Structural properties of universal minimal dynamical systems for discrete semigroups
Bohuslav
Balcar;
Frantisek
Franek
1697-1724
Abstract: We show that for a discrete semigroup $S$ there exists a uniquely determined complete Boolean algebra $B(S)$ - the algebra of clopen subsets of $M(S)$. $M(S)$ is the phase space of the universal minimal dynamical system for $S$ and it is an extremally disconnected compact Hausdorff space. We deal with this connection of semigroups and complete Boolean algebras focusing on structural properties of these algebras. We show that $B(S)$ is either atomic or atomless; that $B(S)$ is weakly homogenous provided $S$ has a minimal left ideal; and that for countable semigroups $B(S)$ is semi-Cohen. We also present a class of what we call group-like semigroups that includes commutative semigroups, inverse semigroups, and right groups. The group reflection $G(S)$ of a group-like semigroup $S$ can be constructed via universal minimal dynamical system for $S$ and, moreover, $B(S)$ and $B(G(S))$ are the same.
$C^*$-extreme points in the generalised state spaces of a $C^*$-algebra
Douglas
R.
Farenick;
Phillip
B.
Morenz
1725-1748
Abstract: In this paper we study the space $S_{H}(A)$ of unital completely positive linear maps from a $C^{*}$-algebra $A$ to the algebra $B(H)$ of continuous linear operators on a complex Hilbert space $H$. The state space of $A$, in this notation, is $S_{\mathbb {C}}(A)$. The main focus of our study concerns noncommutative convexity. Specifically, we examine the $C^{*}$-extreme points of the $C^{*}$-convex space $S_{H}(A)$. General properties of $C^{*}$-extreme points are discussed and a complete description of the set of $C^{*}$-extreme points is given in each of the following cases: (i) the cases $S_{{\mathbb {C}}^{2}}(A)$, where $A$ is arbitrary ; (ii) the cases $S_{{\mathbb {C}}^{r}}(A)$, where $A$ is commutative; (iii) the cases $S_{{\mathbb {C}}^{r}}(M_{n})$, where $M_{n}$ is the $C^{*}$-algebra of $n\times n$ complex matrices. An analogue of the Krein-Milman theorem will also be established.
A Characterization of Minimal Locally Finite Varieties
Keith
A.
Kearnes;
Ágnes
Szendrei
1749-1768
Abstract: In this paper we describe a one-variable Mal'cev-like condition satisfied by any locally finite minimal variety. We prove that a locally finite variety is minimal if and only if it satisfies this Mal'cev-like condition and it is generated by a strictly simple algebra which is nonabelian or has a trivial subalgebra. Our arguments show that the strictly simple generator of a minimal locally finite variety is unique, it is projective and it embeds into every member of the variety. We give a new proof of the structure theorem for strictly simple abelian algebras that generate minimal varieties.
Reversion of power series and the extended Raney coefficients
Charles
Ching-An
Cheng;
James
H.
McKay;
Jacob
Towber;
Stuart
Sui-Sheng
Wang;
David
L.
Wright
1769-1782
Abstract: In direct as well as diagonal reversion of a system of power series, the reversion coefficients may be expressed as polynomials in the coefficients of the original power series. These polynomials have coefficients which are natural numbers (Raney coefficients). We provide a combinatorial interpretation for Raney coefficients. Specifically, each such coefficient counts a certain collection of ordered colored trees. We also provide a simple determinantal formula for Raney coefficients which involves multinomial coefficients.
Correlation dimension for iterated function systems
Wai
Chin;
Brian
Hunt;
James
A.
Yorke
1783-1796
Abstract: The correlation dimension of an attractor is a fundamental dynamical invariant that can be computed from a time series. We show that the correlation dimension of the attractor of a class of iterated function systems in $\mathbf {R}^N$ is typically uniquely determined by the contraction rates of the maps which make up the system. When the contraction rates are uniform in each direction, our results imply that for a corresponding class of deterministic systems the information dimension of the attractor is typically equal to its Lyapunov dimension, as conjected by Kaplan and Yorke.
On trace identities and universal eigenvalue estimates for some partial differential operators
Evans
M.
Harrell II;
Joachim
Stubbe II
1797-1809
Abstract: In this article, we prove and exploit a trace identity for the spectra of Schrödinger operators and similar operators. This identity leads to universal bounds on the spectra, which apply to low-lying eigenvalues, eigenvalue asymptotics, and to partition functions (traces of heat operators). In many cases they are sharp in the sense that there are specific examples for which the inequalities are saturated. Special cases corresponding to known inequalities include those of Hile and Protter and of Yang.
Bodies with similar projections
G.
D.
Chakerian;
E.
Lutwak
1811-1820
Abstract: Aleksandrov's projection theorem characterizes centrally symmetric convex bodies by the measures of their orthogonal projections on lower dimensional subspaces. A general result proved here concerning the mixed volumes of projections of a collection of convex bodies has the following corollary. If $K$ is a convex body in ${\mathbb {R}}^{n}$ whose projections on $r$-dimensional subspaces have the same $r$-dimensional volume as the projections of a centrally symmetric convex body $M$, then the Quermassintegrals satisfy $W_{j}(M)\ge W_{j}(K)$, for $0\le j < n-r$, with equality, for any $j$, if and only if $K$ is a translate of $M$. The case where $K$ is centrally symmetric gives Aleksandrov's projection theorem.
The homotopy groups of the $L_2$-localized Toda-Smith spectrum $V(1)$ at the prime 3
Katsumi
Shimomura
1821-1850
Abstract: In this paper, we try to compute the homotopy groups of the $L_2$-localized Toda-Smith spectrum $V(1)$ at the prime 3 by using the Adams-Novikov spectral sequence, and have almost done so. This computation involves non-trivial differentials $d_5$ and $d_9$ of the Adams-Novikov spectral sequence, different from the case $p>3$. We also determine the homotopy groups of some $L_2$-localized finite spectra relating to $V(1)$. We further show some of the non-trivial differentials on elements relating so-called $\beta$-elements in the Adams-Novikov spectral sequence for $\pi _*(S^0)$.
Small cancellation groups and translation numbers
Ilya
Kapovich
1851-1875
Abstract: In this paper we prove that C(4)-T(4)-P, C(3)-T(6)-P and C(6)-P small cancellation groups are translation discrete in the strongest possible sense and that in these groups for any $g$ and any $n$ there is an algorithm deciding whether or not the equation $x^{n}=g$ has a solution. There is also an algorithm for calculating for each $g$ the maximum $n$ such that $g$ is an $n$-th power of some element. We also note that these groups cannot contain isomorphic copies of the group of $p$-adic fractions and so in particular of the group of rational numbers. Besides we show that for $C^{\prime \prime }(4)-T(4)$ and $C''(3)-T(6)$ groups all translation numbers are rational and have bounded denominators.
Boundary value maps, Szegö maps and intertwining operators
L.
Barchini
1877-1900
Abstract: We consider one series of unitarizable representations, the cohomological induced modules $A_{\mathfrak {q}}(\lambda )$ with dominant regular infinitesimal character. The minimal $K$-type $(\tau , V)$ of $A_{\mathfrak {q}}(\lambda )$ determines a homogeneous vector bundle $V_{\tau } \longrightarrow G/K$. The derived functor modules can be realized on the solution space of a first order differential operator $\mathcal {D}_{\mathfrak {l}}^{\lambda }$ on $V_{\tau }$. Barchini, Knapp and Zierau gave an explicit integral map $\mathcal {S}$ from the derived functor module, realized in the Langlands classification, into the space of smooth sections of the vector bundle $V_{\tau } \longrightarrow G/K$. In this paper we study the asymptotic behavior of elements in the image of $\mathcal {S}$. We obtain a factorization of the standard intertwining opeartors into the composition of the Szegö map $\mathcal {S}$ and a passage to boundary values.
On the extremal structure of the unit balls of Banach spaces of weakly continuous functions and their duals
Zhibao
Hu;
Mark
A.
Smith
1901-1918
Abstract: A sufficient and then a necessary condition are given for a function to be an extreme point of the unit ball of the Banach space $C(K,(X,w))$ of continuous functions, under the supremum norm, from a compact Hausdorff topological space $K$ into a Banach space $X$ equipped with its weak topology $w$. Strongly extreme points of the unit ball of $C(K,(X,w))$ are characterized as the norm-one functions that are uniformly strongly extreme point valued on a dense subset of $K$. It is shown that a variety of stronger types of extreme points (e.g. denting points) never exist in the unit ball of $C(K,(X,w))$. Lastly, some naturally arising and previously known extreme points of the unit ball of $C(K,(X,w))^{*}$ are shown to actually be strongly exposed points.
Higher-dimensional virtual diagonals and ideal cohomology for triangular algebras
Alan
L. T.
Paterson;
Roger
R.
Smith
1919-1943
Abstract: We investigate the cohomology of non-self-adjoint algebras using virtual diagonals and their higher-dimensional generalizations. We show that infinite dimensional nest algebras always have non-zero second cohomology by showing that they cannot possess 2-virtual diagonals. In the case of the upper triangular atomic nest algebra we exhibit concrete modules for non-vanishing cohomology.
Principal Eigenvalues with Indefinite Weight Functions
Zhiren
Jin
1945-1959
Abstract: Both existence and non-existence results for principal eigenvalues of an elliptic operator with indefinite weight function have been proved. The existence of a continuous family of principal eigenvalues is demonstrated.
Singular set of some Kähler orbifolds
Thalia
D.
Jeffres
1961-1971
Abstract: We consider some examples of orbifolds with positive first Chern class. Applying a result of Ding and Tian, we show that the singularities must be very mild if the orbifold admits a Kähler-Einstein metric.
Partial regularity of solutions to a class of degenerate systems
Xiangsheng
Xu
1973-1992
Abstract: We consider the system $\displaystyle \frac{\partial u }{\partial t}-\Delta u=\sigma \left ( u\right ) \left | \nabla \varphi \right | ^2$, $\mathrm {div}\left ( \sigma \left ( u\right ) \nabla \varphi \right ) =0$ in $Q_T\equiv \Omega \times \left ( 0,T\right ]$ coupled with suitable initial-boundary conditions, where $\Omega$ is a bounded domain in $\mathbf {R}^N$ with smooth boundary and $\sigma \left ( u\right )$ is a continuous and positive function of $u$. Our main result is that under some conditions on $\sigma$ there exists a relatively open subset $Q_0$ of $Q_T$ such that $u$ is locally Hölder continuous on $Q_0$, the interior of $Q_T\backslash Q_0$ is empty, and $u$ is essentially bounded on $Q_T\backslash Q_0$.
Classification problems for shifts on modules over a principal ideal domain
Fabio
Fagnani;
Sandro
Zampieri
1993-2006
Abstract: In this paper we study symbolic dynamics over alphabets which are modules over a principal ideal domain, considering topological shifts which are also submodules. Our main result is the classification, up to algebraic and topological conjugacy, of the torsion-free, transitive, finite memory shifts.
Menas' Result is Best Possible
Arthur
W.
Apter;
Saharon
Shelah
2007-2034
Abstract: Generalizing some earlier techniques due to the second author, we show that Menas' theorem which states that the least cardinal $\kappa$ which is a measurable limit of supercompact or strongly compact cardinals is strongly compact but not $2^{\kappa }$ supercompact is best possible. Using these same techniques, we also extend and give a new proof of a theorem of Woodin and extend and give a new proof of an unpublished theorem due to the first author.
Periodic solutions to nonlinear one dimensional wave equation with $X$-dependent coefficients
V.
Barbu;
N.
H.
Pavel
2035-2048
Abstract: This paper deals with $t$-periodicity and regularity of solutions to the one dimensional nonlinear wave equation with $x$-dependent coefficients
Wavelet transform and orthogonal decomposition of space $L^2$ on the Cartan domain $BDI(q=2)$
Qingtang
Jiang
2049-2068
Abstract: Let $G=\left ({\mathbb {R}}^{*}_{+}\times SO_{0}(1, n)\right ) \ltimes {\mathbb {R}}^{n+1}$ be the Weyl-Poincaré group and $KAN$ be the Iwasawa decomposition of $SO_{0}(1, n)$ with $K=SO(n)$. Then the ``affine Weyl-Poincaré group'' $G_{a}=\left ({\mathbb {R}}^{*}_{+}\times AN\right ) \ltimes {\mathbb {R}}^{n+1}$ can be realized as the complex tube domain $\Pi ={\mathbb {R}}^{n+1}+iC$ or the classical Cartan domain $BDI(q=2)$. The square-integrable representations of $G$ and $G_{a}$ give the admissible wavelets and wavelet transforms. An orthogonal basis $\{ \psi _{k}\}$ of the set of admissible wavelets associated to $G_{a}$ is constructed, and it gives an orthogonal decomposition of $L^{2}$ space on $\Pi$ (or the Cartan domain $BDI(q=2)$) with every component $A_{k}$ being the range of wavelet transforms of functions in $H^{2}$ with $\psi _{k}$.
Twin trees and $\lambda_{\Lambda}$-gons
Curtis
D.
Bennett
2069-2084
Abstract: We define a natural generalization of generalized $n$-gons to the case of $\Lambda$-graphs (where $\Lambda$ is a totally ordered abelian group and $0<\lambda \in \Lambda$). We term these objects $\lambda _{\Lambda }$-gons. We then show that twin trees as defined by Ronan and Tits can be viewed as $(1,0)_{\Lambda }$-gons, where $\Lambda = Z \times Z$ is ordered lexicographically. This allows us to then generalize twin trees to the case of $\Lambda$-trees. Finally, we give a free construction of $\lambda _{\Lambda }$-gons in the cases where $\Lambda$ is discrete and has a subgroup of index $2$ that does not contain the minimal element of $\Lambda$.
Duality of restriction and induction for $C^*$-coactions
S.
Kaliszewski;
John
Quigg;
Iain
Raeburn
2085-2113
Abstract: Consider a coaction $\delta$ of a locally compact group $G$ on a $C^*$- algebra $A$, and a closed normal subgroup $N$ of $G$. We prove, following results of Echterhoff for abelian $G$, that Mansfield's imprimitivity between $A\times _{\delta |}G/N$ and $A\times _\delta G\times _{\hat {\delta } ,r}N$ implements equivalences between Mansfield induction of representations from $A\times _{\delta |}G/N$ to $A\times _\delta G$ and restriction of representations from $A\times _\delta G\times _{\hat {\delta } ,r}N$ to $A\times _\delta G$, and between restriction of representations from $A\times _\delta G$ to $A\times _{\delta |}G/N$ and Green induction of representations from $A\times _\delta G$ to $A\times _\delta G\times _{\hat {\delta } ,r}N$. This allows us to deduce properties of Mansfield induction from the known theory of ordinary crossed products.
Another note on Weyl's theorem
Robin
Harte;
Woo
Young
Lee
2115-2124
Abstract: ``Weyl's theorem holds" for an operator $T$ on a Banach space $X$ when the complement in the spectrum of the ``Weyl spectrum" coincides with the isolated points of spectrum which are eigenvalues of finite multiplicity. This is close to, but not quite the same as, equality between the Weyl spectrum and the ``Browder spectrum", which in turn ought to, but does not, guarantee the spectral mapping theorem for the Weyl spectrum of polynomials in $T$. In this note we try to explore these distinctions.