The Loeb Space of Denumerable Infinite Dimensional Probability Product Measure Spaces

Let {(Xi, Si, μi) : i ℃ N} be a sequence of probability measure spaces and (*Xi, L(*Si), L(*μi)) be the Loeb measure space with respect to (Xi, Si, μi) for i ℃ N. Let X =× Xi, S = ×Si,μ = ×μi. We prove that × L(*Si) CL(*S) and in embedding meaning.

FRAME MULTIRESOLUTION ANALYSIS AND INFINITE TREES IN BANACH SPACES ON LOCALLY COMPACT ABELIAN GROUPS

We extend the concept of frame multiresolution analysis to a locally compact abelian group and use it to define certain weighted Banach spaces and the spaces of their antifunctionals. We define analysis and synthesis operators on these spaces and establish the continuity of their composition. Also, we prove a general result to characterize infinite trees in the above Banach spaces of antifunctionals. This paper paves the way for the study of corresponding problems associated with some other types of Banach spaces on locally compact abelian groups including modulation spaces.

STATIONARY DISTRIBUTIONS OF INFINITE PARTICLE SYSTEMS WITH NONCOMPACT STATE SPACE.

This paper deals with the preblem of existence and uniqueness of the stationary distributions (abbr., s. d.'s) for the processes constructed in [4] .The main results are stated in § 1. For the reader's convenience we first restate the existence theorems (Theorem 1 and 2) of the processes given in [4]. Then two existence theorems (Theorem 3 and 4) and a uniqueness theorem (Theorem 5) for the s. d.'s of the processes are presented. The last result (Theorem 6), as an application of the previous ones, is about the Schlgl model which comes from nonequilibrium statisticali physics. The details of the proofs of Theorem 3—6 are given in § 2—4.

AN EXTREMUM THEORY OF THE RESIDUAL FUNCTIONAL IN SOBOLEV SPACES W^(m,p)(Ω)

In the present paper the concept and properties of the residual functional in Sobolev space are investigated.The weak compactness,force condition,lower semi-continuity and convex of the residual functional are proved.In Sobolev space,the minimum principle of the residual functional is proposed.The minimum existence theoreomfor J(u)=0 is given by the modern critical point theory.And the equivalence theorem or five equivalence forms for the residual functional equation are also proved.

k-MAJOR ORDER IN REAL INFINITE-DIMENSIONAL LINEAR SPACE AND ITS APPLICATION

In this paper,the k major cone and strict k major cone in real infinite dimensional linear space are introduced,through which the k major order is defined,and their properties are also discussed.Therefore,with the help of them any two elements in real infinite dimensional linear space can be compared.

Self-maps of p-local infinite projective spaces

Abstract Denote by z（p） （resp. Zp） the p localization （resp. p completion） of z. Then we have the canonical inclusion Z（p）→ zp. Let S2n-1（p） be the p-local （2n- 1）-sphere and let B2n（p） be a connected p-local space satisfying S2n-l（p）≌ΩB2n（p）, then H＊B2n（p）,Z（p）） = Z（p）[U] with ｜u｜ = 2n. Define the degree of a self-map f of B2n（p） to be k E Z（p） such that f＊（u） = ku. Using the theory of integer-valued polynomials we show that there exists a self-map of B2n（p） of degree k if and only if k is an n-th power in Zp.

The Maximum Dissipative Extension of Schrodinger Operator

In the present paper we study the maximum dissipative extension of Schrodingeroperator.introduce the generalized indefinite metvic space and get the representation ofmaximum dissipative extension of Schrodinger operator in natural boundary space.make preparation for the further study longtime chaotic behaxior of infinite dimensiondynamics system in nonlinear Schrodinger equation.

A microwave photonic filter based on multi-wavelength fiber laser and infinite impulse response

A microwave photonic filter(MPF) based on multi-wavelength fiber laser and infinite impulse response(IIR) is proposed. The filter uses a multi-wavelength fiber laser as the light source, two sections of polarization maintaining fiber(PMF) and three polarization controllers(PCs) as the laser frequency selection device. By adjusting the PC to change the effective length of the PMF, the laser can obtain three wavelength spacings, which are 0.44 nm, 0.78 nm and 1.08 nm, respectively. And the corresponding free spectral ranges(FSRs) are 8.46 GHz, 4.66 GHz and 3.44 GHz, respectively. Thus changing the wavelength spacing of the laser can make the FSR variable. An IIR filter is introduced based on a finite impulse response(FIR) filter. Then the 3-d B bandwidth of the MPF is reduced, and the main side-lobe suppression ratio(MSSR) is increased. By adjusting the gain of the radio frequency(RF) signal amplifier, the frequency response of the filter can be enhanced.

Actads

In this paper,I introduce a new generalization of the concept of an operad,further generalizing the concept of an opetope introduced by Baez and Dolan(1998),who used this for the definition of their version of non-strict n-categories.Opetopes arise from iterating a certain construction on operads called the+-construction,starting with monoids.The first step gives rise to plain operads,i.e.,operads without symmetries.The permutation axiom in a symmetric operad,however,is an additional structure resulting from permutations of variables,independent of the structure of a monoid.Even though we can apply the+-construction to symmetric operads,there is the possibility of introducing a completely different kind of permutations on the higher levels by again permuting variables without regard to the structures on the previous levels.Defining and investigating these structures is the main purpose of this paper.The structures obtained in this way are what I call n-actads.In n-actads with n>1,the permutations on the different levels give rise to a certain special kind of n-fold category.I also explore the concept of iterated algebras over an n-actad(generalizing an algebra and a module over an operad),and various types of iterated units.I give some examples of algebras over 2-actads,and show how they can be used to construct certain new interesting homotopy types of operads.I also discuss a connection between actads and ordinal notation.