Pseudo algebraically closed fields over rings

We prove that for almost allσ ∈G ℚ the field has the following property: For each absolutely irreducible affine varietyV of dimensionr and each dominating separable rational mapϕ:V→ there exists a point a ∈ such thatϕ(a) ∈ ℤ^r. We then say that is PAC over ℤ. This is a stronger property then being PAC. Indeed we show that beside the fields other fields which are algebraic over ℤ and are known in the literature to be PAC are not PAC over ℤ.     

On non-closure of range of values of elliptic operator for a plane angle

Summary Let a plane angle of opening α∈(π, 2π). LetP _D andP _N the Dirichlet and Neumann problems associated to the Poisson equation in . ForP _D andP _N it is proved non existence of solution in L _p ( ) whenp=2/(1±π/α). In other words, the ranges of elliptic operators naturally associated toP _D andP _N are not-closed in L _p ( ) forp=2/(1±π/α).

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Sunto Sia } un angolo piano di apertura α∈(π, 2π). SianoP _D eP _N i problemi di Dirichlet e di Neumann associati all'equazione di Poisson in . PerP _D eP _N si prova non esistenza di soluzioni in L _p ( ) quandop=2/(1±π/α). Vale a dire i ranges degli operatori ellittici naturalmente associati aP _D eP _N sono non-chiusi in π^--AgBrα _K L _p ( ) perp=2/(1±π/α).

     

Asymptotic properties for median cross-validated nearest neighbor median estimate in nonparametric regression

Consider the nonparametric regression model , whereg is an unknown function to be estimated on [0, 1], are the fixed design points in the interval [0, 1] and is a triangular array of row iid random variables having median zero. The nearest neighbor median estimator is taken as the estimator of the unknown functiong(x). Median cross validation (mev) criterion is employed to select the smoothing parameterh. Leth _π ^* be the smoothing parameter chosen by mev criterion. Under mild regularity conditions, the upper and lower bounds ofh _π ^* , the rate of convergence and the weak consistency of the median cross-validated estimate are obtained. Project supported by the National Natural Science Foundation of China and the Doctoral Foundation of Education of China.     

Eigenvalue estimate on a compact Riemann manifold

LetM be a compact Riemann manifold with the Ricci curvature ≽ - R(R = const. > 0) . Denote by d the diameter ofM. Then the first eigenvalue λ₁ ofM satisfies . Moreover if , then      

The Yang-Mills fields on the Minkowski space

Let the coordinatex=(x ⁰,x ¹,x ²,x ³) of the Minkowski spaceM ⁴ be arranged into a matrix

Sui fibrati con struttura quaternionale generalizzata

Summary Quaternion generalized fiber bundles are studied, both isomorphic to global tensorial product ordinary quaternion fiber bundles right and left respectively) and quite general ones. A cohomology class is considered which represents the obstruction in order the fiber bundle be a tensorial product. Several properties and a splitting principle are proved for bundles . On this ground and founding on a convenient bundle B_E → X associated to jaz (that we call Bonan's bundle and for which ɛ( =ɛ(B_E)) relations are stated among Stiefel-Whitney classes of , B_E and the class ɛ.

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Entrata in Redazione il 14 agosto 1974.

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Lavoro eseguito con contributo del C.N.R., nell'ambito del Gruppo Nazionale per le Strutture Algebriche e Geometriche e loro Applicazioni.     

Absolutely continuous states of exit measures for super-Brownian motions with branching restricted to a hyperplane

The exit measures of super-Brownian motions with branching mechanism from a bounded smooth domain D in ℝ^d+1 are known to be absolutely continuous with respect to the surface area on ϑD if whereas in the case they are singular. However, if the branching is restricted to a singular hyperplane, it is proved that they have absolutely continuous states for alld≥1. Project supported by the National Natural Science Foundation of China (Grant No. 16931063).     

Equivariant D-modules attached to nilpotent orbits in a semisimple Lie algebra

Let _u be a compact Lie algebra and let _u be its complexification. Let ζ^−1/2 be the inverse on the set of regular elements of _u of a square root of the discriminant of . Generalizing a result of W. Lichtenstein in the case _u = (n, ℂ) or (nℝ), we prove that ∂(q).ζ^1/2 is non zero for all harmonic polynomialsq ∈S( ) \ {0}. This fact is deduced from results about equivariantD-modules supported on the nilpotent cone of .     

Stability of the topological pressure for piecewise monotonic maps underC 1-perturbations

Assume thatX is a finite union of closed intervals and consider aC ¹-mapX→ℝ for which {c∈X: T′c=0} is finite. Set . Fix ann ∈ ℕ. For ε>0, theC ¹-map is called an ε-perturbation ofT if is a piecewise monotonic map with at mostn intervals of monotonicity and is ε-close toT in theC ¹-topology. The influence of small perturbations ofT on the dynamical system (R(T),T) is investigated. Under a certain condition on the continuous functionf:X → ℝ, the topological pressure is lower semi-continuous. Furthermore, the topological pressure is upper semi-continuous for every continuous functionf:X → ℝ. If (R(T),T) has positive topological entropy and a unique measure μ of maximal entropy, then every sufficiently small perturbation ofT has a unique measure of maximal entropy, and the map is continuous atT in the weak star-topology.     

Small into-isomorphism from L∞(A,μ) into L∞(B,υ)

In this paper we shall assert that if T is an isomorphism of L_∞(Ω₁, A, μ) into L_∞(Ω₂, B, υ) satisfying the condition ‖T‖·‖T ⁻¹‖⩽1+ɛ for ɛ∈ , then is close to an isometry with an error less than 6ε in some conditions.     

Field dependence of magnetoresistance in epitaxial single-crystal La2/3Ca1/3MnO3-δ thin film

Epitaxial La_2/3Ca_1/3MnO₃ thin films were prepared on NdGaO₃(1lO) substrates by d.c. magnetron sputtering method. The measurements of magnetoresistance ρ(H) upon magnetic field at different temperatures were carried out in the field range of 0–8 T. It is found that ρ(H) obeys the following relations: when the temperature (T) is higher than the Curie temperature below and whenT is far WowT _c. It is suS8ested that the negative magnetoresistive effect is mainly due to enhancement of the magnetoconductance. Project supported by the National Natural Science Foundation of China (Grant No. 19504012) and the Chinese Academy of Sciences.     

Zur Topologie der Dixmier-Abbildung

Summary  LetG be the coadjoint group of a finite-dimensional complex Lie algebrag. Forg solvable, the Dixmier-map is known to be a homeomorphism of the orbit space /G onto the space χ of primitive ideals in the enveloping algebra U(G) [6,15]. For , the Dixmier-map is known to be a bijection (and in general not a homeomorphism) with the space χ^l of all completely prime primitive ideals [7, 16]. Here we derive from a result ofW. SOERGEL [18], that this map issheet- wise a homeomorphism onto the image. Here a sheet is a maximal irreducible subset consisting of orbits of a fixed dimension; obviouslyg decomposes into finitely many sheets [3]. The results of this paper hold more generally forg semisimple, if one restricts to a sheet of polarizable orbits, where a Dixmier-map can be defined. Relative to a fixed polarization (a parabolic subalgebra)p ⊂g let I be the annihilator of the generic module induced from p. The „relative enveloping algebra“ ) has been studied e.g. bySOERGEL [19, 18]. Its center Z is described here by a relative Harish-Chandra isomorphism of the normalization with a suitable ring of group invariants (3.2). We study here the extension ofU by . We suggest that this very mild central extension ofU generates good properties and is very suitable for the study of the Dixmier-map (cf.4.3,5.6). In particular, we conjecture in case : Every minimal primitive ideal of is generated by a maximal ideal of the center. This would generalize for a well known theorem ofM. Duflo (casep Borel, where ). AsJ. Dixmier communicated in a letter, the main result here is exactly what he had hoped for when he first introduced a notion of sheets many years ago.

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Added in proof: This conjecture will be proved in a subsequent paper.     

H p (Rn) is equidistributed withL p (Rn)

Let 0<p<∞. LetH ^p (R ⁿ) be the real variable Hardy spaces defined by Stein and Weiss. Let L^p(R ⁿ) be the usual Lebesgue space. It is shown that forf∈L ^p there is an with the distribution functions of |f| and identical and . The converse is trivially true. Research partially supported by NSF Grant #MCS77-02213.     

A characterization of wavelet families arising from biorthogonal MRA’s of multiplicityd

In this article we give a necessary and sufficient condition for a pair of wavelet families in L²(ℝ ⁿ ), to arise from a pair of biorthogonal MRA’s. The condition is given in terms of simple equations involving the functions ψ^ℓ and . To work in greater generality, we allow multiresolution analyses of arbitrary multiplicity, based on lattice translations and matrix dilations. Our result extends the characterization theorem of G. Gripenberg and X. Wang for dyadic orthonormal wavelets in L²(ℝ),and includes, as particular cases, the sufficient conditions of P. Auscher and P.G. Lemarié in the biorthogonal situation.     

A Chebotarev Theorem for finite homogeneous extensions of shifts

We derive a Chebotarev Theorem for finite homogeneous extensions of shifts of finite type. These extensions are of the form :X×G/H→X×G/H where (x,gH)=(σx, α(x)gH), for some finite groupG and subgroupH. Given a σ-closed orbit τ, the periods of the -closed orbits covering τ define a partition of the integer |G/H|. The theorem then gives us an asymptotic formula for the number of closed orbits with respect to the various partitions of the integer |G/H|. We apply our theorem to the case of a finite extension and of an automorphism extension of shifts of finite type. We also give a further application to ‘automorphism extensions’ of hyperbolic toral automorphisms. Financially supported by Universiti Kebangsaan Malaysia     

Computation of generalized real radicals of polynomial ideals

For an ordered field (K,T) and an idealI of the polynomial ring , the construction of the generalized real radical ofI is investigated. When (K,T) satisfies some computational requirements, a method of computing is presented. Project supported by the National Natural Science Foundation of China (Grant No. 19661002) and the Climbing Project.     

On generalized local time for the process of brownian motion

We prove that the functionals of a d-dimensional Brownian process are Hida distributions, i.e., generalized Wiener functionals. Here, δ_Γ(·) is a generalization of the δ-function constructed on a bounded closed smooth surface Γ⊂R ^d , k≥1 and acting on finite continuous functions φ(·) in R ^d according to the rule where ι(·) is a surface measure on Γ.     

Asymptotic expansion of the log-likelihood function based on stopping times defined on a Markov process

Consider the parameter space Θ which is an open subset of ℝ ^k ,k≧1, and for each θ∈Θ, let the r.v.′sY _n ,n=0, 1, ... be defined on the probability space (X,A,P _θ) and take values in a Borel setS of a Euclidean space. It is assumed that the process {Y _n },n≧0, is Markovian satisfying certain suitable regularity conditions. For eachn≧1, let υ _n be a stopping time defined on this process and have some desirable properties. For 0 < τ _n → ∞ asn→∞, set h _n →h ∈R ^k , and consider the log-likelihood function of the probability measure with respect to the probability measure . Here is the restriction ofP _θ to the σ-field induced by the r.v.′sY ₀,Y ₁, ..., . The main purpose of this paper is to obtain an asymptotic expansion of in the probability sense. The asymptotic distribution of , as well as that of another r.v. closely related to it, is obtained under both and . This research was supported by the National Science Foundation, Grant MCS77-09574. Research supported by the National Science Foundation, Grant MCS76-11620.     

Chaos and asymptotical stability in discrete-time recurrent neural networks with generalized input-output function

We theoretically investigate the asymptotical stability, local bifurcations and chaos of discrete-time recurrent neural networks with the form of

Exponential generating functions and complexity of Lie varieties

Suppose that % MathType!End!2!1! is a variety of Lie algebras, and letc _n( % MathType!End!2!1!) be the dimension of the linear span of all multilinear words onn distinct letters in the free algebraF( % MathType!End!2!1!,X) of the variety % MathType!End!2!1!. We consider an exponential generating function % MathType!End!2!1!, called the complexity function. The complexity function is an entire function of a complex variable provided the variety of Lie algebras is nontrivial. In this paper we introduce the notion of complexity for Lie varieties in terms of the growth of complexity functions; also we describe what the complexity means for the codimension growth of the variety. Our main goal is to specify the complexity of a product of two Lie varieties in terms of the complexities of multiplicands. The main observation here is thatC( % MathType!End!2!1!),z) behaves like a composition of three functionsC( % MathType!End!2!1!),z), exp(z), andC( % MathType!End!2!1!),z). Partially supported by grant RFFI 96-01-00146; the author is grateful to the University of Bielefeld for hospitality, where he was DAAD-fellow.