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Cooperation among kin is common in animal societies. Kin groups may form by individuals directly discriminating relatives based on kin recognition cues, or form passively through natal philopatry and limited dispersal. We describe the genetic landscape for a primitively eusocial wasp, Polistes dominula, and ask whether individuals choose cooperative partners that are nearby and/or that are genetic relatives. Firstly, we genotyped an entire sub-population of 1361 wasps and found genetic structuring on an extremely fine scale: the probability of finding genetic relatives decreases exponentially within just a few meters of an individual’s nest. At the same time, however, we found a lack of genetic structuring between natural nest aggregations within the population. Secondly, in a separate dataset where ~2000 wasps were genotyped, we show that wasps forced experimentally to make a new nest choice tended to choose new nests near to their original nests, and that these nests tended to contain some full sisters. However, a significant fraction of wasps chose nests that did not contain sisters, despite sisters being present in nearby nests. Although we cannot rule out a role for direct kin recognition or natal nest-mate recognition, our data suggest that kin groups may form via a philopatric rule-of-thumb, whereby wasps simply select groups and nesting sites that are nearby. The result is that most subordinate helpers obtain indirect fitness benefits by breeding cooperatively. [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 2019 ICC Cricket World Cup was held from May 30 to July 14 in England and Wales with a new tournament format being used. The competition was reduced to ten teams, in a single pool, with each team scheduled to play every one of the other nine teams. The purpose of such a tournament is to identify the strongest or most dominant teams, but this was thwarted in the 2019 Cricket World Cup by the scoring system, in particular the use of net run rate to rank teams tied on points awarded for wins, draws and ‘no result’ matches. Concepts from discrete mathematics, in particular graph theory, are applied to create a ranking system better suited to this type of tournament. The dominance graph for matches between the top five teams shows that the team which came fourth, under the current system, was dominated by the team placed in fifth position. This paper shows how the use of net run rate may be replaced by calculating one stage, two stages and if necessary three stage dominances, to rank teams. These calculations are easily done using a dominance matrix. Recommendations to ensure that the strongest teams play in the final match, for the cup, are made and it is suggested that the group stage of the FIFA Football World Cup use the scoring method explained in this paper. [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]

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 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]

In the modern era, graph theory is considered a useful tool for quantification and simplification of various dynamic components in complex systems. By representing elements as nodes and their connections as edges, graph theory can transform anything from urban planning to computer data into a meaningful mathematical language. Nowadays, numerous practical applications have been designed and developed based on graph theory. Graph theory is a branch of discrete mathematics that aims to describe and solve problems with discrete structures using points and edges. One of the problems concerning graphs is the Eulerian path problem. This research demonstrates that this problem can also be investigated from the perspective of Operations Research (OR). In a more general sense, the Eulerian path problem is a routing problem. This paper presents a pure mathematical model to describe the relationship between the variables of the Eulerian path problem. One of the features of the proposed mathematical model is its solvability by most optimization software. Finally, several numerical examples are provided to enhance the understanding of this model, and they are solved using the proposed approach. All the analyses in this research are conducted using one of the most advanced optimization software, MATLAB. The proposed mathematical model provides a systematic and efficient approach to discover Eulerian paths in specific graphs, contributing to the advancement of graph theory and its practical applications. [ABSTRACT FROM AUTHOR]

Within the existing taxonomy of mathematical proofs, visual proofs distinguish themselves by using representation as a tool for demonstration. This paper explores visual proofs in mathematics and their relationship with architectural representation. Most notably, stereotomy and graphic statics exhibit qualities of visual proofs by demonstrating how forms are generated. [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]

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]

Given a graph G with vertex set V (G) and edge set E (G) , for the bijective function f (V (G)) → { 1 , 2 , ⋯ , | V (G) | } , the associated weight of an edge x y ∈ E (G) under f is w (x y) = f (x) + f (y) . If all edges have pairwise distinct weights, the function f is called an edge-antimagic vertex labeling. A path P in the vertex-labeled graph G is said to be a rainbow x − y path if for every two edges x y , x ′ y ′ ∈ E (P) it satisfies w (x y) ≠ w (x ′ y ′) . The function f is called a rainbow antimagic labeling of G if there exists a rainbow x − y path for every two vertices x , y ∈ V (G) . We say that graph G admits a rainbow antimagic coloring when we assign each edge x y with the color of the edge weight w (x y) . The smallest number of colors induced from all edge weights of antimagic labeling is the rainbow antimagic connection number of G, denoted by r a c (G) . This paper is intended to investigate non-symmetrical phenomena in the comb product of graphs by considering antimagic labeling and optimizing rainbow connection, called rainbow antimagic coloring. In this paper, we show the exact value of the rainbow antimagic connection number of the comb product of graph F n ⊳ T m , where F n is a friendship graph with order 2 n + 1 and T m ∈ { P m , S m , B r m , p , S m , m } , where P m is the path graph of order m, S m is the star graph of order m + 1 , B r m , p is the broom graph of order m + p and S m , m is the double star graph of order 2 m + 2 . [ABSTRACT FROM AUTHOR]

Creativity is a complex, multi-faceted concept encompassing a variety of related aspects, abilities, properties and behaviours. If we wish to study creativity scientifically, then a tractable and well-articulated model of creativity is required. Such a model would be of great value to researchers investigating the nature of creativity and in particular, those concerned with the evaluation of creative practice. This paper describes a unique approach to developing a suitable model of how creative behaviour emerges that is based on the words people use to describe the concept. Using techniques from the field of statistical natural language processing, we identify a collection of fourteen key components of creativity through an analysis of a corpus of academic papers on the topic. Words are identified which appear significantly often in connection with discussions of the concept. Using a measure of lexical similarity to help cluster these words, a number of distinct themes emerge, which collectively contribute to a comprehensive and multi-perspective model of creativity. The components provide an ontology of creativity: a set of building blocks which can be used to model creative practice in a variety of domains. The components have been employed in two case studies to evaluate the creativity of computational systems and have proven useful in articulating achievements of this work and directions for further research. [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]

This theoretical paper sets forth two aspects of predication, which describe how students perceive the relationship between a property and an object. We argue these are consequential for how students make sense of discrete mathematics proofs related to the properties and how they construct a logical structure. These aspects of predication are (1) populating the way students generate sets of examples of the property, and (2) testing membership how one tests whether or not a given object has a specific property. Using data from two teaching experiments in which undergraduate students read proofs of theorems about the discrete concept of multiple relations, we illustrate the nature of these aspects of predication and demonstrate how they help explain student interpretations of the proofs. We argue that these particular properties from number theory likely have correlates in many other discrete mathematics topics because of the role of computation/algorithms for defining and testing properties as well as the role of iteration and recursion in populating examples. We anticipate that these constructs will be useful to teachers and researchers of discrete mathematics to foster and assess student understanding of various mathematical properties. They provide tools for thinking about what it means to understand properties in a rich and coherent way that supports understanding complex lines of inference and generalizations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following discrete fractional p-Laplacian equations: ...; where λ is the parameter and f (a; u(a)) satisfies no symmetry assumption. As a result, a specific positive parameter interval is determined by some requirements for the nonlinear term near zero, and then infinitely many homoclinic solutions are obtained by using a special version of Ricceri's variational principle. [ABSTRACT FROM AUTHOR]

The paper discusses a certain type of competitions based on distance interaction of a participant with simulation models of concepts from discrete mathematics and computer science. One of them is the "Construct, Test, Explore" (CTE) competition, developed by the authors, the other is the Olympiad in Discrete Mathematics and Theoretical Informatics (DM&TI). The tasks presented in this paper are generally devoted to the concept of a graph isomorphism. Most of the tasks are verified automatically. [ABSTRACT FROM AUTHOR]

The new applications of Clifford's geometric algebra surveyed in this paper include kinematics and robotics, computer graphics and animation, neural networks and pattern recognition, signal and image processing, applications of versors and orthogonal transformations, spinors and matrices, applied geometric calculus, physics, geometric algebra software and implementations, applications to discrete mathematics and topology, geometry and geographic information systems, encryption, and the representation of higher order curves and surfaces. [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]

Robots are widely used in present day situations. Since a robot consists of number of systems with different mechanisms, it needs to be planned and programmed systematically. Discrete Mathematics helps us in designing and functioning of a robotic system. For instance Robotic arm is a type of linkage and the study of which is a part of Discrete Geometry. Scheduling tasks to be completed by a set of machines use the concept of Bin-packing problem, which is a part of Discrete Optimization. Discussing functions like these, motion and path planning of the robotic system is the main concern of any industry to adapt new and modern technologies in its functioning. Literature surveys reveal that there a worldwide attempts to develop robotic systems with minimum errors and maximum efficiency. This Paper includes the usage of the Concepts of Discrete Mathematics to have a path planning for a Robotic System to work with high efficiency. [ABSTRACT FROM AUTHOR]

Wang et al. proposed a method for obtaining elliptic curves with embedding degree 1 for securing critical infrastructures, and presented several elliptic curves generated by their method with torsion points of 160 bits and 189 bits orders. They also presented some experimental results and claimed that their implementation of an elliptic curve generated with their method is faster than an implementation for embedded devices presented by Bertoni et al. In this paper, we point out that the security and efficiency claims given by Wang et al. are flawed. Specifically, we show that it is possible to solve finite field discrete logarithm problems defined over their elliptic curves in practice. On the elliptic curves with torsion points of 160 bits orders generated by Wang et al., their instances of finite field discrete logarithm problems are solved in around 4 hours by using a standard desktop PC. On the torsion points of 189 bits orders, their instances are solved in around 10 days by using two standard desktop PCs. The hardness of the finite field discrete logarithm problems is one of the most important bases of security; therefore, their elliptic curves should not be used for cryptographic purposes. [ABSTRACT FROM AUTHOR]

Introduction: In this paper we perform a replication analysis of “Effect of a cash transfer programme for schooling on prevalence of HIV and herpes simplex type 2 in Malawi: a cluster randomised trial” by Sarah Baird and others published in “The Lancet” in 2012. The original study was a two-year cluster randomized intervention trial of never married girls aged 13–22 in Malawi. Enumeration areas were randomized to either an intervention involving cash transfer (conditional or unconditional of school enrollment) or control. The study included 1708 Malawian girls, who were enrolled at baseline and had biological testing for HIV and herpes simplex virus type 2 (HSV-2) at 18 months. The original findings showed that in the cohort of girls enrolled in school at baseline, the intervention had an effect on school enrollment, sexual outcomes, and HIV and HSV-2 prevalence. However, in the baseline school dropout cohort, the original study showed no intervention effect on HIV and HSV-2 prevalence. Methods: We performed a replication of the study to investigate the consistency and robustness of key results reported. A pre-specified replication plan was approved and published online. Cleaned data was obtained from the original authors. A pure replication was conducted by reading the methods section and reproducing the results and tables found in the original paper. Robustness of the results were examined with alternative analysis methods in a measurement and estimation analysis (MEA) approach. A theory of change analysis was performed testing a causal pathway, the effect of intervention on HIV awareness, and whether the intervention effect depended on the wealth of the individual. Results: The pure replication found that other than a few minor discrepancies, the original study was well replicated. However, the randomization and sampling weights could not be verified due to the lack of access to raw data and a detailed sample selection plan. Therefore, we are unable to determine how sampling influenced the results, which could be highly dependent on the sample. In MEA it was found that the intervention effect on HIV prevalence in the baseline schoolgirls cohort was somewhat sensitive to model choice, with a non-significant intervention effect for HIV depending on the statistical model used. The intervention effect on HSV-2 prevalence was more robust in terms of statistical significance, however, the odds ratios and confidence intervals differed from the original result by more than 10%. A theory of change analysis showed no effect of intervention on HIV awareness. In a causal pathway analysis, several variables were partial mediators, or potential mediators, indicating that the intervention could be working through its effect on school enrollment or selected sexual behaviors. Conclusions: The effect of intervention on HIV prevalence in the baseline schoolgirls was sensitive to the model choice; however, HSV-2 prevalence results were confirmed. We recommend that the results from the original published analysis indicating the impact of cash transfers on HIV prevalence be treated with caution. [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]

Recursive reasoning is a powerful tool used extensively in problem solving. For us, recursive reasoning includes iteration, sequences, difference equations, discrete dynamical systems, pattern identification, and mathematical induction; all of these can represent how things change, but in discrete jumps. Given the school mathematics curriculum's later emphasis on calculus—the mathematics of change in continuous contexts—it is surprising that the curriculum seems to neglect recursive thinking after the early grades. Research shows that recursion supports the learning of algebra among younger students, but the lack of similar research with older students is concerning. In this paper we suggest possible affordances from teaching recursive modeling, including a basic model of the spread of contagious diseases. We also discuss different ways to present these models at various points in the curriculum that might develop connections between mathematics and the real world, and support students' learning of mathematics. This leads to what we, as mathematicians, think would be interesting research questions for mathematical educators. [ABSTRACT FROM AUTHOR]

Discrete mathematics brings interesting problems for teaching and learning proof, with accessible objects such as integers (arithmetic), graphs (modeling, order) or polyominoes (geometry). Many problems that are still open can be explained to a large public. The objects can be manipulated by simple dynamic operations (removing, adding, 'gluing', contracting, splitting, decomposing, etc.). All these operations can be seen as tools for proving. In this paper we particularly explore the field of 'discrete optimization'. A theoretical background is defined by taking two main axes into account, namely, the epistemological analysis of discrete problems studied by contemporary researchers in discrete optimization and the design of adidactical situations for classrooms in the frame of the Theory of Didactical Situations. Two problems coming from ongoing research in discrete optimization (the Pentamino Exclusion and the Eight Queens problems) are developed and transposed for the classroom. They underscore the learning potentialities of discrete mathematics and epistemological obstacles concerning proving processes. They emphasize the understanding of a necessary condition and a sufficient condition and problematize the difference between optimal and optimum. They provide proofs involving partitioning strategies, greedy algorithms but also primal–dual methods leading to the concept of duality. The way such problems can be implemented in the classroom is described in a collaborative study by mathematicians and mathematics education researchers (Maths à Modeler Research Federation) through the Research Situations for the Classroom project. [ABSTRACT FROM AUTHOR]

In recent decades, there has been considerable research that explores the teaching and learning of combinatorics. Such work has highlighted the fact that understanding and justifying combinatorial formulas can be challenging for students, and there is a need to identify ways to support students' combinatorial reasoning. In this paper, we contribute to research that explores effective ways to foster students' combinatorial reasoning by highlighting the role that shifts in representational registers can play in supporting students' combinatorial reasoning and activity. In particular, we present a case in which two students, aged 12 and 14, reasoned about, and developed a formula for, binomial coefficients. This occurred in a teaching experiment designed to examine their generalizing activity. In these interviews, the students first solved problems about determining the volume of a cube and then shifted to a combinatorial interpretation of this task, leveraging binomial coefficients as a way to consider growth in higher dimensions. The students demonstrated sophisticated reasoning about combinatorial tasks and were ultimately able to generate and understand a formula for binomial coefficients. In framing this case, we focus on the students' use of different representations, highlighting the power of being able to transition between representational registers as a way of supporting students' combinatorial reasoning. In this way, our case demonstrates the value of representational registers specifically as a mechanism by which to potentially improve the teaching and learning of combinatorics. [ABSTRACT FROM AUTHOR]

The role of logic in mathematics education has been widely discussed from the seventies and eighties during the "modern maths period" till now, and remains still a rather controversial issue in the international community. Nevertheless, the relevance of discrete mathematics and algorithmic thinking for the development of heuristic and logical competences is both one of the main points of the program of Tamás Varga, and of some didactic teams in France. In this paper, we first present the semantic perspective in mathematics education and the role of logic in the Hungarian tradition. Then, we present insights on the role of research problems in the French tradition. Finely, we raise some didactical issues in algorithmic thinking at the interface of mathematics and computer science. [ABSTRACT FROM AUTHOR]

Minimal balanced collections are a generalization of partitions of a finite set of n elements and have important applications in cooperative game theory and discrete mathematics. However, their number is not known beyond n = 4. In this paper we investigate the problem of generating minimal balanced collections and implement the Peleg algorithm, permitting to generate all minimal balanced collections till n = 7. Secondly, we provide practical algorithms to check many properties of coalitions and games, based on minimal balanced collections, in a way which is faster than linear programming-based methods. In particular, we construct an algorithm to check if the core of a cooperative game is a stable set in the sense of von Neumann and Morgenstern. The algorithm implements a theorem according to which the core is a stable set if and only if a certain nested balancedness condition is valid. The second level of this condition requires generalizing the notion of balanced collection to balanced sets. [ABSTRACT FROM AUTHOR]

This paper introduces lessons designed to incorporate applications to teaching high school mathematics in an undergraduate Discrete Mathematics course. Because many prospective high school teachers do not take courses that are specifically designed for teachers, providing materials aimed at teacher preparation that can be easily integrated into courses that serve a general mathematics major is one strategy for addressing mathematics teacher preparation. We developed lessons using four guiding features: choosing appropriate content; making school mathematics connections; incorporating active learning; and providing robust lesson notes. Our interview-based findings document how the lessons were used by instructors and analyze the mathematical ideas and understandings that arose from the use of the lessons at two different sites. [ABSTRACT FROM AUTHOR]

This paper investigates a well-known complex combinatorial problem known as the vehicle routing problem with time windows (VRPTW). Unlike the standard vehicle routing problem, each customer in the VRPTW is served within a given time constraint. This paper solves the VRPTW using an improved artificial bee colony (IABC) algorithm. The performance of this algorithm is improved by a local optimization based on a crossover operation and a scanning strategy. Finally, the effectiveness of the IABC is evaluated on some well-known benchmarks. The results demonstrate the power of IABC algorithm in solving the VRPTW. [ABSTRACT FROM AUTHOR]

The topic of recurrence relations has recently been introduced in many discrete mathematics textbooks. Recurrence relations are efficient modelling and problem-solving techniques used in mathematics. Combinatorial problems are often used to introduce recurrence relations. However, many textbooks consider problems that can be reduced only to the recurrence relations of the first or second order. To overcome this problem, in this paper, we propose a general combinatorial problem of path tiling with tiles of different sizes and colours. For especially selected tile sizes and for certain tile colours, the problem can be reduced to solving a recurrence relation of the third, fourth, or higher orders. Furthermore, through actual examples, we show that various cases can be derived from the roots of a characteristic equation: different real roots, different real and complex roots, and repeated real roots with different multiplicities. We hope that the presented problems will provide teachers with the opportunity to present this topic with better content. [ABSTRACT FROM AUTHOR]

Classical null hypothesis significance tests are not appropriate in corpus linguistics, because the randomness assumption underlying these testing procedures is not fulfilled. Nevertheless, there are numerous scenarios where it would be beneficial to have some kind of test in order to judge the relevance of a result (e.g. a difference between two corpora) by answering the question whether the attribute of interest is pronounced enough to warrant the conclusion that it is substantial and not due to chance. In this paper, I outline such a test. [ABSTRACT FROM AUTHOR]

In this paper we investigate the changes in the functional connectivity intensity, and some related properties, in healthy people, across the life span and at resting state. For the explicit computation of the functional connectivity we exploit a recently proposed model, that bases not only on the correlations data provided by the acquisition equipment, but also on different parameters, such as the anatomical distances between nodes and their degrees. The leading purpose of the paper is to show that the proposed approach is able to recover the main aspects of resting state condition known from the available literature, as well as to suggest new insights, perspectives and speculations from a neurobiological point of view. Our study involves 133 subjects, both males and females of different ages, with no evidence of neurological diseases or systemic disorders. First, we show how the model applies to the sample, where the subjects are grouped into 28 different groups (14 of males and 14 of females), according to their age. This leads to the construction of two graphs (one for males and one for females), that can be realistically interpreted as representative of the neural network during the resting state. Second, following the idea that the brain network is better understood by focusing on specific nodes having a kind of centrality, we refine the two output graphs by introducing a new metric that favours the selection of nodes having higher degrees. As a third step, we extensively comment and discuss the obtained results. In particular, it is remarkable that, despite a great overlapping exists between the outcomes concerning males and females, some intriguing differences appear. This motivates a deeper local investigation, which represents the fourth part of the paper, carried out through a thorough statistical analysis. As a result, we are enabled to support that, for two special age groups, a few links contribute in differentiating the behaviour of males and females. In addition, we performed an average-based comparison between the proposed model and the traditional statistical correlation-based approach, then discussing and commenting the main outlined discrepancies. [ABSTRACT FROM AUTHOR]

In software engineering, real world projects are very important for software engineering students. Real world projects help software engineering students to be prepared for real world development. However, directly involving in real world projects is difficult and its benefits are questionable. Therefore, simulation of real world projects can be a solution for this situation. In this paper, the power of cryptography is used to simulate real world projects. Simulation of a project is designed and applied, and its results are discussed. [ABSTRACT FROM AUTHOR]

A synchronizer with a phase counter (sometimes called asynchronous phase clock) is an asynchronous distributed algorithm, where each node maintains a local "pulse counter" that simulates the global clock in a synchronous network. In this paper, we present a time-optimal self-stabilizing scheme for such a synchronizer, assuming unbounded counters. We give a simple rule by which each node can compute its pulse number as a function of its neighbors' pulse numbers. We also show that some of the popular correction functions for phase clock synchronization are not self-stabilizing in asynchronous networks. Using our rule, the counters stabilize in time bounded by the diameter of the network, without invoking global operations. We argue that the use of unbounded counters can be justified by the availability of memory for counters that are large enough to be practically unbounded and by the existence of reset protocols that can be used to restart the counters in some rare cases where faults will make this necessary. [ABSTRACT FROM PUBLISHER]

Post's functional completeness theorem for Boolean functions plays an important role in discrete mathematics. In paper [A functional completeness theorem for De Morgan functions, Discrete Appl. Math. 162 (2014) 1-16, doi: 10.1016/j.dam.2013.08.006.] a functional completeness criterion for De Morgan functions is established by the present author and Yu. Movsisyan. Namely, the concepts of closed, complete and precomplete classes of De Morgan functions are introduced there and a functional completeness theorem for De Morgan functions is proven. As a result it is shown that there are five precomplete classes of De Morgan functions. Four of those are defined as sets of functions preserving some finitary relations. However, the fifth class - the class of zigzag De Morgan functions, is not defined by relations. In this paper, we prove that zigzag De Morgan functions can be defined as De Morgan functions preserving an atmost 16-ary relation. [ABSTRACT FROM AUTHOR]

In this paper, we break down and compressed the issues that understudies are inclined to mistake in the educating of discrete science, for example, judgment of suggestions, interpretation of recommendations, development of essential ordinary frame, and operation of aggregate, exceptional components in fractional request and give the comparing measures to taking care of every issue. [ABSTRACT FROM AUTHOR]

Research shows that though some students can easily find the correct solutions to the problem(s) at hand, the algorithms that these students create are not always ones that would, when implemented, produce the correct solution(s). Towards shedding light on this phenomenon, the present study explicates a mechanism—accounting for features of the solution—by which students create algorithms. Via this mechanism, students notice particular features of the solution they found, then create specific rules (instructions) within their algorithm, which guarantee that the algorithm outputs an object that possesses the noticed features. Thus, an algorithm can produce the solution only if the object it outputs has all the features of the solution. In this paper, I explicate the accounting for mechanism within the collaborative work of two groups of students on a contextualised graph theory task, which invited the groups to create an algorithm for finding an optimal seating arrangement. Both groups found an optimal seating arrangement and seemingly employed the accounting for mechanism to create their respective algorithms. However, only one group's algorithm could actually produce the optimal arrangement. The questions explored in this study are: (1) What sorts of features of their solutions did the two groups account for? and (2) What differences (if any) between the features accounted for by the respective groups, might explain why only one group's algorithm could produce the optimal arrangement? [ABSTRACT FROM AUTHOR]

Abstract This paper studies the N-player gambler’s ruin problem where two players are involved in an even-money bet during each round. The objective is to solve for each player’s final placing probabilities given their initial wealths, as well as the expected time until ruin. Multigraphs were constructed to model the transitions between chip states. Linear systems were constructed based on these graphs. Solutions for the placing probabilities of each player were obtained using these linear systems. A numerical algorithm is developed to solve the general N-player gambler’s ruin for any positive integer chip total. Expected time until ruin for any initial state can be solved using absorbing Markov chains. The solution leads to exact values and equities in this model depend on the exact number, not just proportion, of chips each player holds. Compared with the usual lattice model, the multigraph model uses smaller matrices in computations, yielding considerable savings in the number of operations and program run time. [ABSTRACT FROM AUTHOR]

How can we teach inquiry? In this paper, I offer practical techniques for teaching inquiry effectively using activities built from routine textbook exercises with minimal advanced preparation, including rephrasing exercises as questions, creating activities that inspire students to make conjectures, and asking for counterexamples to reasonable, but false, conjectures. [ABSTRACT FROM PUBLISHER]

One of the key aspects of computational systems biology is the investigation on the dynamic biological processes within cells. Computational models are often required to elucidate the mechanisms and principles driving the processes because of the nonlinearity and complexity. The models usually incorporate a set of parameters that signify the physical properties of the actual biological systems. In most cases, these parameters are estimated by fitting the model outputs with the corresponding experimental data. However, this is a challenging task because the available experimental data are frequently noisy and incomplete. In this paper, a new hybrid optimization method is proposed to estimate these parameters from the noisy and incomplete experimental data. The proposed method, called Swarm-based Chemical Reaction Optimization, integrates the evolutionary searching strategy employed by the Chemical Reaction Optimization, into the neighbouring searching strategy of the Firefly Algorithm method. The effectiveness of the method was evaluated using a simulated nonlinear model and two biological models: synthetic transcriptional oscillators, and extracellular protease production models. The results showed that the accuracy and computational speed of the proposed method were better than the existing Differential Evolution, Firefly Algorithm and Chemical Reaction Optimization methods. The reliability of the estimated parameters was statistically validated, which suggests that the model outputs produced by these parameters were valid even when noisy and incomplete experimental data were used. Additionally, Akaike Information Criterion was employed to evaluate the model selection, which highlighted the capability of the proposed method in choosing a plausible model based on the experimental data. In conclusion, this paper presents the effectiveness of the proposed method for parameter estimation and model selection problems using noisy and incomplete experimental data. This study is hoped to provide a new insight in developing more accurate and reliable biological models based on limited and low quality experimental data. [ABSTRACT FROM AUTHOR]

Discrete mathematics is one of the most important parts of the introduction to programming. Although known as the basis of writing good software and its usage is necessary, the students have many difficulties while gaining the competency to apply it in software projects. In order to make it a little bit easier for learners, an additional course Discrete Mathematics and Programming was conducted in the Informatics major in the New Bulgarian University. Its objectives, conduction, some exemplary tasks and homework are presented in this paper. The difficulties in the learning process and different approaches to reduce them are discussed. [ABSTRACT FROM AUTHOR]

Course-based Undergraduate Research Experiences (CUREs) have been well developed in the hard sciences, but math CUREs are all but absent from the literature. Like biology and chemistry, math programs suffer from a lack of research experiences and many students are not able to participate in programs like REUs (Research Experiences for Undergraduates). CUREs are a great alternative, but the current definition of CURE (see [1]) has potential barriers when applied to mathematics (e.g. time, nov- elty of project). Our solution to these barriers was to develop a math CURE pathway in which students complete "Math CUREs" in targeted courses. After finishing the pathway (or part of the pathway), students complete a research project in at least one of the following areas: Lie theory, representation theory, or combinatorics. The focus of this paper is the math CURE implemented in a discrete mathe- matics course for math and computer science majors. We share our experiences with the development and implementation of this CURE over several iterations as well as the impact of the CURE on students experiences through participant survey data obtained from this CURE. [ABSTRACT FROM AUTHOR]