Papers

Let $R$ be a $\phi$-ring and put $N=Nil(R)$. We determine the $\phi$-weak global dimension of every $\phi$-Prüfer ring. Its only possible values are $0$, $1$, and $\infty$: the value is $0$ when $R/N$ is a field, it is $1$ when $R$ is strong and $R/N$ is a nonfield Prüfer domain, and it is infinite when $R$ is not strong. No nilpotence assumption on $N$ is needed. The proof is based on the canonical kernel $K=ker(R\to\phi(R))$. In the local non-strong case, $K$ is a nonzero torsion divisible module over the valuation domain $R/N$, and its self-Tor groups satisfy \[ Tor^R_{2n}(K,K)=0\quad(n\geq 1), \qquad Tor^R_{2n+1}(K,K)\neq 0\quad(n\geq 0). \] It follows that $fd_RK=\infty$, and a filtered-colimit argument transfers this obstruction to cyclic quotients by nonnil principal ideals. We also prove the parallel trichotomy for the $\phi$-global dimension of $\phi$-Dedekind rings. For strong $\phi$-rings we obtain a change-of-rings theorem without assuming that the quotient map $R\to R/N$ splits. If $D=R/N$, $Q=Frac(D)$ and $T=T(R)$ is the total quotient ring, then \[ R\cong T\times_Q D. \] Using Ferrand's flat patching theorem and Milnor's projective patching theorem, we prove that every torsion $D$-module $M$, regarded as an $R$-module through $R\to D$, satisfies \[ fd_R(M)=fd_D(M), \qquad pd_R(M)=pd_D(M). \] Consequently, for every strong $\phi$-ring, \[\phi-wgd(R)=wgld(R/N), \qquad \phi-gld(R)=gld(R/N). \]

We study triples of distinct nonconstant polynomials $a,b,c$ over $\mathbb{R}$, $\mathbb{Q}$ or $\mathbb{Z}$ for which $$ ab+1,\qquad ac+1,\qquad bc+1,\qquad abc+1 $$ are all polynomial squares. We show that, up to permutation and a common invertible affine change of variable, there is exactly one such triple of quadratic polynomials in $\mathbb{R}[X]$, the one arising from a recent construction of Jurasić, and that none exists in $\mathbb{Q}[X]$ or $\mathbb{Z}[X]$. Moreover, for arbitrary nonconstant entries over $\mathbb{R}$ or $\mathbb{Q}$, we prove that the degree pattern $(2n,2m,2l)$ with $1\le n\le m\le l$ satisfies either $l=m$ or $l>n+m$. For the case $l=m$, we derive factorization systems and reducibility restrictions. In the equal-degree case over $\mathbb{Q}$, we further show that every entry has at least two distinct irreducible factors. As an application, we show that in a natural regular parametrized family of polynomial Diophantine triples in $\mathbb{Q}[X]$ the fourth expression $abc+1$ is never a square when the parameter is constant or the first entry is a square.

Let $X$ be a nonsingular projective curve of genus $g\ge2$ over an algebraically closed field of odd characteristic $p$. Giulietti and Korchmáros established a quadratic genus bound with constant $900$ forcing zero $p$-rank for curves of even genus. We extend this bound to odd genus: for every $g\ge2$, positive $p$-rank implies $|\Aut(X)|<900g^2$. Starting from Montanucci's two-short-orbit reduction, we combine ramification estimates and a minimum-genus argument to reduce a counterexample to an almost-simple group acting on a cover of the projective line with two branch points. The main ingredients are an intrinsic description of the second ramification group via Hasse–Arf and genus bounds for intermediate quotients associated with parabolic subgroups. These, together with order and local-structure estimates, exclude every possible simple socle. The proof uses the classification of finite simple groups.

We continue the study of the convex hull $D\subset\mathbb{R}^{d+n}$ of a disjunction of $n+1$ polytopes in $\mathbb{R}^d$, managed by $n$ binary indicator variables. Previously, $D$ had been characterized for arbitrary polytopes when $d\in\{1,2\}$, for polytopes sharing a common constraint matrix (under a technical condition, including hyper-rectangles), and for two generalized cross polytopes ($n=1$). Here we give a complete description of $D$ for arbitrary $n$ and $d$ when all $n+1$ polytopes are generalized cross polytopes. Every vertical facet is described by a normalized support inequality, and such an inequality is facet describing if and only if its equality graph, a bipartite graph on the polytopes and the coordinates, is connected; all vertical facets are generated explicitly from feasible spanning trees. We characterize when such a facet is one of the previously known pairwise inequalities, and we show that pairwise inequalities suffice when $d\leq3$, but not in general. We establish a bijection between the vertical facets and the maximal cells of a regular subdivision of $\Delta_n\times\Delta_{d-1}$\,, which yields the sharp bound $n+1+2^d\binom{n+d-1}{n}$ on the number of facets of $D$, attained for generic data. We also give a polynomial-time separation algorithm that returns a violated facet, which is maximally violated among the normalized support inequalities. Computational experiments show that the facets that are not pairwise inequalities close a large part of the gap left by the pairwise inequalities.

The Dobrushin coefficient of each attention-rollout factor satisfies $\kappa(\frac12(I+A))\le\frac12(1+\kappa(A))$, and multiplying these inequalities over layers bounds the coefficient of the whole rollout. We characterise exactly when the layerwise bound is tight: equality holds if and only if some token pair attaining $\kappa(A)$ is mutually self-dominant - each of the two attends to itself at least as strongly as the other attends to it. The condition is far from automatic: uniformly random stochastic matrices satisfy it only 24-30% of the time. When tested on the head-averaged attention of each individual input and restricted to content tokens - image patches, words or tabular features, excluding cls, register and separator tokens - the condition holds for essentially every input at every layer of DINOv2 (three model sizes), RoBERTa and DistilBERT. In the supervised models DeiT-B and ViT-B/16 it holds for 91% and 64% of input-layer pairs respectively, with all failures occurring late in depth. In FT-Transformer trained on two standard tabular benchmarks it holds for only 11-44% of input-layer pairs. The special tokens account for almost all failures in DINOv2 and the language models: when they are included, the condition holds for only 82-97% of input-layer pairs.

Feedback can improve the reliability of communication over additive white Gaussian noise (AWGN) channels. Classical feedback codes are interpretable, or easy to understand, but often rely on linear estimation, while deep-learned feedback codes can achieve strong performance but require many learned parameters and are often black boxes that are difficult to interpret. In this work, we propose a new posterior-mean feedback coding framework for AWGN channels with feedback. The proposed scheme uses posterior-mean refinement to construct nonlinear feedback codes under both noiseless passive feedback and noisy active feedback, with maximum a posteriori (MAP) decoding at the receiver. This scheme is analytically described, but a few learned parameters related to power allocation, and hence we consider this an “interpretable,” or human-understandable code. We further develop a projection-based design to improve robustness under noisy feedback and support larger message sizes. Numerical results show that the proposed schemes achieve strong finite-blocklength performance and outperform several analytical and learned feedback coding baselines, while using only a small number of learned design parameters.

We study the free-energy gradient flow associated with the depth-$N$ deep linear network on the space of invertible real $d\times d$ matrices. The loss observes only the diagonal, \[ E_d(W)=\frac12\sum_{i=1}^d(W_{ii}-1)^2, \] and the regularizer is the Boltzmann entropy of the balanced factorization fiber computed by Menon and Yu. This is a natural higher-dimensional version of a matrix-completion problem posed by Menon as a test of whether entropy selects among a noncompact family of minimizers. For diagonal completion, every width $d\geq2$, depth $N>2$, and inverse temperature $\beta>0$, we prove that the free energy is unbounded below and has exactly $2^d$ full-rank critical points, one in each diagonal sign chamber. Every critical point is diagonal and hyperbolic. Its unstable dimension is $\binom d2$ and its stable dimension is $d(d+1)/2$. We also prove that a full-rank trajectory cannot converge to a finite rank-deficient matrix. Consequently, almost every full-rank initial condition has an unbounded forward orbit. Thus finite-temperature fiber entropy does not provide an equilibrium selection principle for diagonal completion.

Let $X$ be a curve of genus $g\ge2$ over an algebraically closed field of odd characteristic $p$, and let $G=\mathrm{Aut}(X)$. A theorem of Giulietti and Korchmaros shows that if $G$ fixes no point and a $p$-subgroup $S\le G_P$ satisfies $|S|>\frac{p}{p-1}g$, then $X$ is one of the classical hyperelliptic, Hermitian, or Ree curves. We determine the sharp boundary and the first gap below it. At equality, besides the classical curves, exactly three types occur: the ordinary curves $y^p-y=x+c/x$, the zero-$p$-rank curves $y^3=x^p-x$ for $p\equiv2\pmod3$, and, in characteristic $3$, the ordinary genus-six family $y^3-y=c\left(x+\frac1{x^3-x}\right)$. As a key ingredient, we classify zero-$p$-rank curves whose full automorphism group fixes no point and which admit a $p$-subgroup of order larger than the genus. We then prove that, when $|S|>p$, the boundary classification already follows from the weaker inequality $|S|>p(g-p+1)/(p-1)$. For $|S|\ge p^3$, the forbidden interval above $m=(p-1)|S|/p$ has width at least $p(p-1)/2$, and below $m+(p^2-1)/2$ at most one further genus can occur, with a rigid ramification structure. Finally, we classify the first admissible layer: it occurs only for $|S|=p^2$, and the resulting ordinary curves admit a complete description in terms of two-dimensional Artin–Schreier spaces.

We study relatively minimal Lefschetz fibrations over closed oriented surfaces whose monodromy extends over a fixed 3–dimensional handlebody. If $X$ is the total space and $\ell$ is the number of critical points, we prove that $\sigma(X)=-\ell$. Such fibrations with $\ell>0$ exist precisely for fiber genus $g\geq3$ and base genus $h\geq2$, and one can take $\ell=1$ for every such pair. We also show that hyperelliptic handlebody monodromy forces the fibration to be a surface bundle and that every meridian twist, separating or nonseparating, has commutator length two in the handlebody group for $g\geq3$.

We formulate a novel optimal control problem for data-driven dynamic resource allocation in multicore real-time systems. The proposed formulation accounts for hard deadline constraints and resource bounds. Under mild assumptions, we prove several results on the feasibility and existence of optimal solution for the proposed formulation. Building on these analyses, we design an algorithm to solve the multi-core dynamic resource allocation problem. We demonstrate the practical use of this new algorithm by experimentally evaluating it on a real multicore platform using four benchmarks from the literature.

We study spectral zeta densities associated with complex powers of principally scalar Laplace-type operators on pseudo-Riemannian scattering spaces of mixed signature and arbitrary dimension. The construction extends, beyond the even-dimensional Lorentzian case, the local analysis of Feynman complex powers developed by Dang and Wrochna. Namely, for $\varepsilon > 0$, we consider complex powers $(P\pm i\varepsilon)^{-\alpha}$ and prove that the diagonal trace densities admit meromorphic continuation. We relate their poles to local geometric invariants, in particular to the pseudo-Riemannian scalar curvature. Our proof combines uniform Feynman resolvent estimates with a direct analysis of the Hadamard parametrix in pseudo-Riemannian signature. Beyond the Lorentzian case, the use of causal convexity is replaced by a geometric no-return argument.

We establish sharp mean value estimates for various exponential sums lacking translation-dilation invariance properties. For example, suppose that $k\ge 1$ and $s>0$. Then, for each $\varepsilon>0$, one has \[ \int_{[0,1)^k}\biggl| \sum_{1\le h\le X}\sum_{1\le z\le X}e(hz\alpha_1+\ldots +hz^k\alpha_k)\biggr|^{2s}\,{\rm d}\boldsymbol \alpha \ll X^\varepsilon\bigl( X^s+X^{2s-k(k+3)/2}\bigr), \] confirming the main conjecture for a new family of exponential sums. Our methods are based on the application of very recent work of the author employing the nested efficient congruencing method.

This letter develops a physics-informed one-shot beam tracking method for stacked intelligent metasurface (SIM)-enabled low-Earth-orbit (LEO) links with $K$ users and an $L$-layer SIM with $N$ meta-atoms per layer. The predictable orbital motion is propagated through the line-of-sight (LoS) geometry to forecast the finite-interval variation of the user signal-to-noise ratio (SNR) profile, and an analytical multilayer Jacobian relates this variation to the $LN$ SIM phase coefficients. Beam tracking is formulated as normalized SNR-profile preservation and solved through a single regularized Gauss–Newton correction that requires one Jacobian evaluation and a $K\times K$ linear solve, thereby avoiding iterative reoptimization. A tracking-error bound separates the regularization residual from the nonlinear Taylor remainder and establishes a second-order local error under a full-row-rank Jacobian. Numerical results show that the proposed method retains approximately $94\%$ and $92\%$ of the iterative-reoptimization rate for $L=2$ and $L=4$, respectively, while running about $350\times$ and $260\times$ faster at $N=225$, and confirm the predicted second-order residual error behavior.

Let $G$ be a connected graph with minimum degree at least two, positive Lin-Lu-Yau Ricci curvature, and no $K_{2,t}$ minor, where $t\ge 3$ is an integer. We first establish tight upper bounds on the maximum degree of $G$ by showing that $\Delta(G)\le 2t+3$ for $t=4$, $\Delta(G)\le 2t+4$ for $t\in\{5,6\}$, and $\Delta(G)\le 2t+2$ for all $t\ge7$. We then prove that $G$ has at most $10$ vertices for $t=3$, and at most $(t+1)\left(\Delta(G)^2(\Delta(G)-1)/2+1\right)$ vertices for all $t\ge 4$.

We study zeroth-order optimization of convex and strongly convex functions with Hölder smoothness of order $\beta>2$ from values corrupted by random noise with bias independent of the method's random directions, a Lipschitz systematic error and a bounded adversarial error. We measure the number of function evaluations (calls), of sequential rounds of parallel calls, and the largest tolerable systematic and adversarial errors. For strongly convex functions with a bounded Hessian, we prove a lower bound of order $d^2\varepsilon^{-\beta/(\beta-1)}$ on the number of calls, where $d$ is the dimension and $\varepsilon$ the accuracy, and find the sharp power of $d$ under the stronger Hilbert–Schmidt smoothness for all $\beta>2$. An accelerated finite-difference method attains them in $O(\sqrt{\kappa}\log(1/\varepsilon))$ rounds for condition number $\kappa$, like Nesterov's method with exact gradients, and tolerates the optimal systematic and, up to logarithms, adversarial errors; rate-optimal methods need $\Omega(\log d)$ rounds. The same number of calls is attained when the Hessian grows at most linearly with the gradient and, for centered noise, without an upper bound on the Hessian, also in logarithmically many rounds. For convex functions, our lower bound improves the known one polynomially in $d$ and is tight at the curvature of the hard instances; the systematic level is again optimal. These bounds extend to general norms and constraints; on the $\ell_1$ ball, mirror descent needs calls nearly linear in $d$. Experiments confirm the theory.

We study the problem of computing the tightest \(\alpha\)-potential approximation of a general-sum game over continuous action spaces, within a prescribed class of potential functions and when each player has access only to its own utility function. The difficulty is twofold: the approximation error involves a worst-case search over an infinite set of unilateral deviations, and the required utility information is distributed across players. For a linear-in-parameters potential class, we use an exact finite-tuple reformulation that separates the problem into a global outer search over deviation tuples and distributed convex inner problems. We develop a primal–dual inner oracle tailored to this structure and establish a uniform one-sided accuracy guarantee. This oracle can be combined with global outer search to obtain an end-to-end guarantee on the outer optimization error. We also develop a projected zeroth-order outer method as a computationally lighter alternative for higher-dimensional problems. Numerical experiments illustrate the accuracy–computation tradeoff between the two outer-search methods and show that the proposed optimization framework can improve upon analytical \(\alpha\)-potential constructions.

When $k\ge 2$ and $s>0$, we establish that for each $\varepsilon>0$, one has \[ \int_{[0,1)^{k-1}}\biggl| \sum_{1\le x\le X}e(\alpha_2x^2+\ldots +\alpha_k x^k)\biggr|^{2s}\,{\rm d}\boldsymbol \alpha \ll X^{s+\varepsilon}+X^{2s-(k^2+k-2)/2}, \] confirming the main conjecture for a new family of exponential sums beyond those of Vinogradov (translation-dilation invariant) type. Our methods are based on very recent work of the author employing the nested efficient congruencing method. Consequently, analogues of our results hold also in the setting of number fields and function fields.

The theory of finite translation planes is intimately tied to the theory of spreads of PG(3,q), where a spread is a partition of PG(3,q) into $q^2+1$ pairwise disjoint lines. A traditional method of generating spreads begins with the regular spread and identifies sets of lines therein that can be replaced with other lines to create a new spread, such as reguli, Bruen chains, and nests. This work provides computational algorithms to enumerate all possible replaceable sets consisting of reguli, chains and nests in the regular spreads of PG(3,q) for $q \le 11$. Additional results include a refinement to Baker and Ebert's (q-1)-nests which expands that infinite family, and counterexamples to the long-standing conjecture that a t-nest of reguli in PG(3,q) must be replaceable if $t \le q$.

In the context of functional data analysis, we propose a Bayesian test to assess whether the mean function of a Gaussian population is identically zero. By leveraging the structure of the reproducing kernel Hilbert space (RKHS) associated with the covariance function of the underlying process, we approximate the mean by a finite linear combination of kernel sections. This finite-dimensional representation arises from the natural assumption that the mean lies in the corresponding RKHS, enabling a parametric Bayesian approach to hypothesis testing through a spike-and-slab prior on the coefficients. Given the absence of a canonical reference measure on infinite-dimensional spaces, the likelihood is expressed via a Radon-Nikodym derivative between induced Gaussian measures. Since the number of kernel sections is unknown, we employ reversible jump Markov chain Monte Carlo (RJMCMC) to explore the trans-dimensional parameter space and estimate the posterior distribution. From the resulting posterior samples we construct estimators of the posterior probability of the null hypothesis and the Bayes factor, providing quantifiable evidence against the null and supporting informed decisions under uncertainty. Furthermore, to validate this methodology we establish near-optimal posterior contraction rates for the proposed test under mild conditions, demonstrate its empirical performance on simulated data sets, and illustrate its application with real climate data.

The polynomial maximization method estimates signal parameters in non-Gaussian noise from a finite description of the noise by its moments and cumulants, without a density. Known accuracy results assume that these noise characteristics are known, or use a design in which estimating them jointly costs no accuracy. In practice they are estimated from the residuals of a preliminary fit, and the effect of this substitution on accuracy is unknown. We treat the feasible scheme as a two-step estimator and, for a general smooth signal model in a fixed design, find the increment of its asymptotic covariance in closed form: $\Sigma_{\rm fe}-\Sigma_{\rm or}=c_2(1-g_{\rm or}) Q^{-1}\bar{d}\bar{d}^{\top}Q^{-1}$, a positive semidefinite matrix of rank at most one. Here $c_2$ is the noise variance, $g_{\rm or}\le 1$ the oracle variance reduction factor, and $Q$ and $\bar{d}$ the second-moment matrix and mean of the signal gradient over the design. Estimating the fractional moments that form the weights costs no first-order accuracy, because that block is orthogonal; the whole increment comes from estimating the sign-preserving centering means. The increment lies along the single direction $Q^{-1}\bar{d}$, so linear combinations of the parameters orthogonal to it keep oracle accuracy. In the trace metric the variance reduction factor becomes $g_{\rm fe}=g_{\rm or}+(1-g_{\rm or})R_D$, where $R_D=\|Q^{-1}\bar{d}\|^2/{\rm tr} Q^{-1}$ depends on the design alone. We give four sufficient conditions for first-order oracle equivalence; noise symmetry is not among them. Monte Carlo experiments with a harmonic signal confirm the direction and size of the increment. A covariance estimate that ignores the increment undercovers where non-Gaussianity gives the largest gain, symmetric noise included.

We prove a canonical quantization-commutes-with-reduction theorem for CR functions. Under the geometric hypotheses of Hsiao–Ma–Marinescu, if the zero moment level is nonempty and the strictly pseudoconvex reduction is globally CR embeddable of real dimension at least three, the CR Guillemin–Sternberg map is a bounded isomorphism at every real Sobolev order. The central new step is an algebraic and complex-analytic elimination of its finite-dimensional Fredholm defect: multiplicativity removes the kernel, while the conductor of the smooth range, integral dependence, and holomorphic removability produce global lifts. Conductor nonvanishing follows from the reduced Szegő singularity; we also prove it using global CR peak functions. The map is an isomorphism of Fréchet algebras, the reduction is connected, and the corresponding maps for holomorphic line bundles positive near the zero level are isomorphisms in every nonnegative tensor degree. Invariant holomorphic sections vanish in every negative degree; when the reduced base has positive complex dimension, the section maps are isomorphisms in every integer degree.

We develop and analyze several families of fully discrete, high-order methods in the finite difference time domain (FDTD) framework for a nonlinear dispersive electromagnetic model for ultrashort pulse propagation in a nonlinear optical medium. The model consists of the time domain Maxwell's equations coupled to two ordinary differential equations; the nonlinear Kerr-Debye equation and the linear dispersive Lorentz equation, in one space dimension. The proposed methods combine staggered spatial discretizations of $2M$th-order, for integer $M>0$, with second-order explicit leapfrog or implicit trapezoidal time integrators. To handle the stiff relaxation in the nonlinear Kerr-Debye equation, we adopt a modified exponential integrator, introduced in \cite{PengJCP2020}, which ensures asymptotic-preserving (AP) behavior in the limit of vanishing relaxation time. We prove that the resulting schemes satisfy a discrete analogue of the continuous energy identity and preserve the positivity (PP) of the nonlinear susceptibility. For the schemes based on the leapfrog time integrator, we establish energy stability under a CFL condition, while the schemes based on the trapezoidal time integrator are unconditionally energy stable. High-order temporal accuracy in the trapezoidal time integrator based schemes is further achieved via Richardson extrapolation yielding $2M$th-order accurate methods in both space and time. Numerical experiments demonstrate the theoretical convergence rates and the ability of the proposed schemes to capture nonlinear wave phenomena such as soliton propagation, self-steepening, and odd harmonic generation. This work extends previous AP-PP and energy stable approaches in a discontinuous Galerkin framework to the FDTD setting, offering a computationally efficient, flexible and robust tool for simulating electromagnetic wave propagation in nonlinear dispersive optical media.

Let $S$ be a closed oriented surface of genus at least $2$. Let $X$ be a $(2,1)$-dimensional anti-de Sitter spacetime homeomorphic to $S\times[-1,1]$ with smooth, spacelike, strongly convex boundary. By Gauss' formula, the induced metric on each of the two components of $\partial X$ has curvature $K<-1$. In this paper we address the inverse problem. We show that, for any two smooth metrics $h_-, h_+$ on $S$ with curvature $K<-1$, there exists a unique, smooth anti-de Sitter metric on $X:=S\times [-1,1]$ with respect to which the boundary is locally strongly convex and has induced metric isotopic to $h_-$ on its past component and isotopic to $h_+$ on its future component. Existence has already been proven by Tamburelli. We use this result to solve affirmatively a conjecture of Mess: given any two measured laminations $l_-, l_+\in\mathcal{ML}_S$ that together fill $S$, there exists a unique $(2,1)$-dimensional, globally hyperbolic, maximal, Cauchy compact (GHMC) anti-de Sitter spacetime such that the measured bending laminations of the past and future boundary components of its convex core are respectively $l_-$ and $l_+$. Here, existence has already been proven by Bonsante–Schlenker.

Different low-rank compression methods can produce compressed LLMs that respond differently to the same post-compression recovery procedure, and relative advantages observed between methods at the compression endpoint may shrink, grow, or even reverse after recovery. We ask whether this recovery heterogeneity reflects functional structure beyond scalar loss evolution, and how that structure evolves throughout recovery. Our results establish that this heterogeneity reflects a reproducible compression-induced functional structure, which we formalize as recovery pressure. To characterize this structure consistently throughout recovery, we develop a standardized functional characterization within each backbone that is applicable across heterogeneous low-rank methods. The primary backward characterization reveals reproducible module-wise structure across independent probes, while a complementary forward-only characterization recovers related structure without loss or backpropagation. We further find that recovery pressure measured at the endpoint is associated with subsequent recovery response; during recovery, its module-wise structure is reorganized non-uniformly, and localized updates induce distributed responses beyond directly updated modules. Further evidence indicates that tracking this evolving structure provides a complementary functional view of recovery progress alongside scalar loss.

Let $E\subset\mathbb R^2$ be analytic with $t=\dim_{\mathrm H}E>1$. We prove $\dim_{\mathrm H}\Delta E\ge 4/5+\eta_0$ for an absolute, non-explicit constant $\eta_0>0$, and \[ \dim_{\mathrm H}\Delta E\ge \frac{t^2-2t+5}{2t^2-7t+10} \quad\text{for }1<t\le\frac{11-\sqrt{41}}{4}, \] which reaches $7/8$ at the right endpoint. For pinned distances, $\dim_{\mathrm H}\Delta_xE\ge q_*=0.7813\ldots$ for every $x\in\mathbb R^2$ outside an exceptional set of Hausdorff dimension at most one, with stronger bounds depending on $t$ and further exceptional-set estimates. If $E$ is compact and supports an $s$-Frostman measure $\mu$, $s>1$, then \[ \underline{\dim}_{\mathrm B}\Delta_xE\ge \max\left\{\frac45+\eta_0,\min\left\{\frac{3s+1}{5},\frac78\right\}\right\} \] for $\mu$-almost every $x$. The same bound holds for pinned Hausdorff dimension under a common dyadic mass law, allowing oscillation with scale. We retain information about both endpoints in the Kolmogorov complexity estimates of Stull and Fiedler–Stull. Applying the partition estimates with the endpoints reversed yields $4/5$ and the rational bound. Equality forces both information profiles to have excess information on a common interval of scales. Encoding one source measure for all pins converts this constraint into a configuration of points and circles with many rich perpendicular bisectors. Lower bounds of Katz–Tardos and Ren–Wang contradict the upper bound of Fu–Gan–Ren; compactness then gives $\eta_0>0$. This rigidity argument has a precedent in the work of Héra, Shmerkin and Yavicoli on Furstenberg sets, using Bourgain's projection theorem. A separate finite-scale selection proves the pinned lower box bound.

We give a proof of the $\Gamma$-liminf inequality for the Maz'ya–Shaposhni\-kova limit, as $s\to0^+$, of the fractional Sobolev seminorm, based on spherical rearrangements. The key point is that the rearrangements of a weakly convergent sequence are monotone radial profiles, which converge strongly by Helly's theorem; since the energy does not increase under rearrangement (in the limit), this reduces the liminf inequality to a pointwise convergence statement for radially decreasing functions. As an application, we treat truncated energies, in which the interaction is restricted to a ball of radius $r_s\to\infty$. Assuming $r_s^{-2s}\to\lambda\in[0,1]$, we prove that the scaled energies $\Gamma$-converge, with respect to weak $L^2$-convergence, to $$ (1-\lambda)N\omega_N\|u\|_{L^2}^2, $$ where $\omega_N$ is the measure of the unit ball of $\mathbb R^N$. This identifies the contribution of the long-range interactions and gives a continuous interpolation between the classical Maz'ya–Shaposhnikova limit ($\lambda=0$) and the vanishing-energy regime ($\lambda=1$).

We study large deviations of the spectral radius of $X_n=n^{-1/2}(\xi_{ij})_{i,j=1}^n$, where the entries are iid, centered, variance-one, and subgaussian, with zero pseudovariance in the complex case. We write $\rho(X_n)$ for its spectral radius, i.e. the largest modulus among its eigenvalues. For every fixed $r>1$, we prove $$ \liminf_{n\to\infty}\frac1n\log\mathbb{P}\{\rho(X_n)>r\} \ge -\frac\beta2\bigl(r^2-1-2\log r\bigr), $$ where $\beta=1$ for real entries and $\beta=2$ for complex entries. We prove a matching upper bound for real symmetric laws with Gaussian-dominated even moments and for complex laws satisfying the sharp Gaussian Laplace-transform bound. These classes include discrete distributions. In the complex sharp class, we also identify the exponential rate for an eigenvalue to enter a fixed disk outside the unit disk. We then turn to lower deviations and establish a quadratic-speed bound. Under the additional assumption that the entry law $\mu$ has a bounded density, we prove $$ \mathbb{P}\{\rho(X_n)\le r\}\le e^{-c_{\mu,r}n^2} $$ for every fixed $0<r<1$ and all sufficiently large $n$, where $c_{\mu,r}>0$. For the matching upper-tail classes with a bounded density, these bounds give a full speed-$n$ large deviation principle, with the displayed upper-tail rate for $r\ge1$ and infinite rate for $r<1$. The upper-tail proof uses a change of measure that preserves the entry support and creates an outlying eigenvalue. The lower-tail proof develops a weighted comparison for adaptive orthonormal observations and applies it to Arnoldi residuals.

Generative flows sample from an unnormalized target by training a flow to be balanced, and the training loss is the signal a practitioner watches. We ask what that signal is worth: whether a small loss certifies an accurate sampler, whether the loss can be driven to zero, and how fast gradient descent does so. The loss decides the first. Losses that compare the two sides of the balance by their difference bound, in total variation, the error of the sampler the flow implies, with explicit constants that do not involve the policy; flow-matching losses that compare them through a ratio admit no such bound, already on a single cycle, whenever their generator is continuous at balance. On graphs, the backward policy decides the other two. Once it is frozen, balance becomes invariance under the backward chain, so that existence is free on finite graphs, and one constant — the norm of that chain's Green operator, which plays the role of an inverse spectral gap — fixes the order of the curvature of the loss around the balanced flow, from above and below, and sets a floor under the rate at which training converges near it. The mechanism is that gradient descent diffuses the flow along the backward policy. For the squared-logarithm generator of detailed and trajectory balance, training the balance loss on states converges globally on every finite path-connected graph, from every positive initialization. The constant can be infinite while backward trajectories are short on average, and exact flow matching can then fail. The bounds and rates are tested by exact computation on enumerable state spaces, and every theorem carries a certification status computed from a Lean~4 development.

We study the homogenization of an elliptic eigenvalue problem in a periodically perforated domain, where the spectral parameter appears both in the equation, through a bulk density $q$, and in a Steklov condition on the boundaries of the holes, through a surface density $\rho$, and where the resulting bulk–surface spectral measure $q\,dy+\rho\,dS_y$ is indefinite. We show that the asymptotic behavior of the spectrum depends essentially on whether the combined mean $m_{\rm eff}=\int_{Y^*}q\,dy+\int_\Gamma\rho\,dS_y$ is positive, negative, or zero. In all three cases, we prove convergence of eigenvalue clusters with multiplicity and of eigenspaces and, under additional regularity, $O(\varepsilon)$ estimates for the shifted or rescaled eigenvalues, spectral projectors, and aligned eigenspaces.

Clinical trials with survival endpoints lose information when participants are censored before death is observed. We develop target-preserving estimators that use posttreatment disease history, such as recurrence or progression, to recover information lost to censoring for marginal survival and restricted mean survival effects. The difficulty is that the intermediate event is downstream of treatment: naive adjustment can change the causal estimand, and the useful information enters only through the observed coarsening. We derive observed-data influence functions with and without recurrence history and obtain an exact gain identity. The identity shows that efficiency improvement is driven by the censoring hazard, the split of the alive risk set into recurrence states, and the residual-survival separation between those states. In the no-covariate illness-death model, the Aalen-Johansen estimator realizes the recurrence-augmented efficient score after standardization to the marginal target. With covariates, correctly specified Cox-Breslow transition hazards provide a root-N plug-in benchmark, while a hazard-induced one-step estimator gives rate robustness and, under primitive transition and censoring learner rates, canonical inference with a second-order product remainder. A semi-synthetic metastatic breast cancer study calibrated from digitized progression-free survival and overall survival curves illustrates the gain identity. The framework applies broadly to censored time-to-event studies with informative intermediate histories.