic problem, we use a diagonal similarity transform to ensure the semi-discrete generator must have a positive semi-definite Hermitian part, and then solve the resulting ODE system by the optimal linear combination of Hamiltonian simulation (LCHS). For the hyperbolic problem, we rewrite the semi-discrete equation as an equivalent first-order system and solve it by Hamiltonian simulation. We implement our quantum algorithms with explicit block-encoding constructions and circuit implementations, as well as demonstrating the end-to-end complexity bounds together with spatial and quadrature error estimates. We conduct classical numerical experiments on the convection-diffusion equation, inhomogeneous heat equation, and Klein-Gordon equation to validate our structure-preserving analysis and algorithmic constructions.

Implication algebras were introduced by Abbott as algebraic models of the operation of Boolean implication in the classical propositional calculus. In this work, we study additional operators and constants on bounded implication algebras by introducing monadic and cylindric implication algebras. It is demonstrated that the category $\mathbf{MIA}$ of monadic implication algebras is isomorphic to the category $\mathbf{MBA}$ of monadic Boolean algebras and moreover, that the category $\mathbf{CIA}$ of $I$-dimensional cylindric implication algebras is isomorphic to the category $\mathbf{CBA}$ of $I$-dimensional cylindric Boolean algebras. As an application of the obtained categorical isomorphisms, we provide spectral duality results for $I$-dimensional cylindric implication algebras along the lines of Bezhanishvili and Holliday's spectral duality for Boolean algebras combined with McDonald's extension of their duality to monadic and $I$-dimensional cylindric Boolean algebras.

Stone demonstrated that the category $\mathbf{DLATT_{0,1}}$ of bounded distributive lattices is dually equivalent to the category $\mathbf{Spec}$ of spectral spaces and Priestley showed that $\mathbf{DLatt_{0,1}}$ is dually equivalent to the category $\mathbf{Priest}$ of Priestley spaces so that $\mathbf{Spec}$ is equivalent $\mathbf{Priest}$. Cornish strengthened this by showing that $\mathbf{Spec}$ and $\mathbf{Priest}$ are in fact isomorphic. In this study, we investigate the duality theory of various lattice expansions of certain bounded distributive lattice-ordered algebras, known as De Morgan algebras. In particular we obtain spectral duality results for the category $\mathbf{S4DM}$ of De Morgan algebras equipped with a closure operator, which we call S4 De Morgan algebras, as well as for the category $\mathbf{DMGrp}$ of De Morgan groupoids. This is achieved by an appropriate adaptation of Bimbó's Priestley-style duality for general De Morgan algebras together with Urquhart's Priestley-style duality for relevance algebras under the isomorphism between $\mathbf{Priest}$ and $\mathbf{Spec}$.

We study a heat transfer equation in spatial dimensions $n = 1$, $2$, and $3$. A group classification with respect to the functional parameter $k = k(T)$ is done and symmetry algebras are presented. Recursion operators are found in the case $n = 1$ and infinite hierarchies of symmetries are constructed. We also find a number of exact solution in all the three cases.

Recently, the mapping of the replicator equation onto Bayes' theorem has been recognised, leading to an analogy between evolutionary dynamics and Bayesian learning. However, this analogy holds only for pure selection in infinite populations and breaks down when mutations – a central mechanism of evolution – are introduced. Here I propose that evolution by natural selection, at least for populations of haploid replicators in static environments, is best understood not as a learning process but as a process of causal inference. Each mutation event constitutes a natural experiment in which the parent serves as the control and the mutant offspring as the treated unit. Natural selection screens the causal effect of the mutation on fitness, retaining mutations with non-negative effects. I formalise this view within the Neyman-Rubin potential-outcomes framework. I first develop the general theory using a generic fitness outcome and show how the core identification assumptions in causal inference (Stable Unit Treatment Value Assumption, Consistency, Unconfoundedness, Positivity) map onto evolutionary biology. Using the unnormalised quasispecies equation, I prove that the intergenerational change in mean fitness decomposes exactly into a selection term – recovering Fisher's Fundamental Theorem – plus a mutation term that corresponds to a fitness-weighted average of the cumulated effect of all mutations over all parental genotypes. I show that this decomposition extends, under suitable assumptions, to the generalised replicator-mutator equation and that the frequencies of populations of matched parents-offspring update in proportion to the average causal effect of mutations on fitness.

We investigate a recursive variant of the classical Frank-Wolfe algorithm for minimizing a convex differentiable function over a convex compact domain. Unlike the traditional setting, we assume that both the problem domain and the objective function are initially unknown and must be learned from data. To address this, we integrate estimators into the optimization process, allowing the algorithm to iteratively refine approximations of the problem domain and the objective function. Our approach maintains the projection-free nature of the classical Frank-Wolfe algorithm while adapting to the uncertainty inherent in data-driven settings. We establish convergence guarantees for the recursive method, showing that the optimization error scales with the accuracy of the learned estimators. Two experiments support our theoretical findings, demonstrating that the proposed method achieves convergence behavior comparable to that of the classical Frank-Wolfe algorithm under exact knowledge of the problem domain and objective function, while offering significant computational savings.

We study integrability of generalized almost complex structures on the six-dimensional sphere $\mathbb{S}^{6}$. Two notions of integrability are considered: integrability with respect to brackets determined by an affine connection $[\cdot ,\cdot ]_{\nabla}$ (in particular the Levi-Civita connection), and the Courant integrability for strong generalized almost complex structures. After recalling the necessary background on the generalized tangent bundle and on spherical combinations of the canonical generalized structures determined by an almost Hermitian triple $(J,g,\omega )$, we derive local coordinate criteria for $[\cdot ,\cdot ]_{\nabla }$-integrability of weak generalized structures. Applying these formulae to the nearly Kähler structure on $\mathbb{S}^{6}$ induced by the octonionic product, we prove that no nontrivial spherical combinations $\mathsf{J}=a\mathsf{J}_{1,J}+b\mathsf{J}_{g}+c\mathfrak{J}_{\omega }$ with smooth coefficients such that $a^{2}+b^{2}+c^{2}=1$ (except $\pm \mathsf{J}_{g}$) are integrable with respect to $[\cdot ,\cdot ]_{\nabla ^{LC}}$. We then turn to Courant integrability: we give local characterizations for Courant integrability of strong generalized almost complex structures, prove a gluing result for local Courant algebroids and $b$-field transforms, and use it to exhibit obstructions to constructing, via certain gluing procedures, smooth strong generalized complex structures on $\mathbb{S}^{6}$.

We construct a natural Legendrian cycle $\mathcal{N}_\mathcal{S}$ associated with any $F_nW^{2,n}$-set $\mathcal{S}$, that is, a closed set locally described as a finite union of graphs of $(C^0\cap W^{2,n})$-regular functions with integer multiplicity. The construction relies on the fact that $\mathcal{S}$ is countably $\mathcal{H}^n$-rectifiable of class $C^2$ and, at $\mathcal{H}^n$-almost every point $p\in\mathcal{S}$, the proximal unit normal bundle at $p$, denoted by $\operatorname{nor}(\mathcal{S},p)$, consists of exactly two antipodal vectors $\{u,-u\}$, even in the presence of overlapping $W^{2,n}$-graphs. As a consequence, we prove Reilly-type variational formulae for the higher-order mean curvature integrals of $\mathcal{S}$, extending the classical results of Reilly to this non-smooth setting.

We identify two constructions from different mathematical traditions. In linear logic and realisability, logical types are generated rather than fixed in advance: one begins with a universe of realisers equipped with execution, uses orthogonality to test their interactions, and takes types to be the biorthogonally closed subsets. In enriched Isbell duality, a quantitative relation induces an adjunction whose fixed points form a category, its nucleus. These constructions proceed by different means; we show that, in the present setting, they produce the same objects. The shared datum is minimal: an associative product, called execution, and a real-valued measurement, with no compatibility assumed between them. The failure of the measurement to be additive is at once the relation defining orthogonality and the quantitative relation whose Isbell nucleus we form, and the types cut out by orthogonality are exactly the fixed points of the associated adjunction. The identification pays off in both directions. The most natural product of types fails to be associative; repairing this failure forces a different notion of type, sensitive to both sides of a composite, on which the induced product is associative and, when execution has units, carries two residuals. What emerges is a noncommutative Lambek calculus, derived directly from execution and orthogonality rather than imposed. In the reverse direction, each such type, read on the categorical side, generates a quantitative relation of its own, and with it a derived adjunction and a further generation of types; these derived types are again types of the original situation, computed by the residuals of the Lambek calculus. We also prove a coherence theorem for the threefold arrangements of this construction and, in the finite-dimensional case, give explicit formulas for the product.

We propose a shifted-symplectic formulation of a classical continuous analogue of the symmetry TFT paradigm. Let $G$ be an algebraic or Lie group acting by topological defects on an $n$-dimensional classical topological sigma model with target an $(n-1)$-shifted symplectic derived stack $(X,\omega)$ via the AKSZ construction. We argue that the corresponding $(n+1)$-dimensional bulk theory should be the AKSZ theory with target the shifted cotangent stack $T^*[n] (\mathrm B G)$, equivalently the $(n+1)$-dimensional BF theory for $G$. We characterize the Dirichlet and Neumann boundary conditions, and more general topological boundaries, in terms of shifted Lagrangians in $T^*[n] (\mathrm B G)$. We realize the gauging of the $G$-symmetry in the original theory as inserting a topological domain wall between the corresponding topological boundaries in the BF bulk, and introduce the notion of Hamiltonian reduction, syplectic reduction, and Lagrangian reduction in the shifted symplectic setting. We also discuss prequantum refinements of continuous SymTFTs. In this refinement, higher gerbes on $\mathrm B G$ encode classical analogues of 't Hooft anomaly data by decorating the shifted cotangent bulk and its Lagrangian boundary conditions. Finally, in dimension three we compare the infinitesimal BF model $\mathrm B(\mathfrak g\ltimes\mathfrak g^\vee)$ with the factorizable double $\mathrm B(\mathfrak g\oplus \mathfrak g)$. The resulting topological boundaries are described by Lagrangian Lie subalgebras, and the factorizable case relates the SymTFT dictionary to $r$-matrices and Belavin–Drinfeld data.

Let $\mathcal{I}$ be a $\sigma$-ideal on the Cantor space $2^{\omega}$. A set $X\subseteq 2^\omega$ is called everywhere $\mathcal{I}$ if for any $S\in [\omega]^\omega$ the set $X\!\restriction\! S\in\mathcal{I}(2^S).$ We will discuss relations between such families in the class of perfect and Borel sets.

Diffusion-based generative models have achieved remarkable success in high-dimensional data generation; however, they fundamentally rely on isotropic diffusion processes that destroy meaningful geometric structures in the forward process. For complex, multimodal, and highly correlated distributions such as biologically constrained genetic data, isotropic noise merges distinct modes and distorts intrinsic dependencies. This forces the reverse process to recover structure from heavily degraded signals, leading to slow convergence, mode averaging, and biologically implausible samples. To address this, we introduce the Geometry-aware Ornstein-Uhlenbeck (GOU) process, a structured drift design that embeds data geometry into forward and backward dynamics. By employing a variance-aware anisotropic drift, GOU contracts low-variance directions rapidly while preserving high-variance directions longer, maintaining key multimodal structures as stable channels over time. Crucially, we show that GOU's backward initialization error is governed by local rather than global variance. This geometry-adaptive initialization improves convergence rates by reducing initial mismatch and preserving cluster-level structures. Synthetic and real-world genetic experiments demonstrate that GOU significantly improves mode separation, correlation preservation, and statistical validity over standard isotropic models.

We compute an upper bound on the ground state energy of a dilute two-dimensional Fermi gas with repulsive short-range interactions. Our bound can be viewed as the two-dimensional analogue of a formula derived by Huang and Yang in the three-dimensional case. It captures the first three terms in an asymptotic expansion for small $\varrho a^2$, where $\varrho$ denotes the density and $a$ the scattering length of the interaction potential.

We present a framework to generalize the standard b-ary Gray code to get the k-bonacci ones obtained in [5] as well as many others by using theoretical tools that allow to make calculations on lists. We introduce the notion of Z-Gray product, from which we deduce sequences of lists of finite words avoiding a predefinite list Z of factors and which satisfy a power-associativity property as well a generalizations of the classical flipping digit property.

We prove an upper bound for the number of subgroups of an abelian $p$-group, with constants that are sharp on homocyclic blocks. If $A_\lambda$ has type $\lambda$, the main exponent is $\eta(\lambda)=\sum_i\lfloor(\lambda'_i)^2/4\rfloor$, where the parts of $\lambda'$ are the column heights of the Ferrers diagram of $\lambda$. The leading coefficient is the number of central choices in the blocks of columns of odd height. The proof starts from Delsarte's formula, separates the blocks by column height, and estimates each homocyclic block by successive Durfee squares. We also give three applications: a comparison theorem for $p$-groups of nilpotency class less than $p$, diagonal summatory estimates with the shape fixed, and formulae for leading terms at fixed rank for finite abelian $p$-groups.

This paper proposes a stochastic control framework for the operation of electric-vehicle (EV) charging stations equipped with on-site photovoltaic (PV) generation and battery storage. To preserve scalability for large fleets, vehicles are aggregated into a finite number of classes according to their residual charging demand, yielding a compact state description and avoiding vehicle-level optimization. The charging-station dynamics are modeled in discrete time and capture stochastic arrivals, charging-induced class transitions, stochastic departures, PV generation, battery operation, and power exchange with the grid. Based on the corresponding expectation model, we formulate a finite-horizon smart-charging problem that jointly optimizes class-wise charging actions and energy-management variables to balance electricity-purchase cost and transitions toward lower residual-demand classes. The resulting stage problem is a linear program, solved in shrinking-horizon form and implemented online after integer discretization of the first charging action. We also derive a robust counterpart that preserves feasibility under interval uncertainty on the first moments of arrivals, departures, and PV generation. To validate scalability and robustness, we conduct an extensive simulation campaign across nine configurations, combining three real-world electricity price patterns with three EV-arrival profiles. Numerical results show that, at comparable but slightly lower service levels, the proposed controller reduces the cost per kWh by up to 17.5% relative to a service-greedy First-In-First-Out baseline; the total daily cost decreases even more substantially because the controller avoids economically unattractive charging. Overall, the results demonstrate a tunable Pareto trade-off between economic efficiency and charging-service quality for large-scale charging hubs.