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arXiv2026

PPIM: Pennes Physics-Informed Mamba for Heat-Source-Conditioned 3D Bioheat Simulation

Three-dimensional bioheat simulation aims to predict transient temperature distributions in biological tissue and is commonly modeled using the Pennes bioheat equation, which combines thermal diffusion, perfusion-mediated heat loss, and external heat generation. In this study, we consider a controlled 3D Pennes bioheat simulation under a localized heat-source condition inspired by microwave ablation (MWA). To evaluate neural approximation performance, we compare three neural partial differential equation (PDE) solvers under the same controlled simulation: a spatial Fourier-feature physics-informed neural network (PINN), a generic PINNMamba temporal subsequence model, and Pennes Physics-Informed Mamba (PPIM). PPIM builds on the temporal subsequence model by incorporating conditioned heat-source input and Pennes-aware state-space model (SSM) decay initialization. All three neural models are trained under the same conditions with the same Pennes residual, and an explicit finite-difference method (FDM) solution is used only as the numerical reference. In a representative 600~s run, PPIM achieved the lowest MAE, relative $L_1$ error, and relative $L_2$ error among the evaluated neural solvers. Error maps further showed that the remaining PPIM errors were more concentrated near the heat-source region than across the rest of the domain. These results indicate that PPIM is effective for approximating the FDM reference final temperature field in this controlled simulation. The source code is available at https://github.com/muvYun/PPIM.

cs.LG↗
arXiv2026

Two Adjoint Perspectives on Fokker-Planck Optimization: A Microscopic-Macroscopic Correspondence

The Fokker-Planck equation admits both a macroscopic Eulerian description through probability densities and a microscopic Lagrangian description through stochastic trajectories. Consequently, optimization problems constrained by the Fokker-Planck equation can be formulated from either perspective. Surprisingly, the corresponding adjoint equations appear to be fundamentally different: the macroscopic adjoint is governed by the backward Kolmogorov equation, whereas the microscopic adjoint evolves pathwise along stochastic trajectories. In this note, we reconcile these two formulations by establishing their correspondence in the continuum setting. We further show that, although their discrete gradients no longer coincide after discretization, both provide consistent numerical approximations of the continuum gradient. Explicit convergence rates are established for both discretization strategies.

math.NA↗
arXiv2026

Multiplicative comparisons of Rényi entropies for weighted Bernoulli sums

We establish improved multiplicative bounds relating the Rényi entropies of different orders for weighted sums of independent Bernoulli random variables. In particular, we prove a logarithmic bound between the zeroth-order and infinity-order Rényi entropies, which yields a polynomial improvement over the square-root bound of Jain, Sah, and Sawhney. Additionally, we obtain explicit constant-factor bounds for comparisons among Rényi entropies of nonzero orders.

math.PR↗
arXiv2026

Counterexamples to Charpin's Conjecture on BCH codes

We construct an infinite family of $q$-ary primitive narrow-sense BCH codes whose minimum distance strictly exceeds the Bose distance; in fact, the gap between the two can be arbitrarily large as the length of the code tends to infinity. The key idea is to embed these BCH codes in a suitably large punctured generalized Reed--Muller code, whose codeword weights obey divisibility conditions supplied by Ax's theorem. This divisibility forces the minimum distance of the BCH codes far above the Bose distance. In particular, our family disproves a longstanding conjecture of Charpin asserting that this difference is at most four.

cs.IT↗
arXiv2026

The half-rate linear programming bound for binary codes is $\frac12-\frac1π$

In their work on sphere packing and the conformal bootstrap, Afkhami-Jeddi, Cohn, Hartman, de Laat, and Tajdini conjectured the exact high-dimensional exponent of the Cohn--Elkies sphere-packing linear program. OpenAI's Chapter 1 subsequently proved their conjecture by establishing that both Fourier sign-uncertainty radii are $(1/π+o(1))\sqrt d$. We prove the binary coding analogue: the half-rate point of the asymptotic binary Delsarte linear program is $1/2-1/π$; equivalently, \[ R_D\!\left(\frac12-\frac1π\right)=\frac12. \] We also formulate the two Krawtchouk sign-uncertainty problems and determine both of their asymptotics. If $A^{\mathrm K}_{\pm}(n)$ denotes the first radial layer after which an origin-vanishing Krawtchouk $(\pm1)$-eigenfunction can be nonnegative, then \[ \frac{A^{\mathrm K}_{\pm}(n)}n\longrightarrow \frac12-\frac1π. \] The common lower bound is the Hamming space counterpart of the mass-concentration principle in the Chapter 1 proof. The upper bound has a different source. It is the binary-code counterpart of the final spherical-code construction in OpenAI's Chapter 2. Gay, Jeronimo, and Liu improved the resulting binary bound and suggested the functional $Φ$ used here, but explicitly evaluated only a few low levels of the corresponding hierarchy. We construct and evaluate compatible binary certificates at every level, attaining the upper bound in the limit. The construction uses an $N$-qubit generalization of the pure-state channel of Alrabiah and Guruswami.

math.CO↗
arXiv2026

A $(\log n)^{1/4}$ Bound for the Komlós Problem

Let $A\in\mathbb{R}^{m\times n}$ have columns of Euclidean norm at most one. We prove that $\operatorname{disc}(A)\le2395\left(1+\log_+\frac n9\right)^{1/4}+2\sqrt2$. Here $\log_+t=\max\{0,\log t\}$. Building on Bansal and Jiang's affine spectral independence framework, we remove the $(\log\log n)^{7/4}$ factor from their bound. The fourth root comes from balancing the logarithmic decrease in the alive dimension against the fourth power of the row thresholds. Historical exponential sums control the covariance budget across size classes with summable thresholds. An exact threshold-sum certificate gives the coefficient $2395$, and rounding at most eight remaining fractional coordinates costs $2\sqrt2$. The finite construction also gives partial colourings from any prescribed starting point and at any prescribed depth, preserving existing signs. We formalize the partial- and full-colouring theorems in Lean, including the finite trajectory, exact threshold sum and final rounding, with Bansal--Jiang Theorem A.4 as the sole external research theorem assumption.

math.CO↗
arXiv2026

A Compositional Kernel Model for Feature Learning

We study a compositional variant of kernel ridge regression in which the predictor is applied to a coordinate-wise reweighting of the inputs. Formulated as a variational problem, this model provides a tractable setting for studying feature learning in compositional architectures. From the perspective of variable selection, we show how relevant variables are recovered while noise variables are eliminated. We prove that both global minimizers and stationary points discard noise coordinates when the noise variables are Gaussian distributed. A central finding is that $\ell_1$-type kernels, such as the Laplace kernel, succeed in recovering features contributing to nonlinear effects at stationary points, whereas Gaussian kernels recover only linear ones.

cs.LG↗
arXiv2026

The Levin Method for the Summation of One-dimensional and Multidimensional Infinite Series

The Levin method transforms the evaluation of a highly oscillatory integral into the solution of a first-order linear ODE for a slowly varying auxiliary function. This ODE is typically approximated by collocation, after which the integral value is recovered from the auxiliary function at the endpoints. The present work develops a new extension of the Levin method for the summation of one-dimensional and multidimensional infinite oscillatory series. The summation problem is transformed into the solution of a functional equation involving transformed arguments of the unknown function. The resulting approach is particularly attractive in the multidimensional setting, where the range of existing numerical methods is relatively limited.

math.NA↗
arXiv2026

Countable Graphs with Finite Path-width: Characterisation and Universality

We study path-width and the closely related parameter line-width in countably infinite graphs. Our first result characterises the graphs of finite path-width: they are the graphs that do not have infinitely many vertices of infinite degree, do not have infinitely many pairwise disjoint infinite paths, and contain no subdivision of some finite tree of maximum degree 3. We then investigate universality under the subgraph relation for graphs of bounded path-width or line-width. In particular, we prove that there exists a universal graph with line-width $\mathcal{O}(k^2)$ for the class of graphs with line-width at most $k$. In contrast, we show that no graph of finite path-width is universal for the class of locally finite graphs with path-width $1$. Finally, we show that for each $k\geq 2$, every universal graph for the class of graphs with path-width at most $k$ has line-width at least $k + 1$.

math.CO↗
arXiv2026

Iterative Semantic Decoding for Short Block Codes

This paper proposes an iteratively enhanced semantic receiver for natural-language text transmission over noisy wireless channels using multiple short block codes. At the transmitter, each sentence is permuted by a character-level interleaver, partitioned into segments, and independently encoded by short block codes. At the receiver, we develop an iterative decoder consisting of a channel decoder and a language model, where a de-interleaver between them disperses the burst decoding errors within each segment across the sentence. In each iteration, the language model denoises the channel decoding output, and the denoised characters verified to be consistent with the channel observations are fed back to the channel decoder as semantic information for the next iteration. Simulation results on the Stanford Natural Language Inference (SNLI) corpus over the additive white Gaussian noise (AWGN) channel show that the proposed receiver achieves approximately 1.5 dB block error rate (BLER) gain over conventional short-block coding, while maintaining BLEU and ROUGE scores above 99% at SNRs beyond 1.0 dB.

cs.IT↗
arXiv2026

Set Theory in the Foundation of Math; Internal Classes and External Sets

Usual math sets have special types: countable, compact, open, occasionally Borel, rarely projective, etc. Each such set is described by a single set theory formula with parameters unrelated to formulas. Exotic expressions involving sets related to formulas of unbounded quantifier depth appear mostly in esoteric or foundational studies. Recognizing the internal to math (formula-specified) and external (parameter-based) aspects of math objects greatly simplifies foundations. I postulate that external sets (not internally specified, constituting the domain of quantifiable variables) are hereditarily countable and independent of purely formula-defined classes, i.e. with finite algorithmic information about them. Variables for classes are not explicitly quantified. This opens a way to eliminate all non-integer quantifiers in set theory sentences. The restrictions seem to require almost no changes in math papers, only reinterpreting some formalities.

cs.LO↗
arXiv2026

Tannenbaum's gain-margin optimization meets Polyak's heavy-ball algorithm

This paper highlights an apparent, yet relatively unknown link between algorithm design in optimization theory and controller synthesis in robust control. Specifically, quadratic optimization can be recast as a regulation problem within the framework of $\mathcal{H}_\infty$ control. From this vantage point, the optimality of Polyak's fastest heavy-ball algorithm can be ascertained as a solution to a gain margin optimization problem. The approach is independent of Polyak's original and brilliant argument, and relies on foundational work by Tannenbaum, who introduced and solved gain margin optimization via Nevanlinna--Pick interpolation theory. The link between first-order optimization methods and robust control sheds new light on the limits of algorithmic performance of such methods, and suggests a framework where similar computational tasks can be systematically studied and algorithms optimized. In particular, it raises the question as to whether periodically scheduled algorithms can achieve faster rates for quadratic optimization, in a manner analogous to periodic control that extends the gain margin beyond that of time-invariant control. This turns out not to be the case, due to the analytic obstruction of a transmission zero that is inherent in causal schemes. Interestingly, this obstruction can be removed with implicit algorithms, cast as feedback regulation problems with causal, but not strictly causal dynamics, thereby devoid of the transmission zero at infinity and able to achieve superior convergence rates.

eess.SY↗
arXiv2026

Almost Sharp Equivalence between Approximate Message Passing and Low-Degree Polynomials

We prove a sharp lower bound for growing-degree polynomial estimation in the Gaussian planted submatrix model. The observation is $$ \boldsymbol{Y}= \fracλ{\sqrt{n}} \boldsymbolθ \boldsymbolθ^{\top}+\boldsymbol{W}, $$ where the coordinates of $\boldsymbolθ$ are independent $\mathsf{Ber}(ρ)$ variables and $\boldsymbol{W}$ is symmetric with independent standard Gaussian upper-triangular entries. For every fixed $λ>0$ and $ρ\in(0,1)$, we give an explicit finite-dimensional bound implying that every sequence of polynomial estimators of degree $D(n)=o(n^{1/60})$ has normalized mean-square error with limit inferior at least $ρ-q_{\mathsf{amp}}/λ$, the limiting error of Bayes approximate message passing (AMP). This extends the constant-degree result of Montanari and Wein~\cite{montanari2025equivalence} for the Bernoulli prior. Combined with their polynomial approximation of fixed-iteration AMP, the bound identifies the exact limiting low-degree MMSE whenever $D(n)\to\infty$ within this range. It therefore resolves the Bernoulli rank-one case of the growing-degree AMP-equivalence question discussed in~\cite{wein2025computational, maleki2026high}. The proof constructs a low-degree certificate using \emph{conditional} joint cumulants of the signal coordinates and their products. Specifically, we condition on an auxiliary Gaussian channel $\boldsymbol{R}$ calibrated to the AMP fixed point. This retains signal dependence that is lost in unconditional cumulant bounds and produces the cancellations needed for quantitative control as the degree grows. Most of the arguments in this paper were generated using GPT-6 Astra.

math.ST↗
arXiv2026

Harmonic higher weight distributions, Simonis' approach of MacWilliams identity and moments

We present a combinatorial proof of Simonis type MacWilliams identity for harmonic higher weight distributions of linear codes. Furthermore, we investigate the statistical moments of the harmonic higher weight enumerators for random linear codes. Defining the enumerators via rank functions of the generator matrices of linear codes, we prove that its expectation vanishes for all non-trivial harmonic functions due to the inherent symmetry of random matrices, and we also derive an explicit, non-trivial formula for the covariance.

math.CO↗
arXiv2026

Stabilizing the Rayleigh--Ritz procedure by randomization

Extracting approximate eigenpairs from a prescribed subspace is of fundamental importance in eigenvalue computation. While projecting the target eigenvector onto the subspace yields satisfactory accuracy, extracting an approximate eigenpair that attains a comparable convergence rate has remained a long-standing open problem. Although the standard Rayleigh--Ritz procedure is widely used for this purpose, it may suffer from deteriorated convergence of Ritz values and may even fail to produce convergent Ritz vectors. In this paper, we address this long-standing open problem by introducing a randomized Rayleigh--Ritz procedure whose output converges at a rate similar to the ideal projection. Our analysis requires only the simplicity of the target eigenvalue and extends naturally to nonlinear eigenvalue problems.

math.NA↗
arXiv2026

A Temperature-Coupled Cahn-Hilliard-Stokes-Heat Model for Thermally Driven Phase Separation

We study a diffuse-interface model for thermally driven phase separation in viscous incompressible mixtures. The system couples a convective Cahn-Hilliard equation for the order parameter with a Stokes subsystem for the velocity-pressure field and a heat equation for the temperature. Temperature enters the bulk free energy through a Landau-type coefficient, while the phase field affects the flow through concentration-dependent density and viscosity. The model serves as a proxy for temperature-triggered condensation-like phase separation; humidity, latent heat, vapor pressure, and capillary forcing are absorbed into the choice of the threshold temperature $Θ_S$. We motivate the chemical potential through a temperature-dependent Landau free energy and use a regularized auxiliary formulation to prove local-in-time existence of weak solutions. For the numerical analysis, we employ a first-order sequential finite-element discretization of a simplified quasi-static formulation. The heat equation is advanced by implicit diffusion, the variable-coefficient Stokes problem is treated by a Taylor-Hood discretization, and the Cahn-Hilliard bulk derivative is evaluated at the previous time level, so each algebraic subproblem is linear. An isothermal diffusive test confirms mass conservation to roundoff and exhibits monotone discrete-energy decay for the tested parameters. Time-step and mesh-refinement studies show first-order temporal and approximately second-order spatial behavior. The remaining computations provide qualitative, parameter-specific illustrations; no global discrete energy law is claimed for the non-isothermal sequential scheme.

math.AP↗
arXiv2026

Energy-Consistent Splitting and Decomposition Approaches for Coupled port-Hamiltonian ODEs

Operator splitting provides an attractive approach for the numerical integration of (coupled) port-Hamiltonian systems, as it allows the underlying system structure to be exploited at the level of the individual subproblems. However, the choice of the decomposition is not unique and may strongly affect both the computational efficiency and the preservation of the energy behavior of the original system. In this work, we investigate this interplay systematically and introduce energy consistency as a criterion for assessing splitting methods for port-Hamiltonian ordinary differential equations. We derive sufficient conditions under which a splitting based on a given decomposition inherits the energy behavior of the continuous system and use these conditions to analyze several decomposition strategies for coupled port-Hamiltonian systems. In particular, we compare decompositions that preserve the structure with approaches that exploit lower-dimensional subsystem dynamics or separated time scales. The analysis is complemented by numerical experiments using Strang splitting and its multiple-time-stepping extension. A scalable electro-thermal benchmark with fast electrical and slow thermal dynamics is employed to assess accuracy, energy behavior, and computational efficiency. The results demonstrate that preserving the port-Hamiltonian structure of the subflows is essential for energy-consistent splitting, whereas decompositions that exploit subsystem structure or time-scale separation can provide substantial computational advantages. In particular, the time-scale decomposition yields significant efficiency gains for systems with pronounced multirate characteristics, while structure-destroying decompositions may lead to undesirable energy behavior.

math.NA↗
arXiv2026

Spectra of Non-Self-Adjoint Almost Mathieu Matrices and the Scottish Flag Operator

For $N\geq 3$ and a potential phase $\vartheta\in\mathbb{R}$, we study the non-self-adjoint almost Mathieu matrix obtained by multiplying the discrete Laplacian by a complex phase with angle $φ\in\mathbb{R}$, $A_N(φ,\vartheta)=e^{iφ}(S+S^{-1})/2+\operatorname{diag}(\cos(2πj/N+\vartheta))_{j\in\mathbb{Z}/N\mathbb{Z}}$, where $S e_j=e_{j+1}$ is the periodic shift on $\mathbb{C}^N$. We derive a Chambers formula and isolate the part $Q_{N,φ}$ of the characteristic polynomial that depends only on $N$ and $φ$, but not on $\vartheta$ or on a change of boundary conditions for the shift operator. We then show, for every $N$, that the zeros of $Q_{N,φ}$ lie on the two perpendicular lines $e^{iφ/2}\mathbb{R}\cup e^{i(φ/2+π/2)}\mathbb{R}$. For even $N$, the same property holds for the matrices $A_N(φ,\vartheta)$ with $\vartheta\in 2π\mathbb{Z}/N$, and we compute their limiting eigenvalue measure explicitly. For $φ\in[-π,π]$, the eigenvalue distribution approximates elliptic-integral densities with masses $1-|φ|/π$ and $|φ|/π$, and maximal radii $2|\cos(φ/2)|$ and $2|\sin(φ/2)|$, respectively. At $φ=π/2$, the central polynomial $Q_{N,φ}$ factors into positive quartic factors. This proves that the Scottish flag matrix, after Trefethen and Chapman, has its spectrum on the two diagonal lines of the saltire.

math.SP↗