Papers

In the noncommutative polydisc, we first prove a positive sum of squares formula for a non-negative hereditary rational nc-function. The number of summands is finite. This result is used to derive a finite-dimensional realization formula for contractive nc-rational functions, where the colligation matrix is contractive. It is unitary if and only if the function is inner. Finally, we apply these results to generalize Carathéodory's classical theorem - approximating holomorphic self-maps of the unit disc by finite Blaschke products - to the setting of holomorphic functions on the noncommutative polydisc. This is in sharp contrast with the commutative situation where Carathéodory's approximation is known for Schur classes only in the unit disc and the unit bidisc.

Order theory is increasingly relevant in applications where data is naturally structured as a partially ordered set (poset), often requiring meaningful notions of distance over posets. In this paper, we introduce a new family of extended metrics on path-connected and fence-connected posets that do not require additional structure. Unlike many existing distances, these metrics are not induced by valuations, but instead arise as a type of shortest-path distance determined by both path length and the number of alternations. For discrete posets, we show that these metrics converge to a type of shortest-fence metric. Our main result establishes that these metrics characterize most discrete path-connected posets up to isomorphism, and up to duality for modular posets. Finally, we prove that this family defines interleaving distances when posets are viewed as thin categories.

Modern semiparametric estimation often relies on flexible black-box machine learning methods to estimate nuisance functions, raising a fundamental question: how do nuisance estimation errors propagate into inference for low-dimensional target parameters? The dominant paradigm, exemplified by double machine learning (DML), yields error bounds in which nuisance estimation errors enter multiplicatively. While widely adopted, it remains unclear whether this multiplicative-rate dependence is optimal for black-box models. In this paper, we start by revisiting the partial linear model $Y = \mu_0(X)+T\cdot\beta_0+\varepsilon$ under a structure-agnostic setting, where the nuisance function $\mu_0$ is estimated using a generic machine learning model, with approximation error $\delta^a_\mu$ and stochastic error $\delta_\mu^s$. We show that the standard DML rate is not optimal in the regime where the auxiliary function $\mathbb{E}[T|X=x]$ cannot be consistently estimated. We propose a new estimator for $\beta_0$ that achieves a sharper rate of $n^{-1/2}+\delta^a_\mu+(\delta_\mu^s)^2$ and establish a matching lower bound demonstrating its optimality. Our results reveal a new principle: the first-order stochastic error of nuisance estimation can be eliminated without imposing any additional assumptions. This also leads to a revised tuning strategy favoring under-smoothing, where $\delta^a_\mu\asymp(\delta_\mu^s)^2$, rather than the classical bias-variance trade-off $\delta^a_\mu \asymp \delta_\mu^s$. Under mild additional conditions, the estimator is asymptotically normal with minimal asymptotic variance. The proposed method extends to a broad class of semi-parametric linear functional estimation problems, including average treatment effect estimation. Our results imply that popular orthogonal score methods in semiparametric estimation with black-box nuisance learners can be substantially improved.

In this paper, we study the structure of double cyclic codes of length $(\gamma,\delta)$ over $\mathbb F_q+u\mathbb F_q, u^2=0$. We also study the dual of double cyclic code of length $(\gamma,\delta)$ and give a minimal spanning set of double cyclic codes. Moreover, we study the necessary and sufficient conditions for a double cyclic code to be reversible and reversible-complement double cyclic code and with the help of these codes, we constructed DNA codes over $\mathbb F_4+u\mathbb F_4, u^2=0$. We also constructed some optimal codes to support our results.

We construct an explicit equivariant quantum Satake map for Grassmannians, which enables us to express their torus-equivariant quantum cohomology in terms of that of projective space. We then consider the exterior algebra of the latter, which admits a canonical identification with a Clifford algebra. We describe the resulting action in several complementary ways: first, from a geometric perspective via push-pull maps, and second, in terms of the shuffle product, which also arises in the simplest cohomological Hall algebra associated with the $A_1$-quiver. Exploiting the Clifford algebra structure, we derive new recurrence relations among equivariant Gromov-Witten invariants, yielding a new method for their computation in terms of Wick's Theorem. As an application, we provide combinatorial proofs of Graham positivity for both equivariant quantum Pieri rules, and in one case extend these results to quantum triple Schubert calculus.

This paper develops a unified probabilistic framework based on distributional derivatives and Dirac delta representations for the analysis of conditional and transportation-related quantities. General identities are established for arbitrary random variables, encompassing absolutely continuous, discrete, and mixed distributions. The proposed approach yields unified formulas for conditional expectations, conditional distributions, hazard functions, and improper distributions, revealing a common localization mechanism underlying these classical concepts. The framework is further combined with copula methods to investigate transportation and dispersion functionals through dependence structures. Exploiting the extremal properties of the Fréchet–Hoeffding bounds together with expectation inequalities induced by $\Delta$-antitonic functions, sharp bounds are derived for absolute difference moments under fixed marginals. These results lead to concise derivations of quantile representations for the Wasserstein distance and a corresponding upper transportation functional, as well as survival-function representations and bounds for generalized absolute difference moments. As a particular case, new representations are obtained for the bivariate Gini mean difference and the associated bivariate Gini index. Applications are given to Wasserstein-type functionals arising in the normal approximation of standardized counting distributions, including Poisson, Binomial, and Negative Binomial models, for which explicit quantile representations are derived. Overall, the results establish explicit links among conditional structures, dependence modeling, dispersion measures, normal approximation, and optimal transport, providing a unified perspective on several fundamental constructions in probability and mathematical statistics.

We study deconfined quantum critical points (DQCP) associated with Lieb-Schultz-Mattis (LSM) anomalies in one-dimensional spin chains. Our starting point is a structural characterization of the LSM anomaly in the Lyndon-Hochschild-Serre spectral sequence: $\omega_{\mathrm{LSM}}\in E_\infty^{1,2}= H^1(\mathbb Z_{\mathrm{trans}},H^2(G_{\mathrm{int}},\mathrm{U}(1)))\subseteq H^3(G_{\mathrm{int}}\rtimes_{\rho}\mathbb Z_{\mathrm{trans}},\mathrm{U}(1))$. Physically, this class decorates a translation defect with a projective representation of the internal symmetry $G_\mathrm{int}$. We show that gauging the internal symmetry in the presence of an $E_\infty^{1,2}$-type anomaly necessarily produces a non-invertible dual symmetry. This gives a general mechanism for type-II DQCP: in contrast to type-I examples with $E_\infty^{2,1}$-type anomalies which are dual to ordinary group-like symmetry breaking, type-II transitions are dual to spontaneous breaking of a non-invertible symmetry. We illustrate the mechanism using a spin-$1/2$ chain with an anomalous $D_8$ LSM symmetry. We construct a dimer-to-ferromagnet DQCP candidate, provide numerical evidence for a critical theory with central charge $c\approx 1$, and show, using both category theory and explicit lattice constructions, that gauging the internal symmetry yields the non-invertible $\mathrm{Rep}(H_8)$ dual symmetry.

Asymptotic e-values are emerging as a powerful alternative to asymptotic p-values, particularly in post-hoc inference and multiple testing, where significance levels may be data-dependent. Existing asymptotic e-values, however, suffer from the “missing factor,” a scaling inefficiency resulting in overly conservative inference. Drawing on the framework of near-optimal concentration inequalities developed by Bentkus in the 2000s, we introduce Bentkus-type asymptotic e-values and prove that they successfully eliminate the missing factor. We also demonstrate both theoretically and empirically that Bentkus-type e-values consistently deliver sharper inference than existing alternatives, leading to tighter post-hoc confidence intervals and higher rejection rates in multiple testing procedures.

We determine asymptotic results for the average size of Selmer groups arising from certain isogenies related to Jacobians of hyperelliptic curves of genus $g\geq 2$. Our results come from two different routes. The first route is through the geometry-of-numbers methods pioneered by Bhargava, where we obtain new parametrisations coming from Vinberg theory arising from representations related to the Dynkin diagrams of type $B$ and $C$. The second route uses recent results by Koymans–Smith, which relate the average sizes of Selmer groups to Tamagawa ratios through a formula by Greenberg–Wiles.

Time-dependent high-dimensional partial differential equations (PDEs) with spatially localised and dynamically evolving solutions pose a fundamental challenge for physics-informed neural networks (PINNs), as uniform collocation sampling becomes increasingly ineffective in high-dimensional spatiotemporal domains. In this work, a deep adaptive sampling framework for PINNs is extended to the time-dependent setting by treating space and time as a unified domain without any explicit time marching. A normalising flow neural network model effectively learns the distribution induced by the PDE residual and generates new collocation points concentrated in regions where the solution is most difficult to learn. Unlike conventional adaptive strategies that require explicit time stepping or moving meshes, high-residual regions are automatically identified and tracked across both space and time, driven purely by the PDE residual distribution. The effectiveness of the proposed strategy is assessed on a range of benchmark problems, from sharp and moving features in two spatial dimensions to localised structures in up to eight spatial dimensions.

We introduce a new algorithm for computing persistent homology of Vietoris-Rips filtrations, which offers a considerable improvement both in terms of time and memory over the existing implementations of the persistence pairing algorithm. The key innovation, called active enumeration, is made possible by a new theoretical framework of Reduction to Zero Differentials (hence RedZeD) in which to view persistent homology.

This paper develops a stochastic-geometry framework for high-altitude platform station (HAPs) networks in which platforms execute cyclic patrol trajectories anchored to designated service regions. We introduce two small-circle ring Cox process models on the spherical Earth. In the small-circle ring Poisson Cox process (SCR-PCP), platforms form one-dimensional Poisson point processes on localized patrol rings, whereas in the small-circle ring binomial Cox process (SCR-BCP), each ring contains a fixed number of uniformly distributed platforms. We establish the isotropy of both models and derive spatial statistics, including the distributions of the nearest-anchor, nearest-ring, and nearest-HAPs distances, together with the joint serving distance and serving ring angle distribution required for SCR-BCP analysis. Building on these results, we derive coverage probability expressions under nearest-HAPs association by decomposing aggregate interference into same-ring and other-ring components and characterizing their conditional Laplace transforms. To account for the flight dynamics of patrol-based HAPs, we integrate a steady circular flight propulsion model with the communication analysis and introduce a coverage energy efficiency (CEE) metric. This yields an analytical condition for the energy-optimal patrol radius that balances coverage performance against the propulsion cost of circular flight. Numerical results reveal fundamental differences between intensity-driven (SCR-PCP) and finite-fleet (SCR-BCP) deployments and demonstrate that patrol geometry, platform density, and cruising velocity should be jointly optimized to achieve energy-efficient HAPs operation.

A stabilisation of a $4$-manifold $X$ is the connected sum of $X$ with some number of copies of $S^2\times S^2$. If two smooth surfaces in a $4$-manifold are topologically isotopic, we investigate whether they must moreover be smoothly isotopic in some stabilisation of $X$. We prove this result holds whenever the surfaces are trivial in the $\mathbb{Z}/2$-homology of $X$. We also produce a large class of fundamental groups of the ambient $4$-manifold for which the result holds; this class includes free products of classical knot groups and, in particular, free groups.

A cactus is a graph in which every edge lies on at most one cycle. In 2024, Zhang and Huang generalized this concept to the $k$-cactus, defined as a graph in which every edge lies on at most $k$ cycles. It is known that any cactus on $n$ vertices has at most $\lfloor\frac{3}{2}(n-1)\rfloor$ edges. However, the upper bound on the size of $k$-cacti was known only for $k\le 4$. In this note we consider general $k$. We prove that every $n$-vertex $k$-cactus has $O\!\left(\frac{\log k}{\sqrt{\log\log k}}\,n\right)$ edges for all sufficiently large $k$, and a construction shows this is optimal up to a factor of $\sqrt{\log\log k}$.

Let $\Omega\subset\C$ be a bounded domain whose boundary is an oval of a real algebraic curve. We study when the exterior Cauchy transform \[ \ct_\Omega(z)=\frac1\pi\int_\Omega \frac{dA(\zeta)}{z-\zeta} \] is algebraic. The boundary formula identifies this transform with a Cauchy-type integral on the normalization $X$ of the relevant irreducible component of the Schwarz correspondence $P(z,w)=0$. The main point is that $X$ is fixed while only the divisor $\pi^{-1}(z)$ of moving poles varies. Thus the natural monodromy is point-pushing on a punctured fixed surface and becomes trivial on absolute homology after the moving punctures are filled; in particular, the usual Picard–Lefschetz transvection picture does not produce absolute cycles in this problem. The principal theorem is a residue criterion: if the lifted boundary is separating, i.e., if it bounds an integral two-chain on $X$ disjoint from the fixed polar divisor, then the exterior Cauchy transform is algebraic and is given by an explicit residue sum with chain multiplicities. This implies, in particular, algebraicity for every smooth oval on a rational real algebraic curve and for separating ovals in positive genus. We also record the corresponding complete-real-locus statement for dividing real curves, with the necessary affine-plane caveats. Nonseparating ovals are treated as a conjectural period problem: we formulate a period-rank test which can detect possible Abelian contributions. The examples include the ellipse, the nodal cubic logarithm, a smooth Weierstrass cubic as an elliptic-period test case, and a conditional positive-genus construction illustrating algebraic transforms beyond quadrature domains.

In this paper we present a sharp mixed inequality relating successive minima and quotient covering radii of origin-symmetric convex bodies with respect to lattices. The inequality interpolates between the (classical) covering-density lower bound and the lower bound in Minkowski's second theorem.

In this paper, we investigate the problem of finding a zero of the sum of a maximal monotone operator $A$ and a cocoercive operator $\Bm$ in a Hilbert space. This formulation naturally captures stochastic optimization problems with decision-dependent distributions, often referred to as performative prediction. We propose and analyze continuous-time second-order dynamics governed by a distributionally evaluated forward-backward splitting operator. We establish the existence and uniqueness of the equilibrium point under a general uniform monotonicity assumption. In this setting, employing a vanishing viscous damping coefficient, we prove the strong convergence of the trajectories to the equilibrium, accompanied by fast asymptotic convergence rates for the velocities. Furthermore, when the regularizing operator is strongly monotone, we consider a constant Polyak-type damping coefficient and we establish global exponential convergence rates for the dynamical system.

We construct a two sided categorical action of the Hall algebra of semistable coherent sheaves of fixed slope on a curve X on the derived category of certain Quot spaces in tilted hearts on X. Following the philosophy in arXiv:2207.08926v2, the action is induced by correspondence stacks that parameterize extensions of such quotients by semistable sheaves. In the process, we compare different moduli spaces on X: Quot spaces, Bradlow pairs, and stable pairs in the sense of arXiv:2207.08926v2.

We prove the existence of the C-type renormalisation two-cycle, helping to establish the universality of the C-type route to chaos in families of non-invertible maps of the plane. Families of two-dimensional non-invertible maps, with at least two parameters and critical points of fold type, exhibit a distinct type of critical scaling, the C-type. An accumulation of parameter values leads to an infinite collection of coexisting attracting cycles of periods $4^n$ or $2\cdot 4^n$. Asymptotically, period quadrupling is accompanied by parameter-space scaling and state-space scaling governed by particular universal constants. Kuznetsov et. al. explained this phenomenon in terms of a stationary orbit of period two of the renormalisation group (RG) transformation for period-doubling. We prove the existence of the corresponding renormalisation two-cycle in a Banach space of analytic maps and gain rigorous bounds on the corresponding universal state space scaling constants. This result provides a further step in proving a series of outstanding conjectures concerning distinct universality classes for period-doubling. It extends the recent results for unidirectionally-coupled maps (the FS-type) to bidirectionally-coupled maps, and generalises the framework from fixed points to periodic orbits of the corresponding renormalisation operators. It also provides a further step in establishing the conjectured picture that the C-type universality class is born from the FS-type class via a period-doubling bifurcation in the dynamics of the RG transformation itself. The proof relies on rigorous computations to establish that a variant of Newton's method for the two-cycle is a contraction map. The C-type scaling regularity is known to occur in a number of dynamical systems of interest, perhaps most notably in biologically-plausible models of nephron blood pressure autoregulation.

Lossless pixel-level image transmission is a fundamental regime beyond semantic communications, because exact recovery requires both accurate symbol probability modeling and reliable delivery over noisy channels. This paper proposes DDM-SSCC, a discrete-diffusion-model-based separate source-channel coding framework for lossless image transmission. Different from raster-order autoregressive coding, the proposed source codec adapts a diffusion language model to pixel-token restoration and performs synchronized reverse arithmetic coding under bidirectional attention, allowing multiple masked tokens to be coded within one reverse denoising step. This progressive restoration process also yields a more favorable source representation for noisy transmission, since newly restored tokens can serve as bidirectional context in subsequent denoising steps. To bridge the gap between generation-oriented masked denoising and lossless arithmetic coding, we further introduce a Halton-guided denoising order, a mask-ratio-aware cosine schedule, and a lightweight temperature calibration module. These designs respectively improve spatial coverage, adapt the denoising pace to context reliability, and calibrate the probability tables used by arithmetic coding. Experiments on CIFAR10, DIV2K-LR-X4, and Kodak over additive white Gaussian noise and Rayleigh fading channels show that DDM-SSCC achieves better exact-recovery performance than representative lossless and semantic communication baselines, while ablation studies verify the effectiveness of the proposed denoising order, schedule, and calibration modules.

Physics-informed neural networks (PINNs) have been widely applied to solve Navier-Stokes equations by enforcing outputs and gradients of deep models to satisfy target equations. However, conventional PINNs only constrain the boundary terms by means of the $L^2$-norm when addressing the equations with non-homogeneous boundary conditions. This single constraint strategy may cause inaccurate boundary simulation, further resulting in the decline of prediction accuracy. To resolve this critical issue, this paper proposes an improved physics-informed neural network by correcting the error of the boundary value, which is called Tr-PINNs. Based on the results of nonhomogeneous Stokes problem, the algorithm error analysis of Tr-PINNs is established. The efficacy of the Tr-PINNs algorithm is demonstrated via numerical experiments, which further demonstrate that the Tr-PINNs algorithm achieves a remarkable improvement in computational accuracy.

This addendum extends the spectral lower bounds established in the original paper to the Neumann problem. Under the formal positivity assumption on the elliptic tuple, we prove that the same bounds hold for Neumann boundary conditions as for the Dirichlet case.

In 1989, Erdős and Hajnal conjectured that for any graph $H$, there is a constant $c=c(H)>0$ such that every $n$-vertex graph $G$ with no induced copies of $H$ contains a clique or an independent set of size at least $n^{c}$. This conjecture, known as the Erdős-Hajnal conjecture, is a central open problem in combinatorics and listed as one of the top 10 Erdős problems by Bloom on the Erdős problem website https://www.erdosproblems.com/. In a recent breakthrough, Nguyen, Scott and Seymour proved that Erdős-Hajnal conjecture holds for the case when $H$ is the five-vertex path, which, combined with known results, implies that Erdős-Hajnal conjecture holds for every five-vertex graph. In this paper, we extend the iterative sparsification framework recently developed by Nguyen, Scott and Seymour. We introduce a generalized niceness condition relaxing their nice condition, a novel intermediate property concerning combs and a general structural lemma (which may be of independent interest) that is sufficient to deduce the Erdős-Hajnal conjecture. This framework simultaneously recovers the recent result on the five-vertex path (PLMS 2026) and the classical result on the bull graph by Chudnovsky and Safra (JCTB 2008) as special cases, thereby unifying these two previously independent strands, and further proves the conjecture for two new cases: the E-graph (which contains the five-vertex path) and the Bird graph (which contains both the five-vertex path and the bull). These are the first two six-vertex graphs whose validity does not follow from the known operations (see Alon-Pach-Solymosi, Combinatorica 2001, and Nguyen-Scott-Seymour, TAMS 2026) that preserve the Erdős-Hajnal property.

We consider Bayesian discrimination among multiple quantum states and establish a dimension-free one-shot upper bound on the minimum probability of error in terms of the sum of pairwise errors. This resolves a conjecture of Audenaert and Mosonyi [J. Math. Phys. 55 (2014)] and improves the multiple quantum Chernoff bound of Li [Ann. Statist. 44 (2016)] by removing its dimension-dependent prefactor. In the asymptotic many-copy regime, our bound proves the achievability of the multiple quantum Chernoff distance for arbitrary separable Hilbert spaces, thereby settling the previously open infinite-dimensional case, and further yields constant-factor sharp asymptotics for the optimal error probability. In binary quantum hypothesis testing, we prove that the minimum error probability is characterized, up to universal constants, by a trace harmonic-mean quantity. Consequently, the optimal binary quantum error probability is within a factor of two of the optimal classical error probability for the associated Nussbaum-Szkoła distributions, complementing the lower bound of Nussbaum and Szkoła [Ann. Statist. 37 (2009)].

We establish and prove several results for the smooth Casselman-Jacquet submodule and quotient functors for real reductive groups. In addition to exactness, surjectivity, transitivity and globalization results, we establish a stability property for the intersection of Jacquet subspaces. Our approach is based on a family of seminorms on the Casselman-Jacquet quotient module. As an application, we establish a full version of the real Bernstein-Zelevinsky filtrations for smooth Fréchet representations of moderate growth.

Using André's lemma and the excision square for perfectoidization coming from $p$-complete arc descent, we prove new structural results about perfectoid rings and perfectoidization. The main result is a tameness theorem for torsion in perfectoid rings: if $R$ is a perfectoid ring and $I\subset R$ is an ideal, then the $I$-torsion in $R$ is $I_{\mathrm{perfd}}$-almost zero. This yields an excision-type decomposition of $R$ along its $I$-torsion part. We also study (semi)perfectoid rings and perfectoid ideals and take the opportunity to make some structural remarks about them.

We develop a method for bounding the sensitivity of solutions to stochastic differential equations (SDEs) to changes in the drift, $F$, and diffusion, $\sigma$, by using a combination of information-theoretic uncertainty quantification bounds, functional inequalities, and judiciously chosen coupled auxiliary SDEs. The method is capable of producing non-asymptotic bounds which are well behaved in the $T\to \infty$ limit and does not require the perturbations to $F$ and $\sigma$ to be small. Our approach applies to expectations of both time-averaged and exponentially discounted observables and also produces sensitivity bounds for linear parabolic PDEs. When applied to stationary solutions and Lipschitz observables, our results produce bounds on the $1$-Wasserstein distance between invariant measures which have optimal scaling in each error term. The present method significantly expands on prior information-theoretic SDE sensitivity bounds, which are only applicable to perturbations of the drift.

We prove the existence of a bounded Sobolev extension operator $E:W^{p,1}\left( B,P \right) \rightarrow W^{p,1}\left( \ell^{2} ,P \right)$ using a completely new method, where $B\subset \ell^{2}$ is the unit ball and $P$ is any non-trivial centered Gaussian measure on $\ell^{2}$. This solves an open problem posed in the literatures.

Let $d$ be the smallest positive integer, not divisible by $3$, for which there exists an algebraic number over $\mathbb{Q}$ of degree $d$ whose some three algebraic conjugates sum to zero. Employing the classification of vertex-transitive graphs on 16 vertices of degree 6, we prove that $d\neq 16$. This, combined with results obtained by Dubickas, Smyth and Stong \cite{DubickasSmyth2006}, Dubickas and Jankauskas \cite{DubickasJankauskas2015} and Virbalas \cite{Virbalas2025a}, implies that $d=20$.

We study the de Rham cohomology of three families of completely solvable almost abelian solvmanifolds (called basic, complex, and hypercomplex) constructed from a monic integer polynomial with positive distinct roots whose product equals 1, following the work of Andrada and Barberis. Under two algebraic restrictions on such polynomials (the full rank and quasi full rank conditions) we compute the Betti numbers and Poincaré polynomials of these manifolds. Moreover, we study the Dolbeault cohomology of the complex solvmanifolds by identifying them with generalized Nakamura manifolds recently introduced by Cattaneo and Tomassini. Assuming a suitable condition on the lattice, we compute their Hodge numbers, which exhibit a combinatorial structure related to Pascal's triangle in the full rank setting, and are described by explicit generating polynomials in the quasi full rank case.