Papers

Recently, the wave-particle turbulence simulation (WPTS) has been proposed as a novel framework for non-equilibrium turbulence modeling and simulation. In this work, for the first time the complete model equations of WPTS are explicitly derived from the perspective of wave-particle decomposition, and the physical mechanism of each term is clearly interpreted. To extend its applicability to wall-bounded flows, the WPTS coupled with wall model is developed, and the introduction of wall model substantially alleviates the near-wall grid-resolution constraint. In the bulk region, the wave component resolves the large-scale structures, whereas the particle component accounts for subgrid-scale modeling through the non-equilibrium transport mechanism. As a result, the coupled method enables accurate predictions of the flat-plate transition on coarse-grid. In particular, the computed skin-friction coefficient and mean velocity profiles in the fully turbulent region agree well with the reference data from direct numerical simulation, and the accuracy is markedly superior to that of the gas-kinetic scheme (GKS) under the identical grid. These findings underscore the considerable promise of the multi-scale WPTS method for transitional flow simulations.

This paper investigates the structural and quantitative behaviors of complementable operators on Hilbert spaces, focusing on their norm characteristics and geometric profiles. We establish a comprehensive framework of norm inequalities and lower-bound relationships between a bounded linear operator and its generalized Schur complement (bilateral shorted operator). Under explicit operator factorization and range inclusion criteria, we define the exact conditions under which a bounded linear operator contracts or expands vectors relative to its Schur complement. Furthermore, we explore the lower boundedness and stability configurations of $(M, N, \lambda)$-complementable operators, proving that a bounded-below Schur complement acts as a sufficient condition to propagate injectivity and lower-bounded stability to the global operator. These structural results are subsequently applied to the network-theoretic setting of the parallel sum of two bounded linear operators. With some specific orthogonality conditions, we derive a novel norm decomposition identity, sharp two-sided global operator bounds, and algebraic restrictions on Douglas reduced solutions via Moore-Penrose inverses.

We study recovery of a low-rank tensor $\mathcal{T}$ and a low-rank matrix $M$ from sparse, noisy observations. $\mathcal{T}$ and $M$ share one mode. We relax tensor rank using the nuclear norm of the mode-1 unfolding. This unfolding carries the coupling. It also has an exact proximal operator. We couple $\mathcal{T}$ and $M$ through a learned linear operator $G$. We prove a minimizer exists for the ridge-stabilized penalized objective. We prove that a proximal alternating linearized minimization (PALM) scheme converges to a critical point, for the algorithm as implemented, by verifying the hypotheses of a known nonconvex block-coordinate convergence theorem against our objective and identifying which conditions come from this problem's structure. For the matrix-only sub-problem, we state a proven sampling bound from matrix completion theory. For the coupled problem, we prove a sample-complexity result for a sequential sub-case: a separately-known coupling operator recovers $M$ from $\mathcal{T}$'s recovery accuracy alone, with no observations of $M$ needed. For the fully joint, alternately-estimated case, we state a conjecture and test it empirically, including a low-density regime where coupling does not help. We report multi-seed synthetic experiments with mean and standard deviation across sampling densities, an asymmetric-density experiment, and convergence curves, and we explain why recovery error stays high at low density. We apply the framework to workforce-skill and small-business-health estimation. Every application-specific choice is a proposed design, not a validated result; we have not run the framework on deployed data.

We study the difference between the canonical height and the naive height for polynomial maps on $\mathbb{P}^{1}$. While explicit upper bounds on this height gap generally depend on the degree $d$ for rational maps, we establish a refined bound for polynomial maps that is essentially independent of $d$. As an application, we determine the set of rational preperiodic points for polynomial maps defined over $\mathbb{Q}$ of height $0$ and classify their realizable portraits.

In this article, we study the existence and uniqueness of a transport equation with discontinuous coefficients that comes from the solution of a balance law with a time-dependent source term. We also give a simplified proof providing an explicit formula for the solution of the balance law using a variational formulation. This system can be considered as a generalized version of the one-dimensional model for the large-scale structure of the formation of the universe.

By using Lawlor's curvature criterion, we prove that a class of real tensor varieties are regular area-minimizing cones except for one case, which gives a partial generalization of their minimality as in \cite{HKV23}. Such area-minimizing tensor varieties have not been found beyond matrix varieties. Moreover, we provide a new derivation for the important Lawlor ODE.

The Combinatorial Discretizable Distance Geometry Problem combines a finite binary lateration process with additional distance constraints. When predecessor sets are not consecutive, these pruning constraints interact in ways that make the symmetry-based counting methods available for molecular instances insufficient. We establish an exact, dimension-uniform counting theorem for seed-fixed feasible realizations under strict discretization and a generic feasible framework assumption. Our approach identifies partial reflections through seed-free connected components of the lateration skeleton. These reflections are encoded by binary masks, while a labeled constraint matrix detects combinations that preserve all pruning distances. The feasible branch choices split into constrained choices generated by compatible partial reflections and unconstrained choices outside the predecessor closure of the pruning endpoints. The resulting count is obtained from graph operations and rank computations over the binary field, without enumerating the lateration tree. Completeness follows from the characterization of generic realizations of the relevant joined graphs by partial reflections. The theorem applies in every Euclidean dimension, including dimension one.

In the setting of a Heisenberg group, we first studied the sharp weak estimate for the $n$-dimensional fractional Hardy operator from $L^p$ to $L^{q,\infty}$. Next, we studied the sharp bounds for the $m$-linear $n$-dimensional integral operator with a kernel on weighted Lebesgue spaces. As an application, the sharp bounds for Hardy, Hardy-Littlewood-Pólya, and Hilbert operators on weighted Lebesgue spaces were obtained. Finally, according to the previous steps, we also found the estimate for the Hausdorff operator on weighted $L^p$ spaces.

We study an explicit model of a conic Laplacian on \(\mb P^1\) with cone angle \(2\pi n\) at \(0\) and \(\infty\). Using Fourier decomposition, we reduce the eigenvalue equation to a family of associated Legendre equations and describe the corresponding asymptotic boundary data. The classical Legendre connection formulas induce an explicit gluing map between the boundary data at the two conic points. We compute the Friedrichs spectrum and eigenfunctions and show that the Weyl function, and hence the associated \(S\)-matrix, can be recovered explicitly from this boundary connection map.

We show that relativistic theories with purely relaxational excitation spectra, such as kinetic theory and transient hydrodynamics, naturally endow the dispersion plane $\{i\omega,ik\}$ with a Lorentzian geometric structure analogous to that of the Minkowski plane $\{t,x\}$. In this picture, timelike future-directed, timelike past-directed, and spacelike directions correspond respectively to relaxation-like, unstable-like, and evanescent-like modes. Under mild structural assumptions on the underlying theory, causality constrains dispersion relations to follow spacelike trajectories on the plane. This geometric viewpoint recasts longstanding problems in relativistic matter physics as elementary geometric ones that can often be solved graphically. As applications, we derive universal constraints on dispersion relations, deviations from time dilation, the observer dependence of spectral hierarchies, the regime of validity of hydrodynamics in boosted frame, the maximal allowed diffusivity and viscosity of relativistic media, and the presence of non-hydrodynamic branch cuts in kinetic theory.

We prove global well-posedness of the hard-sphere Boltzmann equation in $\mathbb{R}^d$ for small initial data in the critical Besov space $B_{1,1}^{d-1}L_v^1$ and in the critical Sobolev space $W^{d-1,1}L_v^1$. The proof is based on new bilinear estimates for the nonlinear collision operator, which rely on transversality considerations.

To meet the extended ultra-low latency and high reliability (xURLLC) requirements for autonomous driving sys-tems, multiple access schemes must operate reliably in high-mobility and complex propagation environments. Recently, rate-splitting multiple access (RSMA) has emerged as a promis-ing multi-user transmission framework, showing robustness in dynamic situations where imperfect and outdated channel state information (CSI) is prevalent. Moreover, the advanced sensing, localization, and on-board computation capabilities of autonomous driving vehicles facilitate the construction of a channel knowledge map (CKM), which is a key enabler for environment-aware communications in future 6G networks. In this work, we propose a CKM-aware finite-blocklength (FBL) RSMA for a downlink autonomous driving system. The location-dependent large-scale channel information provided by CKM is exploited to guide the common/private power split and common-rate allocation in RSMA. Specifically, we derive a new and tight closed-form bound for the private-stream ergodic rate. Combined with the closed-form expression for the common-stream ergodic rate, an optimization design of splitting ratios is formulated to maximize the min-rate performance among multiple users. Numerical results show that the proposed scheme outperforms the considered space-division multiple access (SDMA) and non-orthogonal multiple access (NOMA) schemes in high-mobility scenarios. Comparisons across different CKMs further highlight the benefit of accurate channel information, demonstrating how environment-aware channel knowledge complements flexible in-terference management of RSMA in high-mobility short-packet communications.

Let \[ F(a,b;c;x)={}_2F_1(a,b;c;x), \qquad \Phi_{m,\varepsilon}(\lambda;a,b,c) = \int_0^1 x^{m+\lambda}F(a,b;c;\varepsilon x)\,dx, \quad \varepsilon=\pm1. \] We study these moments by deriving and solving a first-order recurrence in $m$. This recurrence leads to formulas for higher powers of the denominator and for denominators of the form $(dn+m+1)^K$, with applications to product-binomial series and moments of complete elliptic integrals. Differentiation with respect to $c$ gives corresponding recurrences for harmonic-number weights, whose initial values are described using Bell polynomials, logarithms, zeta values, and Dirichlet $L$-values. Finally, comparison with a terminating ${}_3F_2(1)$ formula also gives finite hypergeometric and binomial–harmonic identities.

A theorem of Payan says that a cubelike graph cannot have chromatic number exactly three. A nearby question, usually discussed as Payan's finite $(0,2)$-graph question, asks whether a finite graph in which every two distinct vertices have either zero or two common neighbours can have chromatic number exactly three. The finite hypothesis is meaningful: infinite three-chromatic $(0,2)$-graphs can be constructed \cite{Payan1992}. We prove that every finite three-colourable $(0,2)$-graph is bipartite. Thus, no finite $(0,2)$-graph has chromatic number exactly three.

This article describes an approach that uses a single figure to illustrate all six trigonometric functions, their fundamental identities, and the law of cosines. We name this method OFFER (one-figure-facilitates-every-relationship) for the convenience of mathematics teaching. The article also describes the SPORT (story-picture-observe-review-tell) approach to effective mathematics learning.

For integers \(k\le s<t\), the hypergraph Erdős–Rogers function \(f^{(k)}_{s,t}(n)\) is the largest integer \(m\) such that every \(n\)-vertex \(K_t^{(k)}\)-free \(k\)-graph contains a set of \(m\) vertices spanning no copy of \(K_s^{(k)}\). We prove that, for every fixed \(s\ge4\), \[ f^{(4)}_{s,s+1}(n)=(\log n)^{o(1)}, \] thereby resolving a problem posed by Conlon, Fox and Sudakov. The key input is a new \(3\)-uniform estimate: for every fixed \(s\ge3\), \(f^{(3)}_{s,s+1}(n)=O(\frac{\log n}{\log\log n})\), which improves the logarithmic upper bound of Dudek and Mubayi. The proof develops a probabilistic pair-coloring construction based on a robust auxiliary palette and hypergraph containers. As a further consequence, we obtain \(f^{(k)}_{k+1,k+2}(n)=(\log_{(k-3)} n)^{o(1)}\) for every fixed \(k\ge5\), making substantial progress towards a conjecture of Mubayi and Suk.

We study the inverse spectral problems for the Steklov spectrum on compact surfaces with boundary. We prove that among flat annular surfaces, the lateral surface of a conical frustum is uniquely determined by its Steklov spectrum. As a consequence, each circular annulus is uniquely determined among all planar domains, providing the first example of a non-simply connected Euclidean domain with this property. Furthermore, we show that any family of Steklov isospectral flat annular surfaces is compact in the $C^\infty$ topology, extending previous results for simply connected planar domains. These results are established through a detailed analysis of the spectral zeta function and the zeta-regularized determinant associated with the Dirichlet-to-Neumann operator for annular surfaces.

We consider the quantum analogue of a disk with $n$ pairwise disjoint circular holes. We establish a uniform upper bound for the associated spectral constant that is independent of the geometry of the holes, together with a sharper asymptotic bound when the holes are well-separated. The latter generalizes recent estimates for the quantum annulus due to Crouzeix and Pascoe.

Kellner expressed higher congruences for the Wilson quotient $W_p$ of an odd prime $p$ in terms of power sums of Fermat quotients, and recursively constructed certain $p$-independent polynomials $\psi_\nu$ occurring in these congruences. In this note, we give a simple $p$-adic proof of these congruences using the $p$-adic logarithm and $p$-adic exponential function. We also provide an explicit generating function for the polynomials $\psi_\nu$ in terms of complete Bell polynomials. This gives a direct derivation of Kellner's polynomials and allows us to resolve two conjectures stated by Kellner.

We study the weakly coupled elliptic Anderson model with spatial white noise on the four-dimensional torus, which provides a basic example of a critical SPDE requiring renormalization at arbitrarily high orders. With coupling $\lambda |\log\varepsilon|^{-\frac12}$ where $\lambda>0$ is sufficiently small, we prove that the Green's function of the corresponding random Schrödinger operator, suitably centered and rescaled, converges to a centered Gaussian random field with explicit covariance. The main difficulty is that, for such critical models, one must expand up to order $|\log\varepsilon|$, while the perturbative expansion contains factorially many pairings and a growing number of renormalization terms. To overcome this, we construct a truncated renormalized parametrix and prove sharp high-order bounds for its remainder. A central ingredient is a multiscale analysis based on a new version of Hepp trees, combined with new estimates for summations over permutations. These estimates reveal a precise balance between logarithmic losses from scale summation and factorial gains from the structure of primitive pairings. The methods developed here are intended as a first step toward a general theory for critical SPDEs with weak couplings.

For every $n\ge 2$, we prove that there exists an exponent $p_n$ such that, for every even log-concave probability measure $\mu$ on $\mathbb R^n$, all nonempty symmetric convex sets $K,L\subseteq\mathbb R^n$, and all $\lambda\in[0,1]$, $$ \mu(\lambda K+(1-\lambda)L)^{p_n} \ge \lambda\mu(K)^{p_n}+(1-\lambda)\mu(L)^{p_n}, $$ where $$ p_n\ge \frac{c}{n^2\ln n} $$ for some absolute constant $c>0$.

We study a class of diffusion processes on $\mathbb{R}^n$ interacting through a symmetric matrix $X\in\mathbb{R}^{n\times n}$. When eigenvectors of $X$ are Haar-uniform on the orthogonal group, we derive a dynamical mean-field limit for the empirical law of sample paths, extending the classical Sompolinsky–Zippelius characterization for $X\sim\mathrm{GOE}$. The limit takes the form of a generalized Langevin equation with correlated Gaussian noise and memory, whose correlation and response kernels relate to those of the original dynamics through convolution equations involving the free cumulants of the eigenvalue distribution of $X$. For the overdamped Langevin diffusion associated with $\mu(\boldsymbol{\theta})\propto \exp\!\big(\frac12\boldsymbol{\theta}^{\top}X\boldsymbol{\theta}\big)\prod_{i=1}^n\nu(\mathrm{d}\theta_i)$, we analyze the mean-field limit under a rapid-mixing assumption. The correlation and response kernels admit time-translation-invariant approximants satisfying a fluctuation-dissipation relation. The generalized Langevin equation admits a Markovian approximation coupled to an auxiliary multivariate OU process and converges to a replica-symmetric prediction for the empirical coordinate law under $\mu$. This auxiliary correlation structure is characterized through the infinitesimal generator of a Markov semigroup for the lifted path-history process. Consequently, the free energy converges to a replica-symmetric limit under an explicit high-temperature condition, which for an Ising model is $\|X\|_{\mathrm{op}}<1/2$. By recent dynamical universality results, the same free-energy characterization holds for deterministic models without random disorder when $X$ satisfies a set of deterministic delocalization conditions.

This paper aims to analyze a numerical scheme for semilinear subdiffusion problems with singular initial data beyond the $L^\infty$ framework. The main difficulty lies in the stronger singular behavior of the nonlinear term compared with previous analyses. Since the singular initial datum is too rough to guarantee a uniform $L^\infty$ bound for the solution, the usual Lipschitz framework in the base space is no longer sufficient. The analysis must instead be carried out in weaker fractional Sobolev-type spaces, where nonlinear composition is more delicate and the term $f(u(t))$ may exhibit an amplified singularity relative to that of $u(t)$. To overcome this difficulty, we exploit the smoothing properties of the subdiffusion solution operators and formulate suitable nonlinear assumptions in fractional operator spaces. These smoothing estimates allow part of the singularity to be transferred from the nonlinear term to the solution operators, where it can be controlled. Under these assumptions, we establish well-posedness and regularity results for the mild solution and derive a pointwise-in-time error estimate for the exponential convolution quadrature method. Numerical experiments confirm the predicted convergence rates.

We prove that, over an arbitrary field and whenever $q$ is not a root of unity, the Baseilhac–Kolb root vectors form a PBW basis of the $q$-Onsager algebra for every total order on the positive roots of $\widehat{\mathfrak{sl}}_2$. This removes the previous transcendence hypothesis. We establish twelve PBW bases in the alternating generators and show that they persist under arbitrary scalar central specialisation of the alternating central extension. We determine the centraliser of the negative alternating subalgebra and that of the first imaginary alternating generator, and deduce that the four single-family alternating polynomial subalgebras are maximal commutative. Together, these results settle four conjectures of Terwilliger and, in characteristic different from $2$, a conjecture of Baseilhac and Belliard.

We answer the automorphism question raised by Naskręcki and Pokora for their $(24_4, 32_3)$-configuration on the Schur quartic. The stabilizer of either $24$-line half is isomorphic to $W(D_4)\rtimes C_3$, where $C_3$ acts by even triality; the full projective automorphism group has order $1152$. The $D_4$ model also gives an intrinsic coloring and shortens the incidence check.

In this paper, we prove the quantitative propagation of chaos results that allow us to derive multi-species cross-diffusion equations from moderately interacting stochastic particle system. The quantitative propagation of chaos result in $L^1$-norm is obtained by the relative entropy method, and the proof is carried out in two steps. In the first step, we quantify the relative entropy between the joint distribution of the particle system and the tensorised solution of the PDE at the intermediate level. In the second step, we establish a rigorous convergence rate to the multi-species cross-diffusion equations by analyzing the $L^2$-distance between the solution of the intermediate-level PDE and that of the limiting PDE. Furthermore, combining the strong $L^1$-convergence for the propagation of chaos with the $L^p$-estimates $(2\le p<\infty)$ for the marginal distribution of multi-species particle system, we derive the corresponding $L^q$-result $(1<q<\infty)$ via interpolation.

Modelling rarefied gas flow using the Boltzmann equation is vital in many areas. Because of the high dimensionality and coexistence of multiple characteristic scales, conventional solution strategies for this equation incur prohibitively high computational costs and are inadequate for rapid response in engineering design simulations. Based on proper generalised decomposition (PGD), we propose an a priori, asymptotic-preserving reduced-order method for the high-dimensional parameterised Shakhov kinetic model equation. Separated representations reduce the kinetic problem to a set of low-dimensional problems. We incorporate synthetic equations into the PGD iteration so that, in the hydrodynamic limit, the iteration reduces to solving the Navier-Stokes equations, whose solution naturally exhibits a low-rank structure. Furthermore, treating the rarefaction parameter as an additional coordinate yields a solution over the whole range of rarefaction in one offline computation. It enables fast online queries to any point in the parameter space. Numerical examples using this method show reasonable precision and a significant reduction in computational cost.

Chan and Pak introduced the relative linear extension ratio $\rho(P,x)=e(P)/e(P-x)$, where $e(P)$ is the number of linear extensions of a finite poset $P$, and let $\nu(c,d)$ be the least number of elements of a poset that realizes $\rho(P,x)=d/c$. They proved that $\nu(c,d)\le d/c+O(\log d\log\log d)$ for $d\ge 3c$, and asked whether the hypothesis $d\ge 3c$ can be relaxed to $d\ge(1+\varepsilon)c$ or removed. We prove the fixed-gap form of this question: for every fixed $\varepsilon>0$, $\nu(c,d)\le \frac{d}{c}+O_{\varepsilon}(\log d\log\log d)$ whenever $d\ge(1+\varepsilon)c$, and the implied constant is absolute once $d\ge 2c$. The new ingredient is a one-element overlap construction: if $x$ is minimal in $P$ and $y$ is minimal in $Q$, then there is a poset $R$ with $|R|=|P|+|Q|-1$ and an element $z$ such that $\rho(R,z)=\rho(P,x)+\rho(Q,y)-1$. Together with the continued-fraction construction of Chan and Pak and Rukavishnikova's tail bound for sums of partial quotients, this removes the factor $3$ in their range. We also show that the fixed-gap hypothesis is essentially optimal for this construction. In the range $1 < d/c < 2$, with $h=d-c$, the size bound the construction can certify is at least $\lfloor c/h\rfloor$, so the method reaches the stated error term only when $h$ is at least of order $c/(\log c\log\log c)$. The remaining obstruction to removing the hypothesis is a short-interval problem for sums of partial quotients, which we describe. The deductive part of the argument has been checked with the Lean proof assistant.

We prove the existence of the dual of a polarizable log abelian variety.

We introduce a unified entropic framework for real fluids that encompasses the van der Waals, Berthelot, Redlich Kwong, and Peng Robinson equations of state within a common thermodynamic description. The corresponding microscopic interactions are then explored using Geometrothermodynamics, GTD, through the scalar curvature $mathcal{R}$ of the equilibrium manifold. We show that curvature singularities accurately reproduce macroscopic critical phenomena, while vanishing curvature $\mathcal{R}=0$ identifies specific thermodynamic states where attractive and repulsive intermolecular forces effectively balance. Furthermore, we introduce a set of dimensionless critical-amplitude ratios $Q^i_{j}$, which reveal universal geometric features of the critical regime. Although individual critical amplitudes exhibit a logarithmic dependence on the system size, these invariant ratios organize different molecular species according to the strength of criticality and encode universal scaling features, suggesting their potential as robust classification parameters. Finally, employing Bayesian inference and Markov Chain Monte Carlo, MCMC methods, we statistically reconstruct the zero-curvature curves. The posterior distributions support the consistency of the geometric scaling behavior, demonstrating that the GTD manifold encodes non-trivial information about the underlying thermodynamical models.