Series: Progress in Mathematics, Vol. 295
2011, X, 180 p.
Hardcover, ISBN 978-3-0348-0162-1
Due: August 31, 2011
Lagrangian systems constitute a very important and old class in dynamics. Their origin dates back to the end of the eighteenth century, with Joseph-Louis Lagrange’s reformulation of classical mechanics. The main feature of Lagrangian dynamics is its variational flavor: orbits are extremal points of an action functional. The development of critical point theory in the twentieth century provided a powerful machinery to investigate existence and multiplicity questions for orbits of Lagrangian systems. This monograph gives a modern account of the application of critical point theory, and more specifically Morse theory, to Lagrangian dynamics, with particular emphasis toward existence and multiplicity of periodic orbits of non-autonomous and time-periodic systems.
1 Lagrangian and Hamiltonian systems.- 2 Functional setting for the Lagrangian action.- 3 Discretizations.- 4 Local homology and Hilbert subspaces.- 5 Periodic orbits of Tonelli Lagrangian systems.- A An overview of Morse theory.-Bibliography.- List of symbols.- Index.
2011, XIV, 342 p. 10 illus.
Hardcover, ISBN 978-0-387-72765-3
Due: August 29, 2011
The study of Hopf algebras spans many fields in mathematics including topology, algebraic geometry, algebraic number theory, Galois module theory, cohomology of groups, and formal groups and has wide-ranging connections to fields from theoretical physics to computer science. This text is unique in making this engaging subject accessible to advanced graduate and beginning graduate students and focuses on applications of Hopf algebras to algebraic number theory and Galois module theory, providing a smooth transition from modern algebra to Hopf algebras.
After providing an introduction to the spectrum of a ring and the Zariski topology, the text treats presheaves, sheaves, and representable group functors. In this way the student transitions smoothly from basic algebraic geometry to Hopf algebras. The importance of Hopf orders is underscored with applications to algebraic number theory, Galois module theory and the theory of formal groups. By the end of the book, readers will be familiar with established results in the field and ready to pose research questions of their own.
An exercise set is included in each of twelve chapters with questions ranging in difficulty. Open problems and research questions are presented in the last chapter. Prerequisites include an understanding of the material on groups, rings, and fields normally covered in a basic course in modern algebra.
Preface.- Some Notation.- 1. The Spectrum of a Ring.-2. The Zariski Topology on the Spectrum.-3. Representable Group Functors.-4. Hopf Algebras. -5. Larson Orders.-6. Formal Group Hopf Orders.-7. Hopf Orders in KC_p.-8. Hopf Orders in KC_{p^2}.-9. Hopf Orders in KC_{p^3}.-10. Hopf Orders and Galois Module Theory.-11. The Class Group of a Hopf Order.-12. Open Questions and Research Problems.-Bibliography.-Index.
Series: Springer Briefs in Mathematics, Vol. 2
2011, IX, 66 p.
Softcover, ISBN 978-1-4614-0200-8
Due: July 29, 2011
Inequalities based on Sobolev Representations deals exclusively with very general tight integral inequalities of Chebyshev-Gruss, Ostrowski types and of integral means, all of which depend upon the Sobolev integral representations of functions. Applications illustrate inequalities that engage in ordinary and weak partial derivatives of the involved functions. This book also derives important estimates for the averaged Taylor polynomials and remainders of Sobolev integral representations. The results are examined in all directions and through both univariate and multivariate cases. This book is suitable for researchers, graduate students, and seminars in subareas of mathematical analysis, inequalities, partial differential equations and information theory.
Part 1. -Univariate Integral Inequalities based on Sobolev representations.- Introduction.- Background. -Main Results. -Applications.- References. Part 2.- Multivariate Integral Inequalities deriving from Sobolev representations.-Introduction.-Background.-Main Results.- Applications, References.
Series: Algorithms and Combinatorics, Tentative volume 200
2012, XX, 430 p. 190 illus., 15 in color.
Hardcover, ISBN 978-1-4614-0109-4
Due: March 22, 2012
The field of geometric graph theory is a fairly new discipline that began to burgeon in the past couple decades after many critical mathematical developments were made in graph theory. It was developed to solve problems in combinatorial and computational geometry, and has quickly become a useful tool for both discrete mathematicians and computer scientists.
This contributed volume contains twenty-five original survey and research papers on important recent developments in geometric graph theory. These contributions were thoroughly reviewed and written by active researchers in this field.
1 E. Ackerman: The maximum number of tangencies among convex regions with [a?] triangle-free intersection graph.- 2 G. Aloupis - B. Ballinger - S. Collette - S. Langerman - A. Por - D.R.Wood: Blocking coloured point sets.- 3 M. Al-Jubeh - G. Barequet - M. Ishaque - D. Souvaine - Cs. D. Toth - A. Winslow: Constrained tri-connected planar straight line graphs.- 4 D. Bokal - M. Bracic - E. Czabarka - L.A. Szekely: Custom-taylored lower bounds for the crossing number.- 5 S. Buzaglo - R. Pinchasi - G. Rote: Topological hyper-graphs.- 6 J. Cano Vila - L. F. Barba - J. Urrutia - T. Sakai: On edge-disjoint empty triangles of point sets.- 7 J. Cibulka - J. Kyncl - V. Meszaros - R. Stolar - P. Valtr: Universal sets for straight-line embeddings of bicolored graphs.- 8 G. Di Battista - F. Frati - W. Didimo - G. Liotta: The crossing angle resolution in graph drawing.- 9 A. Dumitrescu: Mover problems (tentative title).- 10 S. Felsner (?) .- 11 R. Fulek - N. Saeedi - D. Sarioz: Convex obstacle numbers of outerplanar graphs and bipartite permutation graphs.- 12 R. Fulek - M. Pelsmajer - M. Schaefer - D. Stefankovic: Hanani-Tutte, monotone drawings, and level-planarity .- 13 R. Fulek - A. Suk: On disjoint crossing families in geometric graphs.- 14 F. Hurtado - Cs. D. Toth: Geometric graph augmentation: a generic perspective.- 15 M. Kano - K. Suzuki: Discrete geometry on red and blue points in the plane lattice.- 16 Gy. Karolyi: on Ramsey numbers of geometric graphs.- 17 A. V. Kostochka - K. G. Milans: Coloring clean and K_4-free circle graphs.- 18 T. Mueller: ?.- 19 A. Raigorodskii: Coloring distance graphs and graph diameters.- 20 G. Salazar: on rectilinear crossing numbers, halving lines, etc.- 21 M. Schaefer: Realizability of graphs and linkages.- 22 M. Sharir - A. Sheffer: on counting triangulations.- 23 K. Swanepoel: Favorite distances in high dimensions.- 24 M. Tancer: Intersection patterns of convex sets via simplicial complexes, a survey.- 25 U. Wagner: Minors, embeddability, and extremal problems for hypergraphs
Series: Springer Briefs in Mathematics, Vol. 0
2012, 120 p.
Softcover, ISBN 978-1-4419-9874-3
Due: August 6, 2012
Encountering problems that they were not able to solve, mathematicians, as a rule, introduced new structures, extending the existing ones and making possible to solve previously ”unsolvable” problems. For instance, irrational, real and complex numbers were introduced to make possible solving all algebraic equations. The stimulus to develop the theory of hypernumbers and extrafunctions comes from physics. Exploring mysteries of the microworld, physicists often encounter situations when their formulas acquire infinite values due the divergence of the used series and integrals. At the same time, all measured values are naturally finite. A popular way to eliminate this discrepancy between theory and experiment was an artificial manipulation with formulas, which often allowed to get rid of infinite values. Another natural way to deal with such situation is to learn how to rigorously work with infinities and come to finite values given by measurements. Mathematicians suggested several approaches to a rigorous operation with infinite values by introducing infinite numbers. The most popular of them are: transfinite numbers of Cantor, nonstandard analysis of Robinson, and surreal numbers. However, all these constructions, which contributed a lot to the development of mathematics, especially, Cantor’s set theory, brought very little to the realm of physics. However, there is a mathematical theory called the theory of distributions, which allowed physicists to rigorously operate with functions that take infinite values. An example of such a function is the Heaviside-Dirac’s delta-function. Although at first physicists were skeptical about this theory, now it has become one of the most efficient tools of theoretical physics. In this book, we represent another rigorous mathematical approach to operation with infinite values. At first, the concepts of real and complex numbers are extended in such a way that the new universe of numbers called hypernumbers includes infinite quantities. It is necessary to remark that in contrast to nonstandard analysis, there are no infinitely small hypernumbers. This is more relevant to the situation in physics, where infinitely big values emerge from theoretical structures but physicists have never encountered infinitely small values. The next step of extending the classical calculus based on real and complex functions is introduction of extrafunctions, which generalize not only the concept of a conventional function but also the concept of a distribution. This made possible to solve previously “unsolvable” problems. For instance, there are linear partial differential equations for which it is proved that they do not have solutions not only in conventional functions but even in distributions. At the same, it is proved that all these and many other equations have solutions in extrafunctions. Extrafucntions have been also efficiently used for a rigorous mathematical definition of the Feynman path integral, as well as for solving some problems in probability theory, which is also important for contemporary physics.
-1. Introduction: How mathematicians solve ”unsolvable” problems.-2. Hypernumbers(Constructive definition and algebraic properties of hypernumbers, Axiomatic definition and topological properties of hypernumbers).-3. Extrafunctions and hyperdistributions.-4. How to differentiate any real function.-5. How to integrate any continuous real function. -6. Integration in infinite dimensional spaces and Feynman integral.-7. Conclusion: New opportunities.- Appendix.- References.