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 .     

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 .     

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.     

Fractional integrodifferentiation in Hölder classes of arbitrary order

Hölder classes of variable order (x) are introduced and it is shown that the fractional integralI ₀₊ has Hölder order (x)+ (0 < , ₊, +₊ < 1, ₊ = sup (x)).     

Weak Mixing of Random Walks on Groups

Let G be a locally compact -compact group with right Haar measure m and a regular probability measure on G. We say that is weakly mixing if for all gL (G) and all fL ¹(G) with fdm=0 we have n ^{–1 n} _k=1| ^k *f,g|0. We show that is weakly mixing if and only if is ergodic and strictly aperiodic. To prove this we use and prove some results about unimodular eigenvalues for general Markov operators.     

Nonnegative Hall Polynomials

The number of subgroups of type and cotype in a finite abelian p-group of type is a polynomialg with integral coefficients. We prove g has nonnegative coefficients for all partitions and if and only if no two parts of differ by more than one. Necessity follows from a few simple facts about Hall-Littlewood symmetric functions; sufficiency relies on properties of certain order-preserving surjections that associate to each subgroup a vector dominated componentwise by . The nonzero components of (H) are the parts of , the type of H; if no two parts of differ by more than one, the nonzero components of – (H) are the parts of , the cotype of H. In fact, we provide an order-theoretic characterization of those isomorphism types of finite abelian p-groups all of whose Hall polynomials have nonnegative coefficients.     

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.     

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.     

Bestimmung der Eigenwerte und Eigenvektoren einer Matrix mit Hilfe des Quotienten-Differenzen-Algorithmus

Summary Two previous papers (in Vol. V) describe theory and some applications of the quotient-difference (=QD-) algorithm. Here we give an extension which allows the determination of the eigenvectors of a matrix. Letx ₍₀₎ ¹ , ...,x ₍₀₎ ⁿ be a coordinate system in whichA has Jacobi form (such a system may be constructed with methods ofC. Lanczos orW. Givens). Then the QD-algorithm allows the construction of a sequence of coordinate systemsx ₍₂₎ ¹ , ...,x ₍₂₎ ⁿ , (=0, 1, 2, ...) which converge for to the system of the eigenvectors ofA.     

Convergence proofs and error estimates for an iterative method for conformal mapping

Summary The iterative method as introduced in [8] and [9] for the determination of the conformal mapping of the unit disc onto a domainG is here described explicitly in terms of the operatorK, which assigns to a periodic functionu its periodic conjugate functionK u. It is shown that whenever the boundary curve ofG is parametrized by a function with Lipschitz continuous derivative then the method converges locally in the Sobolev spaceW of 2-periodic absolutely continuous functions with square integrable derivative. If is in a Hölder classC ²⁺, the order of convergence is at least 1+. If is inC ^l+1+ withl1, 0<<1, then the iteration converges inC ^l+. For analytic boundary curves the convergence takes place in a space of analytic functions.For the numerical implementation of the method the operatorK can be approximated by Wittich's method, which can be applied very effectively using fast Fourier transform. The Sobolev norm of the numerical error can be estimated in terms of the numberN of grid points. It isO(N ^1–l–) if is inC ^l+1+, andO (exp (–N/2)) if is an analytic curve. The number in the latter formula is bounded by logR, whereR is the radius of the largest circle into which can be extended analytically such that'(z)0 for |z|<R. The results of some test calculations are reported.     

Necessary and sufficient conditions for the convergence of convolution products of non-identical distributions on finite abelian semigroups

Given a sequence of probability measures ( _n ) on a finite abelian semigroup, we present necessary and sufficient conditions which guarantee the weak convergence of the convolution products _{k,n k+1}*···* _n (k<n), asn for allk0. These conditions are verifiable in the sense that they are based entirely on the individual measures in the sequence ( _n ).     

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 extensions of the Hewitt-Savage zero-one law

Summary Let denote the class of infinite product probability measures = ₁× ₂× defined on an infinite product of replications of a given measurable space (X, A), and let denote the subset of for which (A) =0 or 1 for each permutation invariant event A. Previous works by Hewitt and Savage, Horn and Schach, Blum and Pathak, and Sendler (referenced in the paper) discuss very restrictive sufficient conditions under which a given member , of belongs to . In the present paper, the class is shown to possess several closure properties. E.g., if and _{0 n} for some n 1, then ₀× ₁× ₂×.... While the current results do not permit a complete characterization of they demonstrate conclusively that is a much larger subset of than previous results indicated. The interesting special case X={0,1} is discussed in detail.Research supported by the National Science Foundation under grant No. MCS75-07556     

Sur certaines algèbres associées à une mesure de Guelfand

Let be a Guelfand measure (cf. [A, B]) on a locally compact groupG DenoteL ₁ (G)=*L ₁(G)* the commutative Banach algebra associated to . We show thatL ₁ (G) is semi-simple and give a characterization of the closed ideals ofL ₁ (G). Using the -spherical Fourier transform, we characterize all linear bounded operators inL ₁ (G) which are invariants by -translations (i.e. such that ¹(( _x f) )=( _x ((f)) for eachxG andfL ₁ (G); where _x f(y)=f(xy); x,y G). WhenG is compact, we study the algebraL ₁ (G) and obtain results analogous to ones obtained for the commutative case: we show thatL ₁ (G) is regular, all closed sets of its Guelfand spectrum are sets of synthesis and establish theorems of harmonic synthesis for functions inL _p (G) (p=1,2 or +).

     

On the eigenvalues of the matrix pencilA+μB

Summary The number of independent invariants ofn×n matricesA, B and their products on which the eigenvalues () of the matrix pencilA+B depend is determined by means of the theory of algebraic invariants and combinatorial analysis. Formulas are displayed for coefficients for the calculation of () forn5.

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Zusammenfassung Wir bestimmen die Anzahl der unabhängigen Invarianten dern×n MatrizenA, B und ihrer Produkte, von denen die Eigenwerte () der MatrixbüschelA+B abhängen, mittels der Theorie der algebraischen Invarianten und mittels kombinatorischer Analyse. Formeln für Koeffizienten zur Berechnung von () werden angegeben fürn5.

     

The convolution equation of Choquet and Deny on [IN]-groups

Let be a probability measure on a locally compact groupG. A real Borel functionf onG is called -harmonic if it satisfies the convolution equation *f=f. Given that isnonsingular with its translates, we show that the bounded -harmonic functions are constant on a class of groups including the almost connected [IN]-groups. If is nondegenerate and absolutely continuous, we solve the more general equation *= for positive measure on those groups which are metrizable and separable.Supported by Hong Kong RGC Earmarked Grant and CUHK Direct Grant     

On a Parabolic Variational Inequality That Generalizes the Equation of Polytropic Filtration

We obtain conditions for the existence and uniqueness of a solution of a parabolic variational inequality that is a generalization of the equation of polytropic elastic filtration without initial conditions. The class of uniqueness of a solution of this problem consists of functions that increase not faster than e ^–t , > 0, as t –.     

Ultrafilter translations

We develop a method for extending results about ultrafilters into a more general setting. In this paper we shall be mainly concerned with applications to cardinality logics. For example, assumingV=L, Gödel's Axiom of Constructibility, we prove that if > then the logic with the quantifier there exist many is (,)-compact if and only if either is weakly compact or is singular of cofinality<. As a corollary, for every infinite cardinals and , there exists a (,)-compact non-(,)-compact logic if and only if either < orcf<cf or < is weakly compact.Counterexamples are given showing that the above statements may fail, ifV=L is not assumed.However, without special assumptions, analogous results are obtained for the stronger notion of [,]-compactness.     

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.