Papers

We design and analyze a nonoverlapping additive Schwarz preconditioner for interior penalty discontinuous Galerkin discretizations of anisotropic elliptic problems. The preconditioned method coupled with a Krylov subspace iteration is shown to be independent of the highly discontinuous (and anisotropic) jump coefficients as well as the subdomain size. To increase efficacy, various auxiliary spaces are considered to reduce the size of the coarse grid operator. We demonstrate how to modify the additive Schwarz preconditioner such that it is applicable to the nonsymmetric IPDG schemes. Several numerical experiments verify the theory and validate the robustness of the preconditioner.

Let $R$ be a commutative noetherian local ring. We extend the work of the author and Takahashi by investigating the structure of irreducible monomorphisms originating from free modules in the category of finitely generated $R$-modules. In the case where $R$ is complete, we further study the structure of irreducible monomorphisms and epimorphisms to and from injective modules in the category of artinian $R$-modules.

We study a two-regime summary of a real-valued loss distribution. The two representative levels and the boundary between them are chosen by minimizing a convex residual loss. When the distribution has an atom at the boundary, assigning that atom to the lower or upper regime can give different optimized costs. Taking the better whole-atom assignment yields the lower-semicontinuous cutoff profile. We prove epi-convergence of this profile under a loss-adapted \(\psi\)-weak topology and obtain convergence of minimum values and outer stability of optimal cutoff sets. Under a unique finite atom-free limiting cutoff, positive regime masses, and unique conditional centers, the two fitted levels and a canonical one-jump representation converge as well. We also examine the resulting summary from the viewpoint of monetary risk axioms, establish explicit failures of external monotonicity and subadditivity, and give finite-support algorithms with conditional error control. A four-scenario model illustrates the cutoff geometry and the role of ties and atoms.

We give an explicit tournament on 14 vertices containing every 7-vertex tournament as an induced sub-tournament, and verify this property by exhaustive computation. This improves the previously known upper bound of 15. Together with the known lower bound, it follows that the minimum order of a 7-universal tournament is either 13 or 14.

We prove that plane waves are the only gravitational pp-wave spacetimes which are distinguishing; in particular, they are the only strongly causal ones. This is used to prove a null version of Anderson's rigidity theorem, conjectured by Costa e Silva and Flores: every geodesically complete, transversally complete, strongly causal, Ricci-flat Brinkmann spacetime of dimension $4$ is isometric to a quotient of a plane wave spacetime.

In the year $1977,$ while developing the theory of modules arising in the context of Serre's problem on projective modules in the paper “On Stably Free Modules" (see \cite{AS}), A. A. Suslin constructed matrices inductively using unimodular rows. Our aim is to connect these matrices to the current and applicable field of coding theory. To this end, we construct Suslin-Rao codes, by using looking at certain unimodular all one vectors over the finite field $\mathbb{F}_2$. Comparative analysis with standard families such as Hamming, Simplex and BCH codes shows that the Suslin-Rao codes achieve better performance than Hamming and Simplex codes in terms of code rate and error correction capability, though BCH codes may outperform them for large code lengths. Bit error rate (BER) versus signal-to-noise ratio (SNR) simulations further support the practical applicability of these codes in noisy channels. The recursive nature of the construction allows efficient encoder design, making the proposed codes attractive for real-time communication systems and hardware implementation.

The one-dimensional planar compressible magnetohydrodynamic system is studied on unbounded domains. The longitudinal viscosity obeys a power law of the specific volume, which permits it to degenerate as the density approaches vacuum, and the heat conductivity follows a power law of the temperature. For any fixed non-negative exponent in the heat conductivity term, the existence and uniqueness of global strong solutions are established for arbitrarily large initial data, provided that the positive exponent for viscosity is sufficiently small. The specific volume and temperature possess time-independent upper and lower bounds. Moreover, for each admissible viscosity exponent, the solution converges uniformly in space to the equilibrium state as time tends to infinity.

We determine the minimax dimension dependence of noisy zeroth-order optimization for globally strongly convex functions with higher-order smoothness in multilinear operator norm. Each query returns one function value with fresh independent Gaussian noise. For dimension $d$, query budget $T$, and any fixed finite smoothness order $\beta>2$, the minimax expected objective error satisfies $\mathcal{E}_{\beta}(T,d)=\Theta_{\beta}\left(\min\left\{1,(d^2/T)^{(\beta-1)/\beta}\right\}\right)$, with dimension-independent strong-convexity, gradient-smoothness, higher-order-smoothness, noise, and minimizer-radius bounds. Thus, for sufficiently small target error $\varepsilon$, the necessary and sufficient number of noisy values is $\Theta_{\beta}(d^2\varepsilon^{-\beta/(\beta-1)})$, with no logarithmic loss. The new lower bound matches the dimension and budget dependence of the upper bound of Akhavan et al. (2024), and holds for arbitrary randomized adaptive algorithms with unrestricted query points. Its key step localizes a normalized hypercube with one radial cutoff: a query can emphasize one hidden coordinate, but the aggregate information about all coordinates remains small. A self-contained multiscale spherical estimator attains the matching upper bound by cancelling lower-order bias terms. Constants in the rates depend only on the fixed smoothness order under the stated normalization.

We prove a joint stochastic zeroth-order lower bound for Goldstein stationarity on a Euclidean query ball, even when the ball is guaranteed to contain a stationary point. In dimension $d$, let $f=\mathbb{E}[F(\cdot;\xi)]$, assume $\mathbb{E}[\operatorname{Lip}(F(\cdot;\xi))^2]\le L_0^2$, and bound the initial objective gap over the ball by $\Delta$. For neighborhood radius $\delta>0$ and residual tolerance $\varepsilon>0$, our smooth hard family requires $\Omega(dL_0^2\Delta/(\delta\varepsilon^3))$ scalar evaluations for success probability $1/2$, against randomized adaptive algorithms that may retain and repeatedly query each sampled function. The result holds for $\varepsilon\le cL_0$, $\Delta\ge C\delta\varepsilon$, and $d\ge C[1+\log(2+\Delta L_0^2/(\delta\varepsilon^3))]$, on a constructed ball of radius $\Theta(\Delta/\varepsilon)$. A sequence of localized regions forces repeated direction estimation, and an adaptive Gaussian posterior argument controls sample reuse. The result establishes a joint dimension and accuracy obstruction on solvable bounded-domain instances. The gap is local to the query ball; a matching minimax characterization at a common radius and the corresponding unrestricted global-gap lower bound remain open here.

We prove that every $p$-absolutely Cesàro bounded operator with constant $K>1$ is $q$-absolutely Cesàro bounded for $1\leq q<pK/(K-1)$. This range is optimal, even for positive weighted shifts. On Banach spaces of type $s\in (1,2]$, we prove that every absolutely Cesàro bounded operator is $q$-absolutely Cesàro bounded for every $1\leq q<s$, with a logarithmic estimate at $q=s$. We also prove the polynomial growth bound $\| T^n \| =O(n^{1/s-\varepsilon})$ for some $\varepsilon>0$. On $L^p$-spaces, $1<p<\infty$, we extend the implication from absolute Cesàro boundedness to $p$-absolute Cesàro boundedness from positive to individually eventually positive operators. Finally, for every $1\leq p<\infty$, we construct a $p$-absolutely Cesàro bounded operator $T$ on $\ell^1$ such that $\sigma(T)\subseteq\mathbb D\cup\{1\}$ but $\limsup_{n\to\infty}\|T^n(I-T)\|=\infty$. We also observe that the construction yields a single operator which is $p$-absolutely Cesàro bounded for every $1\leq p<\infty$.

We associate to finite block size data \[ \bm n=(n_d)_{d\ge1},\qquad n_1>0, \] a positive definite cyclotomic form and a Householder reflection representation. Writing \[ N=\operatorname{lcm}\{d:n_d\ne0\},\qquad \wt=\sum_d d^2n_d, \] the form on the cyclic group $C_N$ is \[ B_{\bm n}(g,h)=\frac1{\wt}\sum_{(g^{-1}h)^d=1}d^2n_d. \] Its labelled Fourier spectrum reconstructs the complete multiplicity sequence by Möbius inversion. If the $n_d$ are the Wedderburn multiplicities of a split semisimple algebra $S=\prod_d M_d(k)^{n_d}$, the divisibility widths are dimensions of characteristic sums of Wedderburn blocks, and the form together with $\dim_kS$ recovers the isomorphism type of $S$. The main theorem is an arithmetic rigidity result. For $N\ge2$, the reflection group is finite if and only if it preserves a full rank lattice, equivalently \[ \supp(\bm n)=\{1,N\},\qquad n_1=N^2n_N. \] In this case $B_{\bm n}=\frac12(I+J)$ and the group is $W(A_N)\cong S_{N+1}$. In semisimple algebra terms, finiteness is equivalent to \[ S\cong k^{N^2m}\times M_N(k)^m \] for some $m\ge1$. For $N=2$, every nonfinite case is a dense copy of $D_\infty$ in the corresponding orthogonal group. For contraction algebras of threefold flops, Toda's width formulas identify the total scale and primitive Fourier mode with $\dim A_{\con}$ and $\dim A_{\con}^{\ab}$. This yields infinitude for all irreducible flops of length at least three and, in length two, \[ \Gamma_f\text{ finite}\quad\Longleftrightarrow\quad \dim A_{\con}=2\dim A_{\con}^{\ab}. \] The final part gives a complementary BPS/Riemann–Hilbert realization of the same reflection matrices.

We establish sharp high-probability statistical rates for nonsmooth stochastic convex optimization under heavy-tailed noise, closing the confidence-dependence gap in both general convex and strongly convex settings. We consider composite objectives with a Lipschitz convex loss and a known proximal regularizer, accessed through fresh unbiased stochastic subgradients. The centered noise has conditional norm and directional $p$th moments bounded by $L^p$ and $s^p$, respectively, where $1<p\le2$ and $0<s\le L$. For $T$ oracle calls and failure probability $0<\delta<1/10$, set $q=1-1/p$, $\ell=\log(3/\delta)$, and $A=s(L^2/s^2)^q$. With initial distance bound $D$, the minimax objective error is $\Theta(D(A+s\ell^q)/T^q)$ in the nonsaturated statistical regime. For a $\mu$-strongly convex regularizer, the corresponding quadratic-regime rate is $\Theta((A^2+s^2\ell^{2q})/(\mu T^{2q}))$. These rates require domination of the deterministic and finite-variance terms; matching lower bounds require $d\ge\lceil L^2/s^2\rceil$, while the upper bounds hold in every finite dimension. The new upper bounds attain known statistical lower-bound orders, replacing the previous confidence coefficient $s^{1/p}L^{1-1/p}$ by $s$, and its square by $s^2$ under strong convexity. The key is an online inequality that separates directional fluctuations from the total second moment created by rejection sampling. Localization preserves this separation under strong convexity without additional horizon logarithms. The results characterize oracle complexity with unrestricted internal computation.

We determine the dimensions of all derivative loci and all higher multiplicity loci of an elementary symmetric polynomial over an algebraically closed field of positive characteristic. For the degree-d polynomial in n variables, set Q = p^{v_p(n-d+1)}. The common zero locus of its order-q derivatives has dimension d - q - 1 + 1_{Q>d-q}, whereas the locus of zeros of multiplicity at least s has dimension d - s + 1_{Q>d}. The order-two case settles the dimension question in a conjecture of Orzel, with a necessary correction when d is a power of p. We classify all components attaining the larger dimension and prove that they are generically smooth. Along the coordinate components in the smaller-dimension cases, we determine the local rings and obtain generic multiplicity exactly Q. Finally, we identify the full translation stabilizer scheme and derive a Frobenius normal form. These results distinguish dimension jumps from infinitesimal thickening and give a uniform description of their dependence on the characteristic.

The para-Krawtchouk polynomials arose in the search for spin chains with perfect state transfer; the para-Racah, $q$-para-Racah and para-Bannai–Ito polynomials followed, obtained through singular truncations of families of the Askey scheme and used in turn to design spin chains. All are orthogonal on bi-lattices; the prefix goes back to the para-Krawtchouk case, whose spectrum is that of the parabose oscillator. The representations that these polynomials provide of the corresponding algebras — Hahn, Racah, $q$-Racah, Bannai–Ito and complementary Bannai–Ito — are shown to be reducible. Each decomposes into a direct sum of two irreducible modules of the same algebra, one attached to each of the two interlaced sub-lattices, and the parameters of the submodules are given. The para polynomials are thereby expressed in terms of the classical polynomials that the submodules carry, and the pairing $\lambda_n=\lambda_{N-n}$ of their eigenvalues is accounted for; the three features of the bi-lattice setting on which the reduction rests are identified. The properties of the para families are then presented in a unified way in the light of this decomposition, together with the constructions through which these families arise and the settings in which they have been put to use.

Stochastic quantization formulates $d$-dimensional Euclidean QFT as the equilibrium limit of a $(d+1)$-dimensional Langevin system. Using the supersymmetric formulation of stochastic systems, we apply techniques from Morse theory to study Langevin dynamics. We interpret stochastic quantization as the absolutization of a relative QFT by a bulk cohomological topological Equilibrium TFT ("EqmTFT"), and relate its enriched Neumann boundary data to the $d$-dimensional BV/BRST complex. Analytic continuation reveals connections to the A-model and non-relativistic and Carrollian limits.

Smoothed Particle Hydrodynamics (SPH) is well suited to a range of problems, particularly those involving large deformations in fluid dynamics. In recent years, in addition to the advancement of SPH formulations, the development of differentiable solvers has also progressed. However, unified frameworks that flexibly accommodate diverse numerical schemes, including advanced and implicit methods, while supporting continuous extension and updating remain limited. In this study, a high-accuracy and composable differentiable SPH solver, CraftSPH, has been developed. In CraftSPH, major computational operations are implemented as independent modules, which can be combined according to the intended purpose of constructing a solver. This design enables different computational schemes to be intuitively constructed from combinations of common components and allows newly developed high-accuracy discretization methods to be flexibly incorporated and extended. Furthermore, each module supports automatic differentiation, enabling applications to physical-parameter estimation and to optimization that combines SPH solvers with deep learning models. Accordingly, CraftSPH provides an SPH computational framework in which high-accuracy discretization, explicit and implicit computations, and automatic differentiation can be handled in a unified and extensible manner. To demonstrate the applicability of CraftSPH to a wide range of problems, forward analyses are conducted for Poiseuille flow, two- and three-dimensional dam-break problems, and a rising-bubble problem. In addition, the performance of automatic differentiation is evaluated through inverse problems involving parameter estimation for the Taylor-Green vortex and lid-driven cavity flow.

The representation number of a graph is the least positive integer $k$ for which its vertices can be arranged in a word, each occurring $k$ times, so that two distinct letters alternate precisely when the corresponding vertices are adjacent. We prove that every bipartite graph on $N\ge9$ vertices has representation number at most $\lceil N/4\rceil$. Together with the known representation number of crown graphs, this settles the conjecture that crowns maximise the representation number among bipartite graphs of the same order. The proof develops a construction of Mozhui and Krishna by reducing the choice of a representing word to an ordering problem for neighbourhoods. We characterise the obstructions to this ordering and use probability estimates to exclude them for all sufficiently large balanced bipartitions. Two finite assertions complete the argument, each established by a checked Boolean unsatisfiability certificate. A refinement of the ordering argument treats the remaining odd part sizes directly. The main theorem and the crown-extremality corollary, including the finite certificate arguments, have also been formalised and verified in Lean 4.

We establish matching polynomial query bounds for low-rank approximation from exact matrix–vector products. Given an unknown matrix $A\in\mathbb{R}^{m\times n}$, at each step a randomized algorithm chooses either $v\in\mathbb{R}^n$ and receives $Av$, or $u\in\mathbb{R}^m$ and receives $A^\top u$. The choice may depend measurably on all previous queries and replies and on the algorithm's private randomness; each vector product costs one query. The output is a rank-$k$ right projector with Schatten-$p$ residual at most $1+\varepsilon$ times optimal. Write $N=\min\{m,n\}$ and let $Q_p^*$ denote the worst-case query budget for success probability $2/3$ on every input. For every $1\le k<N$ and sufficiently small $\varepsilon$, our lower bounds, combined with existing Krylov upper bounds, give $Q_p^*=\widetilde{\Theta}\!\left(\min\{N,k\min\{p^{1/6}\varepsilon^{-1/3},\varepsilon^{-1/2}\}\}\right)$ $(2\le p<\infty)$, $Q_\infty^*=\widetilde{\Theta}\!\left(\min\{N,k\varepsilon^{-1/2}\}\right)$. These bounds have universal constants and allow $p$ to vary with the problem parameters, identifying the transition at $p\varepsilon\asymp1$. A complementary result for each fixed $1\le p<2$ gives $\widetilde{\Theta}_p(\min\{N,k\varepsilon^{-1/3}\})$, with constants and an accuracy threshold that may depend on $p$. Together, the results recover this fixed-norm rate for every fixed finite $p$, supplying the multiplicative rank dependence missing from previous lower bounds. Tildes suppress logarithmic factors. The proof extends adaptive Wishart deferred decisions to a rectangular factor with a $k$-dimensional nullspace. Posterior overlap gives a short fixed-norm argument, while persistence of small compression eigenvalues controls growing $p$ and the spectral endpoint. Exact range recovery handles target costs of order $k$; the Wishart family covers the remaining regimes.

For an $r$-uniform hypergraph $H$, let $\nu(H)$ be the maximum number of edges no two of which share $r-1$ vertices, and $\tau(H)$ the minimum number of $(r-1)$-sets such that every edge contains one of them. Aharoni and Zerbib conjectured that $\tau(H)\le\lceil\frac{r+1}{2}\rceil\,\nu(H)$, which for $r=3$ generalizes Tuza's conjecture on triangles. We prove that the fractional relaxation $\tau^*(H)$ of $\tau(H)$ satisfies $\tau^*(H)\le\frac{r+1}{2}\,\nu(H)$. This constant is best possible for every $r$, and for $r\ge4$ it improves on the previous bound of roughly $3r/4$.

Given a point set $[n]$ and a collection $B$ of $k$-subsets of $[n]$, we create a monomial ideal whose Betti numbers completely determine whether the pair $([n],B)$ is a Steiner system $S(t,k,n)$. In particular, our construction characterizes Steiner triple systems and Steiner quadruple systems solely on the basis of Betti numbers of monomial ideals. Moreover, these monomial ideals allow us to reformulate the renowned prime power conjecture as follows: If a monomial ideal $M$ of $S = K[x_1, \dots, x_{q^2+q+1}]$ generated by squarefree monomials of degree $q+1$ has Betti numbers $\mathrm{b}_s(S/M) = {{q^2+q+1} \choose s} \text{ for all } s=0,\ldots,q$ then $q$ is a prime power.

Sudakov minoration asks whether a large family of separated random linear forms must have a large expected maximum. We present a proof candidate for this principle for all unconditional log-concave vectors. The argument starts with a classical reduction to sparse coordinate supports. It then selects one feasible threshold witness for each label, removes a common coordinate core, and controls the remaining overlaps. The main analytic step is an exponential-moment estimate for a softened witness payoff. We obtain it by combining a one-dimensional clipping inequality with triangular transport, using log-concavity of the source and sign symmetry of the target. A Bernoulli comparison restores the signs at the end. The exposition includes historical context, a guide to the proof, all intermediate arguments, and explicit choices of constants. Two consequences concern covering numbers in the moment metric and concentration of bounded functions of the magnitudes.

This paper considers the generic behaviour of sheaves supported on Schubert varieties in finite flag varieties. Our main results are that, for any fixed prime $p$, as the rank of the associated algebraic group grows, almost every intersection cohomology sheaf admits $p$-torsion in its stalks or costalks, and almost every indecomposable parity sheaf is not perverse. For certain families of partial flag varieties we also determine explicit asymptotics for this growth.

The polynomial Bohnenblust–Hille inequality controls the coefficient $\ell_{2m/(m+1)}$-norm of an $m$-homogeneous polynomial by its supremum norm, with a constant independent of the dimension. We study the associated Bohnenblust–Hille ratio from a probabilistic point of view, by placing normalized surface measure on the Euclidean coefficient sphere. Our results reveal a sharp contrast between the extremal behavior governing the classical Bohnenblust–Hille constants and the typical scale seen in coefficient directions. For arbitrary prescribed monomial supports, the ratio is eventually at most $1$ almost surely, and it tends to zero in spherical measure exactly when the number of monomials tends to infinity. On the full complex polynomial spaces we determine its typical asymptotic scale uniformly in the dimension; in the critical regime $n_m/m\to1$ this gives $2\sqrt2/\sqrt{m\log m}$, in contrast with the nonvanishing extremal scale. For real polynomials, the corresponding ratio tends to zero in spherical measure exactly when the number of variables tends to infinity.

Networking defenses against traffic analysis fill idle slots with cover traffic, yet the physical layer carrying it still hands upper layers the erasure-only model of a silent-idle link, leaving the cover content unexploited. We shape that content: the idle-slot waveform is drawn from the codebook complement under a design distribution, the litter distribution, chosen to defeat a public-design passive observer while the upper layer keeps its block z-channel model. The maximum a posteriori (MAP) receiver reduces to a single threshold on the log-likelihood ratio between the best codeword and the prior-weighted litter aggregate; its errors are silent corruptions the upper layer cannot detect and erasures that retransmission absorbs. Receiver design is thus one-dimensional, minimizing erasure under a cap on silent corruption, and the joint design a Stackelberg game against a Kullback-Leibler (KL) indistinguishability metric, solved by an alternating convex relaxation and corroborated by an independent reinforcement-learning policy. On a rate-$3/4$ $(16,12)$ parity-check (LDPC) code against an observer who knows her own channel but whose channel the transmitter knows only statistically, shaped litter cuts the observer's Stein detection exponent by $41$ to $48\%$ relative to uniform cover, flat over a $10$~dB span and in the activity rate. Below the point where the legitimate link turns reliable, no cover distribution admits an operating point; at that point shaping costs about one percent of throughput, and nothing measurable $2$~dB above it.

We prove the Fraser–Sherman-Bennett conjecture on the quasi-equivalence of positroid cluster structures arising from admissibly relabeled plabic graphs. We construct a tilting module from the boundary necklace and obtain a Frobenius categorification of the relabeled cluster structure. By identifying localized characters of rank-one modules with Plücker coordinates, we match the stable images of the mutable face modules with the summands of a reachable cluster-tilting object in the stable category. The induced derived equivalence yields a unimodular transformation of relative index lattices intertwining the full extended exchange matrices. These comparisons show that the identity map on the coordinate ring is a quasi-cluster isomorphism between the two cluster structures.

The para-Bannai–Ito polynomials are orthogonal on a Bannai–Ito bi-lattice that is two-sided — it extends on both sides of a central value $c_0$ — and depend, besides the two grid parameters, on an isospectral deformation parameter. Their Jacobi matrix $J$ is invariant under the reversal of the sites at one value of that parameter only, the persymmetric point, and we determine its continuum limit there. Because the grid is two-sided, the natural object is the shifted operator $Q=J-c_0$, and the limit is taken about the centre of the band: $Q$ contracts to a first–order Dunkl operator, a supercharge, and $H=Q^2$ to the generalized Pöschl–Teller (Scarf I) Hamiltonian, with little $-1$ Jacobi eigenfunctions — supersymmetric quantum mechanics with reflections, the reflection entering the square root of the Hamiltonian. The second grid parameter adds a reflection term to the Hamiltonian itself, and the bi-lattice is, up to a shift, the spectrum of the limiting supercharge. The convergence of the eigenvectors is proved directly on the polynomials. On this grid the persymmetric model transfers states perfectly, under an arithmetic condition on the parameters, but never revives fractionally, revival being the work of the deformation; both properties pass to the limit. These results are for an even number of sites; for an odd number of sites the limit is instead the matrix supersymmetric pair of Pöschl–Teller Hamiltonians, with the reflection in the Hamiltonian as a Dunkl potential. The bispectral pair of the discrete model goes over to the Dunkl realization of the anticommutator algebra of the little $-1$ Jacobi polynomials.

We consider generalised Casimir operators associated with irreducible representations for finite-dimensional, semisimple, complex Lie algebras. Using Klimyk's formula and Okubo's formula, we derive a rational ordinary generating function for the Casimir operator eigenvalues for simple Lie algebras which does not rely on knowledge of the Weyl group. We also directly reproduce generating function expressions for the classical Lie algebras originally due to Perelomov and Popov.

We establish sharp temporal lower bounds for the full spatial $L^2$-norm of energy solutions to $\partial_tu-\Delta u=vu$ on $\mathbb{R}^n$, with $n\geq3$. For global solutions, nonvanishing at a single time implies the lower bound $ce^{-Ct}$ when $v$ is small in the scale-invariant space $L^\infty_tL^{n/2}_x$, and $ce^{-Ct^2}$ throughout the subcritical range $v\in L^\infty_tL^p_x$, with $p>n/2$, without smallness. The linear time exponent is optimal already for the free heat equation, while we show that the quadratic exponent is optimal for bounded complex potentials on $\mathbb{R}^3$. No additional regularity of the potential is required. The proof uses a convexified mixed Carleman estimate whose norms are designed together with the final time truncation. Integration over spatial centers removes the heat-kernel weight and yields quantitative comparisons of the full solution norms. To prove quadratic sharpness in the Euclidean energy class, we construct a time-dependent function $g$ that localizes Meshkov's periodic parabolic example while keeping the induced potential bounded across the zeros of the periodic profile.

Machine-learning policies have attracted increasing interest for ride-hailing fleet control in recent years. Reinforcement learning, in particular, requires a structured simulation environment that specifies observations, actions, rewards, and decision epochs for training and evaluation. For electric fleets, this environment must also capture the interaction among stochastic demand, vehicle operations, and capacitated charging infrastructure. We present OpenHail, an open-source Gymnasium environment for joint control of electric ride-hailing fleets. Its fixed-size observation–action interface exposes request assignment, repositioning, and charging to a single policy. The event-driven simulator represents requests with pickup deadlines, vehicle job queues, battery dynamics, and finite-capacity charging facilities with first-in–first-out queues. A configurable decision-epoch mechanism separates internal simulator events from policy interactions, supporting event-driven, periodic, hybrid, and policy-requested control within the same operational model. The software provides seeded instances, feasible-action utilities, evaluation tools, operational metrics, and baseline policies. The source code is available at https://github.com/tommaso-schettini/openhail.

The para–Racah Hamiltonian is the Jacobi matrix $J$ of the para–Racah polynomials, taken at the persymmetric value of their isospectral deformation parameter: a mirror–symmetric tridiagonal matrix whose spectrum is a quadratic bi–lattice, the squares $(s+a)^2$ and $(s+c)^2$ of two interlaced arithmetic progressions of unit step, and which carries out fractional revival when the grid parameters $a$ and $c$ satisfy a Diophantine condition. As the number of sites grows, $J$ contracts to the trigonometric Pöschl–Teller Hamiltonian on a finite interval. We establish this twice: on the recurrence relation, and on the polynomials themselves, for which it is the confluence Racah~$\to$~Jacobi. Both grid parameters survive: $c-a$ becomes a scalar centrifugal term at the interior point fixed by the mirror — the exact analogue of the para–Krawtchouk origin — while $a+c-1$ sets two confining singular walls at the ends, which turn the spectrum into the quadratic pair $(t+a)^2,(t+c)^2$ with Jacobi eigenfunctions $P_t^{(a+c-1,\,\mp(c-a))}$, the two sublattices converging to two families that differ only by the sign of the second Jacobi parameter. The mirror symmetry of $J$ becomes the reflection about the midpoint; it grades the two families and carries the fractional revival to the limit. Algebraically, the bispectral pair of the discrete model, which generates the Racah algebra $\mathfrak{r}_1$, becomes a pair of generators of the Jacobi algebra $\mathfrak{j}_1$ — one rung down the Askey–Wilson hierarchy — the dynamical algebra of the Pöschl–Teller Hamiltonian.