Papers

We develop positivity certificates for polynomials that are affine in auxiliary parameters, with sum-of-squares multipliers depending only on the principal variables. Under a restricted Archimedean order-unit condition, every strictly positive scalar polynomial admits such a certificate. We construct a convergent restricted moment-SOS hierarchy for polynomial optimization problems with affine parameters and establish a flatness criterion for global optimality detection and global minimizer recovery. The scalar theorem extends to polynomial-matrix constraints with any number of affine parameters. For matrix-valued objectives, pointwise positivity is sufficient with one parameter but can fail with two or more parameters; uniform free positivity provides a general sufficient replacement. Applications include robust linear programming, robust Lyapunov design, min-max polynomial optimization, analysis and control of input-affine dynamical systems. Numerical experiments show that the proposed approach substantially reduces semidefinite block sizes and solution times while incurring little or no degradation in bound quality for most test cases.

We prove the $L^p(\mathbb R^n\times\mathbb R^m)$ boundedness of the rough product singular integral $T_\Omega$, its maximal truncation $T_\Omega^*$, and the product Marcinkiewicz integral $\mu_\Omega$ for $1<p<\infty$, assuming that $\Omega\in H^1(S^{n-1}\times S^{m-1})$ satisfies separate cancellation. These results include kernels in $H^1(S^{n-1}\times S^{m-1})$ that lie outside the Orlicz classes required in earlier work. We also obtain equivalent characterizations of the product Hardy space in terms of restricted, conjugate, and spectral Riesz transforms, together with radial maximal functions, non-tangential maximal functions, and area integrals associated with the spectral and ball Poisson extensions. The operator bounds are proved using restricted Riesz transforms and the Calderón–Zygmund rotation method. For the Marcinkiewicz integral, we further establish integrable bounds for the radial profiles of the transformed kernels.

Decoding signals over unknown channels with minimal pilot overhead is a critical challenge in next-generation communications. Existing deep learning approaches typically rely on generic encoders that struggle to model long-range temporal dependencies or efficiently capture the channel's physical properties from scarce data. We argue that standard architectures suffer from agnostic estimation gaps, as they must implicitly learn the constellation geometry that is already known. We introduce the Constellation-Aware Transformer (CAT), a novel architecture that explicitly injects geometric inductive biases into the equalization process. CAT is composed of a stack of custom TransFIRmer blocks, which use an "early interaction" paradigm to co-process received signals and ideal constellation symbols. Each block features a split Feed-Forward Network that applies a Finite Impulse Response (FIR)-inspired filter for deconvolution and a parallel MLP for geometric refinement. We show that this design is structurally aligned with the optimal linear (MIMO Wiener) receiver: its attention can implement a matched-filter bank, and its bidirectional FIR branch provides the non-causal filtering that block MMSE equalization requires. In the semi-supervised setting, CAT needs fewer pilots than VAE and standard Transformer baselines: on two of our three ISI channels, it reaches a lower SER with 64 pilots than they do with 128.

Assuming a suitably uniform Hardy–Littlewood prime tuples hypothesis, we obtain a second-order asymptotic for the proportion of short intervals containing a prescribed number of primes, when the interval length is comparable to the average prime spacing. The leading term is Poisson, but the arithmetic correction differs from the binomial correction in Cramér's independent model and predicts a stronger bias toward counts near the mean. The proof combines inclusion–exclusion with singular-series estimates of Montgomery and Soundararajan and a finite-sieve argument that controls the required alternating sums as the truncation order grows. We also give a refined random model that reproduces the correction and numerical comparisons that support the prediction.

Information diffusion can be controlled by restricting or removing links (edges) in online social networks, as well as in real-world networks. To identify the most influential links to remove while minimizing diffusion, previous studies have proposed upper bounds for spreading processes in SIR and SIS models, using supermodularity and weighted matrices to identify critical links in contact networks. However, in some cases, existing upper bounds are not sufficiently tight to accurately capture the effect of important edges, as in the SDIR model of [14] (Khanh-Cho-Dung, Proceedings of 40th ICOIN, 2026). We therefore propose a tighter upper bound for controlling diffusion in the SDIR model by directly analyzing the dynamics of the two state vectors D and I in a $2N$-dimensional space. This approach yields an improved spectral-radius convergence condition and outperforms the previous method. Simulations on the synthetic Erdos-Renyi network and the real-world Haslemere dataset using a Greedy edge-deletion algorithm demonstrate its effectiveness for influence minimization on social networks.

For a connected graph \(G\), let \[ Q(G)=\frac{T(G;2,0)T(G;0,2)}{T(G;1,1)^2}. \] We obtain quantitative lower bounds for \(Q\) under the graph join operation. If \(A\) and \(B\) are arbitrary simple graphs of orders \(3\le a\le b\), then \(Q(A\vee B)\) admits an explicit lower bound depending only on \(a\) and \(b\), and this bound is strictly greater than \(1\). We further quantify the improvement produced by edges inside the two factors. For every simple graph \(F\), with \(n=|V(F)|+2\ge4\), we prove \[ Q(K_2\vee F)\ge \frac{27}{n^2}\left(\frac32\right)^{n-4}. \] Consequently, every join of at least three nonempty factors, and every complete multipartite graph with at least one edge and no cut edges, satisfies the strict multiplicative Merino–Welsh inequality. The proofs combine orientation estimates with spanning-tree comparisons based on effective resistance and block elimination.

We prove the full termination of the minimal model program for klt pairs with big boundaries. We also prove the finiteness of marked Mori fiber spaces.

In this work, a globally stiffly accurate Implicit-Explicit (IMEX) Runge-Kutta scheme is developed and implemented in the GBS code [Ricci et al., Plasma Phys. Control. Fusion, 2012], for two-fluid plasma turbulence simulations. The stiffest phenomena, governed by shear Alfvén waves and parallel diffusion, are treated implicitly, while the remaining non-stiff terms are advanced explicitly. This splitting enables time steps well beyond the Courant-Friedrichs-Lewy limit, without incurring the full computational cost of a globally implicit formulation. To efficiently solve the implicit subsystem at each time step, a three-dimensional physics-based preconditioner, inspired by techniques developed in the magnetohydrodynamic (MHD) context, is used. The resulting framework is verified through the method of manufactured solutions and exhibits both algorithmic and parallel scalability. Significant advantages in numerical stability and computational efficiency are demonstrated with respect to an adaptive explicit Runge-Kutta scheme.

We study linearized perturbations of the subextremal Kerr-Newman black hole solution of the Einstein-Maxwell equations. We show how to overcome what Chandrasekhar called the "apparent indissolubility of the coupling between the spin-1 and spin-2 fields" by using a matrix integrating factor for suitably transformed first-order systems that describe the radiative degrees of freedom. For the radial ODEs obtained after angular separation, we prove mode stability on the real axis, i.e., the absence of quasinormal modes with real frequencies, using a Whiting-type transform.

Spectral zeta functions provide a standard tool for regularizing determinants of differential operators in quantum physics. In a previous paper (J. Math. Phys. 66: 083505, 2025), we introduced a $p$-adic spectral zeta function for discrete spectra via a locally analytic interpolation function. In this paper we extend the framework to continuous spectra using the inverse Stieltjes transform. Starting from the resolvent of a bounded operator, we construct a generalized distribution, called the $p$-adic spectral distribution, and define bosonic and fermionic zeta functions and functional determinants. We establish their analytic properties, special value formulas, and Stirling expansions, and show that our previous framework is embedded into the present one in the discrete case. In this approach, the bosonic and fermionic cases exhibit an interesting symmetry. As an application, we consider the position operator in $p$-adic quantum mechanics on an arbitrary compact subset of $\mathbb{C}_{p}$.

Given a compact complex surface with a nonflat Ricci-flat Kähler metric, we show that its holomorphic tangent bundle admits an RC-positive Hermitian metric. The proof relies on a characterization of RC-positivity through the anti-self-dual part of the Weyl operator, the Weitzenböck formula on the four dimensional Einstein manifold, and a conformal perturbation on the Ricci-flat metric. As a consequence, we prove that RC-positivity is not preserved after taking tensor, exterior, or symmetric power. Moreover, we show that RC-positivity is a strictly weaker notion than uniform RC-positivity, and that RC-positivity of holomorphic tangent bundle does not necessarily imply rational connectedness of the base manifold.

This paper proposes an adaptive model reduction approach based on nonlinear participation factors (NPFs) which determines the most effective selection of unimportant generators in a power system to be linearized to accelerate time-domain simulation for the entire system. The method enables a dynamic transition between the full-order model, and a hybrid reduced model. It uses modal energies to rank system modes under contingencies, and computes NPFs for highly energized modes based on Normal Form theory. To accelerate the computation of NPFs and make it achievable online, a tensor contraction technique is introduced. The proposed approach is tested on a 48-machine, 140-bus NPCC system using both partitioned and unpartitioned strategies. It demonstrates significant simulation speedup while preserving better accuracy than a linear participation factor-based method if the nonlinear behaviors of the system cannot be ignored, especially when a near-resonance condition is presented.

We prove log-concavity for the determinant-weighted external semi-activity polynomials of all real flat arrangements, strengthening their known trapezoidality. In fact, we establish a quadratic coefficient inequality that, in rank at least two, implies power concavity with an explicit rank-dependent exponent. The proof uses a new mixed-volume representation of the coefficients and the Alexandrov–Fenchel inequality. A more general formula gives a factorization and log-concavity for related mixed-volume sequences. As applications, we establish the conjectured log-concavity for spanning-tree polynomials of Eulerian digraphs, extend it to positive circulation weights, and strengthen the coefficient inequalities for Alexander polynomials of special alternating links.

In an earlier paper, M. Talagrand made a number of daring conjectures. An AI model provided a very beautiful proof of one of them which we explain here.

We give a short, self-contained proof of Bernstein's theorem: every entire minimal graph in $R^3$ is a plane. The only global tool is the Omori-Yau maximum principle, which holds on entire minimal graphs with their induced metric. We apply it exactly once to a single explicit function $F$ built from the surface's angle function and principal curvatures. A pointwise differential inequality for $F$, combined with the Omori-Yau principle, forces it to attain its minimum value everywhere, which yields the planarity of the surface and hence Bernstein's theorem. As a by-product of our approach, we show that the same method, with a different function and with some extra work, also proves in a simple and self-contained way the generalization of Bernstein's theorem given by Osserman that a complete minimal surface in $R^3$ whose normals omit a neighbourhood of some direction must be a plane.

We introduce the $m$-ary recursive tree, a tree structure that starts with a single node. At each step of the growth process, a random node, called the recruiter, is selected, and $m$ new nodes are attached to this recruiter. The main objective is an asymptotic analysis of distances in this structure. We first derive the distribution of the depth of nodes in such a tree as a convolution of independent (though not identically distributed) Bernoulli random variables. This representation yields normal and Poisson approximations with quantified errors for the depth. We also investigate the height of these trees, that is, the longest root-to-leaf distance, via a coupling to a weighted random recursive tree with suitably chosen weights. The coupling transfers the results on a strong law and tightness from the weighted recursive tree to the $m$-ary recursive tree. Furthermore, we establish a bivariate limit law for the Wiener index, that is, the sum of all pairwise distances, and the internal path length, that is, the sum of the depths of all nodes, using the contraction method. As a novel technical feature of our approach, we employ weighted norms instead of Euclidean norms within the contraction method.

For every integer $\nu\geq 2$, we construct a $(\nu+1)$-dimensional, locally compact, geodesically complete CAT(0) space of asymptotic rank $\nu$ which does not satisfy a linear isoperimetric inequality for integral $\nu$-cycles. Its filling function is bounded below by $c v\log v$ for all sufficiently large $v$.

Let $X$ be a fixed complex K3 surface and let $0\ne\alpha\in\overline{Kah_X}$ be a $(1,1)$-class on the boundary of the Kähler cone with $\alpha^2=0$. We build a novel connection between two seemingly unrelated, intensively studied problems: the rigidity of a closed positive current in the class $\alpha$ and the collapse geometry of a sequence of Ricci-flat Kähler metrics in the Kähler class $\alpha+t\kappa$ as $t\to 0^+$. More precisely, we prove that $\alpha$ contains a unique closed positive $(1,1)$-current if and only if the Ricci-flat metrics in class $\alpha+t\kappa$ collapse to a point in the Gromov–Hausdorff sense. Then we exhibit the first example of a nonrigid irrational nef class $\alpha$ on a Kummer surface, which answers a question of Filip–Tosatti and Sibony–Soldatenkov–Verbitsky. We also construct many new examples of rigid nef classes on hyperkähler manifolds, which extend the work of Filip–Tosatti and Sibony–Soldatenkov–Verbitsky.

Differentiable optimization brings the structural guarantees of mathematical optimization to network pipelines, allowing them to be trained end-to-end. However, its application remains challenging for nonconvex constrained problems, as existing differentiable solvers often suffer from limited modeling expressiveness due to their reliance on specialized problem structures, while also incurring substantial computation time and memory overhead in both the forward and backward passes. To address these challenges, we propose lapanda, a matrix-free differentiable solver for nonconvex optimization with general constraints. It reformulates the problem to a sequence of augmented Lagrangian subproblems, each handled by a first-order inner solver through a proximal averaged quasi-Newton algorithm with adaptive linesearch, thus enabling efficient forward optimization. We establish local well-posedness of the solution map and convergence of the outer iterations, and further derive a sensitivity alignment between the original problem and the final subproblem in the backward pass, demonstrating that the subproblem sensitivity, which can be computed efficiently in a matrix-free manner, provides a principled approximation to the exact optimizer sensitivity. We evaluate lapanda on nonconvex constrained Rosenbrock benchmarks, imitation learning with several representative constrained optimal control problems, and embedded robotic obstacle-avoidance tasks. Compared with state-of-the-art differentiable solvers, lapanda delivers substantial reductions in computation time and memory footprint while maintaining reliable constraint satisfaction and learning performance.

Smoothing empirical data can fill gaps between observations, while noise in the surrounding space can move mass away from the manifold supporting the data. We derive Wasserstein error bounds for ambient Gaussian smoothing and for smoothing through an encoder–decoder representation. For a compact connected smooth manifold without boundary and a smooth positive target density, established heat-flow and entropy estimates imply that sufficiently small intrinsic heat smoothing improves the squared intrinsic Wasserstein error. Coupling this estimate with ambient noise yields a bound whose quadratic term includes ambient dimension and a curvature correction. Our main latent bound tracks reconstruction error, tangential distortion, and the decoder's response to orthogonal latent noise. Its quadratic surrogate gives bandwidth rules and explicit conditions under which the latent upper-bound surrogate is smaller than the ambient one. These surrogate comparisons do not assert an ordering of the exact Wasserstein errors. Synthetic experiments examine the predicted scaling and latent geometry on spheres and a product of spheres.

We give a construction of a Hamiltonian diffeomorphism of the equal-area symplectic product $S^2\times S^2$ with two fixed points. The argument starts with a four-fixed-point map, passes to rescaled cotangent coordinates near the antidiagonal, and reduces the local fixed-point equations to two planar saddle problems. A relative modification merges each pair of saddles, and mixed generating functions are used to transfer the modification to the original map. We include detailed explanations of the normal form along a level arc, the Moser equation, uniform existence of mixed generating functions, the order of parameter choices, and the global graph argument. This construction gives a counterexample to the degenerate version of Arnold's conjecture proposed by Arnold\cite{Arnold86} when the manifold is standard $S^2\times S^2$. This work is done with ChatGPT. The author gave the idea to study the limit behavior of $\Phi_\rho$. GPT suggested a key idea of using second generating function to construct this Hamiltonian. Also, most of the computation is done by GPT.

We study the maximum sum rate of uniquely decodable code pairs. A weighted refinement of a complement-gluing construction yields an explicit code of length 672 and sum rate exceeding $1.318639029203$. An iterated projection argument, combined with a conditional entropy estimate and a justified continuous limit, gives an upper bound of $1.480063425539$.

Reduced descriptions of unresolved fluctuations are commonly based on Gaussian processes, although many realistic forcings have finite correlation times and a renewal structure. For memoryless step (Kubo–Anderson) noise, we show that the reduced dynamics admit two exact and complementary descriptions: a kernel representation governed by the Boolean cumulants of the jump distribution, and a frozen-noise representation adapted to stationary probability densities. For exponentially distributed waiting times, the totally time-ordered $G$-cumulants coincide with the Boolean cumulants of the jump law, and the memory kernel is resummed exactly as the Boolean generator $\eta$ evaluated on a resolvent operator. Second-order closure is exact if and only if the jump law is symmetric Bernoulli; otherwise, the leading closure error is controlled by $(b_4/b_2)\lambda^2$. The Boolean hierarchy, however, acts on the kernel, not on the stationary measure. Stationary densities are approximated by replacing the jump law with its $N$-point Gauss quadrature: each surrogate preserves all multi-time correlations up to order $2N-1$ for any waiting-time law, its kernel is a Padé resummation of the Boolean one, and it is itself an exactly solvable renewal problem. For the linear system with linear multiplicative interaction (LIMI/CAM), the frozen-noise representation reduces the dynamics to a random affine recursion and yields exact support boundaries, singularity exponents and Kesten tail indices, confirmed numerically. Rates, moments and closure errors are governed by the Boolean hierarchy, whereas stationary densities are determined by the geometry of the jump distribution. For memoryless renewal noise, the combinatorial structure controlling finite-correlation reductions is Boolean.

Integrated sensing and communication (ISAC) enables joint communication and sensing using a shared waveform, but its signal design is challenging due to the inherent trade-off between the two objectives, particularly in the short blocklength regime. This paper proposes an autoencoder (AE)-based framework for ISAC waveform design in noncoherent settings. We derive a modified Cramér-Rao bound for multi-target delay estimation and analyze the maximum-likelihood decoding rule for noncoherent communication under correlated fading. These results reveal structural connections and trade-offs between communication and sensing objectives in waveform design. Based on this analysis, the AE learns waveform representations that jointly optimize both functionalities, with a tunable parameter controlling the trade-off. Simulation results show that the proposed design outperforms conventional schemes in both communication reliability and sensing accuracy, especially under short blocklength and fading conditions.

For every nonnegative integer $n$, we construct a commutative total ring of quotients of weak global dimension exactly $n$. For $n\geq1$ the examples are reduced and non-coherent. The construction applies to any nonzero local ring $(A,\m)$: one adjoins countably many residue-field coordinates subject to eventual agreement with the residue of the $A$-coordinate. The resulting ring $\T(A)$ satisfies $Q(\T(A))=\T(A)$ and $\wgd\T(A)=\wgd A$. Its prime spectrum, maximal localizations, nilradical, Jacobson radical, and idempotents are described explicitly. For every pair of modules, positive-degree Tor over $\T(A)$ is naturally identified with Tor over $A$ after quotienting by a projective ideal generated by orthogonal idempotents. We prove that $\T(A)$ is coherent if and only if $A$ is a field. Specializing to regular local polynomial rings yields the prescribed dimensions, with explicit Koszul witnesses for the lower bounds. We also examine the minimal spectrum, the distinction between global and local Prüfer conditions, and a finitely presented cyclic module of flat dimension one and projective dimension two.

A Belyĭ function on a smooth projective algebraic curve defined over a number field determines a bipartite graph called a dessin d'enfant. We study the regularity and automorphism groups of dessins with uniform passports. In previous papers, we proved that every passport of the form $[n,n,n]$, $[n,b^{q},n]$, or $[b^{q},b^{q},n]$ of genus at least $2$ admits a dessin with trivial automorphism group. In this paper, we prove the analogous result for passports of the form $[a^{p},b^{q},n]$. The proof is mainly based on a counting argument: we compare a lower bound for the number of permutation representations having the prescribed passport with an upper bound for the number admitting a nontrivial automorphism. Together with our previous results, this shows that every uniform unicellular passport of genus at least $2$ admits a dessin with trivial automorphism group.

For a nonempty acyclic oriented graph $H$, let $p(H)$ be the order of a longest directed path and let $\tau(H)$ be the least positive integer $n$ such that $H$ admits a homomorphism to every tournament of order $n$. For all $p\ge3$ and $g\ge1$, we construct a connected acyclic oriented graph $H$ with underlying girth greater than $g$, absolute and relative oriented clique numbers equal to three, and \[ p(H)=p,\qquad \tau(H)=r_{\mathrm{tr}}(p), \] where $r_{\mathrm{tr}}(p)=2^{\Theta(p)}$ is the tournament Ramsey number for a transitive $p$-vertex tournament. This disproves the conjectured polynomial bounds under bounded absolute or relative oriented clique number. It also shows that a forbidden graph can yield a polynomially $\tau$-bounded class only if its underlying graph is a forest. For fixed $g$, the least order of these examples is bounded by a polynomial in $p$. A separate construction gives maximum in- and outdegree $O(p^2)$, uniformly in $g$. For $p=4$, the least order is $2^{\Theta(g)}$. We also establish polynomial $\tau$-boundedness for every orientation of the two four-vertex trees. The pure-claw case follows from the known $O(p^4)$ bound. We obtain the bound $2p-2$ for mixed claws and one-turn orientations of $P_4$ when $p\ge2$, and bounds $4$ and $3p-2$ for the directed and alternating orientations of $P_4$, respectively. In the alternating case, $\tau(H)=p(H)$ when the underlying graph is triangle-free.

We study the ergodic integrals for linear flows on the infinite staircase, a translation surface with infinite area, and deduce detailed information on the bias properties of local discrepancy functions for irrational numbers of bounded type.

We develop a Hopf-algebraic theory of graph coefficients arising in the quantum Magnus expansion. At the classical level, the expansion is governed by directed trees, whereas its quantum counterpart involves loop graphs, multiple edges, and several types of edge data. We introduce a contraction Hopf algebra of mixed quivers containing these diagrams and closed under contraction. On this algebra we define a character $e$ from normalized linear-extension data, directed-tadpole factors, and a prescription for undirected edges, and let $\omega$ be its convolution inverse. Our main result is a universal closed formula for the connected function $\omega_c$ on every finite mixed quiver, expressed as a finite sum over ordered set partitions of the vertices. We also construct a weighted vertex-splitting Hopf structure on the unlocalized mixed-quiver algebra, with convolution $\star_V$ and canonical edge-weight character $\zeta_V$, such that $\omega_c=\log_{\star_V}\zeta_V$. Compatibility of the two coproducts and an explicit identity between their characters explain this equality. We further prove that the two-vertex contraction identity, tadpole factorization, vanishing on disconnected graphs, and boundary data uniquely determine $\omega_c$. For acyclic quivers, the formula reduces to a permutation formula with coefficients depending only on descent numbers. We derive this formula independently from the quantum Magnus expansion using operator products and Wick contractions, showing that the physical graph coefficients are governed by the convolution structure of the contraction Hopf algebra. Further consequences include a refined quantum Murua formula and orientation-sum identities related to Tutte and chromatic polynomials.

For a line arrangement $\mathcal{A} \colon f=0$ in $\mathbb{P}^2$, let $\nu(\mathcal{A})$ be the maximal dimension of a graded piece of the Jacobian module of $f$. We study the conjecture that $\nu(\mathcal{A})$ depends only on the intersection lattice of $\mathcal{A}$. It is known that $\mathcal{A}$ is free if and only if $\nu(\mathcal{A})=0$, and hence this conjecture is a strengthening of Terao's conjecture for line arrangements. We show that the conjecture holds for arrangements of at most $13$ lines, except possibly for arrangements of exactly $13$ lines whose intersection points have maximal multiplicity $5$.