Papers

We construct arbitrarily large finite sets $A$ of real algebraic integers such that both $|A+A|$ and $|AA|$ are at most $|A|^{1.95835}$. The construction combines truncated ideal-smooth $S$-unit fibres with a coprime additive factor. A tensor-product rank argument, using a two-dimensional local feature at each selected prime ideal, controls the loss in the additive factor, while a direct estimate for an outer parallel body improves the sumset packing bound. The arithmetic input is an unramified pro-$2$ tower over a known degree-ten field, with simultaneous Frobenius cuts controlling the small prime ideals. We use unconditional Tsfasman–Vlăduţ inequalities for the joint class-number–regulator cost. All finite numerical comparisons entering the exponent are certified by outward rational interval arithmetic. No unproved hypothesis is used.

This paper studies data-driven stabilization of unknown discrete-time linear systems using only quantized state measurements that take values in finite sets. The proposed approach consists of two stages. In the controller design stage, we collect quantized data while maintaining a prescribed quantization error bound and use them to formulate a semidefinite program (SDP). We establish a verifiable condition under which any feasible solution to the SDP yields a stabilizing feedback gain, and show that a stabilizing gain can always be obtained when the quantization error is sufficiently small. In the stabilization stage, a Lyapunov-based quantizer update rule is developed to guarantee exponential convergence under quantized state feedback. As a key feature, the number of quantization cells remains finite in both stages and constant during stabilization. Simulation results illustrate the effectiveness of the proposed approach.

Given incomplete measurements of a single physical field in a coupled system with unknown parameters, can we infer its full physical state and identify the underlying parameters? This problem is challenging because multiple coupled fields must be reconstructed simultaneously from limited observations of only one, while the system parameters are unknown. In this work, we propose a machine learning framework for full-field reconstruction and parameter identification of unknown physical systems from sparse observations of a single physical field. Specifically, the cross-attention encoder propagates sparse sensor observations onto a regular grid to construct a sensor-conditioned latent representation, while a Fourier neural operator (FNO) decoder captures global spatial dependencies to reconstruct all coupled physical fields. The network parameters and unknown physical parameters are jointly optimized by minimizing observation losses, governing equation residuals, and boundary/initial condition constraints. The proposed approach is validated on two- and three-dimensional lid-driven cavity flows, a two-dimensional cylinder wake, and a two-dimensional non-ideal magnetohydrodynamics problem, demonstrating the recovery performance of unobserved fields and physical parameters from incomplete observations.

Let $G$ be a simple graph on $n$ vertices, and let $F(G)$ denote the number of its spanning forests. Bencs and Csikvári [Upper bound for the number of spanning forests of regular graphs, European J. Combin. 110 (2023) 103677] proved that every $r$-regular graph $G$ with $r\geq 2$ satisfies $F(G) \leq r^{n}$. They further conjectured that for $r \geq 3$, \[ F(G)^{1/n} \leq \frac{(r - 1)^{r-1}}{(r^2 - 2r - 1)^{r/2-1}}. \] In this paper, we resolve this conjecture in the affirmative.

It has been known that (the square of) Jacobi elliptic function can be expressed by an infinite sum of single solitons of $\sech^2$ shape. In this paper, we show that there exists a similar structure for higher genus cases. It turns out that this is just a consequence of the quasi-periodicity of the Riemann $\theta$-function. More precisely, we show that the second derivative of $\log \theta$ with the real Riemann $\theta$-function of genus $g$ near soliton limit can be well approximated by the sum of real and regular $g$-soliton solutions of Hirota-type in $g$-dimensional real space $\R^g$. This leads to a tessellation of $\R^g$, whose tile is an oblique prism divided into $2^g$ sections by hyper-planes of dominant exponents in the theta function. We apply the results to study quasi-periodic solutions to the KdV and KP equations. We construct the quasi-periodic solutions using the Schottky group, which uniformizes the corresponding Riemann surfaces. We also discuss solitons on quasi-periodic background by pinching some of the homological cycles

We introduce a framework for multi-bit text watermarking with security defined directly through $f$-divergence from the base language model distribution. Unlike prior approaches that focus on average-key distortion-freeness or a particular statistical distance, our formulation supports general $f$-divergences, including total variation and KL divergence, and enforces the guarantee for each realized key and embedded message. We develop a coding-based watermarking scheme that optimally biases next-token distributions subject to a prescribed divergence budget, and characterize the resulting tradeoff between embedding rate, decoding reliability, and statistical security. Experimentally, we compare our method against prior multi-bit watermarking schemes across modern language models and payload regimes. Our approach achieves substantially lower watermark detectability while maintaining competitive message-recovery performance and generation quality. Our results provide a unified view of secure multi-bit watermarking and recover several commonly used security notions as special cases.

We study the field concentration phenomenon between two closely spaced inclusions with imperfect bonding interfaces of low conductivity type. The inclusions are assumed to have finite conductivities. The problem is governed by a system of elliptic equations coupled with Robin-type boundary conditions. While it is known that finite-conductivity inclusions with ideal interfaces yield bounded gradients, in this paper we show that with imperfect bonding interfaces, the gradient of the solution may blow up as $\varepsilon$ (the distance between two inclusions) tends to zero when the bonding parameter $\gamma$ is large, while it is uniformly bounded independently of $\varepsilon$ when the bonding parameter $\gamma$ is small. Moreover, we identify the threshold of $\gamma$ and the optimal blow-up rates under certain symmetry assumptions. Compared to the case when the inclusions are perfect conductors, we find a novel logarithmic blow-up phenomenon at the critical value of $\gamma$.

We study $W$-free planar graphs for $W\in\{W_{2,4},W_{2,5},W_{3,4}\}$, where the quasi-double star $W_{h,k}$ is obtained from a three-vertex path by attaching $h$ leaves to one endpoint and $k$ leaves to the other. We prove that every $W_{2,4}$-free planar graph on $n$ vertices has at most $9n/4$ edges, and the bound is attained whenever $8\mid n$. This determines the planar Turán density of $W_{2,4}$ as $9/4$. We also establish the sharp upper bound $5n/2$ for $W_{2,5}$. Combined with known constructions of planar graphs of maximum degree five, it yields $\ex_{\PP}(n,W_{2,5})=\lfloor5n/2\rfloor$ for every $n\ge15$. These results close the two corresponding coefficient gaps in the bounds of Liu et~al. Our proofs use structural restrictions on high-degree vertices, local deletions, and degree deficits in neighborhoods of radius two. For $W_{3,4}$, we characterize the planar graphs with a dominating vertex that avoid this tree and determine their exact extremal number, $\lfloor(5n-7)/2\rfloor$, for every $n\ge10$. Finally, $W$-free planar triangulations have at most eight, twelve, and eleven vertices for $W=W_{2,4},W_{2,5},W_{3,4}$, respectively; the first two bounds are sharp.

This paper study observability for the Dirichlet heat equation on a bounded interval with a real bounded potential depending on space and time. We prove an observability inequality from every measurable subset of spacetime with positive measure. The constant is uniform over potentials with a prescribed $L^\infty$ bound. By duality, positive spacetime measure is equivalent to null controllability by square-integrable distributed controls. Controls can also be chosen bounded in time with values in $L^2$. If almost every spatial slice of the observation set has a fixed positive lower bound on its measure, the observability constant is bounded above by $C_0e^{C_1/T}$, uniformly in the locations of the slices. The proof combines observation estimates on finite-dimensional spaces transported by the evolution with decay estimates for the distance to those spaces.

Constant curvature is one of the most fundamental intrinsic conditions in submanifold geometry. Investigating the existence and classification of biharmonic submanifolds under intrinsic curvature constraints is generally difficult, since the biharmonic equation is expressed in terms of extrinsic data. In this paper, we obtain a complete classification of biharmonic Lagrangian surfaces with constant Gaussian curvature in 2-dimensional complex space forms. As a consequence, we classify the biharmonic Lagrangian surfaces with flat normal bundle and we prove that pseudo-umbilical biharmonic Lagrangian surfaces are necessarily minimal.

We study the large time asymptotics of the Feynman-Kac semigroups of the symmetric Lévy process on unbounded open sets. Our main result proves the exact exponential asymptotic decay rate for the survival probability given in terms of the bottom of the spectrum of the associated nonlocal Schrödinger operator. We also prove quantitative upper bounds with an explicit polynomial correction. The proof is probabilistic and is done by decomposing the Lévy process into a finite-range jump process collecting the small jumps and an independent compound Poisson process describing the large jumps. Our approach gives a direct link between the spectral properties of nonlocal Schrödinger operators and pointwise decay of survival probabilities, which was previously unknown for jump processes.

A $k$-substring edit in a sequence first deletes a substring of length at most $k$, and then inserts a sequence of length at most $k$ at the same position. A code that can correct a $k$-substring edit is called a $k$-substring edit code. In this paper, we develop a new localization method and use it to construct a $q$-ary $k$-substring edit correcting code with $\log n+8\log\log n+o(\log\log n)$ bits of redundancy for any fixed $q\ge2$ and $k\ge1$, where $n$ is the code length. For the binary alphabet, this improves upon the redundancy $\log n+16k\log\log n+o(\log\log n)$ obtained by Li et al. When the deleted substring and the inserted sequence have different lengths, we further construct codes with redundancy $\log n+O_{q,k}(1)$, which is optimal up to an additive constant. As corollaries, for all fixed $q\ge2$ and $1\le t\le T$, we obtain $q$-ary $(\le t)$-burst-deletion correcting codes and $(t,T)$-localized deletion correcting codes with redundancies $\log n+O_{q,t}(1)$ and $\log n+O_{q,T}(1)$, respectively. To the best of our knowledge, these are the first constructions attaining optimal redundancy up to an additive constant for these two deletion models over the full range of fixed parameters. For $(\le t)$-burst-deletion correction with $t\ge2$, such redundancy had previously been achieved only for $q=t=2$ by Levenshtein in 1967.

We study state-constrained stochastic control problems with a scalar surplus $R \geq 0$. Controls for which the drift and volatility of $R$ vanish at zero define a lower-dimensional Hamilton-Jacobi-Bellman equation. Starting from control-wise state-constraint viscosity inequalities, we prove that the upper and lower limits of the interior value are a subsolution and a supersolution of this boundary equation. Terminal compatibility and comparison identify their common limit. We require one-sided bounds on positive surplus drift and generator growth, together with domination by boundary generators. For compact or coercive controls, these follow from a local lower bound and continuity of the boundary control set. For a smooth stochastic target value $w$, the transformation $R = Y - w(t,X)$ flattens the viable epigraph, and, under the stated hypotheses, the boundary control value supplies the Dirichlet datum. Applications include set-valued boundary controls, unbounded drift with quadratic costs, and redundant hedging instruments with a nonconstant target boundary.

We study the offline gap between deterministic calibration distance C and its fractional relaxation L for binary unit-weight sequences under total absolute-change cost. We sharpen the offline comparison C <= L + O(sqrt(T)) (Qiao and Zheng, 2024, Theorem 2) to the sharp worst-case order Theta(T^(1/3)). If Delta_T is the supremum of C - L over length-T inputs, then T^(1/3)/1000 <= Delta_T <= 41T^(1/3) for T >= 216. The upper bound holds for every input, while each T >= 216 has a rational lower-bound input. For every input with m distinct forecasts, C <= L + m, and the unrestricted-sample worst-case sparse order is Theta(m). For rational forecasts and accuracy, with binary-encoded multiplicities of separately assignable unit identities, a grid-free polynomial-bit-time procedure returns B <= L <= U, U - B < eta, and an exactly calibrated compact repair of cost at most U + m <= L + m + eta.

Feller random measures generalize the Feller diffusion (the CIR process) by giving it memory. They arise as the scaling limits of nearly unstable Hawkes processes, and include the rough CIR process and its hyper-rough and discontinuous relatives. We show that every Feller random measure is the occupation measure of a Dawson–Watanabe superprocess whose spatial motion is a killed Lévy subordinator, integrated over the branching time. This is a continuum analogue of the Hawkes–Oakes cluster representation. It yields existence, stability in the parameters, and stochastic equations driven by an explicit noise. The structure of this noise depends on whether the kernel has an atom at the origin. Without an atom, the noise is a Brownian motion time-changed by the distribution function of the measure. This leads to a martingale characterization and to sharp results on densities and their Hölder regularity. With an atom, the noise is a compensated inverse-Gaussian process time-changed by the compensator of that distribution function, and the measure is purely atomic.

In this paper, a complete characterization of the boundedness, compactness, norm and essential norm of the differential Hilbert operator $\mathcal{H}_2:A^2_\alpha\to A^2_\beta$ is obtained. More precisely, $\mathcal H_2:A^2_\alpha \to A^2_\beta$ is bounded if and only if $-1<\alpha<0$ and $\beta\geq\alpha+2$ and it is compact if and only if $-1<\alpha<0$ and $\beta>\alpha+2$. The norm and essential norm of $ \|\mathcal H_2\|_{A^2_\alpha\to A^2_{\beta}}$ are also investigated. In particular, when $\beta=\alpha+2$, \[ \|\mathcal H_2\|_{A^2_\alpha\to A^2_{\alpha+2}} =\|\mathcal H_2\|_{\mathrm e,A^2_\alpha\to A^2_{\alpha+2}} =\frac{\pi\sqrt{(\alpha+2)(\alpha+3)}} {\sin(\pi(\alpha+2)/2)},\qquad -1<\alpha<0. \] When $\beta>\alpha+2$, $\|\mathcal H_2\|_{\mathrm e,A^2_\alpha\to A^2_{\beta}}=0$. Furthermore, for every $1\leq p<\infty$, the operator $\mathcal H_2$ belongs to the Schatten class $\mathcal S_p$ if and only if it is compact.

In this paper, we provide an algebraic proof of the condition for the existence of modular form solutions to the Kaneko-Zagier differential equation of weight $k$. Unlike previous representation-theoretic approaches relying on $SL_2(\mathbb{Z})$, our method employs the connection matrices of the principal congruence subgroup $\Gamma(N)$. By deriving a condition for simultaneous triangularizability from the commutators of the representation matrices and applying Galois theory, we prove that the associated two-dimensional representation of $\Gamma(N)$ is irreducible when the denominator $m$ of the fraction $(k+1)/6 = n/m$ satisfies $m=1$ or $m \ge 7$. Consequently, we identify the weights that admit no modular form solutions and obtain a complete classification of the dimensions of the spaces of such solutions.

Partly motivated by enumeration of parking functions and their variants, there is a long-standing interest in lattice paths refined by several statistics, including the notable $\mathsf{run}$ and $\mathsf{return}$. We consider bounded lattice paths, which are bounded by a given lattice path and are related to vector parking functions, for which we generalize $\mathsf{run}$ to composition runs, parameterized by a composition. By constructing involutions on these paths, we establish symmetries relating composition runs to some generalized return statistics. As an application, we settle an open problem of Dai, Fu, and Qiu on rational Dyck paths. The symmetries between generalized $\mathsf{run}$ and $\mathsf{return}$ are then transferred to vector parking functions, which suggest a new notion of prime decomposition. In the special case of $(a, b)$-parking functions, we compute the generating function refined by $\mathsf{run}$ and $\mathsf{pri}$, which is new even for classical parking functions.

We study the hitting time index, a graph invariant defined in terms of expected hitting times of simple random walks on finite connected graphs, through its relation with the Kirchhoff index. Using the commute time identity and Tetali's formula, we decompose the hitting time index into a Kirchhoff-index term and a nonnegative asymmetry term expressed by degree-weighted effective resistance sums. We derive lower and upper bounds for the asymmetry term, together with their equality conditions, and obtain a lower bound for the hitting time index in terms of the range of the degree-weighted resistance sums. For trees, we obtain lower and upper bounds for the asymmetry term, with the upper bound expressed in terms of the Wiener index, and characterize the equality case in the lower bound. In particular, we show that among all trees of fixed order, the hitting time index is minimized uniquely by the star graph. Finally, we apply the decomposition to complete bipartite graphs, paths, and graphs obtained by conjoining complete graphs, and recover several known hitting-time formulas.

Let $s(n):=\sum_{d\mid n,\,d < n} d$ denote the sum of the proper divisors of $n$. Distinct positive integers $n,m$ form an amicable pair if $s(n)=m$ and $s(m)=n$. Let $A(x)$ count the positive integers not exceeding $x$ that belong to an amicable pair. We report on a proof, found by ChatGPT Astra, that $$ A(x)\le x\exp\{-(1/2+o(1))\log x\,\log_3 x/\log_2 x\}$$ as $x\to\infty$. Here $\log_k$ denotes the $k$th iterate of the natural logarithm.

We give non-asymptotic operator-norm tail bounds for random multi-matrix models with quadratic plus globally Lipschitz interactions. More precisely, let $\mu_f$ be the probability measure on $(\mathbb{M}_n)_{\operatorname{sa}}^m$ with density proportional to $\exp(-n^2(\lVert x \rVert_2^2/2+f(x)))$. If $f$ is $L$-Lipschitz with respect to $\lVert \cdot \rVert_1$, we construct a coupling of $X\sim\mu_f$ and an $m$-tuple $Z$ of independent GUE matrices such that $\lVert X-Z \rVert_\infty\leq L$ almost surely, where $\lVert \cdot \rVert_\infty$ is the maximum of the operator norms of the matrix tuple. This is an operator-norm analog of the $L^\infty$-Wasserstein bound of Khudiakova, Maas, and Pedrotti (2025). We give a new proof based on the variational formula of Boué and Dupuis (1998), the associated Hamilton–Jacobi–Bellman equation, and the duality between entropy and pressure.

In this paper we consider the 4th order (or biharmonic) nonlinear Schrödinger (NLS) equation in dimensions 1, 2, and 3, where the potential term is expressed as a power non-linearity (for any positive power) and the dispersion operator has the fourth order combined with the lower second order. The fourth order NLS equation has recently attracted attention of researchers since quartic solitons have been experimentally obtained in optics, and thus, mathematical theory of solutions to the 4th order NLS equation in physical dimensions $N=1,2,3$ is timely to develop. In this work we show the local well-posedness of the 4th order or bi-harmonic NLS equation on a weighted subset of a Sobolev space as well as in $H^s$ spaces.

For a finite group $G$, we define a constant spectral $G$-Mackey functor as a genuine $G$-spectrum in which the restrictions between categorical fixed points are equivalences. We further identify the full subcategory of constant spectral $G$-Mackey functors with the category of modules over a certain $\mathbb{E}_1$-ring spectrum $R_G$. When $G=C_{p^n}$ is a cyclic $p$-group, we provide an explicit construction of the $\mathbb{E}_1$-ring spectrum $R_{C_{p^n}}$. The proof uses a reconstruction theorem for genuine $C_{p^n}$-spectra, which might be of independent interest. As an application, we compute the $\mathbb{Z}$-graded $C_{p^n}$-equivariant mod $p$ Steenrod algebra.

This work is devoted to a systematic study of slope stability for $\mathbf Z_p$-towers of the projective line over finite fields of characteristic $p$, ramified only at infinity. Our aim is to understand to what extent geometric stability of the genus sequence, or of the ramification break sequence, implies arithmetic stability of the Newton slopes of the zeta functions of the curves in the tower. We prove positive results and construct counterexamples that clarify the boundary between slope stability and its failure, with sharp bounds in several cases. For towers with eventual minimal ramification break ratios, a condition stronger than genus stability, Kosters and Zhu proved slope stability under an additional degree-gap condition. We construct towers with minimal break ratios that are not slope stable, answering negatively both their question of whether the gap can be removed and their broader question of whether genus stability implies slope stability. Under the minimal-break hypothesis alone, we prove exact partial slope stability on growing intervals approaching the first block, with uniform comparisons over explicit boundary annuli and finite flat spectral pieces. Further counterexamples distinguish partial from complete first-block propagation, first-block from two-block propagation, and full classical stability from a uniform annular slope law below one. We obtain improved bounds for the classical degree-gap constant, the partial-stability loss, and the sufficient starting levels, all independent of the constant-field degree. The proof uses a separate precision theorem for the semilinear Frobenius on finite contact blocks.

When auction bidders misperceive their rivals, a seller can change not only the amount of information rent but also the states in which it is delivered. We ask when this payment-design opportunity survives a finite bidding interface, nonnegative-payment restrictions, and comparison with conventional auctions under the same beliefs. We formulate payment design as a linear program in nonnegative bidder rents. A single state can safely carry a report's rent when it is both cheapest for its intended type and least attractive to potential mimics. We characterize this alignment, bound the cost of a restricted payment architecture when alignment is imperfect, and use shadow prices when the rankings conflict. Own-type projection yields a unique optimal expected-payment rule for a fixed allocation. Contiguous bid bands implement truthful reporting at every value, but create pooling losses and within-band surplus. Under a separate smooth-belief model, feasible truthful rules and revenue upper bounds support comparisons with a first-price equilibrium under the same beliefs. A calibration using 1,000 eBay proof-set auction records illustrates a partition-dependent reversal: first price dominates all exactly truthful, no-subsidy rules on ten equal-probability bands in one scenario, while a richer contingent rule on ten equal-dollar bands outperforms it. An operator should choose interface resolution, payment dependence, and integrity tolerances jointly. Validate the probabilities that support incentives, compare formats under the same beliefs, and distinguish redesign from deployment. Any receipt advantage must cover implementation costs before it can imply a profit advantage.

We prove the Prym-Green conjecture in every even genus from 20 to 30. The genus 24 case disproves the suggestion of Chiodo-Eisenbud-Farkas-Schreyer that the conjecture might fail in every genus divisible by 8. Our proof uses rational nodal curves with cyclic symmetry to decompose the relevant Koszul differential into a direct sum of smaller maps indexed by characters. We carry out the resulting rank computations over finite fields using the block Wiedemann algorithm.

We prove a uniform scalar curvature bound, depending only on the dimension, for Kähler–Ricci shrinkers. We also prove that every noncompact Kähler–Ricci shrinker has a unique tangent cone at infinity, naturally homeomorphic to the base of its polarized Fano fibration.

For fixed $n$ and $\epsilon>0$, we prove boundedness of projective birational morphisms $f\colon X\to\mathbb P^n$ over $\mathbb C$ for which $(X,B)$ is an $\epsilon$-lc pair, $B\geq 0$, $K_X+B$ is nef over $\mathbb P^n$, and $\mathrm{deg}(f_*B)$ is bounded. The nonzero coefficients of $B$ need not have a positive lower bound, and $(\mathbb P^n,f_*B)$ need not be log canonical. The $f$-exceptional prime divisors on $X$, equipped with their reduced induced scheme structures, form a bounded family of projective schemes, including when they are nonnormal. We also obtain a uniform positive lower bound for the log canonical thresholds of pullbacks of all effective target divisors of bounded degree. The boundedness statement proved in this paper was originally conjectured by Caucher Birkar.

We prove an asymptotically sharp lower bound for the eigenvalues $\lambda_k$ of the Dirichlet Laplacian on bounded open sets in the plane with area $A>0$. The proof employs a trace identity for the commutator between a certain multiplication operator and its adjoint. As a corollary, we obtain $(2/3)\cdot 4\pi k/A\leq \lambda_k$ for every $k$, which is two thirds of the bound conjectured by Pólya.

Let $Q_n=[0,1]^n$, $n\ge2$. Denote by $M_n$ be the maximal volume of a pyramid in $Q_n$ whose base is a hyperplane section of $Q_n$, and by $M_n^0$ the corresponding maximum when the section contains the center of $Q_n$. We prove that $nM_n<(n+1)M_{n+1}^0$.