arXiv Math @arxiv_math@qoto.org

In this paper, we show a physics-free derivation of a Landau-Zener type integral introduced by Kholodenko and Silagadze.

Given $k$ graphs $G_{1}, \ldots, G_{k}$, their intersection is the graph $(\cap_{i\in [k]}V(G_{i}), \cap_{i\in [k]}E(G_{i}))$. Given $k$ graph classes $\mathcal{G}_{1}, \ldots , \mathcal{G}_{k}$, we call the class $\{G: \forall i \in[k], \exists G_{i} \in \mathcal{G}_{i} \text{ such that } G=G_{1}\cap \ldots \cap G_{k}\}$ the graph-intersection of $\mathcal{G}_{1}, \ldots , \mathcal{G}_{k}$. The main motivation for the work presented in this paper is to try to understand under which conditions graph-intersection preserves $χ$-boundedness. We consider the following two questions: Which graph classes have the property that their graph-intersection with every $χ$-bounded class of graphs is $χ$-bounded? We call such a class intersectionwise $χ$-guarding. We prove that classes of graphs which admit a certain kind of decomposition are intersectionwise $χ$-guarding. We provide necessary conditions that a finite set of graphs $\mathcal{H}$ should satisfy if the class of $\mathcal{H}$-free graphs is intersectionwise $χ$-guarding, and we characterize the intersectionwise $χ$-guarding classes which are defined by a single forbidden induced subgraph. Which graph classes have the property that, for every positive integer $k$, their $k$-fold graph-intersection is $χ$-bounded? We call such a class intersectionwise self-$χ$-guarding. We study intersectionwise self-$χ$-guarding classes which are defined by a single forbidden induced subgraph, and we prove a result which allows us construct intersectionwise self-$χ$-guarding classes from known intersectionwise $χ$-guarding classes.

We present a strategy for proving an asymptotic upper bound on the number of defects (non-hexagonal Voronoi cells) in the $n$ generator optimal quantizer on a closed surface (i.e., compact 2-manifold without boundary). The program is based upon a general lower bound on the optimal quantization error and related upper bounds for the Löschian numbers $n$ (the norms of the Eisenstein integers) based upon the Goldberg-Coxeter construction. A gap lemma is used to reduce the asymptotics of the number of defects to precisely the asymptotics for the gaps between Löschian numbers. We apply this strategy on the hexagonal torus and prove that the number of defects is at most $O(n^{1/4})$ -- strictly fewer than surfaces with boundary -- and conjecture (based upon the number-theoretic Löschian gap conjecture) that it is in fact $O(\log n)$. Incidentally, the method also yields a related upper bound on the variance of the areas of the Voronoi cells. We show further that the bound on the number of defects holds in a neighborhood of the optimizers. Finally, we remark on the remaining issues for implementation on the 2-sphere.

Elementary school students sometimes exhibit deplorable behaviors when faced with arithmetic problems. These behaviors may result from certain teaching practices and could be mitigated by a few actions that are relatively easy to implement in the classroom. We propose four avenues focusing on the choice and formulation of problems and the clarification of elements of the didactical contract.

We establish the order of the maximum length of an increasing sequence, bounded by $n$, in which the largest prime divisor of the elements form a decreasing sequence.

We examine connections between the mathematics behind methods of drawing geographical maps due, on the one hand to Marinos and Ptolemy (1st-2nd c. CE) and Delisle and Euler (18th century). A recent work by the first two authors of this article shows that these methods are the best among a collection of geographical maps we term ``conical''. This is an example of an instance where after practitioners and craftsmen (here, geographers) have used a certain tool during several centuries, mathematicians prove that this tool is indeed optimal. Many connections among geography, astronomy and geometry are highlighted.The fact that the Marinos-Ptolemy and the Delisle-Euler methods of drawing geographical maps share many non-trivial properties is an important instance of historical continuity in mathematics.

We present the foundational theory of condensed sets and basic condensed algebra after having introduced key concepts from category theory and homological algebra. In the later sections, we indicate the relevance of condensed mathematics to classical fields such as functional analysis and ergodic theory. We include many pointers to the literature, where most of the given results and proofs can be found.

Let $\mathbb{N}$ and $\mathcal{P}$ be the sets of natural numbers and primes, respectively. Motived by an old problem of Erd\H os and Kalmár, we prove that for almost all $y>1$ the lower asymptotic density of integers of the form $p+\lfloor y^k\rfloor~(p\in \mathcal{P},k\in \mathbb{N})$ is positive.

In this paper, we introduce a new class of polynomials, called probabilistic q-Bernstein polynomials, alongside their generating function. Assuming Y is a random variable satisfying moment conditions, we use the generating function of these polynomials to establish new relations. These include connections to probabilistic Stirling numbers of the second kind and higher-order probabilistic Bernoulli polynomials associated with Y. Additionally, we derive recurrence and differentiation properties for probabilistic q-Bernstein polynomials. Utilizing Leibniz's formula, we give an identity for the generating function of these polynomials. In the latter part of the paper, we explore applications by choosing appropriate random variables such as Poisson, Bernoulli, Binomial, Geometric, Negative Binomial, and Uniform distributions. This allows us to derive relationships among probabilistic q-Bernstein polynomials, Bell polynomials, Stirling numbers of the second kind, higher-order Frobenius-Euler numbers, and higher-order Bernoulli polynomials. We also present p-adic q-integral and fermionic p-adic q-integral representations for probabilistic q-Bernstein polynomials.

In this paper we obtain a complete characterization of reducing, invariant, and hyperinvariant subspaces for the completely non-unitary component of a power partial isometry. In particular, precise characterization of reducing, invariant, and hyperinvariant subspaces of a truncated shift operator has been achieved.

In the current paper, we investigate the fifth order modified KP-I eqaution, namely \begin{equation*} \partial_t u-\partial_{x}^{5}u-\partial_{x}^{-1}\partial_{y}u+\partial_{x}(u^3)=0. \end{equation*} This equation is $L^2$ critical and we prove on $\mathbb{R}\times\mathbb{R}$ that it is globally well posed in the natural energy space if the $L^2$ norm of the initial data is less the $L^2$ norm of the ground state associated to this equation. We also find a subspace of the natural energy space associated to this equation where we have local well-posedness, nevertheless if the initial data is sufficiently localized we obtain blow-up. On $\mathbb{R}\times \mathbb{T},$ we prove global well-posedness in the energy space for small data.

In this paper, we demonstrate that several classes of functions, specifically n-multiplicative isomorphisms, derivations, elementary maps, and Jordan elementary maps on a class of algebras that includes Jordan algebras with idempotents, J(a)-axial algebras and M(a,b)-axial algebras, are additive under appropriate conditions, which may be referred to as Martindale-type conditions. Furthermore, we answer the question left open in the recent article titled "Multiplicative isomorphisms and derivations on axial algebra."

We construct a description of graded derivations in group algebras. Using this result for arbitrary graduation of the group algebra, we describe all possible structures of DG algebras. The corresponding examples are given. The description is given in terms of characters on a groupoid analogous to the groupoid of inner action.

We point out some minor errors in a paper by the first author, and explain why they do not affect the main results in the paper.

We prove expressive completeness results for convex propositional and modal team logics, where a logic is convex if, for each formula, if it is true in two teams $t$ and $u$ and $t\subseteq s\subseteq u$, then it is also true in $s$. We introduce multiple propositional/modal logics which are expressively complete for the class of all convex propositional/modal team properties. We also answer an open question concerning the expressive power of classical propositional logic extended with the nonemptiness atom NE: we show that it is expressively complete for the class of all convex and union-closed properties. A modal analogue of this result additionally yields an expressive completeness theorem for Aloni's Bilateral State-based Modal Logic. In a specific sense, one of the novel propositional convex logics extends propositional dependence logic and another, propositional inquisitive logic. We generalize the notion of uniform definability, as considered in the team semantics literature, to formalize the notion of extension pertaining to the convex logics.

We present a new class of numerical methods for solving stochastic differential equations with additive noise on general Riemannian manifolds with high weak order of accuracy. In opposition to the popular approach with projection methods, the proposed methods are intrinsic: they only rely on geometric operations and avoid coordinates and embeddings. We provide a robust and general convergence analysis and an algebraic formalism of exotic planar Butcher series for the computation of order conditions at any high order. To illustrate the methodology, an explicit method of second weak order is introduced, and several numerical experiments confirm the theoretical findings and extend the approach for the sampling of the invariant measure of Riemannian Langevin dynamics.

In this paper, we study representations from the four-punctured sphere group into isometry groups of Gromov-hyperbolic spaces. We prove that the set of simple-stable representations (in analogy with Minsky's notion of primitive-stability) and the set of Bowditch representations (inspired by Bowditch's work on the free group of rank two, generalized by Tan, Wong and Zhang) are equal. Along the way, we study the combinatorics of simple closed curves on the four-punctured sphere and prove a result which quantifies the redundancy of subwords of certain given lengths within simple words.

We provide a superalgebraic analogue of Markov numbers, which are defined as the Grassmann integer solutions to the equation $x^2 + y^2 + z^2 + (xy + yz + xz)ε= 3(1 + ε)xyz$, as well as applications to the Decorated Super Teichmüller spaces associated to the once-punctured torus and certain annuli. We conclude with further directions for study.

Graded vector bundles over a given $\mathbb{Z}$-graded manifold can be defined in three different ways: certain sheaves of graded modules over the structure sheaf of the base graded manifold, finitely generated projective graded modules over the algebra of global functions on the base graded manifold, or locally trivial graded manifolds with a suitable linear structure. We argue that all three approaches are the same. More precisely, the respective categories are proved to be equivalent.