Simplicity and the stable rank of some free product C*-algebras
Kenneth
J.
Dykema
1-40
Abstract: A necessary and sufficient condition for the simplicity of the C$^{*}$-algebra reduced free product of finite dimensional abelian algebras is found, and it is proved that the stable rank of every such free product is 1. Related results about other reduced free products of C$^{*}$-algebras are proved.
An algorithm for calculating the Nielsen number on surfaces with boundary
Joyce
Wagner
41-62
Abstract: Let $f:M\to M$ be a self-map of a hyperbolic surface with boundary. The Nielsen number, $N(f)$, depends only on the induced map $f_{\#}$ of the fundamental group, which can be viewed as a free group on $n$ generators, $a_1,\dotsc,a_n$. We determine conditions for fixed points to be in the same fixed point class and if these conditions are enough to determine the fixed point classes, we say that $f_{\#}$ is $W$-characteristic. We define an algebraic condition on the $f_{\#}(a_i)$ and show that ``most'' maps satisfy this condition and that all maps which satisfy this condition are $W$-characteristic. If $f_{\#}$ is $W$-characteristic, we present an algorithm for calculating $N(f)$ and prove that the inequality $|L(f)-\chi(M)|\le N(f)-\chi(M)$ holds, where $L(f)$ denotes the Lefschetz number of $f$ and $\chi(M)$ the Euler characteristic of $M$, thus answering in part a question of Jiang and Guo.
Rotating an interval and a circle
Alexander
Blokh;
Michal
Misiurewicz
63-78
Abstract: We compare periodic orbits of circle rotations with their counterparts for interval maps. We prove that they are conjugate via a map of modality larger by at most 2 than the modality of the interval map. The proof is based on observation of trips of inhabitants of the Green Islands in the Black Sea.
Maximal function estimates of solutions to general dispersive partial differential equations
Hans
P.
Heinig;
Sichun
Wang
79-108
Abstract: Let $u(x,t)=(S_\Omega f)(x,t)$ be the solution of the general dispersive initial value problem: \begin{displaymath}\partial _tu-i\Omega(D)u=0, \quad u(x,0)=f(x), \qquad (x,t)\in \mathbb{R}^n \times \mathbb{R}\end{displaymath} and $S^{**}_\Omega f$ the global maximal operator of $S_\Omega f$. Sharp weighted $L^p$-esimates for $S^{**}_\Omega f$ with $f\in H_s(\mathbb{R}^n)$ are given for general phase functions $\Omega$.
Large time asymptotics of solutions to the generalized Benjamin-Ono equation
Nakao
Hayashi;
Pavel
I.
Naumkin
109-130
Abstract: We study the asymptotic behavior for large time of solutions to the Cauchy problem for the generalized Benjamin-Ono (BO) equation: $u_{t} + (|u|^{\rho -1}u)_{x} + \mathcal{H} u_{xx} = 0$, where $\mathcal{H}$ is the Hilbert transform, $x, t \in {\mathbf{R}}$, when the initial data are small enough. If the power $\rho$ of the nonlinearity is greater than $3$, then the solution of the Cauchy problem has a quasilinear asymptotic behavior for large time. In the case $\rho=3$ critical for the asymptotic behavior i.e. for the modified Benjamin-Ono equation, we prove that the solutions have the same $L^{\infty }$ time decay as in the corresponding linear BO equation. Also we find the asymptotics for large time of the solutions of the Cauchy problem for the BO equation in the critical and noncritical cases. For the critical case, we prove the existence of modified scattering states if the initial function is sufficiently small in certain weighted Sobolev spaces. These modified scattering states differ from the free scattering states by a rapidly oscillating factor.
Classifying 2-groups by coclass
M.
F.
Newman;
E.
A.
O'Brien
131-169
Abstract: Now that the conjectures of Leedham-Green and Newman have been proved, we probe deeper into the classification of $p$-groups using coclass. We determine the pro-$2$-groups of coclass at most 3 and use these to classify the 2-groups of coclass at most 3 into families. Using extensive computational evidence, we make some detailed conjectures about the structure of these families. We also conjecture that the 2-groups of arbitrary fixed coclass exhibit similar behaviour.
Monomial Bases for $H^*(\mathbf{C}P^\infty\times\mathbf{C}P^\infty)$ over $\mathcal A(p)$
M.
D.
Crossley
171-192
Abstract: We consider the polynomial algebra $H^*(\mathbf{C} P^\infty \times \mathbf{C} P^\infty; \mathbf{F}_p)$ as a module over the mod $p$ Steenrod algebra, $\mathcal{A}(p)$, $p$ being an odd prime. We give a minimal set of generators consisting of monomials and characterise all such `monomial bases'.
Reducibility of some induced representations of $p$-adic unitary groups
Fiona
Murnaghan;
Joe
Repka
193-210
Abstract: In this paper we study reducibility of those representations of quasi-split unitary $p$-adic groups which are parabolically induced from supercuspidal representations of general linear groups. For a supercuspidal representation associated via Howe's construction to an admissible character, we show that in many cases a criterion of Goldberg for reducibility of the induced representation reduces to a simple condition on the admissible character.
Turnpike property for extremals of variational problems with vector-valued functions
A.
J.
Zaslavski
211-231
Abstract: In this paper we study the structure of extremals $\nu\colon[0,T]\to R^n$ of variational problems with large enough $T$, fixed end points and an integrand $f$ from a complete metric space of functions. We will establish the turnpike property for a generic integrand $f$. Namely, we will show that for a generic integrand $f$, any small $\varepsilon>0$ and an extremal $\nu\colon[0,T]\to R^n$ of the variational problem with large enough $T$, fixed end points and the integrand $f$, for each $\tau\in[L_1, T-L_1]$ the set $\{\nu(t)\colon t\in[\tau,\tau+L_2]\}$ is equal to a set $H(f)$ up to $\varepsilon$ in the Hausdorff metric. Here $H(f)\subset R^n$ is a compact set depending only on the integrand $f$ and $L_1>L_2>0$ are constants which depend only on $\varepsilon$ and $|\nu(0)|$, $|\nu(T)|$.
Properties of some families of hypergeometric orthogonal polynomials in several variables
J.
F.
van Diejen
233-270
Abstract: Limiting cases are studied of the Koornwinder-Macdonald multivariable generalization of the Askey-Wilson polynomials. We recover recently and not so recently introduced families of hypergeometric orthogonal polynomials in several variables consisting of multivariable Wilson, continuous Hahn and Jacobi type polynomials, respectively. For each class of polynomials we provide systems of difference (or differential) equations, recurrence relations, and expressions for the (squared) norms of the polynomials in question.
Density doubling, double-circulants, and new sphere packings
Alexander
Vardy
271-283
Abstract: New nonlattice sphere packings in dimensions 20, 22, and 44-47 that are denser than the best previously known sphere packings were recently discovered. We extend these results, showing that the density of many sphere packings in dimensions just below a power of 2 can be doubled using orthogonal binary codes. This produces new dense sphere packings in $\mathbb R^n$ for $n = 25,26,\dots,31$ and $55,56,\dots,63$. For $n = 27,28,29,30$ the resulting packings are denser than any packing previously known.
Existence and uniqueness for a degenerate parabolic equation with $L^1$-data
F.
Andreu;
J.
M.
Mazón;
S.
Segura
de León;
J.
Toledo
285-306
Abstract: In this paper we study existence and uniqueness of solutions for the boundary-value problem, with initial datum in $L^{1}(\Omega )$, \begin{equation*}u_{t} = \mathrm{div} {\hbox{$\mathbf a$}}(x,Du) \quad \text{in } (0, \infty ) \times \Omega, \end{equation*} \begin{equation*}-{\frac{{\partial u} }{{\partial \eta _{a}}}} \in \beta (u) \quad \text{on } (0, \infty ) \times \partial \Omega,\end{equation*} \begin{equation*}u(x, 0) = u_{0}(x) \quad \text{in }\Omega ,\end{equation*} where a is a Carathéodory function satisfying the classical Leray-Lions hypothesis, $\partial / {\partial \eta _{a}}$ is the Neumann boundary operator associated to ${\hbox{$\mathbf a$}}$, $Du$ the gradient of $u$ and $\beta$ is a maximal monotone graph in ${\mathbb{R}}\times {\mathbb{R}}$ with $0 \in \beta (0)$.
The Krein-Milman theorem in operator convexity
Corran
Webster;
Soren
Winkler
307-322
Abstract: We generalize the Krein-Milman theorem to the setting of matrix convex sets of Effros-Winkler, extending the work of Farenick-Morenz on compact C$^*$-convex sets of complex matrices and the matrix state spaces of C$^*$-algebras. An essential ingredient is to prove the non-commutative analogue of the fact that a compact convex set $K$ may be thought of as the state space of the space of continuous affine functions on $K$.
Operator valued weights without structure theory
Tony
Falcone;
Masamichi
Takesaki
323-341
Abstract: A result of Haagerup, generalizing a theorem of Takesaki, states the following: If ${\mathcal{N}}\subset {\mathcal{M}}$ are von Neumann algebras, then there exists a faithful, normal and semi-finite (fns) operator valued weight $T \colon {\mathcal{M}}_{+} \rightarrow \widehat {{\mathcal{N}}_{+}}$ if and only if there exist fns weights $\tilde \varphi$ on ${\mathcal{M}}$ and $\varphi$ on ${\mathcal{N}}$ satisfying $\sigma ^{\varphi }_{t}(x) = \sigma ^{\tilde \varphi }_{t}(x) \, \forall x \in {\mathcal{N}} , t \in \mathbb{R}$. In fact, $T$ can be chosen such that $\tilde \varphi = \varphi \circ T$; $T$ is then uniquely determined by this condition. We present a proof of the above which does not use any structure theory.
Multi-separation, centrifugality and centripetality imply chaos
Jiehua
Mai
343-351
Abstract: Let $I$ be an interval. $I$ need not be compact or bounded. Let $f:I\rightarrow I$ be a continuous map, and $(x_0, x_1, \cdots, x_n)$ be a trajectory of $f$ with $x_n\leq x_0<x_1$ or $x_1<x_0\leq x_n$. Then there is a point $v\in I$ such that $\min\{x_0, \cdots, x_n\}<v=f(v)<\max\{x_0, \cdots, x_n\}$. A point $y\in I$ is called a centripetal point of $f$ relative to $v$ if $v<f(y)<y$, and $y$ is centrifugal if $f(y)<y<v$ or $v<y<f(y)$. In this paper we prove that if there exist $k$ centripetal points of $f$ in $\{x_0, \cdots, x_{n-1}\}, k\geq 1$, then $f$ has periodic points of some odd ($\not= 1$) period $p\leq (n-2)/k+2$. In addition, we also prove that if $(x_0, x_1, \cdots, x_{n-1}$) is multi-separated by Fix($f$), or there exists a centrifugal point of $f$ in $\{x_0, \cdots, x_{n-1}\}$, then $f$ is turbulent and hence has periodic points of all periods.
Scrambled sets of continuous maps of 1-dimensional polyhedra
Jiehua
Mai
353-362
Abstract: Let $K$ be a 1-dimensional simplicial complex in $R^3$ without isolated vertexes, $X = |K|$ be the polyhedron of $K$ with the metric $d_K$ induced by $K$, and $f:X\rightarrow X$ be a continuous map. In this paper we prove that if $K$ is finite, then the interior of every scrambled set of $f$ in $X$ is empty. We also show that if $K$ is an infinite complex, then there exist continuous maps from $X$ to itself having scrambled sets with nonempty interiors, and if $X = R$ or $R_+$, then there exist $C^\infty$ maps of $X$ with the whole space $X$ being a scrambled set.
On sectional genus of quasi-polarized 3-folds
Yoshiaki
Fukuma
363-377
Abstract: Let $X$ be a smooth projective variety over $\mathbb{C}$ and $L$ a nef-big (resp. ample) divisor on $X$. Then $(X,L)$ is called a quasi-polarized (resp. polarized) manifold. Then we conjecture that $g(L)\geq q(X)$, where $g(L)$ is the sectional genus of $L$ and $q(X)=\operatorname{dim}H^{1}(\mathcal{O}_{X})$ is the irregularity of $X$. In general it is unknown whether this conjecture is true or not, even in the case of $\operatorname{dim}X=2$. For example, this conjecture is true if $\operatorname{dim}X=2$ and $\operatorname{dim}H^{0}(L)>0$. But it is unknown if $\operatorname{dim}X\geq 3$ and $\operatorname{dim}H^{0}(L)>0$. In this paper, we prove $g(L)\geq q(X)$ if $\operatorname{dim}X=3$ and $\operatorname{dim}H^{0}(L)\geq 2$. Furthermore we classify polarized manifolds $(X,L)$ with $\operatorname{dim}X=3$, $\operatorname{dim}H^{0}(L)\geq 3$, and $g(L)=q(X)$.
Combinatorial families that are exponentially far from being listable in Gray code sequence
Ted
Chinburg;
Carla
D.
Savage;
Herbert
S.
Wilf
379-402
Abstract: Let $S(n)$ be a collection of subsets of $\{1,...,n\}$. In this paper we study numerical obstructions to the existence of orderings of $S(n)$ for which the cardinalities of successive subsets satisfy congruence conditions. Gray code orders provide an example of such orderings. We say that an ordering of $S(n)$ is a Gray code order if successive subsets differ by the adjunction or deletion of a single element of $\{1,\ldots,n\}$. The cardinalities of successive subsets in a Gray code order must alternate in parity. It follows that if $d(S(n))$ is the difference between the number of elements of $S(n)$ having even (resp. odd) cardinality, then $|d(S(n))| - 1$ is a lower bound for the cardinality of the complement of any subset of $S(n)$ which can be listed in Gray code order. For $g \ge 2$, the collection $B(n,g)$ of $g$-blockfree subsets of $\{1,\ldots,n\}$ is defined to be the set of all subsets $S$ of $\{1,\ldots,n\}$ such that $|a-b| \ge g$ if $a,b \in S$ and $a \ne b$. We will construct a Gray code order for $B(n,2)$. In contrast, for $g > 2$ we find the precise (positive) exponential growth rate of $d(B(n,g))$ with $n$ as $n \to \infty$. This implies $B(n,g)$ is far from being listable in Gray code order if $n$ is large. Analogous results for other kinds of orderings of subsets of $B(n,g)$ are proved using generalizations of $d(B(n,g))$ . However, we will show that for all $g$, one can order $B(n,g)$ so that successive elements differ by the adjunction and/or deletion of an integer from $\{1,\ldots,n\}$. We show that, over an $A$-letter alphabet, the words of length $n$ which contain no block of $k$ consecutive letters cannot, in general, be listed so that successive words differ by a single letter. However, if $k>2$ and $A>2$ or if $k=2$ and $A>3$, such a listing is always possible.
On the unitary dual of $Spin(2n,\mathbb C)$
Alfredo
O.
Brega
403-415
Abstract: In this paper we begin a systematic study of the unitarity question for genuine representations of the group $Spin (2 n, \mathbb{C})$. The main result is that we find a class of unitary representations (mostly isolated) analogous to the special unipotent representations defined by D. Barbasch and D. Vogan. In particular the full unitary dual of $Spin (2 n, \mathbb{C})$ should be obtainable from this set by complementary series.
Ergodic sequences in the Fourier-Stieltjes algebra and measure algebra of a locally compact group
Anthony
To-Ming
Lau;
Viktor
Losert
417-428
Abstract: Let $G$ be a locally compact group. Blum and Eisenberg proved that if $G$ is abelian, then a sequence of probability measures on $G$ is strongly ergodic if and only if the sequence converges weakly to the Haar measure on the Bohr compactification of $G.$ In this paper, we shall prove an extension of Blum and Eisenberg's Theorem for ergodic sequences in the Fourier-Stieltjes algebra of $G.$ We shall also give an improvement to Milnes and Paterson's more recent generalization of Blum and Eisenberg's result to general locally compact groups, and we answer a question of theirs on the existence of strongly (or weakly) ergodic sequences of measures on $G.$