Papers

We give a mathematical proof of the optimal convergence rates for the D2Q9 BGK lattice Boltzmann method with the half-way bounce-back rule for the incompressible Stokes equations in a flat channel. The convergence rates are second-order for the velocity and first-order for the pressure as the lattice spacing $h$ tends to zero, in agreement with formal analyses and numerical experiments, whereas the available rigorous convergence theorems only yield an $O(h^{1/2})$ bound for the velocity error. A key step in the proof is a decomposition of the leading boundary consistency error into macroscopic and kinetic components. These components are absorbed by suitably constructed Stokes and discrete Knudsen layer correctors, respectively. Incorporating these correctors into the prediction function used in previous rigorous analyses, we obtain a refined prediction function with consistency errors of sufficiently high order. Combined with the known weighted $L^2$-stability estimate, this gives the optimal convergence rates.

Let $(X,L)$ be a polarized complex abelian surface with $L^2=2d$. We establish a Reider-type criterion for Property $N_p$. If $d\geq7$, then $L$ satisfies Property $N_0$ if and only if there is no elliptic curve $E\subseteq X$ with $L\cdot E\leq2$, with one explicitly described exception. If $p\geq1$ and $d\geq(p+2)^2+1$, then $L$ satisfies Property $N_p$ if and only if there is no elliptic curve $E\subseteq X$ with $L\cdot E\leq p+2$. The numerical bounds on $d$ are optimal for $p=0,1$. These results improve upon earlier work of Küronya–Lozovanu, Ito, and Rojas. We also construct a polarized abelian surface whose basepoint-freeness threshold is irrational.

In this paper, we establish an elementary functional identity \begin{align*}\sum_{k=1}^{n}\bigg(\frac{f(a,d_k)}{f(a,x)}\bigg)^{n-1}\prod_{i=1,i \neq k}^{n} \frac{g(x,d_i)}{g(d_k,d_i)}=1\end{align*} based on a pair of functions $f(x, y)$ and $g(x,y)$ satisfying $$f(x, a) g(b, c)+f(x, b) g(c, a)+f(x, c) g(a, b)=0.$$ The latter may serve as a prototype for the classical Weierstrass theta identity. As a direct application, we establish a general transformation formula for finite sums. Some special algebraic, triangular and elliptic series transformations are also presented.

The spaces $\mathcal Q_\alpha(\mathbb R^n)$ for the critical index range $\alpha\in(0,\min\{1,n/2\})$ form a scale-invariant family lying strictly between the space of constant functions and $\mathrm{BMO}(\mathbb R^n)$. A long-standing open problem, posed by Essén, Janson, Peng and Xiao in 2000, asks to characterize the quasisymmetric mappings $\varphi:\ \mathbb R^n\to\mathbb R^n$ for which the composition operator $\mathcal C_\varphi(f)=f\circ\varphi^{-1}$ is bounded on $\mathcal Q_\alpha(\mathbb R^n)$. This paper establishes the full intrinsic characterization by introducing a new geometric quantity, the capacity gauge $\|\varphi\|_{\mathscr G_\beta}$, which measures the distortion of $\varphi$ through pullback volume ratios along the dyadic tree. We prove that for any quasisymmetric mapping $\varphi$ (with a mild Muckenhoupt \(A_\infty(\mathbb R)\) assumption on the Jacobian determinants of \(\varphi\) and \(\varphi^{-1}\) when \(n=1\)), the composition operator $\mathcal C_\varphi$ is bounded on $\mathcal Q_\alpha(\mathbb R^n)$ if and only if $\|\varphi\|_{\mathscr G_{1-\frac{2\alpha}{n}}}<\infty,$ with quantitative equivalence $$ \|\mathcal C_\varphi\|_{\mathcal Q_\alpha(\mathbb R^n)\to\mathcal Q_\alpha(\mathbb R^n)}^2 \simeq \|\varphi\|_{\mathscr G_{1-\frac{2\alpha}{n}}}. $$ This sharp, necessary and sufficient characterization refines earlier sufficient criteria of Koskela, Xiao, Zhang and Zhou in 2017, which were formulated in terms of local or global self-similar Minkowski dimension of the exceptional sets of the Jacobian. As applications, we obtain the composition stability of the finite capacity gauge, the invariance of $\mathcal Q_\alpha$-removability under quasisymmetric mappings with finite capacity gauge, and the propagation of $\mathcal Q_\alpha(\mathbb R^n)$-regularity and initial-data stability for transport equations driven by quasisymmetric flows.

Low-altitude wireless networks (LAWNs) integrate terrestrial and aerial platforms to provide ubiquitous communication, sensing, and localization services for unmanned aerial vehicles (UAVs) and electric vertical takeoff and landing (eVTOL) aircraft. However, dynamic air-ground and air-air channels, abrupt blockages, and heterogeneous interference hinder the realization of this goal. Nevertheless, fluid antenna (FA), a cutting-edge multiple-input multiple-output (MIMO) technique, overcomes these challenges by reconfiguring antenna positions to unlock additional spatial degrees-of-freedom. In this paper, towards bringing low-altitude FA networks into reality, we study the fast and high-performance FA reconfiguration for low-altitude FA networks with multi-agent reinforcement learning (MARL). Specifically, we present an electromagnetic digital twin (EM-DT)-assisted MARL framework. To fill the sim-to-real gap, we introduce a two-stage transfer learning framework. Our case study shows that joint FA positions and beamforming optimization can enhance the system sum-rate by 118.5%, compared to the fixed position baseline. This gain comes from the dynamic millisecond timescale reconfiguration of FA arrays and the adaptive steering of beams toward aerial users with mobility.

The Khintchine inequality provides a lower bound on the expected absolute value of a weighted sum of independent Rademacher random variables. In the classical setting, when the coefficient vector has unit Euclidean norm, this lower bound is a constant, with equality attained only by a simple family of extremal configurations. A refinement due to De, Diakonikolas, and Servedio, referred to as the linear Khintchine inequality in this work, strengthens this estimate by establishing a lower bound that depends linearly on the distance of the coefficient vector from the extremal set. In this paper, we present a refined analysis of this dependence on the weight vector. Our results reveal a phase transition in the rate of improvement: when the dimension exceeds six, the lower bound undergoes an abrupt change as the weight vector deviates from the minimizer. Additionally, we improve the slope constant in linear Khintchine inequality. As a consequence, we establish an improved lower bound on the degree-1 Fourier weight for linear threshold functions $\mathbf W^{\leq1}[f]\geq0.55100929$, marking progress towards a conjecture of Benjamini, Kalai, and Schramm.

We study symplectic embeddings between four-dimensional toric domains using embedded contact homology. Extending Hutchings' criterion for embeddings between convex toric domains, we obtain obstructions for embeddings from convex toric domains into concave toric domains, from semi-weakly convex toric domains into convex toric domains, and from concave toric domains into concave toric domains. The obstructions are formulated in terms of factorizations of convex, concave, and semi-weakly convex generators together with combinatorial constraints relating their ECH indices and actions. As applications, we recover the sharp obstruction for symplectic embeddings of a polydisk into a union of cylinders, and obtain sharp results for embeddings of certain quadrilateral toric domains into balls and ellipsoids. We also show that these obstructions are in some cases strictly stronger than those coming only from ECH capacities.

We prove that there exist hyperbolic integer homology spheres with arbitrarily large Dehn surgery number. Previously, no integer homology sphere was known to have a surgery number larger than $2$. Our approach uses Froyshov's invariant $q_3$ of integer homology spheres, which is defined in terms of mod 2 instanton homology. We show that if $W: Y \to Y'$ is a cobordism between integer homology spheres with no $2$-torsion in its first homology, then $-b^+(W) \le q_3(Y') - q_3(Y) \le b^-(W)$. We also extend both $q_3$ and the inequality to rational homology spheres.

We construct regular, spherically symmetric, compact energy-cutoff equilibria for gravitational Vlasov–Poisson systems sourced by positive and negative mass distributions in the context of stellar dynamics. Motivated by Bondi's notion of negative mass and its relation to the weak field limit of Einstein gravity with a signed mass–energy source, we show that the internal structure of the steady states depends qualitatively on the type of negative mass present. In the Bondi convention, the signs of inertial, passive gravitational, and active gravitational mass are reversed together, preserving the equivalence principle. The resulting equilibria consist of a central overlap core containing both mass species, surrounded by a finite positive mass shell and an exterior vacuum. Thus, although a pure Bondi negative mass gas cannot form a compact steady state, Bondi negative mass can be spatially confined by a suitable positive mass distribution. In the gravitational-charge convention, only the passive and active gravitational masses change sign, so that the two species obey opposite free-fall laws and unlike masses repel. The resulting equilibria are spatially segregated into a negative mass core, a vacuum gap, a positive mass shell, and an exterior vacuum. For cutoff exponent $n=-1/2$, the radial matching is explicit, while for the regular cutoff $n=1/2$ the matter regions satisfy Lane–Emden-type equations. These results provide a kinetic framework for investigating the distribution of negative mass in astrophysical and cosmological settings.

In this paper, we study the degree of irrationality of properly elliptic surfaces with a section. We prove $\min\{\chi(\mathcal O_S),\,2\operatorname{gon}(C)\} \leq \operatorname{irr}(S) \leq 2\operatorname{gon}(C)$. The lower bound is obtained from the canonical bundle formula and the Cayley–Bacharach property. We show that this bound is sharp. We also study the behavior of the degree of irrationality in moduli. A very general properly elliptic surface with a section over a curve of genus at least two has degree of irrationality at least four, whereas special families with $\chi(\mathcal O_S)=1$ or $2$ have degree two. Finally, we investigate properly elliptic surfaces with $\chi(\mathcal O_S)=0$, proving a generic lower bound of four and showing that $\operatorname{irr}(C\times E)=4$ for every hyperelliptic curve $C$ of genus at least two and every elliptic curve $E$. Our paper also includes special properly elliptic surfaces without a section. For Dolgachev surfaces, we exclude degree two for a very general member and construct special examples of degrees two and three.

Eigenspaces of hypercube graph adjacency matrices are well understood. We extend this to subgraphs obtained by removing a vertex or antipodal pair of vertices, motivated by a question on zonotopes.

The problem of sampling from a probability distribution arises in many applications such as posterior sampling in hierarchical Bayesian inverse problems and Gaussian processes for machine learning. Markov chain Monte Carlo (MCMC) algorithms are often used for sampling from a target probability distribution, but implementations can be computationally expensive, especially for large-scale problems. In certain applications, the target distribution naturally factorizes into a lower dimensional marginal distribution and a conditional distribution that allows exact sampling. We describe an MCMC algorithm called MarCo that exploits such a structure and generates a Markov chain via Metropolis-Hastings sampling from the marginal distribution, followed by sampling from the exact conditional distribution. By design, MarCo constructs a Markov chain on the joint space that inherits the convergence behavior of the marginal MCMC algorithm. This provides multiple theoretical and computational advantages. We prove that MarCo can achieve improved mixing time upper bounds compared to direct sampling from the joint distribution. Moreover, compared to one-block methods that also exploit marginal-conditional structure, we use the framework of Peskun-Tierney ordering to show that MarCo has a larger right spectral gap and smaller asymptotic variance, thus leading to superior convergence properties. Numerical results illustrate the performance benefits of MarCo and are provided for various problems, including a semi-blind image deblurring example.

We investigate the extremal statistics of planar $\beta=2$ Coulomb gases with radial external fields. For the rightmost eigenvalue and the spectral radius, we establish sharp Berry–Esseen bounds for their convergence to the Gumbel distribution, with explicit rates \[ \frac{25\log\log n}{4e\log n} \quad\text{and}\quad \frac{2\log\log n}{e\log n}, \] respectively. In addition, we derive sharp asymptotic equivalences for the large and moderate deviations of both statistics across all relevant scales. Analogous results hold for the smallest modulus.

For a real binary form with no real roots, Julia's zero and the hyperbolic zero are two SL(2,R)-equivariant points in the upper half-plane. We show that they admit closely related equilibrium characterizations, with radial weights given respectively by the hyperbolic tangent and hyperbolic sine of the distances to the roots. This common framework gives geometric criteria for coincidence of the two zero maps. They always agree for binary quartics; for binary sextics they agree exactly when the three upper-half-plane roots form an equilateral hyperbolic triangle, or are collinear with one root the hyperbolic midpoint of the other two. We also obtain a corresponding result for collinear binary octics. The two equilibrium laws further reveal a sharp difference in the influence of distant roots. A strict majority of roots confined to a compact set keeps Julia's zero in a compact set, and the threshold one-half is optimal. In contrast, an escaping minority can force the hyperbolic zero to escape at linear scale. We also illustrate the computational advantages of the explicit formula for the hyperbolic zero.

Early discovery of at least one valid design satisfying a target requirement is a central objective in failure-prone closed-loop inverse design. A natural batch baseline ranks candidates by a product-form marginal valid-hit score, but selecting the highest-ranked candidates independently can produce redundant recommendations under predictive uncertainty and waste the experiment budget. We introduce ARC-SC(Anchored Risk-Constrained Scenario Coverage), a batch acquisition method that preserves strong marginal candidates as anchors and allocates the remaining batch positions by maximizing complementary coverage over predictive target scenarios under a risk-support constraint. In frozen-oracle closed-loop simulations on superconductivity and JARVIS materials-property benchmarks, ARC-SC yields a statistically supported improvement in first-hit discovery and remains competitive with directionally favorable first-hit performance on more challenging design space. These results establish ARC-SC as a POF-anchored, scenario-aware batch strategy for improving early valid-target discovery under structured experimental failure.

We study deep loci for the reduced-word Berenstein-Fomin-Zelevinsky cluster atlas on type-A double Bruhat cells. As defined in CGSS24, for a seed collection on a cluster variety X, the deep locus is the complement of the union of all corresponding cluster tori. We focus on double Bruhat cells G(u,v) for G = SL(n). Our main results are several theorems describing the structure of these deep loci, including their interaction with the Poisson structure, the relationships between the deep loci of different cells, and a tracking formula that allows us to compute the reduced-word deep locus of a cell from a lower-dimensional one. The paper also includes explicit low-rank calculations for double Bruhat cells in SL(3) and Borel double Bruhat cells in SL(4). Finally, we discuss the relationship between these results and some general properties of deep loci.

The rigorous justification of the Vlasov-Navier-Stokes system remains an outstanding open problem, both as a mean-field limit of an $N$-particle fluid-interacting system and as a hydrodynamic limit derived from multiphase Boltzmann equations. Inspired by the work of Bernard, Desvillettes, Golse, and Ricci [Commun. Math. Sci., 15(6), 1703-1741, 2017], we present a rigorous derivation of the incompressible Vlasov-Navier-Stokes system from the two-component Boltzmann system for elastic hard-sphere collisions. To this end, we establish the well-posedness of the rescaled multicomponent Boltzmann system for small initial data. Specifically, we obtain estimates for the solution that hold uniformly with respect to both the thermal speed ratio $\varepsilon$ and the mass ratio $\eta$. Furthermore, under the Vlasov-Navier-Stokes scaling assumption $O(1)\varepsilon^3 \leqslant \eta \leqslant o(1)\varepsilon^2$ as $\varepsilon \rightarrow 0$, we establish the weak convergence of the solution to the fluid-kinetic coupled limit system.

Direct numerical simulation (DNS) of compressible transitional and turbulent flows requires numerical methods that combine high-order accuracy, robustness, and computational efficiency to resolve a broad range of spatial and temporal scales. This paper presents a massively parallel hybridizable discontinuous Galerkin (HDG) solver for DNS of the compressible Navier-Stokes equations on GPU-accelerated high-performance computing systems. The proposed solver combines high-order HDG discretization with robust shock capturing, diagonally implicit Runge-Kutta (DIRK) time integration, and an efficient Newton-GMRES solution strategy accelerated by additive Schwarz preconditioning and reduced-basis approximation. A distributed implementation of these methods based on GPU-aware MPI, Kokkos, and CUDA/HIP libraries enables scalable execution on heterogeneous computing platforms. The solver is demonstrated on three canonical benchmark problems covering a wide range of Mach-number flow regimes: subsonic transitional flow over the Eppler 387 airfoil, the supersonic Taylor-Green vortex, and hypersonic boundary-layer transition. Numerical results are compared with available experimental measurements and published DNS data, showing good agreement across distinct flow regimes. The results demonstrate the ability of the proposed solver to resolve laminar-turbulent transition, strong compressibility effects, shock-associated flow structures, and fully three-dimensional turbulent dynamics.

We study the holomorphy of the ground state and its energy with respect to the coupling constant for bosonic quadratic Hamiltonians. We prove general theorems establishing this holomorphy, thereby providing a rigorous justification for formal perturbative expansions. This theory is applied to several concrete models, including the single pair interaction model and the Pauli-Fierz model in the dipole approximation. Importantly, for the Pauli-Fierz model with a realistic ultraviolet cutoff, the perturbation expansions in the elementary charge converge at the physical charge value, even when mass renormalization is taken into account. Furthermore, we explicitly determine the radii of convergence of the ground states and their energies for both the single pair interaction model and the fibers of the translation-invariant Pauli-Fierz model.

Matching Vector families (MVFs) are defined by two ordered lists of vectors in $\mathbb{Z}_m^n$ whose inner products satisfy specific residue patterns modulo an integer $m$. Most famously, restricted MVFs are used to construct the best-known constant-query Locally Decodable codes (LDCs). We prove an upper bound of $2^{O\left(\sqrt{n\log n \log m}\right)}$ on the size of $3$-restricted MVFs in $\mathbb{Z}_m^n$ for $m \leq \sqrt{n}$, substantially improving on the previous best bound of $2^{O(n/\log n)}$ by Bhowmick, Dvir and Lovett (STOC'13, SICOMP'14). Our proof relies on a new polynomial method argument that controls collisions in sumsets of matching vectors.

We study $\ell_0$ isoperimetry for a convex body $K\subset \mathbb{R}^n$, $n\ge2$. For a Borel set $S\subset K$, let $\partial_0^K S$ be the set of points in $K \setminus S$ that can be reached from $S$ by changing at most one coordinate (i.e. the $\ell_0$ boundary of $S$). Suppose that, for some unconditional convex body $Q \subset \mathbb{R}^n$, numbers $r,R>0$, and possibly different centers $x_0,y_0$, \[ x_0+rQ \subset K\subset y_0+RQ. \] Writing $s=\text{vol}(S)/\text{vol}(K)$, we prove that whenever $0<s \le 1/2$, \[ \frac{\text{vol}(\partial_0^K S)}{\text{vol}(S)} \ge \frac{cr}{nR} \min\left\{1,\frac{\log(e/s)}{n}\right\}, \] where $c > 0$ is an absolute constant. Consequently, the associated $\ell_0$-isoperimetric coefficient is at least $cr/(n^2R)$. Previous direct lower bounds were only known for $\ell_2$ and $\ell_\infty$ regularity whereas our lower bound holds directly for any $Q$-regularity, where $Q$ is an unconditional convex body. Compared to $\ell_2$ and $\ell_\infty$ regularity, our lower bound result improves upon the previously best known lower bounds, for any $s$, by a factor of $n$. As an application of our result, we give improved mixing time bounds for the Coordinate Hit and Run walk (CHAR). Our proof of the lower bound is based on a modification of the method of canonical paths applied to a continuous Hamming graph over our convex body. Our construction of canonical paths can be viewed as a suitable coordinate discretization of certain mass transport maps from $S$ to $S^c$. We also give complementary upper-bounds for any $Q$-regularity, with an overall factor of $n$ gap between the two.

This paper studies the $C^1$ existence and dynamical stability of temporal periodic solutions involving strong transonic shocks for the quasi–one–dimensional full compressible Euler equations in diverging nozzles. We reveal the physical structural dissipation mechanism of strong transonic shocks hidden in the Rankine–Hugoniot conditions. This allows us to estimate the resonance at the shock boundary caused by the left acoustic waves in the subsonic downstream region. With this dissipation, we complete the proof by developing a fraction–step linearized iterative method.

This paper explores the use of a reduced manifold space initialization method for the generation of both time and fuel optimal many-revolution, low-thrust transfers between three-body periodic orbits within a family. Libration point orbit families are approximated by multi-segment Chebyshev polynomials and 1-dimensional Fourier series. Control is mapped into the family space, and a 1-dimensional targeting scheme in the family space dynamics is used to generate an initial guess for the phase space, directly accounting for winding. An integral collocation-based direct method is then used to first continue a feasible trajectory into the phase space, and then optimize for a particular cost function. The methodology is first applied to the distant retrograde orbit family. Modifications for time-regularized dynamics are then discussed, with an ensuing example of many-revolution transfers in the $L_2$ halo family.

In this paper, we firstly study Strichartz estimates $\|e^{it D^\alpha} u_0\|_{L_t^q(\mathbb{R};L_x^r(\mathbb{R}^d))}\leq C(d,\alpha,q,r,s)\|u_0\|_{\dot{H}^s}.$ We show some counterexamples for $(q,r) = (2,\infty)$. Then we consider the embedding $X^{s,b}_\alpha \hookrightarrow L_t^q(\mathbb{R};L_x^r(\mathbb{R}^d))$, where $\|u\|_{X_\alpha^{s,b}}:=\|\langle\xi\rangle^s\langle \tau-|\xi|^\alpha\rangle^b\hat{u}(\tau,\xi)\|_{L^2_{\tau,\xi}}$. We present the necessary and sufficient conditions for this embedding.

In this article, we first investigate the necessary and sufficient conditions on the ranges of $s,s_1,s_2\in R$, $1\leq p_1,p_2,q\leq \infty$, $\theta_1,\theta_2\geq 0$ for the validity of the fractional Gagliardo–Nirenberg inequalities $$\|D^su\|_{L^q(R^d)}\lesssim \|D^{s_1}u\|_{L^{p_1}(R^d)}^{\theta_1}\|D^{s_2}u\|_{L^{p_2}(R^d)}^{\theta_2}.$$ Secondly, we consider the anisotropic Gagliardo–Nirenberg inequalities $$\|u\|_{L^q(R^d)}\lesssim \|u\|_{L^{p_0}(R^d)}^{\theta_0} \prod_{j=1}^n\|D_{x_j}^{s_j}u\|_{L^{p_j}(R^d)}^{\theta_j}.$$ We show the sharp conditions of these inequalities except “one" case. Finally, we establish the profile decomposition associated with the above inequalities when $p_j = 2$, $0\leq j\leq n$. Using these results, we establish existence of extremizers and construct soliton solutions of relevant dispersive equations.

We study how motivic and étale motivic cohomology of a smooth and proper variety changes under extensions of algebraically closed base fields $k$. With finite coefficients, motivic cohomology is independent of $k$, and étale motivic cohomology is independent of $k$ away from the characteristic, but depends on $k$ at the characteristic. We then consider the fiber of $F-1$ for $F$ the Frobenius, on motivic cohomology and étale motivic cohomology for varieties defined over finite fields. We show that this fiber is independent of $k$ with finite coefficients. In addition, we give some results and examples with integral coefficients.

We establish a version of the classical results concerning descriptions of isometries on $C^*$-algebras and noncommutative $L_p$-spaces due to Kadison (1951) and Yeadon (1981) in the setting of Haagerup–Schultz algebras. Precisely, we show that (not necessarily surjective) isometries on such algebras are necessarily implemented by partial isometries and trace-preserving Jordan $^*$-monomorphisms.

Exact cumulants of entanglement entropies of random state ensembles have traditionally been studied within the random matrix framework. In this work, we propose an alternative approach based on the intrinsic connection to integrable systems. The central idea is to embed entropic quantities into tau functions satisfying Toda-type lattice equations, which in turn yield linear difference equations for their averages. Directly solving the difference equations recovers exact entropy formulas in the literature. The integrable systems approach bypasses the case-by-case, ensemble-dependent derivations required by random matrix methods. The approach also suggests a possible route towards unified and more efficient higher-order cumulant calculations by exploring integrable hierarchies.

We study the continuum scaling limit of the Langevin dynamics for three-dimensional U(1) lattice Yang–Mills theory. The model is defined on the discrete 3D torus with a general class of plaquette actions that are suitably normalized, including Wilson, Manton, and Villain actions. Under the weak-coupling scaling and in the DeTurck gauge, we prove that the rescaled logarithmic field converges to the solution of the one-form stochastic heat equation in probability. In particular, the limiting dynamics are universal and do not depend on the higher-order details of the plaquette action.

We study strictly outer actions of locally compact groups with a compact open subgroup on von Neumann factors. For amenable groups, we prove 2-cohomology vanishing and obtain classification results using a description of the central sequence algebra and Rohlin-type observations. We also characterize the inclusions of factors associated with group actions, and extend M. Choda's result to this locally compact setting.