Papers

We prove a purely classical, coding-theoretic characterization of Hausdorff and packing dimensions in \(\mathbb R^n\). A ball code assigns names to closed balls of \(\mathbb R^n\): it is a partial map from a prefix-free set of finite binary strings to balls. For every nonempty \(E\subseteq\mathbb R^n\), the Hausdorff dimension of \(E\) is the minimum, over all ball codes, of the supremum over \(x\in E\) of the lower asymptotic rate at which \(x\) can be described by names of balls containing it; the packing dimension is obtained in the same way from the upper rates. The Hausdorff characterization is a Euclidean counterpart of Ryabko's coding theorem for combinatorial sources. We obtain the packing characterization from the standard characterization of packing dimension by upper modified box-counting dimension, without encoding \(\mathbb R^n\) into a sequence space. We then derive the point-to-set principle of J.~Lutz and N.~Lutz by replacing a minimizing ball code with a rational one and storing the resulting countable codebook in an oracle.

A company with a fixed artificial intelligence (AI) budget must decide which large language model (LLM) handles each recurring workload. What it lacks is the quality table, how well each model performs on each workload. Given that table, the decision is a multiple-choice knapsack problem and is routine to solve, so estimating it is the difficulty, and that estimation fails in two ways. Models are rarely compared on the same work, and the recorded score is usually a proxy rather than the outcome the company values. Causal and off-policy methods repair the first but condition on the second, while evaluator-validation methods estimate the second but stop short of the decision. Worse, buying more re-evaluation cannot settle the second: randomization governs which requests are scored, not how a score is produced, so the table stays uncertain however much evaluation is purchased. Yet the deployment decision may still be determined even when the table is not. We therefore ask whether one assignment stays optimal across every quality table consistent with the evidence. For the fixed-budget problem, this admits an exact two-solve certificate: solve once at the estimated table and once at a least-favourable table. Agreement certifies the assignment; disagreement identifies the model-workload pairs where further evidence can matter. We propose CASE (causal active sequential experimentation), which targets evaluation to those pairs and repeats the test as evidence accumulates. On a production log, the measurement failure is the larger of the two: correcting assignment exactly still leaves most of the loss, and randomized re-evaluation does not remove it. In our experiments, the available evidence often does not determine the assignment. On paid software tasks, better information about model quality yields more savings than further optimization of the assignment on the same estimates.

Let $u_j $ traverse an orthonormal basis of Hecke–Maass forms for $\mathrm{SL}_2 (\mathbb {Z}) $ with Hecke eigenvalues $\lambda_j (n)$ and Laplace eigenvalue $1/4+t_j^2$. In this paper, we consider the short-interval variant of the twisted spectral large sieve inequality of Luo for $ \lambda_j (n) n^{it_j} $ on the range $t_j \leqslant T$ and prove that the `Eisenstein–Kloosterman' cancellation discovered by Luo is effective on the interval $ T < t_j \leqslant T + M $ as long as $\sqrt{T} < M \leqslant T$. Moreover, our approach yields an improvement of the large sieve inequality of Luo.

In Einstein gravity, every null geodesic of a non-static Bianchi I spacetime satisfying the null energy condition is incomplete in at least one direction within the Bianchi I region. We prove a stronger directional rigidity theorem: a single null geodesic complete in both affine directions forces staticity if the null curvature is nonnegative along it and, at each time, in one transverse null direction. The spatial metric need not be diagonal, and the expansion rates may have either sign. The proof uses caustic-free null congruences whose conserved spatial covectors define fixed screen hyperplanes. The two curvature conditions admit averaged replacements along the geodesic and, when cosmic time is complete, along comoving worldlines. In a non-static metric, at most four oriented directions are optically saturated; every other complete null geodesic has strictly negative averaged null curvature. Periodic and finite-pulse examples attain this bound with NEC-violating sources and nonpositive total energy density. For finite pulses, we derive bounds on metric excursions from three directional curvature budgets at fixed cosmic duration, and on the mode mixing of a free, massless, minimally coupled test scalar field from the averaged null-energy deficit at fixed affine duration.

Let $K$ be a field containing $\mathbb{Q}$, let $\Gamma$ be the ring of all functions from the positive integers to $K$ under Dirichlet convolution, and let $\Gamma_n$ be its truncation to functions supported on $[1,n]$. In [Snellman, Homology Homotopy Appl. 2 (2000), 17-27; arXiv:math/9904143] it was shown that $\Gamma_n$ is a polynomial ring modulo a monomial ideal $I_n$ which is stable after reversing the order of the variables, and the Poincare-Betti series of $\Gamma_n$ was computed in terms of the numbers $C_{n,v}$ of minimal generators of $I_n$ of least support $v$. We prove that $C_{n,v} = \Phi(n,p_v)$, Legendre's sifting function: the number of integers in $[1,n]$ free of prime factors $\le p_v$. This identifies an invariant of a minimal free resolution with a classical object of sieve theory, and it refines to the grading by total degree, with the number of prime factors carried in a variable. As consequences we obtain: a proof of Conjecture 4.6 of the 2000 paper, which was left open there; the asymptotics $C_n \sim \pi(n)^2/2$ of the total number of minimal generators, showing that the lower bound $C_n \ge \binom{\pi(n)+1}{2}$ of that paper is asymptotically sharp, together with the exact order $C_n - \binom{\pi(n)+1}{2} \sim \frac{16}{3} n^{3/2} / \log^3 n$ of the error; and the identification of $C_n$ with the OEIS sequence A182843. We also record errata for the 2000 paper: one stated result is false, one worked example is miscomputed, and two proofs are incomplete. Corrected statements and complete proofs are given.

This paper is a sequel to [9], where the first-named author constructed a family of approximate Hermitian Yang–Mills metrics $H_{0,\epsilon}$ on stable rank-two holomorphic vector bundles arising from double spectral covers over the product of two one-dimensional complex tori. We prove that these approximate metrics give an all-order, exponentially accurate asymptotic description of the exact Hermitian Yang–Mills metrics in the large Kähler limit. More precisely, the mean curvature of $H_{0,\epsilon}$ decays exponentially in every $C^k$-norm. Moreover, if $H_{1,\epsilon}$ denotes the exact Hermitian Yang–Mills metric and \[ H_\epsilon=H_{0,\epsilon}^{-1}H_{1,\epsilon}, \] then, after normalization, for every nonnegative integer $k$, there exist positive constants $C_k$ and $c_k$ such that \[ \|H_\epsilon- Id\|_{C^k}\leq C_k e^{-\frac{c_{k}}{\epsilon}}. \] The main analytic difficulty lies in the global $C^0$-comparison. Obtaining $C^0$-estimates for the coupled nonlinear Hermitian Yang–Mills system is intrinsically difficult; moreover, the equation controls only the contraction of the curvature, and hence only certain combinations of second derivatives, whereas one needs global control of the full matrix-valued metric.

We introduce a notion of pseudo-spherical representations for any Brylinski-Deligne (BD) cover of a split torus over a $p$-adic field. In particular, we do not assume that the cover is tame. We enumerate the pseudo-spherical representations and compute their dimension. When the torus is the maximal split torus of a connected split reductive group $G$, we enumerate a subset of distinguished pseudo-spherical representations, under mild assumptions on the BD datum. We expect these distinguished representations to provide the torus input for a Borel-Casselman Theorem for BD covers of $G$. We have verified this expectation for certain non-tame double covers in a separate work. While investigating the pseudo-spherical representations, we also prove that the restriction of irreducible genuine representations of covering tori to their maximal compact subgroup is multiplicity-free. From this we see that the irreducible genuine representations of this maximal compact subgroup are types in the sense of Bushnell-Kutzko. We conclude by describing the Hecke algebras attached to these types.

In an inhomogeneous dissipative medium, different rays experience different round-trip attenuation, so an instantaneous time mirror may retrace rays without producing a balanced point focus. We solve this problem for a microlocally selected back-propagating transverse-magnetic component generated by a short pulsating modulation of a thin shell surrounding an inhomogeneous Debye medium with Ohmic conductivity. An exact decomposition of the Maxwell–Debye field isolates the pulse-generated return, and a semiclassical mode reduction yields a scalar returned operator whose principal symbol separates the temporal pulse response, shell modulation, and accumulated material loss. Our analysis shows that geometric distortion and amplitude attenuation are both involved in wave propagation, but the geometric distortion cancels between the outgoing and returning branches, whereas physical attenuation remains. A normalized directional energy, computed from pre-modulation measurements using the known lossless backward propagator, contains the same leading attenuation factor as the returned symbol. Its reciprocal defines a smooth, nonnegative, physically admissible modulation that equalizes the selected source-side gain up to a relative $O(h)$ error without requiring a complete attenuation model. With such a modulation, the field generated by a point source has a focal-scale leading profile independent of directional attenuation, with a unique maximum at the source and coherent return time. In every fixed focal window, the maximum is displaced by at most $O(h^2)$ in space and time, and the spatial half-amplitude area is $O(h^2)$.

Over an algebraically closed field of characteristic $p>0$, we classify all matched pairs between a generalized Taft algebra $T_{N,n,\xi}$ and the Radford algebra $R$. If $p\nmid N$, every matched pair is trivial, and the corresponding bicrossed product is the tensor product Hopf algebra. If $p\mid N$, matched pairs are parametrized by $\beta\in\mathbb F_p$, and the bicrossed products are described by explicit cross relations. Two such bicrossed products are isomorphic as Hopf algebras if and only if their parameters are equal; hence there are exactly $p$ isomorphism classes in this case.

We consider the two-dimensional Cahn-Hilliard equation forced by divergence of space-time white noise that represents Kawasaki dynamics in conservative form. The standard heuristic argument shows that the solution is a distribution and thus the product within the nonlinear term is ill-defined, making this stochastic partial differential equation challenging. We prove the global-in-time unique solution theory after multiple Da Prato-Debussche tricks where the last equation is solved in a weak formulation. Besides a minimum amount of the standard renormalization procedure to deal with ill-defined products, our proof consists of applications of deterministic analysis tools and taking advantage of the unique structure of the equation, which is crucial. We also prove the local-in-time unique solution theory of a mild solution, demonstrating that our improved estimate gets us arbitrarily close, although not quite, to being able to prove the global-in-time solution theory with the last equation in not only a weak formulation but also a mild formulation. Additionally, our approach can be readily extended to prove the global-in-time existence, although not path-wise uniqueness, of solution to the singular surface quasi-geostrophic equations with the resulting equation after multiple Da Prato-Debussche tricks in weak formulation.

This paper investigates a stochastic multiscale subdiffusion model driven by fractional Gaussian noise, where the multiscale Abel kernel with variable exponent $\alpha(t)\in(0,1)$ is used to capture multiscale and crossover behavior in anomalous diffusion. The main difficulties of this model lie in the complexity of the multiscale Abel kernel (e.g. non-monotonicity and non-coercivity) and the low regularity caused by the noise. Concerning these issues, we prove the well-posedness and regularity of the mild solutions by means of solution operator approach and a perturbation technique for multiscale Abel kernel. Then both the semidiscrete-in-time and fully-discrete numerical schemes are proposed and analyzed under the low-regularity numerical analysis framework, with proved temporal and spatial convergence rates. Numerical experiments are presented to substantiate the theoretical results.

Motivated by a theorem of A. Schmidt on the part of the complex Lagrange spectrum below $2$, we study the restricted Lagrange spectrum $L_{\frac{1}{2}+i\mathbb{R}}$ arising from the approximation of complex numbers of the form $\frac{1}{2}+i\alpha$, $\alpha\in\mathbb{R}\setminus\mathbb{Q}$ by Gaussian rationals $p/q$ with $p,q\in\mathbb{Z}[i]$, $q\neq 0$. We show that this spectrum admits a description in terms of a dynamical spectrum associated with a real horseshoe. As a consequence, we obtain several fractal properties of $L_{\frac{1}{2}+i\mathbb{R}}$, such as continuity of the dimension function $t\mapsto\dim_H(L_{\frac{1}{2}+i\mathbb{R}}\cap(-\infty,t))$. We also prove that the set of complex numbers $z$ satisfying \begin{equation*} \left\lvert z-\frac{p}{q}\right\rvert\geq\frac{1}{2|q|^2}, \quad\text{for all } p,q\in\mathbb{Z}[i], q\neq 0, \end{equation*} is uncountable. In fact, we show that this inequality holds for every complex number of the form $z=\frac{1}{2}(1+i\theta)$ where $\theta\in\mathbb{R}\setminus\mathbb{Q}$ is a root of one of Schmidt's $C$-minimal forms.

An edge-colored graph is called a rainbow graph if all its edges have distinct colors. The anti-Ramsey number, denoted by $\ar(n,F),$ for a fixed graph $F$ and a positive integer $n$, is the maximum number of colors used in an edge-coloring of the complete graph $K_n$ that contains no rainbow copy of $F$. Meanwhile, the Turán number, denoted by $\ex(n,F),$ for graph $F$ and $n$, is the maximum number of edges in an $n$-vertex graph that does not contain $F$ as a subgraph. For a vertex $v$ and a multiset $\mathcal{H}$ of graphs, the suspension $\mathcal{H} + v$ of $\mathcal{H}$ is the graph obtained by connecting the vertex $v$ to all vertices of $H$ for each $H \in \mathcal{H}$. Let integers $k\ge 1$ and $r\ge 2$ be fixed, and suppose that $\mathcal{H}_{k+1}=\{H_1, H_2, \ldots, H_{k+1}\}+v$ satisfying $H_1, H_2, \ldots, H_{k+1}$ are pairwise vertex-disjoint edge-critical graphs, and $\chi(H_i)=r$ for $i=1,2,\ldots, k+1$.In this paper, we determine $ \ar(n,\mathcal{H}_{k+1}) $ for $k\ge 1$, $r\ge 2$ and sufficiently large $n$. This result unifies and generalizes a result of Liu et al. (arXiv:2411.08475) concerning the friendship graph, and a result of Lu et al. (arXiv:2507.13165) on the intersecting cliques.

Strong topological insulators are classified by integer invariants which admit different real-space expressions. Inspired by recent work studying these invariants for spaces with curved boundaries, we revisit the problem of equivalence of the various expressions, including Fredholm index pairings and Kubo formulas involving half-space projections. Using Roe algebras and a variant of KK-theory suitable for non-separable $C^*$-algebras, we prove a general form of bulk-edge correspondence for the index pairing with a general position-space Dirac operator in the presence of arbitrary boundaries. For spaces coarsely equivalent to $\mathbb{R}^d$, we show that up to multiplicity, these pairings are equal to those obtained with the standard dual Dirac operator, with the multiplicity explicitly given as the topological degree of the symbol function. In addition, we prove a generalized Kubo formula which computes the general index pairing by a real-space formula.

For a finite graph, its energy is the sum of the absolute values of its adjacency eigenvalues. Let n be the product of powers of distinct odd primes, where all exponents are positive integers, the sum of the exponents is odd, and every prime whose exponent is odd is at least five. Among all nonempty sets of proper divisors of n, the energy of the corresponding integral circulant graph is uniquely maximised by the checkerboard divisor set consisting of those divisors for which the sum of the prime-power exponents is even. The maximum energy is one half of the sum of n and the product, over all prime factors, of explicit polynomial quantities. For exponent k and parameter x, the corresponding polynomial is given by two k plus one times x to the k-th power, together with four times the sum from zero to k minus one of j plus one times x to the j-th power with alternating sign determined by the parity of k minus j. For the proof, we use weighted prime-power Ramanujan transforms, which give an exact energy model in tensor form and extend to a polynomial matrix family for real parameters greater than one. A semidefinite parity theorem shows that the matrix obtained from this polynomial family by an explicit positive diagonal normalisation is a real symmetric contraction whose unique eigenvalue of modulus one is the simple eigenvalue determined by the parity of the exponent; this holds for even exponents at least two and parameters at least three, and for odd exponents at least one and parameters at least five. Tensorising this spectral structure and centring the binary divisor tensor yield a sharp one-norm bound for the corresponding tensor operator on sign vectors. In particular, for n equal to p squared times q cubed, with distinct odd primes p and q and with q at least five, this proves Roldan's conjectured maximality in this range.

We study the simplest parabolic-elliptic model of chemotaxis in ${\bf R^N}$, and show the optimal criteria for the existence of infinite time blow-up of solutions with the initial data below the Chandrasekhar singular solution when $N \geq 10$. In particular, we exhibit the differences of the criteria between the cases $N \geq 11$ and $N = 10$. Our argument is based on the study of the Cauchy problem for the transformed equation involving the averaged mass of the solution and the asymptotic expansions of solutions to the Liouville equation.

In resin transfer moulding, complete saturation of the fibre preform is necessary before the resin front reaches the outlet vent(s), to prevent dry-spot formation. In practice, the flow front rarely advances uniformly due to race-tracking effects. We propose a combined estimation and control strategy to address this issue. We use pressure-sensor data collected during filling to estimate the unknown race-tracking strengths via an iterated extended Kalman filter, and to simultaneously optimise auxiliary gate pressures to prevent the resin from arriving at the vent before complete saturation occurs. To make this approach feasible in real time, we replace the expensive finite element-control volume model with neural network surrogate models. The Bayesian approximation error framework is used to account for the discrepancy between the surrogates and the original model. To make the training of the surrogates feasible with time-dependent control actions, we prove that the flow-front geometry depends only on the time-averaged gate pressures. We demonstrate the methodology on a representative forked geometry, showing that the combined estimation-and-control framework can substantially reduce the number of unfilled nodes at the end of filling compared with an uncontrolled baseline. However, the level of improvement depends strongly on the location and number of auxiliary gates.

In this paper, we construct canonical fundamental solutions for linear $q$-difference systems with coefficient matrix $Ax+B$, develop the associated isomonodromy theory, and study the differential limit as $q\to1$. We determine the asymptotic behavior of generic isomonodromic deformations, and derive explicit formulas for their monodromy data in terms of their asymptotic leading terms. We also show that the differential limit as $q\to1$ recovers the canonical fundamental solutions, the monodromy data, and the isomonodromic deformations of the corresponding differential systems.

We resolve three conjectures proposed by Tokushige in 2013 about cross $t$-intersecting families of subsets and integer sequences. For $0<p<1$ and $\mathcal F\subseteq2^{[n]}$, define the $p$-biased measure by $\mu_p(\mathcal F)=\sum_{F\in\mathcal F}p^{|F|}(1-p)^{n-|F|}$. Two families $\mathcal F_1,\mathcal F_2\subseteq2^{[n]}$ are cross $t$-intersecting if $|F_1\cap F_2|\geq t$ for every $F_1\in\mathcal F_1$ and $F_2\in\mathcal F_2$. We prove that, for every $n\geq t\geq 2$ and $0<p_1,p_2\leq1/(t+1)$, such a pair satisfies $\mu_{p_1}(\mathcal F_1)\mu_{p_2}(\mathcal F_2)\leq(p_1p_2)^t$. When $p_1,p_2<1/(t+1)$, equality holds if and only if both families are the same $t$-star $\mathcal S_T=\{F\subseteq[n]:T\subseteq F\}$ for some $T\in\binom{[n]}{t}$. Together with the previously known case $t=1$, this completely resolves Tokushige's measure conjecture. We also prove that every pair whose measure product is close to the maximum must be close to a common $t$-star. More precisely, if $p_1,p_2<1/(t+1)$ and $\mu_{p_1}(\mathcal F_1)\mu_{p_2}(\mathcal F_2)>(1-\varepsilon)^2(p_1p_2)^t$, then there exists $T\in\binom{[n]}{t}$ such that $\mu_{p_i}(\mathcal F_i\mathbin{\triangle}\mathcal S_T)<C\varepsilon$ for $i=1,2$, where $C$ depends only on $t,p_1,p_2$. This improves Tokushige's conjectured $C\sqrt{\varepsilon}$ estimate to $C\varepsilon$. For integer sequences, we prove that if every sequence in $\mathcal H_1\subseteq[m]^n$ agrees with every sequence in $\mathcal H_2\subseteq[m]^n$ in at least $t$ coordinates, then $|\mathcal H_1||\mathcal H_2|\leq m^{2(n-t)}$ for all $n\geq t\geq1$ and $m\geq t+1$. We further obtain a more general result in which a separate agreement requirement is imposed for each possible value. This extends a theorem of Frankl and Kupavskii and recovers their earlier cross intersection–union product theorem.

We study the design of linear time-invariant (LTI) encoder-decoder pairs for transmitting the state of a discrete-time LTI vector source over power-constrained parallel Gaussian channels with feedback. Two types of power constraints are considered. Under individual subchannel power constraints, a necessary and sufficient condition for designing an encoder-decoder pair that achieves bounded estimation error covariance (EEC) is established via two coupled majorization inequalities involving the subchannel signal-to-noise ratios and the antistable poles of the source. Under total channel power constraint, we derive the minimum total power required for a feasible encoder-decoder design by exploiting partial-order progamming under majorization order. An analytical optimal power allocation is obtained for the case of equal noise variances, which admits a water-filling interpretation; for general noise case, a sequential water-filling algorithm is developed. Our results reveal that the difficulty of transmitting a discrete-time LTI source via LTI coding is governed not only by its topological entropy, but also by the evenness of the log-magnitudes of its antistable poles. The design methods for feasible encoder-decoder pairs are also provided.

We study generalized Hamming weights along the one-point code flag of an AJ-Gorenstein curve. We organize these weights in a graded array, the zero diagram, whose entries are generalized coweights: the largest numbers of evaluation points on which subcodes of prescribed dimensions vanish simultaneously. Twisted and Wei duality show that each row of the zero diagram determines both the generalized Hamming weights of a lower block of short codes and the missing weights of a reflected upper block of long codes in the GHW diagram of the complete flag. Our main quantitative result is a uniform coverage theorem. For an AJ-Gorenstein curve of genus $g$ and evaluation length $n>2g$, the proportion of generalized-weight positions determined exactly throughout the complete flag satisfies $\operatorname{Cov}_{\mathrm{full}} \ge \frac{n(n-1)+4g}{n(n+2g-1)}>\frac12$. Thus more than half of all generalized-weight positions in the complete flag are determined uniformly. For the smallest Suzuki curve, the general and Castle-specific mechanisms together determine $2{,}280$ of the $2{,}912$ positions, giving an exact coverage of $78.30\%$.

Diffusion models are increasingly used not only for sampling from learned data distributions, but also for generating samples that optimize task-specific objectives. A common approach is to guide the reverse diffusion process using gradients of an external objective. However, when the data distribution is supported on a structured feasible set, such as a manifold or a constraint set, gradient guidance can move samples away from the learned data geometry. In this paper, we study a simple projected-gradient-guided diffusion update based on the observation that the Stein denoising operator can act as an approximate projection onto the data geometry. The proposed update incorporates the objective gradient inside the denoising step, yielding an inference-time method that uses only a pretrained denoiser and gradient evaluations. We analyze this update as an inexact projected-gradient method for constrained optimization over learned feasible geometries. Our theory covers three settings: linear manifolds, compact convex feasible sets, and compact Riemannian submanifolds. In all these settings, we prove descent and finite-time convergence guarantees. Numerical experiments support the theoretical interpretation and illustrate how the proposed update balances objective descent with preservation of the learned geometry.

We study the simultaneous Busemann-Petty and Shephard volume comparison problem: whether comparison of the volumes of all central hyperplane sections and all orthogonal hyperplane projections determines the ordering of the volumes of two convex bodies. For every $n\geq5$, we construct origin-symmetric convex bodies of revolution $K,L\subset {\mathbb R}^n$ such that every central hyperplane section and every orthogonal hyperplane projection of $K$ has strictly smaller volume than the corresponding section or projection of $L$, while $|K|>|L|$. For $n\leq4$, the affirmative solution of the Busemann–Petty problem shows that the section inequalities alone imply $|K|\leq|L|$. Without origin symmetry, we construct such counterexamples in every dimension $n\geq2$, with one body a nontrivial translate of a Euclidean ball and the other a noncentrally symmetric body of constant brightness.

We study a Brownian directed polymer in a centered Gaussian environment that is white in time and colored in space having long-range spatial correlations. For product-type covariances \(Q(x)\asymp\prod_{j=1}^d(1+|x_j|)^{-\alpha_j}\), with \(\alpha_j\in(0,1)\) and \(\kappa=\sum_j\alpha_j\), we identify the disorder transition at the marginal value \(\kappa=2\). For \(\kappa>2\), weak disorder holds at sufficiently small inverse temperature; for \(\kappa<2\), the quenched free energy $p(\beta)$ satisfies \(-p(\beta)\asymp\beta^{4/(2-\kappa)}\) as \(\beta\downarrow0\). For \(\kappa=2\), strong disorder holds for every \(\beta>0\), while \(p(\beta)=0\) for all sufficiently small \(\beta\), so \(\beta_c=0<\bar\beta_c\). We also consider the radial covariance cases, where $ Q(x)\asymp(1+|x|)^{-\vartheta}$, when \(d\ge3,\vartheta=2\) and \(d=2,\vartheta\ge2\), which was left unanswered in Lacoin~\cite{Lacoin2011}. When $d=2$, we get \(\ln(-p(\beta))\asymp-\beta^{-2}\) for \(\vartheta>2\) and \(\ln(-p(\beta))\asymp-\beta^{-1}\) for \(\vartheta=2\). The proofs consist of replica coupling, Feynman–Kac variational formula, overlap methods, and continuous-space fractional moments with ordered Wiener-chaos changes of measure.

We present an alternative proof of the global well-posedness of nonlinear wave equations in three spatial dimensions under the null condition, in the regularity regime of the classical local existence theory, assuming smallness of the angular derivatives. Unlike previous methods, which rely on decay in time, our approach is based solely on spatial decay. This provides a new perspective on the problem and offers potential applications to the study of Einstein's equations, which we intend to explore in future work.

Physics-informed neural networks (PINNs) have emerged as a meshless alternative to conventional numerical methods for solving partial differential equations (PDEs). However, their limited ability to capture sharp gradients can lead to substantial errors when resolving boundary and interior layers. Here, we introduce an energy-norm-enhanced PINN (ENPINN) that incorporates gradient information and variational structure into the loss function to improve the resolution of layer-dominated solutions. We first examine two related formulations: weak-loss PINNs (WLPINNs), which incorporate test functions into the conventional PINN residual, and gradient-enhanced PINNs (gPINNs), which augment the loss with spatial derivatives of the PDE residual. By analyzing these formulations, we identify their limitations in resolving steep solution gradients and motivate the systematic construction of ENPINN. We establish theoretically how the energy-norm error depends on the ENPINN loss and show that a suitably modified residual-derivative term is essential for accurately capturing boundary layers. We further establish the existence of neural-network approximations with arbitrarily small energy error and derive corresponding derivative bounds, providing a theoretical foundation for the proposed framework. The performance of ENPINN is assessed through systematic comparisons with existing PINN variants for convection-diffusion-reaction problems exhibiting steep gradients. Numerical experiments include a combustion model, a coupled multi-scale system, a two-dimensional Burgers equation with an interior layer, and a three-dimensional time-dependent problem.

For the bilaplacian in spherical and hyperbolic spaces, clamped vibrating and buckling eigenvalues on geodesic disks are shown to satisfy scaling inequalities analogous to the standard scale invariance of Euclidean plate eigenvalues. These results extend curved-space scaling results from the Laplacian to fourth-order eigenvalue problems. In addition, the limiting behavior of the scaled eigenvalues is determined as the disks expand to fill the ambient space. Depending on the geometry and the choice of scaling, the eigenvalues tend to zero, positive or negative infinity, or finite values associated with the bottom of the hyperbolic spectrum. The negative divergence in one spherical buckling limit is related by conformal inversion to an exterior buckling problem.

In this paper, we construct the $(\infty,\infty)$-category $\mathsf{Span}_\infty(\mathcal{C})$ of spans, also known as correspondences, in any given $(\infty,1)$-category $\mathcal{C}$ with finite limits. This yields new models for the span $(\infty,n)$-categories for $n \in \mathbb{N} \cup \{\infty\}$. We characterize the mapping $(\infty, n-1)$-categories in these $(\infty,n)$-categories, and thereby verify that our model agrees with other models for spans. Finally, and most importantly, we prove a new universal property, characterizing functors into span $(\infty,n)$-categories, which specializes to the well-known relation with the twisted arrow categories in dimension $1$. These results will be used in the sequels to construct higher analogs of the classical Hall algebra construction, where "higher" refers to both higher categorical and "higher monoidal" structures, i.e., $\mathsf{E}_k$-algebras in $(\infty, n)$-categories for $n, k>1$.

Although Pollak's minimax formulation is one of the central frameworks in quickest change detection (QCD), the strongest general optimality results available for it are predominantly asymptotic, applying as the average run-length-to-false-alarm constraint, $\gamma$, tends to infinity. Exact results for finite $\gamma$ have previously been available only for special models or restricted regimes. This paper addresses the general finite-$\gamma$ problem over the complete class of randomized, history-dependent stopping rules for i.i.d. change models. The key technical development is a survival-process representation that recasts the optimization of Pollak's minimax criterion over stopping times as an equivalent linear variational optimization over admissible survival processes. Although this formulation is infinite-dimensional, it establishes the existence of an optimizer and provides an exact characterization of the finite-$\gamma$ Pollak minimax value. This characterization, in turn, provides a principled basis for constructing computable stopping rules whose performance can be made arbitrarily close to the optimum. The resulting rules are driven by a recursively updated weighted likelihood-ratio statistic with generally time-varying injections and boundaries. Importantly, this structure is not imposed a priori: the optimization is carried out over the full class of randomized, history-dependent stopping rules, and the Shiryaev–Roberts form emerges naturally from the solution. In particular, the classical Shiryaev–Roberts recursion arises as the time-homogeneous special case. Finally, under a likelihood-ratio floor condition, the framework yields closed-form exact minimax solutions for a nontrivial class of change models.

We study cabling operations on braids, and characterize the matrices that can be realized as crossing matrices of reducible pure braids. We also use cabling to construct enhanced conjugacy invariants of braids.