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The book "Number Theory Through the Eyes of Sophie Germain: An Inquiry Course" by David Pengelley explores the correspondence between Sophie Germain and renowned mathematicians Joseph-Louis Lagrange, Adrien-Marie Legendre, and Carl Friedrich Gauss as they attempted to find a proof for Fermat's Last Theorem. Although Germain's plan was ultimately unsuccessful, it represented significant progress on the problem and provided insights into the theorem. The book takes an inquiry-based approach, using primary source materials to teach undergraduate students elementary number theory. It offers a unique and engaging perspective on the subject, allowing students to discover and prove results themselves. [Extracted from the article]

In this study, we applied graph theory to clinical decision-making for Stress Urinary Incontinence (SUI) treatment. Utilizing discrete mathematics, we developed a system to visually understand the shortest path to the desired treatment outcomes by considering various patient variables. Focusing on women aged 35–50, we examined the effectiveness of Tension-free Vaginal Tape (TVT) surgery and Vaginal Erbium Laser (VEL) treatment for over 15 years. The TVT group consisted of 102 patients who underwent surgery using either the Advantage Fit mid-urethral sling system (Boston Scientific Co., MA, USA) or the GYNECARE TVT retropubic system (Ethicon Inc., NJ, USA). The VEL group included 113 patients treated with a non-ablative Erbium: YAG laser (FotonaSmooth™ XS; Fotona d.o.o., Ljubljana, Slovenia), and there were 112 patients in the control group. We constructed a network diagram analyzing the correlations between health, demographic factors, treatment methods, and patient outcomes. By calculating the shortest path using heuristic functions, we identified significant correlations and treatment effects. This approach supports patient decision making by choosing between TVT and VEL treatments based on individual objectives. Our findings provide new insights into SUI treatment, highlighting the value of a data-driven personalized approach for clinical decision-making. This interdisciplinary study bridges the gap between mathematics and medicine, demonstrating the importance of a data-centric approach in clinical decisions. [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]

Given a class of graphs H , the problem ⊕ Sub (H) is defined as follows. The input is a graph H ∈ H together with an arbitrary graph G. The problem is to compute, modulo 2, the number of subgraphs of G that are isomorphic to H. The goal of this research is to determine for which classes H the problem ⊕ Sub (H) is fixed-parameter tractable (FPT), i.e., solvable in time f (| H |) · | G | O (1) . Curticapean, Dell, and Husfeldt (ESA 2021) conjectured that ⊕ Sub (H) is FPT if and only if the class of allowed patterns H is matching splittable, which means that for some fixed B, every H ∈ H can be turned into a matching (a graph in which every vertex has degree at most 1) by removing at most B vertices. Assuming the randomised Exponential Time Hypothesis, we prove their conjecture for (I) all hereditary pattern classes H , and (II) all tree pattern classes, i.e., all classes H such that every H ∈ H is a tree. We also establish almost tight fine-grained upper and lower bounds for the case of hereditary patterns (I). [ABSTRACT FROM AUTHOR]

In 1954, Allison proposed that hemoglobin S (HbS) gene causes protection against fatal malaria. This would explain the high HbS gene frequency observed in certain regions hyperendemic for malaria, so-called "malaria hypothesis". This in silico study was conducted to examine the feasibility of the hypothesis under more realistic initial conditions, where a mutant gene with heterozygous advantage against malaria (e.g., HbS) was introduced in a group of Neolithic hunter-gatherers who decided to start agriculture nearby water where malaria killed a proportion of population. The tribe population size, number of children born to each woman in each generation, mortality from malaria and sickle cell disease, the protection factor provided by the gene carriers against malaria, the probability of mating between the members of the parent and offspring populations, population growth, and increased fertility in women heterozygous for HbS, were also considered. For effectively confer protection against malaria within the shortest possible period, the mutation needs to be happened in a small population. For a large population, the process would take around 100 generations (~ 2500 years) or more to provide an effective protection. Even then, the probability that the new gene could survive and propagate to future generations is about 35%. Conventional population genetics equations with differential or difference equations, give totally incorrect estimates of the gene frequency in small populations; discrete mathematics should be used, instead. After introduction of the advantageous mutation, the gene frequency increased until a steady state value. This value is far less than the gene frequency reported in certain tribes of Africa. It seems that the malaria hypothesis, per se, could not explain such a high observed gene frequency, unless HbS is associated with lower mortality from other causes too. [ABSTRACT FROM AUTHOR]

We survey the history of the arithmetic derivative and more recent advances in research on this topic. Among other things, we discuss a few generalizations of the original arithmetic derivative and some arithmetic differential equations that are related to Goldbach's conjecture and the twin prime conjecture. Our primary purpose is to give an overview of this field, but we also aim at providing for an introductory course on discrete mathematics or number theory. Therefore, our survey contains ten exercises. [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 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]

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]

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]

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]

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]

Complex systems, as interwoven miscellaneous interacting entities that emerge and evolve through self-organization in a myriad of spiraling contexts, exhibit subtleties on global scale besides steering the way to understand complexity which has been under evolutionary processes with unfolding cumulative nature wherein order is viewed as the unifying framework. Indicating the striking feature of non-separability in components, a complex system cannot be understood in terms of the individual isolated constituents' properties per se, it can rather be comprehended as a way to multilevel approach systems behavior with systems whose emergent behavior and pattern transcend the characteristics of ubiquitous units composing the system itself. This observation specifies a change of scientific paradigm, presenting that a reductionist perspective does not by any means imply a constructionist view; and in that vein, complex systems science, associated with multiscale problems, is regarded as ascendancy of emergence over reductionism and level of mechanistic insight evolving into complex system. While evolvability being related to the species and humans owing their existence to their ancestors' capability with regards to adapting, emerging and evolving besides the relation between complexity of models, designs, visualization and optimality, a horizon that can take into account the subtleties making their own means of solutions applicable is to be entailed by complexity. Such views attach their germane importance to the future science of complexity which may probably be best regarded as a minimal history congruent with observable variations, namely the most parallelizable or symmetric process which can turn random inputs into regular outputs. Interestingly enough, chaos and nonlinear systems come into this picture as cousins of complexity which with tons of its components are involved in a hectic interaction with one another in a nonlinear fashion amongst the other related systems and fields. Relation, in mathematics, is a way of connecting two or more things, which is to say numbers, sets or other mathematical objects, and it is a relation that describes the way the things are interrelated to facilitate making sense of complex mathematical systems. Accordingly, mathematical modeling and scientific computing are proven principal tools toward the solution of problems arising in complex systems' exploration with sound, stimulating and innovative aspects attributed to data science as a tailored-made discipline to enable making sense out of voluminous (-big) data. Regarding the computation of the complexity of any mathematical model, conducting the analyses over the run time is related to the sort of data determined and employed along with the methods. This enables the possibility of examining the data applied in the study, which is dependent on the capacity of the computer at work. Besides these, varying capacities of the computers have impact on the results; nevertheless, the application of the method on the code step by step must be taken into consideration. In this sense, the definition of complexity evaluated over different data lends a broader applicability range with more realism and convenience since the process is dependent on concrete mathematical foundations. All of these indicate that the methods need to be investigated based on their mathematical foundation together with the methods. In that way, it can become foreseeable what level of complexity will emerge for any data desired to be employed. With relation to fractals, fractal theory and analysis are geared toward assessing the fractal characteristics of data, several methods being at stake to assign fractal dimensions to the datasets, and within that perspective, fractal analysis provides expansion of knowledge regarding the functions and structures of complex systems while acting as a potential means to evaluate the novel areas of research and to capture the roughness of objects, their nonlinearity, randomness, and so on. The idea of fractional-order integration and differentiation as well as the inverse relationship between them lends fractional calculus applications in various fields spanning across science, medicine and engineering, amongst the others. The approach of fractional calculus, within mathematics-informed frameworks employed to enable reliable comprehension into complex processes which encompass an array of temporal and spatial scales notably provides the novel applicable models through fractional-order calculus to optimization methods. Computational science and modeling, notwithstanding, are oriented toward the simulation and investigation of complex systems through the use of computers by making use of domains ranging from mathematics to physics as well as computer science. A computational model consisting of numerous variables that characterize the system under consideration allows the performing of many simulated experiments via computerized means. Furthermore, Artificial Intelligence (AI) techniques whether combined or not with fractal, fractional analysis as well as mathematical models have enabled various applications including the prediction of mechanisms ranging extensively from living organisms to other interactions across incredible spectra besides providing solutions to real-world complex problems both on local and global scale. While enabling model accuracy maximization, AI can also ensure the minimization of functions such as computational burden. Relatedly, level of complexity, often employed in computer science for decision-making and problem-solving processes, aims to evaluate the difficulty of algorithms, and by so doing, it helps to determine the number of required resources and time for task completion. Computational (-algorithmic) complexity, referring to the measure of the amount of computing resources (memory and storage) which a specific algorithm consumes when it is run, essentially signifies the complexity of an algorithm, yielding an approximate sense of the volume of computing resources and seeking to prove the input data with different values and sizes. Computational complexity, with search algorithms and solution landscapes, eventually points toward reductions vis à vis universality to explore varying degrees of problems with different ranges of predictability. Taken together, this line of sophisticated and computer-assisted proof approach can fulfill the requirements of accuracy, interpretability, predictability and reliance on mathematical sciences with the assistance of AI and machine learning being at the plinth of and at the intersection with different domains among many other related points in line with the concurrent technical analyses, computing processes, computational foundations and mathematical modeling. Consequently, as distinctive from the other ones, our special issue series provides a novel direction for stimulating, refreshing and innovative interdisciplinary, multidisciplinary and transdisciplinary understanding and research in model-based, data-driven modes to be able to obtain feasible accurate solutions, designed simulations, optimization processes, among many more. Hence, we address the theoretical reflections on how all these processes are modeled, merging all together the advanced methods, mathematical analyses, computational technologies, quantum means elaborating and exhibiting the implications of applicable approaches in real-world systems and other related domains. [ABSTRACT FROM AUTHOR]

The class of nonabelian 2-groups H with cyclic subgroup of index 2 includes the dihedral group, the generalized quaternion group, the semidihedral group, and the modular maximal cyclic group, which have many various applications in discrete mathematics and cryptography. We introduce piecewise-quasiaffine transformations on a group H, and put forward criteria of their bijectivity. For the generalized group of quaternions of order 2m, we obtain a complete classification of orthomorphisms, complete transformations, and their left analogues in the class of piecewise-quasiaffine transformations under consideration. We also evaluate their cardinalities. [ABSTRACT FROM AUTHOR]

In the case of combinatorial theory, problem solving becomes more demanding than in the rest of the topics of discrete mathematics. This is influenced by the complexity of the content, the variety of types of problems to be solved, and the insufficient skills with which many students get started in this subject. To contribute to the development of the ability to solve combinatorial problems, the authors of this communication identified a set of combinatorial heuristic procedures that, together with those of a general type, allow reversing the indicated difficulties. To familiarize students with the use of these procedures, a guide of heuristic questions is presented as a mediating entity. In the final part of the communication, criteria from the students of a teaching group are collected on the application of this guide in solving a combinatorial problem in a practical Discrete Mathematics class. [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]

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]

The development of the ability to solve combinatorial problems in discrete mathematics of the Informatics Sciences Engineering (ICI) career is not complete if the student is not able to carry out the demonstration of combinatorial propositions. On the other hand, mastery of demonstration techniques is an essential tool for learning Computer Science and developing professional skills. However, both solving combinatorial problems and carrying out proofs constitute challenging contents for significant numbers of students. The first part of this communication describes some theoretical background that explains the causes of such difficulty. Immediately afterwards, a set of budgets is offered that were applied in the scientificmethodological work of the group of Discrete Mathematics teachers with the aim of taking advantage of the advantages offered by working with mathematical demonstrations in the development of the ability to solve combinatorial problems. Finally, the results of a test are offered to evaluate the levels of satisfaction of teachers regarding the application of the proposal made in this communication. [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]

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]

The ill-posed problem of phase retrieval in optics, using one or more intensity measurements, has a multitude of applications using electromagnetic or matter waves. Many phase retrieval algorithms are computed on pixel arrays using discrete Fourier transforms due to their high computational efficiency. However, the mathematics underpinning these algorithms is typically formulated using continuous mathematics, which can result in a loss of spatial resolution in the reconstructed images. Herein we investigate how phase retrieval algorithms for propagation-based phase-contrast X-ray imaging can be rederived using discrete mathematics and result in more precise retrieval for single- and multi-material objects and for spectral image decomposition. We validate this theory through experimental measurements of spatial resolution using computed tomography (CT) reconstructions of plastic phantoms and biological tissues, using detectors with a range of imaging system point spread functions (PSFs). We demonstrate that if the PSF substantially suppresses high spatial frequencies, the potential improvement from utilising the discrete derivation is limited. However, with detectors characterised by a single pixel PSF (e.g. direct, photon-counting X-ray detectors), a significant improvement in spatial resolution can be obtained, demonstrated here at up to 17%. [ABSTRACT FROM AUTHOR]

This document is an erratum for a publication titled "More applications of the d-neighbor equivalence: Acyclicity and Connectivity constraints" in the SIAM Journal on Discrete Mathematics. The authors identified an error in their previous publication and provide a correction. The correction involves changing the section and theorem numbers and providing the corrected algorithms for solving partition problems. The authors also mention recent improvements by other researchers in handling acyclicity and removing the dependence on the number of vertices in certain upper bounds. [Extracted from the article]

Graph labeling admits a distinct place in Discrete Mathematics and Combinatorics. In 1998, Hikoe Enomoto et al. brought out the idea of super edge-magic (SEMC) labelling of graphs. In this article, we shall provide a complete solution to an open problem, regarding double stars St, t, proposed by W. D. Wallis & A. M. Marr [9], by exhibiting a super edge-magic labeling of St, t, for all possible values of t and t. We will also provide a SEMC labelling of union of cycles Cr being disjoint and various trees and exhibit super edge-magicness of some cases of Usmanian Ladders and planar graphs involving chains of C5. [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]

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 editorial serves as an introduction to Part 2 of the Special Issue Mathematics and the Life Sciences – a collection of articles showcasing ideas, examples, pedagogical frameworks, and curricular materials aiming to bridge the stubbornly persistent gap at the undergraduate level between the mathematical and the life sciences. The special issue features authors from public and private institutions of diverse types, sizes, and geographic locations: community colleges, liberal arts colleges, and research-oriented universities. We hope this special issue will serve as a resource to faculty who seek to make changes to their own course(s) or initiate curriculum reforms at their own schools. This Part 2 special issue features course materials and programs based on discrete mathematics, computational approaches, and statistics. It also includes articles on internship programs and co-curricular opportunities. Part 1 focuses on educational initiatives that are appropriate for calculus classes or require calculus as a prerequisite. [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]

In a recent article, Crider recommends ending a course with a memorable learning experience, called an epic finale, instead of a final exam. Here, we give the details of epic finales given in four mathematics courses: Discrete Mathematics, Information and Coding Theory, Real Analysis, and Complex Analysis. We describe how to reconfigure a course to allow for this type of closing activity and give suggestions for how to create an epic finale. [ABSTRACT FROM AUTHOR]

In [ Discrete Mathematics 306 (2005) 153-158], So proposed a conjecture saying that integral circulant graphs with different connection sets have different spectra. This conjecture is still open. We prove that this conjecture holds for integral circulant graphs whose orders have prime factorization of 4 types. [ABSTRACT FROM AUTHOR]

This document is a correction notice for an article titled "A NOTE ON THE POLAR DECOMPOSITION IN METRIC SPACES" published in the Journal of Mathematical Sciences. The correction states that the wrong supplementary file was initially published with the article, but it has now been replaced with the correct file. The original article has been corrected. The article is licensed under a Creative Commons Attribution 4.0 International License, allowing for use, sharing, adaptation, distribution, and reproduction as long as appropriate credit is given to the original author(s) and source. The contact information for the authors is provided at the end of the document. [Extracted from the article]

Exhibiting an explicit Boolean function with a large high-order nonlinearity is an important problem in cryptography, coding theory, and computational complexity. We prove lower bounds on the second-order, third-order, and higher order nonlinearities of some monomial Boolean functions. We prove lower bounds on the second-order nonlinearities of functions tr n (x 7) and tr n (x 2 r + 3) where n = 2 r. Among all monomial Boolean functions, our bounds match the best second-order nonlinearity lower bounds by Carlet [IEEE Transactions on Information Theory 54(3), 2008] and Yan and Tang [Discrete Mathematics 343(5), 2020] for odd and even n , respectively. We prove a lower bound on the third-order nonlinearity for functions tr n (x 15) , which is the best third-order nonlinearity lower bound. For any r , we prove that the r -th order nonlinearity of tr n (x 2 r + 1 − 1) is at least 2 n − 1 − 2 (1 − 2 − r) n + r 2 r − 1 − 1 − O (2 n 2 ). For r ≪ log 2 ⁡ n , this is the best lower bound among all explicit functions. [ABSTRACT FROM AUTHOR]

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]

Combinatorial proof serves both as an important topic in combinatorics and as a type of proof with certain properties and constraints. We report on a teaching experiment in which undergraduate students (who were novice provers) engaged in combinatorial reasoning as they proved binomial identities. We highlight ways of understanding that were important for their success with establishing combinatorial arguments; in particular, the students demonstrated referential symbolic reasoning within an enumerative representation system, and as the students engaged in successful combinatorial proof, they had to coordinate reasoning within algebraic and enumerative representation systems. We illuminate features of the students' work that potentially contributed to their successes and highlight potential issues that students may face when working with binomial identities. [ABSTRACT FROM AUTHOR]

For a connected graph G, let diam(G) and d(u, v) denote the diameter of G and distance between u and v, respectively. A radio labeling of a graph G is a mapping f : V(G) Æ {0, 1, 2, ... } such that |f(u) – f(v)| ≥ diam (G) + 1 – d(u, v) for every pair of distinct vertices u, v of G. The span of f is defined as span (f) = max { f(u) – f (v) : u, v ϵ V(G)}. The radio number rn(G) of G is defined as rn(G) = min{span(f) : f is a radio labeling of G}. In this paper, we determine the radio number for middle graph of dandelion, which generalized Bantva's results (Electronic Notes in Discrete Mathematics, 63 (2017), 93-100). [ABSTRACT FROM AUTHOR]

A set D ⊆ V (G) of a graph G is a dominating set if every vertex in V (G) ∖ D has a neighbor in D. The minimum cardinality of a dominating set of G is the domination number of G , denoted by γ (G). Let G be a maximal outerplanar graph of order n. The authors, in the article "On dominating sets of maximal outerplanar and planar graphs. Discrete Applied Mathematics, 198(2016):164-169", proved that γ (G) ≤ ⌊ k + n 4 ⌋ when k > 0 , where k is the number of bad vertices. However, we found that this result is incomplete. In this note, we construct a class A n of order n maximal outerplanar graphs such that k = 1 and γ (G) > ⌊ k + n 4 ⌋. In addition, we show that the upper bound ⌊ k + n 4 ⌋ holds for every order n maximal outerplanar graph G satisfying k > 0 and G ∉ A n , which improves the previous conclusion. [ABSTRACT FROM AUTHOR]

This article addresses an interdisciplinary study in which a course on Computational Thinking (CT) was developed in partnership with the Discrete Mathematics (DM) curricular subject to analyze and evaluate the results obtained in the discipline and compare them with those presented from of the CT course intervention, including a control group. The DM is a discipline considered difficult, on the part of the students, because it involves a lot of abstraction and logical reasoning causing a lot of disapproval and low grades. To develop the CT, the andragogical construct proposed in the Cordenonzi doctoral thesis was used. The case study was applied in a binational course - in which students, adults, Brazilians, and Uruguayans, shared the same learning space. Through this research and, based on the qualitative and quantitative analysis of the data, it was found that 95.2% of the sample obtained averages above the reference value, 60, which indicates that the development of the CT significantly collaborated in the resolution of problems. As an important contribution presented from this study was the combination of assessment instruments in two interventions that helped the understanding of the concepts addressed in the MD from the PC, as well as the scarce research on the interconnection of mathematics with PC in the context of higher education. [ABSTRACT FROM AUTHOR]

Quite recently, Mneimneh introduced the remarkable result whereby ∑ k = 0 n H k ( n k ) p k (1 − p) n − k = ∑ i = 1 n 1 − (1 − p) i i for 0 ≤ p ≤ 1 as the main result of a 2023 Discrete Mathematics article, where H k = 1 + 1 2 + ⋯ + 1 k denotes the k th harmonic number. Mneimneh's formula, as above, generalizes a result due to Paule and Schneider and to Spivey, and naturally arises as an expected value of a sum involving permutations chosen uniformly at random. In contrast to Mneimneh's probabilistic methods employed to prove the above formula, we generalize the above result so as to not require the given bounds on p , according to our two much simplified proofs of this generalized version of Mneimneh's formula introduced in this article. Our new and simplified proofs demonstrate how our previously introduced methods concerning harmonic sums may be applied systematically to prove and generalize identities as above. Our first proof is based on a way of extending Zeilberger's algorithm so as to be applicable to non-hypergeometric summands; our second proof is based on a beta-type integral formula. We also consider how Mneimneh's identity may be proved and generalized using the Mathematica package Sigma. [ABSTRACT FROM AUTHOR]

Pachinko is a Japanese mechanical gambling game similar to pinball. Recently, several mathematical models of Pachinko have been proposed. A number of pins are spiked in a field. A ball drops from the top of the playfield and the ball falls down. In the 50-50 model, if the ball hits a pin, it moves to the left or right passage of the pin with an equal probability. An arrangement of pins generates a distribution of the drop probability for all of the columns. This problem was considered by generating uniform distributions. Previous studies have demonstrated that the (1 / 2 a) -uniform distribution is possible for a ∈ { 0 , 1 , 2 , 3 , 4 } and is conjectured so that it is possible for any positive integer a. This study describes the constructive proof for this conjecture. This study also formalizes a natural decision problem yielded by this model while investigating its computational complexity. More precisely, given any drop-probability distribution A and any partial drop-probability distribution B , this study uses non-deterministic polynomial-time (NP) hardness to determine if there exists a pin arrangement that transforms A into B. [ABSTRACT FROM AUTHOR]

Measuring soil water-physical properties is laborious, time-consuming, and expensive. That provokes a lot of scientists to estimate them which action is troubled by the usage of different soil texture classification systems. The study proposes a rapid, reliable, and universally applicable methodology for soil textural transformations between different classification systems. The method of discrete mathematics is applied to make the conversion of particlesize classes from the Kachinsky system, which is used in Bulgaria to the International systems. Three different data sources were used to determine the water-physical properties of soils from textural data - 376 soil profiles from Bulgaria, extraction from the SoilGrids system for the Plovdiv district in Bulgaria and data from CanSIS/NSDB database. The relationship between the dependent variables field capacity (FC), wilting point (WP) and bulk density (BD), and independent variables sand, silt, and clay soil content was sought in the form of a regression equation. The applied stepwise regression procedure produces a close dependence between the soil texture and its water-physical properties. [ABSTRACT FROM AUTHOR]

Math courses graded on homework, tests, and a final examination can result in students putting together enough partial credit to pass the class without gaining a reasonable mastery of the course content. Mastery-based testing can help to address this by allowing the instructor to align course grades with mastery of the learning objectives for the course. This article is a reflection on my experiences implementing mastery-based testing in Discrete Mathematics. I give a brief introduction to the ideas behind mastery-based testing and discuss the construction of two of my mastery tests. The change affected my teaching, helping me to see where students need help understanding the material. Instructor observation and feedback from student evaluations of instruction indicate that student learning was improved with the change. [ABSTRACT FROM AUTHOR]

The text describes the application of one of the most important isometric transformations to the projected metro-rhythmic entities of individual harmonics of the spectrum. It is a direct isometry called central rotation. Central rotation conditions the hemiola structuring of the meter. Hemiolas are identified with regular and irregular geometric figures (primarily triangles) by means of a partition and the composition (index) number of a particular spectral harmonics. The partition and composition of numbers, which are dealt with in discrete mathematics, on the one hand, and, the technique of horizontal hemiolas, characteristic of the polyphony of the sub-Saharan region, on the other, served as a means of creating methods by which the isometric transformation of central rotation would be realized in (musical) time. [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]

We developed an automated 2-tiered Fuhrman’s grading system for clear cell renal cell carcinoma (ccRCC). Whole slide images (WSI) and clinical data were retrieved for 395 The Cancer Genome Atlas (TCGA) ccRCC cases. Pathologist 1 reviewed and selected regions of interests (ROIs). Nuclear segmentation was performed. Quantitative morphological, intensity, and texture features (n = 72) were extracted. Features associated with grade were identified by constructing a Lasso model using data from cases with concordant 2-tiered Fuhrman’s grades between TCGA and Pathologist 1 (training set n = 235; held-out test set n = 42). Discordant cases (n = 118) were additionally reviewed by Pathologist 2. Cox proportional hazard model evaluated the prognostic efficacy of the predicted grades in an extended test set which was created by combining the test set and discordant cases (n = 160). The Lasso model consisted of 26 features and predicted grade with 84.6% sensitivity and 81.3% specificity in the test set. In the extended test set, predicted grade was significantly associated with overall survival after adjusting for age and gender (Hazard Ratio 2.05; 95% CI 1.21–3.47); manual grades were not prognostic. Future work can adapt our computational system to predict WHO/ISUP grades, and validating this system on other ccRCC cohorts. [ABSTRACT FROM AUTHOR]

This research is motivated by John Swales' genre analysis of academic texts and rhetorical studies of visuals in scientific texts. Swales' approach still generates insights into the rhetorical structure of research genres across disciplines, yet few studies explore their nonverbal aspects. Rhetorical studies of visuals examine images and ignore links with surrounding texts and may overlook ways that the nonverbal contributes to the discipline's intellectual project. ESP genre research has begun multimodal analysis of academic genres but its focus on conference presentations has left a gap in the study of verbal-visual links in other academic genres. This paper addresses this gap in multimodal genre scholarship. After analyzing a corpus of 30 RAs from discrete mathematics to identify the functions of visuals and possible visual–move associations, we found that visuals in two-thirds of the corpus perform three functions: ontological, argumentative, and epistemological. Our data also indicates three multimodal rhetorical moves initiated primarily by visuals. Implications of our findings are that visual moves go beyond textual considerations, they can disrupt the RA's chronological structure, and novice writers in the field would benefit from understanding the crucial associations between the visual representations, disciplinary knowledge, and the rhetorical structure of RAs in this field. • Visuals in discrete mathematics RAs perform three functions: ontological, argumentative, epistemological. • Visuals in discrete mathematics RAs support arguments by performing multiple functions. • Visuals in discrete math RAs initiate three moves: Define visually, Represent the operation outcome, Embody the operation. • Novice writers must understand how visuals in discrete mathematics RAs connect to rhetorical structure. [ABSTRACT FROM AUTHOR]

Purpose: During resections of brain tumors, neurosurgeons have to weigh the risk between residual tumor and damage to brain functions. Different perspectives on these risks result in practice variation. We present statistical methods to localize differences in extent of resection between institutions which should enable to reveal brain regions affected by such practice variation. Methods: Synthetic data were generated by simulating spheres for brain, tumors, resection cavities, and an effect region in which a likelihood of surgical avoidance could be varied between institutions. Three statistical methods were investigated: a non-parametric permutation based approach, Fisher’s exact test, and a full Bayesian Markov chain Monte Carlo (MCMC) model. For all three methods the false discovery rate (FDR) was determined as a function of the cut-off value for the q-value or the highest density interval, and receiver operating characteristic and precision recall curves were created. Sensitivity to variations in the parameters of the synthetic model were investigated. Finally, all these methods were applied to retrospectively collected data of 77 brain tumor resections in two academic hospitals. Results: Fisher’s method provided an accurate estimation of observed FDR in the synthetic data, whereas the permutation approach was too liberal and underestimated FDR. AUC values were similar for Fisher and Bayes methods, and superior to the permutation approach. Fisher’s method deteriorated and became too liberal for reduced tumor size, a smaller size of the effect region, a lower overall extent of resection, fewer patients per cohort, and a smaller discrepancy in surgical avoidance probabilities between the different surgical practices. In the retrospective patient data, all three methods identified a similar effect region, with lower estimated FDR in Fisher’s method than using the permutation method. Conclusions: Differences in surgical practice may be detected using voxel statistics. Fisher’s test provides a fast method to localize differences but could underestimate true FDR. Bayesian MCMC is more flexible and easily extendable, and leads to similar results, but at increased computational cost. [ABSTRACT FROM AUTHOR]