Weakly repelling fixpoints and the connectivity of wandering domains
Walter
Bergweiler;
Norbert
Terglane
1-12
Abstract: It is proved that if a transcendental meromorphic function $f$ has a multiply-connected wandering domain, then $f$ has a fixpoint $z_0$ such that $|f'(z_0)|>1$ or $f'(z_0)=1$. Entire functions with a multiply-connected wandering domain have infinitely many such fixpoints. These results are used to show that solutions of certain differential equations do not have wandering domains at all.
Invertibility preserving linear maps on $\mathcal{L}(X)$
A.
R.
Sourour
13-30
Abstract: For Banach spaces $X$ and $Y$, we show that every unital bijective invertibility preserving linear map between $\mathcal L(X)$ and $\mathcal L(Y)$ is a Jordan isomorphism. The same conclusion holds for maps between $\mathbb CI+ \mathcal K(X)$ and $\mathbb CI+\mathcal K(Y)$.
The space of invariant functions on a finite Lie algebra
G.
I.
Lehrer
31-50
Abstract: We show that the operations of Fourier transform and duality on the space of adjoint-invariant functions on a finite Lie algebra commute with each other. This result is applied to give formulae for the Fourier transform of a ``Brauer function''---i.e. one whose value at $X$ depends only on the semisimple part $X_s$ of $X$ and for the dual of the convolution of any function with the Steinberg function. Geometric applications include the evaluation of the characters of the Springer representations of Weyl groups and the study of the equivariant cohomology of local systems on $G/T$, where $T$ is a maximal torus of the underlying reductive group $G$.
Measure-Valued Solutions to Initial-Boundary Value Problems for Certain Systems of Conservation Laws: Existence and Dynamics
Hermano
Frid
51-76
Abstract: A framework for studying initial-boundary value problems for systems of conservation laws, in what concerns to the existence of measure-valued solutions and their asymptotic behavior, is developed here with the helpful introduction of a class of flux maps which allow a rather complete treatment of these questions including systems of practical importance as those arising in multiphase flow in porous media. The systems of this class may, in general, admit umbilic points, submanifolds where genuine nonlinearity fails, as well as elliptic regions. We prove the existence of measure-valued solutions by using the vanishing viscosity method and, also, finite difference schemes. The main result about the dynamics of the measure-valued solutions is that for certain special boundary values, given by constant states, the time-averages of these m-v solutions converge weakly to the Dirac measure concentrated at those states, for a.e. space variable. The rate of convergence of the time-averages of the expected values can be estimated by properties of the flux maps only.
Polynomial structures for nilpotent groups
Karel
Dekimpe;
Paul
Igodt;
Kyung
Bai
Lee
77-97
Abstract: If a polycyclic-by-finite rank-$K$ group $\Gamma$ admits a faithful affine representation making it acting on $\Bbb R^{K}$ properly discontinuously and with compact quotient, we say that $\Gamma$ admits an affine structure. In 1977, John Milnor questioned the existence of affine structures for such groups $\Gamma$. Very recently examples have been obtained showing that, even for torsion-free, finitely generated nilpotent groups $N$, affine structures do not always exist. It looks natural to consider affine structures as examples of polynomial structures of degree one. We introduce the concept of a canonical type polynomial structure for polycyclic-by-finite groups. Using the algebraic framework of the Seifert Fiber Space construction and a nice cohomology vanishing theorem, we prove the existence and uniqueness (up to conjugation) of canonical type polynomial structures for virtually finitely generated nilpotent groups. Applying this uniqueness to a result going back to Mal$'$cev, it follows that, for torsion-free, finitely generated nilpotent groups, each canonical polynomial structure is expressed in polynomials of limited degree. The minimal degree needed for obtaining a polynomial structure will determine the ``affine defect number''. We prove that the known counterexamples to Milnor's question have the smallest possible affine defect, i.e. affine defect number equal to one.
Nonnegative Radix Representations for the Orthant $R^n_+$
Jeffrey
C.
Lagarias;
Yang
Wang
99-117
Abstract: Let $A$ be a nonnegative real matrix which is expanding, i.e. with all eigenvalues $|\lambda| > 1$, and suppose that $|\det(A)|$ is an integer. Let ${\mathcal D}$ consist of exactly $|\det(A)|$ nonnegative vectors in $\R^n$. We classify all pairs $(A, {\mathcal D})$ such that every $x$ in the orthant $\R^n_+$ has at least one radix expansion in base $A$ using digits in ${\mathcal D}$. The matrix $A$ must be a diagonal matrix times a permutation matrix. In addition $A$ must be similar to an integer matrix, but need not be an integer matrix. In all cases the digit set $\mathcal D$ can be diagonally scaled to lie in $\Z^n$. The proofs generalize a method of Odlyzko, previously used to classify the one--dimensional case.
Gelfand-Kirillov dimension in Jordan Algebras
C.
Martinez
119-126
Abstract: In this paper we study Gelfand-Kirillov dimension in Jordan algebras. In particular we will relate Gelfand-Kirillov (GK for short) dimensions of a special Jordan algebra and its associative enveloping algebra and also the GK dimension of a Jordan algebra and the GK dimension of its universal multiplicative enveloping algebra.
Complex convexity in Lebesgue-Bochner Function Spaces
Patrick
N.
Dowling;
Zhibao
Hu;
Douglas
Mupasiri
127-139
Abstract: Complex geometric properties of continuously quasi-normed spaces are introduced and their relationship to their analogues in real Banach spaces is discussed. It is shown that these properties lift from a continuously quasi-normed space $X$ to $L^p(\mu , X)$, for $0 < p < \infty$. Local versions of these properties and results are also considered.
Subgroups of $GL(n^2, \bold C)$ containing $PSU(n)$
V.
P.
Platonov;
D.
Z.
Ðokovic
141-152
Abstract: Let ${\operatorname{PSU}}(n)$ be the image of the unitary group ${\operatorname{U}}(n)$ under the representation $x\to axa^{-1}$ on the space $M_n({\mathbf C} )$ of $n$ by $n$ complex matrices. We classify all connected Lie subgroups of ${\operatorname{GL}}(n^2,{\mathbf C} )$ containing ${\operatorname{PSU}}(n)$. We use this result to obtain a description of all abstract overgroups of ${\operatorname{PSU}}(n)$ in ${\operatorname{GL}}(n^2,{\mathbf C} )$. We apply this classification to solve the problem of describing all invertible linear transformations of $M_n({\mathbf C} )$ which preserve the set of normal matrices. Our results can be applied to solve many other problems of similar nature.
Some New Homogeneous Einstein Metrics on Symmetric Spaces
Megan
M.
Kerr
153-171
Abstract: We classify homogeneous Einstein metrics on compact irreducible symmetric spaces. In particular, we consider symmetric spaces with rank$(M)> 1$, not isometric to a compact Lie group. Whenever there exists a closed proper subgroup $G$ of Isom$(M)$ acting transitively on $M$ we find all $G$-homogeneous (non-symmetric) Einstein metrics on $M$.
On the variances of occupation times of conditioned Brownian motion
Biao
Zhang
173-185
Abstract: We extend some bounds on the variance of the lifetime of two--dimensional Brownian motion, conditioned to exit a planar domain at a given point, to certain domains in higher dimensions. We also give a short ``analytic'' proof of some existing results.
Failure of normality in the box product of uncountably many real lines
L.
Brian
Lawrence
187-203
Abstract: We prove in ZFC that the box product of $\omega_1$ many copies of $\omega+1$ is neither normal nor collectionwise Hausdorff. As an addendum to the proof, we show that if the cardinality of the continuum is $2^{\omega_1}$, then these properties also fail in the closed subspace consisting of all functions which assume the value $\omega$ on all but countably many indices.
A Concordance Extension Theorem
Joel
L.
Jones
205-218
Abstract: Let $p\:E\to B$ be a manifold approximate fibration between closed manifolds, where $\dim(E)\ge 4$, and let $M(p)$ be the mapping cylinder of $p$. In this paper it is shown that if $g\colon B\times I\to B\times I$ is any concordance on $B$, then there exists a concordance $G\:M(p)\times I \to M(p)\times I$ such that $G|B\times I=g$ and $G|E\times\{0\}\times I= id_{E\times I}$. As an application, if $N^n$ and $M^{n+j}$ are closed manifolds where $N$ is a locally flat submanifold of $M$ and $n\ge 5$ and $j\ge 1$, then a concordance $g\colon N\times I\to N\times I$ extends to a concordance $G\:M\times I\to M\times I$ on $M$ such that $G|N\times I=g$. This uses the fact that under these hypotheses there exists a manifold approximate fibration $p\colon E\to N$, where $E$ is a closed $(n+j-1)$-manifold, such that the mapping cylinder $M(p)$ is homeomorphic to a closed neighborhood of $N$ in $M$ by a homeomorphism which is the identity on $N$.
A problem of Foldes and Puri on the Wiener process
Z.
Shi
219-228
Abstract: Let $W$ be a real-valued Wiener process starting from 0, and $\tau (t)$ be the right-continuous inverse process of its local time at 0. Földes and Puri [3] raise the problem of studying the almost sure asymptotic behavior of $X(t)=\int _0^{\tau (t)} {\text{\bf 1}\hskip -1.25pt\mathrm{l}}_{\{ | W(u)| \le \alpha t\} }du$ as $t$ tends to infinity, i.e. they ask: how long does $W$ stay in a tube before ``crossing very much" a given level? In this note, both limsup and liminf laws of the iterated logarithm are provided for $X$.
Second Root Vectors for Multiparameter Eigenvalue Problems of Fredholm Type
Paul
Binding;
Tomaz
Kosir
229-249
Abstract: A class of multiparameter eigenvalue problems involving (generally) non self-adjoint and unbounded operators is studied. A basis for the second root subspace, at eigenvalues of Fredholm type, is computed in terms of the underlying multiparameter system. A self-adjoint version of this result is given under a weak definiteness condition, and Sturm-Liouville and finite-dimensional examples are considered.
Uniform harmonic approximation of bounded functions
Stephen
J.
Gardiner
251-265
Abstract: Let $\Omega$ be an open set in $\mathbb R^n$ and $E$ be a relatively closed subset of $\Omega$. We characterize those pairs $(\Omega,E)$ which have the following property: every function which is bounded and continuous on $E$ and harmonic on $E^0$ can be uniformly approximated by functions harmonic on $\Omega$. Several related results concerning both harmonic and superharmonic approximation are also established.
Global smooth solutions for a class of parabolic integrodifferential equations
Hans
Engler
267-290
Abstract: The existence and uniqueness of smooth global large data solutions of a class of quasilinear partial integrodifferential equations in one space and one time dimension are proved, if the integral kernel behaves like $t^{-\alpha }$ near $t=0$ with $\alpha > 2/3$. An existence and regularity theorem for linear equations with variable coefficients that are related to this type is also proved in arbitrary space dimensions and with no restrictions for $\alpha$.
Powers in Finitely Generated Groups
E.
Hrushovski;
P.
H.
Kropholler;
A.
Lubotzky;
A.
Shalev
291-304
Abstract: In this paper we study the set $\G^n$ of $n^{th}$-powers in certain finitely generated groups $\G$. We show that, if $\G$ is soluble or linear, and $\G^n$ contains a finite index subgroup, then $\G$ is nilpotent-by-finite. We also show that, if $\G$ is linear and $\G^n$ has finite index (i.e. $\G$ may be covered by finitely many translations of $\G^n$), then $\G$ is soluble-by-finite. The proof applies invariant measures on amenable groups, number-theoretic results concerning the $S$-unit equation, the theory of algebraic groups and strong approximation results for linear groups in arbitrary characteristic.
On the Well-Posedness of the Kirchhoff String
Alberto
Arosio;
Stefano
Panizzi
305-330
Abstract: Let us consider the Cauchy problem for the quasilinear hyperbolic integro-differential equation \begin{displaymath}% {% \begin{array}{ll} u_{tt}-m \left(\g{ \int_{_{\p{\Omega}}}} |\bigtriangledown_{x}u|^{2} \, dx \right) \bigtriangleup_{x}u= f(x,t) \: & \,(x\in \, \Omega, \, t \G 0),\qquad\qquad\qquad u(\cdot ,t)_{|\partial\Omega} =0 &\,(t\, \geq \,0), \end{array} } \end{displaymath} where $\; \Omega \;$ is an open subset of $\; \Reali^{n} \;$ and $\, m \,$ is a positive function of one real variable which is continuously differentiable. We prove the well-posedness in the Hadamard sense (existence, uniqueness and continuous dependence of the local solution upon the initial data) in Sobolev spaces of low order.
Defect zero blocks for finite simple groups
Andrew
Granville;
Ken
Ono
331-347
Abstract: We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a $p$-block with defect 0, completing an investigation of many authors. The only finite simple groups whose defect zero $p-$blocks remained unclassified were the alternating groups $A_{n}$. Here we show that these all have a $p$-block with defect 0 for every prime $p\geq 5$. This follows from proving the same result for every symmetric group $S_{n}$, which in turn follows as a consequence of the $t$-core partition conjecture, that every non-negative integer possesses at least one $t$-core partition, for any $t\geq 4$. For $t\geq 17$, we reduce this problem to Lagrange's Theorem that every non-negative integer can be written as the sum of four squares. The only case with $t<17$, that was not covered in previous work, was the case $t=13$. This we prove with a very different argument, by interpreting the generating function for $t$-core partitions in terms of modular forms, and then controlling the size of the coefficients using Deligne's Theorem (née the Weil Conjectures). We also consider congruences for the number of $p$-blocks of $S_{n}$, proving a conjecture of Garvan, that establishes certain multiplicative congruences when $5\leq p \leq 23$. By using a result of Serre concerning the divisibility of coefficients of modular forms, we show that for any given prime $p$ and positive integer $m$, the number of $p-$blocks with defect 0 in $S_n$ is a multiple of $m$ for almost all $n$. We also establish that any given prime $p$ divides the number of $p-$modularly irreducible representations of $S_{n}$, for almost all $n$.
Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators
F.
Gesztesy;
B.
Simon
349-373
Abstract: New unique characterization results for the potential $V(x)$ in connection with Schrödinger operators on $\mathbb{R}$ and on the half-line $[0,\infty )$ are proven in terms of appropriate Krein spectral shift functions. Particular results obtained include a generalization of a well-known uniqueness theorem of Borg and Marchenko for Schrödinger operators on the half-line with purely discrete spectra to arbitrary spectral types and a new uniqueness result for Schrödinger operators with confining potentials on the entire real line.
Negative Flows of the potential KP-hierarchy
Guido
Haak
375-390
Abstract: We construct a Grassmannian-like formulation for the potential KP-hierarchy including additional ``negative'' flows. Our approach will generalize the notion of a $\tau$-function to include negative flows. We compare the resulting hierarchy with results by Hirota, Satsuma and Bogoyavlenskii.
The spaces of index one minimal surfaces and stable constant mean curvature surfaces embedded in flat three manifolds
Manuel
Ritoré;
Antonio
Ros
391-410
Abstract: It is proved that the spaces of index one minimal surfaces and stable constant mean curvature surfaces with genus greater than one in (non fixed) flat three manifolds are compact in a strong sense: given a sequence of any of the above surfaces we can extract a convergent subsequence of both the surfaces and the ambient manifolds in the $C^k$ topology. These limits preserve the topological type of the surfaces and the affine diffeomorphism class of the ambient manifolds. Some applications to the isoperimetric problem are given.