Papers

Absolutely maximally entangled (AME) states and, more generally, $k$-uniform states in $(\C^q)^{\otimes n}$ are central objects in multipartite entanglement theory, with applications to quantum secret sharing, quantum masking, and quantum error correction. In the extremal case $k=\lfloor n/2\rfloor$, Scott (2004) proved a sharp nonexistence bound showing that AME states cannot exist once the number of parties $n$ exceeds a threshold of order $2q^{2}$ (with a parity dependence on $n$), where $q$ is the local dimension. Recently, Ning et al.\ studied defective AME states (i.e., $k=\lfloor n/2\rfloor-l$ with $l>0$), gave explicit Scott-type bounds for defects $l=1,2$ and conjectured a general $(2l+2)q^{2}+o(q^{2})$ behavior. In this paper, we solve this conjecture and establish a fully explicit Scott-type upper bound for AME states with arbitrary defect $l\ge 0$, yielding Scott's bound for $l=0$ and Ning et al.'s bounds for $l=1,2$ as special cases. Equivalently, this gives nonexistence bounds for one-dimensional pure quantum error-correcting codes near the quantum Singleton regime. The proof uses a truncated MacWilliams linear-programming system and an explicit infeasibility certificate. As a direct application, we derive explicit asymptotic upper bounds on $k/n$ for fixed local dimension $q$, improving the implicit upper bounds given by Ning et al.

We introduce a novel exclusion process with two conservation laws, mass and energy, designed to mimic the essential features of continuous systems like interacting oscillators within the framework of interacting particle systems. This distinguishes our model from conventional multi-species processes where only particle numbers are conserved. As a basis for our fluctuation analysis, we first show that applying nonlinear fluctuating hydrodynamics (NFH) to this model reveals a wide variety of universality classes depending on the parameter choices. The main objective of this work is to study the stationary fluctuations of these conserved quantities. For a suitable choice of parameters, we rigorously show that the fluctuation fields converge to uncoupled stochastic Burgers equations (SBE) in the scaling limit. The proof relies on the second-order Boltzmann-Gibbs principle that we establish for this model, along with the spectral gap estimate and the equivalence of ensembles. Of independent interest is our general proof of the diagonalizability of the Jacobian matrix for the macroscopic current with distinct real eigenvalues. While this property is often taken as given in the physics literature, we establish it rigorously for multi-component systems even when the eigenvectors cannot be explicitly computed, offering a firm mathematical foundation for a broad class of models.

In our previous work, we generalized the concept of Galois points for irreducible plane curves to the case of reduced plane curves. We also introduced the concept of simultaneous Galois points, which is an equivalent concept to Galois points, and studied their number when the irreducible components are nonsingular. In this paper, we consider the remaining cases where the irreducible components are of degree $d=2$ or $3$. For the case of $d=2$, we establish a generalized version of the theorem in our previous paper. For the case of $d=3$, we classify simultaneous Galois points into the first and second kinds. We give a necessary condition for the former and provide examples for the latter.

For a nonnegative integer $k$ and a rational number $r\in\mathbb{Q}^+$, we define the generalized Gaussian binomial coefficient $\qbinom{r+k}{k} = \frac{(q^{r+1}; q)_k}{(q; q)_k}$. When $r=a/b$ with $a,b$ coprime positive integers and $b\geq 2$, expanding $\qbinom{r+k}{k}$ via the finite $q$-binomial theorem produces fractional powers of $q$, so that $\qbinom{r+k}{k}$ is a Puiseux series in $q$ with nonnegative exponents; concretely it lies in $\mathbb{Q}[[q^{1/b}]]$. The notion we single out is the integer trace of this expansion, the subseries consisting of those terms $c_r(d)\,q^d$ whose exponent $d$ is an integer, with all fractional powers discarded. This projection is not standard, and there is no a~priori reason for the surviving coefficients to behave coherently as $r$ varies. Nonetheless, ordering the family by the coefficientwise partial order leads to the \emph{$\tfrac{1}{2}$-Conjecture}: among all $r\in\mathbb{Q}^+$, the value $r=\tfrac{1}{2}$ maximizes the integer trace, in the sense that the coefficients of $\qbinom{1/2+k}{k}$ dominate those of $\qbinom{r+k}{k}$ coefficientwise for every $r$. That so elementary a definition should single out $\tfrac{1}{2}$ this cleanly came as a surprise to us. We prove the conjecture in several special cases and provide further computational evidence.

We study the exact spectrum of the XX spin chain with constrained non-diagonal boundary fields, which can be analyzed by solving the associated Bethe Ansatz equations. In these equations, the number of Bethe roots has a definite parity, and all Bethe roots are located at the zeros of a unary function. We investigate the possible positions of the Bethe roots. Based on numerical observations, we analyze the Bethe root configurations for the ground state and the first excited state. Our results show that elementary excitations are characterized by the cooperative change of a pair of Bethe roots. Furthermore, we obtain an analytical expression for the ground state energy in the thermodynamic limit.

Let $\mathcal P(H)$ be the semigroup obtained by endowing the family of all non-empty subsets of a semigroup $H$ with the setwise operation naturally induced by $H$ on its power set, and denote by $\mathcal P_\text{fin}(H)$ the subsemigroup of $\mathcal P(H)$ consisting of all non-empty finite subsets of $H$. We obtain (as a corollary of a theorem of independent interest) that if $H$ is a group and $K$ is a semigroup, then $\mathcal P(H) \cong \mathcal P(K)$ implies $H \cong K$. The finitary analogue of this statement is considerably more difficult, and we prove it only for $H$ an additive subgroup of the rationals. Most notably, the proof of the second result relies, in a rather circuitous way, on a special case of the Evertse–Schlickewei–Schmidt theorem.

We consider singular problems for a general class of quasilinear hyperbolic systems that involve two a priori independent stiff parameters. We argue that such situations may lead to the rapid development of small-amplitude spatial oscillations of small wavelength starting from arbitrarily smooth initial data. Despite this phenomenon we provide sufficient conditions on initial data that secure the uniform control of solutions and show strong convergence in the singular limit of small stiff parameters. We apply our general theory to the rapidly rotating shallow-water system with bottom topography, and to hyperbolic systems stemming from a constraint-relaxation strategy applied to dispersive models for the propagation of water waves, specifically the Benjamin-Bona-Mahony, Boussinesq-Peregrine and Serre-Green-Naghdi equations.

In this paper, we study the space of compactly supported embeddings between Euclidean spaces, $\mathrm{Emb}_c(\mathbb{R}^j, \mathbb{R}^n)$. By utilizing hairy graphs, we construct elements in the homotopy groups $\pi_{\bullet}(\overline{\mathrm{Emb}}_c(\mathbb{R}^j, \mathbb{R}^{n})) \otimes \mathbb{Q}$ corresponding to certain uni-trivalent graphs in the model. We then prove that these elements are nontrivial. Consequently, we show that the rational homotopy groups of $\mathrm{Emb}_c(\mathbb{R}^{n-2}, \mathbb{R}^n)$ are infinite-dimensional in infinitely many degrees when $n \ge 5$ is odd.

The theory of generalized locally Toeplitz (GLT) sequences is an apparatus for computing the spectral and singular value distribution of sequences of matrices that possess a (possibly hidden) Toeplitz-like structure. These sequences, which are known as GLT sequences, arise in several applications, including the discretization of differential equations. Associated with any GLT sequence is a special function called symbol. In this paper, we prove that, if $\{A_n\}_n$ is a GLT sequence with symbol $\kappa$ and $P_n$ is any block Jacobi or block Gauss-Seidel preconditioner for $A_n$ with a fixed number of blocks independent of $n$, then $\{P_n\}_n$ is a GLT sequence with symbol $\kappa$, just like $\{A_n\}_n$. This result allows us to predict a remarkable efficiency of block Jacobi/Gauss-Seidel preconditioning for GLT sequences, which is in fact illustrated through numerical experiments. It also allows us to extend the Fasino-Tilli theorem on the zero distribution of Hankel matrix sequences generated by $L^1$ functions to a larger class of matrix sequences called generalized locally Hankel (GLH) sequences.

We consider Bernoulli bond percolation with fixed retention parameter $p$ on the random recursive tree, coupled through the natural growth process. We prove that the root cluster has a strictly positive probability of remaining a largest cluster at every time. Equivalently, in the associated Simon-type Chinese restaurant process, the first table has a positive probability of remaining a largest table forever. We further show that two associated quantities, the limit as $n\to\infty$ of the probability that the root cluster is a largest cluster at time $n$ and the probability that it remains a largest cluster for all times, are strictly increasing and continuous functions of $p$ on $(0,1]$, and both tend to zero as $p\downarrow0$.

We found sets of exact analytic quasi-stationary states of a massive scalar field in a dyonic Kerr-Sen black hole~(DKSBH) background in the maximally extended spacetime region. A central novelty is the use of horizon-regular ingoing Eddington-Finkelstein coordinates, which enables a direct and unambiguous imposition of the ingoing boundary condition at the horizon. The exact radial solutions are in the form of confluent Heun functions. Imposing regularity at spatial infinity enforces a series truncation condition, yielding an exact quantization of the quasi-stationary frequencies. The spectrum exhibits a rich multi-branch structure, which we show splits into two distinct classes: modes that are insensitive to the black hole spin and charges and modes that explicitly depend on them. We uncover a clear asymmetry between co-rotating and counter-rotating configurations, driven by the spin-angular momentum coupling, as well as a systematic shift of the spectrum induced by electric and magnetic charges. The physical branches exhibit a universal behavior: modes with positive real frequency possess positive imaginary parts and therefore grow exponentially in time, whereas modes with negative real frequency are damped and decay. This suggests that positive-energy excitations in the region behind the outer horizon including the inner region of the inner horizon which contains the closed-timelike-curve, exponentially destabilize the background spacetime, supporting Hawking's chronology protection conjecture. In addition, the purely imaginary modes contain no oscillatory component and hence do not propagate through the spacetime, preventing traveling excitations along closed timelike curves and remaining consistent with the conjecture.

Waterproof Editor provides an educational environment specifically targeted to teaching with proof assistants or programming languages. It arose from Waterproof, educational software targeted at helping students acquire the skill of giving mathematical proofs. Its original features such as enabling rich formatting and providing clear input areas are now abstracted away in an npm package and can be used in different educational contexts. We invite interested parties to use this component in their educational software, and offer to assist with this.

We study a two-dimensional semiflexible membrane model whose formal Hamiltonian is given by $H[\phi]=\sum_x \bigl(\|\nabla \phi_x\|^2 +N^\lambda |\Delta \phi_x|^2\bigr)$, interpolating between the discrete Gaussian free field (DGFF) and the membrane model (MM). We analyze its finite-volume covariance in the bulk as $N\to\infty$, and identify distinct regimes depending on the parameter $\lambda$. For $\lambda<0$, the covariance of the model agrees with that of the DGFF up to a negligible error, while for $\lambda>2$, it agrees with the rescaled MM covariance up to a negligible error. In the intermediate regime $\lambda\in[0,2]$, we identify a different crossover behavior: in the microscopic range $\|x-y\|\ll N^{\lambda/2}$, the leading asymptotics no longer resolve the precise distance between the two points. In this microscopic regime, we further determine the leading logarithmic coefficient of the bulk covariance. These results provide a unified description of the crossover from gradient-dominated to curvature-dominated behavior in this class of models.

Continual learning struggles to balance retaining past knowledge with absorbing new tasks. Stefan-CL elegantly resolves this stability-plasticity dilemma through the physics of melting. It frames consolidated knowledge as a protected "solid" and unused capacity as an adaptable "liquid." As the network learns, this boundary expands, governed by a "latent heat" tuning dial. By mathematically freezing the learned interior, Stefan-CL cuts forgetting to near zero, matching memory-heavy baselines without storing raw data, forging a beautiful, physics-grounded path for AI.

We study an MA(1)-process with uniform innovations conditioned to stay positive. Representing the model as a Markov chain, we prove the existence of the limiting finite-dimensional distributions under this conditioning and identify the limiting process explicitly as a Doob $h$-transform. In the non-expanding case, i.e. when the coupling parameter $\theta$ satisfies $\theta\in[-1,1)$, we compute the relevant generating functions, extract sharp persistence asymptotics, and give explicit formulas for the eigenfunction $h$ and the persistence exponent. The resulting transition kernel of the limiting process is therefore fully explicit and displays a phase-dependent structure in the parameters. This provides a rare solvable example of a Markov chain on a continuous state space conditioned on persistence.

We apply the class field theory and Minkowski bound to obtain an upper bound estimate for the number of solutions to the restricted ramifications when the Galois group is solvable. Together with suitable conditions on the solvable group and the ordering of number fields, we could prove an upper bound on specific field-counting problems, hence the infinite moment of the class groups. In particular, for non-Galois cubic fields ordered by the product of ramified primes, we could show that the $\mathbb{Z}/3\mathbb{Z}$-moment is infinite with the results on the $\mathbb{Z}/3\mathbb{Z}$-moment of quadratic number fields and the field-counting on cubic fields ordered by the generalized discriminant.

Sharpness-Aware Minimization (SAM) has established itself as a powerful and widely adopted optimizer for training machine learning models. By explicitly minimizing the sharpness of the loss landscape, SAM often improves generalization while delivering strong empirical performance. However, SAM and its variants, like most training algorithms, are sensitive to the choice of learning rate, which is typically selected through extensive hyperparameter tuning or predefined schedulers. In this work, motivated by recent advances on the effectiveness of stochastic Polyak step sizes for Stochastic Gradient Descent (SGD), we derive Polyak schedulers tailored to SAM-style updates, yielding novel adaptive algorithms in both deterministic and stochastic settings. In the smooth setting, we prove linear convergence for strongly convex objectives and an $\mathcal{O}(1/T)$ convergence rate for convex objectives in the deterministic case. In the stochastic setting, we establish analogous convergence guarantees up to a neighborhood of the optimum. Numerical experiments demonstrate that the proposed Polyak schedulers achieve performance comparable to or better than carefully tuned SAM baselines, while substantially reducing the need for learning-rate tuning.

Pivoted Cholesky factorizations construct low-rank approximations of symmetric positive definite matrices by sequentially selecting pivots from the residual diagonal. Classical greedy and randomized rules, such as randomly pivoted Cholesky, target the algebraic trace-norm error of the residual. In many applications, however, the matrix enters a nonlinear matrix functional whose value, not the trace-norm error, determines solution quality, and residual-based rules ignore this structure. We derive a pivot rule that maximizes the exact one-step change of such a functional under Cholesky-consistent rank-1 updates, for a functional combining log-determinant, quadratic, and trace terms. This functional arises as the variational free energy in Gaussian process regression, where the matrix is a kernel matrix. The resulting per-step gain admits a closed-form additive decomposition into complexity, data-fit, and trace contributions, and is used directly as a pivot-selection criterion. We refer to the resulting method as $\Delta$-VFE pivoted Cholesky. At each iteration, the criterion is evaluated on a batch of $s$ candidate pivots sampled proportionally to the residual diagonal via incremental Woodbury updates, at a total cost of $\mathcal{O}(snr^2)$ for an $n\times n$ matrix and target rank $r$. This matches the asymptotic complexity of randomly pivoted Cholesky up to the batch factor $s$. Cholesky-consistent rank-1 updates yield monotonically non-decreasing functional values, and the proposed rule maximizes the per-step gain among them. Numerical experiments show improved objective values and predictive accuracy at low to moderate ranks compared to classical and randomly pivoted Cholesky, while preserving trace-norm approximation quality.

For integers $k ,\ell \geq 2$ let $m(k,\ell)$ denote the maximum of $|\mathcal{F}| |\mathcal{G}|$ where the maximum is taken over all pairs of cross-intersecting families, $\mathcal{F}$ being a $k$-graph with covering number $\ell$ and $\mathcal{G}$ a $\ell$-graph with covering number $k$ (see the paper for the definitions). Erdos and Lovasz initiated the study of the one family version. That is, they provided lower and upper bounds on the maximal size $m(k)=|\mathcal{F}|$ where $\mathcal{F}$ is an intersecting k-graph with covering number $k$. In many similar situations $m(k,k)=m(k)^2$ holds. However, as our results show $m(k,k)/m(k)^2$ is tending to infinity as $k$ grows(Th.1.5) . For $k>k_0$ we establish the exact value $m(k,k)=(k^{k-1}+k-1)^2$(Th.1.6). As to smaller values we prove $m(3,3)=121$ (Th.1.7) and determine $m(2,k) $ for all $k\geq 2$ (Th.1.8).

This paper is devoted to construction and convergence analysis of the linear explicit immersed finite element (IFE) function. For the interface elements, the proposed IFE functions precisely satisfy the interface conditions on the actual interface. The IFE functions are constructed in an explicit form and can be obtained directly without solving any auxiliary problems or local linear systems. Although the constructed IFE functions are non-polynomial, we establish rigorous theoretical analysis showing that they achieve optimal approximation properties and satisfy the essential trace inequalities. And the constants in the analysis are independent of how the interface cuts through the elements. Based on these IFE functions, an immersed discontinuous Galerkin numerical scheme is developed. Several numerical experiments are implemented to confirm that both the IFE functions and the numerical method achieve optimal convergence rates in the $H^1$ and $L^2$ norms. Furthermore, the numerical results indicate that the condition numbers of the stiffness matrices are robust with respect to the interface location.

This paper studies an inverse coefficient problem for a time-fractional damped wave equation on a finite time interval. The aim is to determine two spatially varying coefficients, namely the fractional damping coefficient and the zeroth-order potential, from the associated Dirichlet-to-Neumann (DtN) map. We first prove the well-posedness of the forward problem for boundary data with sufficient regularity to admit a suitable lifting. The main result is a uniqueness theorem showing that if two coefficient pairs give rise to the same DtN map, then the corresponding coefficients coincide almost everywhere in the domain. The proof is based on a convolution-type integral identity that eliminates the unknown boundary traces, the construction of high-frequency beam solutions adapted to the singular kernel of the Caputo derivative, and a detailed asymptotic analysis of the resulting fractional convolution terms. This result extends classical uniqueness results for hyperbolic inverse problems to a fractional-order model with damping and provides a theoretical basis for related applications involving memory and viscoelastic effects.

A central problem in extremal set theory is to determine or estimate $m(n,k,t),n>2k\geq 2t$, the maximum size of an intersecting $k$-graph and covering number at least $t$(see the paper for the definitions). For $t=1$ and $2$ the classical Erdős-Ko-Rado Theorem and the Hilton-Milner Theorem provide the answer.The complete solution for $t=3$ was only achieved recently . There are some partial results for $t=4,5$ but for the general case even to determine the asymptotic appears to be hopelessly difficult . Denoting by $\widetilde{m}(n,k,t)$ the maximum of $|\mathcal{F}||\mathcal{G}|$ for a pair of cross-intersecting $k$-graphs with covering number at least $t$, $\widetilde{m}(n,k,t)\geq {m}(n,k,t)^2$ is obvious. Pyber showed that equality holds for $t=1$. The same was shown for $t=2$ in a wide range(cf.[7]). Quite surprisingly our results show that the inequality is strict for $t\geq 3$ and for $n>n_0(k,t)$, Theorem 1.7 determines the exact value of $\widetilde{m}(n,k,t)$ for $k>2t$ and $n$ sufficiently large.

A new class of asynchronous adaptive first-order optimization methods is introduced, comprising asynchronous variants of several popular algorithms. Versions of these methods using momentum and/or inexact normalization are also considered. The convergence of methods in the class on non-convex functions is analyzed in a fully stochastic setting, and is shown to be (up to logarithmic factors) of order O(1/sqrt{t}) under reasonable assumptions. Numerical experiments suggest that such asynchronous adaptive algorithms are very relevant in heterogeneous large-scale machine learning systems.

Let $f(N,k)$ be the minimum size of a subset of $\{0,\ldots,N-1\}$ meeting every $k$-term arithmetic progression of positive difference contained in that interval. Improving on a bound of Truss, we prove that \[ f(n^2,n)\ge n+n^{1-o(1)} \] as $n\to\infty$. Improving on a bound of Xu, for primes $p$ we prove that \[ f(p^2,p)\le2p-(\sqrt2-o(1))\sqrt{\frac p{\log p}}. \]

This paper investigates the long time well-posedness of the Benjamin-Ono (BO) equation with quasi-periodic initial data in analytic space. We demonstrate that the Lax operator of the BO equation exhibits a trivial spectrum, and elucidate the fundamental mathematical obstacles to establishing global well-posedness. Furthermore, by employing Tao's gauge transformation in conjunction with the Birkhoff Normal Form method, we extend the lifespan of the solutions to $O(\epsilon^{-3})$.

We establish a quantitative version of strong localization of the Kobayashi-Eisenman volume element and the quotient invariant near plurisubharmonic peak points of domains in $\mathbb{C}^n$. As an application of this strong localization result, we derive the non-tangential asymptotic limit of the Kobayashi-Eisenman volume element at exponentially flat infinite type boundary points of domains in $\mathbb{C}^{n+1}$.

This paper investigates an isoperimetric-type problem posed by L. A. Santaló, concerning convex surfaces in hyperbolic 3-space that minimize total mean curvature among all convex surfaces with fixed surface area. This problem asks for a characterization of the minimizers and for the optimal form of a Minkowski-type inequality in hyperbolic 3-space. In this work, we propose a conjectural description of the minimizers under certain regularity assumptions. We also construct a new family of convex surfaces as as potential minimizer candidates and establish a property of the singular points of any minimizer.

We revisit the convergence guarantees of the Extragradient (EG) method for unconstrained biaffine min-max optimization. It is known that EG with a fixed stepsize achieves a $\Theta(T^{-1/2})$ last-iterate convergence rate, which is slower than the optimal $\mathcal{O}(T^{-1})$ rate attainable by incorporating additional mechanisms such as anchoring. Motivated by recent advances showing that dynamic stepsizes alone can significantly accelerate gradient descent, we ask whether dynamic stepsizes can similarly accelerate the last-iterate convergence of EG. We present the first positive result in this direction. Specifically, we provide a deterministic dynamic stepsize schedule that accelerates the convergence rate of EG to $\mathcal{O}(T^{-2/3+\varepsilon})$ for any $\varepsilon > 0$. We also show that this rate is nearly tight when the extrapolation and update steps of EG use the same stepsize. We then show that allowing different stepsizes for the extrapolation and update steps further improves the convergence rate to the near-optimal $\mathcal{O}(T^{-1+\varepsilon})$. Our analysis reduces stepsize scheduling to an optimization problem, whose solution leads to a stepsize schedule that follows (a discretization of) a power-law distribution. Our proposed stepsize schedules and analysis extend to other methods, such as Optimistic Gradient (OG), and suggest broader applicability to general min-max optimization problems.

We study a high-velocity inverse scattering problem for nonlinear Schrödinger equations with spatially dependent nonlinearities in dimensions $d\ge3$. We consider the whole mass-supercritical and energy-subcritical range, including the endpoint cases. By introducing a moving frame adapted to highly boosted initial data, we construct the scattering operator for a class of large incoming states generated by Galilean boosts. The key observation is that, although the boosted data become large in Sobolev norms, the nonlinear interaction becomes effectively weak at high velocity due to rapid spatial separation. Using the resulting high-velocity asymptotics, we derive a reconstruction formula for the X-ray transform of the coefficient. As a consequence, we prove that the scattering operator uniquely determines both the nonlinearity exponent and the spatial coefficient. Our results extend previous work of Watanabe to all dimensions $d \ge 3$, include the endpoint nonlinearities, and replace the repulsiveness and radial monotonicity assumptions on the coefficient by suitable decay conditions.

The $k$-th power $G^k$ of a graph $G$ is the graph on the same vertex set where the edge set consists of those pairs of distinct vertices of $G$ that are at distance at most $k$ from each other. A. Abiad, G. Coutinho, and M. A. Fiol [On the $k$-independence number of graphs, Discrete Mathematics 342 (2019), 2875–2885] proposed extensions of the classical ratio (for regular graphs) and inertia bounds to the independence number of $G^k$ for $k\ge 2$. Continuing a line of work comparing these two parameters with other known bounds, we show that the $\vartheta$-function of L. Lovász and the weighted inertia bound of A. R. Calderbank and P. Frankl, when applied directly to $G^k$, perform at least as well as the ratio and inertia bounds of Abiad-Coutinho-Fiol, respectively. In particular, $\vartheta(G^k)$ provides a polynomial-time computable upper bound on the independence number of $G^k$ that is at least as strong as the ratio bound when the latter applies (i.e.,\ when the graph $G$ is regular).