Papers

This work presents a novel neural-network compression approach for polyconvex envelopes of isotropic functions. The approach relies on a classical sufficient criterion for polyconvexity and is particularly suited for the representation of determinant-constrained energy densities arising in non-linear elasticity. Compared with existing compression methods based on the necessary and sufficient characterisation of polyconvex isotropic functions, the proposed framework reduces computational costs, due to the domain reduction through the restriction to the positive octant in the singed singular value space. The underlying neural-network architecture employs input-convex neural networks (ICNNs) with non-negative weight constraints to enforce the required convexity and monotonicity properties. The additional symmetry and inequality conditions characterising the polyconvex envelope are incorporated weakly through the loss function during training. Although the employed criterion is only sufficient and thus generally yields only a lower bound on the polyconvex envelope, numerical experiments based on the classical Saint Venant–Kirchhoff energy demonstrate that the proposed approach produces accurate approximations in practice while offering a computationally more efficient alternative to existing methods.

We study residual finite-dimensionality for ultragraph algebras, both in the algebraic and in the C-star-algebraic settings. We introduce graph-theoretic RFD conditions for ultragraphs, extending the conditions that characterize RFD graph C-star-algebras. Using the boundary ultrapath branching system, we construct finite-dimensional branching-system representations associated to terminal boundary sets and no-exit cycles. These representations are used to prove that, whenever an ultragraph satisfies the graph-theoretic RFD conditions, its ultragraph Leavitt path algebra LK(G) is RFD, for every field K, and its ultragraph C-star-algebra RFD. For ultragraphs satisfying Condition (RFUM2), we prove converses in both settings. The analytic converse uses the groupoid model and the density of periodic points, while the algebraic converse is proved directly by finite-dimensional linear algebra. Thus, for RFUM2 ultragraphs, RFD of LK(G), RFD of C(G), and the graph-theoretic RFD conditions are equivalent. This gives, in particular, a common combinatorial description linking the algebraic and analytic theories, recovers the graph C-start-algebra characterization, and yields an algebraic characterization for Leavitt path algebras of graphs. We also construct an RFD ultragraph algebra which is genuinely outside the graph-algebra class in both settings.

Vineyards, or time-varying families of persistence diagrams, are widely used in topological data analysis (TDA) pipelines to track how topological features change and evolve as a parameter varies. When the parameter traces a closed loop, a vineyard can exhibit monodromy: diagram points permute over the course of a full traversal, which obstructs feature tracking and can complicate downstream analysis of such data. Chambers et al. considered the periodic vineyards that arise from the radial persistence transform, which maps the manifold to a family of persistence diagrams, where each diagram fixes a base point and considers the filtration that is based on Euclidean distance to that point, and showed that monodromy and knotting can occur. Other recent work by Arya et al. considers geometric conditions that exclude monodromy in two dimensions, in an effort to better understand when this effect happens. That said, understanding when and why monodromy occurs is a fundamental open problem with direct practical consequences for many data analysis pipelines. In this work, we study this question for 1-manifolds in $\mathbb{R}^2$, using a surprising connection with tools from singularity theory, and provide a classification for the causes of monodromy in vineyards. More precisely, we prove that the vineyard of a sufficiently small loop $\gamma$ cannot exhibit monodromy unless it contains a specific singularity of the distance function. The central geometric object in our analysis is the symmetry set, which is the locus of centers of spheres tangent in more than one point to the manifold; this object classifies singularities of the distance function, and in our setting, dictates precisely when monodromy occurs. This characterization opens the door to the development of algorithmic criteria for detecting and utilizing (or avoiding) monodromy in TDA pipelines.

Portfolio diversification - a cornerstone of modern investment management - can be formulated as a Maximum Independent Set (MIS) problem on asset correlation graphs. Solving this problem at scale is computationally challenging, motivating the exploration of quantum algorithms for practical financial optimization. We propose an end-to-end pipeline leveraging qReduMIS, a recursive hybrid quantum-classical algorithm. Rather than using quantum optimization to directly produce a final solution, qReduMIS leverages independent set measurements from the Quantum Approximate Optimization Algorithm (QAOA) to identify frozen nodes - vertices likely to belong to optimal solutions - thereby guiding and unblocking subsequent (provably optimal) classical reductions on the remaining graph. We benchmark qReduMIS on real financial data from four major market indices with up to 225 assets, executing experiments on Quantinuum's 98-qubit trapped-ion Helios system, with QAOA circuits acting on kernels of up to 78 qubits and 1016 two-qubit gates. While standalone QAOA fails to find the optimal solution for two of the largest indices (S&P 100 and Nikkei 225), qReduMIS achieves success probabilities of $0.40$ and $0.95$, respectively, with average approximation ratios $\geq 0.96$ across all four indices. We perform a systematic benchmark on the Quantinuum H2-1 noisy emulator over 73 asset correlation graphs of varying size showing that, for $p=2$ QAOA layers, the optimal time-to-solution scaling exponent of qReduMIS is $3.2$ times smaller than that of standalone QAOA.

We investigate and establish an analytic connection between the exact solutions describing the dynamics of a two-level system driven by periodic external fields, focusing on the cases of linear driving and the so-called rotating-wave approximation, or circular driving. In both cases, the exact solutions can be obtained by mapping the Schrodinger equation onto Heun equations: the confluent Heun equation for linear driving and the Heun equation for the rotating-wave case. In particular, we demonstrate a direct analytic connection between the exact solutions for linear driving and those for the rotating-wave case. This result is obtained by analyzing local solutions expressed in terms of hypergeometric functions, which, in the case of the confluent Heun equation, can be derived by considering path-multiplicative Floquet solutions involving a bilateral series. This series leads to two continued-fraction expansions that can be perturbatively solved by imposing a suitable consistency condition. The connection between the linear-driving and rotating-wave solutions is established through a perturbative procedure that allows us to recover not only the rotating-wave approximation itself, but also the correct Stark and Bloch-Siegert shifts, as well as the so-called high-frequency approximation.

We study a weak form of formality for differential graded algebras, called $A_3$-formality, and show that the differential graded $\mathbb{F}_2$-algebras of continuous cochains of all pro-$2$ Demushkin groups are $A_3$-formal. We prove this result by an explicit computation of the Benson–Krause–Schwede canonical class using the classification of pro-$2$ Demushkin groups by Demushkin, Serre, and Labute. Compared to the case of odd primes, the new idea is to interpret the data of the canonical class as defining systems of higher Massey products.

We study quiver Hecke algebras of untwisted affine type $A$ with an arbitrary choice of parameters and establish a duality with the Iwahori-Hecke algebra of the symmetric group. The parameter $t$ of the Iwahori-Hecke algebra is explicitly determined by the parameters defining the quiver Hecke algebra. This duality provides a deformation of the imaginary Schur-Weyl duality introduced by Kleshchev and Muth. Furthermore, we prove that the characters of simple modules in the imaginary strata are computed in terms of the dual canonical basis and Kazhdan-Lusztig polynomials, and the characters of standard modules coincide with the PBW vectors of the corresponding quantum group under certain assumptions. In addition, we examine other untwisted affine types, where the quiver Hecke algebra is known to be independent of the choice of parameters and the imaginary Schur-Weyl duality with the symmetric group has been established. As in type $A$, we apply this duality to the computation of characters of simple and standard modules.

We describe the Poisson geometry of the Madelung transform between quantum mechanics and hydrodynamics for generic wave functions. We prove that for arbitrary oriented manifolds this transform, being regarded as a momentum map, naturally defines prequantum bundles for coadjoint orbits of semidirect extensions of diffeomorphism groups. Furthermore, we show that the Madelung framework provides a natural infinite-dimensional version of the convexity results for Hamiltonian torus actions, thus giving a partial answer to Atiyah's question. In particular, for wave functions without zeros our results provide a Kähler map between the infinite-dimensional Fubini–Study and Fisher–Rao geometries, thus extending previous results to non-simply-connected manifolds. Furthermore, for wave functions with noncritical zeros, the Madelung transform is shown to be a symplectomorphism to the coadjoint orbits with Morse–Bott densities. The latter, in turn, furnishes a novel momentum map point of view on the Wallstrom quantization condition for the hydrodynamical form of quantum mechanics. We also comment on the relation between the Madelung setting and the Marsden–Weinstein symplectic structures on knots and membranes.

In this paper, we develop a physics-informed convolutional neural network (PICNN) assisted physics-preserving method for a thermodynamically consistent model of incompressible and immiscible two-phase flow in porous media. Following the physics-preserving prediction-correction scheme of Li et al. \cite{li2025class}, the prediction step is performed by a PICNN trained with finite-volume residuals, where the interfacial fluxes are evaluated by the two-point flux approximation (TPFA) using two-point difference quotients of neighboring cell-centered unknowns to approximate interfacial normal gradients. The PICNN output is further corrected by a post-processing procedure to obtain energy-stable, mass-conservative, and bounds-preserving solutions. Numerical results show that the finite-volume residuals trained PICNN can replace the traditional prediction solver within the physics-preserving framework. Compared with conventional physics-informed neural networks (PINNs), the PICNN better captures local spatial interactions between each control volume and its neighboring cells, while the finite-volume residuals accommodate discontinuous permeability fields and interfacial flux continuity.

Estimating peak prevalence is a central problem in epidemic modeling because it determines the period of greatest infectious burden and is closely linked to health-care demand. In multistage SIR models, however, peak prevalence is generally less tractable than in the classical model with exponentially distributed infectious periods. Motivated by the use of weighted infectious-stage aggregates as surrogates for prevalence, we investigate the relationship between the prevalence peak and the maximum of a weighted stage functional in deterministic SI$(k)$R epidemic models. We show that this relationship depends critically on how the stage-progression rate is scaled as the number of infectious stages increases. Under naive scaling, in which the progression rate remains fixed, the weighted peak is asymptotically equivalent to the prevalence peak and the commonly used factor-two approximation fails. Under Erlang scaling, which preserves the mean infectious period, the multistage model converges to a delay formulation in which prevalence and the weighted stage functional become unweighted and triangularly weighted moving averages of incidence. This limiting representation provides a theoretical basis for the factor-two approximation and identifies the regimes in which it is accurate. It also explains why this approximation deteriorates as epidemic waves become more sharply peaked. We derive analytical error bounds and develop curvature-based and parameter-based corrections that substantially improve accuracy. Numerical studies confirm these improvements across a broad range of epidemiological parameters. Overall, the results show when weighted-stage peaks can be used reliably as proxies for peak prevalence and how the resulting estimates can be refined when the standard approximation loses accuracy.

We investigate the box dimension of the graphs of a class of continuous periodic functions $G_\delta(x)=\sum_{n=1}^{\infty}g(n^{2}x)n^{-1-\delta}$ with 1-periodic Lipschitz functions $g$ and $0<\delta\le 1$, which generalizes the result of the classical Riemann function corresponding to $g(x)=\sin(2\pi x)$ and $\delta=1$. More precisely, we first prove that the lower box dimension of the graph of $G_{\delta}$ is no less than $\frac74-\frac{\delta}{2}$ when the Fourier coefficients of $g$ satisfy an arithmetic non-vanishing condition related to the distribution of quadratic residues. This result is new and non-trivial even when $g$ has a finite Fourier expansion, highlighting the intrinsic arithmetic complexity of the series. Secondly, if $g'$ is Lipschitz continuous on $\mathbb{R}$, we show that the upper box dimension does not exceed \(\frac74-\frac{\delta}{2}\), which extends earlier work of Chamizo and Córdoba and reveals deep connection between the regularity of $g$ and the fractal dimension of the associated Riemann-type series. In the end, we give some illustrative examples and propose some further problems.

The success of deep learning models in classification and regression is widely attributed to the low-dimensional structure that real-world data tend to exhibit, despite their high-dimensional representation. This work attempts to provide a mathematical framework for binary classification on low-dimensional data, building on Cover's (1965) function-counting theory. With our framework, we aim to address the question of how the low-dimensional structure of the data affects the classification capabilities of learning models. Cover's theory relies on a general position assumption that blinds it to the underlying data structure. We refine this assumption to account for the low-dimensionality of the data and derive dichotomy counts that reflect the data structure. We further extend Cover's separation capacity and problem of generalization to the low-dimensional setting, enabling the impact of the underlying data structure on both to be analyzed.

Wajnryb gave a finite presentation of the handlebody group $\mathcal{M}(V_g)$ and showed that it is generated by five elements for $g\geq 2$. Using his relations, we show that $\mathcal{M}(V_g)$ is generated by three elements if $g\geq 2$.

We reformulate binary classification on a manifold M as a Yang-Mills-Higgs variational problem. Labelled data is encoded as a functor from the fundamental groupoid of M to the one-object groupoid B(Z_2), whose monodromy class in H^1(M, Z_2) is a topological obstruction to realising the classifier by a sign function. The classifier-section and the connection jointly minimise a Yang-Mills-Higgs energy subject to hard data conditions: the matter sector carries the classification content, while the Yang-Mills sector is bounded below in each topological class by the Bogomolny inequality and selects the gauge background. This recovers the companion paper's harmonic interpolation as the contractible-base, flat-connection reduction. Two structural payoffs follow. First, the curvature 2-form of the selected connection supplies a dictionary with transformer attention: by the plaquette formula it is a pairwise, antisymmetric, algebra-valued form on tangent directions, which we match - as a stipulated dictionary, not a derived identity - to the antisymmetric component of the attention bilinear; the abelian/non-abelian split of curvature corresponds to the single-head/multi-head split of attention. Second, XOR on the torus is realised by the covariantly harmonic section of the double-Mobius bundle, which the variational selector picks out: solving the regularised capacitance system on all four flat Z_2-bundles gives a strict energy ordering favouring the double-Mobius class, whereas an MLP trained on the same data finds a structurally different boundary that ignores the toroidal identifications. Worked examples run an example ladder (circle, torus, S^2 monopole, S^4 instanton). The connection is throughout either flat or pinned to the Bogomolny moduli, so curvature is data-decoupled by construction; whether freeing it makes curvature respond to label proximity is posed as an open question.

We propose a dictionary between binary classification in machine learning and classical potential theory on vector bundles. Classifiers are parallel sections of vector bundles over the data space; training labels become Dirichlet boundary conditions; the kernel of an RKHS interpolant is the Green's function of an elliptic operator; and backpropagation is the flat-geometry limit of an exact geometric problem. The central identification: for L = (Delta + kappa^2)^nu with nu > n/2, the L-harmonic interpolation problem - find the minimum-H^nu-norm classifier satisfying Lf = 0 away from the data with prescribed values at the training points - is precisely what RKHS interpolation and kernel ridge regression already solve. Kimeldorf-Wahba (1971) and Lindgren-Rue-Lindstrom (2011) supply the analytic content; what we add is the potential-theoretic reading: the kernel is the Green's function of L, the coefficients are electrostatic capacitances, and the decision boundary is the zero equipotential. The condition nu > n/2 excludes the Dirichlet energy (nu = 1) in dimension n >= 2, where point data has zero capacity. The hard-margin SVM is a margin-constrained variant. For finite data on any smooth manifold, flat O(2) solutions always exist. Over non-contractible bases the geometry becomes load-bearing: the parity invariant on S^1 and the Euler class on S^2 are obstructions no classifier can evade. Code is available on GitHub.

Static radio maps provide location-dependent propagation priors, but they cannot capture short-term blockage caused by moving objects. Direct sensing-assisted beam prediction is also limited because a beam index discards SINR margins, MCS thresholds, BS alternatives, and communication-equivalent neighboring beams. This paper proposes QuaMoE-DRF, a quality-aware multimodal dynamic radio map forecasting framework for proactive beam and rate adaptation in ISAC networks. Its core representation is a future beam-SINR field. We show that the full multi-BS beam-SINR field is sufficient for finite-codebook threshold-rate BS, beam, MCS, goodput, and outage decisions. For tractability, the implemented model learns a compact reference-BS local field, complemented by BS-level supervision, joint BS–beam supervision, and latent network context; we also clarify that this compact projection alone is not sufficient for BS association. QuaMoE-DRF fuses static geometry, event-like motion observations, structured sensing states, and wireless history through a quality-aware mixture-of-experts module motivated by inverse-variance fusion under heteroscedastic modality errors. It jointly predicts communication-oriented map channels and proactive BS, beam, and MCS decisions. On a dynamic multi-BS and multi-UE urban benchmark, QuaMoE-DRF achieves 402.5 Mbps effective rate, 0.0417 outage probability, and 0.1836 map RMSE, improving the effective rate by 5.67% and reducing outage by 8.35% over the strongest completed effective-rate baseline. The current validation uses labels from a compact blockage/path-loss simulator, with ray tracing used only for calibration and sanity checking.

We study a dynamic averaging process on finite regular graphs with bounded, time-varying load arrivals. At each discrete time $t$, an edge is chosen uniformly at random, a load $0\le w_t \le 1$ is introduced, and the total load of its two endpoints together with $w_t$ is divided equally between them. Starting from the flat configuration, we obtain a pairwise concentration bound governed by the effective resistance between vertices and use generic chaining to derive a general upper bound on the expected gap between the largest and smallest loads. As a consequence, we show that every $d$-regular graph has expected gap $O_d(\sqrt n)$, uniformly in time and over all deterministic arrival sequences. Applying our general bound to the discrete two-dimensional torus yields the sharp $O(\log n)$ upper bound, improving the best previously known bound. For the cycle, whenever the arriving loads are bounded away from zero, we prove that the expected gap is $\Omega(\sqrt n)$ for all sufficiently large times. Together with our upper bound, this confirms the conjecture of Alistarh, Nadiradze, and Sabour that the expected gap on the cycle is of order $\sqrt n$.

We prove that, for every bounded-degree graph $\Lambda$ admitting a finitely cobounded coarse quasi-action by a group, there is a finitely generated group which does not coarsely embed into $\Lambda$. More generally, for every countable family $(\Lambda_i)$ of such graphs, there is a finitely generated group that does not coarsely embed into any $\Lambda_i$. This resolves two conjectures of Simon Thomas: neither a universal Cayley graph nor a universal quasi-isometry class of finitely generated groups exists. As another consequence, we show that no locally compact second countable group coarsely contains every finitely generated group. The proof uses an exponential upper bound on the number of finite graphs admitting an $(L,M)$-regular map into $\Lambda$, together with a superexponential supply of high-girth $3$-regular graphs, yielding a sequence of finite high-girth obstruction graphs. A graphical small-cancellation labeling, using a variation of Osajda's labeling theorem following Esperet and Giocanti, then realizes this sequence isometrically inside the Cayley graph of a finitely generated group.

We describe a dense subset of the Gromov boundary of the grand arc graph of an infinite-type surface as a space of geodesic laminations, analogous to Klarreich's description of the Gromov boundary of the curve complex. After showing that the grand arc graph satisfies a bounded geodesic image theorem, we also prove that the boundary is not compact.

Microwave linear analog computers (MiLACs) have recently gained attention for future gigantic multiple-input multiple-output (MIMO) systems by enabling beamforming with greatly reduced hardware and computational cost. However, channel estimation for MiLAC-aided multiuser systems remains an open problem. Conventional channel estimation requires many radio-frequency (RF) chains to access full-dimensional received signals, followed by massive digital processing, which undermines the advantages of MiLAC-aided systems in reducing the number of RF chains and computational complexity. In this paper, we propose computationally efficient channel estimation and beamforming schemes for MiLAC-aided multiuser multiple-input single-output (MU-MISO) systems with a limited number of RF chains. We consider the general case where different user groups experience different channel correlation matrices. By exploiting the rank deficiency of these matrices, the proposed schemes use MiLAC to compress the full-dimensional received signals in the analog domain, making them compatible with the available RF chains while preserving the essential channel information. Then, in the digital domain, only low-dimensional channel estimation is performed based on these compressed observations, substantially reducing computational cost. We further show how regularized zero-forcing beamforming (R-ZFBF) can be efficiently realized from the low-dimensional channel estimates through a cascade of two MiLACs, which offers greater computational flexibility than a single MiLAC. Numerical results show that the proposed schemes reduce computational complexity up to $1540\times$ and $16108\times$, for channel estimation and beamforming, respectively, while achieving performance comparable to digital baselines.

Integrated sensing and communication (ISAC) unifies high-precision sensing and wireless data transmission. In this paper, we investigate the design of ISAC systems enabled by flexible intelligent metasurface (FIM) and aim to minimize the Cramér-Rao bound (CRB) with quality of service (QoS) constraints using deep reinforcement learning (DRL). Specifically, we formulate the joint design of beamforming matrix and FIMs surface shape to reduce the CRB subject to transmit power, QoS and the FIMs surface shape constraints. However, the non-convex formulation makes optimization problem difficult to solve. To tackle this issue, we develop a deep deterministic policy gradient (DDPG) actor critic DRL scheme for the joint design, guided by a constraint aware reward to progressively improve sensing performance. Numerical results demonstrate that jointly optimizing the beamforming matrix and the FIMs surface shape substantially decreases CRB while ensuring communication quality compared with existing rigid arrays.

We prove a sharp Veronese rigidity theorem for closed immersed submanifolds of the Euclidean unit ball under intrinsic harmonic-structure assumptions. For an isometric immersion $F:(\Sigma,g)\looparrowright\overline B(1)$, define the maximal normal curvature by \[ \kappa(F):= \sup_{x\in\Sigma} \sup_{\substack{v\in T_x\Sigma\\ |v|_g=1}} |A_x(v,v)|. \] If $\Sigma^{2n}$ is almost Hermitian with harmonic fundamental two-form, or $\Sigma^{4n}$ is almost quaternion-Hermitian with harmonic fundamental four-form, $n\ge2$, then \[ \kappa(F)\ge \sqrt{\frac{2n}{n+1}} . \] In the equality case the harmonic form is parallel and the immersion is, up to a totally geodesic inclusion, the standard complex or quaternionic Veronese embedding of projective spaces. The key input is a Bochner–Gauss mechanism that turns the Bochner curvature term of the harmonic form into a sharp algebraic estimate for the shape operators.

We investigate the influence of routing strategies and speed limit policies on optimal solutions in traffic emission models. Building on a first-order macroscopic traffic model coupled with an advection-diffusion model, we formulate single- and multi-objective optimization problems to simultaneously maximize traffic efficiency and minimize air pollution. We compare three control scenarios: optimizing only the routing strategy, optimizing only the speed limit policy, and optimizing both simultaneously. Numerical experiments on a small road network demonstrate that speed limit policies consistently achieve larger reductions in emissions and greater gains in traffic efficiency than routing strategies. Multi-objective optimization reveals the trade-off between the two goals and confirms that including speed limits in the control set yields Pareto-optimal solutions that are strictly superior to those obtained by routing control only. Our results provide quantitative guidance for traffic management seeking to balance mobility and environmental objectives.

In this paper, we classify all solitons of the Gauss curvature flow in the three-dimensional Heisenberg group $\mathrm{Nil}_3$ that are invariant under a one-parameter group of ambient isometries. By means of the four canonical types of Killing vector fields and the three families of invariant surfaces (vertical translations, horizontal translations, and helicoidal motions), we analyze the twelve resulting types of possible solitons. In some cases, there do not exist any invariant solitons; in others, we find explicit parametrizations, or describe their geometric properties.

Bergeron, Garsia, Haiman and Tesler conjectured in 1999 that, for all partitions $\mu,\lambda\vdash n$, the polynomial $(-1)^{|\mu|-\ell(\mu)}\langle \nabla m_\mu, s_\lambda\rangle$ has nonnegative integer coefficients, where $\nabla$ is the Bergeron–Garsia nabla operator, which acts diagonally on the modified Macdonald basis, and $m_\mu$ is the monomial symmetric function. In this article, we prove this conjecture, and more generally that $(-1)^{|\mu|-\ell(\mu)}\langle\nabla^r m_\mu,s_\lambda\rangle\in\mathbb{N}[q,t]$ for all $r\geq 1$. We establish a recursion showing that $(-1)^{|\mu|-\ell(\mu)}m_\mu$ has an expansion with coefficients in $\mathbb{Q}_{\geq 0}[q]$ in the symmetric functions $C_\alpha(1)$, where $C_a$ denotes the operator introduced by Haglund, Morse and Zabrocki. Combining this expansion with the compositional shuffle theorems of Carlsson–Mellit and Mellit, and with the Schur positivity of LLT polynomials, completes the proof. The same method, using the $e$-positivity of column LLT polynomials after the substitution $q\mapsto q+1$, also gives an $e$-positive analogue.

We resolve (for all sufficiently large $n$) a conjecture of Pilz on the symmetric difference $A\Delta (2A)\Delta \cdots\Delta (nA)$ for finite sets $A\subseteq \mathbb{N}$ of positive integers. We show that this set always has cardinality at least $n$ for large $n$.

Let $p$ be an odd prime and let $V_{k,a_p}$ be the two-dimensional crystalline representation of the Galois group of ${\mathbb Q}_p$ of weight $k \geq 2$ and parameter $a_p \in \bar{\mathbb{Q}}_p$. We study the semi-simplification $\bar{V}_{k,a_p}$ of the mod $p$ reduction of $V_{k,a_p}$ when the slope (valuation of $a_p$) is a positive fraction $< p-1$ using the mod $p$ local Langlands correspondence. We describe the $\textit{exact shape}$ of $\bar{V}_{k,a_p}$ for all such slopes and all (sufficiently large, depending on the slope) weights $k$, as long as certain Jordan-Hölder factors of dimension $p-1$ do not intervene in the computation (when $k$ is odd), though we also provide some criteria which further determine the shape of $\bar{V}_{k,a_p}$ in some of these exceptional cases. To keep this paper a reasonable length, we assume that for certain bad congruence classes of $k$ mod $p$, the slope is less than the representative - taken in the range $[1,p-1]$ - of the congruence class of $k-2$ mod $(p-1)$, which is generically the case if the slope is small. Finally, a folklore conjecture predicts that the reduction $\bar{V}_{k,a_p}$ is $\textit{irreducible}$ for fractional slopes if $k$ is even. We deduce this conjecture for all fractional slopes $< p-2$ and all (sufficiently large, even) weights $k$ under the aforementioned slope assumption.

Reconstructing images on Cartesian grids from nonuniform Fourier samples is a basic computational task in Fourier reconstruction methods for computed tomography. This paper proposes a direct solver for the 2D type-II nonuniform discrete Fourier transform~(NUDFT) based on the face-splitting product of hierarchically semiseparable~(HSS) matrices. We define the face-splitting product of two HSS trees and prove that the face-splitting product of two HSS matrices admits an HSS representation on the resulting tree. The proof is constructive, providing explicit formulas for the HSS generators and bounds on the off-diagonal ranks at each level. For the NUDFT, a Fourier change of basis expresses the transformed 2D matrix as a face-splitting product of two transformed 1D NUDFT matrices. This identity yields an algebraic construction of a 2D HSS approximation from two 1D HSS approximations, for which we establish a Frobenius-norm approximation error bound. Combining this construction with recompression, URV factorization, and a two-dimensional inverse fast Fourier transform gives a direct solver for the associated least-squares problem. For fixed accuracy, square frequency grids, and balanced HSS trees with fixed leaf column sizes, the solver has an offline complexity of $O(M \log^{2} N + N^{3 / 2} \log^{3} N)$ and an online complexity of $O(M + N \log^{3} N)$ per right-hand side, where $M$ and $N$ are the numbers of sample points and unknown coefficients, respectively. Numerical experiments on random and polar grids illustrate the computational scaling and reconstruction accuracy of the method, and demonstrate a substantial reduction in lsqr iteration counts when the direct solver is used as a preconditioner.

Order polytopes for generalized snake posets were recently studied by von Bell et al. (2022), and are known to be unimodularly equivalent to strength-one flow polytopes for acyclic directed graphs strongly dual to generalized snake posets. Lee, Vindas-Meléndez, and Wang (2026) conjectured that the Ehrhart $h^*$-polynomials of these order polytopes are real-rooted. We prove this conjecture using a connection between these $h^*$-polynomials and non-nesting rook polynomials, which were recently introduced by Alexandersson and Jal (2024+) in connection with $P$-Eulerian polynomials for width two posets.

This paper studies the stochastic performance of a downlink multiple-input multiple-output integrated sensing and communication (ISAC) system over Rician fading channels. Rician fading is important in line-of-sight (LoS)-dominated deployments, where a deterministic propagation component can strongly affect sensing and communication reliability. The base station (BS) simultaneously serves a user and senses a target. The BS-user channel contains LoS and non-line-of-sight components. The user LoS angle may be fixed or random, and the target angle may follow an arbitrary distribution potentially correlated with the user angle. Compared with Rayleigh fading, the deterministic LoS component introduces angle-dependent terms and leads to generally independent but non-identically distributed random vectors, requiring new analysis. We analyze two beamforming strategies: subspace joint beamforming (SJB), optimal for the shared waveform structure, and linear beamforming (LB), a practical alternative using separate sensing and communication beamformers. For both schemes, we derive communication outage probability (OP) and sensing OP based on the Cramer–Rao bound (CRB). We also identify special cases with simpler expressions. For LB, we derive upper and lower bounds on sensing OP and a tractable approximation. We characterize large-system and high-power scaling laws. LB without dirty paper coding (DPC) is interference-limited at high power due to radar self-interference. Results show the Rician K-factor affects communication more strongly than sensing, with non-monotonic behavior across regimes. LB with DPC achieves the best overall performance in strong LoS environments and is the only scheme achieving ultra-high communication reliability in Rayleigh fading, while SJB provides a robust lower-complexity alternative across operating conditions.