Self-Similarity and Probability: Parameters Describing the Geometry of Cantor Sets

A deterministic self-similar Cantor set F defines a naturalprobability space, the elements of which are neighbourhoodsof various sizes of all points in F. Considering random variableson this probability space, some porosity and density parametersare explicitly calculated. 1991 Mathematics Subject Classification28A80, 60D05.     

On the existence of conically self-similar free-vortex solutions to the Navier-Stokes equations

In this paper a proof of existence and non-existence of theconically self-similar free-vortex solutions to the Navier-Stokesequations, originally found by Yih et al. (1982, Phys. Fluids.25, 2147-2158), is presented. This proof clearly establishesthat these solutions do not have any kind of singularity atthe symmetry axis. This analysis gives considerably improvedexistence and non-existence bounds and it is shown that thesebounds are close to optimal in the low-swirling limit. Thisapproach links the questions of existence and non-existencefor the swirling case and for the non-swirling case. The proof,which is an extension of techniques developed by Serrin (1972,Phil. Trans. R. Soc. Lond. 271, 325-360), is based on Schauder'sFixed Point Theorem and is, therefore, non-constructive. Therefore,the paper ends with a brief discussion of the question of howto compute the conically self-similar free-vortex solutionsto the Navier-Stokes equations.     

Multifractal products of cylindrical pulses

 New multiplicative and statistically self-similar measures μ are defined on ℝ as limits of measure-valued martingales. Those martingales are constructed by multiplying random functions attached to the points of a statistically self-similar Poisson point process defined in a strip of the plane. Several fundamental problems are solved, including the non-degeneracy and the multifractal analysis of μ. On a bounded interval, the positive and negative moments of diverge under broad conditions. First received: 14 September 1999 / Resubmited: 27 June 2001 / Revised version: 30 May 2002 / Published online: 30 September 2002 Mathematics Subject Classification (2002): 28A80, 60G18, 60G44, 60G55, 60G57 Key words or phrases: Random measures – Multifractal analysis – Continuous time martingales – Statistically self-similar Poisson point processes     

The Hausdorff Dimension of the Boundary of a Self-Similar Tile

An effective method is given for computing the Hausdorff dimensionof the boundary of a self-similar digit tile T in n-dimensionalEuclidean space: where 1/c is the contraction factor and is the largest eigenvalueof a certain contact matrix first defined by Gröchenigand Haas.     

Normal and Non-Normal Points of Self-Similar Sets and Divergence Points of Self-Similar Measures

Let K and µ be the self-similar set and the self-similarmeasure associated with an IFS (iterated function system) withprobabilities (S_i, p_i)_i=1,...,N satisfying the open set condition.Let ={1,...,N}^N denote the full shift space and let : K denotethe natural projection. The (symbolic) local dimension of µat is defined by lim_n (log µK_|n/log diam K_|n), where for = (₁, ₂,...) . A point for which the limit lim_n (log µK_|n/log diam K_|n) doesnot exist is called a divergence point. In almost all of theliterature the limit lim_n (log µK_|n/log diam K_|n) is assumedto exist, and almost nothing is known about the set of divergencepoints. In the paper a detailed analysis is performed of theset of divergence points and it is shown that it has a surprisinglyrich structure. For a sequence (_n)_n, let A(_n) denote the setof accumulation points of (_n)_n. For an arbitrary subset I ofR, the Hausdorff and packing dimension of the set and related sets is computed. An interesting and surprisingcorollary to this result is that the set of divergence pointsis extremely ‘visible’; it can be partitioned intoan uncountable family of pairwise disjoint sets each with fulldimension. In order to prove the above statements the theory of normaland non-normal points of a self-similar set is formulated anddeveloped in detail. This theory extends the notion of normaland non-normal numbers to the setting of self-similar sets andhas numerous applications to the study of the local propertiesof self-similar measures including a detailed study of the setof divergence points.     

Smoothness of the Lq-Spectrum of Self-Similar Measures with Overlaps

Let µ be the self-similar measure for a linear functionsystem S_jx=x+b_j (j=1,2,...,m) on the real line with the probabilityweight . Under the condition that satisfies the finite type condition, the L^q-spectrum (q) of µ is shown to bedifferentiable on (0,); as an application, µ is exactdimensional and satisfies the multifractal formalism.     

On singularity of energy measures on self-similar sets

We provide general criteria for energy measures of regular Dirichlet forms on self-similar sets to be singular to Bernoulli type measures. In particular, every energy measure is proved to be singular to the Hausdorff measure for canonical Dirichlet forms on 2-dimensional Sierpinski carpets.Partially supported by Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Encouragement of Young Scientists, 15740089.Mathematics Subject Classification (2000): 28A80 (60G30, 31C25, 60J60)     

On Aronson's Upper Bounds for Heat Kernels

Let L be a uniformly elliptic operator in divergence form onR^d, and let p(t,x,y) be the fundamental solution to the heatequation for L. A new proof is given of Aronson's upper bound: 2000 Mathematics Subject Classification35J15, 60J60.     

Characterizing Automorphism Groups of Ordered Abelian Groups

A proof is given of the following theorem, which characterizesfull automorphism groups of ordered abelian groups: a groupH is the automorphism group of some ordered abelian group ifand only if H is right-orderable. 2000 Mathematics Subject Classification20K15, 20K20, 20F60, 20K30 (primary); 03E05 (secondary).     

Strict contractivity of the 2-Wasserstein distance for the porous medium equation by mass-centering

We show that the Euclidean Wasserstein distance between two compactly supported solutions of the one-dimensional porous medium equation having the same center of mass decays to zero for large times. As a consequence, we detect an improved -rate of convergence of solutions of the one-dimensional porous medium equation towards well-centered self-similar Barenblatt profiles, as time goes to infinity.

     

Disintegration-of-measure techniques for commuting multivariable weighted shifts

We employ techniques from the theory of disintegration of measuresto study the Lifting Problem for commuting n-tuples of subnormalweighted shifts. We obtain a new necessary condition for theexistence of a lifting, and generate new pathology associatedwith bringing together the Berger measures associated to eachindividual weighted shift. For subnormal 2-variable weightedshifts, we then find the precise relation between the Bergermeasure of the pair and the Berger measures of the shifts associatedto horizontal rows and vertical columns of weights. 2000 MathematicsSubject Classification 47B20, 47B37, 47A13, 28A50 (primary),44A60, 47-04, 47A20 (secondary).     

Positive Definite Functions and Operator Inequalities

We construct several examples of positive definite functions,and use the positive definite matrices arising from them toderive several inequalities for norms of operators. 1991 MathematicsSubject Classification 42A82, 47A63, 15A45, 15A60.     

Latin Squares and the Hall-Paige Conjecture

The Hall–Paige conjecture deals with conditions underwhich a finite group G will possess a complete mapping, or equivalentlya Latin square based on the Cayley table of G will possess atransversal. Two necessary conditions are known to be: (i) thatthe Sylow 2-subgroups of G are trivial or non-cyclic, and (ii)that there is some ordering of the elements of G which yieldsa trivial product. These two conditions are known to be equivalent,but the first direct, elementary proof that (i) implies (ii)is given here. It is also shown that the Hall–Paige conjecture impliesthe existence of a duplex in every group table, thereby provinga special case of Rodney's conjecture that every Latin squarecontains a duplex. A duplex is a ‘double transversal’,that is, a set of 2n entries in a Latin square of order n suchthat each row, column and symbol is represented exactly twice.2000 Mathematics Subject Classification 05B15, 20D60.     

Entropy of Absolute Convex Hulls in Hilbert Spaces

The metric entropy of absolute convex hulls of sets in Hilbertspaces is studied for the general case when the metric entropyof the sets is arbitrary. Under some regularity assumptions,the results are sharp. 2000 Mathematical Subject Classification41A46 (primary), 60G15 (secondary).     

Riesz Transforms on a Discrete Group of Polynomial Growth

Let G be a finitely generated, discrete group of polynomialgrowth. This paper provides a proof of the boundedness in L_p(G),1 < p < , of some higher-order Riesz transform operatorsassociated with a probability density on G. 2000 MathematicsSubject Classification 60B15, 43A15.     

Factorization in finite-codimensional ideals of group algebras

Let G be a -compact, locally compact group and I be a closed2-sided ideal with finite codimension in L¹(G). It is shownthat there are a closed left ideal L having a right boundedapproximate identity and a closed right ideal R having a leftbounded approximate identity such that I = L + R. The proofuses ideas from the theory of boundaries of random walks ongroups. 2000 Mathematics Subject Classification: primary 43A20;secondary 42A85, 43A07, 46H10, 46H40, 60B11.     

Self-similar solutions to the isothermal compressible Navier-Stokes equations

^** Email: guo_zhenhua{at}iapcm.ac.cn^*** Email: jiang{at}iapcm.ac.cn We investigate the self-similar solutions to the isothermalcompressible Navier–Stokes equations. The aim of thispaper is to show that there exist neither forward nor backwardself-similar solutions with finite total energy. This generalizesthe results for the incompressible case in Neas, J., Rika, M.& verák, V. (1996, On Leray's self-similar solutionsof the Navier-Stokes equations. Acta. Math., 176, 283–294),and is consistent with the (unproved) existence of regular solutionsglobally in time for the compressible Navier–Stokes equations.     

Weak Laws of Large Numbers in Some Non-Commutative Spaces

We prove weak laws of large numbers for freely independent,uniformly bounded and non-identically distributed random variablesbelonging to non-commutative Ciach and, in particular, LorentzL^p-spaces. 2000 Mathematics Subject Classification 60B11, 60F25,46L54, 47N30.     

Shift-Generated Random Permutations and The M/D/1 Queue

We consider a simple shift procedure which transforms any mappingf:{0, ..., n}{0, ..., n} into a permutation of 0, ..., n. Westudy this shift under the assumption that the values f(0),..., f(n) are uniformly and independently chosen random numbersin {0, ..., n}. The exact and asymptotic distributions of therequired shift sizes are derived, and a connection to queueingtheory is exhibited. 1991 Mathematics Subject Classification60C05, 60K25.