Numbers and Functions: Theory, Formulation, and Python Codes (1版)

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【簡介】 This unique volume covers two fundamental elements of computational methods - numbers and functions. It provides an in-depth discussion of the behaviors of numbers, including both real and complex numbers. The discussion leads to the important closure properties of numbers, ensuring solution consistence and existence, and also possible failure in numerical computations in science and engineering. This text then introduces types of functions that take numbers as independent variables and produce a single number. Approaches for constructing inverse functions are also provided. Frequently used basis functions are introduced, with detailed studies on their properties and behaviors. Techniques are presented for constructing basis functions and their use in function approximation in computational methods. 【目錄】

Mathematics for Engineers Vol.1 Numbers and Equations, Vectors and Matrices, Functions in one Variable, Differential Calculus

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【簡介】 The Karlsruhe University of Applied Sciences started an English-language Bachelor's program in Electrical Engineering in 2021. In order to support our international students with appropriate material, this three-volume textbook was written. Taking into account that different students will attend different mathematics courses, we decided to design three volumes of a series of mathematics books, each part for a single semester. Mathematical terms are clearly motivated, systematically equated and visualized in many animations. Mathematical proofs are almost completely renounced. Instead, a lot of applications support not only the application of mathematics but also contribute to a better understanding of mathematics. Important formulas and statements are clearly highlighted in order to increase the readability of the books. More than 300 images and sketches support the character of modern textbooks. The color-coded layout provides a clear overview of the presentation of the content, e.g. by adding new terms and definitions in light grey, important statements and sentences in grey. 【目錄】 Numbers, Equations and Systems of Equations Vectors and Vector Calculus Matrices and Determinants Basic Functions Complex Numbers Limits and Continuity Differential Calculus

PERFECT NUMBERS AND FIBONACCI SEQUENCES

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In this book, we first review the history and current situation of the perfect number problem, including the origin story of the Mersenne primes, and then consider the history and current situation of the Fibonacci sequence. Both topics include results from our own research. In the later sections, we define the square sum perfect numbers, and describe for the first time the secret relationships connecting the square sum perfect numbers, the Fibonacci sequence, the Lucas sequence, the twin prime conjecture, and the Fermat primes. Throughout, we raise various interesting questions and conjectures. Sample Chapter(s) Preface Chapter 1: The History of Perfect Numbers Contents: Preface The History of Perfect Numbers: What are Perfect Numbers? Euclid's Elements Nicomachus Sums of Squares and Cubes Ibn al-Haytham Mersenne Numbers and Mersenne Primes Fermat and Descartes The Euclid–Euler Theorem Ivan Pervushin The Lucas–Lehmer Primality Test The Great Internet Mersenne Prime Search Questions Related to Perfect Numbers: Properties of Even Perfect Numbers Open Questions Odd Perfect Numbers Touchard's Theorem Deficient and Abundant Numbers Weird Numbers and Semiperfect Numbers Ore Numbers and the Harmonic Mean Variations on Ore Numbers Amicable Numbers Multiply Perfect Numbers Three Further Generalizations S-perfect Numbers The Golden Ratio Conjecture The Fibonacci Sequence: Fibonacci, Leonardo of Pisa The Rabbit Problem General Terms and Limits Connection with Continued Fractions Three Identities Equations Between Binomial Coefficients Divisibility Sequences Zeckendorf's Theorem From Base 2 to Base 3 Hilbert's Tenth Problem Lucas Numbers and Lucas Sequences: The Lucas Numbers Criteria for Fibonacci Numbers Prime Divisors of Fibonacci Numbers Fibonacci Congruences A More General Congruence Narayana Sequence Congruences Pythagorean Triples Diophantine m-tupples Generating Functions Lucas Sequences Pisano Period π(n) and πL(n) Prime Divisors of Lucas Numbers Perfect Numbers and Fibonacci Primes: Square Sum Perfect Numbers Some Lemmas Proof of Theorems A New Conjecture Concerning Perfect Numbers Affine Square Sum Perfect Numbers Square Sum Perfect Numbers and the Twin Prime Conjecture Fermat Primes and GM Numbers The abcd Equation Applications of Elliptic Curves Results with Lucas Sequences Appendices: The First 100 Fibonacci Numbers and Their Prime Factorizations The First 100 Lucas Numbers and Their Prime Factorizations Prime Factorization of the First 100 Lucas Numbers Bibliography Index Readership: It is the first monograph to connecting perfect number and Fibonacci sequence, the two old mathematical problems, with rich history, it will attract number theorists and combinatorists in the world. Also good for maths teachers and fans.