The Stefan problem with small surface tension
Avner
Friedman;
Fernando
Reitich
465-515
Abstract: The Stefan problem with small surface tension $ \varepsilon$ is considered. Assuming that the classical Stefan problem (with $\varepsilon = 0$) has a smooth free boundary $\Gamma$, we denote the temperature of the solution by ${\theta _0}$ and consider an approximate solution $ {\theta _0} + \varepsilon u$ for the case where $ \varepsilon \ne 0$, $\varepsilon$ small. We first establish the existence and uniqueness of $u$, and then investigate the effect of $u$ on the free boundary $\Gamma$. It is shown that small surface tension affects the free boundary $\Gamma$ radically differently in the two-phase problem than in the one-phase problem.
Iteration of a composition of exponential functions
Xiaoying
Dong
517-526
Abstract: We show that for certain complex parameters ${\lambda _1},\ldots,{\lambda _{n - 1}}$ and ${\lambda _n}$ the Julia set of the function $\displaystyle {e^{{\lambda _1}{e^{^{{.^{{.^{{.^{{{^{{\lambda _{n - 1}}}}^{{e^\lambda }^{_{{n^z}}}}}}}}}}}}}}}}$ is the whole plane $\mathbb{C}$. We denote by $\Lambda$ the set of $n$-tuples $({\lambda _1},\ldots,{\lambda _n}),{\lambda _1},\ldots,{\lambda _n} \in \mathbb{R}$ for which the equation $\displaystyle {e^{{\lambda _1}{e^{^{{.^{{.^{{.^{{{^{{\lambda _{n - 1}}}}^{{e^\lambda }^{_{{n^z}}}}}}}}}}}}}}}} - z= 0$ has exact two real solutions. In fact, one of them is an attracting fixed point of $\displaystyle {e^{{\lambda _1}{e^{^{{.^{{.^{{.^{{{^{{\lambda _{n - 1}}}}^{{e^\lambda }^{_{{n^z}}}}}}}}}}}}}}}},$ which is denoted by $q$. We also show that when $({\lambda _1},\ldots,{\lambda _n})\, \in \Lambda$, the Julia set of $\displaystyle {e^{{\lambda _1}{e^{^{{.^{{.^{{.^{{{^{{\lambda _{n - 1}}}}^{{e^\lambda }^{_{{n^z}}}}}}}}}}}}}}}}$ is the complement of the basin of attraction of $q$. The ideas used in this note may also be applicable to more general functions.
Complete coinductive theories. II
A. H.
Lachlan
527-562
Abstract: Let $T$ be a complete theory over a relational language which has an axiomatization by $\exists \forall $-sentences. The properties of models of $T$ are studied. It is shown that existential formulas are stable. A theory of forking and independence based on Boolean combinations of existential formulas in $\exists \forall $-saturated models of $ T$ is developed for which the independence relation is shown to satisfy a very strong triviality condition. It follows that $T$ is tree-decomposable in the sense of Baldwin and Shelah. It is also shown that if the language is finite, then $T$ has a prime model.
On the existence of conformal measures
Manfred
Denker;
Mariusz
Urbański
563-587
Abstract: A general notion of conformal measure is introduced and some basic properties are studied. Sufficient conditions for the existence of these measures are obtained, using a general construction principle. The geometric properties of conformal measures relate equilibrium states and Hausdorff measures. This is shown for invariant subsets of ${S^1}$ under expanding maps.
Continuity of translation in the dual of $L\sp \infty(G)$ and related spaces
Colin C.
Graham;
Anthony T. M.
Lau;
Michael
Leinert
589-618
Abstract: Let $X$ be a Banach space and $ G$ a locally compact Hausdorff group that acts as a group of isometric linear operators on $X$. The operation of $x \in G$ on $X$ will be denoted by ${L_x}$. We study the set ${X_c}$ of elements $\mu \in X$ such that $x \mapsto {L_x}\mu$ is continuous with respect to the topology on $G$ and the norm-topology on $X$. The spaces $X$ studied include $M{(G)^{\ast} },{\text{LUC}}{(G)^{\ast} },{L^\infty }{(G)^{\ast} },{\text{VN}}(G)$, and ${\text{VN}}{(G)^{\ast} }$. In most cases, characterizations of ${X_c}$ do not appear to be possible, and we give constructions that illustrate this. We relate properties of ${X_c}$ to properties of $G$. For example, if ${X_c}$ is sufficiently small, then $G$ is compact, or even finite, depending on the case. We give related results and open problems.
Inequalities for the integral means of holomorphic functions and their derivatives in the unit ball of $\bold C\sp n$
Ji Huai
Shi
619-637
Abstract: In this paper, the following two inequalities are proved: $\displaystyle \int_0^1 {{(1 - r)}^{a\vert\alpha\vert + b}}M_p^a(r,D^{\alpha} f)... ...alpha \vert = m} \int_0^1 {(1 - r)}^{am + b}M_p^a(r,D^{\alpha }f)\,dr \right\}$ where $\alpha = ({\alpha _1}, \ldots,{\alpha _n})$ is multi-index, $0 < p < \infty,0 < a < \infty$ and $- 1 < b < \infty$. These are a generalization of some classical results of Hardy and Littlewood. Using these two inequalities, we generalize a theorem in $[9]$. The methods used in the proof of Theorem 1 lead us to obtain Theorem 7, which enables us to generalize some theorems about the pluriharmonic conjugates in $ [8]$ and $[2]$.
Change of variable results for $A\sb p$- and reverse H\"older ${\rm RH}\sb r$-classes
R.
Johnson;
C. J.
Neugebauer
639-666
Abstract: We study conditions under which the map $\displaystyle {T_{h,\gamma }}w(x)= w(h(x))\vert h^{\prime}(x){\vert^\gamma }$ maps the Muckenhoupt class ${A_p}$ into ${A_q}$ and the reverse Hölder class $R{H_{{r_1}}}$ into $ R{H_{{r_2}}}$.
Cardinal representations for closures and preclosures
F.
Galvin;
E. C.
Milner;
M.
Pouzet
667-693
Abstract: A cardinal representation of a preclosure $\varphi$ on a set $E$ is a family $\mathcal{A} \subseteq \mathcal{P}(E)$ such that for any set $X \subseteq \cup \mathcal{A},\varphi (X) = E$ holds if and only if $ \vert X \cap A\vert= \vert A\vert$ for every $A \in \mathcal{A}$. We show, for example (Theorem 2.3) that any topological closure has such a representation, but there are closures which have no cardinal representation (Theorem 11.2). We prove that, if $k$ is finite and a closure has no independent set of size $k + 1$, then it has a cardinal representation, $\mathcal{A}$, of size $\vert\mathcal{A}\vert \leq k$ (Theorem 2.4). This result is used to give a new proof of a theorem of D. Duffus and M. Pouzet [4] about gaps in a lattice of finite breadth. We do not know if a closure which has no infinite independent set necessarily has a cardinal representation, but we do prove this is so for the special case of a closure on a countable set (Theorem 2.5). Even in this special case, nothing can be said about the size of the cardinal representation; however, if the closure is algebraic, then there is a finite cardinal representation (Theorem 2.6). These results do not hold for preclosures in general, but if a preclosure on a countable set has no independent set of size $ k + 1$ ($k$ finite), then it has a cardinal representation $ \mathcal{A}$ of size $ \vert\mathcal{A}\vert \leq k$ (Theorem 2.7).
Quantitative stability of variational systems. I. The epigraphical distance
Hédy
Attouch;
Roger J.-B.
Wets
695-729
Abstract: This paper proposes a global measure for the distance between the elements of a variational system (parametrized families of optimization problems).
Equivariant fixed point index and fixed point transfer in nonzero dimensions
Carlos
Prieto;
Hanno
Ulrich
731-745
Abstract: Dold's fixed point index and fixed point transfer are generalized for certain coincidence situations, namely maps which change the "equivariant dimension." Those invariants change the dimension correspondingly. A formula for the index of a situation over a space with trivial group action is exhibited. For the transfer, a generalization of Dold's Lefschetz-Hopf trace formula is proved.
Regular points for ergodic Sina\u\i measures
Masato
Tsujii
747-766
Abstract: Ergodic properties of smooth dynamical systems are considered. A point is called regular for an ergodic measure $\mu$ if it is generic for $\mu$ and the Lyapunov exponents at it coincide with those of $\mu$. We show that an ergodic measure with no zero Lyapunov exponent is absolutely continuous with respect to unstable foliation $ [$L$]$ if and only if the set of all points which are regular for it has positive Lebesgue measure.
On weak convergence in dynamical systems to self-similar processes with spectral representation
Michael T.
Lacey
767-778
Abstract: Let $(X,\mu,T)$ be an aperiodic dynamical system. Set ${S_m}f = f + \cdots + f \circ {T^{m - 1}}$, where $ f$ is a measurable function on $X$. Let $Y(t)$ be one of a class of self-similar process with a "nice" spectral representation, for instance, either a fractional Brownian motion, a Hermite process, or a harmonizable fractional stable motion. We show that there is an $f$ on $X$, and constants ${A_m} \to + \infty$ so that $\displaystyle A_m^{ - 1}{S_{[mt]}}f\mathop \Rightarrow \limits^d Y(t),$ the convergence being understood in the sense of weak convergence of all finite dimensional distributions in $t$.
Chebyshev polynomials in several variables and the radial part of the Laplace-Beltrami operator
R. J.
Beerends
779-814
Abstract: Chebyshev polynomials of the first and the second kind in $ n$ variables ${z_1},{z_2}, \ldots,{z_n}$ are introduced. The variables ${z_1},{z_2}, \ldots,{z_n}$ are the characters of the representations of $SL(n + 1,{\mathbf{C}})$ corresponding to the fundamental weights. The Chebyshev polynomials are eigenpolynomials of a second order linear partial differential operator which is in fact the radial part of the Laplace-Beltrami operator on certain symmetric spaces. We give an explicit expression of this operator in the coordinates ${z_1},{z_2}, \ldots,{z_n}$ and then show how many results in the literature on differential equations satisfied by Chebyshev polynomials in several variables follow immediately from well-known results on the radial part of the Laplace-Beltrami operator. Related topics like orthogonality, symmetry relations, generating functions and recurrence relations are also discussed. Finally we note that the Chebyshev polynomials are a special case of a more general class of orthogonal polynomials in several variables.
Regularity properties of commutators and layer potentials associated to the heat equation
John L.
Lewis;
Margaret A. M.
Murray
815-842
Abstract: In recent years there has been renewed interest in the solution of parabolic boundary value problems by the method of layer potentials. In this paper we consider graph domains $D = \{ (x,t):x > f(t)\}$ in ${\mathcal{R}^2}$, where the boundary function $ f$ is in ${I_{1/2}}({\text{BMO}})$. This class of domains would appear to be the minimal smoothness class for the solvability of the Dirichlet problem for the heat equation by the method of layer potentials. We show that, for $1 < p < \infty$, the boundary single-layer potential operator for $D$ maps ${L^p}$ into the homogeneous Sobolev space $ {I_{1/2}}({L^p})$. This regularity result is obtained by studying the regularity properties of a related family of commutators. Along the way, we prove ${L^p}$ estimates for a class of singular integral operators to which the $ {\text{T1}}$ Theorem of David and Journé does not apply. The necessary estimates are obtained by a variety of real-variable methods.
Multibasic Eulerian polynomials
Dominique
Foata;
Doron
Zeilberger
843-862
Abstract: Eulerian polynomials with several bases are defined. Their combinatorial interpretations are given as well as congruence properties modulo some ideals generated by cyclotomic polynomials.
Heegaard diagrams of $3$-manifolds
Mitsuyuki
Ochiai
863-879
Abstract: For a $ 3$-manifold $M(L)$ obtained by an integral Dehn surgery along an $n$-bridge link $L$ with $n$-components we define a concept of planar Heegaard diagrams of $M(L)$ using a link diagram of $L$. Then by using Homma-Ochiai-Takahashi's theorem and a planar Heegaard diagram of $ M(L)$ we will completely determine if $M(L)$ is the standard $3$-sphere in the case when $L$ is a $2$-bridge link with $2$-components.
On lifting Hecke eigenforms
Lynne H.
Walling
881-896
Abstract: A classical Hilbert modular form $f \in {\mathcal{M}_k}({\Gamma _0}(\mathcal{N},\mathcal{I}),{\chi _\mathcal{N}})$ cannot be an eigenform for the full Hecke algebra. We develop a means of lifting a classical form to a modular form $F \in { \oplus _\lambda }{\mathcal{M}_k}({\Gamma _0}(\mathcal{N},{\mathcal{I}_\lambda }),{\chi _\mathcal{N}})$ which is an eigenform for the full Hecke algebra. Using this lift, we develop the newform theory for a space of cusp forms ${\mathcal{S}_k}({\Gamma _0}(\mathcal{N},\mathcal{I}),{\chi _\mathcal{N}})$; we also use theta series to construct eigenforms for the full Hecke algebra.