springer.com/NEWSonline Springer News 7/2011 Mathematics 89 M. Abate, Università di Pisa, Italia; F. Tovena, R. P. Agarwal, Texas A&M University, Kingsville, V. Anandam, University of Madras, Chennai, India Università Tor Vergata, Roma, Italia TX, USA; K. Perera, Florida Institute of Technology, Melbourne, Fl, USA; S. Pinelas, Universidade dos Harmonic Functions and Curves and Surfaces Açores, Ponta Delgada, Portugal Potentials on Finite or Infinite An Introduction to Complex Networks The book provides an introduction to Differential Geometry of Curves and Surfaces. The theory of Analysis Random walks, Markov chains and electrical curves starts with a discussion of possible defini- networks serve as an introduction to the study tions of the concept of curve, proving in particular This textbook introduces the subject of complex of real-valued functions on finite or infinite the classification of 1-dimensional manifolds. We analysis to advanced undergraduate and graduate graphs, with appropriate interpretations using then present the classical local theory of param- students in a clear and concise manner. Key probability theory and current-voltage laws. The etrized plane and space curves (curves in n-dimen- features of this textbook: effectively organizes the relation between this type of function theory and sional space are discussed in the complementary subject into easily manageable sections in the form the (Newton) potential theory on the Euclidean material): curvature, torsion, Frenet’s formulas and of 50 class-tested lectures, uses detailed examples spaces is well-established. The latter theory has the fundamental theorem of the local theory of to drive the presentation, includes numerous exer- been variously generalized, one example being curves. Then, after a self-contained presentation cise sets that encourage pursuing extensions of the the axiomatic potential theory on locally compact of degree theory for continuous self-maps of the material, each with an “Answers or Hints” section, spaces developed by Brelot, with later ramifica- circumference, we study the global theory of plane covers an array of advanced topics which allow tions from Bauer, Constantinescu and Cornea. A curves, introducing winding and rotation numbers, for flexibility in developing the subject beyond network is a graph with edge-weights that need not and proving the Jordan curve theorem for curves the basics, provides a concise history of complex be symmetric. This book presents an autonomous of class C2, and Hopf theorem on the rotation numbers. An Introduction to Complex Analysis theory of harmonic functions and potentials number of closed simple curves. will be valuable to students in mathematics, engi- defined on a finite or infinite network, on the lines The local theory of surfaces begins with a neering and other applied sciences. Prerequisites of axiomatic potential theory. Random walks and comparison of the concept of parametrized (i.e., include a course in calculus. electrical networks are important sources for the immersed) surface with the concept of regular (i.e., advancement of the theory. embedded) surface. We then develop the basic Features differential geometry of surfaces in R3: defini- 7 Provides a rigorous introduction to complex Features tions, examples, differentiable maps and func- analysis 7 Arranges the material effectively in 50 7 Number of examples to illustrate the main tions, tangent vectors (presented both as vectors class-tested lectures 7 Uses ample illustrations theory 7 Historical perspectives included to tangent to curves in the surface and as derivations and examples to explain the subject 7 Provides show the development of potential theory in on germs of differentiable functions; we shall problems for practice various forms 7 Self-contained text for an easy consistently use both approaches in the whole reading book) and orientation. Next we study the several From the contents notions of curvature on a surface, stressing both Preface.-Complex Numbers.-Complex Numbers Contents the geometrical meaning of the objects introduced II .- Complex Numbers III.-Set Theory in the 1 Laplace Operators on Networks and Trees.- 2 and the algebraic/analytical methods needed to Complex Plane.-Complex Functions.-Analytic Potential Theory on Finite Networks.- 3 Harmonic study them via the Gauss map, up to the proof of Functions I.-Analytic Functions II.-Elementary Function Theory on Infinite Networks.- 4 Gauss’ Teorema Egregium. Functions I.- Elementary Functions II.- Mappings Schrödinger Operators and Subordinate Structures by Functions.- Mappings by Functions II.- Curves, on Infinite Networks.- 5 Polyharmonic Functions Fields of interest Contours, and Simply Connected Domains.- on Trees. Mathematics, general; Differential Geometry; Complex Integration.- Independence of Path.- Geometry Cauchy–Goursat Theorem.- Deformation Fields of interest Theorem.- Cauchy’s Integral Formula.- Cauchy’s Potential Theory; Functions of a Complex Vari- Target groups Integral Formula for Derivatives.- Fundamental able; Partial Differential Equations Research Theorem of Algebra.- Maximum Modulus Principle.- Sequences and Series of Numbers.- Target groups Discount group Sequences and Series of Functions.- Power Series.- Research P Taylor’s Series.- Laurent’s Series. Discount group Fields of interest P Functions of a Complex Variable; Analysis Target groups Upper undergraduate Discount group P Due November 2011 Due July 2011 Due July 2011 2012. XIII, 393 p. 66 illus., 1 in color. (UNITEXT / La 2011. X, 190 p. (Lecture Notes of the Unione Matematica Matematica per il 3+2) Softcover 2011. XIV, 346 p. 92 illus. Hardcover Italiana, Volume 12) Softcover 7 approx. $89.95 7 $74.95 7 $49.95 ISBN 978-88-470-1940-9 ISBN 978-1-4614-0194-0 ISBN 978-3-642-21398-4 90 Mathematics Springer News 7/2011 springer.com/NEWSonline G. A. Anastassiou, The University of Memphis, TN, G. A. Anastassiou, The University of Memphis, TN, J. Barmak, University of Buenos Aires, Argentina USA USA Algebraic Topology of Finite Advances on Fractional Approximation by Topological Spaces and Inequalities Multivariate Singular Applications Integrals Advances on Fractional Inequalities use primarily This volume deals with the theory of finite topo- the Caputo fractional derivative, as the most Approximation by Multivariate Singular Integrals logical spaces and its relationship with the homo- important in applications, and presents the first is the first monograph to illustrate the approxi- topy and simple homotopy theory of polyhedra. fractional differentiation inequalities of Opial type mation of multivariate singular integrals to the The interaction between their intrinsic combinato- which involves the balanced fractional deriva- identity-unit operator. The basic approximation rial and topological structures makes finite spaces tives. The book continues with right and mixed properties of the general multivariate singular a useful tool for studying problems in Topology, fractional differentiation Ostrowski inequalities integral operators is presented quantitatively, Algebra and Geometry from a new perspective. In in the univariate and multivariate cases. Next the particularly special cases such as the multivariate particular, the methods developed in this manu- right and left, as well as mixed, Landau fractional Picard, Gauss-Weierstrass, Poisson-Cauchy and script are used to study Quillen’s conjecture on the differentiation inequalities in the univariate and trigonometric singular integral operators are poset of p-subgroups of a finite group and the multivariate cases are illustrated. Throughout examined thoroughly. This book studies the rate of Andrews-Curtis conjecture on the 3-deformability the book many applications are given. Fractional convergence of these operators to the unit operator of contractible two-dimensional complexes. This differentiation inequalities are by themselves as well as the related simultaneous approximation. self-contained work constitutes the first detailed an important and great mathematical topic for The last chapter, which includes many examples, exposition on the algebraic topology of finite research. Furthermore they have many applica- presents a related Korovkin type approximation spaces. It is intended for topologists and combina- tions, the most important ones are in establishing theorem for functions of two variables. Relevant torialists, but it is also recommended for uniqueness of solution in fractional differential background information and motivation is advanced undergraduate students and graduate equations and systems and in fractional partial included in this exposition, and as a result this students with a modest knowledge of Algebraic differential equations. Also they provide upper book can be used as supplementary text for several Topology. bounds to the solutions of the above equations. advanced courses. The results presented apply to many areas of pure and applied mathematics, such Features Features a mathematical analysis, probability, statistics and 7 It is the first complete exposition of the topic 7 Use primarily the Caputo fractional deriva- partial differential equations. This book is appro- 7 Has applications to the study of two long- tive, as the most important in applications, and we priate for researchers and selected seminars at the standing conjectures It is self-contained present first fractional differentiation inequali- graduate level. ties of Opial type where we involve the so called Contents balanced fractional derivatives