Ergodic theory for Markov fibred systems and parabolic rational maps
Jon
Aaronson;
Manfred
Denker;
Mariusz
Urbański
495-548
Abstract: A parabolic rational map of the Riemann sphere admits a non atomic $ h$-conformal measure on its Julia set where $h =$ the Hausdorff dimension of the Julia set and satisfies $1/2 < h < 2$. With respect to this measure the rational map is conservative, exact and there is an equivalent $\sigma$-finite invariant measure. Finiteness of the measure is characterised. Central limit theorems are proved in the case of a finite invariant measure and return sequences are identified in the case of an infinite one. A theory of Markov fibred systems is developed, and parabolic rational maps are considered within this framework.
On the Cauchy problem for reaction-diffusion equations
Xuefeng
Wang
549-590
Abstract: The simplest model of the Cauchy problem considered in this paper is the following $(\ast)$ \begin{displaymath}\begin{array}{*{20}{c}} {{u_t} = \Delta u + {u^p},} & ... ... & {\phi \geq 0,\phi \,\not\equiv\,0.} \end{array} \;\end{displaymath} It is well known that when $1 < p \leq (n + 2)/n$, the local solution of $ (\ast)$ blows up in finite time as long as the initial value $\phi$ is nontrivial; and when $p > (n + 2)/n$, if $\phi$ is "small", $(\ast)$ has a global classical solution decaying to zero as $t \to + \infty $, while if $\phi$ is "large", the local solution blows up in finite time. The main aim of this paper is to obtain optimal conditions on $\phi$ for global existence and to study the asymptotic behavior of those global solutions. In particular, we prove that if $n \geq 3$, $ p > n/(n - 2)$, $\displaystyle 0 \leq \phi (x) \leq \lambda {u_s}(x) = \lambda {\left( {\frac{{2... ... \frac{n} {{n - 2}}} \right)} \right)^{1/(p - 1)}}\vert x{\vert^{ - 2/(p - 1)}}$ (${u_s}$ is a singular equilibrium of $ (\ast)$) where $0 < \lambda < 1$, then $(\ast)$ has a (unique) global classical solution $u$ with $0 \leq u \leq \lambda {u_s}$ and $\displaystyle u(x,t) \leq {(({\lambda ^{1 - p}} - 1)(p - 1)t)^{ - 1/(p - 1)}}.$ (This result implies that ${u_0} \equiv 0$ is stable w.r.t. to a weighted ${L^\infty }$ topology when $n \geq 3$ and $p > n/(n - 2)$.) We also obtain some sufficient conditions on $\phi$ for global nonexistence and those conditions, when combined with our global existence result, indicate that for $\phi$ around ${u_s}$, we are in a delicate situation, and when $ p$ is fixed, ${u_0} \equiv 0$ is "increasingly stable" as the dimension $ n \uparrow + \infty$. A slightly more general version of $(\ast)$ is also considered and similar results are obtained.
The Alexander and Markov theorems via diagrams for links in $3$-manifolds
Paul A.
Sundheim
591-607
Abstract: Let $M$ be a $3$-manifold with an open book decomposition. We obtain a new proof that a link in $M$ has a braided form and that two braided forms are related by a sequence of two Markov moves for $ M$ by generalizing Morton's approach for links in ${S^3}$.
Baire class $1$ selectors for upper semicontinuous set-valued maps
V. V.
Srivatsa
609-624
Abstract: Let $T$ be a metric space and $X$ a Banach space. Let $F:T \to X$ be a set-valued map assuming arbitrary values and satisfying the upper semicontinuity condition: $\{ t \in T:F(t) \cap C \ne \emptyset \}$ is closed for each weakly closed set $C$ in $X$. Then there is a sequence of norm-continuous functions converging pointwise (in the norm) to a selection for $F$. We prove a statement of similar precision and generality when $X$ is a metric space.
Applying coordinate products to the topological identification of normed spaces
Robert
Cauty;
Tadeusz
Dobrowolski
625-649
Abstract: Using the $ {l^2}$-products we find pre-Hilbert spaces that are absorbing sets for all Borelian classes of order $ \alpha \geq 1$. We also show that the following spaces are homeomorphic to $\Sigma^\infty$, the countable product of the space $\Sigma = \{(x_n) \in R^\infty: (x_n)$ is bounded}: (1) every coordinate product $\prod_C H_n$ of normed spaces $H_n$ in the sense of a Banach space $ C$, where each $ H_n$ is an absolute $F_{\sigma\delta}$-set and infinitely many of the $ H_n$'s are ${Z_\sigma }$-spaces, (2) every function space $\tilde{L}^p = \cap_{p\prime <p}L^{p\prime}$ with the ${L^q}$-topology, $0<q<p \leq \infty$, (3) every sequence space ${\tilde l^p} = { \cap _{p < p\prime}}{l^{p\prime}}$ with the $l^q$-topology, $ 0 \leq p < q < \infty$. We also note that each additive and multiplicative Borelian class of order $\alpha \geq 2$, each projective class, and the class of nonprojective spaces contain uncountably many topologically different pre-Hilbert spaces which are $ Z_\sigma$-spaces.
Chebyshev type estimates for Beurling generalized prime numbers. II
Wen-Bin
Zhang
651-675
Abstract: Let $N(x)$ be the distribution function of the integers in a Beurling generalized prime system. The Chebyshev type estimates for Beurling generalized prime numbers in the general case $\displaystyle N(x) = x\sum\limits_{\nu = 1}^n {{A_\nu }} {\log ^{{\rho _\nu } - 1}}x + O(x{\log ^{ - \gamma }}x)$ is a long standing question. In this paper we shall give an affirmative answer to the question by proving that the Chebyshev type estimates $\displaystyle 0 < \mathop {\lim \inf }\limits_{x \to \infty } \frac{{\psi (x)}}... ...quad \mathop {\lim \sup }\limits_{x \to \infty } \frac{{\psi (x)}}{x} < \infty$ hold even under weaker condition $\displaystyle \int_1^\infty {{x^{ - 1}}} \left\{ {\mathop {\sup }\limits_{x < \... ...n {{A_\nu }} {{\log }^{{\rho _\nu } - 1}}y} \right\vert} \right\}\,dx < \infty$ with $ \rho_n=\tau \geq 1$, $ 0<\rho_1<\rho_2 <\cdots < \rho_n$, and $A_n > 0$. This generalizes a result of Diamond and a result of the present author.
Computing the equations of a variety
Michela
Brundu;
Mike
Stillman
677-690
Abstract: Let $X \subset {\mathbb{P}^n}$ be a projective variety or subscheme, and let $ \mathcal{F}$ be an invertible sheaf on $X$. A set of global sections of $\mathcal{F}$ determines a map from a Zariski open subset of $X$ to $ {\mathbb{P}^r}$. The purpose of this paper is to find, given $X$ and $ \mathcal{F}$, the homogeneous ideal defining the image in ${\mathbb{P}^r}$ of this rational map. We present algorithms to compute the ideal of the image. These algorithms can be implemented using only the computation of Gröbner bases and syzygies, and they have been implemented in our computer algebra system Macaulay. Our methods generalize to include the case when $X$ is an arbitrary projective scheme and $\mathcal{F}$ is generically invertible.
Varieties of topological geometries
Hansjoachim
Groh
691-702
Abstract: A variety of topological geometries is either A. a projective variety $ \mathcal{L}(F)$ over some topological field $F$, or B. a matchstick variety $\mathcal{M}(X)$ over some topological space $ X$. As a main tool for showing this, we prove a structure theorem for arbitrary topological geometries.
The structure of random partitions of large integers
Bert
Fristedt
703-735
Abstract: Random partitions of integers are treated in the case where all partitions of an integer are assumed to have the same probability. The focus is on limit theorems as the number being partitioned approaches $\infty$. The limiting probability distribution of the appropriately normalized number of parts of some small size is exponential. The large parts are described by a particular Markov chain. A central limit theorem and a law of large numbers holds for the numbers of intermediate parts of certain sizes. The major tool is a simple construction of random partitions that treats the number being partitioned as a random variable. The same technique is useful when some restriction is placed on partitions, such as the requirement that all parts must be distinct.
A generalization of the Airy integral for $f''-z\sp nf=0$
Gary G.
Gundersen;
Enid M.
Steinbart
737-755
Abstract: It is well known that the Airy integral is a solution of the Airy differential equation $ f'' - zf = 0$ and that the Airy integral is a contour integral function with special properties. We show that there exist analogous special contour integral solutions of the more general equation $f'' - {z^n}f = 0$ where $n$ is any positive integer. Related results are given.
Kloosterman sums for Chevalley groups
Romuald
Dąbrowski
757-769
Abstract: A generalization of Kloosterman sums to a simply connected Chevalley group $G$ is discussed. These sums are parameterized by pairs $(w,t)$ where $w$ is an element of the Weyl group of $G$ and $t$ is an element of a $ {\mathbf{Q}}$-split torus in $G$. The $ SL(2,{\mathbf{Q}})$-Kloosterman sums coincide with the classical Kloosterman sums and $ SL(r,{\mathbf{Q}})$-Kloosterman sums, $r \geq 3$, coincide with the sums introduced in [B-F-G,F,S]. Algebraic properties of the sums are proved by root system methods. In particular an explicit decomposition of a general Kloosterman sum over ${\mathbf{Q}}$ into the product of local $ p$-adic factors is obtained. Using this factorization one can show that the Kloosterman sums corresponding to a toral element, which acts trivially on the highest weight space of a fundamental irreducible representation, splits into a product of Kloosterman sums for Chevalley groups of lower rank.
The finite part of singular integrals in several complex variables
Xiaoqin
Wang
771-793
Abstract: A divergent integral can sometimes be handled by assigning to it as its value the finite part in the sense of Hadamard. This is done by expanding the integral over the complement of a symmetric neighborhood of a singularity in powers of the radius, and throwing away the negative powers. In this paper the finite part of a singular integral of Cauchy type is defined, and this is then used to describe the boundary behavior of derivatives of a Cauchy-type integral. The finite part of a singular integral of Bochner-Martinelli type is studied, and an extension of the Plemelj jump formulas is shown to hold.
Totally monotone functions with applications to the Bergman space
B.
Korenblum;
R.
O’Neil;
K.
Richards;
K.
Zhu
795-806
Abstract: Using a theorem of S. Bernstein [1] we prove a special case of the following maximum principle for the Bergman space conjectured by B. Korenblum [3]: There exists a number $\delta \in (0,1)$ such that if $f$ and $g$ are analytic functions on the open unit disk ${\mathbf{D}}$ with $\vert f(z)\vert \leq \vert g(z)\vert$ on $\delta \leq \vert z\vert < 1$ then $ {\left\Vert f \right\Vert _2} \leq {\left\Vert g \right\Vert _2}$, where $ {\left\Vert {} \right\Vert _2}$ is the ${L^2}$ norm with respect to area measure on ${\mathbf{D}}$. We prove the above conjecture when either $f$ or $g$ is a monomial; in this case we show that the optimal constant $\delta$ is greater than or equal to $1/\sqrt 3 $.
Computing the Mordell-Weil rank of Jacobians of curves of genus two
Daniel M.
Gordon;
David
Grant
807-824
Abstract: We derive the equations necessary to perform a two-descent on the Jacobians of curves of genus two with rational Weierstrass points. We compute the Mordell-Weil rank of the Jacobian of some genus two curves defined over the rationals, and discuss the practicality of using this method.
A subcategory of TOP
Alan
Dow;
Stephen
Watson
825-837
Abstract: We consider the smallest class of topological spaces which contains the converging sequence and which is closed under the operations of taking arbitrary sums, quotients and finite products. We show that if there is a model of set-theory in which there is a measurable cardinal then there is a model in which this class does not contain all topological spaces. In addition, we prove that it is consistent that this class does contain all topological spaces--in fact much more, a large cardinal is needed to produce a model of set theory in which this class is proper.
${\rm T}1$ theorems for Besov and Triebel-Lizorkin spaces
Y.-S.
Han;
Steve
Hofmann
839-853
Abstract: We give simple proofs of the $T1$ theorem in the general context of Besov spaces and (weighted) Triebel-Lizorkin spaces. Our approach yields some new results for kernels satisfying weakened regularity conditions, while also recovering previously known results.
Quantitative rectifiability and Lipschitz mappings
Guy
David;
Stephen
Semmes
855-889
Abstract: The classical notion of rectifiability of sets in ${{\mathbf{R}}^n}$ is qualitative in nature, and in this paper we are concerned with quantitative versions of it. This issue arises in connection with $ {L^p}$ estimates for singular integral operators on sets in ${{\mathbf{R}}^n}$. We give a criterion for one reasonably natural quantitative rectifiability condition to hold, and we use it to give a new proof of a theorem in [D3]. We also give some results on the geometric properties of a certain class of sets in ${{\mathbf{R}}^n}$ which can be viewed as generalized hypersurfaces. Along the way we shall encounter some questions concerning the behavior of Lipschitz functions, with regard to approximation by affine functions in particular. We shall also discuss an amusing variation of the classical Lipschitz and bilipschitz conditions, which allow some singularities forbidden by the classical conditions while still forcing good behavior on substantial sets.
Combinatorics of triangulations of $3$-manifolds
Feng
Luo;
Richard
Stong
891-906
Abstract: In this paper, we study the average edge order of triangulations of closed $ 3$-manifolds and show in particular that the average edge order being less than $ 4.5$ implies that triangulation is on the $3$-sphere.
A structural criterion for the existence of infinite central $\Lambda(p)$ sets
Kathryn E.
Hare;
David C.
Wilson
907-925
Abstract: We classify the compact, connected groups which have infinite central $\Lambda (p)$ sets, arithmetically characterize central $\Lambda (p)$ sets on certain product groups, and give examples of $ \Lambda (p)$ sets which are non-Sidon and have unbounded degree. These sets are intimately connected with Figà-Talamanca and Rider's examples of Sidon sets, and stem from the existence of families of tensor product representations of almost simple Lie groups whose decompositions into irreducibles are rank-independent.
On the existence and uniqueness of solutions of M\"obius equations
Xingwang
Xu
927-945
Abstract: A generalization of the Schwarzian derivative to conformal mappings of Riemannian manifolds has naturally introduced the corresponding overdetermined differential equation which we call the Möbius equation. We are interested in study of the existence and uniqueness of the solution of the Möbius equation. Among other things, we show that, for a compact manifold, if Ricci curvature is nonpositive, for a complete noncompact manifold, if the scalar curvature is a positive constant, then the differential equation has only constant solutions. We also study the nonhomogeneous equation in an $n$-dimensional Euclidean space.
Higher-dimensional analogues of Fuchsian subgroups of ${\rm PSL}(2,\germ o)$
L. Ya.
Vulakh
947-963
Abstract: The problem of classification of $2 \times 2$ indefinite Hermitian matrices over orders in Clifford algebras is considered. The unit groups of these matrices are analogous to maximal arithmetic Fuchsian subgroups of $ {\text{PSL}}(2,\mathfrak{o})$ where $ \mathfrak{o}$ is an order in a quadratic number field.
Rational orbits on three-symmetric products of abelian varieties
A.
Alzati;
G. P.
Pirola
965-980
Abstract: Let $A$ be an $n$-dimensional Abelian variety, $n \geq 2$; let ${\text{CH}_0}(A)$ be the group of zero-cycles of $ A$, modulo rational equivalence; by regarding an effective, degree $ k$, zero-cycle, as a point on ${S^k}(A)$ (the $k$-symmetric product of $A$), and by considering the associated rational equivalence class, we get a map $\gamma :{S^k}(A) \to {\text{CH}_0}(A)$, whose fibres are called $\gamma$-orbits. For any $n \geq 2$, in this paper we determine the maximal dimension of the $\gamma$-orbits when $k = 2$ or $3$ (it is, respectively, $1$ and $2$), and the maximal dimension of families of $ \gamma$-orbits; moreover, for generic $A$, we get some refinements and in particular we show that if $ \dim (A) \geq 4$, $ {S^3}(A)$ does not contain any $\gamma$-orbit; note that it implies that a generic Abelian four-fold does not contain any trigonal curve. We also show that our bounds are sharp by some examples. The used technique is the following: we have considered some special families of Abelian varieties: $ {A_t} = {E_t} \times B$ ($ {E_t}$ is an elliptic curve with varying moduli) and we have constructed suitable projections between $ {S^k}({A_t})$ and $ {S^k}(B)$ which preserve the dimensions of the families of $\gamma $-orbits; then we have done induction on $n$. For $n = 2$ the proof is based upon the papers of Mumford and Roitman on this topic.
New invariant Einstein metrics on generalized flag manifolds
Andreas
Arvanitoyeorgos
981-995
Abstract: A generalized flag manifold (or a Kählerian $ C$-space) is a homogeneous space $G/K$ whose isotropy subgroup $K$ is the centralizer of a torus in $ G$. These spaces admit a finite number of Kähler-Einstein metrics. We present new non-Kahler Einstein metrics for certain quotients of $U(n)$, $SO(2n)$ and ${G_2}$. We also examine the isometry question for these metrics.