Counting Lattice Paths Using Fourier Methods: Applied and Numerical Harmonic Analysis
Autor Shaun Ault, Charles Kiceyen Limba Engleză Paperback – 31 aug 2019
This monograph introduces a novel and effective approach to counting lattice paths by using the discrete Fourier transform (DFT) as a type of periodic generating function. Utilizing a previously unexplored connection between combinatorics and Fourier analysis, this method will allow readers to move to higher-dimensional lattice path problems with ease. The technique is carefully developed in the first three chapters using the algebraic properties of the DFT, moving from one-dimensional problems to higher dimensions. In the following chapter, the discussion turns to geometric properties of the DFT in order to study the corridor state space. Each chapter poses open-ended questions and exercises to prompt further practice and future research. Two appendices are also provided, which cover complex variables and non-rectangular lattices, thus ensuring the text will be self-contained and serve as a valued reference.
Counting Lattice Paths Using Fourier Methods is ideal for upper-undergraduates and graduate students studying combinatorics or other areas of mathematics, as well as computer science or physics. Instructors will also find this a valuable resource for use in their seminars. Readers should have a firm understanding of calculus, including integration, sequences, and series, as well as a familiarity with proofs and elementary linear algebra.
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Specificații
ISBN-13: 9783030266950
ISBN-10: 3030266958
Pagini: 136
Ilustrații: XII, 136 p. 60 illus., 1 illus. in color.
Dimensiuni: 155 x 235 mm
Greutate: 0.22 kg
Ediția:1st ed. 2019
Editura: Springer International Publishing
Colecția Birkhäuser
Seriile Applied and Numerical Harmonic Analysis, Lecture Notes in Applied and Numerical Harmonic Analysis
Locul publicării:Cham, Switzerland
ISBN-10: 3030266958
Pagini: 136
Ilustrații: XII, 136 p. 60 illus., 1 illus. in color.
Dimensiuni: 155 x 235 mm
Greutate: 0.22 kg
Ediția:1st ed. 2019
Editura: Springer International Publishing
Colecția Birkhäuser
Seriile Applied and Numerical Harmonic Analysis, Lecture Notes in Applied and Numerical Harmonic Analysis
Locul publicării:Cham, Switzerland
Cuprins
Lattice Paths and Corridors.- One-Dimensional Lattice Walks.- Lattice Walks in Higher Dimensions.- Corridor State Space.- Review: Complex Numbers.- Triangular Lattices.- Selected Solutions.- Index.
Textul de pe ultima copertă
This monograph introduces a novel and effective approach to counting lattice paths by using the discrete Fourier transform (DFT) as a type of periodic generating function. Utilizing a previously unexplored connection between combinatorics and Fourier analysis, this method will allow readers to move to higher-dimensional lattice path problems with ease. The technique is carefully developed in the first three chapters using the algebraic properties of the DFT, moving from one-dimensional problems to higher dimensions. In the following chapter, the discussion turns to geometric properties of the DFT in order to study the corridor state space. Each chapter poses open-ended questions and exercises to prompt further practice and future research. Two appendices are also provided, which cover complex variables and non-rectangular lattices, thus ensuring the text will be self-contained and serve as a valued reference.
Counting Lattice Paths Using Fourier Methods is ideal for upper-undergraduates and graduate students studying combinatorics or other areas of mathematics, as well as computer science or physics. Instructors will also find this a valuable resource for use in their seminars. Readers should have a firm understanding of calculus, including integration, sequences, and series, as well as a familiarity with proofs and elementary linear algebra.
Caracteristici
Introduces a unique technique to count lattice paths by using the discrete Fourier transform Explores the interconnection between combinatorics and Fourier methods Motivates students to move from one-dimensional problems to higher dimensions Presents numerous exercises with selected solutions appearing at the end