Papers

In this paper, we use the Green's function method to study the stability of solutions to micropolar equations in the whole space. An essential difficulty in estimating the Green's function lies in treating a coupled system of equations with variable coefficients, which is completely different from previous Green's function analyses for scalar equations. We propose a dominant factor extraction method to overcome this difficulty. Based on sharp Green's function estimates for the micropolar system, we prove that if the initial data (m_0, \omega_0) satisfy \|(m_0, \omega_{0})\|_{L^1\cap L^{\infty}}\leq c_0\mu^{3/4} for some small constant c_0 independent of the viscosity coefficient \mu, then the corresponding solution remains of order O(\mu^{3/4}) under the effect of the Couette flow. Meanwhile, we establish the optimal decay estimates of solutions in the general L^p norm. More importantly, the dominant factor extraction method developed in this work provides a general framework for the stability analysis of complex coupled fluid systems.

In this paper, we investigate weighted $L^p\rightarrow L^q$ restriction estimates for the Fourier extension operator associated with planar curves of non-vanishing curvature. More precisely, we establish estimates of the form \begin{equation} \|Ef\|_{L^q(X)}\leq C_{\epsilon}R^\epsilon\|f\|_p, \end{equation} where $X$ is an $\alpha$-dimensional set. As an application of this result, we derive a new $L^p$ decay estimate for circular means of Fourier transforms of fractal measures in $\mathbb{R}^2$. We also construct examples that give several upper bounds for the decay exponent when $p\in[1,2]$.

We establish interior $C^{1,\alpha}$ regularity estimates, for some $\alpha > 0$, for fractional $p$-caloric functions when $p$ is in the range $p \in (2,2/(1-s))$. As a consequence, we deduce improved regularity in time, which yields interior continuous differentiability in time for $p \in [1/(1-s), 2/(1-s))$.

In this paper, we study the existence of nonnegative bound states for the higher-order logarithmic Schrödinger equation $$ -\alpha\Delta u+V(x)u-\beta u\ln|u| = \gamma u(\ln|u|)^m, \qquad u\in H^1(\mathbb R^N), $$ where $\alpha,\beta,\gamma>0$, $m$ is an odd positive integer and $V$ is a positive bounded potential converging to a constant at infinity. The simultaneous presence of the first- and higher-order logarithmic terms leads to a nonsmooth variational structure and additional compactness difficulties. We introduce a family of power-law approximations and derive estimates that are uniform as the approximation exponent tends to the logarithmic limit. Under a structural condition on the autonomous nonlinearity, we obtain uniqueness, up to translations, of the nonnegative autonomous profile with connected positivity set and a corresponding energy gap below the two-profile threshold. The small-amplitude behavior exhibits a maximum-principle/compact-support dichotomy: the case $m=1$ yields positivity, whereas the higher odd orders fall into the compact-support regime for the autonomous profile. We then establish a profile decomposition for constrained Palais-Smale sequences and construct a barycenter-based min-max level below the splitting threshold. This prevents loss of mass through multiple profiles and allows us to pass to the logarithmic limit. Under the stated structural and energy conditions, we obtain a nontrivial nonnegative bound state of the original equation.

We solve a problem posed by Audenaert and Kittaneh in 2012 by proving that the Aujla–Bourin subadditivity inequality can be strengthened to an optimal form. We also answer questions posed by Bourin and the second author: (1) we significantly extend both the reciprocal Lie–Trotter theorem of Audenaert and Hiai and Kato's limit theorem; (2) we determine the best constants in inequalities for sums of contractions and positive block matrices, including the sharp triangle inequality $$ |A+B+C| \leq \frac{3}{4} I + |A|+|B|+|C| $$ for three contractions $A,B,C$; and (3) we prove a remarkable eigenvalue inequality for the symmetric modulus, thereby determining the best dimension-free lower bound in a question studied by Bourin, Lee, and Zhang.

Tri-hybrid multiple-input multiple-output (MIMO) architectures have been proposed as a promising solution for enabling energy-efficient communications systems in large-scale antenna arrays by replacing conventional antenna arrays in hybrid beamforming (HBF) systems with low-cost dynamic metasurface antennas (DMAs). In this work, we investigate beamforming design for integrated sensing and communications (ISAC) systems based on tri-HBF architectures, with the objective of jointly enhancing the communications sum rate and sensing sum mutual information. Specifically, we formulate a weighted multi-objective optimization problem that balances communications throughput and sensing mutual information, subject to a total transmit power consumption constraint and the physical limitations inherent to tri-HBF architectures. By exploiting the problem structure, we develop an efficient iterative algorithm with closed-form updates to solve the non-convex optimization problem with low complexity. Numerical results are provided to evaluate the performance of the proposed tri-HBF architecture in various setups. The results demonstrate that the proposed tri-HBF architecture can achieve significant improvement in energy efficiency (EE) compared to the conventional hybrid MIMO and DMA-only configurations, at the cost of minor degradations in sum rate and sensing mutual information.

This paper investigates properties concerning transposed triple products in rings. Motivated by Cline's formula, we characterize symmetric rings by means of group invertible elements and EP elements. We prove that a unital ring $R$ is symmetric if and only if $abc\in R^{\sharp}$ implies $acb\in R^{\sharp}$ for all $a,b,c\in R$. In particular, we give an answer to the problem posed in \cite[Problem 2.9]{MW1}. An example is provided to illustrate that for a symmetric ring $R$, $abc=e$ does not generally yield $acb=e$. For $\ast$-rings, we introduce the notion of pro-symmetric rings: a ring $R$ is pro-symmetric if $abc\in P(R)$ implies $acb\in P(R)$ for all $a,b,c\in R$. We show that every pro-symmetric ring is symmetric. Several counterexamples are constructed to distinguish these classes of rings, and their mutual inclusion relations are also discussed.

Let $R$ be a $\ast$-ring with $a,b\in R$. A ring $R$ is said to be $\ast$-reversible if $ab=0$ implies $b^{\ast}a=0$. In this paper, we first establish several new characterizations of $\ast$-reversible rings and reversible rings. In particular, we prove that $\ast$-reversible rings coincide with $\ast$-symmetric rings. Using these characterizations, we introduce two new classes of generalized $\ast$-reversible rings: pro-$\ast$-reversible rings and nil-$\ast$-reversible rings. A ring $R$ is called pro-$\ast$-reversible if $ab\in P(R)$ implies $b^{\ast}a\in P(R)$, and $R$ is nil-$\ast$-reversible if for every $c\in N(R)$, $cb=0$ yields both $b^{\ast}c=0$ and $cb^{\ast}=0$. The basic properties and characterizations of pro-$\ast$-reversible and nil-$\ast$-reversible rings are investigated. The interrelationships among all these ring classes are considered. The related examples to distinguish these rings are constructed.

We give a computer-assisted proof that the universal real Grothendieck constant satisfies $K_G^{\mathbb R}\le 1.7813$. The construction combines an explicit odd Hermite threshold of degree $11$ with a signed correlation polynomial of degree $51$. A sufficient inverse-majorant inequality is certified by enclosing the scalar coefficient head through degree $301$ and bounding the entire remaining tail using a weighted Gaussian trace estimate. The finite integral is bounded on a complete interval partition; its spatial exterior is controlled analytically. Exact parameters and source code that regenerate all accepted numerical inputs accompany the paper. The bound improves both the explicit bound $1.7818666069360661$ in the recent literature and the subsequently reported, system-tested value $1.7813319810625639$.

Let $S$ be a set of vertices in a connected graph $G$. A vertex outside $S$ is represented by the multiset of its distances to the vertices of $S$. The outer multiset dimension $\operatorname{odim}(G)$ is the minimum cardinality of an $S$ for which these representations distinguish all vertices outside $S$. We determine $\operatorname{odim}(C_s \square C_t)$ for all $s,t\geq 3$, answering a problem of Klavžar, Kuziak, and Yero. The values range from $3$ to $8$. The proof combines a half-turn argument giving a universal four-landmark lower bound when both factors have length at least four, explicit three- and four-landmark constructions for the infinite families, and exact finite enumeration on the remaining strip. The collision classification behind the infinite four-landmark construction is certified by exact quantifier elimination in linear integer arithmetic; source code and all finite upper certificates accompany the paper.

So conjectured that, for a fixed positive integer n, the ordinary adjacency spectrum of an integral circulant graph of order n determines its divisor set. We prove this for graphs of order p to the power a times q, where p and q are primes with p less than q and a is at least one. To handle coincident eigenvalues arising from distinct greatest-common-divisor classes, we use a spectral counting measure. For connected graphs, an exact identity recovers the part of the divisor set consisting of one and q, when present, together with the counting measure for a graph of order p to the power a minus one times q. Strong induction and decomposition into connected components then recover the full divisor set, including the case p equals two and disconnected graphs.

Löfberg and Qi introduced the second order Zarankiewicz number \(z_2\), the recursive-line Zarankiewicz number \(z_{RL}\), and the signed Zarankiewicz number \(z_{SL}\) for doubly simple biquadratic forms. It was shown that \[ z_2(m,n)\ge z_{SL}(m,n)\ge z_{RL}(m,n) \] for all \(m\) and \(n\). However, there was no evidence that there exist particular \(m\) and \(n\) such that \(z_{SL}(m,n)>z_{RL}(m,n)\). The motivation for introducing \(z_{SL}\) was as follows: during the study of the exceptional case \(m=15\), \(n=6\), Löfberg and Qi showed that \[ z_2(15,6)=z_{SL}(15,6)=60, \] but the exact value of \(z_{RL}(15,6)\) was unknown then. In this paper we show that \[ z_{RL}(15,6)=60. \] This eliminates the motivation for introducing \(z_{SL}\). Whether \(z_{SL}(m,n)=z_{RL}(m,n)\) in general remains an open problem. Recently, Lebedev presented an explicit construction separating the augmented Zarankiewicz number \(z_A\) from the limited augmented Zarankiewicz number \(z_L\) at \(m=n=1893\). We hope that the separation problem for \(z_{SL}\) and \(z_{RL}\) can also be solved. We also present the exact values of \(z_{RL}(m,6)\) for \(6\le m\le 16\).

This paper develops a design-based asymptotic theory for two-stage sampling over a continuous spatial domain. The target parameters are integral totals, or smooth functions of such totals, defined over a fixed bounded territory partitioned into an increasingly fine collection of primary sampling units. Within this fixed-area framework, we derive the order of the variance components of the Horvitz-Thompson estimator when secondary sampling units are points selected from continuous sub-regions. We establish design consistency of the estimator, as well as consistency of variance estimators under explicit regularity conditions on inclusion probabilities, sampling densities, and the study variable. The results are extended to plug-in estimators of smooth scaleinvariant functions of totals. For high-entropy first-stage designs, we further show that a Hájek-type variance estimator based only on first-order inclusion probabilities is consistent, and we prove asymptotic normality of both total and plug-in estimators. A simulation study inspired by forest inventory applications illustrates the finite-sample performance of the proposed estimators and variance estimators.

We analyze a nonlocal operator in space and time, called the multidimensional fractional material derivative. We derive its pointwise representation, which allows us to study its other properties. We define an inverse operator, called fractional material integral, and derive its pointwise representation. Furthermore, we analyze a class of linear partial differential equations, which corresponds to deterministic descriptions of the scaling limits of multidimensional Lévy walks, in which transport is driven by a multidimensional fractional material derivative with a speed vector integrated with respect to a suitable probability measure and a distributional source term. Using Fourier-Laplace transform techniques and a direct convolution-kernel construction, we prove the existence and uniqueness of exponentially integrable measure solutions for measure data. Moreover, we identify a necessary and sufficient condition on the source term for conservation of unit mass and provide separate sufficient conditions for non-negativity and weak convergence to $\delta_0$. We also prove weak continuity and self-similarity of the kernel and continuous dependence on the directional distribution.

Topology optimization (TopOpt) is a standard tool for structural conceptual design, providing optimal structures with great design freedom. However, it has a high computational costs due to the repeated evaluation of high-fidelity finite element models and frequently produces complex geometries that require extensive interpretation for manufacturability. As an alternative, this work presents a parameterized optimization framework based on Reduced Order Modeling (ROM) and preconditioning techniques. We employ the Empirical Interscale Finite Element Method (EIFEM) coupled with the Discrete Empirical Interpolation Method (DEIM) to construct localized, parameter-dependent reduced operators. From an optimization perspective, this approach can be interpreted as a preconditioning strategy in which inexact gradients are used to accelerate convergence. Furthermore, parameterizing the design space in terms of explicit geometric features, such as inclusion radii or lattice widths, significantly improves the manu facturability of the resulting optimized designs. We assess the performance of the method in terms of computational time, design topology and structural performance with TopOpt as baseline for three differ ent unit cell geometries and three different benchmarks. While the parametric ROM framework requires an initial offline training phase, our comparative analysis demonstrates that it drastically accelerates the online optimization loop and that it can produce even stiffer design in some of our experiments.

We prove a multiplication formula for the Wang-Wei-Zhang cluster character on a Hom-finite $2$-Calabi–Yau Frobenius extriangulated category $\mathcal C$ with a cluster-tilting object under the assumption that its stable category has constructible cones with respect to the induced cluster-tilting object. This formula generalizes those obtained by Wang-Wei-Zhang for $\mathcal C$ in the one-dimensional case and by Keller-Plamondon-Qin for (stably) $2$-Calabi-Yau Frobenius or triangulated categories. As a consequence, we express the frozen term in an Auslander-Reiten mesh with the character of a minimal cosyzygy middle term. For acyclic quivers with principal coefficients, we compute these frozen multiplicities in Higgs categories and obtain explicit principal coefficient mesh relations.

The paper studies the weighted dual Minkowski problem for the weighted $q$th dual curvature measure of convex bodies in $\mathbb R^n$ first posed by Huang, Lutwak, Yang, and Zhang. Most of the results are in the G-invariant setting, i.e. when the convex bodies are invariant under a closed subgroup $G$ of $O(n)$. They generalize previous results on the existence of solutions to the dual Minkowski problem for origin-symmetric convex bodies. It is proved: (i) a full characterization of the G-invariant weighted dual curvature measure for q\in(0,1], (ii) a necessary and sufficient condition for the existence of solutions to the G-invariant weighted dual Minkowski problem for q\in(1,n), (iii) a sufficient condition for the G-invariant weighted dual curvature measure for q>1, and (iv) an extension of Henk-Pollehn's dual curvature measure concentration property.

Blow-up in the self-dual Chern–Simons–Higgs equation has a rich multiscale structure, but the detailed theory has largely focused on the simple regime, where one entire profile resolves each concentration point. By contrast, non-simple blow-up is analytically more delicate: several interacting regular cores must be resolved simultaneously as they collapse toward a single vortex while remaining separated on their own scales. Because of this difficulty, the existence of such solutions has remained a long-standing open problem. To the best of our knowledge, we prove the first existence result for non-simple blow-up solutions of this equation, covering in particular the previously unresolved finite-height Chern–Simons regime. We also show that non-simple clusters have equal limiting core masses, a common profile type, and regular-polygon arrangements. We determine their possible total masses and derive a mass-gap criterion that rules out such blow-up for certain vortex configurations. Our quantitative profile and moment estimates for general finite-height clusters guide the design of the approximate solution and clarify the core interactions underlying the finite-height construction. A separate construction on the unit disk produces two mean-field cores along a sequence with varying Dirichlet traces.

This paper proposes an Expectation–Maximization algorithm with Metropolis–Hastings sampling for parameter estimation in a Mixed Fractional Merton Jump Diffusion model. The model combines fractional Brownian motion to capture long-range dependence with a compound Poisson jump process to describe abrupt movements in financial returns. The latent jump process is inferred during the E-step using Markov Chain Monte Carlo sampling, while the M-step updates the model parameters by maximizing the expected complete-data likelihood. The consistency and asymptotic normality of the proposed estimator are established under suitable regularity conditions. The methodology is applied to the Helsinki Stock Index (OMXH25). The proposed estimation framework provides a reliable and computationally efficient approach for modeling financial time series exhibiting both long-memory dependence and jump behavior.

In this paper, we study the multifractal analysis of Birkhoff averages for parabolic rational maps. We establish a conditional variational principle and prove the real analyticity and strict monotonicity of the Birkhoff spectrum, as well as the existence and uniqueness of the measure attaining the supremum in the conditional variational principle, on a certain region. To this end, we prove the existence and uniqueness of an expanding equilibrium measure and the real analyticity of the pressure function on a suitable domain. For parabolic systems, our approach using thermodynamic formalism provides a unified framework for establishing the conditional variational principle and investigating finer properties of the Birkhoff spectrum, including its real analyticity, strict monotonicity, and the existence and uniqueness of a measure attaining the supremum on a certain region.

We study continuous-time distributed optimization on the three-dimensional rotation group over undirected tree graphs, where agents with heterogeneous local costs seek a common attitude minimizing the geodesically strongly convex sum of their costs on a geodesically convex operating ball. We propose a nonsmooth protocol with fixed gains that combines local Riemannian gradient descent with signum-based consensus feedback computed from the Lie-group logarithms of relative rotations between neighboring agents. Since geodesic strong convexity alone does not guarantee invariance, as demonstrated by a certified counterexample, we impose an inward-pointing boundary condition and prove that every Filippov solution with all agents initialized in the ball remains there. We construct a nonsmooth intrinsic disagreement Lyapunov function and derive a uniform dissipation bound through a cluster-contraction argument exploiting the tree structure. With bounded local gradients, this bound gives an explicit gain-ratio condition under which we establish exact finite-time consensus for every such solution and obtain a settling-time estimate requiring no structural graph constant beyond the number of agents. After consensus, we identify the sliding dynamics as a scaled Riemannian gradient flow of the summed objective. When the minimizer lies in the interior of the ball, the dynamics converge to it exponentially in geodesic distance. Simulations illustrate our theoretical results.

Switching methods can be seen as certain local transformations that do not alter their basic parameters of a combinatorial structure. Efforts have been devoted in the literature to relate and unify the switching theories for codes and designs, and also for Hadamard matrices and graphs. The combinatorial structures we consider in this paper are graphs and designs. We show an extension of known switching method for constructing 2-designs to divisible designs, and then provide some examples of its application. Moreover, we prove several equivalences between switching methods for graphs and designs, and as a byproduct, we obtain a new switching method to obtain 2-designs.

Rump asked in 2016 whether every finite quasi-linear cycle set is retractible. This was recently proved by Dietzel. In this paper, we reinterpret Rump's question in terms of automorphisms of certain pointed magmas, and propose a more general conjecture on the automorphisms of these magmas. We provide some theoretical and computational evidence in support of our conjecture.

Bonamy-Knor-Lužar-Pinlou-Škrekovski (2017) define $K_n^t$ to be the complete graph of $n-1$ vertices but with an extra vertex that's adjacent to $t$ vertices of the complete graph part. They propose a stronger conjecture which asserts that if $G$ is a finite simple $2$-connected graph of order $n \ge 10$ not isomorphic to $K_n$, $K_n^2$, nor $K_n^{n-2}$, then the Szeged-Wiener gap of $G$ is $\eta(G) \ge 2n$. We improve upon their work to tighten the bounds on the Szeged-Wiener gap, allowing us to prove this conjecture in the affirmative. Afterwards, we construct graphs attaining equality for each $n \ge 10$ and pose a problem for interested readers to determine a necessary and sufficient condition for equality.

We study Lyapunov-type stability for invariant algebraic divisors of codimension-one holomorphic foliations on complex projective spaces. Motivated by the notion of $L$-stability introduced in \cite{LeonScardua2018} for plane singularities, we define a projective stability condition which is compatible with the logarithmic setting and controls leafwise holonomy along the regular part of the invariant divisor, while discarding any stability requirement inside prescribed neighborhoods of its singular locus. Our main result then is a global projective counterpart of the local classification in \cite{LeonScardua2018}: for a maximal invariant algebraic divisor, projective $L$-stability together with non-dicritical non-nodal generalized-curve singularities on a generic plane section forces the ambient foliation to be globally logarithmic. As an application we obtain a topological rigidity consequence on $\mathbb P^2$ for logarithmic foliations under mild generic conditions. The proof of the main theorem is based on propagating the consequences of the stability hypothesis through the reduction tree by means of an explicit Dulac transport argument, together with the classification of $L$-stable groups of germs of one-dimensional complex diffeomorphisms.

For an invertible linear time-invariant system, the Morse normal form reveals that there exist an output injection and an output transformation such that the resulting system possesses a regular relative degree. Based on this fact, this paper develops a method to determine all admissible output injections and output transformations that achieve this property. In contrast to the recursive and implicit computation methods in the existing literature, the proposed determination method for the output injections is explicit and easy to use, since the solutions are obtained by solving linear matrix equations. A rigorous proof of the proposed method is provided, and an illustrative example is presented to demonstrate the applicability of the approach.

We prove that a generically trivial torsor under a reductive group scheme over an unramified regular local ring is trivial.

Gromov–Wasserstein optimal transport (GW-OT) extends classical optimal transport by introducing structure-aware transport cost. This is particularly relevant for spatial transcriptomics, where dynamical reconstruction should preserve tissue structure in addition to matching expression patterns. While static formulations have been widely used for such structure-aware alignment, a general dynamic formulation for reconstructing continuous trajectories is still missing. We introduce Travelling Pair Dynamical Alignment and Trajectory Estimation (TP-DATE), a theoretical and computational framework to generalize GW-OT dynamically in a simulation-free manner. We formulate a broad class of static and dynamic Quadratic-form OT (QOT) through path actions and prove the static dynamic equivalence. We further develop travelling-pair flow matching, which allows interacting conditional paths and marginalizes their interactions into a single vector field. On synthetic and real spatial transcriptomics data, TP-DATE better preserves spatial structure and improves continuous 3D dynamics reconstruction.

In 2025, the affine dual curvature measures $\widetilde{C}_m^a(K,\cdot)$ of convex body $K$ were introduced by Cai, Leng, Wu, and Xi, and the even Minkowski problem for the affine dual curvature measures was solved. In this paper, the necessary and sufficient condition of affine dual Minkowski problem for general measures with $m>1$ are proposed.

Let $F$ be the unramified quadratic extension of $\mathbb{Q}_p$, and let $P_n(T)$ denote the polynomials arising in the $p$-adic Fourier theory of Schneider and Teitelbaum. We determine explicitly the norms of the functions $P_n(x\Omega)$ on the Banach spaces of locally $F$-analytic functions of fixed order, where $\Omega$ is a Lubin–Tate period. Our computation is based on a recent valuation formula of Ardakov and Berger for the values $P_n(\Omega)$. As an application, we obtain an explicit normalization of the basis constructed by Bannai and Kobayashi, whose elements all have norm one.