AN EIGENSPACE CHARACTERIZATION OF LORENTZ-IMPROVING MEASURES ON COMPACT GROUPS

Abstract

A measure μ on a compact group is called Lorentz-improving if for some 1 > p > ∞ and 1 → q ₁ > q ₂ ∞ μ *L (p, q ₂) ? L(p, q ₁). Let T _μ denote the operator on L ² defined by T _μ(f) = μ * f. Lorentz-improving measures are characterized in terms of the eigenspaces of T _μ, if T _μ is a normal operator, and in terms of the eigenspaces of |T _μ| otherwise. This result generalizes our recent characterization of Lorentz-improving measures on compact abelian groups and is modelled after Hare's characterization of L ^p -improving measures on compact groups.     

|T |+‐resplendent models and the Lascar group

In this paper we show that in every |T |⁺‐resplendent model N , for every A ? N such that |A | ≤ |T |, the group Autf(N/A ) of strong automorphisms is the least very normal subgroup of the group Aut(N/A ) and the quotient Aut(N/A )/Autf(N/A ) is the Lascar group over A . Then we generalize this result to every |T |⁺‐saturated and strongly |T |⁺‐homogeneous model. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Directed Graph Pattern Matching and Topological Embedding

Pattern matching in directed graphs is a natural extension of pattern matching in trees and has many applications to different areas. In this paper, we study several pattern matching problems in ordered labeled directed graphs. For the rooted directed graph pattern matching problem, we present an efficient algorithm which, given a pattern graphPand a target graphT, runs in time and spaceO(|E_P| |V_T| + |E_T|). It is faster than the best known method by a factor ofmin{|E_T|, |E_P| |V_T|}. This algorithm can also solve the directed graph pattern matching problem without increasing time or space complexity. Our solution to this problem outperforms the best existing method by Katzenelson, Pinter and Schenfeld by a factor ofmin{|V_P| |E_T|, |V_P| |E_P| |V_T|}. We also present an algorithm for the directed graph topological embedding problem which runs in timeO(|V_P| |E_T| + |E_P|) and spaceO(|V_P| |V_T| + |E_P| + |E_T|). To our knowledge, this algorithm is the first one for this problem.     

Properties of Subtree-Prune-and-Regraft Operations on Totally-Ordered Phylogenetic Trees

We study some properties of subtree-prune-and-regraft (SPR) operations on leaflabelled rooted binary trees in which internal vertices are totally ordered. Since biological events occur with certain time ordering, sometimes such totally-ordered trees must be used to avoid possible contradictions in representing evolutionary histories of biological sequences. Compared to the case of plain leaf-labelled rooted binary trees where internal vertices are only partially ordered, SPR operations on totally-ordered trees are more constrained and therefore more difficult to study. In this paper, we investigate the unit-neighbourhood U(T), defined as the set of totally-ordered trees one SPR operation away from a given totally-ordered tree T. We construct a recursion relation for | U(T) | and thereby arrive at an efficient method of determining | U(T) |. In contrast to the case of plain rooted trees, where the unit-neighbourhood size grows quadratically with respect to the number n of leaves, for totally-ordered trees | U(T) | grows like O(n³). For some special topology types, we are able to obtain simple closed-form formulae for | U(T) |. Using these results, we find a sharp upper bound on | U(T) | and conjecture a formula for a sharp lower bound. Lastly, we study the diameter of the space of totally-ordered trees measured using the induced SPR-metric. Received May 18, 2004     

A Note on Aluthge Transforms of Complex Symmetric Operators and Applications

In this paper, we reprove that: (i) the Aluthge transform of a complex symmetric operator [(T)\tilde] = |T|^\frac12 U|T|^\frac12\tilde{T} = |T|^{\frac{1}{2}} U|T|^{\frac{1}{2}} is complex symmetric, (ii) if T is a complex symmetric operator, then ([(T)\tilde])^*(\tilde{T})^{*} and [(T^*)\tilde]\widetilde{T^{*}} are unitarily equivalent. And we also prove that: (iii) if T is a complex symmetric operator, then [((T^*))\tilde]_s,t\widetilde{(T^{*})}_{s,t} and ([(T)\tilde]_t,s)^*(\tilde{T}_{t,s})^{*} are unitarily equivalent for s, t > 0, (iv) if a complex symmetric operator T belongs to class wA(t, t), then T is normal.     

Hamilton cycles,avoiding prescribed arcs,in close-to-regular tournaments

The local irregularity of a digraph D is defined as i_l(D) = max {|d⁺ (x) − d⁻ (x)| : x ϵ V(D)}. Let T be a tournament, let Γ = {V₁, V₂, …, V_c} be a partition of V(T) such that |V₁| ≥ |V₂| ≥ … ≥ |V_c|, and let D be the multipartite tournament obtained by deleting all the arcs with both end points in the same set in Γ. We prove that, if |V(T)| ≥ max{2i_l(T) + 2|V₁| + 2|V₂| − 2, i_l(T) + 3|V₁| − 1}, then D is Hamiltonian. Furthermore, if T is regular (i.e., i_l(T) = 0), then we state slightly better lower bounds for |V(T)| such that we still can guarantee that D is Hamiltonian. Finally, we show that our results are best possible. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 123–136, 1999     

A remark on the genericity of multiplicity results for forced oscillations on manifolds

We discuss the genericity of some multiplicity results for periodically perturbed autonomous first- and second-order ODEs on manifolds.?In particular, the genericity of the following property is investigated: if the differentiable manifold M is compact, then the equation _π=h(x,)+f(t,x,) on M has |χ(M)| geometrically distinct T-periodic solutions for any small enough T-periodic perturbing function f. Received: January 24, 2000; in final form: January 16, 2001?Published online: March 19, 2002     

On the Polar Decomposition of the Product of Two Operators and Its Applications

Let T = U|T| and S = V|S| be the polar decompositions. In this paper, we shall obtain the polar decomposition of TS as TS = UWV|TS|, where |T||S^*| = W||T||S^*|| is the polar decomposition. Next, we shall show that TS = UV|TS| is the polar decomposition if and only if |T| commutes with |S^*|. Lastly, we shall apply this result to binormal and centered operators. We shall obtain characterizations of these operator classes from the viewpoint of the polar decomposition.     

Generalized Absolute Values and Polar Decompositions of a Bounded Operator

Generalized absolute values as well as corresponding to them generalized polar decompositions of a bounded linear operator T of a Hilbert space H{\mathcal{H}} into a Hilbert space K{\mathcal{K}} are defined, motivated by the inequality |áTx, y?_K|² ￡ á|T|x, x?_Há|T^*|y, y?_K{|\langle{Tx}, {y}\rangle}_{\mathcal{K}}|^2 \leq \langle|T|x, {x}\rangle_{\mathcal{H}}\langle{|T^{*}|y}, {y}\rangle_{\mathcal{K}} . It is shown that there is a natural bijection between generalized absolute values of T and of T* which sends |T| to |T*|. For a bounded nonnegative operator A on H{\mathcal{H}} and a bounded Borel function f: \mathbbR₊ ? \mathbbR₊{f: \mathbb{R}_+ \to \mathbb{R}_+} , equivalent conditions for A and f(|T|) to be generalized absolute values of T are established and corresponding to them generalized absolute values of T* are determined.     

New Algorithm for Ordered Tree-to-Tree Correction Problem

The ordered tree-to-tree correction problem is to compute the minimum edit cost of transforming one ordered tree to another one. This paper presents a new algorithm for this problem. Given two ordered trees S and T, our algorithm runs in O(|S| |T| + min { ²_S|T| + ^2.5_{S T}, ²_T|S| + ^2.5_{T S}) time, where _S denotes the number of leaves of S and _S denotes the depth of S. The previous best algorithms for this problem run in O(|S| |T| min { _S, _S} min { _T, _T}) time (K. Zhang and D. Shasha, SIAM J. Comput.18, No. 6 (1989), 1245–1262) and in O(min {|S|²|T| log₂ |T|, |T|²|S| log₂ |S|}) time (P. N. Klein, in “Algorithms—ESA'98, 6th Annual European Symposium” (G. Bilardi, G. F. Italiano, A. Pietracaprina, and G. Pucci, Eds.), Lecture Notes in Computer Science, Vol. 1461, pp. 91–102, Springer-Verlag, Berlin/New York, 1998). As a comparison, our algorithm is asymptotically faster for certain kind of trees.     

Quantum superdeterminants for OSP q (1-2n)

It is shown that there exists a quantum superdeterminant sdet _q T for the quantum super group OSP _q (1|2n). It is also shown that the quantum superdeterminant sdet _q T is a group-like element and central, and that the square of sdet _q T for OSP _q (1|2n) is equal to 1.     

连续自映射拓扑压的两个注记

令T:X→ X是紧度量空间(X,d)上的连续映射.该文给出了T的拓扑压和T在非游荡集上的限制的拓扑压相等的不依赖于变分原理的一个直接证明.同时,还讨论了半共轭的两个系统的拓扑压之间的关系,证明了拓扑压在一致有限对一条件下是半共轭不变量.     

Binormal operator and *-Aluthge transformation

Let T = U｜T｜ be the polar decomposition of a bounded linear operator T on a Hilbert space. The transformation T = ｜T｜^1/2 U｜T｜^1/2 is called the Aluthge transformation and Tn means the n-th Aluthge transformation. Similarly, the transformation T（＊）=｜T＊｜^1/2 U｜T＊｜＆1/2 is called the ＊-Aluthge transformation and Tn^（＊） means the n-th ＊-Aluthge transformation. In this paper, firstly, we show that T（＊） = UV｜T^（＊）｜ is the polar decomposition of T（＊）, where ｜T｜^1/2 ｜T^＊｜^1/2 = V｜｜T｜^1/2 ｜T^＊｜^1/2｜ is the polar decomposition. Secondly, we show that T（＊） = U｜T^（＊）｜ if and only if T is binormal, i.e., [｜T｜, ｜T^＊｜]=0, where [A, B] = AB - BA for any operator A and B. Lastly, we show that Tn^（＊） is binormal for all non-negative integer n if and only if T is centered, and so on.     

Smallest Limited Vertex-to-vertex Snakes of Unit Triangles

A sequence T=T ₁,T ₂,...,T _n of regular triangles of unit side lengths is called a vertex-to-vertex snake if TT is a common vertex of T and T if |i-j|=1 and is empty if |i-j|<1. A vertex-to-vertex snake of unit triangles is called limited if it is not a proper subset of another vertex-to-vertex snake of unit triangles. We prove that the minimum number of unit triangles which form a limited vertex-to-vertex snake is seven.     

On theT(1) theorem on product domains

The direct proof by R. R. Coifman and Y. Meyer of theT(1) Theorem of G. David and J. L. Journé is based on the following result. LetT be an operator associated to a kernelk(x, y) satisfying

Domination properties of lattice homomorphisms

Let E and F be Banach lattices, T, K : E → F be such that 0 ≤ T ≤ K and T is either a lattice homomorphism or almost interval-preserving. In this paper we will deduce that (1) If K is AM-compact then T also is AM-compact; (2) If either E′ or F has an order continuous norm and K is compact, then T is compact as well; (3) If K is weakly compact then so is T. Some related results are also obtained.      

Absolutely convergent fourier series. An improvement of the Beurling-Helson theorem

We consider the space A(\mathbbT)A(\mathbb{T}) of all continuous functions f on the circle \mathbbT\mathbb{T} such that the sequence of Fourier coefficients [^(f)] = { [^(f)]( k ), k ? \mathbbZ }\hat f = \left\{ {\hat f\left( k \right), k \in \mathbb{Z}} \right\} belongs to l ¹(ℤ). The norm on A(\mathbbT)A(\mathbb{T}) is defined by || f ||_A(\mathbbT) = || [^(f)] ||_{l¹ (\mathbbZ)}\left\| f \right\|_{A(\mathbb{T})} = \left\| {\hat f} \right\|_{l^1 (\mathbb{Z})}. According to the well-known Beurling-Helson theorem, if f:\mathbbT ? \mathbbT\phi :\mathbb{T} \to \mathbb{T} is a continuous mapping such that || e^inf ||_A(\mathbbT) = O(1)\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = O(1), n ∈ ℤ then φ is linear. It was conjectured by Kahane that the same conclusion about φ is true under the assumption that || e^inf ||_A(\mathbbT) = o( log| n | )\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\log \left| n \right|} \right). We show that if $\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right)$\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right), then φ is linear.     

Compact operators on a Banach space into the space of almost periodic functions

LetS be a topological semigroup andAP(S) the space of continous complex almost periodic functions onS. We obtain characterizations of compact and weakly compact operators from a Banach spaceX into AP(S). For this we use the almost periodic compactification ofS obtained through uniform spaces. For a bounded linear operatorT fromX into AP(S), letT ₅, be the translate ofT bys inS defined byT ₅(x)=(Tx) ₅ . We define topologies on the space of bounded linear operators fromX into AP(S) and obtain the necessary and sufficient conditions for an operatorT to be compact or weakly compact in terms of the uniform continuity of the maps→T ₅. IfS is a Hausdorff topological semigroup, we also obtain characterizations of compact and weakly compact multipliers on AP(S) in terms of the uniform continuity of the map S→μ_s, where μ_s denotes the unique vector measure corresponding to the operatorT ₅.     

Self-Affine Sets and Graph-Directed Systems

   Abstract. A self-affine set in R ⁿ is a compact set T with A(T)= ∪ _{d∈ D} (T+d) where A is an expanding n× n matrix with integer entries and D ={d ₁ , d ₂ ,···, d _N } ⊂ Z ⁿ is an N -digit set. For the case N = | det(A)| the set T has been studied in great detail in the context of self-affine tiles. Our main interest in this paper is to consider the case N > | det(A)| , but the theorems and proofs apply to all the N . The self-affine sets arise naturally in fractal geometry and, moreover, they are the support of the scaling functions in wavelet theory. The main difficulty in studying such sets is that the pieces T+d, d∈ D, overlap and it is harder to trace the iteration. For this we construct a new graph-directed system to determine whether such a set T will have a nonvoid interior, and to use the system to calculate the dimension of T or its boundary (if T ^o ≠  ). By using this setup we also show that the Lebesgue measure of such T is a rational number, in contrast to the case where, for a self-affine tile, it is an integer.     

Constructing Almost Excellent Unique Factorization Domains

Abstract

Let (T, M) be a complete local normal integral domain containing the rationals such that |T/M | ≥ c where c is the cardinality of the real numbers. Let p be a non-maximal prime ideal of T such that _{T p} is a regular local ring. We construct a local Unique Factorization Domain (UFD) A such that the M-adic completion of A is T, p is maximal in the generic formal fiber and all fibers of A are geometrically regular except for those over some height one prime ideals.