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Mathematical Methods - Isbn:9781842653418

Category: Mathematics

  • Book Title: Mathematical Methods
  • ISBN 13: 9781842653418
  • ISBN 10: 1842653415
  • Author: S. R. K. Iyengar, Rajendra K. Jain
  • Category: Mathematics
  • Category (general): Mathematics
  • Publisher: Alpha Science International, Limited
  • Format & Number of pages: 300 pages, book
  • Synopsis: This text offers a logical and lucid presentation of both theory and techniques for problem solving to motivate the students in the study and application of mathematics to solve Engineering problems.

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ISBN: 1891389246 - Mathematical Methods For Scientists And Engineers - OPENISBN Project: Download Book Data

Mathematical Methods For Scientists And Engineers

From best-selling author Donald McQuarrie comes his newest text, Mathematical Methods for Scientists and Engineers. Intended for upper-level undergraduate and graduate courses in chemistry, physics, math and engineering, this book will also become a must-have for the personal library of all advanced students in the physical sciences. Comprised of more than 2000 problems and 700 worked examples that detail every single step, this text is exceptionally well adapted for self study as well as for course use. Famous for his clear writing, careful pedagogy, and wonderful problems and examples, McQuarrie has crafted yet another tour de force.

Artwork from this textbook and original animations by Mervin Hanson may be viewed and downloaded by adopting professors and their students. Figures that display the time evolution of an equation and the result of the variation of a parameter have been rendered as QuickTime movies. These movies can be displayed as animations or by using the single-step feature of QuickTime.

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Date Mar 24, 2006

These relations permit us to calculate any desired coefficient am including a0 when the function f (x) is known.

Within the range of 0 ≤ t ≤ 2, this sine series converges to f (t) = t. Outside this range, this series converges to an odd periodic function shown in Fig. 1.9. It converges much faster than the series in (1.19). The first term, shown as dashed line, already provides a reasonable approximation. The difference between the three-term approximation and the given function is hardly noticeable.

= − 1 −inπ 1 bn = i(cn − c−n ) = i (e + einπ ) + (e−inπ − einπ ) i2n 2π n2 1 n = o dd 1 1 n − cos nπ +. sin nπ = 1 2 n πn − n n = even So the Fourier series can be written as n (−1)n π 2n 1 f (t) = − sin nt. cos(2n − 1)t − 2 4 π =1 (2n − 1) n =1.

1.7.4 Differentiation of Fourier Series In differentiating a Fourier series term-by-term, we have to be more careful. A term-by-term differentiation will cause the coefficients an and bn to be multiplied by a factor n. Since it grows linearly, the resulting series may not even converge. Take, for example nπ 2L n (−1)n+1 sin t. t= π =1 n L This equation is valid in the range of −L < t < L, as seen in (1.19). The derivative of t is of course equal to 1. However, a term-by-term differentiation of the Fourier series on the right-hand side 2∞ =∞ n d L n (−1)n+1 nπ nπ sin t 2 t (−1)n+1 cos dt π =1 n L L =1 does not even converge, let alone equal to 1.

(This is because there is no deflection and no moment at either end.) Find the deflection curve of the beam y (x). Solution 1.8.1. The function may be conveniently expanded in a Fourier sine series n∞ nπ y (x) = x. bn sin L =1 The four boundary conditions are automatically satisfied. This series and its derivatives are continuous, therefore it can be repeatedly term-by-term differentiated. Putting it in the equation, we have n∞

= lim eiωt = e−αt. It follows that for t > 0: F −1 = e−αt. For t < 0, the contour can be closed clockwise in the lower half plane as shown in Fig. 2.1b. Since there is no singular point in the lower half plane ∞ l 1 1 1 1 iω t e dω = eiωt dω = 0. 2π i −∞ ω − iα 2π i .h.p. ω − iα.

According to the convolution theorem, g(ω)f (ω ) = F . Since y(ω) = − it follows that: y (t) = F −1 = F −1 F = g (t) ∗ f (t). Therefore y (t) = − 1 2a ∞

Since no other form of wave function can reduce the product of uncertainties below this value, these relations are usually presented as ∆t · ∆E ≥ 2. ∆x · ∆p ≥ 2.

3.1 Functions as Vectors in Infinite Dimensional Vector Space
3.1.1 Vector Space When we construct our number system, first we find that additions and multiplications of positive integers satisfy certain rules concerning the order.

where ∆xj is the width of the subinterval. Regarding ∆xj as the weights, this definition is in accordance with (3.4). If we let n → ∞, this sum becomes an integral b f ∗ (x)g (x)dx. f |g =

Clearly, | φ0 (x)| w dx = For n = 1, let ψ 1 (x) = u1 (x) + a10 φ0 (x). we require ψ 1 (x) to be orthogonal to φ0 (x), φ ∗ 0 (x)ψ 1 (x)w (x) dx φ | 2 ∗ = (x)u1 (x)w(x) dx + a10 φ0 (x)| w(x) dx = 0. 0 Since φ0 is normalized to unity, we have φ ∗ a10 = − 0 (x)u1 (x)w (x)dx. With a10 so determined, ψ 1 (x) is a known function, which can be normalized. Let φ1 (x) = 1 1/2 ψ 1 (x). | 2 ψ 1 (x)| w(x)dx

In this example, we have used the Gram–Schmidt procedure to rearrange the set of linear independent functions into an orthonormal set for the given interval −1 ≤ x ≤ 1 and given weight function w(x) = 1. With other choices of intervals and weight functions, we will get other sets of orthogonal polynomials. For example, with the same set of functions and the same weight function w(x) = 1, if the interval is chosen to be [0, 1]. instead of [−1, 1]. the Gram–Schmidt process will lead to another set of orthogos s nal polynomials known as shifted Legendre polynomial . With Pn (x) s normalized in such a way that Pn (1) = 1, 2x
s Pn (x) = Pn.

Similarly, the second boundary condition requires y = n n (b) y (b) 0. ym (b) ym (b) Clearly, yn (b) p(b) ym (b) ym (b)
n (b).

Corresponding to each λn. the eigenfunction is the Legendre function Pn (x), which is a polynomial of order n. We have met these functions when we constructed an orthogonal set out of in the interval −1 ≤ x ≤ 1, with a unit weight function. The properties of this function will be discussed again in Chap. 4. Since Pn (x) are eigenfunctions of a Sturm–Liouville problem, they are orthogonal to each other in the interval −1 ≤ x ≤ 1 with respect to a unit weight function w(x) = 1. Furthermore, the set (n = 0, 1, 2. ) is complete. Therefore, any piece-wise continuous function f (x) in the interval −1 ≤ x ≤ 1 can be expressed as f (x) = where P |Pn f cn = n |Pn = n∞

To evaluate the integral, let u = 1 − x2. du = −2x dx, so d x1 x u 1 1 1 = = − ln u = − ln(1 − x2 ). dx − 2 −x 2 u 2 2 Thus, p(x) = e

Solution 3.5.1. (a) To solve this inhomogeneous differential equation, let us first look at the related eigenvalue problem, y

In this we have found both linearly independent solutions of this second-order differential equation. In case 2, p = −1, and a1 = 0. Because of the recurrence relation, all odd coefficients are zero, a1 = a3 = a5 = · · · = 0. Therefore we are left with y (x) = a0 1 cos x, x.

Now we will show that Nn (x) so defined is indeed a solution of the Bessel’s equation. By definition, Jα and J−α. respectively, satisfy the following differential equations J x x2 Jα (x) + xJα (x) + 2 − α2 α (x) = 0, J x x2 J−α (x) + xJ−α (x) + 2 − α2 −α (x) = 0. Differentiate with respect to α, ∂ + ∂ + ∂ Jα x2 Jα Jα d2 d x2 2 − 2αJα = 0, − α2 x dx ∂α dx ∂ α ∂α ∂ + ∂ + ∂ J−α x2 J−α J−α d2 d − 2αJ−α = 0. x2 2 − α2 x dx ∂α dx ∂α ∂α Multiplying the second equation by (−1) and subtracting it from the first equation, we have ∂ + + ∂ 2 Jα Jα d n ∂ J−α n ∂ J−α 2d x − (−1) − (−1) x dx2 ∂ α ∂α dx ∂ α ∂α − ∂ Jα x2 n ∂ J−α n − (−1) − α2 2α (Jα − (−1) J−α ) = 0. ∂α ∂α

The higher order of spherical Bessel functions can be generated by the recurrence relation 2l + 1 fl − fl−1. (4.58) fl+1 = x where fl can be jl. nl. hl. or hl. This recurrence relation is obtained by π multiplying (4.34) by /(2x) and set n = l + 1/2.
(1) (2).

Since a0 = 0, from (4.62) we have k = 0 or k = 1. If k = 1, then from (4.63) a1 must be 0. If k = 0, a1 can either be 0, or not equal to 0. So we have three cases: case 1. case 2. case 3. k = 0 and a1 = 0, k = 1 and a1 = 0, k = 0 and a1 = 0.

that al+2 = 0. It follows that al+4 = al+6 = · · · = 0. For example, if l = 0, then a2 = a4 = · · · = 0. The solution, according to (4.60). is simply y = a0. For any l, the solution can be systematically generated from (4.66) and (4.67). Since f (n) = n(n + 1) − l(l + 1). (n + 2)(n + 1) (4.71).



ERIC - Understanding RTI in Mathematics: Proven Methods and Applications, Brookes Publishing Company, 2011

Gersten, Russell, Ed.; Newman-Gonchar, Rebecca, Ed.

Brookes Publishing Company

Edited by National Math Panel veteran Russell Gersten with contributions by all of the country's leading researchers on RTI and math, this cutting-edge text blends the existing evidence base with practical guidelines for RTI implementation. Current and future RTI coordinators, curriculum developers, math specialists, and department heads will get the best, most up-to-date guidance on key facets of RTI in math: (1) conducting valid and reliable universal screening in mathematics; (2) using evidence-based practices to provide a strong general education curriculum for effective Tier 1 instruction; (3) implementing explicit, research-based teaching practices for students who need Tier 2 and 3 instruction; (4) monitoring students' progress with high-quality tools and measures; (5) motivating and engaging struggling students receiving Tier 2 and 3 instruction; (6) teaching students to use an array of visual representations to help them solve math problems; (7) tailoring RTI for every grade level, from kindergarten through high school; and (8) using RTI to target specific mathematical proficiencies and concepts, such as number sense, word problems, algebra, and ratios and proportions. Filled with vignettes, accessible summaries of the most recent studies, and best-practice guidelines for making the most of RTI, this comprehensive research volume is ideal for use as a textbook or as a key resource to guide decision makers. Readers will have the knowledge base they need to strengthen mathematics instruction with proven RTI practices--and help ensure better math outcomes for students at every grade level. An introduction, "Issues and Themes in Mathematics Response to Intervention Research and Implementation," is included. Contents of this book are: (1) Introduction of Response to Intervention in Mathematics (Paul J. Riccomini and Gregory W. Smith); (2) Universal Screening for Students in Mathematics for the Primaryrades: The Emerging Research Base (Russell Gersten, Joseph A. Dimino, and Kelly Haymond); (3) Understanding the R in RTI: What We Know and What We Need to Know About Measuring Student Response in Mathematics (Ben Clarke, Erica S. Lembke, David D. Hampton, and Elise Hendricker); (4) Pursuing Instructional Coherence: Can Strong Tier 1 Systems Better Meet the Needs of the Range of Students in General Education Settings? (Ben Clarke, Chris T. Doabler, Scott K. Baker, Hank Fien, Kathleen Jungjohann, and Mari Strand Cary); (5) Tier 2 Early Numeracy Number Sense Interventions for Kindergarten and First-Grade Students with Mathematics Difficulties (Diane Pedrotty Bryant, Greg Roberts, Brian R. Bryant, and Leann DiAndreth-Elkins); (6) A Two-Tiered RTI System for Improving Word-Problem Performance Among Students at Risk for Mathematics Difficulty (Lynn S. Fuchs, Douglas Fuchs, and Robin F. Schumacher); (7) Effective Instructional Practices in Mathematics for Tier 2 and Tier 3 Instruction (Madhavi Jayanthi and Russell Gersten); (8) Middle School Students' Thinking about Ratios and Proportions (Asha K. Jitendra, John Woodward, and Jon R. Star); (9) Using Visual Representations to Instruct and Intervene with Secondary Mathematics (Bradley S. Witzel, Deborah V. Mink, and Paul J. Riccomini); (10) Double-Dose Algebra as a Strategy for Improving Mathematics Achievement of Struggling Students: Evidence from Chicago Public Schools (Takako Nomi and Elaine Allensworth); (11) The Role of Motivation in Secondary Mathematics Instruction: Implications for RTI (John Woodward); and (12) Practical Considerations in the Implementation of RTI in Mathematics (Lauren Campsen, Alex Granzin, and Douglas Carnine). An index is included. [Foreword by Sharon Vaughn.]

Brookes Publishing Company. P.O. Box 10624, Baltimore, MD 21285. Tel: 800-638-3775; Fax: 410-337-8539; e-mail: custserv@brookespublishing.com; Web site: http://www.brookespublishing.com



Call Center Mathematics - A scientific method for understanding and improving contact centers

Call Center Mathematics - A scientific method for understanding and improving contact centers

This book explains all generic aspects of call and contact centers, from the basic Erlang formula to advanced topics such as skill-based routing and multi-channel environments.

Terms and Conditions:
Ger Koole wrote: This e-book can be printed and copied for personal use, as long as it is distributed as a whole, including the cover pages.
Book Excerpts:

This book is written for everybody who is dedicated to improving call center performance. It offers a scientific method to understanding and improving call centers. It explains all generic aspects of call and contact centers, from the basic Erlang formula to advanced topics such as skill-based routing and multi-channel environments. It does this without using complicated mathematical formulae, but by stressing the meaning of the mathematics. Moreover, there is a companion web page where many calculations can be executed. Next to understanding call center phenomena this book shows how to use this insight to improve call center performance in a systematic way. Keywords are data collection, scenario analysis, and decision support.

This book is also a bridge between call center management and those parts of mathematics that are useful for call centers. It shows the manager and consultant the benefits of mathematics, without having to go into the details of it. It also shows the mathematically educated reader an interesting application area of queueing theory and other fields of mathematics. As such, this book can also be used as additional material in an applied course for mathematics and industrial engineering students. Basic knowledge of call centers is assumed, although a glossary is added in case of omissions.

The first chapters are of a general nature, discussing the benefits of call center mathematics and call center management objectives. Then this book continues with those subjects that are relevant to every call center manager or planner: the Erlang C formula, forecasting, and staffing. The final chapters contain more advanced subjects, such as extensions to the Erlang C model and multiple skills. This book concludes with a glossary, an annotated bibliography, and an appendix with the math of the Erlang C formula.



Mathematical Methods in Psychology

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Mathematical Methods for Physicists, 7th Edition

Mathematical Methods for Physicists, 7th Edition Key Features
  • Revised and updated version of the leading text in mathematical physics
  • Focuses on problem-solving skills and active learning, offering numerous chapter problems
  • Clearly identified definitions, theorems, and proofs promote clarity and understanding

New to this edition:

  • Improved modular chapters
  • New up-to-date examples
  • More intuitive explanations

Now in its 7th edition, Mathematical Methods for Physicists continues to provide all the mathematical methods that aspiring scientists and engineers are likely to encounter as students and beginning researchers. This bestselling text provides mathematical relations and their proofs essential to the study of physics and related fields. While retaining the key features of the 6th edition, the new edition provides a more careful balance of explanation, theory, and examples. Taking a problem-solving-skills approach to incorporating theorems with applications, the book's improved focus will help students succeed throughout their academic careers and well into their professions. Some notable enhancements include more refined and focused content in important topics, improved organization, updated notations, extensive explanations and intuitive exercise sets, a wider range of problem solutions, improvement in the placement, and a wider range of difficulty of exercises.

Graduate students and advanced undergraduates in Physics, Engineering, Applied Mathematics, Chemistry, and Environmental Science/Geophysics; also practitioners and researchers in these fields.

George Arfken



Danilova O


Danilova Olesya Vladislavovna 1. Garifullina Albina Maratovna 2
1 Bashkir State University, student
2 Bashkir State University, associate professor, teacher

This article will acquaint you with mathematical science in economy. Mathematical methods are the most important tool of the analysis of processes and the phenomena in economy.

Article reference:
Danilova O.V. Garifullina A.M. Mathematical methods in economy // Economics and innovations management. 2014. № 1 [Electronic journal]. URL: http://ekonomika.snauka.ru/en/2014/01/3527

The economic problems arising before experts often have difficult character. They depend on a set of various factors which in the majority of the cases contradict each other, they as change over time and influence other problems and processes.

For this reason, for the solution of these problems on an economy and mathematics joint, there are special receptions with the characteristic name "economic-mathematical methods". The channelized scientific is devoted to research of economic systems and processes by means of mathematical models. As the mathematical model reflects a problem in an abstract form, she allows to consider bigger number of various characteristics on which this or that problem depends. The analysis and calculation of mathematical model allow to choose optimum solutions of an objective and to prove this choice. As a whole mathematical modeling becomes the language of the modern economic theory which is equally clear for the scientific all countries of the world.

On the general purposeful purpose economic-mathematical models divide on theoretic-analytical which are used when studying the general properties and regularities of economic processes, and applied which are in turn applied in the solution of specific economic objectives of the analysis, forecasting and management.

Mathematical methods are the most important tool of the analysis of processes and the phenomena in economy, creation of the theoretical models, allowing to display existing communications in economic life. They allow to predict behavior of economic subjects and economic dynamics. As a part of economic-mathematical methods usually allocate such scientific disciplines, as economic cybernetics, mathematical statistics, mathematical economy and econometrics.

By means of use of mathematical methods, including by means of different formulas and schedules, it is possible to diagnose and solve many economic problems that significantly facilitates life to many people. The solution of tasks of the economic analysis mathematical methods probably if they are formulated mathematically, that is real economic interrelations and dependences are expressed with application of the mathematical analysis. As causes the necessity of development of mathematical models.

Application of mathematical methods in practice always demands system approach to research of the set object, and as accounting of interrelations and relations with other objects. For example, with other enterprises and firms. Not less important is improvement of system of information support of management of the enterprise with use of modern electronic computer facilities. And among other things not the last place is taken by development of mathematical models which reflect quantitative indices of system activity of employees of the organization, the processes happening in difficult systems.

Now all economic systems gradually develop and become complicated, thereby there is a change of their structure, and sometimes and the contents caused by scientific and technical progress. All this does outdated many methods applied earlier, they demands their adjustment. But at the same time scientific and technical progress influences and mathematical methods as emergence and improvement of electronic computers made possible wide use of the methods which have been earlier described only theoretically, or applied only for small applied tasks.

For improvement of management by economy in general and commercial activity in particular the increasing attention is given to application of mathematical methods and computer facilities. Thus, mathematical methods are the major methods which are able to give to the economic theory scientific completeness.

The mathematics both in economic science and in economic informatics is applied in the increasing scales. By means of its methods it is possible to display communications which are displayed in economy, to count behavior of various economic subjects and their dynamics. Without mathematics the economy won't be able to exist. At what it becomes economy language which is already clear to scientists of all countries of the world. It is now obvious that it - necessary part of the economic theory. However it is insufficient as also purely economic substantial component becomes more and more difficult, and the unformulated party of the description of economic events will be always present.

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