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Combinatorics
Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012
Broadly speaking, combinatorics is the branch of mathematics that deals with different ways of selecting objects from a set or arranging objects. It tries to answer two major kinds of questions, namely the existence question (does there exist a selection or arrangement of objects with a particular set of properties?) and the enumerative question (how many ways can a selection or arrangement be chosen with a particular set of properties?). But you may be surprised by the depth of problems that arise in combinatorics.
Combinatorics
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
Each of the sections in this chapter leads to new areas of active research, both in the applications to cybersecurity topics and in the study of combinatorial objects themselves. In addition to the examples presented here, algorithm design, network analysis, and high-performance computing are all active areas of research in which combinatorics plays a key role.
Combinatorics
Published in Sriraman Sridharan, R. Balakrishnan, Foundations of Discrete Mathematics with Algorithms and Programming, 2019
Sriraman Sridharan, R. Balakrishnan
Combinatorics is traditionally associated with the permutation and combination of a finite number of objects. Combinatorics deals with the following two types of problems: Existence problems and enumeration problems (see [3, 7]).
What is the problem in problem-based learning in higher education mathematics
Published in European Journal of Engineering Education, 2018
Mathematics is a body of abstract and complex knowledge, which in its purest form is not relating to reality. Plato in Republic VI (1997, 1129–1132) even argued for mathematics solely belonging to the ideal realm. However, even though mathematics is a body of abstract knowledge, PBL is still a fitting curriculum for developing higher level mathematics as students would experience the processes of (re)inventing mathematical knowledge, which sometimes is applicable to the general society, sometimes to the society of professionals, or to the society of researchers. The didactic transposition of mathematics as a scholarly field of knowledge to a PBL curriculum therefore includes that students learn to work research-like in PBL. Examples of the two latter societies, hence pure and applied mathematics are given above, where both problems were from a second-semester PBL project in mathematics. Both are within the overall theme of combinatorics, graph theory, and optimisation within the field of discrete mathematics. Case B from pure mathematics points to the fact that a relevant problem in PBL does not need to be a problem relevant to all of society. This is rarely the case in fundamental research within the natural sciences and mathematics. This means that the problem in PBL in fact only really makes sense and appears relevant within the context of a specific scientific society. To others, it is not considered relevant.
Degenerate 2D bivariate Appell polynomials: properties and applications
Published in Applied Mathematics in Science and Engineering, 2023
Shahid Ahmad Wani, Arundhati Warke, Javid Gani Dar
Polynomial sequences are of interest in enumerative combinatorics, algebraic combinatorics, and applied mathematics. The Laguerre polynomials, Chebyshev polynomials, Legendre polynomials, and Jacobi polynomials are a few polynomial sequences that appear as solutions to particular ordinary differential equations in physics and approximation theory. The most significant polynomial sequences is a class of Appell polynomial sequences [1]. Many applications of the Appell polynomial sequence may be found in theoretical physics, approximation theory, mathematics, and related fields of mathematics. The set of all Appell sequences is closed as a result of umbral polynomial sequence composition. This process turns the collection of all Appell sequences into an abelian group.
Special Issue dedicated to Workshop on Graph Spectra, Combinatorics and Optimization (WGSCO2018)
Published in Optimization, 2020
Rosalind Elster, Tatiana Tchemisova, Gerhard-Wilhelm Weber
The topics of the Workshop reflected the diversity of the scientific interests of Prof. Domingos M. Cardoso and the main lines of research of the Group on Graph Theory, Optimization and Combinatorics, which had been coordinated by him during many years within the Research Unit CIDMA of the Mathematics Department of the University of Aveiro: Algebraic Combinatorics, Algebraic Graph Theory, Algorithms and Computing Techniques, Combinatorial Optimization, Communications and Control Theory, Enumerative and Extremal Combinatorics, Graph Theory, Optimization in Graphs, Graph Spectra and Applications, Linear Optimization, Networks, Nonlinear Optimization, and others.