CATALAN NUMBERS

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Catalan numbers, named after the French-Belgian mathematician Eugène Charles Catalan (1814–1894), arise in a variety of combinatorial problems. They have many interesting properties, a rich history, and numerous arithmetic, number-theoretical, analytical, and combinatorial connections, as well as a variety of classical and modern applications. Considering the long list of open problems and questions related to the classical case, its relatives (Bell numbers, Motzkin numbers, Narayana numbers, etc.) and its generalizations, this book provides a broad perspective on the theory of this class of special numbers that will be of interest to professionals, students, and a general audience. The book begins with the history of the problem, before defining the considered numerical sets. The recurrence equation, closed formula, and generating function are then presented, followed by the simplest properties and number-theoretical properties. Later chapters discuss the relationships between Catalan numbers and other special numbers, as well as their applications and open problems. Sample Chapter(s) Preface Chapter 1: Preliminaries Contents: Preliminaries Pascal's Triangle Catalan Numbers Relatives of Catalan Numbers Catalan Numbers and Their Relatives on Integer Lattice Zoo of Numbers Mini Dictionary Exercises Readership: The target audience for this book is university teachers and students at the undergraduate and postgraduate levels interested in combinatorics, number theory, general algebra, cryptography and related fields, as well as a general audience of amateur mathematicians.

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

SAY OUT ! 英文單字卡牌-Numbers Colors & Nouns

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每副含55張卡片，七種玩法說明書，和透明收納盒。 超級好玩的桌遊卡牌，訓練學生、孩子初階認字速度從這套開始。 適用於課堂學習、課後桌遊，幫助學生、孩子主動認識了解單字，透過大聲喊出卡牌單字，能增進口說能力，鍛鍊反應力、專注力和觀察力，遊戲具有趣味性及多變化，絕對是學習與玩樂兼俱的一款實用教具。 Nouns, Numbers & Colors word list : apple, baby, ball, bear, bird, boy, cake, car, cat, chair, chicken, cow, dog, doctor, door, duck, eye, farmer, fish, flower, girl, hand, head, house, money, nose, pig, rabbit, sheep, sun, table, tiger, tree, window one, two, three, four, five, six, seven, eight,nine, ten eleven, twelve black, blue, yellow, red, purple, white,brown, pink, gray, green, orange SAY OUT BOARD GAMES SAYOUT桌遊基礎字系列有Sight words 1&2, Phonics 1&2 and Nouns & numbers，主要在建立學生的基礎單字認字能力，每副配件55張卡牌，包含57個主要單字，分別出現在有8個單字的紙牌上。遊戲基本原則就是要快速找出每兩張牌相同的單字，藉此培養學生閱讀單字的速度和能力，這裡整理幾種玩法如下: 玩法1 1.將卡牌平分給班上學生(建議不超過10入)。 2.學生自由走動找要玩的對手。 3.找到後兩人同時翻開手上的一張牌，先認出相同單字並大聲唸出的就把對方那張卡牌拿走，接著再找下一個人，玩法一樣。 4.一定時間後看誰手上有最多卡牌為贏家。 玩法#2 1.將卡牌分給班上學生一人一張，剩下的老師留著當作給每一回贏家的獎勵牌。 2.老師用單槍投影顯示一張牌，學生迅速尋找出和他的牌上相同的單字(每個學生的相同單字不一定相同)。 3.第一個喊出正確相同單字的人得分(或是前3個)，老師發給一張牌，該牌做為該學生下一回基準牌。 4.遊戲重複進行直到老師出清手上的牌為止，學生手上牌數最多的即為贏家。 玩法#3 1.將牌洗好後面朝下全部放在桌面(3~4位玩家為佳)。 2.玩家同時翻開一張桌上的牌，然後尋找自己手上牌和任何一個對手相同的單字(每個玩家的相同單字會不同)。 3.第一個喊出正確相同單字的人可以拿走該對手的那張牌，此張牌並做為該玩家基準牌，所有玩家蓋牌後依此玩法繼續進行。 4.遊戲重複進行直到桌上所有牌出清為止，牌數「最多」的人即為赢家。