Cuntz-Krieger algebras and endomorphisms of finite direct sums of type I$_{\infty }$ factors
Berndt
Brenken
3835-3873
Abstract: A correspondence between algebra endomorphisms of a finite sum of copies of the algebra of all bounded operators on a Hilbert space and representations of certain norm closed $\ast$-subalgebras of bounded operators generated by a finite collection of partial isometries is introduced. Basic properties of this correspondence are investigated after developing some operations on bipartite graphs that usefully describe aspects of this relationship.
Generalized subdifferentials: a Baire categorical approach
Jonathan
M.
Borwein;
Warren
B.
Moors;
Xianfu
Wang
3875-3893
Abstract: We use Baire categorical arguments to construct pathological locally Lipschitz functions. The origins of this approach can be traced back to Banach and Mazurkiewicz (1931) who independently used similar categorical arguments to show that ``almost every continuous real-valued function defined on [0,1] is nowhere differentiable". As with the results of Banach and Mazurkiewicz, it appears that it is easier to show that almost every function possesses a certain property than to construct a single concrete example. Among the most striking results contained in this paper are: Almost every 1-Lipschitz function defined on a Banach space has a Clarke subdifferential mapping that is identically equal to the dual ball; if $\{T_{1}, T_{2},\ldots, T_{n}\}$ is a family of maximal cyclically monotone operators defined on a Banach space $X$ then there exists a real-valued locally Lipschitz function $g$such that $\partial_{0}g(x)=\mbox{co}\{T_{1}(x),T_{2}(x),\ldots, T_{n}(x)\}$for each $x\in X$; in a separable Banach space each non-empty weak$^{*}$compact convex subset in the dual space is identically equal to the approximate subdifferential mapping of some Lipschitz function and for locally Lipschitz functions defined on separable spaces the notions of strong and weak integrability coincide.
Szlenk indices and uniform homeomorphisms
G.
Godefroy;
N.
J.
Kalton;
G.
Lancien
3895-3918
Abstract: We prove some rather precise renorming theorems for Banach spaces with Szlenk index $\omega_0.$ We use these theorems to show the invariance of certain quantitative Szlenk-type indices under uniform homeomorphisms.
Variational principles and mixed multifractal spectra
L.
Barreira;
B.
Saussol
3919-3944
Abstract: We establish a ``conditional'' variational principle, which unifies and extends many results in the multifractal analysis of dynamical systems. Namely, instead of considering several quantities of local nature and studying separately their multifractal spectra we develop a unified approach which allows us to obtain all spectra from a new multifractal spectrum. Using the variational principle we are able to study the regularity of the spectra and the full dimensionality of their irregular sets for several classes of dynamical systems, including the class of maps with upper semi-continuous metric entropy. Another application of the variational principle is the following. The multifractal analysis of dynamical systems studies multifractal spectra such as the dimension spectrum for pointwise dimensions and the entropy spectrum for local entropies. It has been a standing open problem to effect a similar study for the ``mixed'' multifractal spectra, such as the dimension spectrum for local entropies and the entropy spectrum for pointwise dimensions. We show that they are analytic for several classes of hyperbolic maps. We also show that these spectra are not necessarily convex, in strong contrast with the ``non-mixed'' multifractal spectra.
Stochastic processes with sample paths in reproducing kernel Hilbert spaces
Milan
N.
Lukic;
Jay
H.
Beder
3945-3969
Abstract: A theorem of M. F. Driscoll says that, under certain restrictions, the probability that a given Gaussian process has its sample paths almost surely in a given reproducing kernel Hilbert space (RKHS) is either $0$ or $1$. Driscoll also found a necessary and sufficient condition for that probability to be $1$. Doing away with Driscoll's restrictions, R. Fortet generalized his condition and named it nuclear dominance. He stated a theorem claiming nuclear dominance to be necessary and sufficient for the existence of a process (not necessarily Gaussian) having its sample paths in a given RKHS. This theorem - specifically the necessity of the condition - turns out to be incorrect, as we will show via counterexamples. On the other hand, a weaker sufficient condition is available. Using Fortet's tools along with some new ones, we correct Fortet's theorem and then find the generalization of Driscoll's result. The key idea is that of a random element in a RKHS whose values are sample paths of a stochastic process. As in Fortet's work, we make almost no assumptions about the reproducing kernels we use, and we demonstrate the extent to which one may dispense with the Gaussian assumption.
Products of polynomials in uniform norms
Igor
E.
Pritsker
3971-3993
Abstract: We study inequalities connecting a product of uniform norms of polynomials with the norm of their product. This subject includes the well known Gel'fond-Mahler inequalities for the unit disk and Kneser inequality for the segment $[-1,1]$. Using tools of complex analysis and potential theory, we prove a sharp inequality for norms of products of algebraic polynomials over an arbitrary compact set of positive logarithmic capacity in the complex plane. The above classical results are contained in our theorem as special cases. It is shown that the asymptotically extremal sequences of polynomials, for which this inequality becomes an asymptotic equality, are characterized by their asymptotically uniform zero distributions. We also relate asymptotically extremal polynomials to the classical polynomials with asymptotically minimal norms.
A Brunn-Minkowski inequality for the integer lattice
R.
J.
Gardner;
P.
Gronchi
3995-4024
Abstract: A close discrete analog of the classical Brunn-Minkowksi inequality that holds for finite subsets of the integer lattice is obtained. This is applied to obtain strong new lower bounds for the cardinality of the sum of two finite sets, one of which has full dimension, and, in fact, a method for computing the exact lower bound in this situation, given the dimension of the lattice and the cardinalities of the two sets. These bounds in turn imply corresponding new bounds for the lattice point enumerator of the Minkowski sum of two convex lattice polytopes. A Rogers-Shephard type inequality for the lattice point enumerator in the plane is also proved.
Genus $0$ and $1$ Hurwitz numbers: Recursions, formulas, and graph-theoretic interpretations
Ravi
Vakil
4025-4038
Abstract: We derive a closed-form expression for all genus 1 Hurwitz numbers, and give a simple new graph-theoretic interpretation of Hurwitz numbers in genus $0$ and $1$. (Hurwitz numbers essentially count irreducible genus $g$ covers of the sphere, with arbitrary specified branching over one point, simple branching over other specified points, and no other branching. The problem is equivalent to counting transitive factorisations of permutations into transpositions.) These results prove a conjecture of Goulden, Jackson and Vainshtein, and extend results of Hurwitz and many others.
A sharp bound on the size of a connected matroid
Manoel
Lemos;
James
Oxley
4039-4056
Abstract: This paper proves that a connected matroid $M$ in which a largest circuit and a largest cocircuit have $c$ and $c^*$ elements, respectively, has at most $\frac{1}{2}cc^*$ elements. It is also shown that if $e$ is an element of $M$ and $c_e$ and $c^*_e$ are the sizes of a largest circuit containing $e$ and a largest cocircuit containing $e$, then $\vert E(M)\vert \le (c_e -1)(c^*_e - 1) + 1$. Both these bounds are sharp and the first is proved using the second. The second inequality is an interesting companion to Lehman's width-length inequality which asserts that the former inequality can be reversed for regular matroids when $c_e$ and $c^*_e$ are replaced by the sizes of a smallest circuit containing $e$ and a smallest cocircuit containing $e$. Moreover, it follows from the second inequality that if $u$ and $v$ are distinct vertices in a $2$-connected loopless graph $G$, then $\vert E(G)\vert$ cannot exceed the product of the length of a longest $(u,v)$-path and the size of a largest minimal edge-cut separating $u$ from $v$.
Peripheral splittings of groups
B.
H.
Bowditch
4057-4082
Abstract: We define the notion of a ``peripheral splitting'' of a group. This is essentially a representation of the group as the fundamental group of a bipartite graph of groups, where all the vertex groups of one colour are held fixed--the ``peripheral subgroups''. We develop the theory of such splittings and prove an accessibility result. The theory mainly applies to relatively hyperbolic groups with connected boundary, where the peripheral subgroups are precisely the maximal parabolic subgroups. We show that if such a group admits a non-trivial peripheral splitting, then its boundary has a global cut point. Moreover, the non-peripheral vertex groups of such a splitting are themselves relatively hyperbolic. These results, together with results from elsewhere, show that under modest constraints on the peripheral subgroups, the boundary of a relatively hyperbolic group is locally connected if it is connected. In retrospect, one further deduces that the set of global cut points in such a boundary has a simplicial treelike structure.
Limits in the uniform ultrafilters
Joni
Baker;
Kenneth
Kunen
4083-4093
Abstract: Let $u(\kappa)$ be the space of uniform ultrafilters on $\kappa$. If $\kappa$ is regular, then there is an $\mathbf x \in u(\kappa)$which is not an accumulation point of any subset of $u(\kappa)$ of size $\kappa$ or less. $\mathbf x$ is also good, in the sense of Keisler.
Convergence of asymptotic directions
Dinh
The Luc;
Jean-Paul
Penot
4095-4121
Abstract: We study convergence properties of asymptotic directions of unbounded sets in normed spaces. The links between the continuity of a set-valued map and the convergence of asymptotic directions are examined. The results are applied to investigate continuity properties of marginal functions and asymptotic directions of level sets.
Conditional stability estimation for an inverse boundary problem with non-smooth boundary in $\mathcal{R}^3$
J.
Cheng;
Y.
C.
Hon;
M.
Yamamoto
4123-4138
Abstract: In this paper, we investigate an inverse problem of determining a shape of a part of the boundary of a bounded domain in $\mathcal R^3$ by a solution to a Cauchy problem of the Laplace equation. Assuming that the unknown part is a Lipschitz continuous surface, we give a logarithmic conditional stability estimate in determining the part of boundary under reasonably a priori information of an unknown part. The keys are the complex extension and estimates for a harmonic measure.
Hyperbolic conservation laws with stiff reaction terms of monostable type
Haitao
Fan
4139-4154
Abstract: In this paper the zero reaction limit of the hyperbolic conservation law with stiff source term of monostable type \begin{displaymath}\partial _{t} u + \partial _{x} f(u) = \frac {1}{\epsilon } u(1-u)\end{displaymath} is studied. Solutions of Cauchy problems of the above equation with initial value $0\le u_{0}(x)\le 1$ are proved to converge, as $\epsilon \to 0$, to piecewise constant functions. The constants are separated by either shocks determined by the Rankine-Hugoniot jump condition, or a non-shock jump discontinuity that moves with speed $f'(0)$. The analytic tool used is the method of generalized characteristics. Sufficient conditions for the existence and non-existence of traveling waves of the above system with viscosity regularization are given. The reason for the failure to capture the correct shock speed by first order shock capturing schemes when underresolving $\epsilon >0$ is found to originate from the behavior of traveling waves of the above system with viscosity regularization.
On the inverse spectral theory of Schrödinger and Dirac operators
Miklós
Horváth
4155-4171
Abstract: We prove that under some conditions finitely many partially known spectra and partial information on the potential entirely determine the potential. This extends former results of Hochstadt, Lieberman, Gesztesy, Simon and others.
Variational problems on multiply connected thin strips III: Integration of the Ginzburg-Landau equations over graphs
Jacob
Rubinstein;
Michelle
Schatzman
4173-4187
Abstract: We analyze the one-dimensional Ginzburg-Landau functional of superconductivity on a planar graph. In the Euler-Lagrange equations, the equation for the phase can be integrated, provided that the order parameter does not vanish at the vertices; in this case, the minimization of the Ginzburg-Landau functional is equivalent to the minimization of another functional, whose unknowns are a real-valued function on the graph and a finite set of integers.
Invariant distributions supported on the nilpotent cone of a semisimple Lie algebra
Thierry
Levasseur
4189-4202
Abstract: Let $\mathfrak{g}$ be a semisimple complex Lie algebra with adjoint group $G$ and $\mathcal{D}(\mathfrak{g})$ be the algebra of differential operators with polynomial coefficients on $\mathfrak{g}$. If $\mathfrak{g}_0$ is a real form of $\mathfrak{g}$, we give the decomposition of the semisimple $\mathcal{D}(\mathfrak{g})^G$-module of invariant distributions on $\mathfrak{g}_0$ supported on the nilpotent cone.
On the commutativity of the algebra of invariant differential operators on certain nilpotent homogeneous spaces
Hidénori
Fujiwara;
Gérard
Lion;
Salah
Mehdi
4203-4217
Abstract: Let $G$ be a simply connected connected real nilpotent Lie group with Lie algebra $\mathfrak{g}$, $H$ a connected closed subgroup of $G$ with Lie algebra $\mathfrak{h}$ and $\beta\in\mathfrak{h}^{*}$ satisfying $\beta ([\mathfrak{h},\mathfrak{h} ])=\{0\}$. Let $\chi_{\beta}$ be the unitary character of $H$ with differential $2\sqrt{-1}\pi\beta$ at the origin. Let $\tau\equiv$ $Ind_{H}^{G}\chi_{\beta}$ be the unitary representation of $G$ induced from the character $\chi_{\beta}$ of $H$. We consider the algebra $\mathcal{D}(G,H,\beta)$ of differential operators invariant under the action of $G$ on the bundle with basis $H\backslash G$ associated to these data. We consider the question of the equivalence between the commutativity of $\mathcal{D}(G,H,\beta)$ and the finite multiplicities of $\tau$. Corwin and Greenleaf proved that if $\tau$ is of finite multiplicities, this algebra is commutative. We show that the converse is true in many cases.
On the relation between upper central quotients and lower central series of a group
Graham
Ellis
4219-4234
Abstract: Let $H$ be a group with a normal subgroup $N$ contained in the upper central subgroup $Z_cH$. In this article we study the influence of the quotient group $G=H/N$ on the lower central subgroup $\gamma_{c+1}H$. In particular, for any finite group $G$ we give bounds on the order and exponent of $\gamma_{c+1}H$. For $G$ equal to a dihedral group, or quaternion group, or extra-special group we list all possible groups that can arise as $\gamma_{c+1}H$. Our proofs involve: (i) the Baer invariants of $G$, (ii) the Schur multiplier $\mathcal{M}(L,G)$ of $G$ relative to a normal subgroup $L$, and (iii) the nonabelian tensor product of groups. Some results on the nonabelian tensor product may be of independent interest.
On Herstein's Lie map conjectures, I
K.
I.
Beidar;
M.
Bresar;
M.
A.
Chebotar;
W.
S.
Martindale III
4235-4260
Abstract: We describe surjective Lie homomorphisms from Lie ideals of skew elements of algebras with involution onto noncentral Lie ideals (factored by their centers) of skew elements of prime algebras ${\mathcal{D}}$ with involution, provided that $\operatorname{char}({\mathcal{D}})\not=2$ and ${\mathcal{D}}$ is not PI of low degree. This solves the last remaining open problem of Herstein on Lie isomorphisms module cases of PI rings of low degree. A more general problem on maps preserving any polynomial is also discussed.