The structure of a nonlinear elliptic operator
P. T.
Church;
E. N.
Dancer;
J. G.
Timourian
1-42
Abstract: Consider the nonlinear Dirichlet problem $(1) - \Delta u - \lambda u + {u^3} = g$, for $u:\Omega \to \mathbb{R}$, $u\vert\partial \Omega = 0$, and $\Omega \subset {\mathbb{R}^n}$ connected and bounded, and let $ {\lambda _i}$ be the $ i$th eigenvalue of $- \Delta u$ on $\Omega$ with $ u\vert\partial \Omega = 0$, $ (i = 1,2, \ldots )$. Define a map ${A_\lambda }:H \to H\prime $ by $ {A_\lambda }(u) = - \Delta u - \lambda u + {u^3}$, for either the Sobolev space $ W_0^{1,2}(\Omega ) = H = H\prime$ (if $n \leq 4)$ or the Hölder spaces $ C_0^{2,\alpha }(\bar \Omega ) = H$ and ${C^{0,\alpha }}(\bar \Omega ) = H\prime$ (if $\partial \Omega $ is ${C^{2,\alpha }}$ ), and define $ A:H \times \mathbb{R} \to H\prime \times \mathbb{R}$ by $A(u,\lambda ) = ({A_\lambda }(u),\lambda )$. Let $ G:{\mathbb{R}^2} \times E \to {\mathbb{R}^2} \times E$ be the global cusp map given by $ G(s,t,v) = ({s^3} - ts,t,v)$, and let $F:\mathbb{R} \times E \to \mathbb{R} \times E$ be the global fold map given by $F(t,v) = ({t^2},v)$, where $E$ is any Fréchet space. Theorem 1. If $H = H\prime = W_0^{1,2}(\Omega )$, assume in addition that $ n \leqslant 3$. There exit $\varepsilon > 0$ and homeomorphisms $ \alpha$ and $ \beta$ such that the following diagram commutes:
Equiconvergence theorems for Fourier-Bessel expansions with applications to the harmonic analysis of radial functions in Euclidean and non-Euclidean spaces
Leonardo
Colzani;
Antonio
Crespi;
Giancarlo
Travaglini;
Marco
Vignati
43-55
Abstract: We shall prove an equiconvergence theorem between Fourier-Bessel expansions of functions in certain weighted Lebesgue spaces and the classical cosine Fourier expansions of suitable related functions. These weighted Lebesgue spaces arise naturally in the harmonic analysis of radial functions on euclidean spaces and we shall use the equiconvergence result to deduce sharp results for the pointwise almost everywhere convergence of Fourier integrals of radial functions in the Lorentz spaces $ {L^{p,q}}({{\mathbf{R}}^n})$. Also we shall briefly apply the above approach to the study of the harmonic analysis of radial functions on noneuclidean hyperbolic spaces.
Generalizations of the wave equation
J. Marshall
Ash;
Jonathan
Cohen;
C.
Freiling;
Dan
Rinne
57-75
Abstract: The main result of this paper is a generalization of the property that, for smooth $u$, ${u_{xy}} = 0$ implies $(\ast)$ $\displaystyle u(x,y) = a(x) + b(y).$ Any function having generalized unsymmetric mixed partial derivative identically zero is of the form $(\ast)$. There is a function with generalized symmetric mixed partial derivative identically zero not of the form $(\ast)$, but $(\ast)$ does follow here with the additional assumption of continuity. These results connect to the theory of uniqueness for multiple trigonometric series. For example, a double trigonometric series is the $ {L^2}$ generalized symmetric mixed partial derivative of its formal $ (x,y)$-integral.
Andr\'e permutations, lexicographic shellability and the $cd$-index of a convex polytope
Mark
Purtill
77-104
Abstract: The $cd$-index of a polytope was introduced by Fine; it is an integer valued noncommutative polynomial obtained from the flag-vector. A result of Bayer and Fine states that for any integer "flag-vector," the existence of the $cd$-index is equivalent to the holding of the generalized Dehn-Sommerville equations of Bayer and Billera for the flag-vector. The coefficients of the $ cd$-index are conjectured to be nonnegative. We show a connection between the $cd$-index of a polytope $ \mathcal{P}$ and any $ CL$-shelling of the lattice of faces of $ \mathcal{P}$ ; this enables us to prove that each André polynomial of Foata and Schützenberger is the $cd$-index of a simplex. The combinatorial interpretation of this $cd$-index can be extended to cubes, simplicial polytopes, and some other classes (which implies that the $cd$-index has nonnegative coefficients for these polytopes). In particular, we show that any polytope of dimension five or less has a positive $ cd$-index.
Super efficiency in vector optimization
J. M.
Borwein;
D.
Zhuang
105-122
Abstract: We introduce a new concept of efficiency in vector optimization. This concept, super efficiency, is shown to have many desirable properties. In particular, we show that in reasonable settings the super efficient points of a set are norm-dense in the efficient frontier. We also provide a Chebyshev characterization of super efficient points for nonconvex sets and a scalarization theory when the underlying set is convex.
On the characterization of a Riemann surface by its semigroup of endomorphisms
A.
Erëmenko
123-131
Abstract: Suppose $ {D_1}$ and ${D_2}$ be Riemann surfaces which have bounded nonconstant holomorphic functions. Denote by $E({D_i})$, $i = 1,2$, the semigroups of all holomorphic endomorphisms. If $ \phi :E({D_1}) \to E({D_2})$ is an isomorphism of semigroups then there exists a conformal or anticonformal isomorphism $\psi :{D_1} \to {D_2}$ such that $ \phi$ is the conjugation by $\psi$. Also the semigroup of injective endomorphisms as well as some parabolic surfaces are considered.
Asymptotic behavior for a coalescence problem
Oscar
Bruno;
Avner
Friedman;
Fernando
Reitich
133-158
Abstract: Consider spherical particles of volume $x$ having paint on a fraction $y$ of their surface area. The particles are assumed to be homogeneously distributed at each time $t$, so that one can introduce the density number $ n(x,y,t)$. When collision between two particles occurs, the particles will coalesce if and only if they happen to touch each other, at impact, at points which do not belong to the painted portions of their surfaces. Introducing a dynamics for this model, we study the evolution of $n(x,y,t)$ and, in particular, the asymptotic behavior of the mass $ xn(x,y,t)dx$ as $t \to \infty$.
Rigidity of invariant complex structures
Isabel
Dotti Miatello
159-172
Abstract: A Kähler solvmanifold is a connected Kähler manifold $ (M,j,\left\langle , \right\rangle )$ admitting a transitive solvable group of automorphisms. In this paper we study the isomorphism classes of Kähler structures $(j,\left\langle , \right\rangle )$ turning $ M$ into a Kähler solvmanifold. In the case when $(M,j,\left\langle , \right\rangle )$ is irreducible and simply connected we show that any Kähler structure on $M$, having the same group of automorphisms, is isomorphic to $(j,\left\langle , \right\rangle )$.
The weighted Hardy's inequality for nonincreasing functions
Vladimir D.
Stepanov
173-186
Abstract: The purpose of this paper is to give an alternative proof of recent results of M. Arino and B. Muckenhoupt [1] and E. Sawyer [8], concerning Hardy's inequality for nonincreasing functions and related applications to the boundedness of some classical operators on general Lorentz spaces. Our approach will extend the results of [1,8] to the values of the parameters which are inaccessible by the methods of these papers.
The Noetherian property in rings of integer-valued polynomials
Robert
Gilmer;
William
Heinzer;
David
Lantz
187-199
Abstract: Let $D$ be a Noetherian domain, $ D\prime$ its integral closure, and $ \operatorname{Int}(D)$ its ring of integer-valued polynomials in a single variable. It is shown that, if $D\prime$ has a maximal ideal $M\prime$ of height one for which $D\prime /M\prime$ is a finite field, then $\operatorname{Int}(D)$ is not Noetherian; indeed, if $ M\prime$ is the only maximal ideal of $D\prime$ lying over $M\prime \cap D$, then not even $ \operatorname{Spec}(\operatorname{Int}(D))$ is Noetherian. On the other hand, if every height-one maximal ideal of $ D\prime$ has infinite residue field, then a sufficient condition for $ \operatorname{Int}(D)$ to be Noetherian is that the global transform of $ D$ is a finitely generated $ D$-module.
Complex geodesics and iterates of holomorphic maps on convex domains in ${\bf C}\sp n$
Peter R.
Mercer
201-211
Abstract: We study complex geodesics $ f:\Delta \to \Omega$, where $\Delta$ is the unit disk in ${\mathbf{C}}$ and $\Omega$ belongs to a class of bounded convex domains in $ {{\mathbf{C}}^n}$ with no boundary regularity assumption. Along with continuity up to the boundary, existence of such complex geodesics with two prescribed values $z$, $ w \in \bar \Omega$ is established. As a consequence we obtain some new results from iteration theory of holomorphic self maps of bounded convex domains in $ {{\mathbf{C}}^n}$.
Zeta regularized products
J. R.
Quine;
S. H.
Heydari;
R. Y.
Song
213-231
Abstract: If ${\lambda _k}$ is a sequence of nonzero complex numbers, then we define the zeta regularized product of these numbers, $\prod\nolimits_k {{\lambda _k}}$, to be $\exp ( - Z\prime (0))$ where $ Z(s) = \sum\nolimits_{k = 0}^\infty {\lambda _k^{ - s}} $. We assume that $ Z(s)$ has analytic continuation to a neighborhood of the origin. If ${\lambda _k}$ is the sequence of positive eigenvalues of the Laplacian on a manifold, then the zeta regularized product is known as $\det \prime \Delta$, the determinant of the Laplacian, and $\prod\nolimits_k {({\lambda _k} - \lambda )}$ is known as the functional determinant. The purpose of this paper is to discuss properties of the determinant and functional determinant for general sequences of complex numbers. We discuss asymptotic expansions of the functional determinant as $\lambda \to - \infty$ and its relationship to the Weierstrass product. We give some applications to the theory of Barnes' multiple gamma functions and elliptic functions. A new proof is given for Kronecker's limit formula and the product expansion for Barnes' double Stirling modular constant.
Sets of determination for harmonic functions
Stephen J.
Gardiner
233-243
Abstract: Let $h$ denote a positive harmonic function on the open unit ball $B$ of Euclidean space ${{\mathbf{R}}^n}\;(n \geq 2)$. This paper characterizes those subsets $E$ of $B$ for which ${\sup _E}H/h = {\sup _B}H/h$ or ${\inf _E}H/h = {\inf _B}H/h$ for all harmonic functions $H$ belonging to a specified class. In this regard we consider the classes of positive harmonic functions, differences of positive harmonic functions, and harmonic functions with a one-sided quasi-boundedness condition. We also consider the closely related question of representing functions on the sphere $\partial B$ as sums of Poisson kernels corresponding to points in $E$.
Chains, null-chains, and CR geometry
Lisa K.
Koch
245-261
Abstract: A system of distinguished curves distinct from chains is defined on indefinite nondegenerate $ {\text{CR}}$ hypersurfaces; the new curves are called null-chains. The properties of these curves are explored, and it is shown that two sufficiently nearby points of any nondegenerate ${\text{CR}}$ hypersurface can be connected by either a chain or a null-chain.
Intersection cohomology of $S\sp 1$-actions
Gilbert
Hector;
Martin
Saralegi
263-288
Abstract: Given a free action of the circle $ {{\mathbf{S}}^1}$ on a differentiable manifold $M$, there exists a long exact sequence that relates the cohomology of $M$ with the cohomology of the manifold $M/{{\mathbf{S}}^1}$. This is the Gysin sequence. This result is still valid if we allow the action to have stationary points. In this paper we are concerned with actions where fixed points are allowed. Here the quotient space $ M/{{\mathbf{S}}^1}$ is no longer a manifold but a stratified pseudomanifold (in terms of Goresky and MacPherson). We get a similar Gysin sequence where the cohomology of $M/{{\mathbf{S}}^1}$ is replaced by its intersection cohomology. As in the free case, the connecting homomorphism is given by the product with the Euler class $ [e]$. Also, the vanishing of this class is related to the triviality of the action. In this Gysin sequence we observe the phenomenon of perversity shifting. This is due to the allowability degree of the Euler form.
On complete manifolds of nonnegative $k$th-Ricci curvature
Zhong Min
Shen
289-310
Abstract: In this paper we establish some vanishing and finiteness theorems for the topological type of complete open riemannian manifolds under certain positivity conditions for curvature. Key tools are comparison techniques and Morse Theory of Busemann and distance functions.
Global phase structure of the restricted isosceles three-body problem with positive energy
Kenneth
Meyer;
Qiu Dong
Wang
311-336
Abstract: We study a restricted three-body problem with special symmetries: the restricted isosceles three-body problem. For positive energy the energy manifold is partially compactified by adding boundary manifolds corresponding to infinity and triple collision. We use a new set of coordinates which are a variation on the McGehee coordinates of celestial mechanics. These boundary manifolds are used to study the global phase structure of this gradational system. The orbits are classified by intersection number, that is the number of times the infinitesimal body cross the line of syzygy before escaping to infinity.
Local integrability of Mizohata structures
Jorge
Hounie;
Pedro
Malagutti
337-362
Abstract: In this work we study the local integrability of strongly pseudoconvex Mizohata structures of rank $n > 2$ (and co-rank $1$). These structures are locally generated in an appropriate coordinate system $({t_1}, \ldots ,{t_n},x)$ by flat perturbations of Mizohata vector fields ${M_j} = \frac{\partial } {{\partial {t_j}}} - i{t_j}\frac{\partial } {{\partial x}}$, $j = 1, \ldots ,n$. For this, we first prove the global integrability of small perturbations of the structure generated by $\frac{\partial } {{\partial \bar z}} + {\sigma _1}\frac{\partial } {{\partial z}}$, $\frac{\partial } {{\partial {\theta _{n - 1}}}} + {\sigma _j}\frac{\partial } {{\partial z}}$, $ j = 2, \ldots ,n$, defined over a manifold $ {\mathbf{C}} \times S$, where $S$ is simply connected.
Inverse scattering for singular potentials in two dimensions
Zi Qi
Sun;
Gunther
Uhlmann
363-374
Abstract: We consider the Schrödinger equation for a compactly supported potential having jump type singularities at a subdomain of $ {\mathbb{R}^2}$. We prove that knowledge of the scattering amplitude at a fixed energy, determines the location of the singularity as well as the jump across the curve of discontinuity. This result follows from a similar result for the Dirichlet to Neumann map associated to the Schrödinger equation for a compactly supported potential with the same type of singularities.
Twists of Hilbert modular forms
Thomas R.
Shemanske;
Lynne H.
Walling
375-403
Abstract: The theory of newforms for Hilbert modular forms is summarized including a statement of a strong multiplicity-one theorem and a characterization of newforms as eigenfunctions for a certain involution whose Dirichlet series has a prescribed Euler product. The general question of twisting Hilbert modular newforms by arbitrary Hecke characters is considered and the exact level of a character twist of a Hilbert modular form is determined. Conditions under which the twist of a newform is a newform are given. Applications include a strengthening in the elliptic modular case of a theorem of Atkin and Li's regarding the characterization of imprimitive newforms as well as its generalization to the Hilbert modular case, and a decomposition theorem for certain spaces of newforms as the direct sum of twists of spaces of newforms of lower level.
Regularity properties of solutions to transmission problems
Luis
Escauriaza;
Jin Keun
Seo
405-430
Abstract: We show that the gradients of solutions to certain elliptic and parabolic transmission problems with internal Lipschitz boundary and constant coefficients at each side of the internal boundary are square integrable along the internal boundary.
Characterization of automorphisms on the Barrett and the Diederich-Forn\ae ss worm domains
So-Chin
Chen
431-440
Abstract: In this paper we show that every automorphism on either the Barrett or the Diederich-Fornaess worm domains is given by a rotation in $w$-variable. In particular, any automorphism on either one of these two domains can be extended smoothly up to the boundary.
Classification of singularities for blowing up solutions in higher dimensions
J. J. L.
Velázquez
441-464
Abstract: Consider the Cauchy problem (P) $\displaystyle \left\{ {\begin{array}{*{20}{c}} {{u_t} - \Delta u = {u^p}} ... ...text{when}}\;x \in {\mathbb{R}^N},} & {} \end{array} } \right.$ where $p > 1$, and ${u_0}(x)$ is a continuous, nonnegative and bounded function. It is known that, under fairly general assumptions on ${u_0}(x)$, the unique solution of $({\text{P}})$, $u(x,t)$, blows up in a finite time, by which we mean that $\displaystyle \mathop {\lim \sup }\limits_{t \uparrow T} \left( {\mathop {\sup }\limits_{x \in {\mathbb{R}^N}} \;u(x,t)} \right) = + \infty .$ In this paper we shall assume that $u(x,t)$ blows up at $x = 0$, $ t = T < + \infty$ , and derive the possible asymptotic behaviours of $ u(x,t)$ as $(x,t) \to (0,T)$, under general assumptions on the blow-up rate.
Imprimitive Gaussian sums and theta functions over a number field
Jacob
Nemchenok
465-478
Abstract: We obtain a reduction formula for an imprimitive Gaussian sum with a numerical character in an algebraic number field, i.e. a formula that expresses that sum as a product of several elementary factors times a primitive, proper, normed Gaussian sum (formulae (16) and (19)). We also introduce Gaussian sums with Hecke characters and derive a similar reduction formula for them. The derivation is based on an inversion formula for a multivariable theta function associated with the number field, twisted with the numerical character.
Mixing properties of a class of Bernoulli-processes
Doris
Fiebig
479-493
Abstract: We prove that stationary very weak Bernoulli processes with rate $ O(1/n)\;({\text{VWB}}\,O(1/n))$ are strictly very weak Bernoulli with rate $ O(1/n)$. Furthermore we discuss the relation between ${\text{VWB}}\;O(1/n)$ and the classical mixing properties for countable state processes. In particular, we show that $ {\text{VWB}}\,O(1/n)$ implies $\phi$-mixing.