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Desingularization Of Nilpotent Singularities In Families Of Planar Vector Fields
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Book Synopsis Desingularization of Nilpotent Singularities in Families of Planar Vector Fields by : Daniel Panazzolo
Download or read book Desingularization of Nilpotent Singularities in Families of Planar Vector Fields written by Daniel Panazzolo and published by American Mathematical Soc.. This book was released on 2002 with total page 108 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this work, we prove a desingularization theorem for analytic families of two-dimensional vector fields, under the hypothesis that all its singularities have a non-vanishing first jet. Application to problems of Singular Perturbations and Finite Cyclicity are discussed in the last chapter.
Book Synopsis Bifurcations of Planar Vector Fields by : Freddy Dumortier
Download or read book Bifurcations of Planar Vector Fields written by Freddy Dumortier and published by Springer. This book was released on 2006-12-08 with total page 234 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book reports on recent work by the authors on the bifurcation structure of singular points of planar vector fields whose linear parts are nilpotent. The bifurcation diagrams of the most important codimension-three cases are studied in detail. The results presented reach the limits of what is currently known on the bifurcation theory of planar vector fields. While the treatment is geometric, special analytical tools using abelian integrals are needed, and are explicitly developed. The rescaling and normalization methods are improved for application here. The reader is assumed to be familiar with the elements of Bifurcation and Dynamical Systems Theory. The book is addressed to researchers and graduate students working in Ordinary Differential Equations and Dynamical Systems, as well as anyone modelling complex multiparametric phenomena.
Book Synopsis Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem by : Robert Roussarie
Download or read book Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem written by Robert Roussarie and published by Springer Science & Business Media. This book was released on 2013-11-26 with total page 206 pages. Available in PDF, EPUB and Kindle. Book excerpt: In a coherent, exhaustive and progressive way, this book presents the tools for studying local bifurcations of limit cycles in families of planar vector fields. A systematic introduction is given to such methods as division of an analytic family of functions in its ideal of coefficients, and asymptotic expansion of non-differentiable return maps and desingularisation. The exposition moves from classical analytic geometric methods applied to regular limit periodic sets to more recent tools for singular limit sets. The methods can be applied to theoretical problems such as Hilbert's 16th problem, but also for the purpose of establishing bifurcation diagrams of specific families as well as explicit computations. - - - The book as a whole is a well-balanced exposition that can be recommended to all those who want to gain a thorough understanding and proficiency in the recently developed methods. The book, reflecting the current state of the art, can also be used for teaching special courses. (Mathematical Reviews)
Book Synopsis Lecture Notes on O-Minimal Structures and Real Analytic Geometry by : Chris Miller
Download or read book Lecture Notes on O-Minimal Structures and Real Analytic Geometry written by Chris Miller and published by Springer Science & Business Media. This book was released on 2012-09-14 with total page 247 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume was produced in conjunction with the Thematic Program in o-Minimal Structures and Real Analytic Geometry, held from January to June of 2009 at the Fields Institute. Five of the six contributions consist of notes from graduate courses associated with the program: Felipe Cano on a new proof of resolution of singularities for planar analytic vector fields; Chris Miller on o-minimality and Hardy fields; Jean-Philippe Rolin on the construction of o-minimal structures from quasianalytic classes; Fernando Sanz on non-oscillatory trajectories of vector fields; and Patrick Speissegger on pfaffian sets. The sixth contribution, by Antongiulio Fornasiero and Tamara Servi, is an adaptation to the nonstandard setting of A.J. Wilkie's construction of o-minimal structures from infinitely differentiable functions. Most of this material is either unavailable elsewhere or spread across many different sources such as research papers, conference proceedings and PhD theses. This book will be a useful tool for graduate students or researchers from related fields who want to learn about expansions of o-minimal structures by solutions, or images thereof, of definable systems of differential equations.
Book Synopsis Bifurcation of Planar Vector Fields and Hilbert's Sixteenth Problem by : Robert H. Roussarie
Download or read book Bifurcation of Planar Vector Fields and Hilbert's Sixteenth Problem written by Robert H. Roussarie and published by Birkhauser. This book was released on 1998 with total page 232 pages. Available in PDF, EPUB and Kindle. Book excerpt: In a coherent, exhaustive and progressive way, this book presents the tools for studying local bifurcations of limit cycles in families of planar vector fields. A systematic introduction is given to such methods as division of an analytic family of functions in its ideal of coefficients, and asymptotic expansion of non-differentiable return maps and desingularisation. The exposition moves from classical analytic geometric methods applied to regular limit periodic sets to more recent tools for singular limit sets. The methods can be applied to theoretical problems such as Hilbert's 16th problem, but also for the purpose of establishing bifurcation diagrams of specific families as well as explicit computations.
Book Synopsis Topological Invariants of the Complement to Arrangements of Rational Plane Curves by : José Ignacio Cogolludo-Agustín
Download or read book Topological Invariants of the Complement to Arrangements of Rational Plane Curves written by José Ignacio Cogolludo-Agustín and published by American Mathematical Soc.. This book was released on 2002 with total page 97 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors analyse two topological invariants of an embedding of an arrangement of rational plane curves in the projective complex plane, namely, the cohomology ring of the complement and the characteristic varieties. Their main result states that the cohomology ring of the complement to a rational arrangement is generated by logarithmic 1 and 2-forms and its structure depends on a finite number of invariants of the curve (its combinatorial type).
Book Synopsis From Representation Theory to Homotopy Groups by : Donald M. Davis
Download or read book From Representation Theory to Homotopy Groups written by Donald M. Davis and published by American Mathematical Soc.. This book was released on 2002 with total page 50 pages. Available in PDF, EPUB and Kindle. Book excerpt: A formula for the odd-primary v1-periodic homotopy groups of a finite H-space in terms of its K-theory and Adams operations has been obtained by Bousfield. This work applies this theorem to give explicit determinations of the v1-periodic homotopy groups of (E8,5) and (E8,3), thus completing the determination of all odd-primary v1-periodic homotopy groups of all compact simple Lie groups, a project suggested by Mimura in 1989. The method is different to that used by the author in previous works. There is no homotopy theoretic input, and no spectral sequence calculation. The input is the second exterior power operation in the representation ring of E8, which we determine using specialized software. This can be interpreted as giving the Adams operation psi^2 in K(E8). Eigenvectors of psi^2 must also be eigenvectors of psi^k for any k. The matrix of these eigenvectors is the key to the analysis. Its determinant is closely related to the homotopy decomposition of E8 localized at each prime. By taking careful combinations of eigenvectors, a set of generators of K(E8) can be obtained on which there is a nice formula for all Adams operations. Bousfield's theorem (and considerable Maple computation) allows the v1-periodic homotopy groups to be obtained from this.
Book Synopsis The Connective K-Theory of Finite Groups by : Robert Ray Bruner
Download or read book The Connective K-Theory of Finite Groups written by Robert Ray Bruner and published by American Mathematical Soc.. This book was released on 2003 with total page 144 pages. Available in PDF, EPUB and Kindle. Book excerpt: Includes a paper that deals the connective K homology and cohomology of finite groups $G$. This title uses the methods of algebraic geometry to study the ring $ku DEGREES*(BG)$ where $ku$ denotes connective complex K-theory. It describes the variety in terms of the category of abelian $p$-subgroups of $G$ for primes $p$ dividing the group
Book Synopsis Dualities on Generalized Koszul Algebras by : Edward L. Green
Download or read book Dualities on Generalized Koszul Algebras written by Edward L. Green and published by American Mathematical Soc.. This book was released on 2002 with total page 90 pages. Available in PDF, EPUB and Kindle. Book excerpt: Koszul rings are graded rings which have played an important role in algebraic topology, noncommutative algebraic geometry and in the theory of quantum groups. One aspect of the theory is to compare the module theory for a Koszul ring and its Koszul dual. There are dualities between subcategories of graded modules; the Koszul modules.
Book Synopsis Extending Intersection Homology Type Invariants to Non-Witt Spaces by : Markus Banagl
Download or read book Extending Intersection Homology Type Invariants to Non-Witt Spaces written by Markus Banagl and published by American Mathematical Soc.. This book was released on 2002 with total page 83 pages. Available in PDF, EPUB and Kindle. Book excerpt: Intersection homology theory provides a way to obtain generalized Poincare duality, as well as a signature and characteristic classes, for singular spaces. For this to work, one has had to assume however that the space satisfies the so-called Witt condition. We extend this approach to constructing invariants to spaces more general than Witt spaces. We present an algebraic framework for extending generalized Poincare duality and intersection homology to singular spaces $X$ not necessarily Witt. The initial step in this program is to define the category $SD(X)$ of complexes of sheaves suitable for studying intersection homology type invariants on non-Witt spaces. The objects in this category can be shown to be the closest possible self-dual 'approximation' to intersection homology sheaves.It is therefore desirable to understand the structure of such self-dual sheaves and to isolate the minimal data necessary to construct them. As the main tool in this analysis we introduce the notion of a Lagrangian structure (related to the familiar notion of Lagrangian submodules for $(-1)^k$-Hermitian forms, as in surgery theory). We demonstrate that every complex in $SD(X)$ has naturally associated Lagrangian structures and conversely, that Lagrangian structures serve as the natural building blocks for objects in $SD(X).Our main result asserts that there is in fact an equivalence of categories between $SD(X)$ and a twisted product of categories of Lagrangian structures. This may be viewed as a Postnikov system for $SD(X)$ whose fibers are categories of Lagrangian structures. The question arises as to which varieties possess Lagrangian structures. To begin to answer that, we define the model-class of varieties with an ordered resolution and use block bundles to describe the geometry of such spaces. Our main result concerning these is that they have associated preferred Lagrangian structures, and hence self-dual generalized intersection homology sheaves.
Book Synopsis The AB Program in Geometric Analysis: Sharp Sobolev Inequalities and Related Problems by : Olivier Druet
Download or read book The AB Program in Geometric Analysis: Sharp Sobolev Inequalities and Related Problems written by Olivier Druet and published by American Mathematical Soc.. This book was released on 2002 with total page 98 pages. Available in PDF, EPUB and Kindle. Book excerpt: Function theory and Sobolev inequalities have been the target of investigatio for decades. Sharp constants in these inequalities constitute a critical tool in geometric analysis. The $AB$ program is concerned with sharp Sobolev inequalities on compact Riemannian manifolds. Important and significant progress has been made during recent years. We summarize the present state ad describe new results.
Book Synopsis Interpolation of Weighted Banach Lattices/A Characterization of Relatively Decomposable Banach Lattices by : Michael Cwikel
Download or read book Interpolation of Weighted Banach Lattices/A Characterization of Relatively Decomposable Banach Lattices written by Michael Cwikel and published by American Mathematical Soc.. This book was released on 2003 with total page 127 pages. Available in PDF, EPUB and Kindle. Book excerpt: Interpolation of Weighted Banach Lattices It is known that for many, but not all, compatible couples of Banach spaces $(A_{0},A_{1})$ it is possible to characterize all interpolation spaces with respect to the couple via a simple monotonicity condition in terms of the Peetre $K$-functional. Such couples may be termed Calderon-Mityagin couples. The main results of the present paper provide necessary and sufficient conditions on a couple of Banach lattices of measurable functions $(X_{0},X_{1})$ which ensure that, for all weight functions $w_{0}$ and $w_{1}$, the couple of weighted lattices $(X_{0,w_{0}},X_{1,w_{1}})$ is a Calderon-Mityagin couple. Similarly, necessary and sufficient conditions are given for two couples of Banach lattices $(X_{0},X_{1})$ and $(Y_{0},Y_{1})$ to have the property that, for all choices of weight functions $w_{0}, w_{1}, v_{0}$ and $v_{1}$, all relative interpolation spaces with respect to the weighted couples $(X_{0,w_{0}},X_{1,w_{1}})$ and $(Y_{0,v_{0}},Y_{1,v_{1}})$ may be described via an obvious analogue of the above-mentioned $K$-functional monotonicity condition. A number of auxiliary results developed in the course of this work can also be expected to be useful in other contexts. These include a formula for the $K$-functional for an arbitrary couple of lattices which offers some of the features of Holmstedt's formula for $K(t,f;L^{p},L^{q})$, and also the following uniqueness theorem for Calderon's spaces $X^{1-\theta }_{0}X^{\theta }_{1}$: Suppose that the lattices $X_0$, $X_1$, $Y_0$ and $Y_1$ are all saturated and have the Fatou property. If $X^{1-\theta }_{0}X^{\theta }_{1} = Y^{1-\theta }_{0}Y^{\theta }_{1}$ for two distinct values of $\theta $ in $(0,1)$, then $X_{0} = Y_{0}$ and $X_{1} = Y_{1}$. Yet another such auxiliary result is a generalized version of Lozanovskii's formula $\left( X_{0}^{1-\theta }X_{1}^{\theta }\right) ^{\prime }=\left (X_{0}^{\prime }\right) ^{1-\theta }\left( X_{1}^{\prime }\right) ^{\theta }$ for the associate space of $X^{1-\theta }_{0}X^{\theta }_{1}$. A Characterization of Relatively Decomposable Banach Lattices Two Banach lattices of measurable functions $X$ and $Y$ are said to be relatively decomposable if there exists a constant $D$ such that whenever two functions $f$ and $g$ can be expressed as sums of sequences of disjointly supported elements of $X$ and $Y$ respectively, $f = \sum^{\infty }_{n=1} f_{n}$ and $g = \sum^{\infty }_{n=1} g_{n}$, such that $\ g_{n}\ _{Y} \le \ f_{n}\ _{X}$ for all $n = 1, 2, \ldots $, and it is given that $f \in X$, then it follows that $g \in Y$ and $\ g\ _{Y} \le D\ f\ _{X}$. Relatively decomposable lattices appear naturally in the theory of interpolation of weighted Banach lattices. It is shown that $X$ and $Y$ are relatively decomposable if and only if, for some $r \in [1,\infty ]$, $X$ satisfies a lower $r$-estimate and $Y$ satisfies an upper $r$-estimate. This is also equivalent to the condition that $X$ and $\ell ^{r}$ are relatively decomposable and also $\ell ^{r}$ and $Y$ are relatively decomposable.
Book Synopsis Radially Symmetric Patterns of Reaction-diffusion Systems by : Arnd Scheel
Download or read book Radially Symmetric Patterns of Reaction-diffusion Systems written by Arnd Scheel and published by American Mathematical Soc.. This book was released on 2003 with total page 86 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, bifurcations of stationary and time-periodic solutions to reaction-diffusion systems are studied. We develop a center-manifold and normal form theory for radial dynamics which allows for a complete description of radially symmetric patterns. In particular, we show the existence of localized pulses near saddle-nodes, critical Gibbs kernels in the cusp, focus patterns in Turing instabilities, and active or passive target patterns in oscillatory instabilities.
Book Synopsis Invariants of Boundary Link Cobordism by : Desmond Sheiham
Download or read book Invariants of Boundary Link Cobordism written by Desmond Sheiham and published by American Mathematical Soc.. This book was released on 2003 with total page 110 pages. Available in PDF, EPUB and Kindle. Book excerpt: An $n$-dimensional $\mu$-component boundary link is a codimension $2$ embedding of spheres $L=\sqcup_{\mu}S^n \subset S^{n+2}$ such that there exist $\mu$ disjoint oriented embedded $(n+1)$-manifolds which span the components of $L$. An $F_\mu$-link is a boundary link together with a cobordism class of such spanning manifolds. The $F_\mu$-link cobordism group $C_n(F_\mu)$ is known to be trivial when $n$ is even but not finitely generated when $n$ is odd. Our main result is an algorithm to decide whether two odd-dimensional $F_\mu$-links represent the same cobordism class in $C_{2q-1}(F_\mu)$ assuming $q>1$. We proceed to compute the isomorphism class of $C_{2q-1}(F_\mu)$, generalizing Levine's computation of the knot cobordism group $C_{2q-1}(F_1)$.Our starting point is the algebraic formulation of Levine, Ko and Mio who identify $C_{2q-1}(F_\mu)$ with a surgery obstruction group, the Witt group $G^{(-1)^q,\mu}(\Z)$ of $\mu$-component Seifert matrices. We obtain a complete set of torsion-free invariants by passing from integer coefficients to complex coefficients and by applying the algebraic machinery of Quebbemann, Scharlau and Schulte. Signatures correspond to 'algebraically integral' simple self-dual representations of a certain quiver (directed graph with loops). These representations, in turn, correspond to algebraic integers on an infinite disjoint union of real affine varieties. To distinguish torsion classes, we consider rational coefficients in place of complex coefficients, expressing $G^{(-1)^q,\mu}(\mathbb{Q})$ as an infinite direct sum of Witt groups of finite-dimensional division $\mathbb{Q}$-algebras with involution.The Witt group of every such algebra appears as a summand infinitely often. The theory of symmetric and hermitian forms over these division algebras is well-developed. There are five classes of algebras to be considered; complete Witt invariants are available for four classes, those for which the local-global principle applies. An algebra in the fifth class, namely a quaternion algebra with non-standard involution, requires an additional Witt invariant which is defined if all the local invariants vanish.
Book Synopsis $q$-Difference Operators, Orthogonal Polynomials, and Symmetric Expansions by : Douglas Bowman
Download or read book $q$-Difference Operators, Orthogonal Polynomials, and Symmetric Expansions written by Douglas Bowman and published by American Mathematical Soc.. This book was released on 2002 with total page 73 pages. Available in PDF, EPUB and Kindle. Book excerpt: The author explores ramifications and extensions of a $q$-difference operator method first used by L.J. Rogers for deriving relationships between special functions involving certain fundamental $q$-symmetric polynomials. In special cases these symmetric polynomials reduce to well-known classes of orthogonal polynomials. A number of basic properties of these polynomials follow from this approach. This leads naturally to the evaluation of the Askey-Wilson integral and generalizations. Expansions of certain generalized basic hypergeometric functions in terms of the symmetric polynomials are also found. This provides a quick route to understanding the group structure generated by iterating the two-term transformations of these functions. Some infrastructure is also laid for more general investigations in the future
Book Synopsis Topological Invariants for Projection Method Patterns by : Alan Forrest
Download or read book Topological Invariants for Projection Method Patterns written by Alan Forrest and published by American Mathematical Soc.. This book was released on 2002 with total page 137 pages. Available in PDF, EPUB and Kindle. Book excerpt: This memoir develops, discusses and compares a range of commutative and non-commutative invariants defined for projection method tilings and point patterns. The projection method refers to patterns, particularly the quasiperiodic patterns, constructed by the projection of a strip of a high dimensional integer lattice to a smaller dimensional Euclidean space. In the first half of the memoir the acceptance domain is very general - any compact set which is the closure of its interior - while in the second half the authors concentrate on the so-called canonical patterns. The topological invariants used are various forms of $K$-theory and cohomology applied to a variety of both $C DEGREES*$-algebras and dynamical systems derived from such a p
Book Synopsis Equivariant Orthogonal Spectra and S-Modules by : M. A. Mandell
Download or read book Equivariant Orthogonal Spectra and S-Modules written by M. A. Mandell and published by American Mathematical Soc.. This book was released on 2002 with total page 108 pages. Available in PDF, EPUB and Kindle. Book excerpt: The last few years have seen a revolution in our understanding of the foundations of stable homotopy theory. Many symmetric monoidal model categories of spectra whose homotopy categories are equivalent to the stable homotopy category are now known, whereas no such categories were known before 1993. The most well-known examples are the category of $S$-modules and the category of symmetric spectra. We focus on the category of orthogonal spectra, which enjoys some of the best features of $S$-modules and symmetric spectra and which is particularly well-suited to equivariant generalization. We first complete the nonequivariant theory by comparing orthogonal spectra to $S$-modules. We then develop the equivariant theory.For a compact Lie group $G$, we construct a symmetric monoidal model category of orthogonal $G$-spectra whose homotopy category is equivalent to the classical stable homotopy category of $G$-spectra. We also complete the theory of $S_G$-modules and compare the categories of orthogonal $G$-spectra and $S_G$-modules. A key feature is the analysis of change of universe, change of group, fixed point, and orbit functors in these two highly structured categories for the study of equivariant stable homotopy theory.
