Approximation to the expectation of a function of order statistics and its applications

ThisprojectissupportedbytheNationalNaturalScienceFoundationofChinaandDoctoralProgramFoundationofHigherEducation.1.IntroductionLetUI,U2,'bei.i.d.randomvariableswithuniformd.f.ontheinterval(0,l),andforeveryn31,writeUt,,15'5Un,.fortheorderstatisticsofUI,'tUn.SupposethatXI1X2,'arei.i.d.observationsfromanondegenerated.f.F,anddenotebyX.,l5'5X.,.theorderstatisticsofXI,'IX,,'Withoutlossofgenerality,wewillassume0相似文献   

(C,α) Summability of Walsh—Fourier Series

The one-dimensional dyadic martingale Hardy spaces H _p are introduced and it is proved that the maximal operator of the (C,) means of a Walsh—Fourier series is bounded from H _p to L _p (1/( + 1) < p < ) and is of weak type (L ₁,L ₁). As a consequence, we obtain the summability result due to Fine; more exactly, the (C,) means of the Walsh—Fourier series of a function f L ₁ converge a.e. to the function in question. Moreover, we prove that the (C,) means are uniformly bounded on H _p whenever 1/( + 1) < p < . We define the two-dimensional dyadic hybrid Hardy space H ₁ and verify that the maximal operator of the (C,,) means of a two-dimensional function is of weak type H ₁ ,L ₁). Consequence, the Walsh—Fourier series of every function f H ₁ is (C,,) summable to the function f.     

On the linear independence of numbers over number fields

In the present paper, the problem of a lower bound for the measure of linear independence of a given collection of numbers ₁, , _n is considered under the assumption that, for a sequence of polynomials whose coefficients are algebraic integers, upper and lower estimates at the point ( ₁, , _n ) are known. We use a method that generalizes the Nesterenko method to the case of an arbitrary algebraic number field.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 506–517, October, 1998.The author wishes to thank Professor Yu. V. Nesterenko for setting the problem and valuable advice and Professor D. Bertrand for fruitful discussions.This research was partially supported by the International Science Foundation (Soros Foundation) under grant No. 507_s.     

Approximation of functions,which are analytic in a simply connected domain and can be represented by a Cauchy type integral,by sequences of rational fractions with poles prescribed by an array

Let G and {_kj} be the domain and the array mentioned in the title (the boundary of the domain is assumed to be rectifiable). One describes a general scheme for the approximation of fonctionsf in the domain G, representable in the form f(z)=(2i)^–1g()(–z)^–1d, where g L_z (G), by a sequence of rational fractions. The characteristic feature of this scheme is the fact that the poles _k of the fraction lie in the k-th row of the array {_kj}. There is given a condition on {_kj}, necessary and sufficient in order that each functionf, of the kind described above, should admit a uniform approximation inside G with the aid of the indicated scheme. In the case when this condition is not satisfied and \G.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 254–273, 1989.     

On Finite Morphisms of Ruled Surfaces

We prove that a surjective morphism :XX between ruled surfaces is finite and it descends to a finite morphism :CC between base curves of X and X. When is restricted to the fibres of X, it has a constant degree, say a, and then deg=a deg. In addition, we have several properties on the inverse image of a minimal section and a fibre of X as well as on the direct images. We also investigate precisely the case when both C and C are elliptic and X is the fibre product C ×_C X especially.     

Limits of balayage measures

LetA be a subset of a balayage space (X,W) and a measure onX. It is shown that for every sequence _n of measures such that lim_nn and lim_{n n} ^A = the limit measure is of the formf+[(1-f)]^A for some (unique) Borel function 0f1_Cb(A). Furthermore, conditions are given such that any such functionf occurs.     

(q, δ)-Numeration Systems with Missing Digits

We consider the (q, ) numeration system, with basis q2 and the set of digits {, +1,,q+–1} where –(q–1)0. We study properties of numbers where some digits do not occur. This is analogous to the Cantor set {0.a₁a₂a_i{0,2}}. We compute an asymptotic equivalent of the nth moment of the Cantor (q, D)-distribution which can be described as the numbers 0. w₁w₂ with w_iD{,,q+–1}, and each such letter can occur with the same probability 1/CardD. Furthermore, we consider n random strings according to the distribution and the expected minimum of them. We find a recursion which we solve asymptotically.This author was supported by the CNRS/NRF-project no 10959. Part of this work was done during the first authors visit to the John Knopfmacher Centre for Applicable Analysis and Number Theory at the University of the Witwatersrand, Johannesburg, South Africa.This author was supported by the CNRS/NRF-project no 10959.     

Hamiltonian Paths in Cartesian Powers of Directed Cycles

The vertex set of the k ^th cartesian power of a directed cycle of length m can be naturally identified with the abelian group ( _m ) ^k . For any two elements u=(u ₁,,u _k ) and v=(v ₁,,v _k ) of ( _m ) ^k , it is easy to see that if there is a hamiltonian path from u to v, then     

The center of the 3-dimensional and 4-dimensional Sklyanin algebras

LetA=A(E, ) denote either the 3-dimensional or 4-dimensional Sklyanin algebra associated to an elliptic curveE and a point E. Assume that the base field is algebraically closed, and that its characteristic does not divide the dimension ofA. It is known thatA is a finite module over its center if and only if is of finite order. Generators and defining relations for the centerZ(A) are given. IfS=Proj(Z(A)) andA is the sheaf ofO _S -algebras defined byA(S _(f))=A[f ^–1]₀ then the centerL ofA is described. For example, for the 3-dimensional Sklyanin algebra we obtain a new proof of M. Artin's result thatSpec L². However, for the 4-dimensional Sklyanin algebra there is not such a simple result: althoughSpec L is rational and normal, it is singular. We describe its singular locus, which is also the non-Azumaya locus ofA.     

On the exceptional set for the equationn=p+k 2

Denote byE(X) the number of integersn X which are not a sum of a prime and a square. Set = 1 – 0.994428b, whereb min(0.9365/c ₃, 0.02578/c ₄),c ₃,c ₄ being given in Lemma 8. The resultE(x) x is proved in this paper.Project Supported by the National Natural Science Foundation of China.     

Para-orthogonal Laurent polynomials and associated sequences of rational functions

On the space, , of Laurent polynomials (L-polynomials) we consider a linear functional which is positive definite on (0, ) and is defined in terms of a given bisequence, { _k } _– . Two sequences of orthogonal L-polynomials, {Q _n (z) ₀ and , are constructed which span in the order {1,z ^–1,z,z ^–2,z ²,...} and {1,z,z ^–1,z ²,z ^–2,...} respectively. Associated sequences of L-polynomials {P _n (z) ₀ , and are introduced and we define rational functions , wherew is a fixed positive number. The partial fraction decomposition and integral representation of,M _n (z, w) are given and correspondence of {M _n (z, w)} is discussed. We get additional solutions to the strong Stieltjes moment problem from subsequences of {M _n (z, w)}. In particular when { _k } _– is a log-normal bisequence, {M _2n (z, w)} and {M _2n+1 (z, w)} yield such solutions.Research supported in part by the National Science Foundation under Grant DMS-9103141.     

On the non-commutative neutrix product (x + r lnx +) ox − −s

The non-commutative neutrix product of the distributionsx ₊ ^r lnx ₊ andx _– ^–s is evaluated forr=0, 1, 2, ands=1, 2, . Further neutrix products are then deduced.     

Asymptotic Estimates for a Variational Problem Involving a Quasilinear Operator in the Semi-Classical Limit

Let be a domain of ^N . We study the infimum ₁(h) of the functional |u| ^p +h ^–p V(x)|u| ^p dx in W ^1,p () for ||u|| _LP()= 1 where h > 0 tends to zero and V is a measurable function on . When V is bounded, we describe the behaviour of ₁(h), in particular when V is radial and 'slowly' decaying to zero. We also study the limit of ₁(h) when h 0 for more general potentials and show a necessary and sufficient condition for ₁(h) to be bounded. A link with the tunelling effect is established. We end with a theorem of existence for a first eigenfunction related to ₁(h).     

On the width of ordered sets and Boolean algebras

Thewidth (chain number) of a partial order P, < is the smallest cardinal such that ¦A¦< 1 + whenever A is an antichain (chain) in P. We prove that, if a partial order (P, <) has width and cf()=, then P contains antichains A_n (n<) such that ¦A ₀¦<¦A₁¦ <...<={¦A_n¦: n < < } and either A₀1 A₂< ... or A₀>A₁ >A₂> ... A similar structure result is obtained for partial orders with chain number if cf()=. As an application we solve a problem of van Douwen, Monk and Rubin [1] by showing that if a Boolean algebra has width , thencf() .This work has been partially supported by NATO grant No. 339/84.Presented by Bjarni Jonsson.     

On the singular graph and the Weyr characteristic of anM-matrix

LetA be anM-matrix in standard lower block triangular form, with diagonal blocksA _ii irreducible. LetS be the set of indices such that the diagonal blockA is singular. We define the singular graph ofA to be the setS with partial order defined by > if there exists a chain of non-zero blocksA _i, A_ij, , A_l.Let ₁ be the set of maximal elements ofS, and define thep-th level _p ,p = 2, 3, , inductively as the set of maximal elements ofS \( _{1 p-1}). Denote by _p the number of elements in _p . The Weyr characteristic (associated with 0) ofA is defined to be (A) = ( ₁, ₂,, _h ), where ₁ + + _p = dim KerA ^p ,p = 1, 2, , and _h > 0, _h+1 = 0.Using a special type of basis, called anS-basis, for the generalized eigenspaceE(A) of 0 ofA, we associate a matrixD withA. We show that(A) = ( ₁, , _h) if and only if certain submatricesD _p,p+1 ,p = 1, , h – 1, ofD have full column rank. This condition is also necessary and sufficient forE(A) to have a basis consisting of non-negative vectors, which is a Jordan basis for –A. We also consider a given finite partially ordered setS, and we find a necessary and sufficient condition that allM-matricesA with singular graphS have(A) = ( ₁, , _h). This condition is satisfied ifS is a rooted forest.The work of the second-named author was partly supported by the National Science Foundation, under grant MPS-08618 A02.     

On semiregular conditional distributions

Let (,A,P) denote some probability space and some sub--algebra ofA. It is shown that there exists a semiregular versionQ (A),A, , of the conditional distributionP(A|), AA, i.e., Q (A), (AA fixed) is andAQ (A),AA ( fixed), is a probability charge satisfyingQ (N)=0, , for allP-zero setsN, if and only ifL ₁(,P|) has a lifting, which exists for any sub--algebra ofA ifL ₁(,A P) is separable. Separability ofL ₁(,A,P) implies also the existence of a strongly semiregular versionQ (A),A, , ofP(A|), A , i.e., Q (A), (AA fixed), is -measurable andAQ (A),A ( fixed), is a probability charge. Furthermore,P can be written as P ₁+(1–)P ₂, 01, whereP ₁ are probability measures onA such thatP ₁(A|),AA, has a semiregular version vanishing for anyP-zero setN andP ₂ is singular with respect to any probability measure onA of the type ofP ₁. In the case 0<<1 the probability measuresP _j ,j=1, 2, are uniquely determined. The decomposition can be carried over to the case, where the additional condition thatQ (N)=0 for all and anyP-zero setN is valid, is omitted respectively semiregularity is replaced by (i) strong semiregularity, or (ii) classical regularity. In the last mentioned case (ii) the decomposition is multiplicative.     

Completeness of indexed ε-calculus

Epsilon terms indexed by contexts were used by K. von Heusinger to represent definite and indefinite noun phrases as well as some other constructs of natural language. We provide a language and a complete first order system allowing to formalize basic aspects of this representation. The main axiom says that for any finite collection S ₁,,S _k of distinct definable sets and elements a ₁,,a _k of these sets there exists a choice function assigning a _i to S _i for all ik. We prove soundness and completeness theorems for this system S _i ^fin . Mathematics Subject Classification (2000): 03B10, 03B65     

On the motion of visco-elastic electrically conducting liquid in the presence of a magnetic field

Zusammenfassung Es wird die Strömung einer viskoelastischen, elektrisch leitenden Flüssigkeit untersucht auf Grund der elektromagnetischen Grundgleichung und der für solche Flüssigkeiten geltenden Bewegungsgleichungen. Im besonderen wird die Strömung zwischen parallelen Platten erörtert. Sie ist charakterisiert durch zwei dimensionslose Kennzahlen, die Hartmannsche Zahl und die Elastizitätszahl. Das elastische Verhalten bewirkt eine Verflachung des Geschwindigkeitsprofils und verändert auch den Druck und die Schubspannungen.

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Notations x ₁,x ₂,x ₃ space coordinates - B magnetic induction vector - E electric intensity - H magnetic field - D displacement vector - J current density vector - _e E electrostatic force - V×B current density due to the motion of the fluid - _e V convection current - _e excess charge density - electrical conductivity - magnetic permeability - dielectric constant - =1/ magnetic diffusivity - V=v ¹ i+v ² j+v ³ k velocity vector - v ₁= ₁/ kinematic viscosity - ₁ coefficient of viscosity - _c coefficient of cross-viscosity - v _c = _c / kinematic cross-viscosity - density of the fluid - relaxation time - S _j ⁱ stress-tensor - d _j ⁱ strain-tensor - x, y, z dimensionless coordinates - L spacing between the planes, taken as the characteristic length - U ₀ characteristic velocity - f non-dimensional velocity - H non-dimensionalx-component of the magnetic field.     

Equivalence Bimodule Between Non-Commutative Tori

The non-commutative torus C ^*(ⁿ,) is realized as the C*-algebra of sections of a locally trivial C*-algebra bundle over S with fibres isomorphic to C ^*n/S, ₁) for a totally skew multiplier ₁ on ⁿ/S. D. Poguntke [9] proved that A is stably isomorphic to C(S) C(^*( Zⁿ/S, ₁) C(S) A M_kl( C) for a simple non-commutative torus A and an integer kl. It is well-known that a stable isomorphism of two separable C*-algebras is equivalent to the existence of equivalence bimodule between them. We construct an A-C(S) A-equivalence bimodule.     

Joint convergence of conditional expectations

Let P_n, nIN{0}, be probability measures on a-fieldA; f_n, nIN{0}, be a family of uniformly boundedA-measurable functions andA _n, nIN, be a sequence of sub--fields ofA, increasing or decreasing to the-fieldA _o. It is shown in this paper that the conditional expectations converge in P_o-measure to with k, n, m , if P_n|A, nIN, converges uniformly to P_n|A and f_n, nIN, converges in P_o-measure to f_o.