Download or read book Newton s Method Applied to Two Quadratic Equations in C2 Viewed as a Global Dynamical System written by John Hamal Hubbard and published by . This book was released on 2008 with total page 160 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

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.

Download or read book Newton s Method Applied to Two Quadratic Equations in C2 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 162 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduction Fundamental properties of Newton maps Invariant 3-manifolds associated to invariant circles The behavior at infinity when $a=b=0$ The Farey blow-up The compactification when $a=b=0$ The case where $a$ and $b$ are arbitrary Bibliography

Download or read book The Projective Heat Map written by Richard Evan Schwartz and published by American Mathematical Soc.. This book was released on 2017-04-20 with total page 210 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book The Valuative Tree written by Charles Favre and published by Springer. This book was released on 2004-08-30 with total page 251 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Cohomological Invariants Exceptional Groups and Spin Groups written by Skip Garibaldi and published by American Mathematical Soc.. This book was released on 2009-06-05 with total page 102 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Twisted Pseudodifferential Calculus and Application to the Quantum Evolution of Molecules written by AndrŽ Martinez and published by American Mathematical Soc.. This book was released on 2009-06-05 with total page 96 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book The Minimal Polynomials of Unipotent Elements in Irreducible Representations of the Classical Groups in Odd Characteristic written by Irina D. Suprunenko and published by American Mathematical Soc.. This book was released on 2009-06-05 with total page 168 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Uniqueness and Stability in Determining a Rigid Inclusion in an Elastic Body written by Antonino Morassi and published by American Mathematical Soc.. This book was released on 2009-06-05 with total page 74 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Unitary Invariants in Multivariable Operator Theory written by Gelu Popescu and published by American Mathematical Soc.. This book was released on 2009-06-05 with total page 105 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 $F_n^\infty$.

Download or read book Small Divisor Problem in the Theory of Three Dimensional Water Gravity Waves written by GŽrard Iooss and published by American Mathematical Soc.. This book was released on 2009-06-05 with total page 144 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 $(\theta,\mu ).$

Download or read book The Topological Dynamics of Ellis Actions written by Ethan Akin and published by American Mathematical Soc.. This book was released on 2008 with total page 166 pages. Available in PDF, EPUB and Kindle. Book excerpt: An Ellis semigroup is a compact space with a semigroup multiplication which is continuous in only one variable. An Ellis action is an action of an Ellis semigroup on a compact space such that for each point in the space the evaluation map from the semigroup to the space is continuous. At first the weak linkage between the topology and the algebra discourages expectations that such structures will have much utility. However, Ellis has demonstrated that these actions arise naturallyfrom classical topological actions of locally compact groups on compact spaces and provide a useful tool for the study of such actions. In fact, via the apparatus of the enveloping semigroup the classical theory of topological dynamics is subsumed by the theory of Ellis actions. The authors'exposition describes and extends Ellis' theory and demonstrates its usefulness by unifying many recently introduced concepts related to proximality and distality. Moreover, this approach leads to several results which are new even in the classical setup.

Download or read book Random Sets and Invariants for Type II Continuous Tensor Product Systems of Hilbert Spaces written by Volkmar Liebscher and published by American Mathematical Soc.. This book was released on 2009-04-10 with total page 124 pages. Available in PDF, EPUB and Kindle. Book excerpt: In a series of papers Tsirelson constructed from measure types of random sets or (generalised) random processes a new range of examples for continuous tensor product systems of Hilbert spaces introduced by Arveson for classifying $E_0$-semigroups upto cocycle conjugacy. This paper starts from establishing the converse. So the author connects each continuous tensor product system of Hilbert spaces with measure types of distributions of random (closed) sets in $[0,1]$ or $\mathbb R_+$. These measure types are stationary and factorise over disjoint intervals. In a special case of this construction, the corresponding measure type is an invariant of the product system. This shows, completing in a more systematic way the Tsirelson examples, that the classification scheme for product systems into types $\mathrm{I}_n$, $\mathrm{II}_n$ and $\mathrm{III}$ is not complete. Moreover, based on a detailed study of this kind of measure types, the author constructs for each stationary factorising measure type a continuous tensor product system of Hilbert spaces such that this measure type arises as the before mentioned invariant.

Download or read book A Proof of Alon s Second Eigenvalue Conjecture and Related Problems written by Joel Friedman and published by American Mathematical Soc.. This book was released on 2008 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt: A $d$-regular graph has largest or first (adjacency matrix) eigenvalue $\lambda_1=d$. Consider for an even $d\ge 4$, a random $d$-regular graph model formed from $d/2$ uniform, independent permutations on $\{1,\ldots,n\}$. The author shows that for any $\epsilon>0$ all eigenvalues aside from $\lambda_1=d$ are bounded by $2\sqrt{d-1}\;+\epsilon$ with probability $1-O(n^{-\tau})$, where $\tau=\lceil \bigl(\sqrt{d-1}\;+1\bigr)/2 \rceil-1$. He also shows that this probability is at most $1-c/n^{\tau'}$, for a constant $c$ and a $\tau'$ that is either $\tau$ or $\tau+1$ (``more often'' $\tau$ than $\tau+1$). He proves related theorems for other models of random graphs, including models with $d$ odd.

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.

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.

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.