Papers

A connected graph $G$ is matching covered if it contains at least one edge and every edge belongs to some perfect matching of $G$. A matching covered graph is a brick if it is 3-connected and after removing any two vertices, the resulting graph has a perfect matching. Bricks are the building blocks of matching covered graphs. A brick $G$ is extremal if the incidence vectors of all perfect matchings are linearly independent over $\mathbb{R}$. Zhou et al. [On planar extremal nonsolid bricks, J. Graph Theory, 2026] proved that for every planar extremal brick $G$, there exists a sequence of planar extremal bricks $G_0,G_1,\dots,G_n$ such that the underlying graph of $G_n$ is an odd wheel, $G_0=G$, and each $G_i$ $(0\leq i < n)$ has a triangle $T_i$ such that $G_i/T_i = G_{i+1}$, where $G_i/T_i$ means contracting the triangle $T_i$ into a single vertex. In this paper, the result of Zhou et al. is generalized to planar PM-compact bricks, a larger class of bricks including extremal bricks. We present a chain theorem for planar PM-compact bricks: for every planar PM-compact brick $G$, there exists a sequence of planar PM-compact bricks $G_0,G_1,\dots,G_n$ such that the underlying graph of $G_n$ is an odd wheel, $G_0=G$, and each $G_i$ $(0\leq i < n)$ contains a triangle $T_i$ such that $G_i/T_i = G_{i+1}$.

We investigate a viscoelastic wave equation with nonlinear variable-exponent damping and a variable-exponent logarithmic source, allowing supercritical growth. The main difficulties arise from the failure of the source to map the energy space into $L^2(\Omega)$, the sign-changing logarithmic potential, and the lack of Sobolev control of the displacement for supercritical damping. Under an exponent condition linking the source and damping exponents, we prove local existence, an energy identity, and a continuation criterion. We prove finite-time blow-up for nontrivial solutions with nonpositive shifted initial energy. We also establish a blow-up criterion based on a sufficiently large initial displacement–velocity pairing, with a threshold depending on the positive part of the shifted initial energy, and derive an explicit upper bound for the lifespan. This criterion allows us to construct blow-up data at every prescribed level of the original energy. For initial data in a suitable stable region, solutions exist globally. For damping within the Sobolev range, a general relaxation condition yields energy decay estimates that include exponential and polynomial rates. Beyond this range, a time-weighted displacement estimate and a splitting of the memory integral give logarithmic decay.

In this article, a new criterion is established to obtain the linear independence of some values of the $p$-adic Hurwitz zeta function at odd integers $s\geq 3$. We generalize a theorem of F.~Beukers, who gave, among others, an alternative proof—using Padé approximation theory—of a result originally due to F.~Calegari on the irrationality of the $p$-adic Riemann zeta value $\zeta_p(3)$ for $p = 2, 3$. To prove our main theorem, we work in a general setting, where we construct and study new Padé approximants of Laurent series satisfying the functional equation $f(z) = f(1-z)$. This complements our previous work on the linear independence of $p$-adic polygamma values.

These notes introduce crystals and crystalline Dieudonné theory through the example of abelian varieties. After discussing $p$-divisible groups, $F$-isocrystals, slope filtrations, and overconvergence, we explain the parabolicity theorem for $F$-isocrystals and some of its applications to monodromy and Hecke orbits. Two appendices formulate research projects on isocrystals over formal schemes and on monodromy beyond the constant-slope setting. The exposition is intended for readers familiar with basic algebraic geometry and the étale and fppf topologies; prior knowledge of Witt vectors is helpful but not required.

We consider a family $(\Lambda_\lambda)_{\lambda \in \Bbbk^\times}$ of weakly symmetric algebras of Euclidean type over an algebraically closed field $\Bbbk$, where $\Bbbk^\times:= \Bbbk\setminus \{0\}$, and for each $\lambda \in \Bbbk^\times$, $\Lambda_\lambda$ is socle equivalent to $\Lambda_1$, which is the trivial extension of a hereditary algebra of type $\tilde{A}$. We compute their Hochschild cohomologies, which shows that $\Lambda_\lambda$ is not derived equivalent to $\Lambda_1$ if $\lambda \neq 1$. This means that a derived equivalence classification of weakly symmetric algebras of Euclidean type given by Bocian–Holm–Skowroński is not complete.

We show that, in ambient dimension at most seven, varifold limits of the minimizing movements scheme introduced by Almgren–Taylor–Wang and Luckhaus–Sturzenhecker are integral Brakke flows. As a consequence, we show that $L^1$-limit points of the scheme, usually known as flat flows, are generalized BV flows. The result is unconditional and, in particular, remains valid even in the absence of energy convergence and in the presence of fattening.

Non-firm connection products can accelerate grid access while reinforcements are delayed, but their planning value depends on evidence supporting the contractual promise. We develop a one-way evidence-to-access framework for calendarized products. Stage A constructs and fixes directional restriction calendars from public residual-load and day-ahead price-timing signals. Stage B audits the submitted calendar and one bounded common repair, tests joint threshold attainability under the shared annual budget, and applies an aggregation-aware Power Transfer Distribution Factor (PTDF) certificate. Stage C treats the resulting ledger as fixed in queue-aware allocation and intervention planning. In the primary policy-package evaluation, the evidence-supported menu returns the same $2.119$~GW allocation as firm-only access, whereas authorizing the uniformly derated fallback raises awards to $6.442$~GW and reduces backlog from $136.076$ to $113.601$~GW-years; all $4.489$~GW of non-firm awards use the fallback. In a target-free matched control, evidence-routed and state-blind fallback menus attain primary objectives equivalent within EUR~1 and return the same accepted applicants, award years, routes, and intervention portfolio. Thus, in the reported testbed, fallback authorization rather than calendar-specific evidence support drives the access-volume effect.

In this paper, we consider the orbital stability of solitary waves for the derivative nonlinear Schrödinger equation $$iq_t+q_{xx}+i\partial_x\bigl(|q|^2q\bigr)=0, \quad (t,x)\in\mathbb{R}\times\mathbb{R}.$$ For the endpoint solitary waves satisfying $c>0$ and $c^2=4\omega$, we prove the orbital stability in $H^1(\mathbb{R})$ without any additional assumptions. The proof uses conservation of mass and the second Hamiltonian $\mathcal{H}_2$ to keep the Bäcklund-transformed solution small in $H^1(\mathbb{R})$ uniformly in time. We establish an inverse Bäcklund compactness result to reconstruct the algebraic solitary wave as the spectral zero approaches the negative real axis.

We study isomorphic embeddings into Musielak-Orlicz sequence spaces \(\ell_\Phi\), where the normalized generating sequence \(\Phi=\{\phi_n\}\) satisfies the \(\delta_2\) condition. Using admissible normalized blocks, we construct an associated limiting family \(\mathcal C_{\Phi,\tau}\) of Orlicz functions. If a Banach space \(X\) with a symmetric basis embeds into \(\ell_\Phi\), then there is a function \(\psi\in\mathcal C_{\Phi,\tau}\) such that the basis of \(X\) is equivalent to the canonical basis of \(\ell_\psi\). An Orlicz sequence space \(\ell_\psi\) embeds if and only if \(\psi\) is equivalent near zero to a member of \(\mathcal C_{\Phi,\tau}\); in particular, \(\ell^p\) embeds exactly when such a member is equivalent to \(t^p\). Assuming differentiability of the generators on a common interval near zero, we also derive the necessary bound \(p\leq b_{\Phi,(c_n),\delta}\), where the upper growth index incorporates the summable errors in the \(\delta_2\) condition. The same index yields cotype \(q\) for every \(q\geq2\) with \(q>b_{\Phi,(c_n),\delta}\). Examples with nonuniform growth and lacunary weights illustrate this bound. For alternating Nakano exponents, the explicit limiting family gives a sharper restriction than the index alone. For general Nakano exponents, we recover the known exact cotype criterion, including attainment of the optimal exponent. For weighted Orlicz spaces, we express the cotype criterion through summable scalar errors and recover the formula using the global upper growth index when \(w\in\mathcal A\).

Let $F(z)=zf(z)$, where $f$ is a normalized scalar holomorphic function on the unit ball $\mathbb B$ of a complex Banach space $X$ or on the unit polydisk $\mathbb U^n$ of $\mathbb{C}^n$. We establish sharp bounds for two Toeplitz determinants involving the second, third and fourth Fréchet derivatives of $F$ when $F$ is starlike of order $\alpha$, $0\leq\alpha<1$. The first bound is sharp for the full range $0\leq\alpha<1$, while the second is sharp for $0\leq\alpha\leq 2/3$. The extremal mappings are explicitly identified, confirming the sharpness of the estimates.

Designing a massive multiple-input multiple-output (MIMO) detector that is tailored to each specific deployment site can improve detection performance, yet most existing detectors, including deep-unfolding methods, adapt mainly within human-prescribed algorithmic structures. This letter proposes SiteEvo, a site-adaptive agentic discovery framework that extends adaptation from parameter tuning to detector-algorithm discovery. Rather than selecting or recombining predefined architectures, the large language model (LLM) agent synthesizes complete executable detector programs under a common design contract by combining its pretrained communication knowledge, site-specific wireless evidence, and experience accumulated during the search. Each proposal defines its own processing structure and trainable parameterization, and the parameters are optimized by a communication-system backend. The resulting high-performance detector is further structurally refined to reduce block-level computational cost under a bit error rate (BER)-preservation constraint and is finally frozen for deployment without online LLM inference. Simulations across three sites demonstrate BER reductions by factors of 2.6 to 6.0 over the best-performing competing detector at each site at 6 dB, while structural refinement reduces block-level computational cost by 16.7-34.1% under a validation-BER constraint.

We complete the classification of hypersurfaces with constant sectional curvature of dimension greater than or equal to three in the product of two space forms.

Let k be a field and X be a geometrically rational variety over k. We investigate the Kontsevich moduli spaces of genus zero, stable maps with curve classes of low degree on X over k, and define their twisted Galois descents. We study the k-rationality of these spaces in detail when X is a projective space/Severi Brauer variety and when X is moduli of genus zero n-marked points. When X is a homogeneous variety over the field of complex numbers, this is a well-known theorem of Kim-Pandharipande.

Let $\{f_i(x)=M_ix+a_i\}_{i\in I}$ be an affine iterated function system on $\mathbb{R}^d$ with attractor $\Lambda$, and let $\mu$ be an associated self-affine measure. Assume that the $M_i$ generate a Zariski dense subgroup of $\mathrm{GL}_d(\mathbb{R})$. Under the strong open set condition, we show that $\dim_{\mathrm{H}}\Lambda$ equals the affinity dimension and that $\dim_{\mathrm{H}}\mu$ equals the Lyapunov dimension. The above results are consequences of the following projection theorem, which requires no separation condition. For every $1\leq n\leq d$ and for Furstenberg measure a.e. $(d-n)$-dimensional subspace $V$, \[ \dim_{\mathrm{H}} (P^V)_*\mu=\min\{n,\,\dim_{\mathrm{H}}\mu\}. \] This projection theorem in fact holds under hypotheses weaker than Zariski density: simplicity of the Lyapunov spectrum and a mild transversality condition (the Oseledets non-concentration property). The source of dimension increment in our proof is the positive dimension of the fiberwise conditional measures, exploited through Wu's one-dimensional entropy increase theorem. A key ingredient in reducing the higher-dimensional problem to this setting is a new structural result: a slab-filling property of the measure.

A recent manuscript of OpenAI asserts that Landau-Siegel zeros do not exist. Here, we give a detailed account of its argument, arranged as a direct proof with explicit constants. There are absolute constants $q_0$ and $\delta_0>0$ such that, for every primitive real Dirichlet character $\chi$ of conductor $q\ge q_0$, every real zero $\beta\in(0,1)$ of $L(s,\chi)$ satisfies $(1-\beta)\log q\ge\delta_0$, and one can take $q_0=10^{10}$ and $\delta_0=3.0\times 10^{-3}$. Combined with a recent computation of Lu, Zaman, and Zhao, this result yields the bound $(1-\beta)\log q\ge 3.0\times 10^{-3}$ for every conductor $q\ge 3$. The proof is an interpolation determinant argument in the biquadratic field generated by $\sqrt{2}$ over the quadratic field attached to $\chi$. A real zero close to one forces $\chi(p)=-1$ for most primes $p$, and the Frobenius congruence at these primes makes a certain nonzero determinant divisible by higher powers of these primes than its Archimedean size allows. The underlying ideas are those of the OpenAI manuscript. The explicit constants, and the presentation, are new.

We derive polynomial bubble duality in barycentric coordinates and construct explicit nodal bases for the first family of finite element differential forms. Logarithmic coordinates give an elementary proof of the Whitney–bubble decomposition. The Hessian of negative entropy defines a metric whose normalized Hodge star maps bubble generators to complementary-degree polynomial forms. This gives a direct geometric description of the known dualities and proves nondegeneracy of their integral pairings. Combining Whitney forms with shifted Lagrange polynomials then gives bases dual to tangential point evaluations in every dimension and order. The construction extends to scalar nodes that are unisolvent on every face. The interpolant is independent of the roots and commutes with face restriction.

We prove that a Hopf algebra over an arbitrary field has a bijective antipode if either its left or right regular module has finite uniform dimension. In particular, this solves Skryabin's conjecture, posed in 2006, that every noetherian Hopf algebra has a bijective antipode. Starting from Skryabin's injectivity theorem for weakly finite Hopf algebras, we associate to each finite-dimensional subcoalgebra an essential left ideal of common denominators that is also a right coideal in the antipode image. An annihilator identity makes this image essential in the original algebra as a left module over the image, yielding finite left uniform dimension from finite right uniform dimension. A second denominator construction inside the antipode image, together with a restricted inverse-antipode identity, proves surjectivity.

In this paper, we investigate the Schatten class membership of the product of a composition operator and the adjoint of another composition operator on the Hardy and Bergman spaces. We provide several sufficient conditions and necessary conditions for these operators to belong to the Schatten classes. Moreover, for some specific symbols including linear fractional maps, finite Blaschke products, and inner functions, we obtain necessary and sufficient conditions.

Emil Artin asked whether the left and right dimensions of an extension of skew fields must agree. Cohn and Schofield gave counterexamples through elaborate constructions. It has been a long-standing open question whether there is any division ring, infinite-dimensional over its center, for which the left and right indices agree whenever at least one is finite. We give the first such examples. Call a division ring index-symmetric if every division subring with at least one finite index has equal finite indices. Using a duality formula for bimodules over division algebras, we prove that every twisted Calabi–Yau division algebra is index-symmetric. The examples include the Weyl skew fields and, more generally, the quotient division rings of rings of differential operators on smooth affine varieties in characteristic zero. Further examples are the quotient division rings of enveloping algebras of finite-dimensional Lie algebras, Ore extensions of twisted Calabi–Yau noetherian domains, quantum affine spaces, and group algebras of torsion-free polycyclic-by-finite groups.

FirstFit colors each arriving vertex with the smallest positive integer absent from its previously colored neighborhood. We prove that this rule can require $n^{\Omega(1)}$ many colors on certain bipartite graph of $n$ vertices even when the vertices arrive in a uniformly random order. This refutes the conjecture of Bosek, Gutowski, Lasoń, and Przybyło [BGLP24] that FirstFit uses a polylogarithmic expected number of colors on every bipartite graph in the random order model. We obtain the lower bound by analyzing the FirstFit coloring process on random bipartite graphs, where vertices arrive to either side with equal probability, and with an independent edge density $n^{-x}$ for some $0 < x < 1$. Our result can also be viewed as a qualitative strengthening of Kučera [Kuč91], who showed that, for every $\epsilon > 0$, there are graphs with chromatic number at most $n^{\epsilon}$ on which FirstFit uses $\Omega(n/\log n)$ colors in expectation under a uniformly random order. Our result shows that such poor behavior of FirstFit already occurs for $\chi(G) = 2$.

Does evolutionary entropy depend on the width of the age classes used to describe a population? We show that the apparent dependence on temporal resolution is a consequence of the loss of information about reproductive timing within discrete classes. When this information is retained, corresponding continuous and discrete representations yield the same cycle entropy, generation time, evolutionary entropy, and reproductive potential in a common unit of time. When within-class timing is unknown, a uniform local reconstruction yields a logarithmic correction. For constant post-maturity hazard and fertility, continuous exponential and discrete geometric descriptions correspond exactly at every time scale. Thus, for a fixed reference duration, evolutionary entropy is a property of the reproductive process, not of its temporal partition. These results provide a mathematical basis for comparing demographic models across different temporal scales by identifying the information required to separate differences in reproductive organisation from effects of the chosen age-class representation.

We study boundary vanishing orders and asymptotic expansions of planar harmonic functions satisfying the sublinear Robin condition $\partial_\nu u=\lambda u-|u|^{p-1}u,$ with $\lambda\geq0$ and $0<p<1$. For nontrivial global solutions on bounded connected Lipschitz domains, we bound the number of nodal domains by the number of Steklov eigenvalues strictly below $\lambda$ and establish finite vanishing order at every boundary point. The boundary scaling gives the critical exponent $\frac{1}{1-p}$. On $C^1$ boundaries, every positive subcritical vanishing order is an integer, and the leading term is a unique homogeneous Neumann profile in conformal coordinates. At a noninteger critical order, the rescaled solution converges in normalized graph coordinates to an explicit homogeneous profile determined by $p$ up to sign. Every supercritical bulk vanishing order $m$ is an integer, with a unique homogeneous Dirichlet profile in conformal coordinates. The boundary trace has vanishing order $\frac{m-1}{p}>m$. Integer critical vanishing orders are excluded if the boundary is $C^1$ with Dini mean oscillation.

In this paper, we study the focusing mass-critical Hartree equation \[ iu_t+\Delta u=-\bigl(|x|^{-2}*|u|^2\bigr)u \] at the ground-state mass $M(Q)$. We prove that, up to the symmetries of the equation, every global minimal-mass non-scattering solution is a solitary wave, while every finite-time minimal-mass blowup solution is a pseudo-conformal image of a solitary wave. This extends the results of \cite{LZ28,Miao-Xu-Zhao3,Wang-Gao} to the scaling-critical space $L^2$ and removes the radial assumption in the global case.

We give new, elementary proofs of interior second derivative estimates for the Monge-Ampère equation and the constant quadratic Hessian equation, based on the two-point maximum principle. The arguments require only one differentiation of the equation. For the Monge-Ampère equation, the inhomogeneous term is assumed to be positive and continuously differentiable. For the quadratic Hessian equation, the estimate holds in every dimension $n\ge2$.

We prove that a commutative integral domain is Gorenstein Prüfer if and only if every overring is Gorenstein flat over the original domain. It is enough to test overrings that are finitely generated as algebras. The proof combines finite duality with a conductor-quotient argument. Under the overring hypothesis, the conductor quotient of a proper overring generated by two elements as a module is an IF-ring, and degree-one duality lifts first syzygies of finitely presented quotient modules to totally reflexive modules. This gives a bound of one for the Gorenstein flat dimension of every module annihilated elementwise by conductor powers. Quadratic integral elements provide such conductors at nonnormal localizations, while normal localizations give principal-ideal neighborhoods. Constructible compactness yields a finite cover, and a bounded complex of localizations combines these bounds over the original domain. Consequently every finitely generated ideal is Gorenstein flat and the domain is coherent. Examples distinguish finite-type from module-finite tests and Gorenstein flatness from ordinary flatness.

Let $F$ be an algebraically closed field of characteristic $7$, and let $V=L(\frac12,0)_F$. For the degree-four singular vector $q=(L(-2)^2-2L(-4))\mathbf1$ in the universal vacuum vertex operator algebra $\bar V$, set $V_q=\bar V/I(q)$. We prove that the natural quotient map induces an isomorphism $A_1(V_q)\cong A_1(V)\cong F[T]/(T(T-4)(T-5))\cong F^3$. Explicit $O_1(V_q)$-relations and four families of triangular PBW straightening relations give the upper bound, while three one-dimensional homogeneous subspaces give the matching lower bound.

For an $r$-uniform hypergraph $\mathcal H$, let $\ex_r(n,\mathcal H)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph containing no copy of $\mathcal H$. The $r$-uniform expansion $T^{(r)}$ of a tree $T$ is obtained by adding $r-2$ new vertices to each edge, with the added sets pairwise disjoint and disjoint from $V(T)$. In particular, the expansion of $K_{1,k}$ is the linear star $\mathcal L_k^r$, whose $k$ edges intersect pairwise in a single common vertex. Chung and Frankl (1987) determined $\ex_3(n,\mathcal L_k^3)$ and all extremal hypergraphs for every $k\ge3$ and sufficiently large $n$. We extend the odd-$k$ part of their result to every $r\ge5$, determining the exact Turán number and the unique extremal hypergraph up to isomorphism for sufficiently large $n$. More generally, we extend this result to expansions of trees for which the minimum degree among the vertices in the smaller color class is odd and at least three. For every $r\ge5$ and sufficiently large $n$, we establish lower and upper bounds on their Turán numbers. When no two vertices attaining this minimum degree in that color class have a common neighbor, we determine the exact Turán number and the unique extremal hypergraph up to isomorphism. The proofs combine the delta-system method with stability arguments and a detailed analysis of extremal constructions.

There exists a one-dimensional local Gorenstein Prüfer domain whose integral closure is not Prüfer. We prove that the recently constructed coherent counterexample to the normalization problem has this stronger Gorenstein property. The argument establishes permanence of divisoriality of finitely generated ideals under ultraproducts and quotients by divided primes. It also supplies uniform matrix factorizations: in an ultraproduct of plane-curve domains, every nonzero ideal generated by $r$ elements is a cycle of a two-periodic totally acyclic complex with free terms of rank $r$. The divided prime defining the one-dimensional example is flat, so these complexes descend through the quotient and localization. A specified two-generated ideal remains noninvertible in every integral overring. Over the normalization, its associated Laurent overring has nonzero first Tor with every injective cogenerator, although both rings are Gorenstein flat over the original domain. Examples distinguish this failure from a nonnormal Gorenstein curve with Prüfer normalization and show that the periodicity conclusion is stronger than the Gorenstein Prüfer property alone.

Let $\mathbb{F}_q$ be a finite field of any characteristic and $\rm GL(n,\mathbb{F}_q)$ the general linear group over $\mathbb{F}_q$. Suppose $W$ denotes the standard representation of $\rm GL(n,\mathbb{F}_q)$ and $\rm GL(n,\mathbb{F}_q)$ acts diagonally on the direct sum of $W$ and its dual space $W^*$. Let $G$ be any subgroup of $\rm GL(n,\mathbb{F}_q)$. Suppose the invariant field $\mathbb{F}_q(W)^G=\mathbb{F}_q(f_1,f_2,\dots,f_k)$, where $f_1,f_2,\dots,f_k\in\mathbb{F}_q[W]^G$ are homogeneous invariant polynomials. We prove that there exist homogeneous polynomials $\ell_1,\ell_2,\dots,\ell_{n}$ in the invariant ring $\mathbb{F}_q[W\oplus W^*]^{G}$ such that the invariant field $\mathbb{F}_q(W\oplus W^*)^{G}$ is generated by $\{f_1,f_2,\dots,f_k,\ell_1,\ell_2,\dots,\ell_{n}\}$ over $\mathbb{F}_q$.

We study spectral Gårding elliptic equations on compact Kähler manifolds. Smooth solvability is equivalent to the existence of a smooth representative in the admissible domain. The concavity theorem for ideal Gårding polynomials gives a fixed concave formulation to which Székelyhidi's second-order argument applies in every degree $2\le d\le n$, extending Lin's top-degree solvability theorem. Combined with the numerical criteria of Fu–Li–Zhang–Zhang and Chen–Nie–Xu, this yields a Nakai–Moishezon type criterion for solvability on compact Kähler manifolds. A solution converts the local ideal Gårding structure into a global Lorentzian structure. Its mixed derivatives give a strict nonlinear Hodge–Riemann hierarchy from the degree of the equation down to degree two, and the homogenization of the intersection polynomial is Lorentzian in nef class directions. Numerical criteria then produce explicit Hodge–Riemann classes, Lefschetz isomorphisms in bidegree $(1,1)$, and weighted Hodge-index inequalities.