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Author: Feliks Przytycki Publisher: American Mathematical Soc. ISBN: 1470435675 Category : Conformal geometry Languages : en Pages : 81
Book Description
This paper is an interval dynamics counterpart of three theories founded earlier by the authors, S. Smirnov and others in the setting of the iteration of rational maps on the Riemann sphere: the equivalence of several notions of non-uniform hyperbolicity, Geometric Pressure, and Nice Inducing Schemes methods leading to results in thermodynamical formalism. The authors work in a setting of generalized multimodal maps, that is, smooth maps f of a finite union of compact intervals Iˆ in R into R with non-flat critical points, such that on its maximal forward invariant set K the map f is topologically transitive and has positive topological entropy. They prove that several notions of non-uniform hyperbolicity of f|K are equivalent (including uniform hyperbolicity on periodic orbits, TCE & all periodic orbits in K hyperbolic repelling, Lyapunov hyperbolicity, and exponential shrinking of pull-backs). They prove that several definitions of geometric pressure P(t), that is pressure for the map f|K and the potential −tlog|f′|, give the same value (including pressure on periodic orbits, “tree” pressure, variational pressures and conformal pressure). Finally they prove that, provided all periodic orbits in K are hyperbolic repelling, the function P(t) is real analytic for t between the “condensation” and “freezing” parameters and that for each such t there exists unique equilibrium (and conformal) measure satisfying strong statistical properties.
Author: Feliks Przytycki Publisher: American Mathematical Soc. ISBN: 1470435675 Category : Conformal geometry Languages : en Pages : 81
Book Description
This paper is an interval dynamics counterpart of three theories founded earlier by the authors, S. Smirnov and others in the setting of the iteration of rational maps on the Riemann sphere: the equivalence of several notions of non-uniform hyperbolicity, Geometric Pressure, and Nice Inducing Schemes methods leading to results in thermodynamical formalism. The authors work in a setting of generalized multimodal maps, that is, smooth maps f of a finite union of compact intervals Iˆ in R into R with non-flat critical points, such that on its maximal forward invariant set K the map f is topologically transitive and has positive topological entropy. They prove that several notions of non-uniform hyperbolicity of f|K are equivalent (including uniform hyperbolicity on periodic orbits, TCE & all periodic orbits in K hyperbolic repelling, Lyapunov hyperbolicity, and exponential shrinking of pull-backs). They prove that several definitions of geometric pressure P(t), that is pressure for the map f|K and the potential −tlog|f′|, give the same value (including pressure on periodic orbits, “tree” pressure, variational pressures and conformal pressure). Finally they prove that, provided all periodic orbits in K are hyperbolic repelling, the function P(t) is real analytic for t between the “condensation” and “freezing” parameters and that for each such t there exists unique equilibrium (and conformal) measure satisfying strong statistical properties.
Author: Jean-François Coulombel Publisher: American Mathematical Soc. ISBN: 1470440377 Category : Education Languages : en Pages : 143
Book Description
This work is devoted to the analysis of high frequency solutions to the equations of nonlinear elasticity in a half-space. The authors consider surface waves (or more precisely, Rayleigh waves) arising in the general class of isotropic hyperelastic models, which includes in particular the Saint Venant-Kirchhoff system. Work has been done by a number of authors since the 1980s on the formulation and well-posedness of a nonlinear evolution equation whose (exact) solution gives the leading term of an approximate Rayleigh wave solution to the underlying elasticity equations. This evolution equation, which is referred to as “the amplitude equation”, is an integrodifferential equation of nonlocal Burgers type. The authors begin by reviewing and providing some extensions of the theory of the amplitude equation. The remainder of the paper is devoted to a rigorous proof in 2D that exact, highly oscillatory, Rayleigh wave solutions uε to the nonlinear elasticity equations exist on a fixed time interval independent of the wavelength ε, and that the approximate Rayleigh wave solution provided by the analysis of the amplitude equation is indeed close in a precise sense to uε on a time interval independent of ε. This paper focuses mainly on the case of Rayleigh waves that are pulses, which have profiles with continuous Fourier spectrum, but the authors' method applies equally well to the case of wavetrains, whose Fourier spectrum is discrete.
Author: Michael Handel Publisher: American Mathematical Soc. ISBN: 1470441136 Category : Education Languages : en Pages : 276
Book Description
In this work the authors develop a decomposition theory for subgroups of Out(Fn) which generalizes the decomposition theory for individual elements of Out(Fn) found in the work of Bestvina, Feighn, and Handel, and which is analogous to the decomposition theory for subgroups of mapping class groups found in the work of Ivanov.
Author: Luigi Ambrosio Publisher: American Mathematical Soc. ISBN: 1470439131 Category : Education Languages : en Pages : 121
Book Description
The aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces (X,d,m). On the geometric side, the authors' new approach takes into account suitable weighted action functionals which provide the natural modulus of K-convexity when one investigates the convexity properties of N-dimensional entropies. On the side of diffusion semigroups and evolution variational inequalities, the authors' new approach uses the nonlinear diffusion semigroup induced by the N-dimensional entropy, in place of the heat flow. Under suitable assumptions (most notably the quadraticity of Cheeger's energy relative to the metric measure structure) both approaches are shown to be equivalent to the strong CD∗(K,N) condition of Bacher-Sturm.
Author: Gonzalo Fiz Pontiveros Publisher: American Mathematical Soc. ISBN: 1470440717 Category : Education Languages : en Pages : 125
Book Description
The areas of Ramsey theory and random graphs have been closely linked ever since Erdős's famous proof in 1947 that the “diagonal” Ramsey numbers R(k) grow exponentially in k. In the early 1990s, the triangle-free process was introduced as a model which might potentially provide good lower bounds for the “off-diagonal” Ramsey numbers R(3,k). In this model, edges of Kn are introduced one-by-one at random and added to the graph if they do not create a triangle; the resulting final (random) graph is denoted Gn,△. In 2009, Bohman succeeded in following this process for a positive fraction of its duration, and thus obtained a second proof of Kim's celebrated result that R(3,k)=Θ(k2/logk). In this paper the authors improve the results of both Bohman and Kim and follow the triangle-free process all the way to its asymptotic end.
Author: Cristian Gavrus Publisher: American Mathematical Soc. ISBN: 147044111X Category : Education Languages : en Pages : 94
Book Description
In this paper, the authors prove global well-posedness of the massless Maxwell–Dirac equation in the Coulomb gauge on R1+d(d≥4) for data with small scale-critical Sobolev norm, as well as modified scattering of the solutions. Main components of the authors' proof are A) uncovering null structure of Maxwell–Dirac in the Coulomb gauge, and B) proving solvability of the underlying covariant Dirac equation. A key step for achieving both is to exploit (and justify) a deep analogy between Maxwell–Dirac and Maxwell-Klein-Gordon (for which an analogous result was proved earlier by Krieger-Sterbenz-Tataru, which says that the most difficult part of Maxwell–Dirac takes essentially the same form as Maxwell-Klein-Gordon.
Author: Jaroslav Nešetřil Publisher: American Mathematical Soc. ISBN: 1470440652 Category : Education Languages : en Pages : 108
Book Description
In this paper the authors introduce a general framework for the study of limits of relational structures and graphs in particular, which is based on a combination of model theory and (functional) analysis. The authors show how the various approaches to graph limits fit to this framework and that the authors naturally appear as “tractable cases” of a general theory. As an outcome of this, the authors provide extensions of known results. The authors believe that this puts these into a broader context. The second part of the paper is devoted to the study of sparse structures. First, the authors consider limits of structures with bounded diameter connected components and prove that in this case the convergence can be “almost” studied component-wise. They also propose the structure of limit objects for convergent sequences of sparse structures. Eventually, they consider the specific case of limits of colored rooted trees with bounded height and of graphs with bounded tree-depth, motivated by their role as “elementary bricks” these graphs play in decompositions of sparse graphs, and give an explicit construction of a limit object in this case. This limit object is a graph built on a standard probability space with the property that every first-order definable set of tuples is measurable. This is an example of the general concept of modeling the authors introduce here. Their example is also the first “intermediate class” with explicitly defined limit structures where the inverse problem has been solved.
Author: S. V. Ivanov Publisher: American Mathematical Soc. ISBN: 1470441438 Category : Education Languages : en Pages : 106
Book Description
The author introduces and studies the bounded word problem and the precise word problem for groups given by means of generators and defining relations. For example, for every finitely presented group, the bounded word problem is in NP, i.e., it can be solved in nondeterministic polynomial time, and the precise word problem is in PSPACE, i.e., it can be solved in polynomial space. The main technical result of the paper states that, for certain finite presentations of groups, which include the Baumslag-Solitar one-relator groups and free products of cyclic groups, the bounded word problem and the precise word problem can be solved in polylogarithmic space. As consequences of developed techniques that can be described as calculus of brackets, the author obtains polylogarithmic space bounds for the computational complexity of the diagram problem for free groups, for the width problem for elements of free groups, and for computation of the area defined by polygonal singular closed curves in the plane. The author also obtains polynomial time bounds for these problems.
Author: Mircea Mustaţă Publisher: American Mathematical Soc. ISBN: 1470437813 Category : Education Languages : en Pages : 78
Book Description
The authors use methods from birational geometry to study the Hodge filtration on the localization along a hypersurface. This filtration leads to a sequence of ideal sheaves, called Hodge ideals, the first of which is a multiplier ideal. They analyze their local and global properties, and use them for applications related to the singularities and Hodge theory of hypersurfaces and their complements.
Author: Feliks Przytycki
Author: Feliks Przytycki
Author: Jean-François Coulombel
Author: Michael Handel
Author: Luigi Ambrosio
Author: Gonzalo Fiz Pontiveros
Author: Cristian Gavrus
Author: Jaroslav Nešetřil
Author: S. V. Ivanov
Author: Mircea Mustaţă
Author: Victor Beresnevich