Papers

We develop applications of a Jensen-type inequality for pairs of positive measure spaces. The framework yields $L^p$ and operator inequalities for positive integral operators, norm and entropy estimates for Fejér means, and probabilistic moment and variance bounds. We also obtain robustness estimates under random and deterministic erasures. These results provide a unified connection between Jensen-type inequalities, harmonic analysis, operator theory, probability, and robust reconstruction.

We consider the Cauchy problems for a class of nonlinear diffusion equations where each initial data is given by a characteristic function of an open set $\Omega$ in the whole Euclidean space. If the boundary $\partial\Omega$ is of class of $C^2$ in a neighborhood of a point $x$ on it, then we extract the mean curvature of $\partial\Omega$ at $x$ from the initial diffusion at $x$. This corresponds to the author's previous result for the two-phase linear diffusion equation. Applications to some overdetermined problems for nonlinear diffusion equations are given.

We give a finite cover argument for nonnegativity of the ordinary monomial coefficients of every stretched Littlewood-Richardson polynomial with partition lengths at most seven and outer size at most thirty. Explicit reductions leave 358,952 residual triples. The computational part consists of exact finite enumeration, local reduction checks and rational Ehrhart polynomial computations. Its three named dependencies are stated precisely below, separately from the mathematical implication they establish.

For a finite simple graph $G$ of positive order, let $s_k(G)$ be the number of its $k$-vertex tree subgraphs. We prove that, among graphs of a fixed order $n$, the ratios $s_k(G)/s_k(K_n)$ form a nonincreasing sequence in $k$. It follows that the complete graph maximizes the mean subtree order: $\mu(G)\leq\mu(K_n)$, with equality if and only if $G$ is complete. This monotonicity theorem also establishes the $s_{n-1}$-to-$s_n$ ratio conjecture posed in recent work on extremal mean subtree order. The proof lifts a known coupling of rooted Cayley-tree shapes of consecutive orders to uniform labeled tree subgraphs of $K_n$ that are nested at every sample point. Event inclusion gives the count inequalities, and a double-sum identity gives the mean bound.

Nested invariant cones (NICs) encode hierarchical order and oscillation structures in many finite and infinite dimensional dynamical systems. We investigate how this hierarchy constrains the Birkhoff center and the supports of invariant probability measures. For eventually compact, dissipative semiflows that are uniformly eventually strongly monotone with respect to NICs, we prove that every connected component $B$ of the Birkhoff center lies in a single cone layer: there exists $j>0$ such that $B$ is unordered with respect to all lower-level cones and strongly ordered with respect to all cones at level $j$ or higher. The same conclusion holds for every connected component of the support of an invariant probability measure. Under an additional transversality condition involving a codimension-$d$ linear subspace, each such component admits a homeomorphic embedding into $\mathbb R^d$. If $d=1$ or $2$ and the restriction of the semiflow to its global attractor extends to a flow, then the system has zero topological entropy, independently of the dimension of the original phase space. We apply the theory to bidirectional cyclic feedback systems and scalar parabolic equations on the circle. In the parabolic case, the natural infinite family of zero-number NICs fails to be uniformly eventually strongly monotone, even for the heat equation; we overcome this obstruction by constructing a finite family of perturbed NICs. In both applications, every connected component of the Birkhoff center admits a planar embedding, and the topological entropy is zero.

We prove the Colomo-Pronko conjecture for alternating sign matrices with a prescribed square of zeros at a corner, for all matrix sizes and freezing parameters. A known multiple-integral formula for the frozen-corner count yields determinant representations built from fixed polynomial kernels. We relate these kernels to the conjectured determinant through an inverse identity for the commutator of a signed Pascal matrix with reversal. In odd dimension, the comparison uses the one-dimensional nullspace and projection along it to eliminate the central coordinate. Combined with the asymptotic analysis of Colomo and Pronko, our result removes the conjectural assumption from their GUE Tracy-Widom fluctuation theorem for the intersection of the frozen boundary with the main diagonal in uniformly random alternating sign matrices. The finite-dimensional algebraic core of the proof has been formalized in Lean 4.

For a set $W$ of nonempty words, let $S(W)$ be the flat semiring formed by the nonempty factors of words in $W$, together with an absorbing zero. We prove that $S(W)$ has a finite identity basis whenever every word in $W$ is linear, with no restriction on the size of $W$ or on the lengths of its words. In the bounded case its variety is generated by the interval semiring $A_m\cong S(a_1\cdots a_m)$, where $m$ is the maximum word length and the letters $a_i$ are distinct. In the unbounded case its variety is generated by the interval semiring on all finite intervals of the nonnegative integers. We give explicit finite bases in both cases. The proofs encode nonzero polynomial evaluations by endpoint graphs and derive the required graph identifications using finitely many splicing identities. In particular, the eleven-element semiring $S(abcd)$ is finitely based, providing a counterexample to the length-bound conjectures for $S(W)$ proposed by Gao, Ren and Zhao.

Sampling a multidomain tensor from limited measurements is fundamental in structured linear inverse problems. Kronecker-structured sampling avoids the full high-dimensional sensing matrix, but design remains difficult: sequential discrete methods can commit the cross-mode budget too early, whereas standard continuous greedy avoids such early commitment at the cost of repeated state-dependent gradient evaluations. We propose spectral-residual continuous greedy (\alg) for frame-potential (FP) tensor sampling. \alg maintains a state-dependent fractional allocation before rounding. Mode-wise Gram matrices provide safe gradient intervals for direction certification, while Shapley values prioritize unresolved gradient queries. SR-CG then applies deterministic rounding followed by path-guided exchange (PGX), which reuses the final fractional state to restrict candidate swaps and accepts only exact FP-decreasing exchanges. We establish a finite-step approximation guarantee that approaches the classical $1-1/e$ factor as direction certification becomes exact and finite-step residual error vanishes. Experiments show fewer exact gradient evaluations and better FP designs, with clearer gains on instances where the cross-mode budget allocation is difficult to determine. \alg also achieves lower average normalized mean-squared error (NMSE) than Greedy-FP at all tested noise levels, although the reconstruction gain is smaller than the FP gain.

For fixed $k\ge3$, we determine the critical law of totally isotropic $r$-spaces for a uniform random map $\Theta_N:\Lambda^k\mathbb {F}_q^N\to\mathbb {F}_q^m$. At the exact balance $m\binom rk=r(N-r)$, the entire null configuration is asymptotically an independent Bernoulli subset of $\operatorname{Gr}(r,\mathbb {F}_q^N)$ in total variation, uniformly in $q$ and $m$. Consequently, the counting measure $\Xi_r$ is asymptotically a Poisson point process, and its total mass $X_{N,r}$ is asymptotically Poisson. An exterior-rank stability theorem shows that near-extremal families decompose into Grassmann clusters with uniformly controlled span deficiency. We obtain quantitative rates and identify the first exterior-dependence scale. We also show that rare null $(r+1)$-spaces force high-order factorial-moment divergence, while the fixed-target bilinear cases $m=1,2$ exhibit non-Poisson critical behavior.

For a commutative ring $R$, we call an $R$-module $B$ Baer if $\Ext_R^1(B,T)=0$ for torsion $R$-module $T$. It is known that every Baer module is projective when $R$ is $\tau_q$-semisimple, i.e., rings whose total rings of quotients are semisimple. And it was conjectured that the converse characterizes $\tau_q$-semisimple rings. We show that the converse fails in two substantially different ways. First, the failure is already systematic in the non-reduced Noetherian case. If $D$ is a Dedekind domain which is not a field and $R=D[\varepsilon]/(\varepsilon^2)$, then every Baer $R$-module is projective, although $R$ is not reduced, and thus is not $\tau_q$-semisimple. Second, the converse fails even among reduced rings, and in fact among reduced coherent Bézout rings. The main tool is a countable-ring criterion. For a countable Noetherian domain $D$, let $\EC(D)$ be the ring of eventually constant sequences over $D$. If $D$ is not a field, then every Baer $\EC(D)$-module is projective, whereas $\EC(D)$ is reduced and has countably infinitely many minimal prime ideals, and thus is not $\tau_q$-semisimple.

Inspecting or servicing a non-cooperative spacecraft is a two-player game: the target can thrust to defeat the inspector's plan. Hamilton-Jacobi-Isaacs reachability answers such games exactly but its cost grows exponentially with state dimension, while learning-based controllers scale yet certify nothing. Convex formulations are the usual escape, typically cast as convex-concave saddle-point problems. That framing fails in the terminal-distance orbital game: with effort regularization, the payoff is convex in both controls, so no pure open-loop saddle point need exist. In its place we derive two exact one-sided guarantees read from the players' terminal reachable sets via support functions: an escape certificate proving the target can hold a guaranteed standoff, and a security strategy bounding the miss distance the inspector can force. Together they bracket the engagement without assuming player rationality, and over two hundred perturbed trials the escape certificate separated capture from escape without error. A projected extragradient method supplies a strategy pair in about 25 ms, certified in place by best-response gaps. The construction carries unchanged to elliptical reference orbits via Yamanaka-Ankersen dynamics, where the orbital phase moves miss distance by nearly a factor of two, reproduced by nonlinear Keplerian propagation. Under receding-horizon play the inspector captures in half of ten representative engagements. Angles-only navigation error, keep-out zones, multiple pursuers, and closed-loop play are treated as extensions.

We study the problem of identifying nontrivial, i.e., nonzero, periodic fixed-points in discrete-time Lur'e feedback systems. Using the circulant matrix constructed from the transfer function of the linear subsystem, whether stable or unstable, we introduce an algebraic framework that allows us to determine when such fixed-points exist. This framework yields a sector bound defined by two vectors, whose slopes correspond to the maximum and minimum positive singular values of the circulant matrix. Assuming that the nonlinear feedback function is memoryless, we show that a necessary condition for the existence of nontrivial $P$-periodic fixed-points is that the intersection of the continuous completion of the nonlinear feedback function with that sector bound contains at least one point other than the origin. Our characterization provides a unified condition valid for all periods $P$, and further enables us to derive upper bounds on the amplitudes of admissible periodic fixed-points with bounded feedback functions. In particular, for relay feedback systems with passive feedback functions, we derive both upper and lower bounds for the amplitudes of such periodic fixed-points.

Two star exponential random graph models (ERGMs) are an interesting special case of both general ERGMs and mean-field Ising models. In this paper, we study two star ERGMs conditioning on the edge density $p$. We begin with an analytic characterization of the replica symmetric region, where the conditional model is close in cut distance to the Erdős–Rényi random graph $G(n,p)$. We prove that this region is twice as large as the corresponding region for the unconditional model. To study refined properties of the conditional model, we then analyze the global Kawasaki algorithm for sampling from it. Within the replica symmetric region, and under the additional condition that $4\beta p(1-p)<0.5$ when $|p-1/2|\lesssim 0.4632$, where $\beta$ is the model parameter, we prove metastable fast mixing of the Kawasaki algorithm. As corollaries, we obtain a weak Poincaré inequality and use it to deduce concentration inequalities of optimal order for conditional subgraph counts. The additional condition $4\beta p(1-p)<1/2$ if $|p-1/2|\lesssim 0.4632$ comes from our proof technique of contractive coupling. This bottleneck did not appear in previous studies applying the contractive coupling technique to unconditional ERGMs.

We establish quantitative comparisons between the Laplace–Beltrami spectra of Kähler metrics of equal area on a compact Riemann surface. An estimate in terms of the oscillation of a potential gives bounds for fractional powers of reciprocal eigenvalues and explicit intervals for eigenvalue ratios. Bounds involving the gradient and Laplacian of the potential give further comparisons. On the Riemann sphere, we obtain estimates for individual eigenvalues, reciprocal sums and counting functions. For a Kähler metric on the unit 2-sphere which is centered in the sense that the unit normal vector field integrates to zero, the Dirichlet energy of the potential yields quantitative improvements of Hersch's bounds for the first positive eigenvalue and the sum of the first three reciprocal eigenvalues.

Let $q$ be odd. The trace conic in $\mathbb{F}_{q^3}$ determines a Singer difference set in $\mathbb{F}_{q^3}^\times / \mathbb{F}_q^\times$ and a natural square-class lift to $\mathbb{F}_{q^3}^\times / \mathbb{F}_q^{\times 2}$. We study the odd multiplicative Fourier coefficients of this lift and prove that they are bounded in absolute value by $2\sqrt{q}$. The proof improves the naive six-puncture Weil bound by exploiting a projective Klein-four symmetry of the associated rank-one local system. The resulting nontrivial cocycle produces a quaternionic action on its four-dimensional cohomology, while Frobenius symmetry reduces the relevant trace to two Weil-scale eigenvalues. As an application, the oriented conic yields an explicit Singer-invariant signing of the point-line incidence graph of $\mathrm{PG}(2, q)$. The corresponding dihedral Cayley graph is a connected Ramanujan double cover. Thus a conic lift already known in finite-geometric constructions has an additional Ramanujan spectral property governed by its odd multiplicative character sums.

Let $R$ be a one-dimensional commutative Noetherian ring with a unique minimal prime $\mathfrak p$ such that $D=R/\mathfrak p$ is a principal ideal domain. We give a complete characterization of the iso-Artinian property in this class, that is, the following conditions are equivalent: $R$ is iso-Artinian; $\mathfrak pR_{\mathfrak p}=0$; $len_R(\mathfrak p)<\infty$; $\mathfrak p/\mathfrak p^2$ is torsion over $D$. As a consequence, we completely answer the Question~3.9 of Daneshvar and Divaani-Aazar: under their hypotheses $Min R\subsetneq Ass R$, the ring is iso-Artinian precisely when its nilradical has finite length, and it is non-iso-Artinian precisely when the conormal module has positive rank.

We determine the asymptotic height of the discrete-time critical beta-splitting tree. Let $L_n^*$ denotes the height of the tree with $n$ leaves, defined as the maximum graph distance from the root to a leaf. Then, \[ \frac{L_n^*}{(\log n)^2} \longrightarrow C_{\mathrm{ht}}:=\min_{\theta>1} \frac{\theta}{2\{\psi(\theta)+\gamma\}} \approx 0.976 \] almost surely and in $L^{p}$ for every fixed $p>1$ as $n \to \infty$. Here $\psi$ is the digamma function and $\gamma$ is Euler's constant. This answers \cite[Open Problem~4]{AldousJansonII} of Aldous and Janson.

Let $X$ be a closed, smooth, orientable, positive definite $4$-manifold with $H_1(X ; \mathbb{Z}) = 0$. Let $p$ be a prime and suppose that $G = \mathbb{Z}_p$ acts smoothly on $X$ (if $p=2$ we also require an assumption on how $G$ acts on $H^2(X ; \mathbb{Z})$). Using equivariant Yang–Mills theory, Hambleton–Lee and Hambleton–Tanase proved (in the simply-connected case) that the fixed point set and tangential isotropy representations coincide with that of an equivariant connected sum of copies of $\mathbb{CP}^2$ on which $G$ acts linearly. We give a new proof of this result using equivariant Seiberg–Witten theory. Furthermore we also determine the weights of all equivariant line bundles on $X$, showing that these likewise coincide with that of an equivariant connected sum of linear actions on $\mathbb{CP}^2$.

Let $R$ be a commutative ring and let $fPD(R)$ be its small finitistic dimension in the sense of Glaz. We study two ways in which Koszul grade can escape a restricted homological test. For a polynomial extension, we prove that if $R$ is Noetherian, the maximal ideals of $R[X]$ detect exactly the Goldman primes of $R$, and this gives the formula $$ fPD(R[X]) =1+\sup\{depth R_{\mathfrak p}\mid \mathfrak p\in GSpec(R)\}.$$ We construct Noetherian local rings $R_m$ with $fPD(R_m)=0$ and $fPD(R_m[X])=m+1$, and we show that Koszul grade can increase by two along an adjacent pair of primes. We then consider cyclic weak $(n,d)$-conditions. For every $n\geq2$ and $h\geq1$, we construct a local idealization $T$ such that every $n$-presented cyclic $T$-module is projective, while $fPD(T)=h$. The ring $T$ has no nonzero proper finitely presented ideals, but it has a finitely generated ideal $I$ with $Ext_T^i(T/I,T)=0$ for $i<h$ and $Ext_T^h(T/I,T)\neq0$. Thus the cyclic condition controls $fPD$ at presentation level one, but at no higher level.

Motivated by the continuous-time critical beta-splitting tree, we investigate chronological trees recording the successive block splits and their times in homogeneous fragmentation processes restricted to $n$ labels. The separation time of a subset is the first time its labels cease to lie in a common block. For each fixed integer $q\geq2$, we study extremal separation times of $q$-element subsets via their associated point process. Under mild conditions, as $n\to\infty$, the last such time has a randomly shifted Gumbel limit after deterministic centering. Above a threshold $\theta_*$, the coefficients of both the leading $\log n$ term and the $\log\log n$ correction in the centering become independent of $q$, as does the limiting law up to deterministic translation. The extremal process converges to a randomly shifted Poisson point process, without and with decorations, for $q \le \theta_*$ and $q>\theta_*$ respectively. Finally, in the same centered time window, the number of size-$q$ blocks (clades) converges to a pure-death process for $q\leq\theta_*$, and to a process whose trajectories are non-monotone with positive probability for $q>\theta_*$.

Generalization of the inductance to the case of non-quasistatic electromagnetic field oscillations appears to be fruitful when considering wireless power transfer and RF metamaterials consisting of thin wire loop meta-atoms. When dealing with large systems of interacting loops carrying currents, efficiency and precision of calculation in the presence of retardation is crucial. In this work, we derive a series expansion of such generalized inductance and propose a way for its efficient numerical approximation. Illustrative examples are provided both for inductance convergence of a pair of two loops and extinction efficiency for scattering by metamaterial samples.

Let $X:M^n\to\mathbb R^{n+1}$ be a complete, connected, two-sided minimal immersion without boundary, where $n=3,4$. We prove that finite $\delta$-index, finite Morse index, and finite total curvature are equivalent for $\delta>((n-1)/n)^2$, without assumptions on properness, volume growth, or topology. The main estimate shows that $\delta$-stability outside a compact set and finite-dimensional $H_c^1(M;\mathbb R)$ imply intrinsic Euclidean volume growth for $\delta>(n-2)/n$. We also deduce the sharp $\delta$-stable Bernstein theorem in this latter range from results of Hong–Li–Wang and Florit-Simon.

Motivated by the extended deformation theory of complex manifolds, we give a combinatorial description of holomorphic polyvector fields on a smooth compact toric variety via Klyachko's filtrations. This yields dimension and formal equivariant-character formulas and recovers the descriptions of Demazure roots and anticanonical sections.

Low-rank tensor factorization provides a flexible framework for completing multidimensional data from incomplete and corrupted observations. However, unweighted spectral regularizers impose a common shrinkage profile across singular components, which may excessively attenuate dominant low-rank components, and factorized variants either lack component-specific weighting or require costly singular value decompositions (SVDs). This paper proposes two weighted Schatten-$p$ tensor factorization models, termed \WSpTFI{} and \WSpTFII{}, under the tensor-tensor product (t-product) framework to address these limitations. \WSpTFI{} is motivated by a factorized weighted tensor Schatten-$p$ norm identity and permits flexible, possibly asymmetric factor exponents. \WSpTFII{} constructs a regularizer from transform-domain column-pair energies, yielding SVD-free main factor updates and a column-pruning mechanism for reducing redundant rank components. This paper further develops an iteratively reweighted alternating direction method of multipliers (ADMM)-type scheme for \WSpTFI{} and an iteratively reweighted least squares (IRLS)–block successive upper-bound minimization (BSUM) scheme for \WSpTFII{}. Theoretical analysis establishes the weighted factorization relation and provides a conditional limiting Karush–Kuhn–Tucker (KKT) characterization for \WSpTFI{}. For \WSpTFII{}, the actual damped quadratic block updates yield a quantitative sufficient-decrease mechanism for the fixed-$\delta$ smoothed factor objective. This implies asymptotic regularity, and every accumulation point of the fixed-dimensional tail is stationary. Experiments on synthetic tensor completion, color-image restoration, hyperspectral inpainting, and printed-circuit-board defect detection demonstrate competitive reconstruction quality and robustness under various degradation conditions.

In this paper, we study the continuous dependence of solutions to the three-dimensional incompressible anisotropic Navier-Stokes equations $(ANS)$ with initial data in the critical space $\dot{B}^{0,\frac12}$. We prove that the data-to-solution map is continuous in this setting. Due to the critical regularity of the problem, the classical Bona-Smith argument \cite{BoSm} (see also the abstract result \cite{ABITZ}) does not appear to apply because a key ingredient (the Lipschitz continuity at lower regularity) does not appear to hold. As a consequence, we develop a new direct Fourier-weighted method within the Besov framework.

Let $A$ be an $n\times n$ symmetric matrix with eigenvalues $\lambda_1\geq\cdots\geq\lambda_n$. Let $\tilde A:=A+E$, where $E$ is a symmetric noise matrix, and denote the eigenvalues of $\tilde A$ by $\tilde\lambda_1\geq\cdots\geq\tilde\lambda_n$. Bounding the perturbation $|\tilde \lambda_i -\lambda_i| $ is a central problem in linear algebra and numerical analysis. In this paper, we prove new perturbation results by exploring the actual interaction between $E$ and the eigenvectors of $A$. In the setting where $E$ does not act adversarially with respect to these vectors (for instance, if $E$ is random), we obtain a considerable improvement over Weyl's inequality. One can routinely extend these results to the Hermitian and rectangular settings. We obtain the new bounds as corollaries of a regional stability result, which provides a sufficient condition for a region on the real line to be stable (containing the same number of eigenvalues) after the perturbation. This result is of independent interest. We prove our regional stability result using contour integral analysis. Our main new technical ingredient here is the double-jump argument, which is robust and could be useful in many other situations involving Neumann series.

Let $\mathbb{F}_q$ be a finite field of odd characteristic with $q \equiv 2 \pmod3$. We study the directed graph defined by the Borwein–Borwein cubic arithmetic–geometric mean (AGM) over $\mathbb{F}_q$. We prove that this graph is a disjoint union of directed cycles. We associate Hessian curves with this AGM. We also show that each edge corresponds to a $3$-isogeny defined over $\mathbb{F}_q$ between the curves associated with its initial and terminal vertices. We then use a counting formula for Hessian curves to derive a lower bound for the number of cycles.

We prove a Hörmander multiplier theorem for the Dunkl transform associated with an arbitrary finite reflection group. Uniform $H^\sigma(\mathbb{R}^N)$ bounds for the normalised dyadic pieces of a measurable symbol, with $\sigma>\mathbf N/2$ and $\mathbf N$ the homogeneous dimension, imply $L^p(d\omega)$ boundedness for $1<p<\infty$ and weak type $(1,1)$. No radiality or reflection-group invariance is assumed, and the dyadic pieces may meet the reflecting hyperplanes. The main new estimate controls the Dunkl kernel $E_\kappa(i\xi,x)$ and its first Euclidean derivatives in $x$, uniformly near arbitrary intersections of reflecting hyperplanes. Here $\xi$ lies in a fixed compact annulus. Writing $w_\kappa$ for the Dunkl weight and $\Omega_\kappa$ for its inhomogeneous counterpart, it gives the bound $C\Omega_\kappa(x)^{-1/2}w_\kappa(\xi)^{-1/2}$. The proof decomposes each root contribution into Hermitian two-dimensional blocks, applies oscillatory changes of variables near the rootwise transition points, and uses a commutator identity to cancel the non-integrable phase derivative. Chamber lifting also yields endpoint bounds for the pullbacks of the componentwise Coifman–Weiss atomic $H^1$ space and the corresponding BMO space on a fixed chamber.

We develop an analytic framework to study hyperbolicity of Jensen polynomials and related inequalities for arithmetic sequences with general asymptotic growth. Our results provide partial converses to the Hermite–Jensen phenomenon of Griffin, Ono, Zagier, and the third author. As applications we prove several conjectures concerning log-concavity, higher-order Turan inequalities, Laguerre inequalities, infinite log-concavity, and Toeplitz determinants for partition-type functions.

In this paper, we study h-Ricci-Bourguignon solitons on the product manifold D2 x R, where D2 is the Poincare disk equipped with its standard hyperbolic metric. We first establish that this manifold admits a gradient Ricci-Bourguignon h-soliton only if the non-zero function h depends solely on the manifold's last coordinate. Next, in the case where h:=1, we explicitly determine the vector field that makes D2 x R a Ricci-Bourguignon soliton. Finally, we provide the necessary and sufficient condition for the dual 1-form of this vector field to define a contact structure, which corresponds to the existence of an explicitly determined function independent of the manifold's second local coordinate, thereby allowing us to compute its Reeb vector field.