Papers

Boundary value problems for systems of conservation laws, including inviscid compressible flows, arise naturally in physical models such as shallow water flow in a canal, where fluid may leave the domain through the boundary. In contrast to the Cauchy problem, the inviscid limit is highly sensitive to the underlying viscous model: different viscous regularizations may select different effective boundary conditions for the limiting system. In this paper, we show that small BV solutions of the one-dimensional compressible barotropic Euler equations arise as inviscid limits of the barotropic Navier-Stokes system for the outflow boundary problem. We work with the physically meaningful effective boundary condition selected by the Navier-Stokes system for the limiting Euler solutions. Moreover, going beyond smooth flows or a fixed class of Riemann solutions, our stability results apply to small BV solutions exhibiting general wave patterns and complicated configurations generated by interactions with the outflow boundary, including reflected shocks.

Modern space missions require trajectory design methods capable of efficiently adapting to diverse mission objectives and operational constraints. Despite the maturity of non-convex trajectory optimization techniques, their broader adoption remains limited by the substantial domain expertise required to tailor problem formulations across varying mission scenarios. Recent advances in large language model (LLM) reasoning and agentic AI coding capabilities offer a promising pathway to improving the accessibility of advanced trajectory optimization techniques. This paper introduces PLUTO (Plain Language Understanding for Trajectory Optimization), an interactive trajectory design framework that integrates agentic AI coding tools within a sequential convex programming (SCP) architecture. The proposed system maps high-level semantic inputs to well-defined mathematical constraints while preserving a structured representation suitable for SCP. These constraints are processed through an auto-convexification pipeline and incorporated directly into executable optimal control formulations, thereby systematically connecting user intent and trajectory generation. Visual constraint verification and scenario parameterization further support an intuitive environment for trajectory design across a wide range of application scenarios. Within an evaluation of 50 natural-language rendezvous mission prompts spanning 10 constraint types, PLUTO generated a trajectory for every prompt, and 94% of the resulting trajectories satisfied both the quantitative and qualitative requirements of the mission intent. Ultimately, this work explores the potential of agentic AI to bridge the gap between advanced optimization algorithms and engineer-focused trajectory design workflows.

We establish curvature-based bounds for the $p$-capacity of convex domains and related capacity–area inequalities. Let $K$ denote the Gauss–Kronecker curvature of $\partial\Omega$. In Euclidean space, if $ 0<K_-\leq K\leq K_+, $ then, for every $1<p\leq \frac{n+1}{2}$, $$ \operatorname{Cap}_p\left(B(K_+^{\frac{1}{1-n}})\right) \leq \operatorname{Cap}_p(\Omega) \leq \operatorname{Cap}_p\left(B(K_-^{\frac{1}{1-n}})\right). $$ The lower comparison remains valid for the full range $1<p<n$. Moreover, the exponent range in the upper comparison is optimal in general: when $n\geq 4$ and $\frac{n+1}{2}<p<n$, the upper comparison may fail. Thus $p=\frac{n+1}{2}$ is the critical exponent for the curvature-radius upper comparison. Under the pinching condition $ \frac{K_-}{K_+}=\rho\in (0, 1], $ we further obtain an explicit scale-invariant capacity–area lower bound for every $1<p<n$. In the special case $n=3$ and $p=2$, $$ \frac{\operatorname{Cap}_2(\Omega)} {\sqrt{|\partial\Omega|}} \geq \frac{4\sqrt{\pi}\,\rho^{1/4}} {1+\sqrt{\rho}}, $$ which implies the Pólya–Szegö inequality whenever $$ \rho\geq \rho_*:= \left( \frac{\pi^2-4-\pi\sqrt{\pi^2-8}}{4} \right)^2 \simeq 0.15484. $$ On a Cartan–Hadamard manifold with $\operatorname{Sec}\leq -k^2<0$, we obtain a $p$-capacity lower bound retaining all principal curvatures of the boundary, together with corresponding Gauss–Kronecker curvature estimates and a rigidity characterization of the equality case by a warped-product exterior geometry.

We develop and analyze a B-spline scaling-function Galerkin method for the tempered fractional partial differential equation governing European and American option prices under the CGMY process. The spatial generator couples exponentially tempered left- and right-sided Riemann–Liouville derivatives of order $Y\in(0,2)$, whose nonlocality and asymmetry present discretization challenges. We derive a corrected Fourier-symbol construction of the tempered connection-coefficient matrices whose interior blocks are Toeplitz, maintain exact fractional scaling, and enable $\mathcal{O}(N\log N)$ matrix–vector products via the fast Fourier transform. In the regime $Y\in(1,2)$, we prove full-line and finite-domain Gårding inequalities, coercive whenever the risk-free rate is positive, via zero-extension to the full-line symbol. On this foundation, we establish elliptic-projection, semidiscrete, and fully discrete error estimates with spatial rate $\mathcal{O}(h^{p-Y/2})$, alongside an unconditional shifted-energy stability estimate for the forced Crank–Nicolson scheme with mesh-independent constants. For positive rates the admissible shift vanishes, governing the plain scheme exactly as implemented. A single backward solve prices an entire strike strip at fixed maturity with smooth, analytically differentiated Greeks. The same discretization prices American puts through a projected linear complementarity solve, with early-exercise premiums and exercise boundaries validated against a resolved deterministic reference and a Longstaff–Schwartz Monte Carlo benchmark. Calibration to S&P~500 option surfaces across low-volatility, crisis, and recovery regimes yields implied-volatility errors near two vol-points on well-identified surfaces, while an accuracy-matched comparison with a stabilized Grünwald–Letnikov baseline delineates precisely when the high-order variational structure pays off.

The basic hypergeometric series ${}_{n+1}\phi_{n}$ whose upper and lower parameters are non-negative integer powers of $q$ satisfying a certain constraint can be, after a normalisation, computed as a positive polynomial. We derive three formulas for this polynomial. The first formula is a generalisation of the MacMahon identity in which an infinite binomial series is identified with a normalised generating function of des and maj statistics of multiset permutations. The second formula is a generating function of multiset permutations for the inversion statistic and a generalisation of the statistic mstc. The third formula is a finite sum containing $2n$ $q$-binomial coefficients.

An $(s,p)$-edge-coloring of a graph $G$ is an edge coloring using $s$ colors such that exactly $p$ colors appear at each vertex. To generalize the notion of proper edge-coloring, these colorings are defined to be equitable: the numbers of edges of each color incident to a vertex are fairly distributed. We find the necessary and sufficient conditions for embedding an equitable $(s_1,p_1)$-edge-coloring of $K_{n_1}$ into an equitable $(s_2,p_2)$-edge-coloring of $K_{n_2}$. We focus on the values of $n_1$, $p_1$, $n_2$, and $p_2$ where $s_1$ is necessarily larger than $p_1$ and $s_2$ is necessarily larger than $p_2$.

This article is the third of a series of three articles whose goal is to generalize the Cartier transform of Ogus and Vologodsky to the logarithmic setting. In the second one, we generalized a topos-theoretic version of this transform, due to Oyama. Let $W$ be the ring of Witt vectors of a perfect field of positive characteristic $p$ and equip $\operatorname{Spf}W$ with the trivial log structure. Let $\mathfrak{S}$ be an fs log $p$-adic formal scheme, locally of finite type and log flat over $\operatorname{Spf}W$ and denote its special fiber by $S.$ For a log smooth morphism of logarithmic schemes $X \rightarrow S,$ we constructed crystalline-like ringed topoi $\mathcal{E}'$ and $\underline{\mathcal{E}}$ and subcategories of crystals of quasi-coherent modules $\mathcal{C}'$ and $\underline{\mathcal{C}},$ equivalent respectively, under some lifting assumption, to modules with Higgs fields and integrable connections, both satisfying certain nilpotence conditions, and a morphism of topoi $\underline{\mathcal{E}} \rightarrow \mathcal{E}'.$ We proved that the pullback functor of this morphism of topoi preserves quasi-coherent crystals and induces a fully faithful functor $\mathcal{C}' \rightarrow \underline{\mathcal{C}}.$ Since the Frobenius morphism is not, in general, flat in the log smooth setting, it is not clear that this functor is essentially surjective. To address this issue, we refine the topoi and crystals mentioned above by endowing them with an indexed structure, inspired by Lorenzon's indexed extension of Cartier descent to smooth logarithmic schemes. Using the Azumaya property of the indexed algebra of logarithmic differential operators, we then obtain an equivalence between the corresponding categories of indexed crystals, thereby generalizing the Cartier transform.

Robotic systems often exhibit unstable modes, along which small perturbations and disturbances can cause unbounded growth unless corrected through feedback. Controlling such systems from high-dimensional visual observations requires representations that preserve these modes. Joint-embedding predictive architectures (JEPAs) provide a natural framework for learning such representations and their dynamics from visual data. However, we demonstrate that next step prediction combined with anti-collapse regularization does not guarantee that controllable unstable modes are preserved: the training loss can be minimized while these modes are collapsed, making stabilization from the learned representation impossible. To address this, we augment world-model training with an action reconstruction objective (i.e., an inverse dynamics loss) that encourages control-aware representations, namely, visual representations that preserve crucial features for control. We prove that exact action reconstruction makes the encoder injective on the finite-horizon reachable subspace. Thus, the encoder cannot discard any state direction reachable by an action sequence within $H$ steps. Moreover, we show that, as $H$ grows, the dominant eigenspace of the finite-horizon controllability Gramian converges to the controllable unstable subspace. We establish our theoretical results for linear systems and demonstrate empirically that our findings extend to nonlinear visual control tasks (CartPole, Walker2D, and PointMaze), highlighting the benefits of control-aware representation learning.

Consider a finite planar graph with positive real edge weights and designated boundary vertices, called nodes. Such a graph is called a circular planar electrical network. A grove is a spanning forest in which every component contains at least one node. The connected components of a grove determine a partition of the nodes. We relate weighted grove counts to invariant theory for the symplectic group. To a planar electrical network $G$, we associate an $\mathrm{Sp}(2n)$-invariant tensor $Z_G$. For $\mathrm{Sp}(2)=\mathrm{SL}(2)$, we expand $Z_G$ in the Temperley–Lieb basis indexed by noncrossing matchings and relate its coefficients to the Kenyon–Wilson grove formulas. For $\mathrm{Sp}(4)$, we give reduction rules for superpositions of two groves and prove that the tensors indexed by $3$-noncrossing matchings form a basis of the space of $\mathrm{Sp}(4)$-invariant tensors. The coefficients of $Z_G$ in this basis are weighted counts of reduced double groves, up to normalization.

Consider an arithmetic progression inside the finite field $\mathbb{F}_{q^n}$. We study the existence of progressions whose terms are all primitive and at least one of them is normal. Using a combination of sieve estimates, ramified bounds, and computational arguments, we improve significantly upon previous literature for progressions of length $m=4$ and obtain partial results whenever $5\leq m\leq 10$. Finally, we provide a general bound which outperforms previous results in the range $m\geq 53$.

We answer a long-standing open question of Woodin by proving that if $\kappa$ is a strongly compact cardinal and $GCH$ holds below $\kappa$, then $GCH$ holds.

For integers $r \ge 3$, $c \ge 2$, and $s \ge 0$, let $\mathsf{M}_r^{\star}(c,s)$ denote the smallest integer such that any set of at least $\mathsf{M}_r^{\star}(c,s)$ points in the plane in general position (no three on a line), colored with $c$ colors, contains a monochromatic $r$-gon (not necessarily convex) with at most $s$ interior points. Define $\lambda_r^{\star}(c)$ to be the smallest integer such that $\mathsf{M}_r^{\star}(c,\lambda_r^{\star}(c)) < \infty$. The case of triangles, that is, $r=3$, has been studied in recent years. In this paper, we establish bounds on $\lambda_r^{\star}(c)$, for $r \ge 4$. Specifically, in the bichromatic case, we prove that $\lambda_r^{\star}(2)\leq \lfloor\frac{r-2}{2}\rfloor$, for every $r\geq3$. For hexagons, we sharpen this bound to $\lambda_6^{\star}(2)\leq1$, that is, every sufficiently large bichromatic planar point set in general position contains a monochromatic hexagon with at most one interior point. Next, for any number of colors $c\geq3$ and $r\geq3$, we show that $$(r - 2) \left \lfloor \frac{c-1}{2} \right \rfloor \leq \lambda_r^\star(c) \leq (r-2)(c-2)+\left\lfloor(r-2)\delta_c\right\rfloor, $$ for some constant $\delta_c \in (0, \frac{1}{2}]$, depending on $c$. The upper bounds combine spiral polygonizations with nested convex-hull arguments and, in the hexagon case, incidence counting in $4$-connected triangulations. The lower bounds follow from colored Horton-set constructions.

Noncommutative polynomials, elements of the free algebra generated by several noncommuting variables $x_1,...,x_g,x_1^*,...,x_g^*$, are of importance in noncommutative algebra, free analysis and real algebraic geometry. We allow matrix coefficients and view these noncommutative polynomials as functions in several size-free matrix variables. The free locus of a noncommutative polynomial $f$ is the sequence of hypersurfaces $\mathscr{Z}_n(f)=\{X\in M_n(\mathbb{C})^g:\det f(X) = 0\}$. When $f$ is Hermitian, the positivity domain of $f$ is the sequence of semialgebraic sets $\mathcal{D}_n(f)=\{X\in M_n(\mathbb{C})^g:f(X)\text{ is positive semidefinite}\}$. We show for well-behaved cases that the Euclidean boundary of the positivity domain is Zariski dense in the free locus. This leads to a certificate for equality of positivity domains of coefficient-irreducible unsignatured atomic quadratics. Finally, we describe the geometric structure of positivity domains of hereditary quadratic polynomials through $\Gamma$-convexity.

Consider the "snaky" corridor $K_4 \subset \mathbb{R}^3$ from Problem 6.64 of arXiv:2511.02864: a tube of unit-square cross-section whose two right-angle turns lie in perpendicular planes, with a straight middle section of clear length $4$. We exhibit explicit rigid motions of connected bodies through $K_4$ and certify their volumes with the problem's own evaluation pipeline, unmodified, at the tight motion tolerances of its final scoring stage (translation $0.005$, rotation $0.05$ degrees). The certificates give $V^*(K_4) \ge 1.8355162$ at grid fineness $0.002$ and $V^*(K_4) \ge 1.8558404$ at grid fineness $0.001$, against the previously reported $1.81$; rerunning the pipeline's published best path under the identical configuration yields $1.8179428$, so every comparison is path against path under one fixed certifier. Three stages produce the record motion: a simple "staggered twist" backbone in which the whole 90-degree twist about the tube axis happens inside the second turn; evolutionary (CMA-ES) refinement, including a mixed-fidelity scheme that re-scores each generation's leading candidates on a grid twice as fine and lets those ranks steer recombination; and a gradient polish carried out directly at the finer carve resolution, guarded by periodic rebuilding of the frozen interpolation structure and by a penalty on control-pose spacing. These guards close three documented failure modes in which an optimizer inflates its own objective: frozen-structure inflation, spacing stretch, and exploitation of a coarse grid by optimizing and selecting on it. A staircase-based predictor with two grid bases, anchored by matching grid sensitivity, forecast the certificates to about $0.001$ before the runs. With the companion note's upper bound $V^*(K_4) \le 3.4083$, the problem now sits in the bracket $1.8558404 \le V^*(K_4) \le 3.4083$. All paths, notebooks and certified results are public.

Let $K_M \subset \mathbb{R}^3$ be the "snaky" corridor with unit-square cross-section: an infinite entry arm parallel to the $x$-axis, a middle arm parallel to the $y$-axis whose clear straight section has length $M$, and an infinite exit arm parallel to the $z$-axis, with the two turns lying in the $x$-$y$ and $y$-$z$ planes. Let $V^*(K_M)$ be the supremum of the volumes of connected bodies that can be moved continuously (by rigid motions) from the entry arm to the exit arm while remaining inside $K_M$. We prove $V^*(K_M) \le 7.8691$ for every $M \ge 3.53$ (Theorem A); by a sharper analysis of the same mechanism, $V^*(K_M) \le 4.2862$ for every $M \ge 2.61$ (Theorem B); and, by replacing the final analytic caps with a certified branch-and-bound computation over the compact family of entry tubes compatible with the forced events, $V^*(K_M) \le 3.4083$ for every $M > 2.2$ (Theorem C; computer-assisted, in double precision with conservative slack). In particular all three bounds apply to the corridor with $M = 4$ used in Problem 6.64 of Georgiev, Gómez-Serrano, Tao and Wagner (arXiv:2511.02864), and uniformly to every "turns far apart" reading of that problem. To the best of our knowledge, this is the first upper bound for the snaky-corridor 3D moving sofa problem; previously not even finiteness of $V^*$ had been recorded. This note proves upper bounds only; combined with the certified lower bound 1.8558404 from the author's separate certification work via the problem's own evaluation pipeline, the state of the problem becomes $1.8558404 \le V^*(K_4) \le 3.4083$. Theorems A and B are elementary and self-contained: a first-crossing (elbow) event forces a definite tilt of the body's own entry tube, and Fubini-type chord estimates for tilted tubes cap the volume. Theorem C certifies, by the same forced-event skeleton, the largest corridor volume that any admissible entry tube can carry.

We study how the two cusp expansions of the Rogers-Ramanujan continued fraction control its ordinary and logarithmic Mellin moments. Ramanujan's factorizations express the logarithmic transform as a combination of products of zeta and quadratic Dirichlet $L$-functions. Its adjacent integer and half-integer values give an invertible change of coordinates for the corresponding odd special values. We prove that every reciprocal-cusp coefficient is a quadratic irrational algebraic integer, with an explicit minimal polynomial, and that every logarithmic coefficient is quadratic irrational. These facts determine the exact poles of three normalized-moment generating functions. The logarithmic one has an absolutely convergent partial fraction expansion with a pole at every positive square. For the ordinary moments, a two-cusp incomplete-gamma expansion gives meromorphic continuation at every positive splitting point. A general Mellin lemma yields the corresponding nonholonomicity statements. We also establish reflection identities, vertical strip decay and an explicit truncation estimate.

We prove that $\mathrm{Mod}(S_g)$ acts faithfully by orientation-preserving homeomorphisms on a $(2g-1)$-sphere when $g \geq 3$ using equivariant blow-up and bi-orderability of the Torelli group.

We show that an ir{\-}reducible cuspidal eta quotient of level~$N$ is uniquely identified by its orders of vanishing at $O((\log_2 N)^{r-1})$ cusps, where $r$ denotes the number of distinct prime divisors of $N$. In particular, all such eta quotients which are Fricke eigenforms of a prime power level, are uniquely identified only by their orders of vanishing at~$\infty$. We also prove that for all $k\in\N$, there exists a primitive and irreducible cuspidal eta quotient of weight $k/2$. It follows from the exhaustiveness of Zagier's list of the primitive holomorphic eta quotients of weight~$1/2$ that the 1st Mersmann level, $\cl{M}_1=12$. Here we show that for all $k\in\N$, the $k$-th Mersmann level is divisible by $2^{2k}$.

In this paper, we study Ricci flows with uniformly bounded scalar curvature. On such Ricci flows we establish uniform $L^2$ Riemannian curvature bounds and $L^4$ Ricci curvature bounds up to dimension 7. These estimates improves $L^{2-\varepsilon}$ curvature bounds established by Bamler and $L^2$ curvature bounds in dimension 4 established by Bamler and Zhang. More generally, we establish uniform $L^2$ curvature bounds on $n$-manifolds satisfying an integral $L^p$ Ricci curvature bound for $p> n/2$, a uniform volume lower bound, and a strong $\varepsilon$-regularity assumption. This result extends the celebrated $L^2$ curvature estimates of Jiang and Naber, which were established under pointwise two-sided Ricci curvature bounds. To overcome the lack of pointwise control, we develop an $L^p$ adaptation of the neck decomposition theorem and establish a superconvexity estimate for the curvature on neck regions. Furthermore, we demonstrate that our structural assumptions are dynamically natural on Ricci flows with bounded scalar curvature. As an application, by combining our estimate with the a priori $L^{4-\varepsilon}$ curvature radius bound of Ricci flow established by Bamler, we obtain uniform $L^2$ curvature bounds on Ricci flows with bounded scalar curvature of dimension $n \le 7$.

Fractional integral operators involving special functions as kernels have long been one of the main subjects of fractional calculus. As a discrete version of classical fractional calculus, $q$-fractional calculus not only brings often new phenomena and new perspectives but also provides new useful areas for describing and studying new problems of applied nature. Recently, Luo and Zhou (see below) developed a methodology for formulating well-behaved $q$-fractional integral operators steming from a basic general theory. Our aim in this paper is to define and introduce a new class of $q$-fractional integral operators whose kernel involves a new multivariate $q$-Mittag-Leffler function. We shall also define the corresponding $q$-fractional derivatives (as the inverses of $q$-fractional integral operators) and define also the $q$-Caputo fractional derivative operators. We discuss in detail the similarities and variations between the $q$-operators and their counterparts in classical theory, particularly in Prabakar fractional calculus. Further, applications of the defined operators in solving certain $q$-fractional differential equations are also considered. We also mention throughout this paper some useful relevances of our established results with certain known important cases. In the concluding remarks, we point out briefly few possible proposals (or directions) for further work on the present subject matter of $q$-calculus.

This paper contributes to the long line of research on the distribution of mesh patterns in permutations. We extend these studies to separable permutations and carry out a comprehensive analysis of mesh patterns of length at most~2. For mesh patterns of length~1, we determine the distributions for all six equivalence classes. In addition, we obtain the joint distribution for the patterns in the class containing the well-known permutation statistic known as the strict fixed point. Furthermore, computer experiments suggest that at most 124 pairs of mesh patterns of type 12 and 21 with identical shading are equidistributed. Using symmetry operations, we partition these pairs into 38 equivalence classes with respect to distribution. By explicitly determining 24 of these distributions (in many cases, in fact, joint equidistributions of the respective pairs), we reduce the number of equivalence classes to~31, which turns out to be the true number of equivalence classes. Enumerating 14 of these classes is left as an open problem.

In previous work, we established sufficient conditions for recovering geometry and topology from non-random information, demonstrating reconstruction with the Arecibo message. Undeciphered historical records present a related problem: one-way communication without access to their makers' intentions or encoding conventions. Here we extend this framework to the Phaistos Disc, Andean khipu, Rongorongo, Indus inscriptions and the Voynich manuscript, using Egyptian layouts and Arecibo as controls. Algorithmic information dynamics (AID) guides structural perturbations, combining classical information measures, predictive coding, compression and algorithmic-probability-based estimates. Phaistos retains 0.714 bits of excess adjacent-sign mutual information after positional controls; its authentic segmentation lies within a broad information basin, and cross-face boundary prediction achieves AUCs of 0.76 and 0.86. Repetition explains its principal recurrence, while the tested spiral embeddings provide no evidence of additional cross-winding structure. Across several corpora, comparative transfer persists beyond exact shared local pairs. Khipu knot forms distinguish authentic attachments after controlling for degree, depth, fibre and twist (adjusted $p=0.009$); constrained assignment recovers 8.75% of concealed parents against 7.87% expected by chance. Row-alignment tests yield no comparable evidence after correction. Calibrated BDM recovers Arecibo's 23-column width (search-corrected $p=0.005$), while executable controls detect spatial dependence and eliminate it through interventions on the generator. These findings connect information-based structural reconstruction with interventional reasoning, distinguishing recoverable organisation from semantic decipherment.

We prove harmonicity for nonshrinking algebraic generalized Ricci solitons and for all algebraic generalized Ricci solitons on solvable Lie algebras. In the solvable setting, we characterize expanding solitons by normal extensions of generalized nilsolitons, and derive a sharp rank bound.

We formulate a root-conjugation criterion for producing generating pairs and detecting Nielsen inequivalence by a polynomial symmetric-trace invariant. The method simultaneously treats mapping class groups and automorphism groups of free groups. We prove that $\mathrm{Mod}(\Sigma_g)$, $g\ge2$, and $\mathrm{Aut}(F_n)$, $\mathrm{SAut}(F_n)$, $\mathrm{Out}(F_n)$ and $\mathrm{SOut}(F_n)$, $n\ge3$, have countably infinitely many Nielsen classes of generating pairs. A sign parameter unifies the full and special automorphism-group constructions, with common generation arguments and trace polynomials. The same homology representations give the corresponding results for $\mathrm{Sp}_{2g}(\mathbb{Z})$, $g\ge2$, and $\mathrm{SL}_n(\mathbb{Z})$ and $\mathrm{GL}_n(\mathbb{Z})$, $n\ge3$. This answers Hirose–Monden's low-genus question and Kapovich's Question 8.1.

In Beyond ECH Capacities, Hutchings used the ECH cobordism map to produce obstructions for embedding one convex toric domain into another. We generalize the methods there to obstruct symplectic embeddings of a disjoint union of convex toric domains into another convex toric domain. We also find additional obstructions coming from intersection positivity and use similar methods to obstruct the existence of isotopies between two given symplectic embeddings.

We study numerical comparisons for hypersurface singularities under finite holomorphic maps between smooth germs. We retain Jelonek's topological proof of the Milnor-number inequality in the isolated case and give an algebraic proof using a cyclic factorization, Brieskorn modules, and trace. For a reduced divisor-theoretic pullback, we prove lexicographic monotonicity of Segre numbers. Componentwise monotonicity follows when the two singular loci have the same dimension and the target Jacobian ideal is equimultiple. For one-dimensional singular loci, we also obtain componentwise comparison under an equality condition on the first nonzero Segre number or a nonramification condition. The arguments use finite-flat pullback of Segre cycles and Hilbert–Samuel multiplicity. Through the identification with generic Lê numbers, the same results hold for these invariants. We distinguish these statements from comparisons in arbitrary prepolar coordinates and formulate the remaining componentwise and ramified cases as conjectures.

Let $\varepsilon_1,\ldots,\varepsilon_n$ be independent Rademacher random variables. We settle the best constant $c$ such that \[ \mathbb{E}\Bigl|\sum_{j=1}^n a_j\varepsilon_j\Bigr| \ge \frac{1}{\sqrt{2}} +c\Bigl|a-\frac{e_1+e_2}{\sqrt{2}}\Bigr| \] holds for every $n\ge 2$ and every unit vector $a\in\mathbb{R}^n$ with $a_1\ge\cdots\ge a_n\ge 0$. We prove that the optimal constant is $c=(2-\sqrt{2})^{3/2}/8$, with equality attained at $a=\frac{1}{2}(1,1,1,1,0,\ldots,0)$ for $n\ge 4$. This completes a line of research by De-Diakonikolas-Servedio, Eskenazis-Nayar-Tkocz and Fang-Wang.

In this paper we further examine the structure of the graded paramodular Hecke algebra $\mathcal H$. We define a commutative subring $\mathcal{H}^\alpha$ of $\mathcal{H}$ as the elements fixed by a certain automorphism $\alpha$, and we show that the elements $T(q^k)$, defined as the sum of the double cosets of degree $k$, are contained in $\mathcal{H}^\alpha$. We prove that the formal power series in the indeterminate $t$ with coefficients $T(q^k)$ is an explicit rational function in $t$ with coefficients from $\mathcal{H}^\alpha$. We also determine the structure of $\mathcal{H}^\alpha$ and its relation to $\mathcal{H}$.

Approximating linear operators on high-dimensional spaces using only forward operator-vector product queries is a common but challenging task in scientific computing and machine learning arising when access to the adjoint action is experimentally unavailable or difficult to construct computationally. Existing query-efficient methods with polylogarithmic query complexity in the error tolerance rely on self-adjointness or access to the adjoint operator, while existing adjoint-free methods based on global low-order regularity have polynomial query complexity. We show that a priori local row-space information can be exploited for query-efficient, adjoint-free operator approximation, and that this information can be supplied by local high-order regularity estimates. Our approach first extends the fixed-sparsity matrix approximation algorithm of (Amsel et al., SIMAX, 2026) to approximating a matrix by one whose rows lie in prescribed subspaces, providing a general template for exploiting local row-space structure from forward queries alone. We then apply this framework to Hilbert-Schmidt integral operators with asymptotically smooth kernels. We establish asymptotic smoothness for Green's functions of canonical second-order elliptic PDEs in $d\leq 3$ dimensions including Poisson's equation in balls and constant coefficient equations on tori. In $d$ spatial dimensions, our algorithm uses $\mathcal{O}\big(\log(\varepsilon^{-1})\log^{d}(\varepsilon^{-1}\delta^{-1/2})\big)$ non-adaptive randomized queries to produce an $\varepsilon$-accurate approximation in Hilbert-Schmidt norm with probability $1-\delta$. This implies the error decreases as $\mathcal{O}\big( \exp(- c q^{1/(d+1)}) \big)$ in the query count $q$, which we observe in numerical experiments using tens to hundreds of queries to approximate solution operators of one-dimensional variable-coefficient second-order elliptic equations.

Essential dimension counts the parameters needed to define a torsor, but it can drop sharply over a field extension, and it does not say how large an extension is needed. For a $G$-torsor $\tau$ over a field $F\supseteq k$ we study the finite-degree profile $\edd d_k(\tau)=\min\{\ed_k(\tau\otimes_FF'):[F':F]\le d\}$, a variant of invariants of Farb–Kisin–Wolfson and Farb–Wolfson, and its jump degrees $d_j(\tau)$, the least degree of an extension over which $\tau$ needs only $j$ parameters. The profile is geometric over every field: $\edd d_k(\tau)\le j$ iff some generically free $G$-variety of dimension $\le j$ has a closed point of degree $\le d$ in the free locus of its twist by $\tau$. We prove a Sylow divisibility theorem for jump degrees and a valuation-theoretic rank–index theorem, $\edd d_k(\tgen)\ge\min\{\rk B:B\le A,\ [A:B]\le d\}$ for abelian $A$ of order prime to $\operatorname{char}k$, which determines the profile of every abelian group over a field with enough roots of unity, for instance $n-\lfloor\log_2d\rfloor$ for $(\Z/2)^n$. For $\Z/p$ in characteristic $0$ we show $\lceil m/2\rceil\le d_1\le[k(\zeta_p+\zeta_p^{-1}):k]$, $m=[k(\zeta_p):k]$, with equality for $m$ even. For $S_5$ in characteristic $0$ we prove $d_1=4$, so Klein's reduction of the generic quintic is optimal; the lower bound comes from Abel–Jacobi rigidity, not from cohomological invariants. For $n\ge6$, $6\mid d_1(S_n)$.