Papers

We develop two parametric data-driven reduced models: a physics-informed neural network (PINN) and a non-intrusive tensorial reduced-order model (TROM), and apply both approaches to the parametrized one-dimensional shallow-water dam-break problem. Neither reduced model requires time integration: both learn a direct parameter-to-solution map from space, time, and dam-break parameters to the physical state, with the PINN providing predictions at arbitrary times and the TROM reconstructing solutions at the stored snapshot times. In addition, we demonstrate that it is essential to introduce shock-aware collocation to improve the robustness of the PINN model.

We propose dynamical low-rank (DLRA) filters for data-assimilation problems based on stochastic differential equations (SDEs). We first consider a Kalman filter applied to a Euler-Maruyama discretization of the SDE and observation processes, and derive a DLRA Kalman-type filter, named DLRA-JMCO, by minimizing jointly the error on the mean and covariance. We then obtain a time continuous DLRA-JMCO filter as the continuous limit of the time discrete one, and introduce a corresponding particle approximation. We also propose a strategy to enrich the DLRA-JMCO filter with a simple Gaussian process to capture the neglected dynamics in the orthogonal to the DLRA subspace. This correcting term allows the DLRA subspace evolution to be directly influenced by the observation operator. We finally propose a particle filter version of the DLRA-JMCO filter. Numerical experiments assess the performance of the proposed filters.

It is a famous property that a longitude of a classical knot lies in the second commutator subgroup of the knot group. We observe that the same property holds for a longitude of a virtual knot.

This paper concerns a variational system of nonlinear elliptic equations that generalizes the classical Brezis-Nirenberg problem. In addition to considering vector-valued unknown functions in place of scalar-valued unknown functions, the problem under consideration generalizes the Brezis-Nirenberg problem in two directions. First, the nonlinearity is the sum of two homogeneous functions of suitable homogeneity degrees. Second, the Sobolev-critical term in the nonlinearity is nonlocal. We establish conditions under which a nontrivial solution to the system under consideration is guaranteed to exist and conditions under which the system under consideration admits only the trivial solution.

High-order numerical methods for conservation laws can generate spurious oscillations near discontinuities. We propose to suppress these oscillations by post-processing the numerical solution with a total variation (TV) denoising step that reduces the total variation of the nodal values. For a general analysis operator, the resulting discrete minimization problem need not satisfy the submodularity property that existing exact max-flow algorithms require, so we instead compute the TV minimizer exactly using a differential inclusion algorithm. In one dimension, the differential inclusion algorithm computes the TV minimizer in finitely many steps, requires no tuning of algorithmic parameters, and preserves the total mass. In two dimensions, we apply the 1D algorithm dimension by dimension. We test the method as a post-processing limiter for Fourier pseudospectral methods and a fifth-order finite difference scheme applied to scalar conservation laws, compressible Euler equations, and the two-dimensional incompressible Euler equations.

We develop a posterior-level discrete-to-continuum theory for Bayesian inverse problems whose finite-dimensional priors arise from local finite-difference regularization and whose likelihoods are of a general convex GLM-type form. In contrast to continuum-first approaches, the continuum prior and posterior are not assumed at the outset but are constructed as limits of the finite-dimensional probability measures used in numerical computation. First, with the total information parameter $\tau$ and the regularization strength $\kappa$ fixed, we prove weak convergence on $L^2$ of the reconstructed discrete Gaussian priors and posterior measures to well-defined continuum laws. We then consider a coupled grid-refinement and small-noise limit in which $N\to\infty$ and $\tau_N,\kappa_N\to\infty$, while $\kappa_N/\tau_N$ remains fixed. Under explicit growth conditions relating $N$ and $\tau_N$, we prove that the reconstructed discrete MAP estimates converge to the unique minimizer $u_*$ of the limiting continuum cost functional, that the reconstructed posterior measures concentrate at $u_*$, and that their centered and rescaled fluctuations converge to $\mathcal N(\vec 0,Q^{-1})$, where $Q$ is the Hessian of the continuum cost functional at $u_*$. Finally, we show that the same deterministic limit and Gaussian fluctuation law are obtained by first constructing the continuum posterior and then taking its small-noise, high-regularization limit.

We introduce an approximate version of matrix product factorization for graphs. A simple graph $G$ on $n$ vertices is said to admit an $\varepsilon$-matrix product factorization if there exist simple graphs $H$ and $K$ on the same vertex set such that $A(H)A(K)$ and $A(G)$ disagree in at most $\varepsilon n^{2}$ entries. This Hamming-type relaxation preserves, outside the error set, the exact interpretation of each edge as having a unique $H$-then-$K$ two-step witness. We establish equivalent matrix, and witness formulations, showing that the sets $N_H(w)\times N_K(w)$ form an approximate disjoint decomposition of the ordered adjacency relation of $G$, and we derive quantitative constraints involving walk counts and the degrees of the factor graphs. We then construct approximate factorizations for several graph families. Every complete graph $K_n$ has matrix-product-factorization distance $O(1/n)$, despite the exact congruence obstruction that permits exact factorization only when $n\equiv 1\pmod 4$. More generally, a blow-up of a fixed graph on $r$ vertices admits an $\varepsilon$-factorization with $\varepsilon\le r/n$, and the construction is exact whenever every non-isolated part has even order. For bipartite graphs, we give one-sided factorizations that realize one orientation of almost all edges. In particular, every tree on $n\ge2$ vertices admits an $\varepsilon$-factorization with $\varepsilon\le 1/n$, although no nontrivial tree is exactly factorizable. These results show that rigid exact obstructions may disappear under a vanishing proportion of entrywise errors.

Let $G$ be a countably infinite discrete amenable group acting minimally on a compact metric space $X$, and let $\pi:X\to X_{\mathrm{eq}}$ be the maximal equicontinuous factor map. We introduce the conditional topomorphic degree $d:=\tdeg_G(X)\in\mathbb N\cup\{\infty\}$, which, when finite, is the least integer such that $\pi$ is an at most $d$-to-one topomorphic extension. We prove that, for every $r\ge2$, Weyl mean $r$-equicontinuity, mean $r$-equicontinuity along some Følner sequence, and $d\le r-1$ are equivalent. For minimal $\mathbb Z$-systems, this settles a conjecture of Breitenbücher, Haupt, and Jäger. We further establish the decomposition formula $d=\sum_{\mu\in\mathcal M_G^e(X)}\iota_\mu\exp\bigl(h_\mu^*(G)\bigr),$ where $\iota_\mu$ is the degree of the factor map from the measure-theoretic maximal compact factor associated with $\mu$ onto $X_{\mathrm{eq}}$, and $h_\mu^*(G)$ is the maximal measure sequence entropy of $\mu$. As further consequences of the decomposition formula, we show that for every finite $N$ with $2\le N\le d$, the system admits an essential IT $N$-tuple. Consequently, $h_{\mathrm{top}}^*(X,G)\ge \log d.$ This strengthens a lower bound of Liu, Wang, and Xu by also detecting compact multiplicities. As an application, we answer a question of Gómez, León-Torres, and Muñoz-López. If $G$ contains a finite-index normal subgroup isomorphic to $\mathbb Z^r$, then, for every $m\ge2$, there exists a free minimal uniquely ergodic zero-entropy finite-alphabet $G$-subshift with maximal topological sequence entropy $\log m$, an essential IT $m$-tuple, and no essential IN $(m+1)$-tuple. For $G=\mathbb Z$, the alphabet may be chosen to have exactly $m$ symbols. Finally, we realize every finite multiplicity profile by a zero-entropy minimal almost one-to-one extension of an irrational circle rotation.

We compute the parametrized (or twisted) ordinary cohomology of the classifying space for real $C_2$-line bundles, $B_{C_2}O(1)$ with coefficients in the constant Mackey functor $\underline{\mathbb{F}}_2$. Parametrized cohomology refines $RO(G)$-graded Bredon cohomology by assembling equivariant cohomology for all local coefficients into a single graded ring. For this reason, our work also encodes a computation of the $RO(C_2)$-graded cohomology of all Thom spaces of real $C_2$-vector bundles over $B_{C_2}O(1)$. Along the way, we prove many general results that can be applied to computations of parametrized cohomology over general bases $B$. In particular, we introduce a collection of characteristic classes, give a definition of orientation for non-homogeneous bundles, import equivariant Steenrod operations to this context, and give general results relating to base change.

We give a closed formula for the number $p^{S_2}(n)$ of self-avoiding polygons (SAPs) of length $n$ on the strip $S_2:=\mathbb{Z}\times\{0,1,2\}$, together with closed formulas for those subtypes of SAPs which are determined by the numbers of vertical steps in their leftmost and rightmost columns. For the subtype whose leftmost and rightmost columns each contain two vertical steps, we also derive an alternative representation as a binomial sum. Our derivation is elementary: it is purely combinatorial and geometric and avoids generating functions. Comparing the two representations yields a new geometric proof of an identity arising in Larsen's treatment \cite{L07} of a problem posed by Gessel \cite{G95}. Finally, we show that this subtype of SAPs is closely connected to the sequence A007909. More precisely, for $m\geq0$, the number of these SAPs whose leftmost and rightmost columns each contain two vertical steps and whose length equals $2m+6$ is given by the term of this sequence with index $m$, which thereby acquires a geometric interpretation alongside the compositions it enumerates.

Extension types are a concept in dependent type theory that has appeared in various contexts. The idea is to have types whose terms are partially determined, e.g. via a strict boundary condition. Standard examples are path types of cubical type theories (paths with fixed endpoints), Riehl and Shulman's name-giving extension types (terms fixed on subshapes), as well as the controlled-unfolding mechanism of cooltt and Agda (terms that are fixed if a condition is met). In each case, the type theory is equipped with a (meta-theoretic) face calculus, or shape layer, that governs their rules, and comes with intended semantics. We unify all these occurrences in a single framework where no new axioms or model constructions are needed, namely two-level type theory. This step, too, is free (semantically): the standard models of HoTT are automatically models of 2LTT, and the theory is conservative over HoTT. Extension types are definable, and the definition validates Riehl and Shulman's entire extension-type calculus: the rules hold strictly, and the postulated axioms, such as relative function extensionality, become theorems. In this way, every model of the base theory (HoTT) gives rise to a model of the same theory with extension types; the only genuine assumptions are which maps count as cofibrations. Conservativity makes the framework a tool for comparing type theories. We prove that cubical gluing, in a suitable formulation, is equivalent to univalence. On this basis, we suggest an approach toward the conjecture that cubical type theories are conservative over book HoTT, one of the central open problems of homotopy type theory. All results of the main body of the paper are auto-formalized in Agda –two-level, in a development that combines HoTT-internal arguments with reasoning that is external to HoTT.

Homogenized continuum models are widely used to describe wave propagation and band-gap behavior in mechanical metamaterials without explicitly resolving their microstructure. Their validity, however, typically relies on the classical separation of scales assumption, according to which the wavelength of the propagating disturbance is much larger than the characteristic size of the unit cell. In finite-size metamaterial samples and at higher frequencies, this assumption progressively breaks down, and the dynamic response becomes strongly influenced by the way the microstructure is truncated at the external boundaries. In this work we introduce a fundamentally new concept in the homogenized description of mechanical metamaterials: the inertial contribution of macroscopic interfaces. We show that different truncations of the same lattice generate boundaries with distinct mass distributions, which lead to measurable differences in the dynamic response of finite-sized specimens. To capture this complex mechanism in a homogenized framework, we extend the relaxed micromorphic model by introducing a kinetic surface energy defined on the boundary of the considered body. This generates an additional inertial term in the boundary conditions that can be seen as the homogenized counterpart of the interface inertia produced by the truncation of the microstructure. As a result, the homogenized model can now distinguish between finite-sized specimens that share identical bulk properties but differ only in the configuration of their interfaces. The proposed formulation preserves the variational structure of the relaxed micromorphic model while enabling the continuum to reproduce boundary-dependent responses observed in fully resolved simulations, particularly in frequency regimes ....... See the PDF for the full abstract.

Let $K$ be a global function field of characteristic~$p$ and let $G$ be a finite abelian group of exponent $p^e$. We show that the multivariate generating function counting sub-$G$-extensions of $K$ with respect to $e$ specific height functions (encoding successive drops in the exponents of the higher ramification groups) is rational.

Within the semi-classical regime of AdS/CFT correspondence, we consider the limit where the bulk dynamical field is linearized metric perturbations satisfying linearized Einstein equations over background pure AdS spacetime. AdS/CFT correspondence gives us a holographic map, which is an isometric embedding map of the GNS Hilbert space of linearized gravity in the bulk (w.r.t. the AdS-invariant vacuum) to the GNS Hilbert space of CFT in the boundary (w.r.t. the Minkowski-invariant vacuum). We assume that the map takes AdS-vacuum in the bulk to CFT-vacuum in the boundary and that it allows AdS-Rindler wedge reconstruction. Then using this map, we show that for a given ball-shaped region in the boundary $A$, the relative entropy of a bulk state w.r.t. the AdS vacuum in the algebra of causal wedge associated to $A$ matches with the relative entropy of the dual CFT state w.r.t. the CFT vacuum in the algebra of CFT observables in $A$ in the code subspace, which is known as Jafferis-Lewkowycz-Maldacena-Suh (JLMS) condition. Furthermore, for localized semi-classical coherent excitations in the causal wedge associated to $A$ which corresponds to perturbed bulk geometry, we show rigorously using modular crossed product construction that the state-dependent part of entropy of the dual CFT state in the dressed Type-II algebra associated to $A$ satisfies vacuum subtracted Hubeney-Rangamani-Takayanagi (HRT) formula.

Let $\mathcal{C}_{\arcsin}$ denote the Ma–Minda subclass of convex functions generated by the normalized arcsine mapping $\varphi(z)=1+\frac{2}{\pi}\arcsin z.$ For this family, we develop a unified coefficient analysis based on subordination techniques, Carathéodory functions and sharp estimates for Schwarz functions. As consequences, we derive sharp estimates for the initial Taylor coefficients, logarithmic coefficients and certain differences involving the logarithmic and inverse logarithmic coefficients. We further determine the exact bounds for the second Hankel determinant $H_{2,2}(f)$ together with the Hankel determinants $H_{2,1}(F_f/2)$ and $H_{2,1}(F_{f^{-1}}/2)$ associated with the logarithmic coefficients of a function and its inverse. Moreover, sharp estimates are obtained for the initial generalized Zalcman functional and the generalized Fekete–Szegö functional. In every case, the corresponding extremal functions are identified, showing that all of the obtained inequalities are best possible.

For graphs \(G\) and \(H\), let \(N(G,H)\) be the number of unlabeled, not necessarily induced copies of \(H\) in \(G\), and let \(f(n,H)\) be the maximum of \(N(G,H)\) over all \(n\)-vertex planar graphs \(G\). Ghosh, Győri, Martin, Paulos, Salia, Xiao and Zamora conjectured that, for every fixed integer \(\ell\ge 2\), \[ f(n,P_{2\ell+1}) =4\ell\left(\frac{n}{\ell}\right)^{\ell+1}+O(n^\ell). \] We prove the conjecture, including the stated error term. Along the way, we also settle the Cox–Martin optimization conjecture.

We study the estimates for the number of exceptions to the representation of integers as the sum of at most two prime numbers. Most of this article is a survey that gives an overview of existing results. We begin with the legendary Hardy-Littlewood circle method and show how it paved the way to a power saving by Montgomery-Vaughan in 1975 and Pintz in 2018. We conclude with a new result that is a fully explicit formula for the major arcs. Another new observation is the non-existence of exceptional zeros under a sparse version of the Hardy-Littlewood conjecture. The survey part of this article aims to be accessible to an audience that has not encountered these techniques before.

Robust Optimization Over Time (ROOT) is a recent branch of evolutionary dynamic optimization that seeks solutions capable of remaining effective across multiple consecutive environments. Unlike the traditional track-the-moving-optimum (TMO) paradigm, which reoptimizes after every environmental change, ROOT explicitly values persistence. Although the field has grown considerably, most contributions remain algorithmic and empirical, leaving several fundamental properties poorly understood from a theoretical perspective. One such property is survival time, defined as the number of future environments in which a deployed solution continues to satisfy a prescribed quality threshold. While survival time is widely used as a measure of temporal robustness, little is known about how its expected value depends on environmental dynamics, deployment quality, or problem characteristics. This paper studies expected survival time for a fixed deployed solution under isotropic Gaussian environmental dynamics. Modeling survival as a discrete first-exit problem, we derive a rigorous lower bound and a computable multi-step upper bound. The analysis shows that expected survival scales as ${\Theta}({\sigma}^-{2})$ in slowly varying environments and approaches its minimum value of one future change in high dimensions. A comprehensive Monte Carlo study validates the theoretical predictions, examines sensitivity to modeling assumptions and parameter uncertainty, and illustrates how the bounds can support deployment decisions after optimization. The resulting framework provides an analytical characterization of deployment lifetime and identifies when a required deployment horizon can be guaranteed, ruled out, or remains analytically unresolved.

Exponentials of (real/complex) vector fields are classically defined via the vector field integration. Take a k-algebra k[x] \subset R\subset k[[x]], where k\supseteq \Q is a local domain. Suppose a derivation \xi is x-adically nilpotent. Define the exp-operator via the Taylor expansion, e^\xi:=\sum \frac{\xi^j}{j!}. It is a formal automorphism, e^\xi\in Aut_k(k[[x]]). When does e^\xi act on R? When does the formal power series e^\xi x\in k[[x]] belong to R? We address this question for the following rings. i. The algebraic power series, R=k\bl x\br, differentially finite (holonomic) power series, D(k[x]), and their higher versions, Picard-Vessiot extensions D^\bullet(k[x]), Picard-Vessiot closure D^\infty(k[x]), and differentially-algebraic power series D^{alg}(k[x]). ii. Power series over normed fields. In particular, power series with coefficients of controlled growth, e.g. analytic/Denjoy-Carleman/Gevrey classes. iii. Germs of smooth functions C^\infty(\R^n,o)/J, for arbitrary ideal J\subset C^\infty(\R^n,o). In case i. the operator e^\xi is transcendental, and the power series e^\xi x is “usually" far from being algebraic. We give various criteria on e^\xi x to belong to k\bl x\br, D(k[x]), D(R), or D^{alg}(k[x]). In case ii. the answer is positive (i.e. e^\xi acts on R ) under rather weak assumptions on R. In case iii. the answer is “totally negative". For any \xi\neq0 the operator e^\xi (defined as before) does not act on the quotients of the ring of germs of smooth functions, C^\infty(\R^n,o)/J.

We introduce the class $\mathscr{P}_{\mathcal{H}_n^0}(\alpha)$ $(0\leq \alpha<1)$ of normalized pluriharmonic mappings in the setting of several complex variables. This class extends the harmonic family $\mathscr{P}_{\mathcal{H}}^{0}(\alpha)$ to the multidimensional framework. We establish sharp coefficient estimates and growth theorems for functions in $\mathscr{P}_{\mathcal{H}_n^0}(\alpha)$, thereby generalizing the corresponding results of Li and Ponnusamy \cite{Li-Ponnusamy-2013a} and Allu and Halder \cite{Allu-Halder-2021}. We further determine the associated Bohr radius and investigate the sections (partial sums) of functions in this class, obtaining quantitative results that describe the behavior of their truncated expansions.

A tree cover of an $n$-point metric space is a collection of $k$ trees such that every pairwise distance is approximately preserved in at least one tree. For fixed $k$, the best known distortion upper bound is $\tilde{O}(n^{1/k})$ and the best known lower bound is $\Omega_k(n^{1/2^{k-1}})$. In this paper, we prove a lower bound of $\Omega_k(n^{1/(k-1)^2})$ for all $k\ge 3$, significantly narrowing the gap towards the upper bound. Our approach makes use of a novel algebraic topological notion that we call the agreement index: the equivariant topological index of a simplicial complex encoding pairwise agreement among tree labels. This notion helps characterize the power and limit of algebraic and topological methods for analyzing tree covers, and we believe it will also find applications to other problems.

For a finite nonempty subset $A$ of an abelian group, let $\sigma(A)=|A+A|/|A|$ and $\delta(A)=|A-A|/|A|$. The classical sum-difference inequalities state that $$\sigma(A)^{1/2}\leq\delta(A)\leq\sigma(A)^2.$$ The exponent $2$ in the second inequality is known to be optimal, whereas it has remained open whether the exponent $1/2$ in the first inequality can be improved. We settle this question by constructing an explicit family of finite sets $A_K\subset\mathbb{Z}$ such that $$\frac{\log\sigma(A_K)}{\log\delta(A_K)}\longrightarrow 2,$$ hence the exponent $1/2$ in the first inequality is also optimal. The construction and its proof were developed with the assistance of Hyra, an AI research agent based on the open-weights Hy3 model.

We prove a conjecture of Regts and Sevenster: a complex-valued graph parameter $f$ with $f(\emptyset)=1$ has exponentially bounded edge-connection rank if and only if it is a mixed partition function. The bound is exact: for a real number $R\ge 1$, the connection ranks satisfy $\operatorname{rk} M_{f,t}\le R^t$ for all $t\ge 0$ if and only if $f$ has a model on a super vector space $\mathbb{C}^{k|2\ell}$ with $k+2\ell\le R$. Consequently the base of exponential growth of the connection ranks is the least number of colours of a model, the two dimensions of a minimal model are determined by $f$, and the parameters with a model on a prescribed $\mathbb{C}^{k|2\ell}$ are characterised. The proof organises fragments modulo the connection kernel into a rigid symmetric tensor category whose morphism spaces have the connection ranks as dimensions; the rank hypothesis and an argument of Schrijver make its additive idempotent completion semisimple, Deligne's theorem provides a fibre functor to super vector spaces, and the resulting super tensor network is identified with the Regts-Sevenster model exactly, circuit signs included. An appendix shows that in a rigid symmetric $\mathbb{C}$-linear category with $\mathrm{End}(\mathbf{1})=\mathbb{C}$, exponentially bounded endomorphism growth makes the trace zeta function of every endomorphism rational, with explicit degree bounds.

We exhibit a configuration of five point charges in Euclidean space whose electrostatic potential admits at least 24 critical points all of which are non-degenerate. Maxwell's conjecture that the field of \(n\) point charges has at most \((n-1)^2\) critical points which are all non-degenerate is therefore false.

We present a novel approach to regression tasks using classification which is motivated by the mechanism used by fruitflies to sense their environment. Specifically, we formulate a general framework for learning nonlinear input-output relationships by replacing complex global surrogate models with a finite library of representative local patterns. Since scientific data often occupy limited and recurring regions of the input space, we generate predictions by measuring similarities between a query and stored patterns, then combining their associated responses through weighted reconstruction. We apply this approach to nonlinear dynamical systems, data-driven regression, and physics-informed learning using suitable embeddings and similarity measures. For dynamical systems, our offline-online workflow extracts patterns from data or governing equations during the offline phase, while online prediction requires only similarity evaluation and response aggregation. This structure helps us reduce computational and memory demands while providing explicit control over the trade-off among accuracy, storage, and inference cost.

We study the Sachdev-Ye-Kitaev (SYK) Hamiltonian $H_{\operatorname{SYK}}$ on $n$ Majorana modes with $k$-body interactions, and prove that $\mathbb{E}\|H_{\operatorname{SYK}}\|_{\operatorname{op}} = (1 - o(1))\cdot\sqrt{2n}/k$ for super-constant $k\leq o(\sqrt{n})$, where the expectation is over the disorder variables in the Hamiltonian. This confirms predictions due to Garcia-Garcia, Jia and Verbaarschot and answers a question posed by Feng, Tian, and Wei. Our results extend to the sparse SYK Hamiltonian. As a corollary, we obtain that a dissipative quantum algorithm due to Basso, Chen, and Dalzell provably computes the ground state energy of the SYK Hamiltonian up to an $O(1)$-multiplicative factor for all $k < \sqrt{n}/4$. Our key technical idea is identifying an explicit, deterministic linear operator $\mathsf{x}$ such that a fixed quadratic form of $\mathsf{x}^{2\ell}$ exactly equals the expected trace moments of the SYK Hamiltonian for every $n$ and $k$. This linear operator can be naturally viewed as a twisted model of bosons on the space of hyperedges of a hypergraph. The problem thus reduces to identifying the spectral edge of $\mathsf{x}$, which we show is dominated by the spectrum of a natural ${n \choose k}$-dimensional matrix from the Johnson scheme and is straightforward to compute using known results. To show that our bound is sharp, we construct a witness state with a large quadratic form on $\mathsf{x}$ and transform it into a certificate of a lower bound on the largest quadratic form on $H_{\operatorname{SYK}}$. Beyond the SYK Hamiltonian, our techniques also apply to the $k-$local quantum spin glass (QSG), yielding $\mathbb{E}\|H_{k-\operatorname{QSG}}\|_{\operatorname{op}}\leq O(\sqrt{n/k})$ for all $1\leq k\leq\sqrt{n}/2$, thus improving the standard $O(\sqrt{n})$ bound by a factor of $\sqrt{k}$.

Spin is the hidden engine behind the zoo of MacWilliams transforms in weight enumerator theories - not only for qubits and qudits, but even for classical codes. From nothing more than a split into trivial and nontrivial errors, we kinematically derive the MacWilliams transform as a Wigner-$D$ rotation between two canonical bases. Within each classical and quantum theory, changing the length $n$ leaves the rotation untouched: the same element simply reappears at spin $n/2$. And at fixed $n$, changing the rotation axis simply moves between the various classical and quantum theories.

Jacobs comprehension categories subsume a large class of categorical models of type dependency, supporting also the description of morphisms between types. We study the relationship between comprehension categories and a particular subclass, which we call Lawvere-Ehrhard comprehension categories. First, we characterize this subclass by comparing a fibration of terms and a fibration of type morphisms associated to a given comprehension category. Next, we provide the construction of the free comprehension category over a fibration. Finally, we construct the free Lawvere-Ehrhard comprehension category over a Jacobs comprehension category.

The Stanley–Gasharov conjecture asserts that every claw-free graph is Schur-positive. Prajapati and, independently, Matherne and Morales identified the same pair of counterexamples, both of which are line graphs, thereby disproving the conjecture. In this paper, we construct two infinite families of counterexamples to the Stanley–Gasharov conjecture, thereby answering a question of Matherne and Morales. Every graph in the first family is a line graph, whereas no graph in the second family is a line graph. Prajapati further showed that the graph $G_2$, which has $12$ vertices and $21$ edges, is the smallest counterexample under the ordering that first compares the numbers of vertices and then the numbers of edges. We show that $G_2$ is also the smallest counterexample under the reverse ordering, which first compares the edge numbers and then the vertex numbers. Similarly, we exhibit a graph $Q$ with $13$ vertices and $27$ edges and show that $Q$ is the smallest counterexample that is not a line graph under each ordering. Our two infinite families are obtained from $G_2$ and $Q$, respectively, by adjoining a clique of order at least $4$ and connecting one of its vertices to a distinguished vertex of the original graph by a single edge.

Consider a material point in finite dimensions moving under the influence of a potential force according to Newton's laws. Suppose the potential energy function is a generalized well, strictly convex transverse to a smooth submanifold $M$ on which it is minimal. If the potential is steep, one expects the particle will oscillate rapidly about this submanifold and, if the initial displacement is not too great, should approximately move along $M$ as if it were ideally constrained (e.g. geodesic). This expectation is true if the initial conditions are very well prepared but may fail otherwise - additional potential forces determined by how the Hessian of the potential varies along $M$ may be present. The origin of this force is that the transversal motion acts as a simple harmonic oscillator with a slowly varying frequency, which approximately conserves action, not energy. In this work, we regard continuum mechanical systems such as the elastic thread or compressible fluid as material points moving in an infinite dimensional space according to Newton's laws for appropriate potential energy functionals. We show how to arrive at ideally constrained systems such as the inextensible thread and incompressible fluid as a limit of a strong potential force, computing also corrections to the naive predictions when the data is not very well prepared. For example, for the thread we find a resistance to bending emerge from a strong resistance to compression/expansion. For the fluid, the effective incompressible dynamics may be driven by a remnant acoustical wavefield. Both of these emergent features are nonlinear and non-local. Finally, we give examples of some limits for which the naive models robustly hold because the additional force is trivial. These include the homogeneous incompressible Euler, anelastic Euler, as well as the lake and great lake equations.