arXiv Math @arxiv_math@qoto.org

In this study, we delve into a novel dynamic system inspired by Montgomery's pair correlation conjecture in number theory. The dynamic system is intricately designed to emulate the behavior of the nontrivial zeros of the Riemann zeta function. Our exploration encompasses bifurcation analysis and Lyapunov exponents to scrutinize the system's behavior and stability, offering insights into both small and large initial conditions. Our efforts extend to unveiling the probability distribution characterizing the dynamics for varying initial conditions. The dynamic system unfolds intricate behaviors, displaying sensitivity to initial conditions and revealing complex bifurcation patterns. Small deviations in the initial conditions unveil significantly different trajectories, reminiscent of chaotic systems. Lyapunov exponents become our lens into understanding stability and chaos within the system. A comparative analysis between the dynamic system's approximate solutions and the actual nontrivial zeros of the Riemann zeta function enhances our comprehension of model accuracy and its potential implications for number theory. This research illuminates the versatility of dynamic systems as analogs for studying complex mathematical phenomena. It provides fresh perspectives on the pair correlation conjecture, establishing connections with nonlinear dynamics and chaos theory. Notably, we delve into the boundedness of solutions for both small and large initial conditions, unraveling the distinctive probability distribution governing the dynamics in each scenario. Furthermore, we introduce an in-depth analysis of the entropy of our dynamic system for both small and large initial conditions. The entropy study enhances our understanding of the predictability and stability of the system, shedding light on its behavior in different parameter regimes.

In high-speed maritime operations, the broaching phenomenon can pose a significant risk when navigating in following/quartering seas. The occurrence of this phenomenon can result in a violent yaw motion, regardless of the steering effort, which, in turn, cause the resulting centrifugal force to capsize a vessel. A necessary condition for the occurrence of broaching is the surf-riding phenomenon. Therefore, the International Maritime Organization (IMO) has set up criteria to include theoretical formulas for estimating the occurrence of surf-riding phenomena. The theoretical equation used in the IMO's second-generation intact stability criteria (SGISC) to estimate the surf-riding threshold is based on Melnikov's method. This paper presents nonlinear equations describing the forward and backward motions of a ship. However, such equations cannot be directly solved; therefore, we proposed the use of and explain various approximate solution methods, including Meknikov's method. Subsequently, the relationship between the theoretical prediction method of the surf-riding threshold rooted in Melnikov's method and the IMO's SGISC is determined.

The "time-and-band limiting" commutative property was found and exploited by D. Slepian, H. Landau and H. Pollak at Bell Labs in the 1960's, and independently by M. Mehta and later by C. Tracy and H. Widom in Random matrix theory. The property in question is the existence of local operators with simple spectrum that commute with naturally appearing global ones. Here we give a general result that insures the existence of a commuting differential operator for a given family of exceptional orthogonal polynomials satisfying the "bispectral property". As a main tool we go beyond bispectrality and make use of the notion of Fourier Algebras associated to the given sequence of exceptional polynomials. We illustrate this result with two examples, of Hermite and Laguerre type, exhibiting also a nice Perline's form for the commuting differential operator.

We introduce a spin field approach, that is compatible with the Cartan moving frame method, to describe the submanifold in a flat space. In fact, we consider a kind of spin field $ψ$, that satisfies a Killing spin field equation (analogous to a Killing spinor equation) written in terms of the Clifford algebra, and we use the spin field to locally rotate the orthonormal basis $\{\hat{e}_\mathtt{I}\}$. Then, the deformed orthonormal frame $\{\tildeψ\hat{e}_\mathtt{I}ψ\}$ can be seen as the moving frame of a submanifold. We find some solutions to the Killing spin field equation and demonstrate an explicit example. Using the product of the spin fields, one can easily generate a new immersion submanifold, and this technique should be useful for studies in geometry and physics. Through the spin field, we find a linear relation between the connection and the extrinsic curvature of the submanifold. We propose a conjecture that any solution of the Killing spin field equation can be locally written as the product of the solutions we find.

We employ Mittag-Leffler type kernels to solve a system of fractional differential equations using fractal-fractional (FF) operators with two fractal and fractional orders. Using the notion of FF-derivatives with nonsingular and nonlocal fading memory, a model of three polluted lakes with one source of pollution is investigated. The properties of a non-decreasing and compact mapping are used in order to prove the existence of a solution for the FF-model of polluted lake system. For this purpose, the Leray-Schauder theorem is used. After exploring stability requirements in four versions, the proposed model of polluted lakes system is then simulated using two new numerical techniques based on Adams-Bashforth and Newton polynomials methods. The effect of fractal-fractional differentiation is illustrated numerically. Moreover, the effect of the FF-derivatives is shown under three specific input models of the pollutant: linear, exponentially decaying, and periodic.

We discuss transformations on matrices that preserve the effective spectrum and/or the effective spectral radius.

This paper is an introductory text to the theory of $q$-deformed Fourier transforms, as first discussed by Rogov and Olshanetsky. We derive the well-known results in detail, present them in a format that suits our needs, and include some new findings on specific aspects of the theory.

The purpose of the present paper is to investigate cohomologies of Reynolds Lie-Yamaguti algebras of any weight and provide some applications. First, we introduce the notion of Reynolds Lie-Yamaguti algebras and give some new examples. Moreover, cohomologies of Reynolds operators and Reynolds Lie-Yamaguti algebras with coefficients in a suitable representation are established. Finally, formal deformations and abelian extensions of Reynolds Lie-Yamaguti algebras are characterized in terms of lower degree cohomology groups.

A susceptible, asymptomatic, infectious, quarantined, and hospitalized (SAIQH) compartmental model on time scales is introduced and a suitable Lyapunov function is defined. Main results include: the proof that the system is permanent; proof of existence of solution; and sufficient conditions implying the dynamic system to have a unique almost periodic solution that is uniformly asymptotically stable. An example is presented supporting the obtained results.

Given two nilpotent endomorphisms, we determine when their lattices of hyperinvariant subspaces are isomorphic. The study of the lattice of hyperinvariant subspaces can be reduced to the nilpotent case when the endomorphism has a Jordan-Chevalley decomposition; for example, it occurs if the underlying field is the field of complex numbers.

This paper is devoted to the complete algebraic classification of complex 5-dimensional nilpotent bicommutative algebras.

Under suitable requirements on a kernel on a locally compact space, we develop a theory of inner (outer) pseudo-balayage of quite general signed Radon measures (not necessarily of finite energy) onto quite general sets (not necessarily closed). Such investigations were initiated in Fuglede's work (Anal. Math., 2016), which was, however, mainly concerned with the outer pseudo-balayage of positive measures of finite energy, whereas treatment of signed measures of infinite energy requires essentially new methods and approaches. To illustrate this, it is enough to note that in the particular case of kernels satisfying the balayage principle, the pseudo-balayage of positive measures coincides with their balayage, whereas for signed measures, these two concepts are drastically different. The theory thereby established enables us to provide necessary and sufficient conditions for the solvability of the inner Gauss variational problem, a point of interest for a number of recent researches in constructive function theory. This study covers many interesting kernels in classical and modern potential theory, which looks promising for further applications.

In a graph $A$, the measure $|M_g^A(f)|=m_g^A(f)$ for each arbitrary edge $f=gh$ counts the edges in $A$ closer to $g$ than $h$. $A$ is termed an edge quasi-$λ$-distance-balanced graph in a metric space (abbreviated as $EQDBG$), where a rational number ($>1$) is assigned to each edge $f=gh$ such that $m_g^A(f)=λ^{\pm1}m_h^A(f)$. This paper introduces and discusses these graph concepts, providing essential examples and construction methods. The study examines how every $EQDBG$ is a bipartite graph and calculates the edge-Szeged index for such graphs. Additionally, it explores their properties in Cartesian and lexicographic products. Lastly, the concept is extended to nicely edge distance-balanced and strongly edge distance-balanced graphs revealing significant outcomes.

Let $[SU(2n),\mathscr{L}]$ denote the bordism class of $SU(2n)$ $(n\ge 2)$ equipped with the left invariant framing $\mathscr{L}$. Then it is well known that $e_\mathbb{C}([SU(2n), \mathscr{L}])=0$ in $\mathbb{O}/\mathbb{Z}$ where $e_\mathbb{C}$ denotes the complex Adams $e$-invariant. In this note we show that replacing $\mathscr{L}$ by the twisted framing by a specific map it can be transformed into a generator of $\mathrm{Im} \, e_\mathbb{C}$. In addition to that we also show that the same procedure affords an analogous result for a quotient of $SU(2n+1)$ by a circle subgroup which inherits a canonical framing from $SU(2n+1)$ in the usual way.

We formulate the Alternating Current Optimal Power Flow Problem (ACOPF) as a Linear Constrained Quadratic Program (LCQP) with many negative eigenvalues ($r$) and linear constraints, making it NP-hard. We propose two algorithms, Feasible Successive Linear Programming (FSLP) and Feasible Branch-and-Bound (FBB), for a global optimal solution. These use optimization strategies like bounded successive linear programming, convex relaxation, initialization, and branch-and-bound to find a globally optimal solution within a predefined $ε$-tolerance. The complexity of FSLP and FBB is $\mathcal{O}\left(N \prod_{i=1}^r\left\lceil\frac{\sqrt{r}(t_u^i-t_l^i)}{2 \sqrtε}\right\rceil\right)$, where $N$ is the complexity of solving subproblems at each FBB node. Variables $t_l$ and $t_u$ are the lower and upper bounds of $t$, respectively, and $-|t|^2$ is the negative quadratic component in the ACOPF objective function. We use penalized semidefinite modeling, convex relaxation, and line search to design a globally feasible branch-and-bound algorithm for the LCQP form of ACOPF, finding an optimal solution within $ε$-tolerance. Initial results show FSLP and FBB can find global optimal solutions for large-scale ACOPF instances, even with large $r$, and outperform other methods in most PG-lib tests.

This paper suggests integrating one-dimensional optimization methods to tackle diverse problems, emphasizing their significance in resolving practical issues and applying mathematical principles to real-world contexts. It focuses on employing the Octave programming language, backed by specific examples, to simplify the practical application of mathematical concepts and improve problem-solving abilities. The research aims to assess the effect on students' comprehension of one-dimensional optimization.

Let $k$ be a field and $X$ be a smooth projective surface over $k$ with a rational point, we discuss the condition of splitting off the top cell for the motivic stable homotopy type of $X$. We also study some outlying examples, such as K3 surfaces.

In this paper we discuss the geometry of homogeneous spaces witch are almost Hermitian submanifolds of flag manifolds. We prove that such spaces are necessarily minimal submanifolds and in the case where these submanifolds are also flag manifolds, they are totally geodesic.

The weak-cop number of a graph, introduced by Lee et al (2023), is a quasi-isometry invariant of graphs and hence of finitely generated groups. While for any $m\in\mathbb{Z}_+\cup\infty$ there exists graphs with weak-cop number $m$, it is an open question whether there exists finitely generated groups whose weak-cop number is different than $1$ and $\infty$. We prove that wreath products of nontrivial groups by infinite groups, as well as Thompson's group $F$, have infinite weak-cop number. The argument for Thompson's group relies on the representation of its elements by Belk and Brown Forest diagrams.

In real-world systems, the relationships and connections between components are highly complex. Real systems are often described as networks, where nodes represent objects in the system and edges represent relationships or connections between nodes. With the deepening of research, networks have been endowed with richer structures, such as directed edges, edge weights, and even hyperedges involving multiple nodes. Persistent homology is an algebraic method for analyzing data. It helps us understand the intrinsic structure and patterns of data by tracking the death and birth of topological features at different scale parameters.The original persistent homology is not suitable for directed networks. However, the introduction of path homology established on digraphs solves this problem. This paper studies complex networks represented as weighted digraphs or edge-weighted path complexes and their persistent path homology. We use the homotopy theory of digraphs and path complexes, along with the interleaving property of persistent modules and bottleneck distance, to prove the stability of persistent path diagram with respect to weighted digraphs or edge-weighted path complexes. Therefore, persistent path homology has practical application value.