Papers

In this work, we prove area-charge inequalities for free boundary MOTS in initial data sets for the Einstein-Maxwell equations with vanishing magnetic fields. In addition, we prove a local rigidity result under the assumption that equality holds.

The purpose of this paper is to establish certain relationships between the theories of means (aggregation functions) and loss functions. Namely, we derive sufficient conditions under which the sequence of the expected losses of a quasi-arithmetic mean of the first $n$ members from a sequence of exchangeable random variables is (strictly) decreasing with respect to $n$. Some new properties of functions which are strongly convex with respect to a nonnegative even error function, such as a Jensen-type inequality, are described. Additionally, non-trivial new examples for such functions are presented as well.

A pure completely contractive covariant representations of a $C^*$-correspondence dilate to a pure isometric covariant representations due to Muhly-Solel. More specifically, we are curious about the following question for pairs: Does a pure completely contractive covariant representations of a product system dilate to a pure isometric covariant representations of a product system? The goal of this study is to find a pure completely contractive covariant representations of a product system that provide an affirmative answer for the previous question.

This paper studies information design when the designer lacks precise knowledge of agents' payoff coefficients. The Calibrated Bayes Correlated Equilibrium (Cal-BCE) is introduced as a solution concept that augments the Bayes correlated equilibrium with a corrector policy preserving incentive compatibility under the designer's structural uncertainty, adapting its revelation principle to this setting. The design problem is nonconvex in general, but under a linear-quadratic-Gaussian structure it admits convex second-order cone and semidefinite reformulations under two-sided probabilistic and conditional value-at-risk (CVaR) constraints, with feasibility guaranteed by a Hadamard invertibility condition. A joint decentralization theorem shows that both designs cap cross-agent action covariances, the CVaR design more tightly at a common tolerance; but because the formulations operate at design-specific feasibility thresholds, the realized ordering is calibration-dependent. Experiments on fifteen sector ETFs confirm the trade-off: the probabilistic design attains higher mean welfare and the CVaR design better tail protection, with neither dominating outright.

We study the problem of cooking the yolk and albumen of a hen's egg to their respective optimal temperatures of $T_Y^* = 65^\circ$C and $T_W^* = 85^\circ$C, subject to the requirement that neither temperature ever exceed its target at any time during cooking, since temporary overshoot still overcooks the egg even if the final reading is correct. We model the egg as a two-domain sphere with distinct thermal diffusivities, and take the Laplace transform of the heat equation in each domain, reducing the problem to a $3 \times 3$ linear system in the transform variable $s$ with hyperbolic-trigonometric solutions. The resulting transform is inverted numerically via Talbot's method and validated against a finite-difference solver. A single boiling phase cannot satisfy the no-overshoot requirement: the thin outer albumen heats far faster than the insulated yolk and necessarily overshoots $T_W^*$ before the yolk approaches $T_Y^*$. We show that a three-phase protocol resolves this: a sous-vide pre-soak at exactly $65^\circ$C (which cannot overshoot since the bath temperature equals the target), a short boil to bring the albumen toward $T_W^*$, and an ice-water bath that arrests the albumen's residual overshoot while residual heat continues raising the yolk to its target. Optimizing the phase durations gives $17.26$ minutes of sous-vide, $66$ seconds of boiling, and an ice bath, achieving both targets at $T^* \approx 20.67$ minutes with neither constraint violated at any time. This compares favorably with the periodic protocol of Di Lorenzo et al. (2025), which requires 32 minutes and misses both targets substantially.

The Hales-Jewett number HJ(t,r) is the least dimension n such that every r-coloring of the grid [t]^n contains a monochromatic combinatorial line. We prove HJ(3,3) >= 22 and HJ(4,2) >= 14, improving the previous records 14 and 12. The engine is an exact reduction: a coloring of [t]^n invariant under coordinate permutations descends to the discrete simplex of letter-count vectors, where a combinatorial line is precisely a corner tuple; the 4,387,586,157,901 lines of [3]^21 thereby compress to 1771 local conditions on 253 cells. We prove that this symmetric class coincides with the class of one-weight colorings, those reading an integer-weighted count of the letters: a radix weight realizes every symmetric coloring, so the symmetric lower-bound problem is a one-dimensional homothety-avoidance problem, the case d=1 of Gallai's theorem. This yields the closed-form bound HJ(t,r) >= ceil((G_r(S)-1)/D_S) in terms of the Gallai homothety numbers G_r(S), together with the new values G_3({0,1,3})=42, G_3({0,1,4})=57, G_2({0,2,3,5})=67, G_2({0,1,5,6})=80, and G_3({0,2,5})=77, giving HJ(3,3) >= 16 from a one-line certificate. Further results: periodic one-weight palettes give HJ^[12](3,3) = HJ^[12](4,2) = infinity for lines with at most twelve active coordinates; the interval number HJ^(1)(3) is exactly 5; a Rado reading gives G_4({0,1,3}) >= 94 and R_4(z+2x=3y) >= 59; and a rainbow companion gives the anti-Hales-Jewett bound ah(3,4) >= 25. A SAT program written for this article pushes the computation further: eighteen exact two-color Gallai numbers of four-point sets, up to G_2({0,1,6,7}) = G_2({0,3,4,7}) = 79; the exact three-color value G_3({0,1,5}) = 70; and the exact Rado numbers R_r(z+kx=(k+1)y) for 2 <= k <= 5 and r in {2,3}. Every displayed certificate is verified by direct enumeration; certificates and verification scripts are available at https://github.com/ysmouhib/hj-certificates.

In this paper, we establish several properties of the block spaces $B_q^{0,v}$. Our primary contribution resolves an open question regarding the relationship between $B_q^{0,v}$ and the Zygmund-type space $L(\log L)^{1+v}$ by demonstrating that $B_q^{0,v}$ is a proper subspace of $L(\log L)^{1+v}$. Furthermore, we prove that the union of all block spaces $B_q^{0,v}$ for $1 < q \le \infty$ is strictly contained within $L(\log L)^{1+v}$.

In this article, we study the number of limit cycles, bifurcating from the period annulus of any quasi-homogeneous polynomial vector fields with a center, under a one-parameter polynomial perturbation. We first recharacterize quasi-homogeneous polynomial vector fields and its global center, then we establish an upper bound formula for the number of the isolated zeros of the $k$th order Melnikov function in terms of $k$, $max \{s_1,s_2\}$ and the degree $n$ of the perturbation by applying adapted Francoise's algorithm in conjunction with combinatorial techniques, where $(s_1,s_2)$ is the weight exponent of the quasi-homogeneous polynomial vector field. This extends relevant results presented in the literature [JDDE,21(2009)133-152] and [JDE, 276(2021)1-24]. As an application, we completely solve the limit cycle bifurcation problem of a perturbated quasi-homogeneous polynomial vector field.

Let $V$ be a vertex operator algebra, and let $W^1$ and $W^2$ be restricted $V$-modules. We construct a generalized $V$-module $\mathcal{H}(W^1, W^2)$ characterized by canonical universal properties. We show that, under suitable hypotheses, $ \mathcal{H}(W^1, W^2)$ realizes the internal Hom object in the tensor category of restricted $V$-modules. Although our construction differs from Li's, we show that it agrees with the natural logarithmic generalization of Li's module $\Delta(W^1, W^2)$. We further establish a canonical isomorphism between $\mathcal{H} \big(W^1,(W^2 )^\prime \big)$ and the $P(z_0)$-dual product $ W^1 \pzbox_{P(z_0)} W^2 $ recently constructed by Du and Huang. Under the $C_1$-cofiniteness condition, we investigate finiteness properties of $ \mathcal H(W^1, W^2)$. As applications, we obtain a natural isomorphism between $ \mathcal H(W^1, W^2)'$ and $ W^1 \boxtimes (W^2)'$, and prove the finiteness of the corresponding fusion rules.

This study provides initial results on the boundedness of the (smooth) bilinear quadratic functional defined by an arbitrary collection of disjoint strips. The square function under consideration is a combination between the well-known Rubio de Francia square function from the linear setting with the bilinear Hilbert transform's singularity structure, which involves modulation invariance.

We introduce codistinguished abelian subcategories of triangulated categories, which generalise Jorgensen's proper abelian subcategories and are dual to Linckelmann's distinguished abelian subcategories. We show that they come equipped with a non-trivial exact structure induced by the ambient triangulated category, and that every exact structure on an abelian category arises in this way. We also show that fully faithful adjunction triples produce new codistinguished abelian subcategories from old.

We formulate an operator-algebraic mechanism by which exact Boolean records can arise from local completely positive quantum operations without being imposed as microscopic structure. The kinematic input is an algebraic process functional assigning probabilities to local normal completely positive operations in a finite operational context. From the predual response of a target algebra to source interventions, relative to a background strategy class, we define an influence algebra; exact events are then, by definition, the projections in its center. The dynamical question is whether nontrivial centers can be generated by coarse-graining rather than inserted through split-record laboratories. We address this question using state-preserving normal unital completely positive coarse-graining channels. If the Cesaro means of such a channel converge to a Choi-Effros infrared range and the range is asymptotically abelian in the GNS seminorm, then the represented infrared algebra is a commutative von Neumann algebra. Its projection lattice is therefore a complete Boolean algebra. We also give a finite-sector block-primitive criterion, motivated by locality and scrambling, which implies this asymptotic abelianness with exponential suppression of off-sector coherences and intra-sector fluctuations. The result is a conservative mathematical statement: classical facts are not identified with arbitrary projections of a Type-III local algebra, but with central projections selected by an asymptotically abelian completely positive infrared limit.

This paper presents a complete two-dimensional theoretical model for the electromagnetic behavior of square-section solid magnetic bars under sinusoidal loading. Through the application of Maxwell's equations within a Cartesian coordinate system and the integration of complex permeability, exact mathematical expressions are derived for mutual impedance, internal magnetic fields, flux, and core losses. Hyperbolic functions are utilized to separate the variables, enabling the accurate representation of edge flux accumulation and the 2D skin effect. In addition to mathematically decoupling eddy current and hysteresis losses, this formulation yields a new apparent permeability parameter. This parameter establishes a fast, reliable method for magnetic steel characterization that bypasses the extensive processing times associated with Finite Element Analysis (FEA). Numerical results over 1 Hz-1 MHz show the apparent relative permeability decreasing from 500 to 300 and a characteristic resistance peak near 700 kHz, marking the transition from volumetric to surface-dominated loss regimes.

Let $G$ be an edge-colored graph on $n$ vertices, and let $\deltac(G)$ denote its minimum color degree. Li and, independently Li, Ning, Xu, and Zhang, proved that every edge-colored graph on $n$ vertices with $\deltac(G) \ge \frac{n+1}{2}$ contains a rainbow triangle. Let $\rt(G)$ denote the number of rainbow triangles in $G$, and define \[ f(n) = \min\{ \rt(G) : |V(G)| = n,\ \deltac(G) \ge (n+1)/2 \}. \] In \cite{LiNingShiZhang2024}, the following open problem was posed: determine all the values of $f(n)$. In this paper, we determine $f(n)$ completely: $f(n) = (n^2-1)/8$ for odd $n\geq 3$, $f(n) = \frac{n^2}{4} - 1$ for all even $n \ge 6,$ and $f(4) = 4$. This resolves an open problem raised in \cite{LiNingShiZhang2024}.

We propose a graph generative model for sequences of extremely sparse, edge-exchangeable networks. Models for sparse graphs often face a trade-off between desirable properties like exchangeability and the ability to capture the sparsity observed in real-world networks. While models based on vertex or edge exchangeability have successfully generated sparse graphs, achieving the "extremely sparse" regime, where the number of edges scales near-linearly with the number of nodes, has remained a challenge. Recently, a novel Completely Random Measure (CRM) was introduced, demonstrating that this rate could be achieved within the vertex-exchangeable framework of Caron and Fox. This paper extends this work by demonstrating that this new CRM can be integrated into the alternative edge-exchangeable framework to achieve extreme sparsity.

We study a homogenization problem for first-order Hamilton-Jacobi equations in the Wasserstein space with a convex Hamiltonian. We show that the solution $U^\varepsilon$, which is the value function of a mean field control problem, converges uniformly as $\varepsilon \to 0$ to the solution of a limiting Hamilton-Jacobi equation whose Hamiltonian is obtained through a suitable cell problem. Furthermore, we establish quantitative rates of convergence. Under general assumptions with multiscale dependence, we prove that the rate of convergence is $O(\sqrt{\varepsilon})$. When the Hamiltonian depends only on the fast variable and the momentum, we establish the sharp convergence rate $O(\varepsilon)$. To the best of our knowledge, this is the first quantitative convergence result extending the optimal rate for first-order Hamilton-Jacobi equations in finite dimensions to the Wasserstein space. Finally, we show that our analysis extends to dynamic optimal transport problems, where the terminal condition imposes a constraint on the final distribution.

In this article, we establish geometric and analytic constraints imposed by the existence of nontrivial solutions to the Vafa-Witten equations on closed 4-manifolds. Using conformal invariance and refined Bochner-type estimates, we first prove an inequality relating the Yamabe constant $Y(g)$ to the $L^{2}$-norm of the self-dual Weyl tensor: $Y(g)\leq 2\sqrt{6}\|W_{g}^{+}\|_{L^2}$; when $Y(g)>0$, this yields a topological lower bound $\int_{M} |W_{g}^{+}|^{2} \geq \frac{4}{3}\pi^{2}(2\chi(M)+3\sigma(M))$. In the equality case, we show that the manifold must be Kähler with nonnegative scalar curvature and that the connection is reducible. As an application, for positive Einstein manifolds with $\operatorname{Ric}=3g$ admitting an irreducible Vafa-Witten solution, we obtain a sharp volume bound and prove the manifold cannot be Kähler. Through dimensional reduction $S^{1}\times N$, we establish a one-to-one correspondence between stable flat connections on a closed 3-manifold $N$ and $S^{1}$-invariant Vafa-Witten solutions, which yields a new estimate for the Yamabe constant $Y(g_{S^{1}\times N})\leq 2\sqrt{6\pi}\big(\int_{N}|\operatorname{Ric}(g_{N})-\frac{1}{3} R_{g_{N}}g_{N}|^2\big)^{1/2}$. Finally, under a regularity assumption that every anti-self-dual connection in the compactified moduli space is regular, we prove an energy gap: there exists $\varepsilon(g,P)>0$ such that any Vafa-Witten solution satisfies either $F_{A}^{+}\equiv0$ or $\|F_{A}^{+}\|_{L^{2}}\geq\varepsilon$.

We show that some well known results concerning compact disjointness preserving operators on Banach lattices can be extended to the more general framework of finitely generated Banach $C(K)$-modules.

Let $M$ be a closed orientable $3$-manifold admitting a metric of nonpositive sectional curvature (an NPC metric), and let $G$ be a finite group acting freely on $M$ by orientation-preserving diffeomorphisms. Previous results showed that $M$ admits a $G$-invariant NPC metric except possibly when $M$ is a graph manifold. In this paper, we resolve the remaining case by proving that $M$ also admits a $G$-invariant NPC metric when $M$ is a graph manifold. This result advances our understanding in dimension $3$ of the question posed by Schoen-Yau about Nielsen realization for NPC $3$-manifolds.

In this paper, based on the fast averaged L1 method, we present an error analysis for time-fractional advection-dispersion equations with a weak singularity at the initial time. An integrating-factor transformation is introduced to convert the tempered fractional derivative into the standard Caputo derivative, which is more suitable for discretization using the fast averaged L1 method. A sum-of-exponentials approximation is then incorporated into the averaged L1 method to reduce computational cost and storage while preserving the desired accuracy. By deriving error estimates for the discrete coefficients and the accumulated truncation errors, we establish the stability and $H^1$-norm convergence analysis, with a convergence order higher than those in the published literature. Numerical examples are tested to validate our theoretical results. The effects of the fractional parameters $\alpha$ and $\lambda$ on the solution are discussed. The memory effect and long-time tail phenomenon, which are known to exist in real systems yet cannot be captured by classical integer-order equations, are again found in the current fractional case.

Standard continuous-depth models, such as Neural Ordinary Differential Equations (NODEs), offer significant advantages in modeling physical systems by learning continuous vector fields rather than discrete temporal steps. However, when applied to complex dynamical systems, standard NODEs frequently struggle with highly nonlinear dynamics. This paper investigates the Frequency-domain Neural ODE (FNODE), an architecture that projects continuous temporal dynamics into the frequency domain using the Fast Fourier Transform (FFT). By operating in the frequency domain, the model provides better generalization to the dynamical system. The architecture is empirically evaluated against discrete models, specifically Gated Recurrent Units (GRUs) and Long Short-Term Memory (LSTMs), and other continuous-depth variants, including Augmented Neural ODE (ANODE), across four distinct dynamical systems: the Lotka-Volterra model, the forced Duffing oscillator, the Van der Pol oscillator, and the Lorenz system. To rigorously assess generalization and robustness, curriculum and ensemble learning are used to evaluate the model's convergence by estimating confidence intervals across different ensemble models. The empirical results demonstrate that the FNODE architecture achieves better generalization while exhibiting remarkable convergence stability.

We compare Chatterjee's rank correlation $\xi$ with Kendall's $\tau$ and Spearman's $\rho$ under positive-dependence assumptions on bivariate copulas. Our main technical contribution is a sharp order-violation bound for two stochastically ordered distribution functions. This local inequality controls each conditional order-violation probability appearing in Kendall's tau by the cross-rank variance functionals that determine Chatterjee's rank correlation. As a consequence, we prove the sharp Kendall bound $\xi(C)\leq \tau(C)$ for every stochastically increasing copula $C$. The bound is best possible: ordinal sums of product copulas attain equality. We also prove that the weaker left-tail decreasing (LTD) and right-tail increasing (RTI) conditions jointly imply the Spearman bound $\xi(C)\leq \rho(C)$, with equality if and only if $C$ is either the independence or comonotonicity copula. Finally, checkerboard examples show that LTD or RTI alone does not imply $\xi(C)\leq\rho(C)$, that LTD and RTI together do not imply $\xi(C)\leq\tau(C)$, and that both bounds are directional for $\xi$.

In this note, we look at the structure and properties of palindromic or Pascal-like Sprugnoli arrays. We show that there are two closely related families of these arrays. We give closed form expressions for the elements of these families, and in each case, we describe the form of the inverse arrays. Finally, we consider the arrays modulo $2$ and the resulting arithmetic sequences.

We establish the generator polynomials of the Euclidean and Hermitian duals of separable constacyclic codes over $\mathcal{S} = \mathbb{F}_q \times (\mathbb{F}_q+v\mathbb{F}_q)$, with $q$ an odd prime power and $v^2=v$, and we derive the generator polynomials of their Gray images, respectively. The generator polynomials of the Euclidean hulls and Hermitian hulls of separable constacyclic codes over $\mathcal{S}$ and their Gray images are presented, respectively. Furthermore, we provide the generator polynomials of the Euclidean sums and Hermitian sums of separable constacyclic codes and their Gray images, respectively. Finally, we propose two methods to yield quantum error-correcting codes (QECCs) from the hulls and sums of separable constacyclic codes over $\mathcal{S}$, and generate new QECCs that outperform the existing ones in terms of parameters.

Baricz, Prabhu K, Singh and Vijesh asked for the optimal hyperbolically completely monotone (HCM) range of the noncentral chi-square density. The problem was motivated by the gap between known infinite divisibility and the stronger generalized-gamma-convolution/HCM classification. We prove that the HCM range is exactly the central line. More generally, for $a,b>0$ and $\theta\ge 0$, the density $p_{a,b,\theta}(x)$ is HCM if and only if $\theta=0$. Thus the noncentral chi-square density satisfies $\chi_{\mu,\lambda}\in\mathrm{HCM}$ if and only if $\lambda=0$. The proof uses the leading small-$u$ HCM signs of $p(uv)p(u/v)$. These signs are governed by complete Bell polynomials whose signed generating function is $e^{bt}{}_0F_1(;a;-b\theta t)$. A positive zero inherited from $J_{a-1}$ rules out nonnegative Taylor coefficients when $\theta>0$. Consequently, Poisson shape-mixtures of HCM gamma densities need not be HCM.

For integers $n\ge d+1$, let $\mathsf{M}_d(n)$ denote the maximum size of a $(d+1)$-uniform family on an $n$-element ground set with VC-dimension at most $d$. For $n\ge2d+2$, the classical construction of Ahlswede and Khachatrian, later generalized by Mubayi and Zhao, gives \[ \mathsf{M}_d(n)\ge \binom{n-1}{d}+\binom{n-4}{d-2}. \] We introduce a two-cover lifting construction and prove the recursive lower bound \[ \mathsf{M}_d(n)\ge \binom{n-1}{d}+\binom{n-4}{d-2}+\mathsf{M}_{d-3}(n-5) \] for every $d\ge 3$ and $n\ge d+3$. Consequently, \[ \mathsf{M}_d(n)\ge \binom{n-1}{d}+\binom{n-4}{d-2}+\binom{n-6}{d-3}. \] Thus the Mubayi–Zhao conjecture on the exact value of $\mathsf{M}_d(n)$ for $n\ge2(d+2)$ is false for any $d\ge 3$. The proof is elementary and proceeds entirely through an explicit analysis of traces.

We show that spatial zero extension across the boundary may fail in parabolic Sobolev spaces $\mathring{\mathcal{H}}^1_p((0,T) \times \Omega)$, which can also be characterized as $$ L_p(0,T;\mathring{W}^1_p(\Omega))\cap W^1_p(0,T; W^{-1}_{p}(\Omega)). $$ More precisely, for any $p\in [1, \infty)$, we construct a function $u\in \mathring{\mathcal{H}}^1_p((0,T)\times \mathbb{R}^d_+)$ whose zero extension does not belong to $\mathcal{H}^1_p((0,T)\times \mathbb{R}^d)$. The obstruction occurs even for a flat boundary and is caused by a self-similar boundary layer concentrated at the initial-boundary corner, which produces a boundary supported normal flux defect after zero extension. We also discuss the suitability of various Sobolev-type spaces as solution spaces for parabolic equations in divergence form.

The recent work of Greaves, Jing, and Zhu gives an operator construction for the $t$-Schur functions and the $t$-Schur measure. Motivated by their construction, we consider the same type of vertex operator on the odd power-sum ring. Its Fourier modes generate a family of symmetric functions indexed by strict partitions, which we call shifted $t$-Schur functions. These functions specialize to Schur $Q$-functions at $t=0$. We derive a two-row formula, a Pfaffian Giambelli formula, a Cauchy identity, and a finite shifted Gessel-type formula. This note is intended as a first step toward further study of the odd-operator analogue of the Greaves–Jing–Zhu construction.

In this article, we derive the first variation formula for the Dirichlet scalar curvature energy functional on Riemannian manifolds. Motivated by this variational framework, we introduce a natural generalization of Einstein metrics, called Dirichlet–Einstein metrics, and we investigate some of their fundamental properties.

According to the property of stochastic variance reduced gradient (SVRG) algorithms, we construct a class of degenerate stochastic delay modified equations (SDMEs). Using the Lindeberg principle and the Markov property, we approximate the SVRG algorithms by the corresponding SDMEs. We obtain order 1 weak approximation in the smooth Wasserstein distance and the theory is validated through numerical experiments.