Papers

Classical models such as smooth strongly convex minimization and strongly convex–strongly concave optimization combine two types of curvature information: curvature remains uniformly separated from zero, and its positive and negative orientations are prescribed in advance. Uniform nondegeneracy disentangles these roles, retaining the separation from zero while allowing the curvature orientation to vary across the domain. We identify a family of descent–ascent inequalities (DAI) that characterizes uniform nondegeneracy and naturally gives rise to Alternating Descent Ascent (ADA), a simple explicit method that alternates full-gradient descent and ascent. ADA converges globally and linearly to the unique stationary point from arbitrary initialization, with dimension-free first-order complexity. The framework recovers the smooth strongly convex setting as a limit and encompasses the strongly convex–strongly concave and linearly constrained settings, highlighting uniform nondegeneracy as a common first-order structure across various classical regimes.

Let $w \colon (0,1) \to \mathbb{R}_{+}$ be a weight. We prove that for an arbitrary Schauder basis $\{e^{i \lambda_n t}\}_{n \in \mathbb{Z}}$ in $L^2(0,1)$ and an arbitrary lacunary defect set $A \subset \mathbb{Z}$ the system $\{w(t) r_n(t)\}_{n \in \mathbb{Z} \setminus A}$ is always complete and never minimal for any weight $w$ satisfying a natural decay condition, which is sharp on the exponential scale. Moreover, we establish a simple combinatorial criterion for systems of the form $ \{ w(t) e^{2\pi i n t} \}_{n \in \mathbb{Z} \setminus A} $ to be complete and minimal in $L^2(0,1)$ for an arbitrary weight $w$ and a finite defect set $A \subset \mathbb{Z}$.

We establish a criterion for the irreducibility of Verma modules over symmetrizable Kac–Moody superalgebras. Our main tool is the Weyl-Kac denominator identity.

The lampshuffler group of $\mathbb{Z}$ is the semidirect product $\operatorname{FSym}(\mathbb{Z}) \rtimes \mathbb{Z}$, which consists of all permutations of $\mathbb{Z}$ that act as a translation outside a finite set. This infinite permutation group naturally contains as subgroups the wreath products $H \wr \mathbb{Z}$ for every finite group $H$. We prove that the Subgroup Membership Problem, and more generally, the Submonoid Membership Problem, are decidable in $\operatorname{FSym}(\mathbb{Z}) \rtimes \mathbb{Z}$. Our proof reduces Subgroup and Submonoid Membership in $\operatorname{FSym}(\mathbb{Z}) \rtimes \mathbb{Z}$ to Subgroup Membership in the wreath products $H \wr \mathbb{Z}$, which was shown to be decidable by Lohrey, Steinberg and Zetzsche (2015).

We study finite Laver algebras, the generalizations of Laver tables on sets of cardinality not of the form $2^n$. We characterize the conditions under which projections and embeddings between Laver algebras exist and show that there are $2^{\aleph_0}$ pairwise non-isomorphic direct limits of finite Laver algebras.

We investigate derivation modules of hyperplane arrangements associated with the monomial groups $G(r,p,\ell)$. Motivated by Mühlherr's separator-based construction for graphic arrangements, we extend this approach by combining separators with integral expressions. For a subarrangement $\mathcal{A}_G$ defined by a graph $G$, we show that, when $p<r$, the derivation module $\mathcal{D}(\mathcal{A}_G)$ is generated by the standard derivations $\eta_0,\dots,\eta_{\kappa(G)}$ together with a finite set of separator-based derivations. This result generalizes the complete graph case and provides new evidence toward the broader applicability of separator methods in the study of hyperplane arrangements.

We construct explicit off-diagonal Ramsey graphs from polynomial families of incident point–hyperplane flags. Universal interpolation translates the exact dimension of ordered clique configurations into a bound on their coefficient–label incidence. At the critical dimension, a polynomial separates the forbidden-pair image from its diagonal; deleting the corresponding edges preserves a triangular rank certificate. Restriction of scalars realizes the rational endpoint without rounding loss. The resulting fixed-$s$ exponent has leading scale $s/(2\log_2s)$, and short Frobenius relations give the explicit example $R(16,t)\ge\Omega(t^{2.0539221767\ldots})$. A finite-fiber construction gives the range $s^2(\log s)^2=o(\log t)$. For fixed parameters, algebraic preprocessing terminates and field initialization, vertex decoding and adjacency take deterministic time polynomial in the extension degree. We also retain an elimination-free variant and formulate the filtering argument for general forbidden configurations with a distinguished edge.

We classify, up to graded isomorphism, graded bimodules over split group algebras of finite abelian subgroups of an arbitrary group. The classification is governed by double cosets, stabilizer characters, and coefficient modules. Over a field, we determine the essential image of the induction functor on Yetter-Drinfel'd modules, giving a negative answer to a question of Dos Santos and Yasumura in the nonabelian case. We also compute the Peirce components and characterize multiplicity-free bimodules through extensions of stabilizer characters.

Normalized gradient descent is a widely studied adaptive optimization method. Most existing analyses focus on the best iterate or a weighted average of the iterates, whereas practical implementations typically return the last iterate. In this paper, we study the last-iterate convergence of normalized gradient descent for convex, $(\nu,M_\nu)$-Hölder-smooth objectives. For a constant stepsize, we establish an upper bound of $\mathcal{O}\bigl((\log^2(T)/T)^{(1+\nu)/2}\bigr)$, which contains a logarithmic overhead relative to the known $\mathcal{O}\bigl(T^{-(1+\nu)/2}\bigr)$ guarantees for the best and weighted-average iterates. For $\nu = 0$, this overhead is known to be unavoidable. We complement this analysis with numerical results based on the performance estimation problem (PEP), investigating the finite-horizon worst-case behavior in the smooth setting and whether the logarithmic overhead reflects an intrinsic limitation of constant-step normalized gradient descent. We then show that a linearly decreasing stepsize yields a last-iterate guarantee of $\mathcal{O}\bigl(T^{-(1+\nu)/2}\bigr)$, matching the order of the best-iterate/weighted-average guarantees without requiring knowledge of $\nu$ and $M_\nu$.

Boscaggin, Feltrin and Zanolin conjectured that the Neumann and periodic problems associated with the equation \begin{equation*} u”+a(t)u^\gamma=0 \end{equation*} with a two-step indefinite weight $a(t)$ have at most one positive solution for every $\gamma\in\mathbb{R}\setminus\left\{-1,0,1\right\}$. We prove this conjecture and complete the uniqueness picture by also including the logarithmic potential case $\gamma=-1$. More generally, for every $p>1$, we consider the equation \begin{equation*} \left(\left\lvert u'\right\rvert^{p-2}u'\right)'+a(t)u^\gamma=0 \end{equation*} under Neumann or periodic boundary conditions and establish a complete existence and uniqueness classification for all $\gamma\in\mathbb{R}\setminus\left\{0,p-1\right\}$. The necessary mean-value condition $\gamma\int_{0}^{T}a(t)\,\mathrm{d} t<0$ is also sufficient for existence when $\gamma\leq-1$ or $\gamma>p-1$, while an additional sharp condition on the mean of the weight is required in the intermediate range $\gamma\in(-1,p-1)\setminus\{0\}$. Finally, a numerical example provides evidence of multiple positive solutions for the Neumann problem whose weight has the same single change of sign but a non-constant internal profile, highlighting the role of the two-step assumption.

We construct a complex structure on $T^2\times S^4$, giving an affirmative answer to the genus-one case of Calabi's problem. The resulting compact complex threefold is non-Kähler and admits no biholomorphic product decomposition.

Regional weighted phase-volume contraction for nonlinear systems is studied using positive densities selected from a finite-dimensional log-affine family. The affine dependence of the density divergence on the design parameters turns density selection into a semi-infinite convex optimization problem that maximizes a certified uniform contraction margin on a compact region. This provides a constructive alternative to testing a preselected density and ensures a well-posed design problem under compact parameter constraints. Finite active-point and minimax characterizations identify the worst-case states governing the optimum, while rigorous sampling bounds allow continuum contraction guarantees to be recovered from finitely many inequalities. For scalar power densities, feasibility is characterized by an exact interval condition. The same structure is preserved in discrete time and admits distance-weighted and conformal-geometric interpretations. Overall, the framework provides computable certificates for detecting regional contraction that may remain hidden from ordinary divergence. A two-parameter limit-cycle example demonstrates this effect when both ordinary phase volume and a natural single-basis density fail.

The variational derivation of two-dimensional von Kármán shell energies from three-dimensional nonlinear elasticity is investigated in the presence of constant pressure loads. On the one hand, the paper extends the existing analysis of pressure-loaded plates to the curved setting of shells. On the other hand, it complements the classical variational derivation of shell theories by incorporating a nonlinear pressure force.

The evolution of weighted phase volume under a fixed dynamical system is investigated using a positive weight raised to an arbitrary real exponent. Sufficient conditions for uniform exponential contraction and expansion of transported weighted volume are obtained in terms of the corresponding weighted divergence. These conditions make it possible to reveal dissipative properties that may remain undetected by the ordinary divergence. Consequences for invariant sets are established, and, under the assumption of a compact absorbing set, the weighted measure of a global attractor is characterized. The approach is extended to invariant submanifolds through a distance-weighted ambient-volume estimate. A differential-geometric interpretation is also provided, in which the weighted divergence is identified with the ordinary divergence associated with a conformally transformed metric. The dependence on the exponent is used to reveal a duality between mutually inverse weights. In addition, sufficient local conditions for exponential attraction to an invariant hypersurface are derived using a sign-changing weight. Weighted-volume decay is interpreted as an integral property of transported sets and, by itself, is not identified with pointwise Lyapunov stability.

Let $Y_{n,r,q}$ be the graph obtained from the Turán graph $T_{n,r}$ by adding $q$ pairwise disjoint edges inside a largest part, and let $c(n,F)$ be the minimum number of copies of $F$ created by adding a single edge to $T_{n,r}$. Fang, Li, Lin and Ma proved that for every color-critical graph $F$ with $\chi (F)=r+1$, there exists a constant $\delta_F>0$ such that for all sufficiently large $n$ and all $1\le q\le \delta_F \sqrt{n}$, the condition $\lambda(G)\ge\lambda(Y_{n,r,q})$ forces at least $q\, c(n,F)$ copies of $F$. The bound $q=O(\sqrt{n}\,)$ is tight up to a constant factor, in contrast to the linear order $n$ of the edge setting of Mubayi, Pikhurko and Yilma, but the exact constant $\delta_F$ remained unknown for any $F$. In this paper, building on a structural result of Fang, Li, Lin and Ma, we determine the threshold $\delta_F$ when $F$ is a clique and an odd cycle. For every $r\ge2$, we denote $\delta_r :=(1-\tfrac1r)\sqrt2$ and prove that for every $\varepsilon>0$, if $n$ is sufficiently large and $1\le q\le(\delta_r-\varepsilon)\sqrt n$, then every $n$-vertex graph $G$ with $\lambda(G)\ge\lambda(Y_{n,r,q})$ contains at least $q\,c(n,K_{r+1})$ copies of $K_{r+1}$, and $\delta_r$ is best possible. For odd cycles, the threshold is $1/\sqrt2$. For every $k\ge1$ and $\varepsilon>0$, if $n$ is sufficiently large and $1\le q\le(1/\sqrt2-\varepsilon)\sqrt n$, then every $n$-vertex graph $G$ with $\lambda(G)\ge\lambda(Y_{n,2,q})$ contains at least $q\,c(n,C_{2k+1})$ copies of $C_{2k+1}$, and $1/\sqrt2$ is best possible. Our results determine both the exact count of copies and the optimal range of $q$. The behavior in the spectral setting differs from the classical edge setting, in which the range of $q$ is of order $n$ and the threshold is $1/r$ for cliques by Lovász and Simonovits, and $1/2$ for odd cycles by Pikhurko and Yilma.

Oxley conjectured that a matroid with a circuit-cocircuit intersection of size $k\geq 4$ has one of size $k-2$. Using the standard polynomial representation of the classical extended binary Golay code, we construct an identically self-dual binary matroid of rank $12$ on $24$ elements. Its circuit-cocircuit intersection sizes are exactly $0,2,4,6,8$, and $12$, disproving the conjecture for $k=12$. The proof is self-contained and does not rely on the full weight distribution of the code or on an enumeration of all circuits.

We consider Walsh's process on a star graph perturbed by the fact that its paths need to additionally pass through semi-permeable soft membranes, each located on a different edge at a small distance from the center, and each having its own permeability coefficient. We describe the limit processes as the distances converge to $0$, in terms of the characteristics of the approximating ones. The main finding is that the limit process is still of Walsh type, but its parameters need to be appropriately modified. The paper is a close companion to A. Bobrowski and A. Pilipenko “On Portenko's approximation of skew Brownian motion” (arXiv:2601.20440), and to A. Pilipenko and G. Vilmart, “Limit theorems for one-dimensional diffusions: reflected, skew, sticky, skew-sticky, and snapping-out limits.”

The Independent Chip Model (ICM) converts the chip stacks of the players remaining in a poker tournament into finishing-place probabilities and prize equities. It is the standard model in tournament solvers, but its definition considers all possible finishing orders, and exact computation has been regarded as intractable for large fields. The same mathematics appears in other fields; for instance, the ICM is a Plackett-Luce ranking model with stacks as weights. We present DE-ICM, a deterministic algorithm that computes the placement probabilities of all $n$ players for all paid places in $O(M n^{2})$ time, where the number $M$ of quadrature nodes is fixed for the admissible inputs and the target accuracy. Its numerical scheme allows the target to be set close to the accuracy of double-precision arithmetic. The algorithm evaluates a classical integral representation in which, conditional on a player's exponential clock, the number of players ahead is Poisson-binomial. A double-exponential quadrature on a single grid evaluates every integral; its truncation errors have closed-form bounds, and the step size, which controls the discretization error, follows from the field size. A numerically stable deconvolution then recovers each player's leave-one-out coefficients in $O(n)$ time from one shared product. On exactly solvable instances with up to 4,000 players, the worst observed relative error of an equity is $4.0 \times 10^{-14}$ and the worst absolute error of a placement probability $1.1 \times 10^{-14}$. The full placement matrix for 1,000 players takes 0.2 seconds on one CPU core.

We prove a closed formula for DT4 invariants of one dimensional stable sheaves on the total space of $\mathcal{O}_{\mathbb{P}^2}(-1,-2)$ for all degree curve classes. During the proof, orientations of moduli spaces are chosen in a canonical way. Combining with Gopakumar-Vafa type invariants proposed and computed by Klemm-Pandharipande, we verify a conjecture proposed by the author, Maulik and Toda.

We study the extent to which the invariants appearing in the Birch and Swinnerton-Dyer conjecture remember the elliptic curve, and we give several affirmative answers to this question. For example, the real period $\Omega_E =\frac{\Gamma(1/4)^2}{2\sqrt{\pi}\,17^{1/4}}$ and the Tate–Shafarevich group $ \Sha(E/\mathbb{Q})\cong \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ remember the elliptic curve $E: y^2=x^3+17x$. Let $K$ be a number field admitting a real embedding, and fix one such embedding $\sigma\colon K\hookrightarrow\mathbb{R}$. We first prove that, for an elliptic curve $E$ over $K$, the real period of $E$ attached to $\sigma$ determines the $K$-isogeny class of $E$. We then prove that an elliptic curve $E$ over $\mathbb{Q}$ is determined up to isomorphism over $\mathbb{Q}$ by its real period, the number of connected components of $E(\mathbb{R})$, and the real period of the quadratic twist $E^{D}$ by a negative square-free integer $D\equiv 1 \bmod 4$ coprime to the minimal discriminant of $E$. Moreover, over $\mathbb{Q}$, both statements already hold when the real periods are known only to sufficiently many decimal places, where the required precision depends on $E$ (and $D$).

We study transverse polynomials along polar arcs of reduced holomorphic plane function germs. We prove that, under bi-Lipschitz right equivalence, matched tangential polar arcs have the same transverse order and their transverse polynomials agree up to nonzero rescalings of the source and target, at every rational scale $1<q<d(\gamma)$, where $d(\gamma)$ is the gradient canyon degree. This range is sharp as the result can fail at both endpoints, even under analytic right equivalence. As an application, we recover the bi-Lipschitz invariance of the augmented Newton polygon proved by Migus–Păunescu–Tibăr. We also give an example showing that the transverse polynomial contains new information not detected by the polar-value invariants and augmented Newton polygons.

We show that whenever $G$ is a finitely generated group with the rapid decay property (property RD), then the norm on the reduced group C*-algebra $C^*_r(G)$ is computable if (and only if) $G$ has solvable word problem. We also give an example of a finitely presented group $G$ with solvable word problem for which the norm on $C^*_r(G)$ is not computable.

Let $\mathfrak g$ be a simple Lie algebra of classical type. We construct a $\mathbb Z[1/2]$-integral form of the Yangian $Y(\mathfrak g)$ generated by divided powers of the positive and negative Drinfeld generators, and establish its triangular decomposition and PBW basis. Under the loop filtration, its associated graded algebra is the current algebra part of Garland's integral form. As applications, over an algebraically closed field of characteristic $p>3$, we identify the small Yangian with the restricted Yangian and classify their finite dimensional simple modules in terms of restricted Drinfeld polynomials.

Answering a question of Thomas Sinclair, we show that the standard presentation of the reduced group C*-algebra $C^*_r(G)$ of an infinite hyperbolic group $G$ is computable. The main ingredient in proving this result is showing that the Gromov boundary $\partial G$ of $G$ admits a computably compact metric presentation for which the natural action of $G$ on $\partial G$ is computable.

We study the high-frequency Fourier coefficients $C_n$ of critical Gaussian multiplicative chaos $M$ on the circle, for logarithmic covariances with an arbitrary admissible smooth, possibly nonstationary remainder. For every fixed finite set of integer offsets, the corresponding vector of neighboring coefficients, multiplied by $\sqrt{\log n}$, converges stably relative to the full Gaussian field without additional centering. Conditionally on that field, the limit consists of Fourier coefficients of one symmetric complex stable random measure of index one with control proportional to $M$. For every $s>1$, the normalized modulated measures also converge stably in $H^{-s}(\mathbb{T})$ to this conditional stable noise. Unconditionally, the same joint approximation remains valid for offsets of size at most $(\log\log n)^{1/64}$ for every fixed admissible covariance. The scalar limit is an isotropic Cauchy mixture; the joint limiting law retains the spatial distribution of $M$, not only its total mass. We also prove an almost-sure integral test along the entire integer frequency sequence. For a class of regular deterministic gauges $H$, the limsup of $\sqrt{\log|n|}\,|C_n|/H(\log\log|n|)$ is zero or infinity according as $\int^\infty dx/H(x)$ converges or diverges. The proof combines index-one compensation with relative excursion estimates and transfers these conclusions from the canonical model by smooth covariance comparison.

The goal of this article is to study optimal high-order Taylor-based polynomial quasi-Trefftz spaces for first-order time-harmonic Maxwell's equations. Compared with standard polynomial spaces, the purpose of quasi-Trefftz spaces is to retain the same high-order best approximation property but more efficiently: their dimension is considerably smaller. To achieve this goal, they approximate only smooth solutions to the governing equation rather than to approximate general smooth functions. While there is a natural definition of Taylor-based polynomial quasi-Trefftz spaces for scalar equations, the situation is fundamentally different for vector-valued equations, as the naive quasi-Trefftz property does not systematically define the smallest space with the desired best approximation property. This article identifies the optimal quasi-Trefftz spaces for first-order time-harmonic Maxwell's equations, proposes two approaches to construct quasi-Trefftz bases, and proves the spaces' best approximation property. The theoretical results, which strongly leverage polynomial Helmholtz decompositions and the exactness of the polynomial de Rham sequences, are validated with numerical experiments.

We establish transference results for multilinear Fourier multiplier operators from Euclidean spaces to product spaces of the form $\mathbb R^{d_1}\times\mathbb T^{d_2}$. We prove strong and weak-type transference principles that preserve the corresponding Euclidean operator bounds. As applications, we obtain bounds for certain multilinear operators, as well as for some multiplier operators arising naturally in PDEs.

We give an example of a proper, regular, integral arithmetic surface over $\mathrm{Spec}\ \Z$ endowed with a section, and whose étale fundamental group admits an infinite pro-$2$ quotient. The construction is a geometric analog of the Golod–Shafarevich towers for rings of integers of number fields.

We study the Cauchy problem for the one-dimensional isothermal compressible Navier–Stokes system with the van der Waals equation of state, and analyze the well-posedness and asymptotic stability of steady-state phase-transition solutions at subcritical temperatures. Working in Lagrangian coordinates, we use variational techniques to construct piecewise smooth steady-state solutions corresponding to two-phase equilibrium states under finite-mass perturbations. Introducing artificial-viscosity regularization, we first establish existence of the global minimizer of the regularized variational problem, unique within each fixed symmetry class; passing to the vanishing artificial-viscosity limit then yields a physically admissible steady-state solution with two phase interfaces. For initial perturbations that are small with respect to energy, piecewise-gradient, total-variation, and integral norms but may exhibit large pointwise jumps across phase boundaries, we derive uniform a priori bounds for weak solutions near the two-phase equilibrium profile. We further prove that these global weak solutions converge to the steady-state two-phase equilibrium in $L^\infty(\mathbb{R})$ as $t\to\infty$, thereby characterizing their large-time asymptotic behavior. Physically, finite mass perturbations break the uniqueness of the uniform steady state and generate infinitely many piecewise constant two-phase equilibria satisfying the Maxwell equal-area rule. Artificial viscosity acts as a selection principle that singles out the physically relevant two-interface steady-state configuration, and this two-phase equilibrium is asymptotically stable under small integral perturbations that allow large-amplitude discontinuities at phase interfaces.

We study four of Manton's five vortex equations on a Riemann surface $X$ in relation to anti-self-dual connections. From a solution of each vortex equation, we construct an equivariant holomorphic bundle over $X$$\times$$\mathbb{P}^1$, $X$$\times$$\Delta$, or $X$$\times$$\mathbb{C}$, together with an invariant Hermitian or pseudo-Hermitian metric whose associated connection is anti-self-dual. The equivariant structures are given by the actions of $SU(2)$, $SU(1,1)$, and $SE(2)$, respectively. Conversely, we show that an equivariant holomorphic bundle equipped with an invariant (pseudo-)Hermitian metric gives rise to a solution of the corresponding vortex equation when the associated connection is anti-self-dual.