This book presents important works by the Scottish mathematician Colin MacLaurin (1698-1746), translated in English for the first time. It includes three of the mathematician 's less known and often hard to obtain works. A general introduction puts the works in context and gives an outline of MacLaurin's career. Each translation is also accompanied by an introduction and analyzed both in modern terms and from a historical point of view.

The first instance of pre-computer fractals was noted by the French mathematician Gaston Julia. He wondered what a complex polynomial function would look like, such as the ones named after him (in the form of z2 + c, where c is a complex constant with real and imaginary parts). The idea behind this formula is that one takes the x and y coordinates of a point z, and plug them into z in the form of x + i*y, where i is the square root of -1, square this number, and then add c, a constant. Then plug the resulting pair of real and imaginary numbers back into z, run the operation again, and keep doing that until the result is greater than some number. The number of times you have to run the equations to get out of an 'orbit' not specified here can be assigned a colour and then the pixel (x,y) gets turned that colour, unless those coordinates can't get out of their orbit, in which case they are made black. Later it was Benoit Mandelbrot who used computers to produce fractals. A basic property of fractals is that they contain a large degree of self similarity, i.e., they usually contain little copies within the original, and these copies also have infinite detail. That means the more you zoom in on a fractal, the more detail you get, and this keeps going on forever and ever. The well-written book 'Getting acquainted with fractals' by Gilbert Helmberg provides a mathematically oriented introduction to fractals, with a focus upon three types of fractals: fractals of curves, attractors for iterative function systems in the plane, and Julia sets. The presentation is on an undergraduate level, with an ample presentation of the corresponding mathematical background, e.g., linear algebra, calculus, algebra, geometry, topology, measure theory and complex analysis. The book contains over 170 color illustrations.

Covers unit A2 1: Pure Mathematics for the CCEA specification The book has been completely re-designed to follow the same layout as the Further Maths book. Answers are included at the rear of the book. Contents: Algebra and Graphs Functions Radian Measur Coordinate Geometry Sequences and Series Binomial Expansion Trigonometric Function Trigonometric Identities and Equations Differentiation Further Differentiation Integration Differential Equations Numerical Methods Problem Solving

Budgeting can be stressful and overwhelming to the average American. You can learn how to cut corners from grocery shopping, to going on that much wanted vacation to saving money on entertainment that it hard to afford on the budget you have yet to make. Don't feel drained at the end of the month; read Life on A budget to begin feeling more energized by your "desire" and "motivation."

In the 5th century the Indian mathematician Aryabhata (476-499) wrote a small but famous work on astronomy, the Aryabhatiya. This treatise, written in 118 verses, gives in its second chapter a summary of Hindu mathematics up to that time. Two hundred years later, an Indian astronomer called Bhaskara glossed this mathematial chapter of the Aryabhatiya.

An english translation of Bhaskara s commentary and a mathematical supplement are presented in two volumes.

Subjects treated in Bhaskara s commentary range from computing the volume of an equilateral tetrahedron to the interest on a loaned capital, from computations on series to an elaborate process to solve a Diophantine equation.

This volume contains an introduction and the literal translation.

The introduction aims at providing a general background for the translation and is divided in three sections: the first locates Bhaskara s text, the second looks at its mathematical contents and the third section analyzes the relations of the commentary and the treatise."

In the 5th century the Indian mathematician Aryabhata (476-499) wrote a small but famous work on astronomy, the Aryabhatiya. This treatise, written in 118 verses, gives in its second chapter a summary of Hindu mathematics up to that time. Two hundred years later, an Indian astronomer called Bhaskara glossed this mathematial chapter of the Aryabhatiya.

An english translation of Bhaskara s commentary and a mathematical supplement are presented in two volumes.

Subjects treated in Bhaskara s commentary range from computing the volume of an equilateral tetrahedron to the interest on a loaned capital, from computations on series to an elaborate process to solve a Diophantine equation.

This volume contains explanations for each verse commentary translated in Volume 1. These supplements discuss the linguistic and mathematical matters exposed by the commentator. Particularly helpful for readers are an appendix on Indian astronomy, elaborate glossaries, and an extensive bibliography.

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This text gives a detailed account of various techniques that are used in the study of dynamics of continuous systems, near as well as far from equilibrium. The analytic methods covered include diagrammatic perturbation theory, various forms of the renormalization group and self-consistent mode coupling. Dynamic critical phenomena near a second order phase transition, phase ordering dynamics, dynamics of surface growth and turbulence form the backbone of the book. Applications to a wide variety of systems (e.g. magnets, ordinary fluids, super fluids) are provided covering diverse transport properties (diffusion, sound).

The area of analysis and control of mechanical systems using differential geometry is flourishing.

This book collects many results over the last decade and provides a comprehensive introduction to the area.

This monograph, divided into four parts, presents a comprehensive treatment and systematic examination of cycle spaces of flag domains. Assuming only a basic familiarity with the concepts of Lie theory and geometry, this work presents a complete structure theory for these cycle spaces, as well as their applications to harmonic analysis and algebraic geometry.

Key features:

* Accessible to readers from a wide range of fields, with all the necessary background material provided for the nonspecialist

* Many new results presented for the first time

* Driven by numerous examples

* The exposition is presented from the complex geometric viewpoint, but the methods, applications and much of the motivation also come from real and complex algebraic groups and their representations, as well as other areas of geometry

* Comparisons with classical Barlet cycle spaces are given

* Good bibliography and index.

Researchers and graduate students in differential geometry, complex analysis, harmonic analysis, representation theory, transformation groups, algebraic geometry, and areas of global geometric analysis will benefit from this work.

The developmentsin the recent yearsof the potential theoryemphasized a classof functions larger than that of excessive functions (i.e. the positive superharmonic functionsfromtheclassicalpotentialtheoryassociatedwiththeLaplaceoperator), namely the strongly supermedian functions. It turns out that a positive Borel function will be strongly supermedian if and only if it is the in?mum of all its excessive majorants. Apparently, these functions have been introduced by J.F. Mertens and then they have been studied mainly by P.A. Meyer, G. Mokobodzki, D. Feyel and recently by P.J. Fitzsimmons and R.K. Getoor. The aimofthis bookisamongothersto developa potential theoryappropriate to this new class of functions. Although our methods are analytical, we present also the probabilistic counterparts from the Markov processes theory. The natural frame in which this theory is settled is given by a sub-Markovian resolvent of kernels on a Radon measurable space. After a possible extension of the space, such a resolvent becomes that one associated with a right process on a Radon topological space, not necessary locally compact and without existing a reference measure. Intimately related to the excessive functions we present certain basic tools of the theory: the Ray topology and compacti?cation, the ?ne carrier and the reduction operation on measurable sets. We examine di?erent types of negligible sets with respect to a ?nite measure ?: the ?-polar, ?-semipolar and ?-mince sets. We take advantage of the cone of potentials structure for both excessive functions and measures

A practical introduction to fundamentals of computer arithmetic

Computer arithmetic is one of the foundations of computer science and engineering. Designed as both a practical reference for engineers and computer scientists and an introductory text for students of electrical engineering and the computer and mathematical sciences, Arithmetic and Logic in Computer Systems describes the various algorithms and implementations in computer arithmetic and explains the fundamental principles that guide them.

Focusing on promoting an understanding of the concepts, Professor Mi Lu addresses:

• Number representations, including the Conventional Radix and
• Signed-Digit Number Systems as well as Floating Point, Residue, and Logarithmic Number Systems
• Ripple Carry Adders and high-speed adders
• Sequential multiplication, parallel multiplication, sequential division, and fast array dividers
• Floating point operations, Residue Number operations, and operations through logarithms

To assist the reader, alternative methods are examined and thorough explanations of the material are supplied, along with discussions of the reasoning behind the theory. Ample examples and problems help the reader master the concepts.

This book is devoted to analytically approximate methods in the nonlinear dynamics of a rigid body with cavities (containers) partly filled by a liquid. The methods are normally based on the Bateman-Luke variational formalism combined with perturbation theory. The derived approximate equations of spatial motions of the body-liquid mechanical system (these equations are called mathematical models in the title) take the form of a finite-dimensional system of nonlinear ordinary differential equations coupling quasi-velocities of the rigid body motions and generalized coordinates responsible for displacements of the natural sloshing modes. Algorithms for computing the hydrodynamic coefficients in the approximate mathematical models are proposed. Numerical values of these coefficients are listed for some tank shapes and liquid fillings. The mathematical models are also derived for the contained liquid characterized by the Newton-type dissipation. Formulas for hydrodynamic force and moment are derived in terms of the solid body quasi-velocities and the sloshing-related generalized coordinates. For prescribed harmonic excitations of upright circular (annular) cylindrical and/or conical tanks, the steady-state sloshing regimes are theoretically classified; the results are compared with known experimental data. The book can be useful for both experienced and early-stage mechanicians, applied mathematicians and engineers interested in (semi-)analytical approaches to the "fluid-structure" interaction problems, their fundamental mathematical background as well as in modeling the dynamics of complex mechanical systems containing a rigid tank partly filled by a liquid.

The study of CR manifolds lie at the intersection of three main mathematical disciplines, partial differential equations, complex analysis in several complex variables, and differential geometry. While the complex analysis and PDEs aspect have been intensly studied in the last fifty years, much effort has been recently made to understand the differential geometric side of the subject.This monograph provides a unified presentation of several differential geometric aspects in the theory of CR manifolds and tangential Cauchy-Riemann equations. It presents topics from the Tanaka-Webster connection, a key contributor to the birth of pseudohermitian geometry, to the major differential geometric acheivements in the theory of CR manifolds, such as Fefferman's metric, pseudo-Einstein structures and the Lee conjecture, CR immersions, subelliptic harmonic maps as a local manifestation of pseudoharmonic maps from a CR manifold, Yang-Mills fields on CR manifolds, to name several. It also aims at explaining how certain results from analysis are employed in CR geometry. results and stimulating unproved statements and comments referring to the most recent aspects of the theory, this monograph is suitable for researchers and graduate students in differential geometry, complex analysis, and PDEs.

This volume contains 21 research and survey papers on recent developments in the field of diophantine approximation, which are based on lectures given at a conference at the Erwin Schrodinger-Institute (Vienna, 2003). The articles are either in the spirit of more classical diophantine analysis or of a geometric or combinatorial flavor. Several articles deal with estimates for the number of solutions of diophantine equations as well as with congruences and polynomials.

Dialogue and Learning in Mathematics Education is concerned with communication in mathematics class-rooms. In a series of empirical studies of project work, we follow students' inquiry cooperation as well as students' obstructions to inquiry cooperation. Both are considered important for a theory of learning mathematics.
Special attention is paid to the notions of dialogue' and critique'. A central idea is that dialogue' supports critical learning of mathematics'. The link between dialogue and critique is developed further by including the notions of intention' and reflection'. Thus a theory of learning mathematics is developed which is resonant with critical mathematics education.

This book examines the critical roles and effects of mathematics education. The exposition draws from the author's forty-year mathematics career, integrating his research in the psychology of mathematical thinking into an overview of the true definition of math. The intention for the reader is to undergo a "corrective" experience, obtaining a clear message on how mathematical thinking tools can help all people cope with everyday life. For those who have struggled with math in the past, the book also aims to clarify that math learning difficulties are likely a result of improper pedagogy as opposed to any lack of intelligence on the part of the student. This personal treatise will be of interest to a variety of readers, from mathematics teachers and those who train them to those with an interest in education but who may lack a solid math background.

Moebius bagels, Euclid's flourless chocolate cake and apple pi - this is maths, but not as you know it. In How to Bake Pi, mathematical crusader and star baker Eugenia Cheng has rustled up a batch of delicious culinary insights into everything from simple numeracy to category theory ('the mathematics of mathematics'), via Fermat, Poincare and Riemann. Maths is much more than simultaneous equations and pr2 : it is an incredibly powerful tool for thinking about the world around us. And once you learn how to think mathematically, you'll never think about anything - cakes, custard, bagels or doughnuts; not to mention fruit crumble, kitchen clutter and Yorkshire puddings - the same way again. Stuffed with moreish puzzles and topped with a generous dusting of wit and charm, How to Bake Pi is a foolproof recipe for a mathematical feast. *Previously published under the title Cakes, Custard & Category Theory*

This book consists of reviewed original research papers and expository articles in index theory (especially on singular manifolds), topology of manifolds, operator and equivariant K-theory, Hopf cyclic cohomology, geometry of foliations, residue theory, Fredholm pairs and others, and applications in mathematical physics. The wide spectrum of subjects reflects the diverse directions of research for which the starting point was the Atiyah-Singer index theorem.

Since the outstanding and pioneering research work of Hopfield on recurrent neural networks (RNNs) in the early 80s of the last century, neural networks have rekindled strong interests in scientists and researchers. Recent years have recorded a remarkable advance in research and development work on RNNs, both in theoretical research as weIl as actual applications. The field of RNNs is now transforming into a complete and independent subject. From theory to application, from software to hardware, new and exciting results are emerging day after day, reflecting the keen interest RNNs have instilled in everyone, from researchers to practitioners. RNNs contain feedback connections among the neurons, a phenomenon which has led rather naturally to RNNs being regarded as dynamical systems. RNNs can be described by continuous time differential systems, discrete time systems, or functional differential systems, and more generally, in terms of non linear systems. Thus, RNNs have to their disposal, a huge set of mathematical tools relating to dynamical system theory which has tumed out to be very useful in enabling a rigorous analysis of RNNs."

This book highlights the emergence of a new mathematical rationality and the beginning of the mathematisation of physics in Classical Islam. Exchanges between mathematics, physics, linguistics, arts and music were a factor of creativity and progress in the mathematical, the physical and the social sciences. Goods and ideas travelled on a world-scale, mainly through the trade routes connecting East and Southern Asia with the Near East, allowing the transmission of Greek-Arabic medicine to Yuan Muslim China. The development of science, first centred in the Near East, would gradually move to the Western side of the Mediterranean, as a result of Europe's appropriation of the Arab and Hellenistic heritage. Contributors are Paul Buell, Anas Ghrab, Hossein Masoumi Hamedani, Zeinab Karimian, Giovanna Lelli, Marouane ben Miled, Patricia Radelet-de Grave, and Roshdi Rashed.