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Conceptual metaphor theory has been criticized due to its emphasis on concepts instead of words and its top-down direction of analysis. In response to these criticisms, this paper employs a new strategy, utilizing established mathematical modeling methods to allow a systematic, quantitative analysis of the entire dataset produced by the Mapping Metaphor project at the University of Glasgow. This dataset consists of 9609 words performing 18916 metaphorical mappings between 414 domains. The data is represented as a network consisting of 414 nodes, the domains, connected by shared words. Words are represented by groups of directed mappings between all domains in which they occur. This is made possible by the use of a directed hypergraph representation, a tool commonly used in discrete mathematics and various areas of computer science but not previously applied to the metaphorical meanings of words. Examining the dataset as a whole, rather than focusing on individual words or metaphors, allows global patterns of behavior to emerge from the data without pre-filtering or selection by the authors. Outcomes of the analysis relating to the distributions of source and target domains within the network, the growth mechanisms at work in the spread of metaphorical meanings and how these relate to existing concepts in CMT are discussed. [ABSTRACT FROM AUTHOR]

This is the first of a series of papers that develop a systematic bridge between constructions in discrete mathematics and the corresponding continuous analogs. In this paper, we establish an equivalence between Forman's discrete Morse theory on a simplicial complex and the continuous Morse theory (in the sense of any known non-smooth Morse theory) on the associated order complex via the Lovász extension. Furthermore, we propose a new version of the Lusternik–Schnirelman category on abstract simplicial complexes to bridge the classical Lusternik–Schnirelman theorem and its discrete analog on finite complexes. More generally, we can suggest a discrete Morse theory on hypergraphs by employing piecewise-linear (PL) Morse theory and Lovász extension, hoping to provide new tools for exploring the structure of hypergraphs. [ABSTRACT FROM AUTHOR]

This editorial introduces the special issue, Resources for Undergraduate Cryptology. We begin by describing possible roles for cryptology in the undergraduate mathematics curriculum together with a brief overview of the subject. We conclude with a brief preview of each paper included in this issue. [ABSTRACT FROM AUTHOR]

This document is an editorial for a special issue of the journal AIMS Mathematics on "Mathematical Coding Theory and its Applications." The issue includes 8 papers that explore various aspects of coding theory, which involves techniques from commutative algebra, discrete mathematics, and the geometry of numbers. The papers cover topics such as codes over rings (including non-commutative and non-unitary rings), codes for alternative metrics, and codes as ideals in special rings. The issue also includes papers on codes over fields, including finite fields and number fields. The guest editor for the special issue is Professor Patrick Solé from the University of Aix-Marseille in France. [Extracted from the article]

The simplification of complex networks is a research field closely related to graph theory in discrete mathematics. The existing methods are typically limited to simplifying the series sub-networks, parallel sub-networks, diagonal sub-networks, and nested simple sub-networks. From the current perspective, there are no available methods that can handle complex sub-networks and nested complex sub-networks. In this paper, we innovatively propose an efficient and automatic equivalence simplification method for arbitrary complex ventilation networks. The method enables, for the first time, the maximum possible equivalence simplification of nested simple sub-networks and nested complex sub-networks. In order to avoid the NP-hard problem caused by the searching of simplifiable sub-networks, it is necessary to analyze the intrinsic topology relationship between simplifiable sub-networks and spanning sub-graphs to optimize the searching process. One of our main contributions is that we present an efficient searching method for arbitrarily nested reducible sub-networks based on the bidirectional traversal process of a directed tree. The method optimizes the searching process for simplifiable node pairs by combining the characteristics of a directed tree with the judgment rules of simplifiable sub-networks. Moreover, by deriving the formula of an equivalent air resistance calculation for complex sub-networks, another one of our main contributions is that we present an equivalent calculation and simplification method for arbitrarily complex sub-networks based on the principle of energy conservation. The basic idea of the method is to calculate the equivalent air resistance using the ventilation network resolution of the constructed virtual sub-networks. We realize the simplification method of arbitrarily complex mine ventilation networks, and we validate the reliability of the simplification method by comparing the air distribution results using the network solution method before and after simplification. It can be determined that, with appropriate modifications to meet specific requirements, the proposed method can also be applicable to equivalent simplification instances of other types of complex networks. Based on the results analysis of several real-world mine ventilation network examples, the effectiveness of the proposed method is further verified, which can satisfactorily meet the requirements for simplifying complex networks. [ABSTRACT FROM AUTHOR]

This paper is devoted to establishing frame inequalities in Hilbert spaces. By using operator theory methods, several two-sided inequalities for frames are presented, which, comparing to previous inequalities on frames and generalized frames, admit new structures. [ABSTRACT FROM AUTHOR]

In this paper, we give some characterizations for the boundedness and compactness of some Stević-Sharma type operators called the polynomial differentiation composition operators from H∞ into the Bloch space on the unit disk. [ABSTRACT FROM AUTHOR]

In this paper, we present some properties of the time scale nabla calculus and how to apply it for proving the dynamic inequalities. Also, we prove some weighted dynamic inequalities of Hardy type with kernels on time scales nabla calculus and also, we study the characterizations of the weights for these inequalities in different spaces and for the exponent p > 1. The Holder inequality, Jensen inequality, and Minkowski inequality are used to prove our results. [ABSTRACT FROM AUTHOR]

In this paper we obtain generalizations of Hardy-type inequalities for convex functions of the higher order by applying Hermite interpolating polynomials. The results for particular cases: Lagrange, (m,n-m) and two-point Taylor interpolating polynomials are also considered. Finally, we derive the Grüss and Ostrowski type inequalities related to these generalizations. [ABSTRACT FROM AUTHOR]

Abstract. This paper will introduce the theory of ϕ-Jensen coefficient. By means of the functional analysis, linear algebra, discrete mathematics and inequality theories with proper hypotheses, the Jensen-type inequality, Marshall-type inequality and the Ky Fan-type inequality are obtained as follows: ... respectively, as well as we also displayed the applications of our main results in business profit management model, and some conditions such that ... hold are obtained, where p is the profit function and e is the cost function. [ABSTRACT FROM AUTHOR]

In this paper, we connect four different branches of Mathematics: Statistics, Probability, Algebra and Discrete Mathematics with the objective of introducing new results on Markov chains and evolution algebras obtained by following a relatively new line of research, already dealt with by several authors. It consists of the use of certain directed graphs to facilitate the study of Markov chains and evolution algebras, as well as to use each of the three objects to make easier the study of the other two. The results obtained can be useful, in turn, to link different scientific disciplines, such as Physics, Engineering and Mathematics, in which evolution algebras are considered very interesting tools. [ABSTRACT FROM AUTHOR]

The Ruler function or the Gros sequence is a classical infinite integer sequence that underlies some interesting mathematical problems. In this paper, we provide four new problems containing this type of sequence: (i) demographic discrete dynamical automaton, (ii) the middle interval Cantor set, (iii) construction by duplication of polygons and (iv) the horizontal visibility sequence at the accumulation point of the Feigenbaum cascade. In all of them, the infinite sequence is obtained through a recursive procedure of duplication. The properties of the ruler sequence, in particular, those relating to recursiveness and self-containing, are used to achieve a deeper understanding of these four problems. These new representations of the ruler sequence could inspire new studies in the field of discrete mathematics. [ABSTRACT FROM AUTHOR]

For a hybrid stochastic system, most existing feedback controllers need to observe modes at continuous times, which is feasible when the system's mode is observable and does not incur any cost. However, in most cases, the mode is not readily apparent, and identifying it always incurs a certain expense. Therefore, in order to reduce control costs, when designing a feedback controller, both the state and the mode should be observed at discrete moments. This paper introduces an intermittent feedback controller for stabilizing an unstable hybrid stochastic system through discrete delayed observations of state and mode. By utilizing M-matrix theory, intermittent control approach, and the comparison principle, we propose sufficient conditions for the stabilization theory of hybrid stochastic systems. An illustrative example is taken to validate the proposed theory. [ABSTRACT FROM AUTHOR]

A discrete quantum structure is a discrete quantum space that is equipped with relations and functions of various finite arities. Discrete quantum spaces are identified with hereditarily atomic von Neumann algebras, their relations with projection operators, and their functions with unital normal *-homomorphisms. The propositional quantum logic of Birkhoff and von Neumann has been extended to a predicate quantum logic by Weaver; we investigate this predicate quantum logic as the internal logic of discrete quantum structures. We extend this predicate quantum logic to include function symbols and an equality symbol. Overall, we recover the basic structures of discrete quantum mathematics from physical first principles. More complicated structures will be recovered similarly in part II of this paper. [ABSTRACT FROM AUTHOR]

In this paper, the joint distribution of the sum and maximum of independent, not necessarily identically distributed, nonnegative random variables is studied for two cases: (i) continuous and (ii) discrete random variables. First, a recursive formula of the joint cumulative distribution function (CDF) is derived in both cases. Then recurrence relations of the joint probability density function (PDF) and the joint probability mass function (PMF) are given in the former and the latter case, respectively. Interestingly, there is a fundamental difference between the joint PDF and PMF. The proofs are simple and mainly based on the following tools from calculus and discrete mathematics: differentiation under the integral sign (also known as Leibniz’s integral rule), the law of total probability, and mathematical induction. In addition, this work generalizes previous results in the literature, and finally presents several extensions of the methodology. [ABSTRACT FROM AUTHOR]

Recently, the problem of multitasking scheduling has raised a lot of interest in the service industries. Hall et al. (Discrete Applied Mathematics, 2016) proposed a shared processing multitasking scheduling model which allows a team to continue to work on the primary tasks while processing the routinely scheduled activities as they occur. With a team being modeled as a single machine, the processing sharing of the machine is achieved by allocating a fraction of the processing capacity to routine jobs and the remaining fraction, which we denote as sharing ratio, to the primary jobs. In this paper, we generalize this model to parallel machines and allow the fraction of the processing capacity assigned to routine jobs to vary from one to another. The objectives are minimizing makespan and minimizing the total completion time of primary jobs. We show that for both objectives, there is no polynomial time approximation algorithm unless P=NP if the sharing ratios are arbitrary for all machines. Then we consider the problems where the sharing ratios on some machines have a constant lower bound. For each objective, we analyze the performance of the classical scheduling algorithms and their variations and then develop a polynomial time approximation scheme when the number of machines is a constant. [ABSTRACT FROM AUTHOR]

A nanotube is a nanometer–scale tube-like structure, it is a kind of nanoparticle, and may be large enough to serve as a pipe through which other nanoparticles can be channeled, or, depending on the material, may be used as an electrical conductor or an electrical insulator. For computing the structural information of nanotubes, the graph entropies have become the information theoretic quantities. The graph entropy measure has attracted the research community due to its potential application in discrete mathematics, biology, and chemistry. In this paper, our contribution is to explore graph entropies for structures of some nanotubes based on novel information function, which is the number of different degree vertices along with the number of edges between various degree vertices. More precisely, we computed entropies of some classes of nanotubes such as titania nanotube TNT 3 [ m , n ] , TNT 6 [ m , n ] and carbon nanotubes HAC 5 C 6 C 7 [ m , n ] by making a relation of degree-based topological indices with the help of information function. [ABSTRACT FROM AUTHOR]

For two graphs G and F , we say that G is weakly F -saturated if G has no copy of F as a subgraph and one can join all the nonadjacent pairs of vertices of G in some order so that a new copy of F is created at each step. The weak saturation number wsat (n , F) is the minimum number of edges of a weakly F -saturated graph on n vertices. In this paper, we examine wsat (n , K s , t) , where K s , t is the complete bipartite graph with parts of sizes s and t. We determine wsat (n , K 2 , t) for all n ⩾ t + 2 which particularly corrects a previous report in the literature. It is also shown that wsat (s + t , K s , t) = ( s + t − 1 2 ) if gcd ⁡ (s , t) = 1 and wsat (s + t , K s , t) = ( s + t − 1 2 ) + 1 otherwise. [ABSTRACT FROM AUTHOR]

This article is a preface to a special issue of the Journal of Difference Equations and Applications dedicated to Saber Elaydi on his 80th birthday. Elaydi is recognized for his significant contributions to the field of difference equations and discrete dynamical systems. He has had a successful academic career, serving as a professor and chair at Trinity University in Texas. Elaydi has also been a valuable teacher and mentor, inspiring students and attracting disciples from around the world. He has written influential books on difference equations and has been involved in organizing conferences and founding journals in the field. Elaydi's research interests cover a wide range of topics, including mathematical biology, neuroscience, and economics. He has made substantial contributions to these areas and has been recognized with prestigious awards. The article expresses appreciation for Elaydi's achievements and wishes him continued success in his mathematical and personal life. [Extracted from the article]

The aim of this study is to examine how algorithmatizing tasks engage mathematics students in algorithmic thinking. Structured, task-based interviews were conducted with eight Year 12 students as they completed a sequence of algorithmatizing tasks involving maximum flow problems. A deductive-inductive analytical process was used to first classify students' mathematical behavior according to four cognitive skills of algorithmic thinking (decomposition, abstraction, algorithmization, and debugging) and then develop sets of subskills to describe how the students engaged these cognitive skills. The findings show how students used algorithmic thinking to solve maximum flow problems and then made progress towards creating a general algorithm before being introduced to the maximum-flow minimum-cut approach, which guarantees a solution. [ABSTRACT FROM AUTHOR]

In a 2023 Discrete Mathematics article, Mneimneh introduced a remarkable formula for a binomial sum of harmonic numbers, defined by H k = ∑ i = 1 k 1 / i. This formula can be extended to the generalized harmonic numbers, where 1 / i is replaced by 1 / i r for a positive integer r. In this paper, we extend Mneimneh's formula to the generalized hyperharmonic numbers h k (ℓ , r) , and as a special case of this, to the generalized harmonic numbers when ℓ = 1. Some recent notes by Campbell in a follow-up 2023 Discrete Mathematics article are also considered. [ABSTRACT FROM AUTHOR]

This interview was given in 2008 by Arkady Zakrevsky (1928–2014), Corresponding Member of the National Academy of Sciences of Belarus (1972), Doctor of Technical Sciences (1967), and Professor (1969). He stood at the origins of the birth of cybernetics in the Soviet Union. He proposed the programming language for logical tasks LYaPAS, on the basis of which a series of computer-aided design systems for discrete devices was created, and methods for implementing parallel algorithms for the logical control of interacting processes. Some monographs: LYaPAS: A Programming Language for Logic and Coding Algorithms (N.-Y., L.: Academic Press, 1969; with M. A. Gavrilov); Boolesche Gleichungen: Theorie, Anwendung, Algorithmen (Berlin: VEB Verlag Technik, 1984; with Dieter Bochmann and Christian Posthoff); Combinatorial Algorithms of Discrete Mathematics (Tallinn: TUT Press, 2008; with Yu. Pottosin, L. Cheremisinova); Optimization in Boolean Space (Tallinn: TUT Press, 2009; with Yu. Pottosin, L. Cheremisinova); Design of Logical Control Devices (Tallinn: TUT Press, 2009; with Yu. Pottosin, L. Cheremisinova); Combinatorial Calculations in Many-Dimensional Boolean Space (Tallinn: TUT Press, 2012); Solving Large Systems Logical Equations (Tallinn: TUT Press, 2013). [ABSTRACT FROM AUTHOR]

This article explores the Erdős-Szekeres conjecture, a mathematical problem concerning geometric patterns in sets of points. The conjecture was proposed in 1935 and has been the focus of extensive research. Various mathematicians, including Tóth, Valtr, and Suk, have made significant contributions to proving the conjecture. Suk's proof is described as a moment of brilliance in applying the right tools. The article concludes with a quote from Erdős emphasizing the importance of an open mind in mathematics. [Extracted from the article]

The problem of path detection in graphs has been proposed from the past up to present, and various solutions have been proposed for this purpose, but it is often not an easy task to implement these methods on a computer. In this paper, a technique for detecting paths in a graph will be introduced using matrix algebra, which makes it possible to implement this rule on a computer. This method can be helpful the optimization of tree-spanning trees in networks. At the end of this study, a numerical example is solved using the proposed method.

A digraph is strong if it has a directed path from x to y for every ordered pair of distinct vertices x , y and it is k -strong if it has at least k + 1 vertices and remains strong when we delete any set of at most k − 1 vertices. A digraph D is a tournament (semicomplete digraph) if there is precisely (at least) one arc between any pair of distinct vertices of D. A subdigraph of a digraph D is called spanning if it contains all vertices of D. In this paper, we show that every 5-strong semicomplete digraph on at least 9 vertices contains a spanning 3-strong tournament. This shows that the conjecture of Bang-Jensen and Jordán [Discrete Mathematics 310 (2010) 1424–1428] holds for k = 3. [ABSTRACT FROM AUTHOR]

A covering path for a planar point set is a path drawn in the plane with straight- line edges such that every point lies at a vertex or on an edge of the path. A covering tree is defined analogously. Let π(n) be the minimum number such that every set of n points in the plane can be covered by a noncrossing path with at most π(n) edges. Let T(n) be the analogous number for non-crossing covering trees. Dumitrescu, Gerbner, Keszegh, and Tóth (Discrete & Computational Geometry, 2014) established the following inequalities: (5n)/9 - O(1) < π(n) < (1 - 1/601080391), n and (9n)/17 - O(1) < tau(n) ≤ [5n/6]. We report the following improved upper bounds: π(n) <= (1 - 1/22), n and t(n) ≤ =[(4n)/5]. In the same context we study rainbow polygons. For a set of colored points in the plane, a perfect rainbow polygon is a simple polygon that contains exactly one point of each color in its interior or on its boundary. Let rho(k) be the minimum number such that every k-colored point set in the plane admits a perfect rainbow polygon of size p(k). Flores-Peñaloza, Kano, Martínez-Sandoval, Orden, Tejel, Tóth, Urrutia, and Vogtenhuber (Discrete Mathematics, 2021) proved that 20k / 19 - O(1) < p(k) < 10k/7+O(1) We report the improved upper bound p(k) < 7k/5 + O(1) To obtain the improved bounds we present simple algorithms that achieve paths, trees, and polygons with our desired number of edges. [ABSTRACT FROM AUTHOR]

We present a detailed description of a Cryptography and Computer Security course that has been offered at Knox College for the last 15 years. While the course is roughly divided into two sections, Cryptology and Computer Security, our emphasis here is on the Cryptology section. The course puts the cryptologic material into its historical context and also includes reference to ciphers used and discussed in literature. Through lectures and assignments on cryptanalysis, the course motivates students to learn a variety of topics in advanced discrete mathematics, including discrete probability, elementary statistics, number theory, and abstract algebra. The students have found this approach motivating and we believe our experiences are relevant to instructors teaching this material in a variety of different courses and institutions. [ABSTRACT FROM AUTHOR]

Following the recent advances in the study of groups of circle diffeomorphisms, we describe an efficient way of classifying the topological dynamics of locally discrete, finitely generated, virtually free subgroups of the group Diff+ω(S¹) of orientation-preserving real-analytic circle diffeomorphisms, which include all subgroups of Diff+ω(S¹) acting with an invariant Cantor set. An important tool that we develop, of independent interest, is the extension of classical ping-pong lemma to actions of fundamental groups of graphs of groups. Our main motivation is an old conjecture by Dippolito [Ann. of Math. (2) 107 (1978), 403-453] from foliation theory, which we solve in this restricted but significant setting: this and other consequences of the classification will be treated in more detail in a companion work (by a slightly different list of authors). [ABSTRACT FROM AUTHOR]