The present work consists of two parts. First, methods for proving non-holonomicity of sequences and functions are discussed. Holonomic sequences, which are defined by linear difference equations with polynomial coefficients, are a popular object of study in combinatorics. The presentation of methods for finding out whether a given sequence is in this class or not takes the reader through number theory, complex analysis, and analytic combinatorics. The second part of the book is devoted to inequalities involving recursively defined sequences. It contains a new computer algebra assisted proving procedure. This approach, which is based on cylindrical algebraic decomposition, allows to prove many classical inequalities mechanically....

ISBN: 978-3-8364-9460-1
Издательство: Книга по требованию
Дата выхода: июль 2011

In mathematics, a combinadic is an ordered integer partition, or composition. Combinadics provide a lexicographical index for combinations. Applications for combinadics include software testing, sampling, quality control, and the analysis of gambling games such as Canada's national 6/49 lottery. For definiteness, we will consider the k-combinations on the set S = {0, 1, ..., n ? 1} of the first n integers starting from 0. Recall that there are C(n,k) = n! / ( k! (n ? k)! ) of these. The index we are looking for is the mapping associating the numbers i = 0, 1, ..., C(n,k) ? 1 with the list of k-combinations having their elements written in ascending order, and themselves ordered lexicographically. By abuse of notation we denote as...

ISBN: 978-6-1306-3984-6
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics a combinatorial explosion describes the effect of functions that grow very rapidly as a result of combinatorial considerations. Examples of such functions include the factorial function and related functions. Pathological examples of combinatorial explosion include functions such as the Ackermann function. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1308-0373-5
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! En mathematiques, la combinatoire, appelee aussi analyse combinatoire, etudie les configurations de collections finies d'objets ou les combinaisons d'ensembles finis, et les denombrements. La combinatoire remonte a l'Antiquite : Plutarque rapporte ainsi un debat entre Chrysippe et Hipparque sur le nombre de facons de combiner dix propositions, le resultat n'ayant ete compris qu'au Xxe siecle. Parmi, les autres precurseurs, on peut citer Bh?skara II au XIIe siecle (nombre de choix de p elements parmi n), Raymond Lulle au XIIIe siecle, Gersonide au debut du XIVe siecle (rapport entre le nombre d'arrangements et le nombre de combinaison), Michael Stifel au XVIe siecle (premiere approche du...

ISBN: 978-6-1325-2227-6
Издательство: Книга по требованию
Дата выхода: июль 2011

The book investigated the applications of combinatorics and graph theory in the analysis of the Feynman Identity of the partition function of the two dimensional Ising Model. Chapter one gives a general introduction to the partition function of the Ising Model and the Feynman Identity in the language of graph theory. Chapter two describes and proves combinatorially the Feynman Identity in a special case when there is only one vertex and multiple loops. Chapter three introduces a new way to calculate the number of cycles in a directed graph, along with its application in the special case discussed in chapter two to derive the analytical expression of the number of non-periodic cycles. Chapter four comes back to the general form of the...

ISBN: 978-3-6391-7821-0
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! En mathematiques, les combinaisons lineaires sont un concept central de l'algebre lineaire et d'autres domaines des mathematiques connexes. La majeure partie de cet article traite des combinaisons lineaires dans le contexte d'espace vectoriel sur un corps commutatif, et indique quelques generalisations a la fin de l'article. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1341-4466-7
Издательство: Книга по требованию
Дата выхода: июль 2011

Combinatorics on words is a branch of mathematics which applies combinatorics to words and formal languages. The study of combinatorics on words arose independently within several branches of mathematics, e.g. number theory, group theory and probability. It has applications to combinatorial enumeration and fractal analysis and appears in problems of theoretical computer science, automata theory and linguistics. While many applications are new, the classical Chomsky–Schutzenberger hierarchy of classes of formal grammars is perhaps the best known result in the field. The development of computerized text and string processing has led to important applications of combinatorics on words. It is involved in the core algorithms for text...

ISBN: 978-6-1302-2386-1
Издательство: Книга по требованию
Дата выхода: июль 2011

Combinatorics is about the art and philosophy with respect to “count without counting”. This is the dissertation of Dr. Chunwei Song at the University of Pennsylvania back in 2004. Herein various aspects of lattice path combinatorics are investigated, and the elegance of this beautiful subject is majestically demonstrated. For historical concerns, content in this volume is untouched from its original form as in the author's submitted doctoral thesis. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-3-8433-6155-2
Издательство: Книга по требованию
Дата выхода: июль 2011

Combinatory logic is a notation introduced by Moses Schonfinkel and Haskell Curry to eliminate the need for variables in mathematical logic. It has more recently been used in computer science as a theoretical model of computation and also as a basis for the design of functional programming languages. It is based on combinators. A combinator is a higher-order function that uses only function application and earlier defined combinators to define a result from its arguments. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1306-2764-5
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! For the table below, storage aspect ratio (SAR) is based solely on pixel count. It does not take into account pixel aspect ratio (pixels may be non- square) and thus the display aspect ratio (the aspect ratio of the actual image that is displayed) may differ. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1340-3111-0
Издательство: Книга по требованию
Дата выхода: июль 2011

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. There are common integrals in quantum field theory that appear repeatedly. These integrals are all variations and generalizations of gaussian integrals to the complex plane and to multiple dimensions. Other integrals can be approximated by versions of the gaussian integral. Fourier integrals are also considered. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1337-3770-9
Издательство: Книга по требованию
Дата выхода: июль 2011

In mathematics, commutativity is the property that changing the order of something does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. The commutativity of simple operations, such as multiplication and addition of numbers, was for many years implicitly assumed and the property was not named until the 19th century when mathematicians began to formalize the theory of mathematics. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1307-2798-7
Издательство: Книга по требованию
Дата выхода: июль 2011

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra.Some specific kinds of commutative rings are given with the following chain of class inclusions: commutative rings ? integral domains ? unique factorization domains ? principal ideal domains ? Euclidean domains ? fields. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1302-0958-2
Издательство: Книга по требованию
Дата выхода: июль 2011

A comparison sort is a type of sorting algorithm that only reads the list elements through a single abstract comparison operation (often a "less than or equal to" operator) that determines which of two elements should occur first in the final sorted list. The only requirement is that the operator obey two of the properties of a total order: if a ? b and b ? c then a ? c (transitivity), for all a and b, either a ? b or b ? a (totalness or trichotomy). It is possible that both a ? b and b ? a; in this case either may come first in the sorted list. In a stable sort, the input order determines the sorted order in this case. A metaphor for thinking about comparison sorts is that you have a set of unlabelled weights and a balance scale. The...

ISBN: 978-6-1307-6779-2
Издательство: Книга по требованию
Дата выхода: июль 2011

Model (or variable) selection is an essential part of any statistical analysis. Since a GEE model does not specify a likelihood structure, traditional model selection criteria are not well defined in the GEE approach. In last decade, modified Akaike’s Information Criterion (mAIC), modified Bayesian Information Criterion (mBIC), and extended Mallow’s Cp are suggested for GEE.Our main goal is to compare the above mentioned model selection criteria of GEE to find the suitable one. For this we conduct an extensive Monte Carlo simulation study to examine the relative performance of these criteria to select the best underlying model. Finally, considering the simulation study results, we apply GCp criterion to maternal morbidity data to...

ISBN: 978-3-6392-0718-7
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! A comparametric equation is an equation that describes a parametric relationship between a function and a dilated version of the same function, where the equation does not involve the parameter. For example, (2t) = 4 (t) is a comparametric equation, when we define g(t) = f(2t), so that we have g = 4 no longer contains the parameter, t. The comparametric equation g = 4 has a family of solutions, one of which is = t2. To see that = t2 is a solution, we merely substitute back in: g = (2t) = (2t)2 = 4t2 = 4 , so that g = 4 . Comparametric equations arise naturally in signal processing when we have multiple measurements of the same phenomenon, in which each of the measurements was acquired using a...

ISBN: 978-6-1311-4425-7
Издательство: Книга по требованию
Дата выхода: июль 2011

ISBN: 978-3-8383-4031-9
Издательство: Книга по требованию
Дата выхода: июль 2011

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. The competitive Lotka–Volterra equations are a simple model of the population dynamics of species competing for some common resource.The form is similar to the Lotka–Volterra equations for predation in that the equation for each species has one term for self-interaction and one term for the interaction with other species. In the equations for predation, the base population model is exponential. For the competition equations, the logistic equation is the basis.The definition of a competitive Lotka-Volterra system assumes that all values in the interaction matrix are positive or 0 ( ij >= 0 for all i,j). If it...

ISBN: 978-6-1311-4434-9
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In mathematics, a quadratic polynomial or quadratic is a polynomial of degree two, also called second-order polynomial. That means the exponents of the polynomial's variables are no larger than 2. For example, x2 ? 4x + 7 is a quadratic polynomial, while x3 ? 4x + 7 is not.The coefficients of a polynomial are often taken to be real or complex numbers, but in fact, a polynomial may be defined over any ring.When using the term "quadratic polynomial", authors sometimes mean "having degree exactly 2", and sometimes "having degree at most 2". If the degree is less than 2, this may be called a "degenerate case". Usually the context will establish which of the two is meant. Данное...

ISBN: 978-6-1311-2687-1
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In the mathematical field of graph theory, a complete graph is a simple graph in which every pair of distinct vertices is connected by an edge. The complete graph on n vertices has n vertices and n(n-1)/2 edges, and is denoted by (from the German komplett). It is a regular graph of degree . All complete graphs are their own cliques. They are maximally connected as the only vertex cut which disconnects the graph is the complete set of vertices. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1308-0309-4
Издательство: Книга по требованию
Дата выхода: июль 2011

Complexity theory aims at the understanding of the inherent hardness of computational problems. This book studies the complexity of the most basic counting problems in algebraic geometry within an algebraic framework of computation. The author gives an efficient parallel method for the two problems of counting the connected and irreducible components of complex algebraic varieties. On the other hand, it is shown that deciding connectedness of varieties is PSPACE-hard. This result is also extend to higher Betti numbers. Furthermore, the problem of counting irreducible components for a fixed number of equations is reduced to a fixed number of variables. The consequences are a polynomial time algorithm in the BSS-model, and a randomised...

ISBN: 978-3-8364-9844-9
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! Complete School is an educational product aimed at secondary or high school students studying Mathematics or English. It is the third title by Australian educational self-publisher Michael Milford and was first released in May 2006 around Australia.Like the previous two titles by the publisher, Complete School is aimed at the secondary or high school market. However it is broader in scope and contains material aimed at younger secondary school students as well as the older students, unlike previous titles which have focussed only on the older students. It also attempts to cover both the English and Mathematics high school subjects. Данное издание не является...

ISBN: 978-6-1311-3405-0
Издательство: Книга по требованию
Дата выхода: июль 2011

High Quality Content by WIKIPEDIA articles! In set theory, a complement of a set A refers to things not in (that is, things outside of), A. The relative complement of A with respect to a set B, is the set of elements in B but not in A. When all sets under consideration are considered to be subsets of a given set U, the absolute complement of A is the set of all elements in U but not in A.If A and B are sets, then the relative complement of A in B, also known as the set-theoretic difference of B and A, is the set of elements in B, but not in A.The relative complement of A in B is denoted B ? A (sometimes written B ? A, but this notation is ambiguous, as in some contexts it can be interpreted as the set of all b ? a, where b is taken from...

ISBN: 978-6-1311-2446-4
Издательство: Книга по требованию
Дата выхода: июль 2011

A complex quadratic polynomial is a quadratic polynomial whose coefficients are complex numbers.In mathematics, a quadratic polynomial or quadratic is a polynomial of degree two, also called second-order polynomial. That means the exponents of the polynomial's variables are no larger than 2. For example, x2 ? 4x + 7 is a quadratic polynomial, while x3 ? 4x + 7 is not.The coefficients of a polynomial are often taken to be real or complex numbers, but in fact, a polynomial may be defined over any ring. Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.

ISBN: 978-6-1311-0600-2
Издательство: Книга по требованию
Дата выхода: июль 2011

The present book offers a first course on Complex Analysis. It deals with the differential and integral properties of functions of a single complex variable. It includes conformal transformations and applications of complex analysis to potential theory and flow problems.The subject matter is presented here in five chapters. It starts with a historical note for the introduction and subsequent development of the subject right from the initial stage to the present modern complex analysis. The first chapter includes the introduction of complex numbers, their geometric representation, De Moivre's theorem, roots and logarithm of a complex number and real and imaginary parts of a complex function.The concepts of differential calculus for...

ISBN: 978-3-8433-8859-7
Издательство: Книга по требованию
Дата выхода: июль 2011