Papers

Let $q$ tend to infinity through odd primes, let $T=T(q)$, and put $\ell=\log(qT/2\pi)$. Suppose that $\ell^s=o(T)$ for some fixed $s>1$ and that \[ \lambda_{\rm bw}:=\min\left\{1, \liminf_{\substack{q\to\infty\\q\ \mathrm{prime}}} \frac{\log(q-1)}{\ell}\right\}>0. \] For the unweighted family of the $q-2$ nonprincipal characters modulo $q$, we prove unconditional lower bounds, relative to the total zero multiplicity in $(T,2T]$, of $2-1/c_{\lambda_{\rm bw}}^*$ for both simple and distinct critical-line zeros and of $\tfrac12(3-1/c_{\lambda_{\rm bw}}^*)$ for all distinct zeros, where \[ c_\lambda^*=\frac{\sqrt2\tan(\lambda/\sqrt2)} {1+(\lambda/\sqrt2)\tan(\lambda/\sqrt2)}. \] When $\log T=o(\log q)$, this gives respectively $0.6725007036\ldots$ and $0.8362503518\ldots$; in particular it covers every fixed $T=(\log q)^A$ with $A>1$. As the growing-height companion to the authors' public mesoscopic shrinking-height theorem, this article is organized around the different localization technology required when $T$ grows. An abstract theorem converts complex-strip Gevrey decay, sampling density, and a local multiset count into stretched-exponential localization in trace and nuclear norm. In the Dirichlet specialization the strip growth is $X^{1/4}$, where $X=\exp(\lambda\ell)$, and a buffer $D_0=(K\ell)^s$ absorbs the remote-zero contribution uniformly in the character. Quantitative finite-sampling end-effect bounds, a finite explicit-formula matrix, character-averaged first and second traces, hyperbolic functional-equation blocks, and a rank–trace inequality then yield the three zero statistics. The shared finite-inertia mechanism and the Montgomery–Taylor constant are not claimed as new. No form of GRH is assumed.

Optimal execution (OE) in multi-asset settings involves complex interactions across assets, particularly through shared capital constraints during portfolio rebalancing. While existing models capture cross-impact and portfolio-level dynamics, they largely overlook the role of explicit cash constraints along the execution trajectory. As a result, the feasibility of execution strategies under limited capital remains poorly understood. In this paper, we extend the classical Almgren-Chriss framework to incorporate intertemporal constraints on expected cash consumption, requiring that the expected cumulative cash spent does not exceed a prescribed budget at every trading period. We show that the resulting multi-asset OE problem can be equivalently formulated as a quadratically constrained quadratic program (QCQP), and further establish that it admits a convex representation under mild conditions. This provides a tractable framework for analyzing execution strategies under dynamic capital constraints. Through controlled synthetic experiments, we show that the proposed cash constraints qualitatively alter OE schedules toward cash-feasible sell-first executions as the constraints become tighter. Furthermore, evaluations in an out-of-sample agent-based market simulator demonstrate that our method substantially reduces peak cash drawdown while maintaining implementation shortfall comparable to existing execution strategies. Our results highlight the importance of explicitly modeling financial feasibility in multi-asset execution and provide a foundation for bridging theoretical OE models with practical capital constraints.

Let $N(H)$ be the edge–vertex incidence matrix of a finite linear $s$-uniform hypergraph whose line graph is a cograph, and define \[ R_s(H)=(s+1)\operatorname{rank}_{\mathbb R}N(H)-s|E(H)|. \] The inequality $R_s(H)\ge0$ holds, with equality characterized by affine planes of order $s$ on every row-dependent component. Under the corresponding finite-net completion hypothesis, connected systems with bounded $R_s$ admit an explicit classification. If $F$ is connected, $R_s(F)=d\le s-1$, the completion hypothesis holds through deficiency $d$, and $s>(d-1)^2$, then either $N(F)$ has full row rank and $|E(F)|=d$, or $F$ lies in a unique affine plane of order $s$. In the latter case, for some $0\le h\le d$, the system contains every line in $s+1-h$ parallel classes and exactly $d-h$ additional lines from the remaining $h$ classes. Conversely, each such system has defect $d$. Bruck's theorem gives the classification for every fixed $d$ and all sufficiently large $s$, while Metsch's completion theorem covers $d=o(s^{1/3})$. As a global consequence, if the total defect is at most $D$ in the completion range, deleting at most $D$ edges leaves a vertex-disjoint union of finite nets and affine planes. The associated column defect and row leverage are determined explicitly as well.

We study morphisms from symmetric powers of exterior powers to determinant-twisted endomorphism representations of general linear groups. At the minimal positive determinant twist, we prove vanishing over every field of characteristic different from two: if the exterior degree r >= 3 is odd, the symmetric degree satisfies d >= 3, and the underlying space has dimension rd, then the corresponding equivariant Hom space is zero. The proof uses a block-exchange sign and a universal root-subgroup identity, so it also applies in small odd characteristics without semisimplicity. We then determine the second-twist spaces for trivectors in characteristic zero. In dimension 3m and degree 2m, they are scalar and one-dimensional for m = 2, and zero for m >= 3. Plethystic conjugation reduces the latter vanishing to an elementary weight-support bound for exterior powers of the ten-dimensional space of ternary cubics. Finally, over fields of characteristic zero or odd characteristic, an exact computer-assisted classification in dimension nine gives a one-dimensional scalar Hom space in characteristic five and zero in characteristic zero and in every odd characteristic other than five.

For the intersection of two disks meeting at angle $2\alpha$, let $C(\alpha)$ be the least constant in the associated spectral-set inequality. We give a self-contained M"obius reduction to the corresponding numerical-range problem on a sector and determine the sharp constant for affine square-zero operators $B=\lambda I+N$, $N^2=0$: $C_{\mathrm{sq0}}(\alpha)=\pi\sin\alpha/(2\alpha)$. A $2\times2$ matrix attains equality and yields an explicit lens lower-bound certificate. At the right angle, we prove the conjectural $\sqrt2$ bound in arbitrary dimension for the full palindromic quadratic family, and give exact rational certificates for several larger parameter families, including a complex post-automorphism disk, the complete imaginary diameter, boundary-reaching phase arcs, and symmetric and asymmetric three-node admissible-kernel problems. We also obtain a strict central bound $|w^2|\leq\kappa_0<\sqrt2$, a two-small-zero extension, an exact two-moment criterion for the remaining boundary layer, and a verified angle-dependent envelope. Every computer-assisted assertion has an exact rational verifier. These results are dimension-free but do not determine the unrestricted fixed-lens constant.

We evaluate the series $\sum_{n\ge1} z^n\big/\!\big(n^2\binom{4n}{n}\big)$, equal to $-\tfrac{z}{4}\,{}_5F_4\!\left(1,1,1,\tfrac43,\tfrac53;\tfrac54,\tfrac32,\tfrac74,2;\tfrac{27z}{256}\right)$, in closed form at an infinite family of algebraic points indexed by a rational angle $\theta=j\pi/N$. Each value equals $c\,\pi^2$ plus a universal rational quadratic form in three logarithms, with $c=-\tfrac13\left(1-\tfrac{2j}{N}\right)^2$. This is the quartic-base case reached but not evaluated by D'Aurizio and Di Trani. The proof is self-contained: an exact integer factor relating two weights, followed by Landen's identity, reduces the integral to a sum of squared logarithms.

Ocean eddies are highly dynamic structures marked by frequent splitting and merging events. They exhibit complex spatio-temporal clustering that traditional Poisson models fail to capture. In this paper, a novel spatio-temporal Hawkes process is introduced to model self-exciting eddy fields on the basis of high-frequency ocean flow data. We formulate a triggering kernel that couples the self-excitation intensity with spatial deformation caused by the strain rate magnitude. To characterize the asymptotic behavior of the eddy field, we derive a Volterra integral equation that governs the expected eddy intensity over time and space. We develop an Expectation-Maximization (EM) algorithm that treats the unobserved parent-child relationships as latent branching structures for parameter estimation. We further extend this framework to accommodate a time-varying, non-homogeneous background intensity, modifying both the EM updates and the analytical Volterra solution accordingly. Finally, we propose a two-stage simulation framework utilizing a cluster representation algorithm. The simulated empirical paths of eddy formation are compared together with numerical solution of the Volterra equation for mean rate as validation of the Hawkes model.

We introduce a bilingual domain-specific language (DSL) for modelling compute kernels within a generic solver for hyperbolic partial differential equations (PDEs). Users express PDE terms, i.e.~the underlying physics, in a familiar native language such as C or C++, while the numerical scheme is specified in a Python-embedded DSL, DSLHyPE. DSLHyPE's compiler lowers the Python description to MLIR and introduces a translation pass that integrates it with native code likewise mapped to MLIR. Our approach keeps the numerical representation and the physics implementation separate for as long as possible, while delegating optimization to the compiler through existing MLIR optimization passes. This separation of concerns benefits researchers developing numerical schemes on top of existing PDE implementations or with applications involving nonlinear systems whose PDE terms must solve PDEs themselves. We demonstrate the feasibility of the approach using a gravitational-wave solver and a matter-evolution solver on x86 processors and H200 GPUs.

Statistically Meaningful Geometry (SMG) is a differential-geometric and information-theoretic framework that lifts over-parameterized models into infinite-dimensional non-parametric Orlicz statistical fiber bundles with an Ehresmann connection, decoupling unobservable vertical gauge noise from horizontal statistically verifiable directions. We prove the First Edge Theorem: Amari's information geometry (IG) and conventional statistics (CS) are not autonomous statistical universes but degenerate boundary layers of the larger gauge-active SMG space. Taking the structural identifiability radius $R\to\infty$ breaks gauge symmetry, collapses vertical fibers, and forces the total space onto the IG manifold; subsequent local asymptotic normality as $N\to\infty$ flattens the remaining curvature, yielding \[ \mathrm{SMG}\xrightarrow{\;R\to\infty\;}\mathrm{IG}\xrightarrow{\;N\to\infty\;}\mathrm{CS}. \] The same fiber-bundle machinery transforms model non-identifiability from a singular collapse into a structured gauge space. Applications include resolving the deep-learning generalization paradox, constructing gauge-invariant gradient descent and holonomy-matched preference alignment for generative AI, and solving weak identification in structural econometrics via intrinsic horizontal geodesic search.

In this note, we revisit one of Erdélyi's product formulas for the univariate confluent hypergeometric function ${}_{1}F_{1}$ and extend it to the multivariate confluent hypergeometric functions $\Phi_2^{(k)}$, $\Psi_2^{(k)}$, as well as to a certain matrix-variate confluent hypergeometric function. A useful connection related to fractional calculus is also given.

This work develops a general-order fractional differentiation operator on monomials from the discrete half-order coefficient structure established in Part I. Starting from the one-step transfer law of the integer coefficient family, the discrete coefficient relation is continued to a real argument and a canonical Gamma-functional form is selected under the stated normalization and regularity requirements. The operator order is then extended beyond one half by dividing the first derivative into equal operator steps, leading from orders 1/m and k/m to general rational and real orders. The resulting coefficient is independently verified through the addition law for operator orders on the monomial system, subject to the requirement that all intermediate expressions in the composition are defined. On the corresponding parameter domain, the construction yields the standard Gamma-ratio monomial formula for fractional differentiation. For nonnegative integer orders it reduces to ordinary repeated differentiation, while for positive non-integer orders it agrees at the monomial level with the left-sided Riemann-Liouville formula with lower limit 0. Its relation to the Caputo operator is also discussed, including the distinction for constants and low-degree polynomial terms. The aim is not to introduce a new classical fractional derivative, but to provide a constructive algebraic-operator route from a discrete coefficient law to the general fractional-order monomial formula, clarifying the roles of functional continuation, Gamma normalization, and operator composition.

Let $F_k(n)$ be the number of unordered representations \[ n=p_1^k+p_2^k+\cdots +p_k^k \] by primes, with repetitions allowed. Erdős stated that $\limsup_{n\to\infty} F_3(n)=\infty$, but his proof appears not to have been published. A complete unconditional proof is given. The principal input is the classical Hecke equidistribution theorem for the CM Fermat cubic; the rest of the argument uses standard estimates for primes in arithmetic progressions and elementary counting. The argument used for \(k=3\) does not extend to the case \(k=4\). Nevertheless, by applying the Green–Tao–Ziegler theorem to the linear forms arising from an admissible binary quartic identity, \(\limsup_{n\to\infty}F_4(n)\ge2\) is shown.

Let $0\leq\mu_1\leq\cdots\leq\mu_n$ and let $\lambda>\sum_{i=1}^n\mu_i$. Let $X_1,...,X_n$ be independent nonnegative random variables satisfying $\mathbb{E}X_i=\mu_i$, and write $D_i := \lambda-\sum_{k=1}^{i-1}\mu_k$ for $1\leq i\leq n$. We prove that $$ \inf_{X_1,...,X_n} \mathbb{P}\left( \sum_{i=1}^nX_i<\lambda \right) = \min_{1\leq i\leq n} \prod_{j=i}^n \left( 1-\frac{\mu_j}{D_i} \right). $$ The bound is sharp and is attained. This proves Samuels' conjecture. Feige's conjecture is thereby resolved, since it follows immediately from the equal-means case. The proof is self-contained.

We consider the quickest change detection from 1-bit quantized observations, where the post-change distribution is a parametric model with unknown parameters and the quantization thresholds are jointly chosen with the detection statistic. We propose an Adaptive-Quantization-Threshold CUSUM (AQuTeCUSUM) algorithm, which estimates the post-change parameter online and adaptively selects the quantization threshold to maximize the induced Kullback-Leibler divergence. Under suitable regularity conditions, we characterize the average run length and worst-case average detection delay of the AQuTe-CUSUM procedure, and show that it is asymptotically optimal in first-order as the average run length goes to infinity. Finally, we assess the performance of AQuTe-CUSUM for two distributions, namely the Gaussian and Poisson.

We consider plane-wave discontinuous Galerkin (PWDG) approximations of the Helmholtz equation with adaptive local propagation directions. The directions are chosen by minimizing a weighted residual on the mesh skeleton. We study two formulations: in Part A the PWDG system is solved for the coefficients at fixed directions, while in Part B the same residual is minimized jointly over coefficients and directions, with the coefficients eliminated by variable projection. Complex angles unify propagating and evanescent Trefftz waves. The discrete system is normalized in the Trefftz-DG norm, and a local Cauchy-trace Gramian is used to remove numerically dependent directions. On straight edges the residual integrals are exact. For all-Dirichlet problems, the residual equals the squared DG error and provides local adaptive indicators. We prove local quadratic growth of the reduced direction functional near an identifiable zero-residual solution of fixed rank. Numerically, an exact circular DtN test shows that the dominant phase direction of a Hankel wave can be recovered accurately even when a small plane-wave fan gives a larger field error. We also separate the effects of the coefficient basis, trace cutoff, and trace-spectrum arithmetic on the high-$p$ DtN floor. Finally, for finite sums of plane waves, residual-based ENRICH–MOVE continuation recovers all directions to roundoff for $M=1,\ldots,19$. At $M=20$ the automatic birth step enters a false basin, whereas a nearby birth again reaches roundoff. The results indicate that direction adaptation is most effective for solutions of low directional complexity.

The artificial compressibility method (ACM) relaxes the incompressibility constraint in the Navier-Stokes equations by introducing a perturbation term proportional to the time derivative of the pressure, scaled by a small positive parameter $\varepsilon$. Consequently, the ACM system inherently involves two small parameters: the fluid viscosity $\nu$ and the artificial compressibility parameter $\varepsilon$. While existing temporal analyses of ACM provide insights into its behavior, rigorous fully discrete error estimates that are robust with respect to both parameters remain a significant gap in the literature. In this paper, we propose a second-order Crank-Nicolson fully discrete ACM scheme and establish its parameter-robust optimal error estimates. To enhance stability and ensure robustness, we incorporate two distinct grad-div stabilization strategies: one facilitates the decoupling of velocity and pressure computations, while the other guarantees robustness with respect to both $\nu$ and $\varepsilon$. For spatial discretization, we employ the Scott-Vogelius finite element pair, which is crucial for the decoupling and error analysis. The resulting parameter-uniform bounds are crucial for ensuring the long-time accuracy of the scheme, circumventing the exponential dependence on the Reynolds number typically introduced by Grönwall's lemma. Numerical experiments are provided to validate the theoretical findings and demonstrate the efficiency of the proposed methods.

We study quasi-irreducibility, expansion, and their cotangent duals for random $C^1$ (local) diffeomorphisms on compact invariant sets of a Riemannian manifold. Quasi-irreducibility is defined through stationary lifts to the projective tangent bundle: every lift must realize the Furstenberg–Kifer formula for the top Lyapunov exponent. We prove that, for ergodic stationary measures, this is equivalent to the absence of equators, namely non-random invariant subbundles on which the top exponent drops. Under simplicity of the first Lyapunov exponent for every stationary measure, quasi-irreducibility is also equivalent to vertical mostly contraction, contraction on average, a vertical spectral gap, and uniqueness of stationary projective lifts. We then apply these criteria to continuity of the top Lyapunov exponent and to expansion on average. In particular, expansion is characterized by positivity of the top Lyapunov exponent on all non-random invariant subbundles; under quasi-irreducibility, it is equivalent to positivity of the top Lyapunov exponent for every stationary measure. Dual statements hold for coquasi-irreducibility, coequators, coexpansion, and continuity of the bottom Lyapunov exponent. We also describe the interplay between expansion and coexpansion and extend the formalism to Grassmannian bundles, obtaining higher-dimensional versions controlling intermediate sums of Lyapunov exponents.

Let $n$ be a natural number, let $q$ be a prime power, and let $U_n(q)$ denote the group of unitriangular $n\times n$ matrices over the finite field $\mathbb{F}_q$ with $q$ elements. Row closed and column closed subgroups $U$ of $U_n(q)$ are special pattern subgroups obtained by deleting entire rows or, respectively, entire columns (apart from the diagonal entries). The supercharacters of the André-Yan supercharacter theory of $U$ are afforded by the orbit modules arising from a monomial action of $U$ on the character group of the Lie algebra of $U$. We classify the corresponding orbits for row closed subgroups $U$, that is, we determine all orbits and identify which of the associated orbit modules are isomorphic and which afford orthogonal supercharacters. The classification for column closed subgroups follows from the mirror map, which reflects matrices across the antidiagonal.

Bayesian targets may converge under model refinement even when the exact sensitivities used by gradient-based samplers do not. We study this probability–sensitivity mismatch and its consequences for Metropolized Hamiltonian proposals. A vanishing-amplitude wiggly-energy model first gives the basic analytic obstruction: the potential perturbation tends to zero while its classical derivative is of order $r_\varepsilon/\varepsilon$. We then show that the same scaling arises naturally in a periodic elliptic inverse problem, where homogenization makes the forward map and Gaussian likelihood converge while differentiation with respect to a microscopic scale parameter retains an $O(1)$ oscillatory contribution. This provides a PDE origin for the single-scale wiggly mechanism. The main construction concerns a more demanding nested Weierstrass hierarchy, interpreted as an analytically tractable prototype for repeated corrector contributions across geometrically separated scales. There all previously resolved scales persist, adjacent classical-force increments grow geometrically like $(ab)^N$, and the limiting rough component may fail to possess a classical derivative. In this self-similar setting the matched Jackson quotient is structurally adapted to the refinement through dilation covariance and exact finite closure. Uniform negative-log-likelihood approximation yields explicit total-variation, Hellinger, and bounded quantity-of-interest bounds. Measurable kick–drift–kick maps remain exact after Metropolis correction, local field convergence propagates to fixed-length proposals and kernels, and the first classical HMC half-kick can have no fixed-step refinement limit. Numerical experiments on scale-structured inverse problems test the resulting resolution-stability mechanism across one- and two-dimensional inverse problems.

A unit-speed searcher starts at the origin of the Euclidean plane and must hit an unknown straight line whose direction and distance from the origin are both unknown. We prove that every deterministic search path has competitive ratio at least $C_{\log}\approx 12.5937096701246675$. The bound is unconditional: the path need not be cyclic, self-similar, spiral-like, or monotone in angle. For each projection direction, we compare the path with a zigzag obtained by sorting its alternating record turns. The resulting completion constraints are interpreted as jobs with scale-dependent deadlines and lower-bounded through a finite-window scheduling argument in logarithmic time. Averaging these directional bounds then uses the exact Euclidean velocity budget. In one dimension, the same method recovers the optimal cow-path constant $9$. Finally, Arb ball arithmetic provides a rigorous numerical enclosure of the constant.

Working in the setting of enriched algebraic theories for a system of arities, we study several aspects of dual adjunctions between enriched algebraic categories, which we call algebraic dual adjunctions. Firstly, we generalize Freyd's theorem on contravariant algebra-valued right-adjoint functors to this setting. Secondly, we establish a biequivalence between a 2-category of algebraic dual adjunctions and a locally discrete 2-category of bifold algebras, in a sense defined in prior work of the author. Thirdly, we define special classes of algebraic dual adjunctions that we call (left- and right-)stable, in which certain free objects are reflexive, and we establish biequivalences between these and special classes of bifold algebras defined in terms of commutants in prior work of the author. Fourthly, we show that every algebra for an enriched algebraic theory canonically induces left- and right-stable algebraic dual adjunctions, and we establish a biequivalence between such algebras and left- (or right-)stable algebraic dual adjunctions, and also between saturated algebras and stable algebraic dual adjunctions. We also discuss examples of algebraic dual adjunctions, including dualization of internal modules, Pontryagin and Binz-Butzmann duality, dualization of internal affine spaces and convex spaces, dualization of semilattices, dualization of complete sup-lattices, and dualization of abelian groups with reference to a theorem of Ehrenfeucht and Łoś.

For non-integer $\theta > 3$ and $\kappa \geq 1$, we show that the smallest $r_0$ such that the mean value estimate \[ \int_{-\kappa}^{\kappa} \Big| \sum_{X < x \leq 2X} e(\alpha x^{\theta}) \Big|^{2r} d\alpha \ll_{\epsilon} \kappa X^{2r - \theta+\epsilon} \] holds for all integers $r \geq r_0$ satisfies $2r_0 \leq \theta^2(1+O(\theta^{-1/2}))$. This is an improvement over the previous bound by Poulias of $2r_0 \leq (\lfloor 2\theta \rfloor + 1)(\lfloor 2\theta \rfloor + 2)$. As a consequence, the bound on the asymptotic order of the minimum number of variables required to prove the expected asymptotic formula for the number $R_{s,\theta}(N)$ of solutions $(x_1,\ldots,x_s) \in \mathbb{N}^s$ to the Diophantine equation \[ \lfloor x_1^{\theta} \rfloor+\cdots+\lfloor x_s^{\theta} \rfloor = N \] is improved by a factor of $4$. We also discuss a certain Diophantine system which arises naturally from our proof, which may have applications to other counting problems and may be of independent interest.

Finite-player nonzero-sum optimal stopping games typically lead to coupled equilibrium systems whose complexity grows rapidly with the number of players. We introduce an independently randomized formulation in which each stopping rule is represented by an adapted, nondecreasing cumulative stopping process. The canonical embedding preserves pure-profile payoffs, and a pure profile is a Nash equilibrium of the original game if and only if its embedding is a Nash equilibrium of the randomized game. We adopt the $\alpha$-potential approach to construct an $\alpha_N$-potential function, with the error $\alpha_N=O(N^{-1})$ under weak-interaction. We also identify an exact-potential subclass with a closed-form threshold equilibrium. For local stopped-status interactions, randomized payoffs admit a local stopped-mass representation, and potential maximization can be formulated as a multidimensional singular-control problem with local gradient constraints and a nonlocal condition for finite jumps. Under suitable regularity assumptions, we study the associated Hamilton-Jacobi-Bellman quasi-variational inequality and its regularity properties. For unknown model coefficients, we propose a bounded-intensity Potential-CT-DDPG learning algorithm. Numerical experiments closely match the analytical benchmark and yield estimated best-response improvements consistent with $N^{-1}$ scaling.

For fixed $\tau > 0$, non-integer $\theta > 2$ and large enough $s$, we investigate the number of solutions to the Diophantine inequality $|x_1^{\theta}+\cdots +x_s^{\theta} - R| < \tau$ as $R \to \infty$. Here, we restrict the variables $x_i$ in the “almost diagonal" range $X-Y < x_i \leq X+Y$ for $i = 1, \ldots, s$, where $X = (R/s)^{1/\theta}$ and $Y \asymp \sqrt{X}$. Let $\omega = (\lfloor s/2 \rfloor(\theta-1))^{-1/2}$. We will show that if $Y = c\sqrt{X}$ for some $c > \omega$ then for sufficiently large $R$ there must always exist solutions, but if $c < \omega$ then there exist arbitrarily large positive $R$ for which there are no solutions. Our result is thus essentially sharp, with the exception of $c = \omega$. This work is analogous to the results of Daemen and Wright in which similar statements are proved for Waring's problem, though our results are likely somewhat stronger than what is possible in that setting. We discuss other related results and further work to be done.

We prove that if two parameterizations of a complete reduced noetherian curve over an algebraically closed field agree modulo a sufficiently large power of the maximal ideal, then the two parameterizations are equivalent. This strengthens some bounds from Greuel and Pfister. In addition, we prove that if two reduced and irreducible curve singularities are isomorphic modulo sufficiently high (and identical) powers of their respective maximal ideals, then the completions of the two curves are isomorphic, and the isomorphism of the full completions agrees with the original isomorphism modulo some lower power of the maximal ideals. We provide a new and better bound on the lower power, strengthening the bound in Hironaka

We unify two matching theories, one for finite subsets of groups and the other for finite-dimensional subspaces in a field extension. To achieve this, we study groups equipped with a compatible finitary matroid structure, termed here independence groups. Applying Rado's independent transversal theorem, we derive necessary and sufficient rank criteria for matchability between finite-rank sets. In the setting of a modular abelian independence group $G$, we develop an analogue of the $e$-transform from additive number theory, derive structural matching criteria, and characterize a global matching property by the absence of a finite-rank submonoid $H$ satisfying $1<\rho(H)<\rho(G)$, where $\rho$ denotes rank. Examples of modular abelian independence groups are given and examined in the matching context. Arising from this matching theory, but formulated without any reference to it, is a product-growth bound that generalizes the Cauchy–Davenport theorem: we define a parameter $\mu(G)$ and prove that $\rho(XY)\geq \min\{\mu(G),\rho(X)+\rho(Y)-1\}$ for all nonempty finite-rank subsets $X,Y$ of $G$. Furthermore, $\rho(XY)$ is shown to be controlled from below by a submonoid of $G$ that stabilizes a flat, a phenomenon reminiscent of Kneser's theorem.

Let $A$ be $m$-accretive in a real Banach space $X$. Given closed sets $K(t)\subset X$, we develop an existence theory for mild solutions of the constrained problem \[ u'(t)+Au(t)\ni f(t,u(t)), \qquad u(t)\in K_A(t):=K(t)\cap\overline{D(A)}, \] where the forcing is Carathéodory on the effective tube $K_A$. We also assume joint measurability of $f$ on the moving graph or a Carathéodory extension to a fixed cylinder. Under a linear-growth hypothesis and the corresponding regular- and exceptional-time subtangential conditions, we reduce the problem to a bounded effective tube with a common integrable bound. % A first central result constructs a closed separable resolvent-invariant subspace preserving distances to the moving sets and the subtangential conditions, permitting use of the Scorza–Dragoni property for the reduced forcing. A second central ingredient is a two-level approximation. Mild solutions $u_\varepsilon$ driven by $w_\varepsilon$ are accompanied by auxiliary paths $v_\varepsilon$ and current-time selectors $x_\varepsilon$ satisfying $x_\varepsilon(t)\in K_A(t)$ and $w_\varepsilon(t)=f(t,x_\varepsilon(t))$ for almost every $t$ in a closed regularity set of almost full measure. For every $t$ there is also $\sigma_\varepsilon(t)\in[(t-\varepsilon)^+,t]$ with $v_\varepsilon(\sigma_\varepsilon(t))\in K_A(\sigma_\varepsilon(t))$. The current-time selectors identify the limiting forcing, while the lagged nodes guarantee viability. % This yields well-posedness under time-dependent constraints for such forcings $f$ when they are locally Lipschitz in the state variable. Further viability results follow under various compactness conditions. Application of the abstract viability theory yields comparison principles, nonautonomous Lyapunov pairs, periodic solutions, and time-dependent bounds for abstract reaction–diffusion systems.

Let $L_u$ be a symmetric Levy process representing an unobserved phase lift, coupling to a quantum system via charge $Q$. Averaging $e^{iL_uQ}$ gives a completely positive dephasing semigroup. If the characteristic exponent lacks an intrinsic charge scale, the Levy-Khintchine theorem forces $\eta(\xi)=D | \xi |^{\alpha}$ ($0<\alpha\le2$), so coherences decay at $\Gamma_{ab}=D | q_a-q_b |^{\alpha}$. For integer charge differences, the process descends to the wrapped variable $\Theta_u=L_u\bmod 2\pi$; for real charges, the lift is the relevant phase. Schoenberg's theorem ensures complete positivity for any finite real spectrum when $0<\alpha\le2$, while $\alpha>2$ yields an analytic obstruction. This boundary is exact, verified numerically here. A Gaussian integrated phase gives quadratic charge dependence, though colored Gaussian noise need not yield a Markov semigroup. Two idealized non-Gaussian mechanisms are analyzed: inverse-power Poisson shot noise ($\alpha=d/p$) and Bochner subordination of Brownian phase by an $\alpha/2$-stable clock. For a bounded reduction walk, we prove Born probabilities for every symmetric bounded proposal law. Simulations of truncated stable laws ($\alpha=0.5,1,1.5,2$) confirm this law-independence. In continuous time, $\alpha$-stable drivers fail to reduce: the exact-flow (Marcus) equation oscillates, and the naive Ito jump equation fails to preserve state. For Gaussian white noise, the Itô model collapses to Born probabilities, whereas the exact-flow (Stratonovich) model does not collapse without a threshold and yields non-Born exit probabilities. The quadratic case recovers Milburn's small-step limit, not his exact Poisson dynamics. Finally, a bi-temporal geometry is examined as a possible compact phase source, but closed timelike curves, nonunitary evolution, and an unstable mode tower prevent a consistent field-theoretic realization.

We show that the subfactor planar algebra with principal graph $D_\infty$ is the Brauer planar algebra with bubble constant $\delta=2$. The Brauer algebra is similar to the Temperley-Lieb algebra, but with virtual crossings. At $\delta=2$, it relates to the category of representations of the orthogonal group $O(2)$. We work over any commutative ring with $1/2$. Its principal graph encodes information about the corresponding monoidal category.

Inspired by recent work of Aldous, Janson, and Pittel on the critical beta-splitting model, we study the full beta-splitting family for beta greater than minus two through a canonical continuous-time embedding into a homogeneous exchangeable fragmentation. In this representation, the frequency of a tagged fragment is described by a subordinator. We express the continuous height of a typical leaf, its occupation probabilities, the discrete height, and the total continuous-time length in terms of the potential measure of this subordinator. Renewal theory yields first-order asymptotics and a central limit theorem for the continuous height. A regenerative-composition representation gives Gaussian limits for the discrete height above and at the critical value, and a non-Gaussian power-law limit below it. We also obtain residue expansions for the potential measure and the mean continuous height using meromorphic potential theory and generalized Nevanlinna functions. Finally, we study the maximum continuous-time height, proving a law of large numbers and a mixed Gumbel limit. At the critical parameter value, this resolves an open problem of Aldous and Janson.