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Basic Global Relative Invariants For Nonlinear Differential Equations
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Book Synopsis Basic Global Relative Invariants for Nonlinear Differential Equations by : Roger Chalkley
Download or read book Basic Global Relative Invariants for Nonlinear Differential Equations written by Roger Chalkley and published by American Mathematical Soc.. This book was released on 2007 with total page 386 pages. Available in PDF, EPUB and Kindle. Book excerpt: The problem of deducing the basic relative invariants possessed by monic homogeneous linear differential equations of order $m$ was initiated in 1879 with Edmund Laguerre's success for the special case $m = 3$. It was solved in number 744 of the Memoirs of the AMS (March 2002), by a procedure that explicitly constructs, for any $m \geq3$, each of the $m - 2$ basic relative invariants. During that 123-year time span, only a few results were published about the basic relative invariants for other classes of ordinary differential equations. With respect to any fixed integer $\, m \geq 1$, the author begins by explicitly specifying the basic relative invariants for the class $\, \mathcal{C {m,2 $ that contains equations like $Q {m = 0$ in which $Q {m $ is a quadratic form in $y(z), \, \dots, \, y{(m) (z)$ having meromorphic coefficients written symmetrically and the coefficient of $\bigl( y{(m) (z) \bigr){2 $ is $1$.Then, in terms of any fixed positive integers $m$ and $n$, the author explicitly specifies the basic relative invariants for the class $\, \mathcal{C {m, n $ that contains equations like $H {m, n = 0$ in which $H {m, n $ is an $n$th-degree form in $y(z), \, \dots, \, y{(m) (z)$ having meromorphic coefficients written symmetrically and the coefficient of $\bigl( y{(m) (z) \bigr){n $ is $1$.These results enable the author to obtain the basic relative invariants for additional classes of ordinary differential equa
Book Synopsis Basic Global Relative Invariants for Homogeneous Linear Differential Equations by : Roger Chalkley
Download or read book Basic Global Relative Invariants for Homogeneous Linear Differential Equations written by Roger Chalkley and published by American Mathematical Soc.. This book was released on 2002 with total page 223 pages. Available in PDF, EPUB and Kindle. Book excerpt: Given any fixed integer $m \ge 3$, the author presents simple formulas for $m - 2$ algebraically independent polynomials over $\mathbb{Q}$ having the remarkable property, with respect to transformations of homogeneous linear differential equations of order $m$, that each polynomial is both a semi-invariant of the first kind (with respect to changes of the dependent variable) and a semi-invariant of the second kind (with respect to changes of the independent variable). These relative invariants are suitable for global studies in several different contexts and do not require Laguerre-Forsyth reductions for their evaluation. In contrast, all of the general formulas for basic relative invariants that have been proposed by other researchers during the last 113 years are merely local ones that are either much too complicated or require a Laguerre-Forsyth reduction for each evaluation.
Book Synopsis The Stable Manifold Theorem for Semilinear Stochastic Evolution Equations and Stochastic Partial Differential Equations by : Salah-Eldin Mohammed
Download or read book The Stable Manifold Theorem for Semilinear Stochastic Evolution Equations and Stochastic Partial Differential Equations written by Salah-Eldin Mohammed and published by American Mathematical Soc.. This book was released on 2008 with total page 120 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main objective of this paper is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations and stochastic partial differential equations near stationary solutions.
Book Synopsis Spinor Genera in Characteristic 2 by : Yuanhua Wang
Download or read book Spinor Genera in Characteristic 2 written by Yuanhua Wang and published by American Mathematical Soc.. This book was released on 2008 with total page 104 pages. Available in PDF, EPUB and Kindle. Book excerpt: The purpose of this paper is to establish the spinor genus theory of quadratic forms over global function fields in characteristic 2. The first part of the paper computes the integral spinor norms and relative spinor norms. The second part of the paper gives a complete answer to the integral representations of one quadratic form by another with more than four variables over a global function field in characteristic 2.
Book Synopsis Torus Fibrations, Gerbes, and Duality by : Ron Donagi
Download or read book Torus Fibrations, Gerbes, and Duality written by Ron Donagi and published by American Mathematical Soc.. This book was released on 2008 with total page 104 pages. Available in PDF, EPUB and Kindle. Book excerpt: Let $X$ be a smooth elliptic fibration over a smooth base $B$. Under mild assumptions, the authors establish a Fourier-Mukai equivalence between the derived categories of two objects, each of which is an $\mathcal{O} DEGREES{\times}$ gerbe over a genus one fibration which is a twisted form
Book Synopsis Heisenberg Calculus and Spectral Theory of Hypoelliptic Operators on Heisenberg Manifolds by : Raphael Ponge
Download or read book Heisenberg Calculus and Spectral Theory of Hypoelliptic Operators on Heisenberg Manifolds written by Raphael Ponge and published by American Mathematical Soc.. This book was released on 2008 with total page 150 pages. Available in PDF, EPUB and Kindle. Book excerpt: This memoir deals with the hypoelliptic calculus on Heisenberg manifolds, including CR and contact manifolds. In this context the main differential operators at stake include the Hormander's sum of squares, the Kohn Laplacian, the horizontal sublaplacian, the CR conformal operators of Gover-Graham and the contact Laplacian. These operators cannot be elliptic and the relevant pseudodifferential calculus to study them is provided by the Heisenberg calculus of Beals-Greiner andTaylor.
Book Synopsis Scattering Resonances for Several Small Convex Bodies and the Lax-Phillips Conjecture by : Luchezar N. Stoyanov
Download or read book Scattering Resonances for Several Small Convex Bodies and the Lax-Phillips Conjecture written by Luchezar N. Stoyanov and published by American Mathematical Soc.. This book was released on 2009 with total page 90 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work deals with scattering by obstacles which are finite disjoint unions of strictly convex bodies with smooth boundaries in an odd dimensional Euclidean space. The class of obstacles of this type which is considered are contained in a given (large) ball and have some additional properties.
Book Synopsis The Dynamics of Modulated Wave Trains by : A. Doelman
Download or read book The Dynamics of Modulated Wave Trains written by A. Doelman and published by American Mathematical Soc.. This book was released on 2009 with total page 122 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors investigate the dynamics of weakly-modulated nonlinear wave trains. For reaction-diffusion systems and for the complex Ginzburg-Landau equation, they establish rigorously that slowly varying modulations of wave trains are well approximated by solutions to the Burgers equation over the natural time scale. In addition to the validity of the Burgers equation, they show that the viscous shock profiles in the Burgers equation for the wave number can be found as genuine modulated waves in the underlying reaction-diffusion system. In other words, they establish the existence and stability of waves that are time-periodic in appropriately moving coordinate frames which separate regions in physical space that are occupied by wave trains of different, but almost identical, wave number. The speed of these shocks is determined by the Rankine-Hugoniot condition where the flux is given by the nonlinear dispersion relation of the wave trains. The group velocities of the wave trains in a frame moving with the interface are directed toward the interface. Using pulse-interaction theory, the authors also consider similar shock profiles for wave trains with large wave number, that is, for an infinite sequence of widely separated pulses. The results presented here are applied to the FitzHugh-Nagumo equation and to hydrodynamic stability problems.
Book Synopsis Volume Doubling Measures and Heat Kernel Estimates on Self-Similar Sets by : Jun Kigami
Download or read book Volume Doubling Measures and Heat Kernel Estimates on Self-Similar Sets written by Jun Kigami and published by American Mathematical Soc.. This book was released on 2009-04-10 with total page 110 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper studies the following three problems. 1. When does a measure on a self-similar set have the volume doubling property with respect to a given distance? 2. Is there any distance on a self-similar set under which the contraction mappings have the prescribed values of contractions ratios? 3. When does a heat kernel on a self-similar set associated with a self-similar Dirichlet form satisfy the Li-Yau type sub-Gaussian diagonal estimate? These three problems turn out to be closely related. The author introduces a new class of self-similar set, called rationally ramified self-similar sets containing both the Sierpinski gasket and the (higher dimensional) Sierpinski carpet and gives complete solutions of the above three problems for this class. In particular, the volume doubling property is shown to be equivalent to the upper Li-Yau type sub-Gaussian diagonal estimate of a heat kernel.
Book Synopsis Toroidal Dehn Fillings on Hyperbolic 3-Manifolds by : Cameron Gordon
Download or read book Toroidal Dehn Fillings on Hyperbolic 3-Manifolds written by Cameron Gordon and published by American Mathematical Soc.. This book was released on 2008 with total page 154 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors determine all hyperbolic $3$-manifolds $M$ admitting two toroidal Dehn fillings at distance $4$ or $5$. They show that if $M$ is a hyperbolic $3$-manifold with a torus boundary component $T 0$, and $r,s$ are two slopes on $T 0$ with $\Delta(r,s) = 4$ or $5$ such that $M(r)$ and $M(s)$ both contain an essential torus, then $M$ is either one of $14$ specific manifolds $M i$, or obtained from $M 1, M 2, M 3$ or $M {14}$ by attaching a solid torus to $\partial M i - T 0$.All the manifolds $M i$ are hyperbolic, and the authors show that only the first three can be embedded into $S3$. As a consequence, this leads to a complete classification of all hyperbolic knots in $S3$ admitting two toroidal surgeries with distance at least $4$.
Book Synopsis The Scaling Limit of the Correlation of Holes on the Triangular Lattice with Periodic Boundary Conditions by : Mihai Ciucu
Download or read book The Scaling Limit of the Correlation of Holes on the Triangular Lattice with Periodic Boundary Conditions written by Mihai Ciucu and published by American Mathematical Soc.. This book was released on 2009-04-10 with total page 118 pages. Available in PDF, EPUB and Kindle. Book excerpt: The author defines the correlation of holes on the triangular lattice under periodic boundary conditions and studies its asymptotics as the distances between the holes grow to infinity. He proves that the joint correlation of an arbitrary collection of triangular holes of even side-lengths (in lattice spacing units) satisfies, for large separations between the holes, a Coulomb law and a superposition principle that perfectly parallel the laws of two dimensional electrostatics, with physical charges corresponding to holes, and their magnitude to the difference between the number of right-pointing and left-pointing unit triangles in each hole. The author details this parallel by indicating that, as a consequence of the results, the relative probabilities of finding a fixed collection of holes at given mutual distances (when sampling uniformly at random over all unit rhombus tilings of the complement of the holes) approach, for large separations between the holes, the relative probabilities of finding the corresponding two dimensional physical system of charges at given mutual distances. Physical temperature corresponds to a parameter refining the background triangular lattice. He also gives an equivalent phrasing of the results in terms of covering surfaces of given holonomy. From this perspective, two dimensional electrostatic potential energy arises by averaging over all possible discrete geometries of the covering surfaces.
Book Synopsis Complicial Sets Characterising the Simplicial Nerves of Strict $\omega $-Categories by : Dominic Verity
Download or read book Complicial Sets Characterising the Simplicial Nerves of Strict $\omega $-Categories written by Dominic Verity and published by American Mathematical Soc.. This book was released on 2008 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt: The primary purpose of this work is to characterise strict $\omega$-categories as simplicial sets with structure. The author proves the Street-Roberts conjecture in the form formulated by Ross Street in his work on Orientals, which states that they are exactly the ``complicial sets'' defined and named by John Roberts in his handwritten notes of that title (circa 1978). On the way the author substantially develops Roberts' theory of complicial sets itself and makes contributions to Street's theory of parity complexes. In particular, he studies a new monoidal closed structure on the category of complicial sets which he shows to be the appropriate generalisation of the (lax) Gray tensor product of 2-categories to this context. Under Street's $\omega$-categorical nerve construction, which the author shows to be an equivalence, this tensor product coincides with those of Steiner, Crans and others.
Book Synopsis The Beltrami Equation by : Tadeusz Iwaniec
Download or read book The Beltrami Equation written by Tadeusz Iwaniec and published by American Mathematical Soc.. This book was released on 2008 with total page 110 pages. Available in PDF, EPUB and Kindle. Book excerpt: The measurable Riemann Mapping Theorem (or the existence theorem for quasiconformal mappings) has found a central role in a diverse variety of areas such as holomorphic dynamics, Teichmuller theory, low dimensional topology and geometry, and the planar theory of PDEs. Anticipating the needs of future researchers, the authors give an account of the state of the art as it pertains to this theorem, that is, to the existence and uniqueness theory of the planar Beltrami equation, and various properties of the solutions to this equation. The classical theory concerns itself with the uniformly elliptic case (quasiconformal mappings). Here the authors develop the theory in the more general framework of mappings of finite distortion and the associated degenerate elliptic equations.
Book Synopsis Newton's Method Applied to Two Quadratic Equations in $\mathbb {C}^2$ Viewed as a Global Dynamical System by : John H. Hubbard
Download or read book Newton's Method Applied to Two Quadratic Equations in $\mathbb {C}^2$ Viewed as a Global Dynamical System written by John H. Hubbard and published by American Mathematical Soc.. This book was released on 2008 with total page 160 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors study the Newton map $N:\mathbb{C}^2\rightarrow\mathbb{C}^2$ associated to two equations in two unknowns, as a dynamical system. They focus on the first non-trivial case: two simultaneous quadratics, to intersect two conics. In the first two chapters, the authors prove among other things: The Russakovksi-Shiffman measure does not change the points of indeterminancy. The lines joining pairs of roots are invariant, and the Julia set of the restriction of $N$ to such a line has under appropriate circumstances an invariant manifold, which shares features of a stable manifold and a center manifold. The main part of the article concerns the behavior of $N$ at infinity. To compactify $\mathbb{C}^2$ in such a way that $N$ extends to the compactification, the authors must take the projective limit of an infinite sequence of blow-ups. The simultaneous presence of points of indeterminancy and of critical curves forces the authors to define a new kind of blow-up: the Farey blow-up. This construction is studied in its own right in chapter 4, where they show among others that the real oriented blow-up of the Farey blow-up has a topological structure reminiscent of the invariant tori of the KAM theorem. They also show that the cohomology, completed under the intersection inner product, is naturally isomorphic to the classical Sobolev space of functions with square-integrable derivatives. In chapter 5 the authors apply these results to the mapping $N$ in a particular case, which they generalize in chapter 6 to the intersection of any two conics.
Book Synopsis Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings by : Wolfgang Bertram
Download or read book Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings written by Wolfgang Bertram and published by American Mathematical Soc.. This book was released on 2008 with total page 218 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed, without any restriction on the dimension or on the characteristic. Two basic features distinguish the author's approach from the classical real (finite or infinite dimensional) theory, namely the interpretation of tangent- and jet functors as functors of scalar extensions and the introduction of multilinear bundles and multilinear connections which generalize the concept of vector bundles and linear connections.
Book Synopsis The Mapping Class Group from the Viewpoint of Measure Equivalence Theory by : Yoshikata Kida
Download or read book The Mapping Class Group from the Viewpoint of Measure Equivalence Theory written by Yoshikata Kida and published by American Mathematical Soc.. This book was released on 2008 with total page 206 pages. Available in PDF, EPUB and Kindle. Book excerpt: The author obtains some classification result for the mapping class groups of compact orientable surfaces in terms of measure equivalence. In particular, the mapping class groups of different closed surfaces cannot be measure equivalent. Moreover, the author gives various examples of discrete groups which are not measure equivalent to the mapping class groups. In the course of the proof, the author investigates amenability in a measurable sense for the actions of the mapping class group on the boundary at infinity of the curve complex and on the Thurston boundary and, using this investigation, proves that the mapping class group of a compact orientable surface is exact.
Book Synopsis Bernoulli Free-Boundary Problems by : Eugene Shargorodsky
Download or read book Bernoulli Free-Boundary Problems written by Eugene Shargorodsky and published by American Mathematical Soc.. This book was released on 2008 with total page 86 pages. Available in PDF, EPUB and Kindle. Book excerpt: Questions of existence, multiplicity, and regularity of free boundaries for prescribed data need to be addressed and their solutions lead to nonlinear problems. In this paper an equivalence is established between Bernoulli free-boundary problems and a class of equations for real-valued functions of one real variable.
