Papers

In his 1994 doctoral thesis, Bossard introduced the notion of the non-universal tree $T_{NU}(X)$ associated with each separable Banach space $X$ which does not contain an isomorphic copy of $C(2^\mathbb{N})$. Together with the order operation defined on well-founded trees, we obtain a method of classifying the complexity of separable Banach spaces by the degree of isomorphism of finite-dimensional subspaces of $C(2^\mathbb{N})$. Despite further refinement of this concept in later years, we are unaware of any specific instances of direct exhibitions of the order of the non-universal tree for a concrete space and basis. We show that if $H$ is any separable Hilbert space, then $o(T_{NU}(H)) = {\omega} + 1$ when taken with respect to the Haar basis for $C(2^\mathbb{N})$, demonstrating that the class of Hilbert spaces is the least complex class with respect to this measurement when considering this basis.

We investigate how a fractional oscillator reacts to external forces when the dynamic law includes a damping term that is explicitly fractional. We show that the corresponding three-parametric fractional differential equation (in the Caputo sense) admits formal solution. Two of the fractional parameters are associated with the intrinsic dissipation mechanism derived from the fractional nature of the oscillator. The third fractional parameter characterizes the damping term against which the oscillator is driven by the external force. In the absence of fractional damping, the two-parametric fractional dynamic law can be associated with the Newton-Scott-Blair model for viscoelastic materials, in which position and external force are linked to the material's strain and stress, respectively. In general, the Laplace transform of the external force defines the profile of the three-parametric solution; specific examples include the absence of external forces as well as constant, sinusoidal and stepped external forces. In the case where the three fractional parameters converge to the same value, it is observed that, under the action of a constant driving force, the system reaches a fixed position after a transient period. With a sinusoidal driving force, the system exhibits persistent oscillatory behavior. These results suggest that the external driving force strongly influences the long-term behavior of the fractionally damped oscillator.

Let $G$ be a compact, connected, simply connected semisimple Lie group with Weyl group $W$ and simple reflections $S$. For a simplicial complex $\mathcal K$ on $S$, form the homotopy colimit $X_{\mathcal K}(G)=\operatorname*{hocolim}_{I\in\mathcal K}G/G_I$ of standard partial flag manifolds. We compute its integral homology. If $\operatorname{Des}_R(w)$ is the right descent set of $w\in W$ and $\ell(w)$ its Coxeter length, then $$ H_n(X_{\mathcal K}(G);\mathbb Z)\cong \bigoplus_{w\in W}\widetilde H_{n-2\ell(w)-1}(\mathcal K_{\operatorname{Des}_R(w)};\mathbb Z). $$ Thus the induced subcomplexes of $\mathcal K$ supply the topological data, while the Weyl group determines which subcomplex occurs and the Schubert-degree shift. The proof gives a chain-level splitting and an integral Morse reduction. We derive homotopy detection, duality and rigidity results, and recover polyhedral products, matroid–Tutte formulas, and the adjoint sphere as special cases.

For a graph $G$, a proper edge coloring of $G$ is called a D-coloring if every diamond subgraph of $G$ is rainbow. Let $\chi'_D(G)$ be the D-chromatic index of $G$, which is the smallest integer $k$ such that $G$ admits a D-coloring with $k$ colors. Let $\Delta$ be the maximum degree of $G$. The only known Brooks-type upper bound on $\chi'_D(G)$ is $\frac{9}{16}\Delta^2 + \frac{1}{2}\Delta$, given by a greedy coloring. In this paper, using a probabilistic method, we obtain the first improvement upon this upper bound by proving that $\chi'_D(G) \le (1-c)\frac{9}{16}\Delta^2$ for some $c > 0$ and sufficiently large $\Delta$.

We introduce a guessing game, “Permutation Wordle,” in which a guesser attempts to recover a setter's hidden permutation of the set $\{1, \ldots, n\}$. In each round, the guesser submits a word over the alphabet $\{1, \ldots, n\}$, and, as in the game Wordle, learns which entries are correct. We describe a natural strategy and prove that it is optimal in a strong sense: for every $r$, it solves at least as many secrets within $r$ rounds as any possible strategy. The number of permutations it solves in exactly $k+1$ rounds is the Eulerian number $A(n,k)$.

The notion of a generalised category with families, introduced by Coraglia and Emmenegger is one of the most general notions introduced to capture categorically the notion of dependent typing. It is shown that these generalised categories with families are biequivalent to comprehension categories. We will show that the notion of a (2-dep,$\Sigma$)-category, an extension of the notion of a (dep,$\Sigma$)-category, introduced by Petrakis, is not biequivalent to generalised categories with families, but instead is equivalent to a direct generalisation of that notion.

In this work, we study the self-similarity of permutational wreath products of the form \(A \wr_X G\), where \(A\) is a finitely generated abelian group and \(G\) is a self-similar group (the permutational wreath product \(A \wr_X G\) is also known as a lamplighter group). In the case where \(G\) is a non-torsion contracting group, we prove that, under certain conditions, the Scott–Röver–Nekrashevych group \(V_m(\mathbb{Z}^d \wr_X G)\) is finitely presented and virtually simple. Moreover, we prove that \(\mathbb{Z}^d \wr_X G\) embeds into a finitely presented simple group. Furthermore, this provides a new family of groups that satisfy the Boone–Higman conjecture.

We introduce the multiorbital bivariate chromatic polynomial F_\Gamma(G;x,y) = \sum_{H\leq G}\frac{1}{|H|}\sum_{h\in H}P_{\Gamma/h}(x,y), which aggregates the orbital bivariate chromatic polynomials associated with all subgroups of a finite group acting on a graph. We derive the equivalent element-wise representation F_\Gamma(G;x,y) = \sum_{g\in G}c_G(g)P_{\Gamma/g}(x,y), where c_G(g) = \sum_{\langle g\rangle\leq H\leq G}\frac{1}{|H|}. The coefficient function depends only on the cyclic subgroup generated by the element and is constant on conjugacy classes. This yields decompositions by cyclic subgroups and conjugacy classes and an interpretation in terms of the incidence algebra of the subgroup lattice. After normalization, the coefficients define a probability distribution obtained by choosing a subgroup uniformly and then an element uniformly from that subgroup. We also establish diagonal multiplicativity for disjoint unions and a weighted cycle-index expression for edgeless graphs. A further contribution concerns the distinguishing power of the orbital bivariate chromatic polynomial. We answer a question of Dohmen and Lange-Geisler affirmatively by exhibiting two non-isomorphic graphs, P_3 and K_2 \mathbin{\dot\cup} K_1 under C_2-actions, with identical orbital bivariate chromatic polynomials. The two actions nevertheless have different multiorbital bivariate chromatic polynomials. Thus the multiorbital polynomial is not determined by the orbital bivariate polynomial, whereas the converse question remains open.

A collection and clarification is given of the key mathematical details of the method of applying the convolution theorem as part of the solution to the two-dimensional magnetic inverse problem: reconstructing current density from magnetic field data. Particular emphasis is given to obtaining a closed-form solution for the Fourier transform of the convolution kernel that features in the Biot-Savart law. Having done this, the Fourier-transformed current density is obtained, via the convolution theorem, as a function of Fourier-transformed magnetic field data and other variables.

The asymptotic McEliece–Rodemich–Rumsey–Welch bound (1977) limits the largest attainable rate of binary codes as a function of the relative distance. After a nearly half-century hiatus, this result was recently improved in two concurrent works, by OpenAI and by O. Alrabiah and V. Guruswami. The two arguments look entirely different, a Delsarte certificate on the one hand, a classical-quantum channel and the pretty good measurement on the other, and they yield the same bound. The purpose of this note is to explain why: in both proofs, a subspace is attached to every codeword and moved with it, and the bound counts how many such subspaces fit in the ambient space, exactly in the first case and in the probabilistic sense of typicality in the second. We also present the OpenAI proof in the language and context of coding theory, as an extension of the spectral method in which the single vector attached to a codeword is replaced by a subspace.

We give a variational derivation of the central configuration equations in pair space, where the relative position vectors between pairs of bodies serve as the primary variables. Vector Lagrange multipliers enforce the linear triangle relations required for these pair vectors to be realizable in the plane. For arbitrary $N$, we show that the constrained Hessian decomposes as $H_{\mathcal C}=L^\Delta+\widetilde L$, where the gap Laplacian $L^\Delta$ is positive semidefinite and the transverse Laplacian $\widetilde L$ is signed and contains all possible negative directions. For non-collinear planar four-body central configurations, the signed part has rank two. This yields a new proof of the known bound that the Morse index is at most two. We then characterize positive definiteness and degeneracy of the Hessian by the eigenvalues of an explicit $2\times2$ matrix or, equivalently, by two generalized eigenvalues of a pencil of symmetric matrices. We illustrate the general criterion using the equal-mass square central configuration. For reflection-symmetric configurations, the generalized eigenvalue problem decomposes into two independent smaller problems. This reduction applies to kite and isosceles trapezoidal configurations. For the trapezoid, however, the reflection acts non-orthogonally on the pair coordinates, giving a different, non-orthogonal reduction. Applying the reflection-symmetry reduction to the rhombus family, we prove that the Hessian is positive definite modulo the rotational zero mode; equivalently, every rhombus central configuration is nondegenerate modulo rotations and has Morse index zero.

The paper considers the generalization of the construction of orthogonal completion to all semiprime linear algebras (systems), its implementation in maximal systems of quotients similar to the symmetric Martindale ring of quotients for known examples of their description in terms of algebras of quotients of graded Lie algebras and technical results for applying the orthogonal completion method to them.

A graph $G$ is weakly $H$-saturated if one can obtain $K_n$ by adding one edge to $G$ at a time, where each additional edge creates at least one new copy of $H$. The minimum number of edges needed for a weakly $H$-saturated graph $G$ of order $n$ is known as the weak saturation number of $H$, written $wsat(n,H)$. A graph $G$ is minimum weakly saturated if $wsat(n,G)=|E(G)|-1$ for some value of $n$. We explore classes of minimum weakly saturated graphs and their connection to the $H$-bootstrap percolation process, as well as weak saturation in a more general setting than the complete graph.

Let $S$ be a complex curve and let $X$ be a complex manifold. Given a relative regular holonomic $\DXS$-module $\shm$ with respect to the projection $X\times S\to S$, we construct an isomorphism between the solution functors respectively on the (relative subanalytic) sheaf of relative tempered holomorphic functions and the sheaf holomorphic solutions on $X\times S$.

In this paper, we prove that for every integer $k\geq2$ and every $c>1$ sufficiently close to $1$, there is $\kappa>0$ such that every sufficiently large subset of $\{1,\ldots,N\}$ of density at least $(\log\log N)^{-\kappa}$ contains \[ x,\quad x+\lfloor n^c\rfloor,\quad x+\lfloor n^c\rfloor^2, \quad\ldots,\quad x+\lfloor n^c\rfloor^k. \] We also prove pointwise almost-everywhere convergence of the associated multiple ergodic averages.

We develop a structure-preserving, data-driven collision operator for spatially homogeneous multi-species kinetic systems from molecular dynamics (MD). The operator consists of diagonal self-collision blocks and ordered off-diagonal cross-species blocks to describe intra- and inter-species momentum and energy exchange. Within a local and point-wise identifiable kernel class, we develop the necessary and sufficient condition for the admissible kernel class satisfying the conservation laws, the H-theorem, and the frame indifference. Unlike the classical Landau operator, the off-diagonal kernels are not restricted to be symmetric under permutation of the two velocity variables. This unique structural freedom captures the distinct responses of different species to unresolved correlations and many-body effects arising from micro-scale particle interactions. The equivalent parameterizable kernel formalization enables us to learn a generalized data-driven collision operator directly from MD, where the low-rank tensor representations and random sampling are used to achieve efficient kernel training and numerical simulation. Numerical experiments show that the learned operator accurately predicts transport coefficients and the non-equilibrium relaxation, while retaining discrete conservation and entropy production. In particular, it captures plasma kinetics in the moderately coupled regime, where the predictions of both the Landau and the data-driven model restricted to velocity-permutation symmetry show significant discrepancies.

We study the algebraic K-theory of quotients of even ring spectra. Building on Efimov's work, we prove that the limit of the K-theories of these quotients is the continuous K-theory of nuclear modules. For quotients by elements in the height $n$ chromatic ideal, we show that this agrees with the K-theory of the ring spectrum, after chromatic localization at heights $\geq n{+}1$.

We study graph products of groups over defining graphs of arbitrary cardinality from two closely related viewpoints. First, we characterize acylindrical hyperbolicity. If the defining graph is irreducible, has at least two vertices, and has a finite star base, then every parabolically full subgroup is either virtually cyclic or acylindrically hyperbolic. For vertex-full subgroups the finite star base condition is also necessary. In particular, the graph product itself is acylindrically hyperbolic exactly when the graph has a finite star base and the group is not virtually cyclic, or equivalently is not the infinite dihedral group. We also give the corresponding classification for reducible defining graphs. Second, motivated by our earlier work on countable right-angled Coxeter groups, we study pure conjugating automorphisms in the topology of pointwise convergence. We prove that they are topologically generated by finite-support factorwise automorphisms and partial conjugations, and establish closedness, density, exact-equality, and discreteness results. The same finite star base condition that appears in the acylindrical hyperbolicity criterion governs closedness and discreteness in the star-connected case; for arbitrary graphs, its natural refinement is the existence of a finite component witness set. Finally, for graph products of abelian groups we obtain a canonical topological semidirect decomposition of the pure conjugating automorphism group.

We provide an example of a projective tensor product whose unit ball contains an exposed tensor which is not elementary. We then transfer the construction to the symmetric projective tensor product and obtain an analogous result in that setting. Finally, we extend both constructions to projective tensor products with an arbitrary number of factors and to symmetric projective tensor products of every degree.

Let $S$ be an oriented surface, possibly of infinite type, endowed with a complete Riemannian metric with pinched negative curvature. We prove that the area form defines a non-trivial class in the second bounded cohomology group of $S$, unless $S$ is diffeomorphic to the disc or the cylinder. This is in sharp contrast with the $n$-dimensional case, $n>2$, where the non-triviality of the volume form in bounded cohomology is related to the Cheeger constant of the manifold. We also discuss how the bounded area class depends on the metric: we prove that, for compact surfaces, it recognizes constant curvature metrics among pinched negatively curved ones, while, even in the case of surfaces of infinite type, it does not distinguish non-isometric hyperbolic structures. More precisely, when $S$ is compact we show that, among the negatively curved structures of fixed area, the ones with constant curvature provide the unique minimizers for the norm of the bounded area class.

Three seemingly distinct extended-body propagators devised by Ansoldi et al., Erbin et al., and Stanford et al.\ respectively arise in completely different settings, though each can be considered a realization of the Polyakov path integral. This paper proves that the Ansoldi propagator corresponds to the characteristic function of the random variable with the probability density function $f(A)=C\,A\,(\alpha^{2}+A^{2})^{-D/2}\mathbf{1}_{A>0}$ in ambient dimension $D$, of scale $\alpha=\sqrt{b/2}$. The exponent is fixed by the dimension and by nothing else in the Ansoldi/geometric comparison; the on-shell two-point amplitude of Erbin et al. and JT gravity agree with its $D=4$ member in the senses made precise below. After an extensive literature review, the author believes that this observation and constructed area measure is novel. The area measure will, however, emerge as the natural candidate for the extension of the earlier non-extended body path space measures to two dimensional simplicial complexes. We further show that the area measure of this paper is exactly the law of the area of a triangle, two of whose sides are drawn from the measure obtained in the companion manuscripts for the one-dimensional Polyakov integral. We close with an argument, rigorous in part, which reduces the measure of tubular worldsheets between two arbitrary boundary curves to that of this triangle.

We study the nonlinear Schrödinger equation with memory $\Le^s u=\mu|u|^{p-1}u$ for $t>0$, where $\Le^s=(\partial_t-i\Delta_x)^s$, $0<s<1$, and the solution is prescribed for all $t\le0$. We identify a Hilbert space of histories and prove local well-posedness for $1<p\le p_*(s)=1+4s/(d+2-2s)$, with global solutions for sufficiently small histories at the critical exponent. We also establish the passage from this memory problem to the classical nonlinear Schrödinger equation as $s\uparrow1$. For every fixed Schwartz history and $1<p\le1+4/d$, including the classical mass critical exponent, the memory solutions converge in $C([0,T];L^2\cap L^{p+1})$ to the classical solution with initial value $u_0(\cdot,0)$, for every finite $T$ below its maximal existence time. Thus the limiting evolution depends on the prescribed past only through its value at time zero. Our construction uses an oscillatory Poisson extension and an exact identity expressing the weighted $L^2$ norm of the lifted initial datum in terms of the history. This identity allows us to apply the boundary estimates of our earlier work to the memory problem. We recover $\Le^s$ as a weighted Dirichlet-to-Neumann map and establish the corresponding extension bounds on $L^2$ for $0<s\le\frac12$, and from $\operatorname{Dom}(\Le^s)$ to $L^2$ for $\frac12<s<1$.

We determine the density of locally soluble fibres in certain families of varieties over a toric base. For this we develop a new form of the hyperbola method for toric bases which allows for counting functions with a growth behaviour of the form $B^a(\log B)^{-\Delta}$.

We prove that every $3$-manifold equipped with a proper metric admitting a convex geodesic bicombing has Assouad–Nagata dimension at most $3$. In particular, every proper CAT$(0)$ $3$-manifold has Assouad–Nagata dimension at most $3$. As a corollary, the isoperimetric results of Lang–Stadler–Urech \cite{LSU} and Peteranderl \cite{Pet}, previously applicable to $3$-dimensional Hadamard manifolds, also apply to proper CAT$(0)$ $3$-manifolds of asymptotic rank at most $2$.

No single optimization method is uniformly best for all problems, and the most suitable optimizer choice can change during a run. Existing approaches that change optimizer during execution typically predetermine part of the strategy: the portfolio is restricted to one algorithm class, the switch occurs once at a fixed time, or the frequency of decisions is treated as a hyperparameter rather than a learned one. We introduce "Reinforcement Learning to Choose Optimizers", which formulates the optimization algorithm choice as a sequential decision-making problem. At each decision, a recurrent policy reads the current run state and decides both which optimizer should be used next and for how long. The portfolio includes both gradient-based and derivative-free optimizers, and each switch passes on the current best solution and a representative step size. A context proxy conditions a gating network over expert heads, and training employs a decoupled actor-critic whose return is expressed in the same empirical runtime distribution metric used at evaluation. Training tasks and portfolio are designed jointly so that no optimizer dominates. On unseen problems, the learned policy outperforms every portfolio optimizer at all but the smallest budgets, and it remains robust under distribution shift.

We classify lens spaces for which the canonical negative-definite plumbing has minimal $b_2$ among all symplectic fillings of the standard contact structure. The classification is given by 4 infinite families of "forbidden subgraphs" in a manner similar to Aceto-McCoy-Park's work on smooth negative-definite fillings of lens spaces. In particular, no classification of this form can be given using a finite list of forbidden configurations, highlighting a difference between symplectic fillings and smooth negative-definite fillings of lens spaces.

Let $F_n$ and $L_n$ denote the Fibonacci and Lucas numbers, respectively, and consider \[ E_k:\qquad y^2=(F_{2k}x+1)(F_{2k+2}x+1)(F_{2k+4}x+1). \] These elliptic curves arise naturally from the regular Diophantine triples \[ \{F_{2k},F_{2k+2},F_{2k+4}\}. \] For odd $k$, we exhibit the rational point \[ Q_k=\left( -\frac{F_{k-1}}{L_kF_{k+1}F_{k+2}}, \frac{F_{2k+1}}{L_kF_{k+1}F_{k+2}} \right). \] For every odd $k\geq 3$, this point is independent of the standard point $P_k=(0,1)$; in particular, $\operatorname{rank}E_k(\mathbb{Q})\geq 2$. Moreover, if $k\geq 3$ is odd and $\operatorname{rank}E_k(\mathbb{Q})=2$, then all integer points on $E_k$ are exactly the points arising from the two known solutions of the Hoggatt-Bergum extension problem. By parametrizing the two conics $L^2-5F^2=\pm4$ and applying an injective specialization criterion, we also show that the corresponding one-parameter elliptic families have generic ranks $2$ in the odd case and $1$ in the even case. Finally, we discuss computational data and propose the heuristic rank distribution $1/4,1/2,1/4$ for ranks $1,2,3$, respectively, with density zero for rank at least $4$.

We give a nibble-free construction proving $R(3,k)\ge(1/200+o(1))k^2/\log k$. We also include Shearer's proof bounding the independence number of a triangle-free graph, which implies $R(3,k)\le (1+o(1))(k^2/\log k)$.

We investigate automorphic forms for congruence subgroups of $\mathrm{SL}(3, \mathbb{Z})$ that are Iwahori at $p$. In particular, we study the cuspidal cohomology of Iwahori congruence subgroups $\mathcal{I}(3, p)$, which are comprised of matrices in $\mathrm{SL}(3, \mathbb{Z})$ that are upper triangular modulo $p$. In order to work with $\mathcal{I}(3, p)$, we generalize key results from Ash, Grayson, and Green [J.\ Number Theory 19 (1984), pp.\ 412-436]. For levels $\mathcal{I}(3, p)$ with $p \leq 227$, we found three levels with nonzero cuspidal classes and were able to compute the action of Hecke operators at two of these levels. These are the first examples of non-essentially-self-dual automorphic representations of trivial cohomological weight that are Steinberg at $p$ appearing at Iwahori level.

We study the hypoellipticity of operators on a product type Lorentzian manifold where the time variable is periodic. In particular, we prove that hypoellipticity holds for such time-periodic equations, with a stronger estimate when the mass parameter or time period lies outside a set of arbitrarily small measure. We consider both compact and noncompact spatial manifolds and provide explicit examples involving the wave operator. The proof relies on a Fourier series decomposition and asymptotics for eigenvalue counting functions.