Banach空间的正规结构与对径点

假设S（X）是Banach空间X的单位球面,作引进了四个新的几何参数：Jε（X）=sup{βε（x）,x∈S（X）},jε（X）=inf{βε（x）,x∈S（X）},Gε（X）=sup{αε（x）,x∈S（X）},gε（X）=inf{αε（x）,x∈S（S）},其中≤ε≤1,βε（x）=sup{min{‖x εy‖,‖x-εy‖,y∈S（X）}},αε（x）=inf{max{‖x εy‖,‖x-εy‖,y∈S（X）}},讨论了这些参数的性质,本主要结果是：如果主要结果是：如果有一个ε,0≤ε≤1,使得Jε（X）＜1 ε/2或gε（X）＞1 ε/3,那末X有一至正规结构。     

ON THE PUTNAM-FUGLEDE THEOREM OF NON-NORMAL OPERATORS

In this paper we have extended the Putnam-Fuglede Theorem of nomal operators anddiscussed the condition for the Putnam-Fuglede Theorem holding.We have proved that ifA and B~* are hyponomal operators and AX=XB,then A~*X=XB~*;that if A and B~* aresemi-hyponomal operators and X is     

Banach 空间中极大单调算子扰动的值域

设X为实Banach空间, T:D(T)(?)X→2X*为极大单调算子, C: D(T)(?)X→X*为有界算子(未必连续),而C(T+J)-1为紧算子．本文在上述假设条件下,通过附加一定的边界条件应用Leray-Schauder度理论研究了下述包含关系:0∈(T+C)(D(T)∩ BQ(0)),0∈(T+C)(D(T)∩ BQ(0));以及S(?)R(T+C), intS(?)intR(T+C)(其中S(?) X*);B+D(?)R(T+C),int(B+D)(?)intR(T+C)(其中 B(?)X*,D(?)X*)的可解性,得出了一些新的结论．     

Exponential Dichotomy on the Real Line and Admissibility of Function Spaces

The purpose of this paper is to give characterizations for uniform exponential dichotomy of evolution families on the real line. We consider a general class of Banach function spaces denoted and we prove that if with and the pair is admissible for an evolution family then is uniformly exponentially dichotomic. By an example we show that the admissibility of the pair for an evolution family is not a sufficient condition for uniform exponential dichotomy. As applications, we deduce necessary and sufficient conditions for uniform exponential dichotomy of evolution families in terms of the admissibility of the pairs and with      

Oscillation in odd-order neutral delay differential equations

Consider the odd-order functional differential equation

Oscillatory properties of solutions of three-dimensional differential systems of neutral type

The purpose of this paper is to obtain oscillation criteria for the differential system

A STRASSEN LAW OF THE ITERATED LOGARITHM FOR PROCESSES WITH INDEPENDENT INCREMENTS

Let X = {X(t), t ＞- 0} be a process with independent increments (PII) such that E|X(t)| = 0, Dx(t) ∧= E[X(t)^2 < ∞, limt→∞ Dx(t)/t = 1,and there exists a majoring measure G for the jump △X of X. Under these assumptions, using rather a direct method, a Strassen‘s law of the iterated logarithm (Strassen LIL) is established. As some special cases, the Strassen LIL for homogeneous PII and for partial sum process of i.i.d, random variables are comprised.     

UNBOUNDED SOLUTIONS OF CONSERVATIVE OSCILLATORS UNDER ROUGHLY PERIODIC PERTURBATIONS

This note is concerned with the equation $$\[\frac{{{d^2}x}}{{d{t^2}}} + g(x) = p(t)\begin{array}{*{20}{c}} {}&{(1)} \end{array}\]$$ where g(x) is a continuously differentiable function of a $\[x \in R\]$, $\[xg(x) > 0\]$ whenever $\[x \ne 0\]$, and $\[g(x)/x\]$ tends to $\[\infty \]$ as \[\left| x \right| \to \infty \]. Let p(t) be a bounded function of $\[t \in R\]$. Define its norm by $\[\left\| p \right\| = {\sup _{t \in R}}\left| {p(t)} \right|\]$ The study of this note leads to the following conclusion which improves a result due to J. E. Littlewood, For any given small constants $\[\alpha > 0,s > 0\]$, there is a continuous and roughly periodic(with respect to $\[\Omega (\alpha )\]$) function p(t) with $\[\left\| p \right\| < s\]$ such that the corresponding equation (1) has at least one unbounded solution.     

HOW BIG ARE THE LAG INCREMENTS OF A 2-PARAMETER WIENER PROCESS? (II)

The author investigated how big the lag increments of a 2-parameter Wiener process is in [1]. In this paper the limit inferior results for the lag increments are discussed and the same results as the Wiener process are obtained. For example, if $\[\mathop {\lim }\limits_{T \to \infty } \{ \log T/{a_T} + \log (\log {b_T}/a_T^{1/2} + 1)\} /\log \log T = r,0 \leqslant r \leqslant \infty \] $ then $\[\mathop {\lim }\limits_{\overline {T \to \infty } } \mathop {\sup }\limits_{{a_T} \leqslant t \leqslant T} \mathop {\sup }\limits_{t \leqslant s \leqslant T} \mathop {\sup }\limits_{R \in L_s^*(t)} |W(R)|/d(T,t) = {\alpha _r},a.s.,\] $ $\[\mathop {\lim }\limits_{\overline {T \to \infty } } \mathop {\sup }\limits_{{a_T} \leqslant t \leqslant T} \mathop {\sup }\limits_{R \in {{\tilde L}_T}(t)} |W(R)|/d(T,t) = {\alpha _r},a.s.,\] $ where $\alpha _r=(r/(r+1))^{1/2}$, $L*_s(t)$ and $\tider L_T(t)$ are the sets of rectangles which satisfy some conditions. Moreover, the limit inferior results of another class of lag increments are discussed.     

EXPONENTIAL STABILITY OF LINEAR SYSTEMS IN BANACH SPACES

In this paper the author proves a new fundamental lemma of Hardy-Lebesgne class $\[{H^2}(\sigma )\]$ and by this lemma obtains some fundamental results of exponential stability of $\[{C_0}\]$-semigroup of bounded linear operators in Banach spaces. Specially, if $\[{\omega _s} = \sup \{ {\mathop{\rm Re}\nolimits} \lambda ;\lambda \in \sigma (A) < 0\} \]$ and $\[\sup \{ \left\| {{{(\lambda - A)}^{ - 1}}} \right\|;{\mathop{\rm Re}\nolimits} \lambda \ge \sigma \} < \infty \]$ , where \[\sigma \in ({\omega _s},0)\]) and A is the infinitesimal generator of a $\[{C_0}\]$-semigroup in a Banach space $X$, then $\[(a)\int_0^\infty {{e^{ - \sigma t}}\left| {f({e^{tA}}x)} \right|} dt < \infty \]$, $\[\forall f \in {X^*},x \in X\]$; (b) there exists $\[M > 0\]$ such that $\[\left\| {{e^{tA}}x} \right\| \le N{e^{\sigma t}}\left\| {Ax} \right\|\]$, $\[\forall x \in D(A)\]$; (c) there exists a Banach space $\[\hat X \supset X\]$ such that $\[\left\| {{e^{tA}}x} \right\|\hat x \le {e^{\sigma t}}\left\| x \right\|\hat x,\forall x \in X.\]$.     

局部可积半群的谱映射定理

设（Ｔ（ｔ）ｔ〉０为Ｂａｎａｃｈ空间Ｘ上的局部可积半群，我们讨论（Ｔ（ｔ）ｔ〉０与它的多值生成元Ａ之间的谱关系。     

Integers without large prime factors in short intervals: Conditional results

Under the Riemann hypothesis and the conjecture that the order of growth of the argument of ζ(1/2 + it) is bounded by $\left( {\log t} \right)^{\frac{1} {2} + o\left( 1 \right)}$\left( {\log t} \right)^{\frac{1} {2} + o\left( 1 \right)} , we show that for any given α > 0 the interval $(X,X + \sqrt X (\log X)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} + o\left( 1 \right)} ]$(X,X + \sqrt X (\log X)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} + o\left( 1 \right)} ] contains an integer having no prime factor exceeding X ^α for all X sufficiently large.     

一些超空间的非双Lipschitz 齐次性

A metric space(X, d) is called bi-Lipschitz homogeneous if for any points x, y ∈ X,there exists a self-homeomorphism h of X such that both h and h-1are Lipschitz and h(x) = y.Let 2(X,d)denote the family of all non-empty compact subsets of metric space(X, d) with the Hausdorff metric. In 1985, Hohti proved that 2([0,1],d)is not bi-Lipschitz homogeneous, where d is the standard metric on [0, 1]. We extend this result in two aspects. One is that 2([0,1],e)is not bi-Lipschitz homogeneous for an admissible metric e satisfying some conditions. Another is that 2(X,d)is not bi-Lipschitz homogeneous if(X, d) has a nonempty open subspace which is isometric to an open subspace of m-dimensional Euclidean space Rm.     

Compact and Loeb Hausdorff spaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {ZF}$\end{document} and the axiom of choice for families of finite sets

Given a set X, $\mathsf {AC}^{\mathrm{fin}(X)}$ denotes the statement: “$[X]^{<\omega }\backslash \lbrace \varnothing \rbrace$ has a choice set” and $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )$ denotes the family of all closed subsets of the topological space $\mathbf {2}^{X}$ whose definition depends on a finite subset of X. We study the interrelations between the statements $\mathsf {AC}^{\mathrm{fin}(X)},$ $\mathsf {AC}^{\mathrm{fin}([X]^{<\omega })},$ $\mathsf {AC}^{\mathrm{fin} (F_{n}(X,2))},$ $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$ and “$\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$has a choice set”. We show:

• (i) $\mathsf {AC}^{\mathrm{fin}(X)}$ iff $\mathsf {AC}^{\mathrm{fin}([X]^{<\omega } )}$ iff $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set iff $\mathsf {AC}^{\mathrm{fin}(F_{n}(X,2))}$.
• (ii) $\mathsf {AC}_{\mathrm{fin}}$ ($\mathsf {AC}$ restricted to families of finite sets) iff for every set X, $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set.
• (iii) $\mathsf {AC}_{\mathrm{fin}}$ does not imply “$\mathcal {K}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set($\mathcal {K}(\mathbf {X})$ is the family of all closed subsets of the space $\mathbf {X}$)
• (iv) $\mathcal {K}(\mathbf {2}^{X})\backslash \lbrace \varnothing \rbrace$ implies $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$ but $\mathsf {AC}^{\mathrm{fin}(X)}$ does not imply $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$.

We also show that “For every setX, “$\mathcal {K}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$has a choice set” iff “for every setX, $\mathcal {K}\big (\mathbf {[0,1]}^{X}\big )\backslash \lbrace \varnothing \rbrace$has a choice set” iff “for every product$\mathbf {X}$of finite discrete spaces,$\mathcal {K}(\mathbf {X})\backslash \lbrace \varnothing \rbrace$ has a choice set”.     

半参数回归模型小波估计的强相合性

考虑半参数回归模型y_i~(n)=X_i~((n)T)β+g(t_i~(n))+ε_i~(n)(1■i■n),其中β∈R~d为未知参数,g(t)为[0,1】上的未知Borel函数,X_i~(n)为R~d上的随机设计,随机误差序列{ε_i~(n)}为鞅差序列,{t_i~(n))为[0,1]上的常数序列．本文用小波的方法得到β、g(t)的估计量分别为■_n、■_n(t),并证明了它们的强相合性．     

Irregular wavelet frames on L~2 (R~n)

In this paper, we present the conditions on dilation parameter {sj}j that ensure a discrete irregular wavelet system to be a frame on L2(Rn), and for the wavelet frame we consider the perturbations of translation parameter b and frame function ψ respectively.     

Constant-Sign Periodic and Almost Periodic Solutions for a System of Integral Equations

We consider the following system of integral equations $${u_{i}(t)=\int\nolimits_{I} g_{i}(t, s)f(s, u_{1}(s), u_{2}(s), \cdots, u_{n}(s))ds, \quad t \in I, \ 1 \leq i\leq n}$$ where I is an interval of $\mathbb{R}$ . Our aim is to establish criteria such that the above system has a constant-sign periodic and almost periodic solution (u ₁, u ₂,…,u _n ) when I is an infinite interval of $\mathbb{R}$ , and a constant-sign periodic solution when I is a finite interval of $\mathbb{R}$ . The above problem is also extended to that on $\mathbb{R}$ $$u_{i} {\left( t \right)} = {\int_\mathbb{R} {g_{i} {\left( {t,s} \right)}f_{i} {\left( {s,u_{1} {\left( s \right)},u_{2} {\left( s \right)}, \cdots ,u_{n} {\left( s \right)}} \right)}ds\quad t \in \mathbb{R},\quad 1 \leqslant i \leqslant n.} }$$      

Profile of Blow-up Solution to Hyperbolic System with Nonlocal Term

This paper is concerned with a nonlocal hyperbolic system as follows utt = △u ＋ （∫Ωvdx ）^p for x∈R^N,t〉0 ,utt = △u ＋ （∫Ωvdx ）^q for x∈R^N,t〉0 ,u（x,0）=u0（x）,ut（x,0）=u01（x） for x∈R^N,u（x,0）=u0（x）,ut（x,0）=u01（x） for x∈R^N, where 1≤ N ≤3, p ≥1, q ≥ 1 and pq 〉 1. Here the initial values are compactly supported and Ω belong to R^N is a bounded open region. The blow-up curve, blow-up rate and profile of the solution are discussed.     

On the divergence of de la Vallée Poussin means of Fourier series

Пусть {λ _{n 1} ^t8 — монотонн ая последовательнос ть натуральных чисел. Дл я каждой функции fεL(0, 2π) с рядом Фурье строятся обобщенные средние Bалле Пуссена $$V_n^{(\lambda )} (f;x) = \frac{{a_0 }}{2} + \mathop \sum \limits_{k = 1}^n (a_k \cos kx + b_k \sin kx) + \mathop \sum \limits_{k = n + 1}^{n + \lambda _n } \left( {1 - \frac{{k - n}}{{\lambda _n + 1}}} \right)\left( {a_k \cos kx + b_k \sin kx} \right).$$ Доказываются следую щие теоремы.

1. Если λ_n=o(n), то существуе т функция fεL(0, 2π), для кот орой последовательность {V_n ^(λ)(?;x)} расходится почти вс юду.
2. Если λ_n=o(n), то существуе т функция fεL(0, 2π), для кот орой последовательность $$\left\{ {\frac{1}{\pi }\mathop \smallint \limits_{ - \pi /\lambda _n }^{\pi /\lambda _n } f(x + t)\frac{{\sin (n + \tfrac{1}{2})t}}{{2\sin \tfrac{1}{2}t}}dt} \right\}$$ расходится почти всю ду

.     

一类连续体上连续映射的周期点

设X是个阶有限的遗传可分解可链连续体, f:X→X是X上的连续自映射, On(x,f)={fi(x):0≤i≤n)是f的一个返回轨道, inf(On(x,f))相似文献