Eigenvalue distribution of some fractal semi-elliptic differential operators: Combinatorial approach

We obtain the sharp order of growth of the eigenvalue distribution function for the operator in the anisotropic Sobolev space , generated by the quadratic form _Q u² d, whereQ² is the unit square and is a probability self-affine fractal measure onQ. The geometry of Supp should be in a certain way consistent with the parameterst ₁ ,t ₂ .     

Finitely subadditive and countably subadditive outer measures

Results are given comparing countably subadditive (csa) outer measures and finitely subadditive (fsa) outer measures, especially relating to regularity and measurability conditions such as (*) condition:A setE (of an arbitrary setX), is measurable ( an outer measure),ES (the collection of measurable sets) iff (X)=(E)+(E). Specific examples are given contrasting csa and fsa outer measures. In particular fsa and csa outer measures derived from finitely additive measures defined on an algebra of sets generated by a lattice of sets, are investigated in some detail.     

Some types of vector measures

Letx be a metrizable locally convex space with a Schauder basis and letB(T) be a -ring generated by the compact subsets of a locally compact Hausdorff spaceT. We prove that any vector measure :B(T)X which has an antiregular relative is antimonogenic (Theorem 16) and that can be uniquely decomposable, = ₁ + ₂, where ₁ is monogenic and ₂ has an antiregular relative (Theorem 19). These results are due to R. A. Johnshon [6] for the case whereX is the real line.     

Dimension and entropy of a non-ergodic Markovian process and its relation to Rademacher Riesz Products

Given a Markov chain (not necessarily stationary or homogeneous) with finite state space and an initial distribution, we can construct a measure on the unit interval [0, 1]. In this work we examine the equality (up to a constant) of the Hausdorff dimension of and of a suitably defined entropy for the Markovian process. The results are applied to the so-called Rademacher-Riesz Products.     

Cycles and measure of bifurcation sets for two-dimensional diffeomorphisms

Summary We describe a large class of one-parameter families , {}, , of two-dimensional diffeomorphisms which arestable for <0, exhibit acycle for =0, and thereafter have a bifurcation set of positive but arbitrarily smallrelative measure for in small intervals [0, ]. A main assumption is that the basic sets involved in the cycle havelimit capacities that are not too large.The second author acknowledges hospitality and financial support from IMPA/CNPq during the period this paper was prepared     

Harmonic analysis in infinite dimension

We study the Hermite transform onL ²() where is a Gaussian measure on a Lusin locally convex spaceE. We are then lead to a Hilbert space () of analytic functions onE which is also a natural range for the Laplace transform. LetB be a convenient Hilbert-Schmidt operator on the Cameron-Martin spaceH of . There exists a natural sequence Cap _n of capacities onE associated toB. This implies the Kondratev-Yokoi theorem about positive linear forms on the Hida test-functions space.     

On regular embedding integrally in R3 of metrics of class C4 of negative curvature

On the x₀y plane let there be specified a complete metric of negative curvature K by means of the line element ds²=dx²+B²(x, y) dy², and, in the strip _a={0xa, -4-bounded function B>0,K-²<0 ( and are constants). Then, the metric in strip _a is embedded in R³ by means of a surface of class C³.Translated from Matematicheskie Zametki, Vol. 14, No. 2, pp. 261–266, August, 1973.     

Über die dem Lebesgueschen Mass isomorphen topologischen Massräume

Homomorphisms of topological measure spaces had been defined in [5] to be measure-preserving and almost everywhere continuous mappings; this induces a concept of isomorphic topological measures. The main result of the present paper is that a locally finite atomfree measure in a completely regular space X with a countable base is isomorphic to the Lebesgue measure in an interval (case of finite measure) or the real line R (case of infinite measure) if and only if X contains a Polish subspace P such that (X-P)=0. A corollary states that any measure satisfying these conditions is carried by a G-subset of X which can be mapped onto a G-subset of R by a --measure-preserving homeomorphism. Measures with atomic components are also treated. Further theorems concern measures in non-metrizable spaces or spaces without a countable base. For example it is proved that all compactifications (in the sense of topological measures) of a tight topological measure space are isomorphic, and they are isomorphic to the space itself if this space admits a complete metric.

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Die Arbeit dieses Autors wurde durch ein Stipendium der Alexander von Humboldt-Stiftung ermöglicht.     

Nonlinear potential theory and PDEs

We consider equations like -div(|u| ^p–2u)=, where is a nonnegative Radon measure and 1u and the measure are reviewed. A link between potential estimates and the boundary regularity of the Dirichlet problem is established.     

Branching processes,random trees,and a generalized scheme of arrangements of particles

It is shown that the conditional distributions of a number of characteristics of a branching process (t), (0)=m, under the condition that the number of total progeny _m in this process is equal to n, coincide with the distributions of the corresponding characteristics of a generalized scheme of arrangement of particles in cells. In the case where the number of offsprings of a particle has the Poisson distribution, the characteristics of the branching process (t), (0)=1, under the condition that ₁=n+1, coincide with the characteristics of a random tree. By using these connections we obtain in this article a series of limit theorems as n for characteristics of random trees and branching processes under the conditions that _m=n.Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 691–705, May, 1977.     

A quadrature formula for the Hankel transform

We present in this paper a quadrature formula for a certain Fourier-Bessel transform and, closely related to this, for the Hankel transform of order >–1. Such formulas originate in the context of a Galerkin-type projection of the weightedL ₂(–, ; ) space ( is the weight function mentioned below) used to get a discrete representation of a certain physical problem in Quantum Mechanics. The generalized Hermitee polynomialsH ₀ (x),H ₁ (x),..., with weight function (x), are used as the basis on which such a projection takes place. It is shown that theN-dimensional vectors representing certain projected functions as well as the entries of theN×N matrix representing the kernel of that Fourier-Bessel transform, approach the exact functional values at the zeros of theNth generalized Hermitee polynomial whenN.These properties lead to propose this matrix as a finite representation of the kernel of the Fourier-Bessel transform involved in this problem and theN zeros of the generalized Hermitee polynomialH _N (x) as abscissas to yield certain quadrature formulae for this integral and for the related Hankel transform. The error function produced by this algorithm is estimated at theN nodes and its is shown to be of a smaller order than 1/N. This error estimate is valid for piecewise continuous functions satisfying certain integral conditions involving their absolute values. The algorithm is presented with some numerical examples.     

Integral representation of linear operators

In this paper we obtain necessary and sufficient conditions in order that a linear operator, acting in spaces of measurable functions, should admit an integral representation. We give here the fundamental results. Let (T_i, _i) (i=1,2) be spaces of finite measure, and let (T,) be the product of these spaces. Let E be an ideal in the space S(T₁, ₁) of measurable functions (i.e., from |e₁||e₂|, e₁ S (T₁, ₁), e₂E it follows that e₁E). THEOREM 2. Let U be a linear operator from E into S(T₂, ₂). The following statements are equivalent: 1) there exists a-measurable kernel K(t,S) such that (Ue)(S)=K(t,S) e(t)d(t) (eE); 2) if 0e_nE (n=1,2,...) and e_n0 in measure, then (Ue_n)(S) 0 ₂ a.e. THEOREM 3. Assume that the function (t,S) is such that for any eE and for s a.e., the ₂-measurable function Y(S)=(t,S)e(t)d ₁(t) is defined. Then there exists a-measurable function K(t,S) such that for any eE we have (t,S)e(t)d ₁(t)=K(t,S)e(t)d ₁(t) ₁a.e.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 47, pp. 5–14, 1974.     

Large Deviation Principle for Additive Functionals of Brownian Motion Corresponding to Kato Measures

Let (B _t ,P ^W _x ) be the Brownian motion. Let be a Radon measure in the Kato class and A _t the additive functional associated with . We prove that A _t /t obeys the large deviation principle.     

Properties of a special class of doubly stochastic measures

Summary A measure on the unit squareI } I is doubly stochastic if(A } I) = (I } A) = the Lebesgue measure ofA for every Lebesgue measurable subsetA ofI = [0, 1]. By the hairpinL L ^–1, we mean the union of the graphs of an increasing homeomorphismL onI and its inverseL ^–1. By the latticework hairpin generated by a sequence {x _n :n Z} such thatx _n-1 < x_n (n Z), x _n = 0 and x _n = 1, we mean the hairpinL L ^–1 , whereL is linear on [x _n-1 ,x _n ] andL(n) =x _n-1 forn Z. In this note, a characterization of latticework hairpins which support doubly stochastic measures is given. This allows one to construct a variety of concrete examples of such measures. In particular, examples are given, disproving J. H. B. Kemperman's conjecture concerning a certain condition for the existence of doubly stochastic measures supported in hairpins.     

The existence of probability measures with specified projections

Let X and Y be locally compact-compact topological spaces, F X×Y is closed, and P(F) is the set of all Borel probability measures on F. For us to find, for the pair of probability measures (_x, _y P (X)×P(Y), a probability measure P(F) such that _X = _X ^–1 , _Y = _Y ^–1 it is necessary and sufficient that, for any pair of Borel sets A X, B Y for which (A× B) F=Ø, the condition _XA+ _YB 1 holds.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 573–576, October, 1973.     

Limiting behavior of weighted central paths in linear programming

We study the limiting behavior of the weighted central paths{(x(), s())} > 0 in linear programming at both = 0 and = . We establish the existence of a partition (B ,N ) of the index set { 1, ,n } such thatx _i() ands _j () as fori B , andj N , andx _N (),s _B () converge to weighted analytic centers of certain polytopes. For allk 1, we show that thekth order derivativesx ^(k) () ands ^(k) () converge when 0 and . Consequently, the derivatives of each order are bounded in the interval (0, ). We calculate the limiting derivatives explicitly, and establish the surprising result that all higher order derivatives (k 2) converge to zero when .     

Convolution Operators with Fractional Measures Associated to Holomorphic Functions

Let be an open set in the complex plane and let be a holomorphic function on . Let K be a compact subset of with nonempty interior such that 0 K. Let be the Borel measure of R ⁴ C ² given by(E = _K E(z, (z))|z|^–2 d(z)where 0 < 2 and d(x ₁ + ix ₂) = dx ₁ dx ₂ denotes the Lebesgue measure on C. Let T be the convolution operator T f = * f. In this paper we characterize the type set E associated to T .     

Ein Masstheoretisches Marginalproblem

Let ((X_i, K_i, _i) iI) be a family of normed measure spaces. We study the extremal points of the convex set F of normed measures on the product of ((X_i, K_i): iI) with the marginal measures _i. We give a construction principle for extremal points. If _i is the Lebesgue measure on [0, 1] and I is countable, we prove by using this principle that the set of extremal points of F is weakly dense in F. Finally we give a necessary and some sufficient conditions for extremal points in the case that I={1,2} and _i is the Lebesgue measure on [0,1] for i=1,2.     

Characterization of regular singular linear systems of difference equations

This paper deals with linear systems of difference equations whose coefficients admit generalized factorial series representations atz=. We are concerned with the behavior of solutions near the pointz= (the only fixed singularity for difference equations). It is important to know whether a system of linear difference equations has a regular singularity or an irregular singularity. To a given system () we can assign a number , called the Moser's invariant of (), so that the system is regular singular if and only if 1. We shall develop an algorithm, implementable in a computer algebra system, which reduces in a finite number of steps the system of difference equations to an irreducible form. The computation ot the number can be done explicitly from this irreducible form.     

The Spectral Theory of Amenable Actions and Invariants of Discrete Groups

Let G denote a semisimple group, a discrete subgroup, B=G/P the Poisson boundary. Regarding invariants of discrete subgroups we prove, in particular, the following:(1) For any -quasi-invariant measure on B, and any probablity measure on , the norm of the operator () on L ²(B,) is equal to (), where is the unitary representation in L ²(X,), and is the regular representation of .(2) In particular this estimate holds when is Lebesgue measure on B, a Patterson–Sullivan measure, or a -stationary measure, and implies explicit lower bounds for the displacement and Margulis number of (w.r.t. a finite generating set), the dimension of the conformal density, the -entropy of the measure, and Lyapunov exponents of .(3) In particular, when G=PSL₂() and is free, the new lower bound of the displacement is somewhat smaller than the Culler–Shalen bound (which requires an additional assumption) and is greater than the standard ball-packing bound.We also prove that ()=_G() for any amenable action of G and L ¹(G), and conversely, give a spectral criterion for amenability of an action of G under certain natural dynamical conditions. In addition, we establish a uniform lower bound for the -entropy of any measure quasi-invariant under the action of a group with property T, and use this fact to construct an interesting class of actions of such groups, related to 'virtual' maximal parabolic subgroups. Most of the results hold in fact in greater generality, and apply for instance when G is any semi-simple algebraic group, or when is any word-hyperbolic group, acting on their Poisson boundary, for example.