Papers

In this paper, we study kernel ridge regression using the data-driven diffusion maps (DM) kernel, which is constructed through algebraic manipulations of the diffusion maps algorithm. Under the assumptions that the data lie on a manifold and are sampled uniformly, we prove that the appropriately scaled DM kernel converges uniformly to the heat kernel on the manifold for sufficiently large times as the dataset size increases and the kernel bandwidth is scaled appropriately. Consequently, the limiting Reproducing Kernel Hilbert Space (RKHS) induced by the DM kernel coincides with the RKHS associated with the heat kernel on the manifold. We further show that the RKHS induced by the DM kernel is isometrically isomorphic to the RKHS of the Gaussian kernel, which is continuously embedded in the RKHS of a Matérn kernel whose norm is equivalent to an appropriate Sobolev norm. This result implies that standard risk bounds for kernel ridge regression applicable to Matérn kernels also apply to the DM kernel. Finally, we provide numerical results that (1) validate the convergence of the heat kernel approximation, (2) demonstrate the greater expressiveness of the DM kernel compared to the Gaussian kernel for supervised learning over a larger class of functions on manifolds with boundary, and (3) demonstrate the advantage of the DM kernel over the Gaussian kernel in learning functions with varying frequencies and co-dimensions.

Let $\mu_t$ be the full set of $t$-th roots of unity. Adjoining $r$ free reciprocal pairs gives a two-parameter family of alphabets; we settle three parts of it. At $r=1$, for every $t\ge2$ and every $\lambda$ with at most $t+2$ parts, $s_\lambda(\mu_t,z,z^{-1})$ is a signed product of exactly three factors over a fixed denominator, or zero, the arguments read off core and quotient. What it sees of $\lambda$ is a multiset of three integers and a sign, and exactly that: two partitions of any sizes share a nonzero value if and only if they agree on that datum. The proof is a Laplace expansion along the $t$ frozen rows with one cancellation lemma, and delivers the sign, of which Littlewood's is one factor. At $t=2$ and every $r$, $s_\lambda(1,-1,z_1^{\pm1},\dots,z_r^{\pm1})$ vanishes exactly when the beta set has constant parity or $\lambda$ is self-complementary of odd width; that direction is a corollary of complementation over an index family of Ayyer and Behrend, the converse an extremal argument in the degree filtration, modulo one rigidity theorem for Schur products. Equivalently: exactly those $V_\lambda$ restrict to $O(N,\mathbb{C})$ $\det$-stably. At odd $t$ and every $r$ it vanishes exactly when a residue class is absent, at no external cost. And for every $t$ and $r$, a reflection of the beta set's excess part with one increment hitting its centre forces vanishing. Three consequences of the first. A vanishing criterion: an empty residue class, or two distinguished classes concentric as intervals, the second only for even $t$. An extension of Ayyer-Kumari's independence criterion: on the reciprocal locus it acquires one further family, classified by core and quotient. And at $t=2$ a $(-1)$-enumeration of plane partitions in a box refined by a parameter that stays free. The factorization is isolated: it fails under each of four deformations of the alphabet, for one reason.

The discretization of second-order (viscous) terms in Discontinuous Spectral Element Methods (DSEMs) typically relies on an auxiliary gradient variable, whose treatment at element interfaces affects the accuracy and stability of the scheme. The Bassi-Rebay (BR1) formulation is attractive for its simplicity and parameter-free character, but suffers from sub-optimal convergence at even polynomial orders and requires an extended five-element stencil. Inspired by Huynh's Flux Reconstruction formulation, we develop a compact, fully-centered scheme for second-order fluxes within the Spectral Difference (SD) method. The proposed approach modifies the reconstruction of the auxiliary gradient using interface-dependent, one-sided continuous fluxes, reducing the stencil from five to three elements while preserving the centered and parameter-free nature of BR1. The formulation is developed in one dimension and extended to multiple dimensions. Temporal eigenanalysis is used to characterize its dissipation and dispersion properties, including the effects of interior penalty terms. Numerical tests consider the linear diffusion equation, an under-resolved localized Dirac's delta, the nonlinear porous medium equation, and implicit large-eddy simulations of the three-dimensional Taylor-Green vortex at $\mathrm{Re}=1600$ and $5000$. The compact scheme restores the expected convergence order for all polynomial degrees, including even orders, and reduces spurious oscillations in under-resolved and nonlinear regimes. It also remains stable in turbulent cases where the standard formulation fails, owing to improved damping of high-wavenumber numerical modes. The proposed approach provides an attractive compact alternative to BR1 for second-order fluxes in the SD method.

We propose Choquet-type extensions for set-pair functions, an $L^p$-integration of a family of such extensions, and the associated min-max extension constants, and apply these to investigate variational problems and establish sharp bounds for many combinatorial and geometric quantities. Specifically, we use Choquet-type extensions to construct equivalent functional representations of fractional optimization problems with set-variables, which find applications in various settings, such as maxcut and bipartiteness ratio on graph limits, and Cheeger constants on Riemannian manifolds. We further establish the $L^p$ integration of a family of Choquet-type extensions, and apply it to derive spectral bounds for conductance and other combinatorial quantities on measure spaces. A monotonicity inequality on min-max extension constants is proposed, which unifies classical estimates and uncovers new bounds for many geometric and combinatorial quantities, such as torsional rigidity, Cheeger constants, Dirichlet $p$-isoperimetric constant, and $p$-Laplacian eigenvalues, in seemingly unrelated underlying structures—-including hypergraphs, graph limits, Riemannian manifolds and metric measure spaces.

This paper introduces a novel block preconditioner for a class of double saddle point systems arising from the finite element discretization of liquid crystal director modeling. We first establish sufficient conditions to guarantee the convergence of the underlying iterative scheme. A spectral analysis of the preconditioned matrix is then performed, leading to an effective strategy for parameter selection. The new preconditioner is used to accelerate the convergence of the flexible version of GMRES. Experimental results confirm the strong performance of the method.

Let $(M^n,g)$ be a closed, connected, oriented Riemannian manifold with negative sectional curvature and $n\ge 2$. It is known, due to Brin and Gromov, that the oriented frame flow is ergodic in every odd dimension $n\ge 3$, $n\neq 7$. We prove that if a closed, connected, oriented negatively curved Riemannian manifold is strictly quarter-pinched, then its oriented frame flow is ergodic.

As a natural extension of Stirling permutations, quasi-Stirling permutations are multipermutations $\pi$ with the property that for any subsequence $\pi_{j_1}\pi_{j_2}\pi_{j_3}\pi_{j_4}$ satisfying $\pi_{j_1}=\pi_{j_3}$ and $\pi_{j_2}=\pi_{j_4}$, we have $\pi_{j_1}=\pi_{j_2}$. Using a bijective construction, Yan, Yang, Huang and Zhu showed that the joint distribution of ascents, descents and plateaux over quasi-Stirling permutations of a multiset $M=\{1^{k_1},2^{k_2},\ldots,n^{k_n}\}$ coincides with that over the multiset $M'=\{1^{k_1+\cdots+k_n-n+1},2,\ldots,n\}$. In this paper, we prove that the same invariance of distribution holds for decreasing runs, and consequently for all decreasing consecutive patterns. To this end, following the Yan-Yang-Huang-Zhu approach, we construct a multiplicity-redistribution bijection that preserves decreasing runs, thereby reducing the computation of joint distribution of decreasing consecutive patterns over quasi-Stirling permutations from $M$ to $M'$. Together with the classical run theorem, our bijection leads to explicit recurrence relations and generating functions for the distribution functions of these statistics over quasi-Stirling permutations.

We develop a general $\infty$-categorical framework for internal Eilenberg-MacLane objects, internal homotopy groups, and an internal notion of covering spaces, and study their behavior under suitable localizations. Applying this to the ordinary motivic localization, we identify internal $n$-Eilenberg-MacLane objects with strongly $\mathbb{A}^1$-invariant sheaves of (abelian when $n\geq 2$) groups, yielding a formal obstruction to the motivic homotopy category being an $\infty$-topos. We prove that taking $\mathbb{A}^1$-localizations induces an equivalence between classical $\mathbb{A}^1$-coverings of a Nisnevich local space and the internal motivic coverings of its motivic localization, providing a streamlined proof of a generalized motivic Van Kampen theorem. Along the way, we establish a Nisnevich-local-to-global $\mathbb{A}^1$-connectivity criterion: a $k$-scheme is $\mathbb{A}^1$-connected if and only if it admits a Nisnevich cover by $\mathbb{A}^1$-connected schemes such that each pairwise intersection has a ($k$-)point. Finally, we show that over a general Qcqs base, passing to the birational motivic homotopy category recovers certain essential topos-theoretic properties absent in the ordinary $\mathbb{A}^1$-setting. In fact, for a scheme with finitely many generic points, we show that the birational motivic homotopy category is a Postnikov-complete $\infty$-topos of cohomological dimension $0$.

We consider discrete-time non-linear stochastic dynamical systems where the driving noise is not i.i.d. but an autoregressive process (“red noise”). Under a standard dissipativity condition, we prove geometrically fast convergence to a stationary distribution in total variation.

Let $\psi({\mathbf x})$, ${\mathbf x} \in{\mathbb R}^{3N}$, be an eigenfunction of the $N$-particle atomic Schrödinger operator. We consider the one-particle density matrix $\gamma(x, y)$ and one-particle kinetic energy density $\varkappa(x, y)$, $x, y\in {\mathbb R}^3$, associated with the eigenfunction $\psi$. Both functions play a central role in quantum chemistry computations of atomic and molecular bound states: the knowledge of the eigenvalue behaviour of the integral operators ${\sf{\Gamma}}$ and ${\sf{K}}$ with kernels $\gamma(x, y)$ and $\varkappa(x, y)$ serves to estimate the errors due to finite-dimensional approximations. We find the following asymptotic formulas for their eigenvalues $\lambda_k({\sf{\Gamma}})>0$ and $\lambda_k({\sf{K}})>0$: \[ \lim_{k\to \infty} k^{\frac{8}{3}} \,\lambda_k({\sf{\Gamma}}) = A^{\frac{8}{3}},\quad \lim_{k\to \infty} k^2\,\lambda_k({\sf{K}}) = B^2, \] where $A$ and $B$ are non-negative constants given explicitly in terms of the eigenfunction $\psi$. These asymptotics are determined by the singularities of the function $\psi$ at pair coalescence points of the particles. To identify and isolate these singularities we use some recent regularity results for $\psi$. At the last step we apply Birman-Solomyak spectral asymptotics results for pseudodifferential operators with homogeneous symbols. In the special case where the eigenfunction $\psi$ is totally antisymmetric, it exhibits enhanced regularity, which leads to a faster decay of the eigenvalues $\lambda_k(\sf{\Gamma})$ and $\lambda_k(\sf{K})$. The asymptotic formulas take the form \[ \lim_{k\to \infty} k^{\frac{10}{3}} \,\lambda_k({\sf\Gamma}) = \big(A_{asym}\big)^{\frac{10}{3}},\quad \lim_{k\to \infty} k^{\frac{8}{3}} \,\lambda_k({\sf K}) = \big(B_{asym}\big)^{\frac{8}{3}}, \] where $A_{asym}$ and $B_{asym}$ are non-negative constants given explicitly in terms of the gradient of $\psi$.

We consider smooth convex minimization over the spectrahedron using Frank-Wolfe-type methods based only on extreme-eigenvector computations. In our recent work \cite{garber2026randomized} we presented the first ambient-dimension-independent linear convergence rate under quadratic growth. However, the method makes an additional strong strict complementarity assumption, it is randomized, its linear rate holds only after a burn-in phase and in expectation, and it requires the objective smoothness constant. We show that these limitations can be removed. Assuming quadratic growth and that all optimal solutions have the same rank, but without assuming strict complementarity, we give a deterministic and parameter-free Frank-Wolfe-type method with a global ambient-dimension-independent linear convergence rate.

Optimization theory is a widely used tool for intelligent decision-making. While classical optimization deals with fixed, time-invariant objective functions, many modern applications operate in dynamic environments where data arrive sequentially, and the learning objective evolves over time, often under decentralized data and communication constraints. Motivated by these trends, we study decentralized optimization from streaming data through a structured time-varying formulation in which the global objective is a temporally weighted average of losses observed across the network. We analyze multi-iteration decentralized first-order methods, including decentralized gradient descent. For strongly convex and smooth losses, we develop guarantees for the Euclidean-norm tracking error through a contraction-mapping viewpoint. The resulting bounds decompose the tracking error into a fixed-point tracking component and a bias term induced by decentralization and data heterogeneity. We specialize our analysis to uniform and exponentially discounted weights, as well as their finite-memory windowed counterparts. The bounds explicitly characterize the roles of the temporal weighting rule, per-step iteration budget, step size, and network connectivity. Uniform weighting yields a vanishing fixed-point tracking contribution of order $\mathcal O(1/t)$, whereas discounted and windowed strategies generally induce non-vanishing tracking floors governed by the discount factor and effective memory, respectively. In all cases, decentralization induces an additional non-zero bias floor under a constant step size. Numerical experiments illustrate the predicted trends.

Ensemble decoding is a promising technique for ultra-reliable low-latency communication, as it trades hardware parallelism for decoding latency by running M diverse belief propagation (BP) decoders in parallel. Conventional ensembles select their output from the candidates. Hence, decoding fails whenever no member finds the correct codeword. In this work, we show that shallow, diverse BP members are poor correctors, but excellent sorters. Based on this observation, we propose a soft ensemble-combining stage (SECS) that combines extrinsic messages after only a few iterations, yielding a reliability ordering whose most reliable positions are nearly error-free. A subsequent re-encoding stage (like ordered-statistics decoding (OSD)) converts this ordering into a near-maximum likelihood (ML) candidate codeword, at a fraction of the BP latency of a fully converged ensemble. We demonstrate the proposed scheme on three short codes: a (63,30) Bose-Ray-Chaudhuri-Hocquenghem code, an overcomplete PG(2,8) code, and a search-designed (105,53) cyclic code. In all cases, the SECS with OSD post-processing closes most of the gap to ML decoding, while significantly reducing the number of required BP iterations.

In this paper, we consider the problem of estimating conditional expectations as an ill-posed inverse problem. We propose a solution based on a generalization of Vapnik's theorem \cite[Theorem 7.2]{Vapnik1998} for solving stochastic ill-posed problems in Hilbert spaces. As an application, we derive a new upper bound for sample errors of conditional expectation estimation.

Belief-propagation (BP) decoding of short and moderate-length low-density parity-check (LDPC) codes is limited by finite-length graph effects: a single decoder trajectory can become trapped or oscillatory even when an alternative trajectory would decode the received word. Existing ensemble-BP decoders create the required diversity through multiple parity-check matrices, automorphisms, modified schedules, subcodes, or altered update rules. We introduce row-boosted ensemble (RBE) decoding as a minimal decoder-side diversity mechanism: all ensemble members share the same parity-check matrix and the same BP kernel, and differ only in a small set of parity-check rows whose outgoing messages are boosted. On the 5G~NR BG1 \((144,96)\) code, RBE with \(32\) members lowers the frame-error rate of BP with \(20\) iterations (BP-20) from \(1.6\times10^{-2}\) to \(1.6\times10^{-3}\) at \(E_\mathrm{b}/N_0=4.0\,\mathrm{dB}\), outperforming saturated-min-sum and affine subcode ensembles of equal size. Increasing the ensemble size yields additional gains, indicating that RBE provides a scalable performance-complexity tradeoff. The gains transfer across 5G~NR block lengths and rates, to non-5G short LDPC codes, and across flooding and layered schedules.

How expressive is prompting a transformer? Answering this question is important for separating the roles of prompting, architecture, and pretraining in transformer models, and for determining whether task-specific behavior must be stored in model weights or can instead be induced at inference time through the prompt. We show, in an approximation-theoretic sense, that pretraining is optional: a single-layer softmax attention network with random, untrained weights can approximate any Hölder function on a compact manifold when steered by an appropriate soft prompt. Guided by the connection between softmax attention and kernel methods, we construct explicit soft prompts (a prompt per target function, independent of the query) as solutions to linear systems matching attention logits to Gaussian kernel exponents, under which the frozen transformer emulates the classical Nadaraya-Watson kernel estimator. The construction requires only a mild rank condition on the weights, which we show holds almost surely under Gaussian initialization. The prompted network inherits the theoretical guarantees of kernel regression, leading to universal approximation theorems with minimax-optimal rates that depend on the intrinsic dimension. We further quantify the cost of prompting, exposing a tradeoff between the norm of the constructed soft prompt tokens, prompt length, and hidden dimension. Numerical experiments corroborate the constructions and predicted rates.

We characterise the periodic orbits of geodesic billiards on spherical lunes on $\mathbb{S}^{n}$. In the case of angle openings of the form $\pi/p$ for positive integer $p$ we fully determine their Dirichlet and Neumann spectra. We then show that lunes with an angle opening smaller than $\pi$ which is not a rational multiple of $\pi$, or those with an angle opening of the form $\pi/p$ for $p$ larger than one satisfy Pólya's conjecture eventually, independently of whether the corresponding geodesic billiards satisfy the nonperiodicity condition or not. For lunes with an angle opening $\pi/p$ we further provide a two-term asymptotic formula for the eigenvalues based on sharp upper and lower bounds, together with a corresponding two-term counting function established using the geoesic billiards approach. Finally,we give an explicit bound on $p$ in terms of the dimension ensuring the corresponding lunes satisfy Pólya's conjecture for all eigenvalues.

We consider the nonlinear Schrödinger equation on a two-dimensional strip with an attractive $\delta$ interaction and power nonlinearity. We investigate the transverse stability and bifurcation of line solitons as the width of the strip varies. We first establish local well-posedness in $H^1$, conservation of mass and energy, and global existence in the $H^1$-subcritical regime. We then identify a critical width $L_*$ at which the line soliton undergoes a transverse instability. More precisely, we prove orbital stability for $L<L_*$ and orbital instability for $L>L_*$. At the critical width, a simple eigenvalue of the linearized operator crosses zero, and we construct, via the Lyapunov-Schmidt reduction, a branch of positive nontrivial stationary solutions bifurcating from the line soliton. We determine the direction of this bifurcation by computing the second-order variation of the width along the branch. Finally, we investigate the orbital stability of the bifurcating solitons and obtain a stability criterion which can be evaluated in the regime of sufficiently small interaction strength.

Chan and Yeung showed that finite groups suffice to determine which homogeneous linear information inequalities are universally valid. We compare two restricted group-characterizable entropy cones: the Abelian cone $\widetilde\Gamma^{\mathrm{Abl}}_n$ and the homomorphic cone $\widetilde\Gamma^{\mathrm{Hom}}_n$, the latter generated by coset systems of normal subgroups. We prove \[ \widetilde\Gamma^{\mathrm{Abl}}_{16}\subsetneq\widetilde\Gamma^{\mathrm{Hom}}_{16}, \] and, if $n_{\rm AH}$ is the least number of variables for which these cones differ, we show $6\le n_{\rm AH}\le16$. The separating functional is a class-restricted entropy inequality: it is valid on the Abelian cone but is not a universal information inequality. It is obtained by lifting the order dual of the Pálfy–Szabó six-cross identity while quantifying errors at inexact subgroup joins. We then construct sixteen normal subgroups of a class-two $2$-group of order $2^{43}$ for which every join error vanishes while the endpoint containment fails by one bit. Since mixed-linear random variables are Abelian, the same example also separates the mixed-linear and homomorphic entropy cones.

We introduce the $\textit{expanded core graph}$, which records minimal unbounded domains and their axial directions, for two classes of hierarchically hyperbolic group structures modeled on compact special groups and mapping class groups. Our main structural result shows that every embedding of a right-angled Artin group, after replacing its standard generators by positive powers, factors through an intermediate RAAG generated by suitably supported axial elements; for the class modeled on compact special groups, this intermediate RAAG is quasi-isometrically embedded. We show that its extension graph embeds into the expanded core graph. This yields a Kim–Koberda-type obstruction to RAAG embeddings and a complete embedding criterion when the rank is at most two. For the standard HHG structure on a mapping class group, the expanded core graph is the disjointness graph of essential curves, while for natural rich-family structures on a RAAG it recovers the extension graph. We also establish permanence results under finite direct products and the standard relatively hyperbolic construction.

Given a finite group G, we develop a generalization of the fundamental classification of Rokhlin G-actions on C*-algebras to the setting of anomalous G-actions with the Rokhlin property. For Rokhlin G-kernels on C*-algebras covered by the classification program, this implies that the induced G-action on some known invariants determines the conjugacy class. Extending a result of Izumi, we show that G-kernels with the Rokhlin property on Kirchberg algebras are classified by their anomaly and the induced module structure on the K-theory groups. We explore K-theoretic obstructions for the existence of Rokhlin G-kernels on general separable C*-algebras and use these to characterise which G-module structures arise as K-groups of Kirchberg algebras admitting a Rokhlin G-kernel with a given lifting obstruction.

This paper introduces a two-variable polynomial invariant for oriented twisted knots, denoted by $Q_{K}^{z}(s,t)$, refining the $Q$-polynomial of N. Kamada and S. Kamada \cite{NaoSei}. We exhibit an infinite family of twisted knots indistinguishable by the $Q$-polynomial but separated by the $Q^z$-polynomial. To prove invariance, we first determine a generating set of oriented Reidemeister moves for twisted knot diagrams, extending the result of Ali \cite{Dan} for oriented virtual knots; this result is new and of independent interest, as it provides the minimal framework needed to verify invariance of any oriented twisted knot invariant. As further applications, we derive an explicit crossing change formula, obtain lower bounds on the Gordian distance between homotopic twisted knots, examine the existence of cosmetic crossings in a twisted knot diagram, and finally prove that $Q^{z}_{K}(s,t)$ is a Vassiliev invariant of order one.

We study constructive discrete multimarginal optimal transport (MOT) among $m$ distributions on $n$ points, where the cost tensor $C$ has $N=n^m$ entries. For additive accuracy $\varepsilon$, let $\kappa=\max\{1,(\max C-\min C)/\varepsilon\}$. Classically, we give two algorithms that return exactly feasible additive-$\varepsilon$ couplings. A deterministic box–simplex method with rounding runs in $O(m^2N\kappa\log N)=\widetilde O(m^3N\kappa)$ time, while an exact reduction to positive packing followed by rank-one completion runs in randomized time $\widetilde O(m^2N\kappa)$. At fixed $\kappa$, both match the $\Omega(N)$ cost of writing a dense coupling, up to factors in $m$ and logarithms. Quantumly, in the general entry-access model, tensor Sinkhorn followed by sparse recovery returns an exactly feasible additive-$\varepsilon$ coupling as a classical list of $\widetilde O(m^2n\kappa^2)$ atoms, using $\widetilde O(m^4\sqrt{Nn}\,\kappa^3)$ coherent cost queries without materializing the tensor. Finally, for fixed $m$ and constant normalized accuracy, we prove sparse-output lower bounds of $\widetilde\Omega(N)$ randomized classical queries, even with unrestricted output size, and $\widetilde\Omega(\sqrt{Nn})$ quantum queries for outputs with at most $n\operatorname{polylog}(n)$ atoms. Hence, for fixed $m$ and normalized accuracy between $(\log n)^{-O(1)}$ and a sufficiently small constant, explicit sparse MOT construction has query complexity $\widetilde\Theta(N)$ classically and $\widetilde\Theta(\sqrt{Nn})$ quantumly.

We revisit the nonlinear two-scale Dynamic Diffusion (DD) finite element formulation mathematically analyzed by Santos et al. (2021) for stationary advection–diffusion–reaction problems. We establish an explicit local Lipschitz estimate for the artificial diffusivity, with a constant of order $h_T$, and use it to prove uniqueness of the discrete solution for sufficiently fine meshes. By separating the approximation error in the energy norm from the contribution associated with artificial diffusion, we derive an optimal first-order a priori energy-norm estimate for continuous piecewise linear finite elements enriched with simplex bubble functions. We further prove that the square root of the nonlinear artificial dissipation is $O(h)$. Consequently, the combined energy-error and artificial-dissipation measure also converges with first order, sharpening the previously available $O(h^{1/2})$ estimate. A smooth manufactured problem, considered with and without reaction, corroborates the predicted convergence rates. A second, strongly advection-dominated problem with sharp outflow layers illustrates the stabilizing behavior of the DD formulation.

We study the boundary regularity of weak solutions to nonlocal nonlinear elliptic equations with bounded measurable coefficients. Our main result establishes that a capacity density condition is equivalent to the validity of a uniform boundary Hölder estimate for solutions with Hölder continuous exterior data. More generally, we derive a fine capacitary estimate on the modulus of continuity that captures how regularity is inherited from the exterior datum to the solution.

Fluid antenna systems (FASs) exploit many candidate ports for spatial diversity, but hardware constraints allow channel observations at only a few active ports. Whether full-port CSI can be recovered without pre-acquired channel statistics remains open. Under the Clarke isotropic scattering model, we show that the channel lies in a low-dimensional spatial modal subspace determined by the scattering environment rather than the total port count. Consequently, recovery becomes feasible when the number of observed ports reaches the modal dimension (i.e., $M\geq r$), even when $M\ll N$. We further establish a sharp feasibility threshold: reliable recovery is impossible below this dimension regardless of SNR, whereas accuracy improves with additional observations above it. By decomposing the recovery error into modal truncation, estimation, and learning components, we derive explicit tradeoffs among RF chains, pilot overhead, transmit power, and training data. These results enable scalable prior-free full-port CSI recovery with few active ports.

We show that if a complete binary tree of height $h$ is isomorphic to a subgraph of the strong product of a graph $H$ and a path, then $\mathrm{pw}(H)$ is $\Omega(h)$. This solves a problem posed by Bose, Dujmović, Javarsineh, Morin, and Wood (2022). The proof was found by OpenAI's GPT-5.6 Sol Pro.

In this paper we provide Monte Carlo and deep neural network approximations for stochastic representations of solutions to linear elliptic partial differential equations with constant diffusion, drift and killing. Building on the modified Walk-on-Spheres algorithm of Beznea et al. (arXiv:2209.01432), we introduce Monte Carlo estimators that explicitly incorporate sampled random times arising in the analyzed stochastic representations. We establish uniform error bounds for these estimators and show that, under suitable assumptions, a prescribed approximation accuracy is achieved with sample complexities growing at most polynomially in both the inverse accuracy and the problem dimension. Furthermore, we prove a deep neural network approximation result for the stochastic representations. Assuming suitable neural network representations of the boundary data and the distance function to the boundary, we use the constructed Monte Carlo to design deep neural networks that approximate the representation uniformly with a number of parameters growing at most polynomially in the inverse accuracy and the problem dimension. These results extend previous complexity analyses to a broader class of elliptic equations involving drift and killing.

In this paper, we consider the initial-boundary value problem of a three-component reaction-diffusion system with dual-dependent motility, which depends on both the chemical concentration and the nutrient level. We systematically establish the global existence, boundedness, and asymptotic behavior of the classical solutions to the system with no-flux boundary conditions through classified discussion and by exploiting specific structural properties of the motility function. In turn, by comparing the results obtained in the distinct cases, we demonstrate that the solution converges to different constant steady states, thereby indicating the crucial role of the dual-dependent motility function in the long-term behavior of the system.

We prove that for every integer $N\geq2$, there exist positive integers $a$ and $b$ such that $N=a+b$ and $\Omega(ab)\leq33$, where $\Omega(n)$ denotes the number of prime factors of $n$, counted with multiplicity. This improves the previous bound of $40$ obtained by Dudek and Dunn. The proof applies the explicit Friedlander–Iwaniec $\Lambda^-\Lambda^2$ lower-bound sieve to a sequence derived from the products $n(N-n)$. The main new ingredient is pre-sieving at the prime $3$, which eliminates the extremal small-prime case in the dimension condition while keeping the resulting remainder terms under explicit control. We complete the proof using analytic estimates for large $N$, finite verification over an intermediate range, and explicit prime-gap data for small $N$.