Microlocal Analysis of the Doppler Transform on R<Superscript>3</Superscript>

The Doppler transform of a vector field $F = (f_1,f_2,f_3)$ on $\mathbb{R}^3$ is defined by \[\displaystyle\mathcal{D}F(x,\omega) = \sum_j\int_\mathbb{R} \omega_j f_j(x+t\omega)\, dt~,\] where $x\in \mathbb{R}^3$ and $\omega \in S^2$ specifies the direction of a line passing through $x$. In practical applications, $\mathcal{D}F$ is known only for a small subset of lines in $\mathbb{R}^3$. In this article, we deal with the case of $\mathcal{D}F$ restricted to all lines passing through a fixed smooth curve. Using techniques from microlocal analysis, we study the problem of recovering the wavefront set of $\mbox{curl}(F)$ from that of the restricted Doppler transform of $F$.     

ERGODIC THEOREMS FOR NON-LIPSCHITZIAN SEMIGROUPS WITHOUT CONVEXITY

§１．ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＨｂｅａＨｉｌｂｅｒｔｓｐａｃｅｗｉｔｈｎｏｒｍ‖·‖ａｎｄｉｎｎｅｒｐｒｏｄｕｃｔ（·，·）ａｎｄｌｅｔＣｂｅａｎｏｎｅｍｐｔｙｓｕｂｓｅｔｏｆＨ．ＡｍａｐｐｉｎｇＴ：Ｃ｜→ＣｉｓｓａｉｄｔｏｂｅＬｉｐｓｃｈｉｔ...     

一类多复变全纯映照子族的增长和偏差定理

在一般复Banach空间X中的单位球B上引入一类全纯映照族M_g.考虑B上满足条件(Df(x))~(-1)f(x)∈M_g的正规化局部双全纯映照f(x)(其中x=0是f(x)-x的k+1阶零点)并得到其增长定理.作为应用,也得到了C~n中单位多圆柱D~n上映照f关于Jacobi矩阵Jf(z)的偏差定理,该结果统一和推广了星形映照许多子族的相应结论.     

The Boson Normal Ordering Problem and Generalized Bell Numbers

For any function F(x) having a Taylor expansion we solve the boson normal ordering problem for $F [(a^\dag)^r a^s]$, with r, s positive integers, $F [(a, a^\dag]=1$, i.e., we provide exact and explicit expressions for its normal form $\mathcal{N} \{F [(a^\dag)^r a^s]\} = F [(a^\dag)^r a^s]$, where in $ \mathcal{N} (F) $ all a's are to the right. The solution involves integer sequences of numbers which, for $ r, s \geq 1 $, are generalizations of the conventional Bell and Stirling numbers whose values they assume for $ r=s=1 $. A complete theory of such generalized combinatorial numbers is given including closed-form expressions (extended Dobinski-type formulas), recursion relations and generating functions. These last are special expectation values in boson coherent states.AMS Subject Classification: 81R05, 81R15, 81R30, 47N50.     

$\mathcal{F}$-Perfect Rings and Modules

Let $R$ be a ring, and let $(\mathcal{F}, C)$ be a cotorsion theory. In this article, the notion of $\mathcal{F}$-perfect rings is introduced as a nontrial generalization of perfect rings and A-perfect rings. A ring $R$ is said to be right $\mathcal{F}$-perfect if $F$ is projective relative to $R$ for any $F ∈ \mathcal{F}$. We give some characterizations of $\mathcal{F}$-perfect rings. For example, we show that a ring $R$ is right $\mathcal{F}$-perfect if and only if $\mathcal{F}$-covers of finitely generated modules are projective. Moreover, we define $\mathcal{F}$-perfect modules and investigate some properties of them.     

非单特征值的广义分歧定理

讨论了抽象算子方程F(λ,u)=0的局部分歧问题,其中F:R×X→Y是一个C~2微分映射,λ是参数,X,Y为Banach空间.利用Lyapunov-Schmidt约化过程及偏导算子F_u(λ~*,O)的有界线性广义逆,在dim N(F_u(λ~*,0))≥codim R(F_u(λ~*,O))=1的条件下,证明了一个广义跨越式分歧定理.当参数空间的维数等于值域余维数时,应用同样的方法又得到了多参数方程的抽象分歧定理.     

Hypercentralizing generalized skew derivations on left ideals in prime rings

Let $\mathcal{R }$ be a prime ring of characteristic different from $2, \mathcal{Q }_r$ the right Martindale quotient ring of $\mathcal{R }, \mathcal{C }$ the extended centroid of $\mathcal{R }, \mathcal{I }$ a nonzero left ideal of $\mathcal{R }, F$ a nonzero generalized skew derivation of $\mathcal{R }$ with associated automorphism $\alpha $ , and $n,k \ge 1$ be fixed integers. If $[F(r^n),r^n]_k=0$ for all $r \in \mathcal{I }$ , then there exists $\lambda \in \mathcal{C }$ such that $F(x)=\lambda x$ , for all $x\in \mathcal{I }$ . More precisely one of the following holds: (1) $\alpha $ is an $X$ -inner automorphism of $\mathcal{R }$ and there exist $b,c \in \mathcal{Q }_r$ and $q$ invertible element of $\mathcal{Q }_r$ , such that $F(x)=bx-qxq^{-1}c$ , for all $x\in \mathcal{Q }_r$ . Moreover there exists $\gamma \in \mathcal{C }$ such that $\mathcal{I }(q^{-1}c-\gamma )=(0)$ and $b-\gamma q \in \mathcal{C }$ ; (2) $\alpha $ is an $X$ -outer automorphism of $\mathcal{R }$ and there exist $c \in \mathcal{Q }_r, \lambda \in \mathcal{C }$ , such that $F(x)=\lambda x-\alpha (x)c$ , for all $x\in \mathcal{Q }_r$ , with $\alpha (\mathcal{I })c=0$ .     

具误差隐格式迭代逼近严格伪压缩映像族公共不动点 (英)

设$K$是实Banach空间$E$中非空闭凸集, $\{T_i\}_i=1^{N}$是$N$个具公共不动点集$F$的严格伪压缩映像, $\{\alpha_n\}\subset [0,1]$是实数列, $\{u_n\}\subset K$是序列, 且满足下面条件 (i)\ 设$K$是实Banach空间$E$中非空闭凸集, $\{T_i\}_i=1^{N}$是$N$个具公共不动点集$F$的严格伪压缩映像, $\{\alpha_n\}\subset [0,1]$是实数列, $\{u_n\}\subset K$是序列, 且满足下面条件 (i)\ 设$K$是实Banach空间$E$中非空闭凸集, $\{T_i\}_i=1^{N}$是$N$个具公共不动点集$F$的严格伪压缩映像, $\{\alpha_n\}\subset [0,1]$是实数列, $\{u_n\}\subset K$是序列, 且满足下面条件 (i)\ 设K是实Banach空间E中非空闭凸集，{Ti}i=1^N是N个具公共不动点集F的严格伪压缩映像，{αn}包括于[0,1]是实数例，{un}包括于K是序列，且满足下面条件（i）0〈α≤αn≤1;（ii）∑n=1∞（1-αn）=＋∞.（iii）∑n=1∞ ‖un‖〈＋∞.设x0∈K,{xn}由正式定义xn=αnxn-1＋（1-αn）Tnxn＋un-1,n≥1,其中Tn=Tnmodn，则下面结论（i）limn→∞‖xn-p‖存在，对所有p∈F；（ii）limn→∞d（xn,F）存在，当d（xn,F）=infp∈F‖xn-p‖；（iii）lim infn→∞‖xn-Tnxn‖=0.文中另一个结果是，如果{xn}包括于[1-2^-n,1]，则{xn}收敛，文中结果改进与扩展了Osilike（2004）最近的结果，证明方法也不同。     

具有凹凸非线性项和变号位势函数拟线性椭圆系统解的多重结果

研究拟线性椭圆系统(?)的非平凡非负解或正解的多重性,这里Ω(?)R~N是具有光滑边界(?)Ω的有界域,1≤qp~*/p~*-q,其中当N≤p时,p~*=+∞,而当1相似文献   

A time fractional functional differential equation driven by the fractional Brownian motion

Let $B^H$ be a fractional Brownian motion with Hurst index $H>\frac12$. In this paper, we prove the global existence and uniqueness of the equation $$ \begin{cases} ^CD_t^{\gamma}x(t)=f(x_t)+G(x_t)\frac{d}{dt}B^H(t),\ \ \ \ &t\in(0,T], \x(t)=\eta(t), \ \ \ \ \ &t\in[-r,0], \end{cases} $$ where $\max\{H,2-2H\}<\gamma<1$, $^CD_t^{\gamma}$ is the Caputo derivative, and $x_t\in \mathcal{C}_r=\mathcal{C}([-r,0],\mathbb{R})$ with $x_t(u)=x(t+u),u\in[-r,0]$. We also study the dependence of the solution on the initial condition.     

Some results on metric n-Lie algebras

We study the structure of a metric n-Lie algebra G over the complex field C. Let G = SR be the Levi decomposition, where R is the radical of G and S is a strong semisimple subalgebra of G. Denote by m(G) the number of all minimal ideals of an indecomposable metric n-Lie algebra and R ⊥ the orthogonal complement of R. We obtain the following results. As S-modules, R ⊥ is isomorphic to the dual module of G/R. The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on G is equal to that of the vector space of certain linear transformations on G; this dimension is greater than or equal to m(G) + 1. The centralizer of R in G is equal to the sum of all minimal ideals; it is the direct sum of R ⊥ and the center of G. Finally, G has no strong semisimple ideals if and only if R⊥■R.     

一列连续函数的遍历优化

设$T:X\rightarrow X$是紧度量空间$X$上的连续映射, $\mathcal{F}=\{f_n\}_{n\geq 1}$是$X$上的一族连续函数. 如果 $\mathcal{F}$是渐近次可加的, 那么$\sup\limits_{x\in \mathrm{Reg}(\mathcal{F},T)}\lim\limits_{n\rightarrow\infty}\frac 1 n f_n (x)=\sup\limits_{x\in X} \limsup\limits_{n\rightarrow\infty}\frac 1 n f_n (x) =\lim\limits_{n\rightarrow\infty}\frac 1 n \max\limits_{x\in X}f_n (x)=\sup\{\mathcal{F}^*(\mu):\mu\in\mathcal{M}_T\}$, 其中$\mathcal{M}_T$表示$T$-\!\!不变的Borel概率测度空间, $\mathrm{Reg}(\mathcal{F},T)$ 表示函数族$\mathcal{F}$的正规点集, $\mathcal{F}^*(\mu)=\lim\limits_{n\rightarrow\infty}\frac 1 n \int f_n \mathrm{d}\mu$. 这把Jenkinson, Schreiber 和 Sturman 等人的一些结果推广到渐近次可加势函数, 并且给出了次可加势函数从属原理成立的充分条件, 最后给出了 一些相关的应用.     

Conclaves of planes in PG (4,2) and certain planes external to the Grassmannian ${\cal G}_{1,4,2} \subset$ PG(9,2)

We show that in $\operatorname{PG}(4,2)$ there exist octets $\mathcal{P} _{8}=\{\pi_{1},\,\ldots\,,\pi_{8}\}$ of planes such that the 28 intersections $\pi_{i}\cap\pi_{j}$ are distinct points. Such conclaves (see [6]) $\mathcal{P}_{8}$ of planes in $\operatorname{PG}(4,2)$ are shown to be in bijective correspondence with those planes $P$ in $\operatorname{PG}(9,2)$ which are external to the Grassmannian $\mathcal{G}_{1,4,2}$ and which belong to the orbit $\operatorname{orb}(2\gamma)$ (see [4]). The fact that, under the action of $\operatorname{GL}(5,2),$ the stabilizer groups $\mathcal{G}_{\mathcal{P}_{8}}$ and $\mathcal{G}_{P}$ both have the structure $2^{3}:(7:3)$ is thus illuminated. Starting out from a regulus-free partial spread $\mathcal{S}_{8}$ in $\operatorname{PG}(4,2)$ we also give a construction of a conclave of planes $P\in\operatorname{orb}(2\gamma)\subset\operatorname{PG}(9,2).$     

Convergence of Solutions of General Dispersive Equations Along Curve

In this paper, the authors give the local L~2 estimate of the maximal operator S_(φ,γ)~* of the operator family {S_(t,φ,γ)} defined initially by ■which is the solution(when n = 1) of the following dispersive equations(~*) along a curve γ:■where φ : R~+→R satisfies some suitable conditions and φ((-?)~(1/2)) is a pseudo-differential operator with symbol φ(|ξ|). As a consequence of the above result, the authors give the pointwise convergence of the solution(when n = 1) of the equation(~*) along curve γ.Moreover, a global L~2 estimate of the maximal operator S_(φ,γ)~* is also given in this paper.     

阶极小渐近基

Let N denote the set of all nonnegative integers and A be a subset of N.Let W be a nonempty subset of N.Denote by F~*(W) the set of all finite,nonempty subsets of W.Fix integer g≥2,let A_g(W) be the set of all numbers of the form sum f∈Fa_fg~f where F∈F~*(W)and 1≤a_f≤g-1.For i=0,1,2,3,let W_i = {n∈N|n≡ i(mod 4)}.In this paper,we show that the set A = U_i~3=0 A_g(W_i) is a minimal asymptotic basis of order four.     

Cocycle Perturbation on Banach Algebras

Let $α$ be a flow on a Banach algebra$\mathcal{B}$, and $t → u_t$ a continuous function from$\boldsymbol{R}$into the group of invertible elements of$\mathcal{B}$such that $u_sα_s(u_t) = u_{s+t}, s, t ∈ \boldsymbol{R}$. Then $β_t$ = Ad$u_t ◦ α_t$, $t ∈ \boldsymbol{R}$ is also a flow on $\mathcal{B}$, where Ad$u_t(B) \triangleq u_tBu^{−1}_t$ for any $B ∈ \mathcal{B}$. $β$ is said to be a cocycle perturbation of $α$. We show that if $α$, $β$ are two flows on a nest algebra (or quasi-triangular algebra), then $β$ is a cocycle perturbation of $α$. And the flows on a nest algebra (or quasi-triangular algebra) are all uniformly continuous.     

3$\times$3阶上三角型算子矩阵的可能谱

Let H1, H2 and H3 be infinite dimensional separable complex Hilbert spaces. We denote by M（D,V,F） a 3×3 upper triangular operator matrix acting on Hi ＋H2＋ H3 of theform M（D,E,F）=（A D F 0 B F 0 0 C）.For given A ∈ B（H1）, B ∈ B（H2） and C ∈ B（H3）, the sets ∪D,E,F^σp（M（D,E,F））,∪D,E,F ^σr（M（D,E,F））,∪D,E,F ^σc（M（D,E,F）） and ∪D,E,F σ（M（D,E,F）） are characterized, where D ∈ B（H2,H1）, E ∈B（H3, H1）, F ∈ B（H3,H2） and σ（·）, σp（·）, σr（·）, σc（·） denote the spectrum, the point spectrum, the residual spectrum and the continuous spectrum, respectively.     

Boundedness of the Cesàro Operator in Hardy Spaces

The Ces\aro operator $\mathcal{C}_{\alpha}$ is defined by \begin{equation*} (\mathcal{C}_{\alpha}f)(x) = \int_{0}^{1}t^{-1}f\left( t^{-1}x \right)\alpha (1-t)^{\alpha -1}\,dt~, \end{equation*} where $f$ denotes a function on $\mathbb{R}$. We prove that $\mathcal{C}_{\alpha}$, $\alpha >0$, is a bounded operator in the Hardy space $H^{p}$ for every $0 < p \leqq 1$.     

p-超循环嵌入子群的一个判别准则

令E是有限群G的一个正规子群,且U是所有有限超可解群的集合.E称为在G中是p-超循环嵌入的,如果E的每个pd-阶的G-主因子是循环的.G的子群H称为在G中是U-Φ-可补充的,如果存在G的一个次正规子群T,使得G=HT,且(H∩T)H_G/H_G≤Φ/(H/H_G)Z_U(G/H_G),其中Z_U(G/H_G)是商群G/H_G的U-超中心.作者证明,如果E的一些p-子群在G中是U-Φ-可补充的,那么E在G中是p-超循环嵌入的.作为应用,得到了有限群是p-超可解的若干判断准则,并且推广了一些已知的结果.     

参数函数$\epsilon$取边际值时的$GCF_\epsilon$集

Let ∈ :N → R be a parameter function satisfying the condition ∈(k) + k + 1 > 0and let T∈ :(0,1] →(0,1] be a transformation defined by T∈(x) =-1 +(k + 1)x1 + k-k∈x for x ∈(1k + 1,1k].Under the algorithm T∈,every x ∈(0,1] is attached an expansion,called generalized continued fraction(GCF∈) expansion with parameters by Schweiger.Define the sequence {kn(x)}n≥1of the partial quotients of x by k1(x) = ∈1/x∈ and kn(x) = k1(Tn-1∈(x)) for every n ≥ 2.Under the restriction-k-1 < ∈(k) <-k,define the set of non-recurring GCF∈expansions as F∈= {x ∈(0,1] :kn+1(x) > kn(x) for infinitely many n}.It has been proved by Schweiger that F∈has Lebesgue measure 0.In the present paper,we strengthen this result by showing that{dim H F∈≥12,when ∈(k) =-k-1 + ρ for a constant 0 < ρ < 1;1s+2≤ dimHF∈≤1s,when ∈(k) =-k-1 +1ksfor any s ≥ 1where dim H denotes the Hausdorff dimension.