arXiv Math @arxiv_math@qoto.org

The study of stochastic variational principles involves the problem of constructing fixed-endpoint and adapted variations of semimartingales. We provide a detailed construction of variations of semimartingales that are not only fixed at deterministic endpoints, but also fixed at first entry times and first exit times for charts in a manifold. We prove a stochastic version of the fundamental lemma of calculus of variations in the context of these variations. Using this framework, we provide a generalization of the stochastic Hamilton-Pontryagin principle in local coordinates to arbitrary noise semimartingales. For the corresponding global form of the stochastic Hamilton-Pontryagin principle, we introduce a novel approach to global variational principles by restricting to semimartingales obtained as solutions of Stratonovich equations.

The Ishango Bone, discovered in 1950 near the Semliki River in present-day Democratic Republic of Congo and dated to over 24,000 years ago, features a series of 16 grouped numerical notches arranged across three columns. The interpretation of these markings has long been debated, ranging from numerical intention to decorative symbolism. This study presents a newly identified structural pattern, uncovering four irregularities in the numerical groupings that, when resolved, reveal a consistent internal logic across all three columns. The analysis identifies repeating sums, dualistic pairing structures, and cross-column symmetry, pointing toward a deliberate and complex mathematical arrangement. These findings suggest the notches may have functioned as a form of reference system, potentially using stone markers for visual or narrative purposes. Hypotheses include links to cosmic cycles, mathematical instruction, or mythological integration. While interpretations remain speculative, this study argues for an interdisciplinary reevaluation of the Ishango Bone's significance in terms of mathematical, cultural, and cosmological understanding. It offers new insights into the symbolic thinking and cognitive capacities of prehistoric societies and challenges conventional assumptions about the origins of numeracy.

In 1969 J. Verhoeff provided the first examples of a decimal error detecting code using a single check digit to provide protection against all single, transposition and adjacent twin errors. The three versions of such a code that he presented are length 3-digit codes with 2 information digits. Existence of a 4-digit code would imply the existence of 10 such disjoint 3-digit codes. This paper presents 3 pairwise disjoint 3-digit codes. The codes developed herein, have the property that the knowledge of the multiset of digits included in a word is sufficient to determine the entire codeword even though their positions were unknown. Thus the codes are permutation-free, and this fulfills his desire to eliminate cyclic errors. Phonetic errors, where 2 digit pairs of the forms X0 and 1X are interchanged, are also eliminated.

In this manuscript, we extend the global gradient estimates for positive solutions to the heat equation under a general compact Finsler $CD(-K,N)$ geometric flow and derive the corresponding Harnack inequality.

We present a mathematical framework for quantifying energy efficiency in intelligent systems by linking energy consumption to information processing capacity. Our proposed watts per intelligence metric integrates algorithmic thermodynamic principles of Landauer with computational models of machine intelligence. By formalising the irreversible energy costs of computation, we derive rigorous lower bounds on energy usage of algorithmic intelligent systems and their adaptability. We introduce theorems that constrain the trade offs between intelligence output and energy expenditure. Our results contribute to design principles for energy efficient intelligent systems.

In 1971, by induction on $n$ and using a two-term linear recurrence relation, Graham and Pollak got a beautiful formula $$\det(D_n)=-(n-1)(-2)^{n-2}$$ on the determinant of distance matrix $D_n$ of a tree $T_n$ on $n$ vertices. The recurrence relations are very crucial when proving this formula by inductive method: in 2006, Yan and Yeh used two-term and three-term recurrence relations; in 2020, Du and Yeh used a homogeneous linear three-term recurrence relation. In this paper, we analyze the subtree structure of the tree and find four-term, five-term, six-term and seven-term homogeneous linear recurrence relations on $\det(D_n)$, as a corollary new proofs of Graham and Pollak's formula can be given.

A red-white coloring of a nontrivial connected graph $G$ is an assignment of red and white colors to the vertices of~$G$. Associated with each vertex $v$ of $G$ of diameter $d$ is a $d$-vector, called the code of $v$, whose $i$th coordinate is the number of red vertices at distance $i$ from $v$. A red-white coloring of $G$ for which distinct vertices have distinct codes is called an ID-coloring of $G$. In 2025, a criterion to determine whether a red-white coloring of a path is an ID-coloring or not was presented by Kono, with the aid of a result shown by Marcelo et al. in 2024. The criterion utilizes the fact that ID-colorings of paths are ``opposite'' of colorings with a certain symmetry. In this paper, we establish a similar criterion that can be applied for cycles whose order is a prime number at least 3. In order to do so, we employ an analogous approaches used for the criterion for paths, i.e., we pay attention to symmetries of given red-white colorings of cycles.

The goal of this paper is to show that Stokes data coming from flat bundles form a locally geometric derived stack locally of finite presentation. This generalizes existing geometricity results on Stokes data in four different directions: our result applies in any dimension, $\infty$-categorical coefficients are allowed, derived structures on moduli spaces are considered and more general spaces than those arising from flat bundles are permitted.

Let $G$ be a finite group. For subgroups $U$ and $V$ let $1_U^G$ and $1_V^G$ be the permutation characters for the action of $G$ on the right cosets of $U$ and $V$, respectively. Let $N$ be a normal subgroup of $G$. Norbert Klingen, in his book, shows that if $1_U^G=1_V^G$, then $1_{NU}^G=1_{NV}^G$. We give a counterexample to an argument in his proof and we give a new proof of this statement.

A status updating system is considered in which multiple processes are sampled and transmitted through a shared channel. Each process has its dedicated server that processes its samples before time stamping them for transmission. Time stamps, however, are prone to errors, and hence the status updates received may not be credible. Our setting models the time stamp error rate as a function of the servers' busy times. Hence, to reduce errors and enhance credibility, servers need to process samples on a relatively prolonged schedule. This, however, deteriorates timeliness, which is captured through the age of information (AoI) metric. An optimization problem is formulated whose goal to characterize the optimal processes' schedule and sampling instances to achieve the optimal trade-off between timeliness and credibility. The problem is first solved for a single process setting, where it is shown that a threshold-based sleep-wake schedule is optimal, in which the server wakes up and is allowed to process newly incoming samples only if the AoI surpasses a certain threshold that depends on the required timeliness-credibility trade-off. Such insights are then extended to the multi-process setting, where two main scheduling and sleep-wake policies, namely round-robin scheduling with threshold-waiting and asymmetric scheduling with zero-waiting, are introduced and analyzed.

Building on work of Harer \cite{Ha86}, we construct a spine for the decorated Teichmüller space of a non-orientable surface with at least one puncture and negative Euler characteristic. We compute its dimension, and show that the deformation retraction onto this spine is equivariant with respect to the pure mapping class group of the non-orientable surface. As a consequence, we obtain a model for the classifying space for proper actions of the pure mapping class group of a punctured non-orientable surface, which is of minimal dimension in the case there is a single puncture.

A family of sets is $s$-intersecting if every pair of its sets has at least $s$ elements in common. It is an $s$-star if all its members have some $s$ elements in common. A family of sets is called $s$-EKR if all its $s$-intersecting subfamilies have size at most that of some $s$-star. For example, the classic 1961 Erdős-Ko-Rado theorem states essentially that the family of $r$-sized subsets of $\{1,2,\ldots,n\}$ is $s$-EKR when $n$ is a large enough function of $r$ and $s$, and the 1967 Hilton-Milner theorem provides the near-star structure of the largest non-star intersecting family of such sets. Two important conjectures along these lines followed: by Chvátal in 1974, that every family of sets that all subsets of its members is 1-EKR, and by Holroyd and Talbot in 2005, that, for every graph, the family of all its $r$-sized independent sets is 1-EKR when every maximal independent set has size at least $2r$. In this paper we present similar 1-EKR results for families of length-$r$ paths in graphs, specifically for sun graphs, which are cycles with pendant edges attached in a uniform way, and theta graphs, which are collections of pairwise internally disjoint paths sharing the same two endpoints. We also prove $s$-EKR results for such paths in suns, and give a Hilton-Milner type result for them as well. A set is a transversal of a family of sets if it intersects each member of the family, and the transversal number of the family is the size of its smallest transversal. For example, stars have transversal number 1, and the Hilton-Milner family has transversal number 2. We conclude the paper with some transversal results involving what we call triangular families, including a few results for projective planes.

In this paper, we investigate the existence of quantum fractional revival in unitary Cayley graphs over finite commutative rings with identity. We characterize all finite local rings that permit quantum fractional revival in their unitary Cayley graphs. Additionally, we present results for the case of finite commutative rings, as they can be expressed as products of finite local rings.

In this paper, we present and explore several key concepts within the framework of Hom-Poisson algebras. Specifically, we introduce the notions of admissible Hom-Poisson algebras, along with the related ideas of matched pairs and Manin triples for such algebras. We then define the concept of a purely admissible Hom-Poisson bialgebra, placing particular emphasis on its compatibility with the Manin triple structure associated with a nondegenerate symmetric bilinear form. This compatibility is crucial for understanding the structural interplay between these algebraic objects. Additionally, we investigate the notion of Hom-$ \mathcal O$-operators acting on admissible Hom-Poisson algebras. We analyze their properties and establish a connection with admissible Hom-pre-Poisson algebras, shedding light on the relationship between these two structures.

The purpose of this article is to bring together the third-order Jacobsthal numbers and 3-parameter generalized quaternions, which are a general form of the quaternion algebra according to 3-parameters. With this purpose, we introduce and examine a new type of quite big special numbers system, which is called third-order Jacobsthal 3-parameter generalized quaternions (shortly, third-order Jacobsthal 3PGQs). Further, we compute both some new equations and classical well-known equations such as: linear recurrence, Binet formulas, generating function, sum formulas, Cassini identity and d'Ocagne identity.

The relationship between solvability of linear diffential-algebraic equations (DAEs) and their transformability into canonical forms has been investigated for more than forty years. After a comparative analysis of numerous DAE frameworks the notions regularity and almost regularity were established only recently. Regular DAEs resulted to be equivalently transformable into so-called standard canonical forms (SCF) with block-structured nilpotent matrix functions featuring certain rank properties. In this paper we prove that for regular DAEs, even a transformation into a strong standard canonical form (SSCF) is possible, i.e. a SCF with a constant nilpotent matrix. We start from block-structured SCF and give a constructive proof.

This paper explores applications of the so-called Freese's technique, a classical approach to study the congruence variety of a given algebra. We leverage this tool to investigate lattices that are admissible as congruence sublattice of a given algebra. In particular, we present a novel characterization of congruence modular varieties, Taylor varieties, and varieties satisfying a non-trivial congruence identity by means of lattice omission.

In this paper we describe a method to estimate a neighborhood containing a periodic orbit of a given system of two ordinary differential equations. By using the theory of integral averages, the system of differential equations can be transformed into an equivalent autonomous system which, by using the Hopf bifurcation theorem, the existence of a periodic solution of this autonomous nonlinear differential equations can be demonstrated. Using of this procedure it is possible to estimate an annular region where the orbit of the periodic solution is located. The method allows to improve the results that the Hopf Bifurcation provides on periodic solutions. In addition, some quantitative characteristics of the solution can be known, such as the amplitude, the period and, a region where the periodic orbit of the original system is located. The method is applied to a three-dimensional system of differential equations that models the competition of two predators and one prey, which under the assumption that the predators are equally voracious, property that in this case leads to a two-dimensional system, where all of the conditions of the method described here are easily applicable.

We present a unified categorical framework that connects the syntactic Henkin construction for the first-order Completeness Theorem with Lawvere's Fixed-Point Theorem. Concretely, we define two canonical functors from the category of first-order theories to the category of their models, and then introduce a canonical natural transformation that links the Henkin-based term models to semantically constructed models arising from compactness or saturation arguments. We prove that every component of this natural transformation is an isomorphism, thereby establishing a strong equivalence between the syntactic and semantic perspectives. Furthermore, we show that this transformation is 2-categorically rigid: any other natural transformation in the same setting is uniquely isomorphic to it. Our framework highlights the shared diagonalization principle underlying both Henkin's and Lawvere's methods and demonstrates concrete applications in automated theorem proving, formal verification, and the design of advanced type-theoretic systems.

The \emph{Alon-Tarsi number} of a graph $G$ is the smallest $k$ so that there exists an orientation $D$ of $G$ with max outdegree $k-1$ satisfying the number of even Eulerian subgraphs different from the number of odd Eulerian subgraphs. In this paper, the Alon-Tarsi number of the $n$-cube is obtained according to its special properties, we obtain the Alon-Tarsi number of Cartesian product of some special bipartite graphs, and get the Alon-Tarsi number of Corona product of graphs. As corollaries, we get the Alon-Tarsi number of Cartesian product and Corona product of hypercube graph and special graphs.