Newton S Method Applied To Two Quadratic Equations In C2 Viewed As A Global Dynamical System
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Newton S Method Applied To Two Quadratic Equations In Mathbb C 2 Viewed As A Global Dynamical System
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Author : John H. Hubbard
language : en
Publisher: American Mathematical Soc.
Release Date : 2008
Newton S Method Applied To Two Quadratic Equations In Mathbb C 2 Viewed As A Global Dynamical System written by John H. Hubbard and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematics categories.
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.
Newton S Method Applied To Two Quadratic Equations In C2 Viewed As A Global Dynamical System
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Author : John Hamal Hubbard
language : en
Publisher:
Release Date : 2008
Newton S Method Applied To Two Quadratic Equations In C2 Viewed As A Global Dynamical System written by John Hamal Hubbard and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Differentiable dynamical systems categories.
Studies the Newton map $N: \mathbb{C} DEGREES2\rightarrow\mathbb{C} DEGREES2$ associated to two equations in two unknowns, as a dynamical system. This title focuses on the first non-trivial case: two simultaneous quadratics, to intersect two conics. It proves among other things: the Russakovksi-Shiffman measure does not change the points of
The Projective Heat Map
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Author : Richard Evan Schwartz
language : en
Publisher: American Mathematical Soc.
Release Date : 2017-04-20
The Projective Heat Map written by Richard Evan Schwartz and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-04-20 with Mathematics categories.
This book introduces a simple dynamical model for a planar heat map that is invariant under projective transformations. The map is defined by iterating a polygon map, where one starts with a finite planar -gon and produces a new -gon by a prescribed geometric construction. One of the appeals of the topic of this book is the simplicity of the construction that yet leads to deep and far reaching mathematics. To construct the projective heat map, the author modifies the classical affine invariant midpoint map, which takes a polygon to a new polygon whose vertices are the midpoints of the original. The author provides useful background which makes this book accessible to a beginning graduate student or advanced undergraduate as well as researchers approaching this subject from other fields of specialty. The book includes many illustrations, and there is also a companion computer program.
The Valuative Tree
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Author : Charles Favre
language : en
Publisher: Springer
Release Date : 2004-08-30
The Valuative Tree written by Charles Favre and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004-08-30 with Mathematics categories.
This volume is devoted to a beautiful object, called the valuative tree and designed as a powerful tool for the study of singularities in two complex dimensions. Its intricate yet manageable structure can be analyzed by both algebraic and geometric means. Many types of singularities, including those of curves, ideals, and plurisubharmonic functions, can be encoded in terms of positive measures on the valuative tree. The construction of these measures uses a natural tree Laplace operator of independent interest.
Uniqueness And Stability In Determining A Rigid Inclusion In An Elastic Body
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Author : Antonino Morassi
language : en
Publisher: American Mathematical Soc.
Release Date : 2009-06-05
Uniqueness And Stability In Determining A Rigid Inclusion In An Elastic Body written by Antonino Morassi and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-06-05 with Mathematics categories.
The authors consider the inverse problem of determining a rigid inclusion inside an isotropic elastic body $\Omega$, from a single measurement of traction and displacement taken on the boundary of $\Omega$. For this severely ill-posed problem they prove uniqueness and a conditional stability estimate of log-log type.
Cohomological Invariants Exceptional Groups And Spin Groups
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Author : Skip Garibaldi
language : en
Publisher: American Mathematical Soc.
Release Date : 2009-06-05
Cohomological Invariants Exceptional Groups And Spin Groups written by Skip Garibaldi and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-06-05 with Mathematics categories.
This volume concerns invariants of $G$-torsors with values in mod $p$ Galois cohomology--in the sense of Serre's lectures in the book Cohomological invariants in Galois cohomology--for various simple algebraic groups $G$ and primes $p$. The author determines the invariants for the exceptional groups $F_4$ mod 3, simply connected $E_6$ mod 3, $E_7$ mod 3, and $E_8$ mod 5. He also determines the invariants of $\mathrm{Spin}_n$ mod 2 for $n \leq 12$ and constructs some invariants of $\mathrm{Spin}_{14}$. Along the way, the author proves that certain maps in nonabelian cohomology are surjective. These surjectivities give as corollaries Pfister's results on 10- and 12-dimensional quadratic forms and Rost's theorem on 14-dimensional quadratic forms. This material on quadratic forms and invariants of $\mathrm{Spin}_n$ is based on unpublished work of Markus Rost. An appendix by Detlev Hoffmann proves a generalization of the Common Slot Theorem for 2-Pfister quadratic forms.
Small Divisor Problem In The Theory Of Three Dimensional Water Gravity Waves
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Author : Grard Iooss
language : en
Publisher: American Mathematical Soc.
Release Date : 2009-06-05
Small Divisor Problem In The Theory Of Three Dimensional Water Gravity Waves written by Grard Iooss and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-06-05 with Science categories.
The authors consider doubly-periodic travelling waves at the surface of an infinitely deep perfect fluid, only subjected to gravity $g$ and resulting from the nonlinear interaction of two simply periodic travelling waves making an angle $2\theta$ between them. Denoting by $\mu =gL/c^{2}$ the dimensionless bifurcation parameter ( $L$ is the wave length along the direction of the travelling wave and $c$ is the velocity of the wave), bifurcation occurs for $\mu = \cos \theta$. For non-resonant cases, we first give a large family of formal three-dimensional gravity travelling waves, in the form of an expansion in powers of the amplitudes of two basic travelling waves. ``Diamond waves'' are a particular case of such waves, when they are symmetric with respect to the direction of propagation. The main object of the paper is the proof of existence of such symmetric waves having the above mentioned asymptotic expansion. Due to the occurence of small divisors, the main difficulty is the inversion of the linearized operator at a non trivial point, for applying the Nash Moser theorem. This operator is the sum of a second order differentiation along a certain direction, and an integro-differential operator of first order, both depending periodically of coordinates. It is shown that for almost all angles $\theta$, the 3-dimensional travelling waves bifurcate for a set of ``good'' values of the bifurcation parameter having asymptotically a full measure near the bifurcation curve in the parameter plane $(\theta,\mu ).$
Unitary Invariants In Multivariable Operator Theory
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Author : Gelu Popescu
language : en
Publisher: American Mathematical Soc.
Release Date : 2009-06-05
Unitary Invariants In Multivariable Operator Theory written by Gelu Popescu and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-06-05 with Mathematics categories.
This paper concerns unitary invariants for $n$-tuples $T:=(T_1,\ldots, T_n)$ of (not necessarily commuting) bounded linear operators on Hilbert spaces. The author introduces a notion of joint numerical radius and works out its basic properties. Multivariable versions of Berger's dilation theorem, Berger-Kato-Stampfli mapping theorem, and Schwarz's lemma from complex analysis are obtained. The author studies the joint (spatial) numerical range of $T$ in connection with several unitary invariants for $n$-tuples of operators such as: right joint spectrum, joint numerical radius, euclidean operator radius, and joint spectral radius. He also proves an analogue of Toeplitz-Hausdorff theorem on the convexity of the spatial numerical range of an operator on a Hilbert space, for the joint numerical range of operators in the noncommutative analytic Toeplitz algebra $F_n^\infty$.
The Minimal Polynomials Of Unipotent Elements In Irreducible Representations Of The Classical Groups In Odd Characteristic
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Author : Irina D. Suprunenko
language : en
Publisher: American Mathematical Soc.
Release Date : 2009-06-05
The Minimal Polynomials Of Unipotent Elements In Irreducible Representations Of The Classical Groups In Odd Characteristic written by Irina D. Suprunenko and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-06-05 with Mathematics categories.
The minimal polynomials of the images of unipotent elements in irreducible rational representations of the classical algebraic groups over fields of odd characteristic are found. These polynomials have the form $(t-1)^d$ and hence are completely determined by their degrees. In positive characteristic the degree of such polynomial cannot exceed the order of a relevant element. It occurs that for each unipotent element the degree of its minimal polynomial in an irreducible representation is equal to the order of this element provided the highest weight of the representation is large enough with respect to the ground field characteristic. On the other hand, classes of unipotent elements for which in every nontrivial representation the degree of the minimal polynomial is equal to the order of the element are indicated. In the general case the problem of computing the minimal polynomial of the image of a given element of order $p^s$ in a fixed irreducible representation of a classical group over a field of characteristic $p>2$ can be reduced to a similar problem for certain $s$ unipotent elements and a certain irreducible representation of some semisimple group over the field of complex numbers. For the latter problem an explicit algorithm is given. Results of explicit computations for groups of small ranks are contained in Tables I-XII. The article may be regarded as a contribution to the programme of extending the fundamental results of Hall and Higman (1956) on the minimal polynomials from $p$-solvable linear groups to semisimple groups.
Twisted Pseudodifferential Calculus And Application To The Quantum Evolution Of Molecules
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Author : Andr Martinez
language : en
Publisher: American Mathematical Soc.
Release Date : 2009-06-05
Twisted Pseudodifferential Calculus And Application To The Quantum Evolution Of Molecules written by Andr Martinez and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-06-05 with Mathematics categories.
The authors construct an abstract pseudodifferential calculus with operator-valued symbol, suitable for the treatment of Coulomb-type interactions, and they apply it to the study of the quantum evolution of molecules in the Born-Oppenheimer approximation, in the case of the electronic Hamiltonian admitting a local gap in its spectrum. In particular, they show that the molecular evolution can be reduced to the one of a system of smooth semiclassical operators, the symbol of which can be computed explicitely. In addition, they study the propagation of certain wave packets up to long time values of Ehrenfest order.