Papers

We develop a unified asymptotic theory for Minnaert resonances generated by a small bubble $D_\varepsilon$ placed near an interface $\Gamma$ of material discontinuity in an acoustic background. The bubble has characteristic size $\varepsilon>0$, with its density and bulk modulus depending on the same small parameter through the Minnaert high-contrast scaling. The analysis is organized by the scaled separation \[ \Theta_\varepsilon := \frac{\operatorname{dist}(D_\varepsilon,\Gamma)}{\varepsilon}, \] and covers the three natural regimes \[ \Theta_\varepsilon\to+\infty, \qquad \Theta_\varepsilon\to\Theta_*\in(0,\infty), \qquad \Theta_\varepsilon\to0: \] 1) In the far-separation regime, the interface is invisible at leading order and the classical full-space Minnaert law is recovered. 2) In the critical regime, the rescaled geometry retains both the bubble and the interface at finite separation, and the Newtonian capacitance is replaced by a flat-interface capacitance determined by the limiting tangent transmission problem. 3) In the near-contact regime, this family converges to a contact-limiting capacitance, yielding the same leading-order Minnaert resonance for all gap scalings satisfying $\operatorname{dist}(D_\varepsilon,\Gamma) = o(\varepsilon)$. These three regimes provide a unified description of how the resonance frequency changes as the bubble approaches the interface, from the classical isolated-bubble limit to the contact limit. The analysis relies on a uniform reduction of the resonance problem to a scalar equation and on a variational characterization of the interface-dependent capacitances; in particular, Mosco convergence of the associated transmission energies yields convergence of the capacitary potentials and minimum energies up to the singular contact configuration.

We study Fredholm determinant solutions to the first two positive and the first negative members of the modified Kadomtsev-Petviashvili (mKP) hierarchy, together with the bilinear two-dimensional Toda lattice (2DTL) equation. By applying isospectral reductions, we explicitly construct Fredholm determinant solutions for the modified Korteweg-de Vries (mKdV), the modified Camassa-Holm, and the focusing nonlinear Schrödinger equations. Furthermore, we demonstrate that in the non-isospectral case, the reduction from the mKP and 2DTL equations to the non-isospectral negative KdV equation yields solutions characterized by the deformed Bessel determinant.

We study K3 surfaces polarized by the lattice $U\oplus D_8(-1)$ through elliptic fibrations and orthogonal modular forms. Using an alternate genus-one fibration and its relative Jacobian, we construct an explicit map from Vinberg's coefficient space to the Hashimoto–Ueda family. We also explain the geometric meaning of certain character forms associated with these moduli spaces and extend the relative-Jacobian construction to the exceptional lattices $U\oplus E_8(-1)\oplus A_k(-1)$ with $k=1, 2$.

We investigate the a posteriori controllability of remodelling within the framework of Geometric Control Theory applied to continuum systems. Remodelling is described through an isochoric Bilby–Kroner–Lee decomposition, where the internal structural transformation is represented by a remodelling tensor and its evolution is governed by a stress-driven constitutive law. Assuming infinitesimal deformation and remodelling stretches, while retaining finite rotations of the principal remodelling directions, we derive a nonlinear state-space representation in terms of principal remodelling stretches and orientation variables. This formulation allows the remodelling process to be interpreted as a finite-dimensional nonlinear control system parametrised by the material position. Equilibrium configurations are characterised, and under-actuated controllability problems are studied by means of the tools of Geometric Control Theory. The three-dimensional setting shows that the nonlinear coupling between remodelling stretches and orientations enables complete steering of the internal state through a reduced number of independent controls, while the planar reduction reveals a loss of controllability despite the presence of nonlinearity. The results provide a first systematic investigation of remodelling controllability and further support the development of Continuum Control Theory as a framework for the analysis and design of controlled internal structural transformations in continuous media.

Let $s(N)$ denote the smallest side length of a square containing $N$ unit squares with arbitrary orientations and pairwise disjoint interiors. Nagamochi's Packing Unit Squares in a Rectangle (2005) states a rectangle packing bound from which he deduces two infinite families of exact values: $s(k^2-1) = k$ and $s(k^2-2) = k$ for every integer $k \geq 2$. We construct a family of counterexamples, local to a corner of the container, to the scoring assertion in Nagamochi's Lemma 1. These counterexamples show that the published proof of the rectangle bound is incomplete, but do not disprove the bound itself. We then give an independent proof of a weaker rectangle bound using a strip measure. This recovers $s(k^2-1) = k$ for every integer $k \geq 2$ and yields an explicit lower bound for $s(N)$ that improves strictly on the area bound for every nonsquare integer $N \geq 8$. Our argument does not establish Nagamochi's full rectangle bound or the identity $s(k^2-2) = k$.

Let $\Sigma_g$ be a closed oriented surface of genus $g\geq2$, and let $\mathscr F_g$ be its split-closed, strictly unobstructed, two-periodic Fukaya category over the complex Novikov field $\Lambda$. We determine the derived Picard group \[ \operatorname{DPic}_{\Lambda}(\mathscr F_g) \cong \bigl(H^1(\Sigma_g;\Lambda^\times) \rtimes\pi_0\operatorname{Diff}^+(\Sigma_g)\bigr) \times\mathbb Z/2\mathbb Z. \] This proves the enhanced form of the conjecture of arXiv:2006.09689. As a corollary, we obtain that the derived Picard group itself determines the genus of the surface.

We study the stability and instability of stratified steady states $\rho_s=\rho_s(y)$ for the two-dimensional incompressible porous media equation on $\mathbb{T}\times(-1,1)$ and $\mathbb{T}\times\mathbb{R}$. On the periodic channel, uniformly decreasing steady states satisfying natural boundary-compatibility conditions are nonlinearly stable under small $H^m$ perturbations, for every integer $m>2$. The solutions converge in $L^2$ to the measure-preserving stratification of the initial density at the rate $t^{-m/2}$. Conversely, every steady state with $\sup\rho_s'>0$ is nonlinearly unstable in $H^m$. On the infinite cylinder, we prove nonlinear instability under the additional gap condition \[ \sup_{\mathbb{R}}\rho_s' >\limsup_{|y|\to\infty}\rho_s'(y). \] In both settings, the instability is generated by positive eigenvalues of the linearized operator converging to $\sup\rho_s'$, which equals both its spectral bound and semigroup growth bound.

We present a determinantal evaluation formula for the very-well-poised ${}_8\Phi_7$ basic hypergeometric Askey-Wilson function in terms of elementary functions. The formula in question is valid for discrete values of the four permutation-symmetric Askey-Wilson parameters; specifically, these consist of all quadruples of parameter values that are given, up to a sign, by powers $q^k$ with $k$ a positive integer or half-integer, in such a way that each of the four types (positive/negative sign, integer/half-integer power) occurs exactly once. The proof hinges on a product formula for an associated Cauchy-type determinant, which is of independent interest and is established here by elementary means using Krattenthaler's `identification of factors' method for determinant evaluations.

For an integer $b\neq0,1$ let $\theta$ be the unique real root of $f_b(x)=x^3-3bx-b^3$ and let $K_b=\mathbb{Q}(\theta)$. We exhibit an explicit set of $b$ of positive density for which $\eta_b=-1/(\theta-(b+1))$ is the fundamental unit of $K_b$, and we determine the set of $b$ for which $\eta_b$ is the square of a unit: it is parametrized by the Pell equation $D^2-3E^2=1$, hence infinite, and under an explicit mild condition on the field discriminant it accounts for all $b$ for which $\eta_b$ is not the fundamental unit. That condition fails for only four $b$ with $|b|\le3000$, and at one of them, $b=3$, the conclusion itself fails. In the order $\mathbb{Z}[\eta_b]$ generated by $\eta_b$, by contrast, $\eta_b$ is the fundamental unit for every $b$, so these exceptions are a phenomenon of the maximal order. As an application we construct infinitely many biquadratic fields whose $3$-class field tower has length greater than $1$.

We study the Assouad dimension, lower dimension, and Assouad spectrum of graphs of the functions \(f(x)=\sum_{n=1}^{\infty}a_n\phi(b_nx+\theta_n)\), where \(\phi\) is a nonconstant \(1\)-periodic \(C^2\) function, \(a_n,b_n>0\), \((\theta_n)\) is an arbitrary real sequence, \(\sum_{n=1}^{\infty}a_n<\infty\), and \(b_{n+1}/b_n\to\infty\). We give sufficient conditions for the graph over every nondegenerate compact interval to have Assouad dimension two or lower dimension one. For \(a_n=b_n^{-\alpha}\), \(0<\alpha<1\), an explicit frequency-gap condition ensures that the Assouad spectrum equals two for \(\alpha\leq\vartheta<1\). Writing \(\beta_*=\limsup_{n\to\infty}\log b_{n+1}/\log b_n\), we show that, if \(\beta_*>(2-\alpha)/\alpha\), the spectrum attains the general upper bound for \(\alpha\)-Hölder graphs whenever \((2-\alpha)/\beta_*\leq\vartheta<\alpha\) and \(\vartheta>0\). When \(\beta_*=\infty\), this determines the complete spectrum. Together with Barański's formulas, our results yield graphs whose lower, Hausdorff, upper box, and Assouad dimensions are four distinct numbers.

We prove combinatorial rigidity for the full renormalization attractor of neutral quadratic polynomials. More precisely, any two bi-infinite renormalization towers with the same combinatorics are conformally conjugate on neighborhoods of their Mother Hedgehogs. The rigidity theorem includes arbitrary irrational combinatorics and respective parabolic enrichments. The proof relies on a comprehensive analysis of neutral cascades, i.e. transcendental dynamical systems arising from the rescaled limits of the first return maps of neutral quadratic polynomials. We establish uniform butterfly bounds for neutral cascades and prove that any two combinatorially equivalent neutral cascades are affinely conjugate. We also prove rigidity for parabolic towers with equivalent backward combinatorics.

In factorial randomized experiments with $K$ binary treatment components, we study which baseline covariates modify component main effects and interactions. We define effect-modifier cells $(j,S)$, $j\in[p]$ and $\emptyset\neq S\subseteq[K]$, through the coefficients $\theta^\ast_{jS}$ of population linear projections of conditional Walsh–Hadamard contrasts $\tau_{S}$ onto standardized covariates. Using baseline-robust direct scores, we develop frequentist inference for all $p(2^K-1)$ cells. First, we prove a debiased central limit theorem with global insensitivity: the residualization baseline enters the cross-fitted score through an exactly conditionally mean-zero perturbation. The influence-function remainder therefore requires only foldwise weighted $L_2$ consistency of the baseline estimator, with no prescribed polynomial rate or nuisance product-rate condition. A separate, explicit penalty-scale condition controls the baseline's effect on the score Lasso. Second, under explicitly stated uniform nuisance, baseline, studentization, and influence-array conditions derived from bounded sparse-regression primitives, we establish a high-dimensional Gaussian approximation over the full grid, calibrated by a Walsh-spectral multiplier bootstrap. This yields simultaneous confidence bands and Romano–Wolf step-down selection of modified cells with strong familywise error control, without effect-heredity assumptions. The factorial design is fixed in the asymptotic analysis, while the covariate dimension may grow; all $2^K-1$ contrasts are covered jointly.

Learning how to replenish inventory and estimating the cost of doing so require different information. We establish information-theoretic lower bounds for offline multistage inventory control with independent bounded demand. For fixed additive error in total expected cost, policy learning requires cubically many scalar observations in the horizon in the worst case, even under full observation and adaptive sample allocation. This matches the known empirical-risk upper order. Stationarity permits a quadratic policy-learning guarantee, yet valuation with inherited stock can remain cubically difficult even when the optimal policy is known. Censoring sharpens this distinction: optimal decisions can be learnable while absolute costs remain unidentified. We give a carrysafe coverage condition under which truncation preserves inventory transitions and policy gaps exactly. Combining this reduction with a matching lower bound yields the sharp censored-learning rate, with raw sample requirements inversely proportional to the usable fraction of sales logs. The same qualified observations can check coverage and fit the policy under a joint error guarantee. The results isolate information limits for decisions and valuation under an explicit conditionally independent logging protocol.

For graphs $H$ and $F$, let $\text{ex}(n,H,F)$ denote the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Gerbner, Győri, Methuku, and Vizer proved that $\text{ex}(n,C_{2\ell},C_{2k})=\Theta(n^\ell)$ for $k>\ell\ge2$. They determined the leading term for $\ell=2$, but for $k>\ell\ge3$ their general lower and upper bounds had different leading constants, leaving open the problem of closing this gap. We solve this problem by showing that, for every $k>\ell\ge2$, \[\text{ex}(n,C_{2\ell},C_{2k})=\left(\frac{(k-1)_\ell}{2\ell}+o(1)\right)n^\ell, \] where $(k-1)_\ell=(k-1)(k-2)\cdots(k-\ell)$.

The consistent histories formulation of quantum mechanics assigns classical probabilities to sequences of coarse-grained events when the corresponding interference terms vanish. In finite models these terms generically need not vanish exactly, and their smallness alone supplies no asymptotic geometry on the space of histories. We construct such a geometry for histories that accumulate quantum records. When the normalized Dowker - Halliwell interference between two histories factorizes as the cumulative fidelity of their records, its decay is controlled by the accumulated squared Bures distance. In the infinite registration limit, the Kakutani - von Neumann dichotomy gives a sharp sector alternative: summable Bures defect yields weakly equivalent record sequences, whereas nonsummable defect produces orthogonal superselection sectors and vanishing normalized interference. The convergence exponent of the Bures defect defines a pseudo-ultrametric on histories. After quotienting its zero distance degeneracy, it becomes a complete ultrametric, requiring neither compactness of the outcome space nor continuity of the record family, and coincides with the gauge invariant sector metric $\tilde d$ introduced in \cite{LesTP}. Its value, the consistency rate, has three equivalent interpretations: it is the sector separation metric, the polynomial growth exponent of the negative logarithm of normalized interference, and, by the Helstrom formula, the corresponding exponent of the negative logarithm of the minimum error probability for discriminating the cumulative records. Classical separation profiles determine the rate: positive long time mean defect gives the maximal value $\delta=1$, while the borderline profile $d_m^2\asymp m^{-1}$ produces distinct superselection sectors at zero polynomial rate, $\delta=0$.

A phase that reaches a fault-tolerant processor in additive pieces can be remembered until the last piece arrives, or executed on arrival: the first option costs classical memory carried across rounds, the second costs magic states committed before the phase is known. We determine the exchange rate $\alpha$, committed $T$ gates per bit of memory forgone, for ancilla-free coordinatewise Clifford+$T$ compilation. The Ramanujan bound gives $\alpha\ge2$ with explicit constants, the square-root barrier of the spectral method. An elementary determinant method, using that quaternion numerators lie on spheres in both real embeddings of $\mathbb{Q}(\sqrt2)$, counts words near any rotation coset below that barrier. With a height dichotomy for the sphere sections it produces and a fibration over rational projections, it gives every $\alpha<5/2$ unconditionally and uniformly over frames, by an exact SMT check of its continuum case analysis; $5/2$ is where this method stops. At Clifford-framed cosets the volume law holds up to subexponential factors: all but a vanishing fraction of $z$-rotations need $T$-count $(3-o(1))\log_2(1/\varepsilon)$, and processes whose committed pieces are near Clifford-framed $z$-rotations, including per-rotation pipelines, have $\alpha\ge3-o(1)$, which a fractional-passthrough family attains under the Ross-Selinger typical-cost hypothesis. Under an equidistribution conjecture supported by exhaustive enumeration to $T$-count 22, $\alpha=3$ in general and memory should be shed in whole rotations. The bounds hold even when the phases cancel to the identity; side information enters through a conditional entropy; probabilistic mixing halves the costs but not the rate. With clean ancillas and a phase-gradient catalyst, table lookups batched over coordinates drive the rate to $O(1/\log\log(1/\varepsilon))$, so the constant-rate law is specific to coordinatewise synthesis.

We give a complete characterization of periodic graph operators whose dispersion polynomials are reducible for every potential. Consequently, the reducibility dichotomy for parameter-dependent Laurent polynomials implies that, for every other periodic graph operator, the dispersion polynomial, and hence the Bloch variety, is irreducible for generic potentials. Our proof proceeds through two reductions. We first use the monodromy of the roots of the dispersion polynomial with respect to the potential parameters to reduce the problem to the splitting case. We then show that the splitting property passes to the principal submatrices of the Floquet matrix, reducing the problem further to the case that the fundamental domain contains two vertices.

In this paper, we employ the concentration-compactness principle to show that if the Dirichlet norm is replaced by the standard Sobolev norm, then the supremum of $$ \int_{\mathbb{B}} |x|^{N \epsilon} \Phi \left( \alpha_N \left( 1+ \epsilon \right) |u|^{ \frac{N}{N-1} } \right) dx $$ over all such functions is uniformly bounded. Furthermore, we also prove the existence of extremals. Finally, we consider the compactness of the sequence of extremals of the inequalities and the limit of this sequence is the extremal of $$ \int_{\mathbb{B}} \Phi \left( \alpha_N |u|^{ \frac{N}{N-1} } \right) dx $$ in $C^1 \left( \mathbb{B} \right)$, where $\alpha_N = N \omega_{N-1}^{ \frac{1}{N-1} } $, $\Phi \left( t \right) := e^t - \sum_{j=0}^{N-2} \frac{t^j}{j!} $ and $\omega_{N-1}$ is the surface of the unit ball in $\mathbb{R}^N$.

We develop a logarithmic framework to study the lattice of subgroups of the additive group of rational numbers. By encoding positive rationals via their prime exponent sequences, one obtains a bijection between $\mathbb{Q}^+$ and finitely supported integral sequences indexed by the prime numbers. This correspondence extends to arbitrary subgroups of $(\mathbb{Q},+)$ through a logarithmic greatest common divisor, yielding a classification of subgroups in terms of eventually nonpositive sequences in $\mathbb{Z}\cup \{- \infty\}$. Within this framework, subgroup inclusion, sum, product, and intersection admit simple coordinatewise descriptions, providing a transparent interpretation of the subgroup lattice and recovering several classical results such as the subgroup classification up to isomorphism. Exploiting this perspective, we obtain a complete characterization of the intersection configurations realizable in subgroups of $(\mathbb{Q},+)$. For rank configurations, realizability is characterized by the decreasing condition together with restrictions on the possible ranks and on the minimal zero sets, and, in the finite-support case, by an additional cardinality condition. For binary configurations, this reduces to saying that every decreasing configuration is realizable in the infinite-support case, whereas in the finite-support case the mentioned cardinality condition is required.

Generating optimization instances that are both feasible and computationally challenging is crucial for benchmarking solvers and training learning-based optimization algorithms. Existing non-LLM generators rely on seed instances or parameter tuning, resulting in high test-time computational cost, while existing LLM generators lack explicit hardness measures. Recent reinforcement learning methods with verifier feedback evaluate only binary correctness, which is misaligned with generating challenging problems. We note that an optimization solver reports the cost of solving at several stages of its pipeline, and leverage this to design a reward that scores both the solvability and the hardness of generated problems, measured by branch-and-bound nodes and post-cut relaxation gaps. Our key idea is a challenger-solver asymmetric self-play approach, where an LLM challenger generates progressively harder instances and the solver verifies feasibility and hardness, so no seed or training MILP instances are required. We fine-tune Gemma-4-12B and Qwen3.5-4B with GRPO and a size curriculum into OptiScribe-12B and OptiScribe-4B, which generate feasible yet challenging MILP problems from natural language instructions. On capacitated facility location and max-cut, OptiScribe-12B raises median SCIP search nodes by 1.7-5x and post-cut gaps by 1.1-1.7x over its base model and improves the feasibility rate on facility location by 9-19 points, while OptiScribe-4B raises median nodes by up to 15.6x. The problems cover a wider difficulty range than public benchmarks of the same size, follow instructions on density and difficulty, and can tune solver settings for families that public libraries lack. These results indicate that optimization-specific rewards, used in self-play mode, can teach LLMs to generate high-difficulty optimization benchmarks. We will release our code and models publicly on acceptance.

Rempe and Rippon gave a general result which enables the construction of a transcendental entire function with a Baker domain, in which the iterates tend to infinity at a prescribed arbitrarily slow rate. Here we construct a transcendental entire function with a Baker domain, in which the iterates are controlled from both above and below in a prescribed arbitrarily slow but regular manner. The proof uses distortion theorems for conformal mappings due to Ahlfors and Warschawski, and an approximation theorem due to Rempe and Rippon.

Space-air-ground integrated networks (SAGINs) are expected to support ubiquitous three-dimensional (3D) communication and sensing services in future sixth-generation (6G) wireless networks. However, heterogeneous mobility and service requirements across the space, air, and ground segments challenge fixed-boresight or fixed-sector antennas in supporting wide-area coverage, maintaining directional alignment over time-varying communication links, and flexibly adjusting observation directions for sensing tasks. To address these limitations, non-fixed flexible antenna architectures, such as fluid antenna system (FAS), movable antenna (MA), and pinching antenna, have garnered significant interest in recent years. Among them, rotatable antenna (RA) technology has emerged as a promising solution by enabling mechanical or electronic boresight adjustment while keeping the antenna positions fixed. This article investigates the roles of RA technology in enabling communication and sensing in SAGINs. Specifically, we first explain how RA boresight control complements platform mobility and supports wide-area coverage, dynamic directional transmission, cross-segment cooperative operation, and cooperative sensing. We then discuss the main design challenges, potential approaches, and future research directions, including channel acquisition and predictive tracking, joint orientation control and access/handover coordination, cross-segment coordination and resource management, as well as deployment tradeoffs and practical implementation. Finally, an RA array prototype and two representative simulation studies are presented to illustrate the feasibility and potential performance gains of RA-enabled SAGINs.

We prove a rank-preserving version of Gaifman's Theorem. Compared to earlier rank-preserving locality theorems (in particular, [Grohe, Kreutzer, Siebertz, JACM 2017]), our theorem is much simpler and yields formulas in exactly the same normal form as Gaifman's original theorem. Furthermore, it holds not only for first-order logic, but also for first-order logic with modulo-counting quantifiers and, more generally, for the first-order logic on weighted structures ngFOW+ that is introduced in this article. As an application of our theorem, we give a simplified proof of the algorithmic meta-theorem of [Grohe, Kreutzer, Siebertz, JACM 2017] stating that first-order properties of nowhere dense structures can be decided in almost-linear time. Our locality theorem for the weight logic ngFOW+ can be seen as an essential step toward such a meta-theorem for this logic.

We obtain optimal Hardy inequalities for fractional powers of the $p$-Laplacian $\Delta_p^\sigma$ on $\mathbb{Z}$, $p\in (1,\infty)$, $\sigma \in (0, 1/p)$. Asymptotically, the optimal weights behave like $|x|^{-p\sigma}$ as $|x|\to \infty$. We show a similar result for the Riesz fractional $p$-Laplacian. In an appendix, we show that for general $p$-Laplace operators on locally summable graphs, $p$-null-criticality implies $p$-optimality at infinity.

To address commons dilemmas under costly social cooperation, this paper develops a tripartite coevolutionary system that integrates social behavior, ecological resource dynamics, and institutional regulation. Institutional intervention is modeled as an adaptive feedback mechanism whose intensity responds to deviations from prescribed cooperation and ecological targets. Through equilibrium analysis, stability analysis, and phase space characterization, we identify the principal operating regimes and transition mechanisms of the coupled system. The results show that the system admits a sustainable operating equilibrium in which cooperation, ecological resources, and institutional regulation coexist. We further establish the structural stability of a desirable operating state characterized by full cooperation, peak resource availability, and persistently minimal intervention. In addition to these stable regimes, the system exhibits bistability and limit cycles, indicating that alternative long term outcomes may arise from different initial conditions and parameter configurations. These nonlinear mechanisms generate a persistent crackdown rebound policy trap, in which intensified intervention temporarily improves cooperation and resource conditions but eventually produces recurrent oscillations and prevents system wide consensus. The proposed framework provides a system level explanation of governance failure and offers analytical insights for designing adaptive institutional regulation in coupled human environment systems.

Let $f\colon\mathbb{N}\to\mathbb{U}$ be a non-pretentious multiplicative function in the sense of Granville and Soundararajan. We study the convergence of the autocorrelation averages of $f$. In particular, we investigate the relationship between the local pretentious behaviour of $f$ on intervals of the form $[x^{\varepsilon},x]$ and its deviation from $1$ on the primes. As an application, we exhibit a class of non-pretentious multiplicative functions for which Elliott's conjecture in its original formulation holds, following up on a result by Klurman, Mangerel and Teräväinen.

We investigate the interplay between the geometry of a planar domain and the spectrum of the nonlinear $p$-Laplacian, subject to a homogeneous magnetic field and Dirichlet boundary conditions. First, we establish a lower bound for the bottom of the spectrum on the whole plane, extending the classical Landau-level estimate to the nonlinear regime. Second, we prove Pólya-type bounds for the first eigenvalue on bounded convex planar domains in terms of the area, perimeter, minimal width and magnetic field strength, and show that they are asymptotically sharp along sequences of suitable thinning domains. Finally, we post some remarks with open problems.

We construct slope-stable ample rank-two vector bundles on every Hirzebruch surface $\mathbb{F}_e$ that admit no smooth strongly pseudoconvex Finsler metric with semipositive Kobayashi curvature. These examples give a negative answer to Kobayashi's problem. The same bundles give counterexamples to the smooth strongly pseudoconvex Finsler formulation of Fulton's problem and to the Hartshorne–Schneider conjecture on the convexity of complements. For $e=1$, these counterexamples have a simply connected Fano base that is not a product.

This paper introduces a hybrid joint-selective optimization (HJSO) framework for large-scale numerical problems in which a small subset of trainable quantities is of primary interest. We partition the full parameter vector into a high-dimensional remaining block and a low-dimensional block of parameters of interest (POIs), perform joint first-order optimization over the full parameter set, and then freeze the remaining variables while applying a reduced-space Levenberg-Marquardt (LM) refinement to the POIs. The method is designed for settings in which the POIs are low-dimensional but strongly influence the quality of the computed solution, while the full parameter space remains too large for full-space second-order methods. The framework is evaluated on three representative problems: a matrix eigenvalue problem, an inverse Bratu problem solved with a physics-informed neural network, and a 100-dimensional nonlinear Black-Scholes problem solved with the DeepBSDE method. In each test, HJSO reaches prescribed POI-error thresholds faster than the corresponding joint first-order baseline and improves the final POI accuracy for the reported solver configurations. The contribution is therefore not a universal optimizer, but a practical reduced-space strategy for problems with known low-dimensional parameters of interest and expensive high-dimensional training variables.

We determine the algebra generated by the normalized parity-twisted crank moments and describe it explicitly as a proper subalgebra of the quasi-modular forms on $\Gamma_0(2)$ closed under $D=q\frac{d}{dq}$. We derive congruences from differential identities and construct a prime-detecting expression involving only the normalized twisted second moment, its products, and its derivatives. Its coefficient of $q^n$ vanishes if and only if $n$ is prime, for every $n\ge2$. We prove that its highest weight, eight, is minimal among prime-detecting elements of this algebra. Finally, we identify the same algebra as the one generated by MacMahon's functions $B_k$, obtaining a corresponding prime-detecting expression in $B_1$.