区组长为奇素数的自反MD设计 (英)

本文利用差方法对自反MD设计SCMD$(4mp, p,1)$的存在性给出了构造性证明, 这里$p$为奇素数, $m$为正整数.     

THE EXISTENCE OF CLOSE GEODESICS ON A COMPLETE RIEM ANNIAN MANIFOLD

This paper studios the existence of closed geodesics in the homotopy class of a given closed curve. Let M be a complete Riemannian manifold without boundary, f∈C~1(S~1, M). Look at S~1 as [0, 2π]/{0, 2π}. The following results are proved:A. The initial value problem of heat equation _if_t=τ(f_i), f_0=f always admits a global solution.B. (Existence of closed geodesics). If there exists a compact set KM such that f(S~1)∩K≠φ andE(f)≤(1/π)l(K)~2,then there exists a closed geodesic homotopic to f. If then the closed geodesic is minimal.C. Some estimates about injective radius are obtained.Some example is found showing that the inequalities in B are sharp.     

数学机械化中的AC=BD模式

介绍了AC=BD模式及其在用机械化方法求解方程和证明定理中的应用.首先证明对可单值化算子D,如果CKerD C KerA,则存在算子B使AC=BD.利用带余除法对于给定的算子A给出了求其(C-D对的算法,使得AC=BD.并将其应用到求解算子方程,可以将一些较为复杂的方程化为简单方程求解.其次,利用对偶算子给出了将非线性非交换算子方程组化为单个方程求解的算法.最后,利用解方程的方法给出了机械化产生并证明定理的模式,并且给出了一些实例.     

On the zeros of a minimal realization

In an earlier work, the authors have introduced a coordinate-free, module-theoretic definition of zeros for the transfer function G(s) of a linear multivariable system (A,B,C). The first contribution of this paper is the construction of an explicit k[z]-module isomorphism from that zero module, Z(G), to V¹/R¹, where V¹ is the supremal (A,B)-invariant subspace contained in kerC and R¹ is the supremal (A,B)-controllable subspace contained in kerC, and where (A,B,C) constitutes a minimal realization of G(s). The isomorphism is developed from an exact commutative diagram of k-vector spaces. The second contribution is the introduction of a zero-signal generator and the establishment of a relation between this generator and the classic notion of blocked signal transmissions.     

Control subspaces of minimal dimension and root vectors

We investigate the following characteristic of a linear operator A in a Banach space X: disc {inf(dim R′:R′?R,R′εCyc A) :RεCyc A}, where Cyc A={R:R is a subspace of X, dim R<∞, span (AⁿR: :n≥0)=X}. The value disc A is equal to the dimension of a cyclic subspace that can be chosen in an arbitrary cyclic finite dimensional subspace. If we consider a dynamical system \(\dot x = Ax + Bu\) with the controllability property, disc A shows to what extent the dimension of the input subspace of control can be diminished without loss of controllability. In this paper we investigate when easy inequality disc A≥(the multiplicity of A) turn into the equality. Some estimates from below of disc A (of the type disc A≥sup dim Ker(A-λI)) are found for some classes of operators e.q. for compact operators, for Toeplitz operators with antianalytic symbols, for strictly lower triangular operators and some other classes.     

ON THE TAYLOR'S JOINT SPECTRUM OF 2n-TUPLE (L_A, R_B)

Let A= (A_1, …, A_n) and B=(B_1, …, B_n) be double commuting n-tuples of operators on Hilbert space H and let L_(A_i), and R_(B_j), decode the left and right multiplications induced by A_i and B_j, respectively. The following results are proven: Sp (L_A, R_B)=Sp(A)×Sp(B), Sp_e(L_A, R_B)=Sp_e(A)×Sp(B) ∪ Sp(A)×Sp_e(B).     

关于亚纯函数的0-1-∞集（英）

Let {an}, {bn} and {pn} be three disjoint sequences with no finite limit points. If it is possible to construct a meromorphic function N in the plane whose zeros, one points and poles are exactly {an}, {bn} and {pn} respectively, where their multiplicities are taken into consideration, then the given triple ({an}, {bn}, {Pn}) is called the zero-one-pole set (of N). In general an arbitrary triad ({an}, {bn}, {pn}) is not a zero-one-pole set of any meromorphic function. This was proved by Rubel and Yang[6] explicitly for entire functions. Ozawa[5] proved the following.     

Lipschitz stability of controlled invariant subspaces

Let (A,B)∈C^n×n×C^n×m and M be an (A, B)-invariant subspace. In this paper the following results are presented: (i) If M∩ImB={0}, necessary and sufficient conditions for the Lipschitz stability of M are given. (ii) If M contains the controllability subspace of the pair (A, B), sufficient conditions for the Lipschitz stability of the subspace M are given.     

ON THE DIOPHANUNE EQUATION $\[\sum\limits_{i = 0}^N {\frac{{{x_i}}}{{{d_i}}}} \equiv 0\]$ (mod 1) AND ITS APPLICATIONS

The number $\[A({d_1}, \cdots ,{d_n})\]$ of solutions of the equation $$\[\sum\limits_{i = 0}^n {\frac{{{x_i}}}{{{d_i}}}} \equiv 0(\bmod 1),0 < {x_i} < {d_i}(i = 1,2, \cdots ,n)\]$$ where all the $\[{d_i}s\]$ are positive integers, is of significance in the estimation of the number $\[N({d_1}, \cdots {d_n})\]$ of solutiohs in a finite field $\[{F_q}\]$ of the equation $$\[\sum\limits_{i = 1}^n {{a_i}x_i^{{d_i}}} = 0,{x_i} \in {F_q}(i = 1,2, \cdots ,n)\]$$ where all the $\[a_i^''s\]$ belong to $\[F_q^*\]$. the multiplication group of $\[F_q^{[1,2]}\]$. In this paper, applying the inclusion-exclusion principle, a greneral formula to compute $\[A({d_1}, \cdots ,{d_n})\]$ is obtained. For some special cases more convenient formulas for $\[A({d_1}, \cdots ,{d_n})\]$ are also given, for example, if $\[{d_i}|{d_{i + 1}},i = 1, \cdots ,n - 1\]$, then $$\[A({d_1}, \cdots ,{d_n}) = ({d_{n - 1}} - 1) \cdots ({d_1} - 1) - ({d_{n - 2}} - 1) \cdots ({d_1} - 1) + \cdots + {( - 1)^n}({d_2} - 1)({d_1} - 1) + {( - 1)^n}({d_1} - 1).\]$$     

Rayleigh Quotient and Residual of a Definite Pair

Let {$A,B$} be a definite matrix pair of order $n$, and let $Z$ be an $l$-dimensional subspace of $C^n$. In this paper we introduce the Rayleigh quotient matrix pair{$H_1,K_1$} and residual matrix pair {$R_A,R_B$} of {A,B} with respect to Z, and used the norm of {$R_A,R_B$} to bound the difference between the eigenvalues of {$H_1,K_1$} and that of {$A,B$}, and to bound the difference between $Z$ and an $l$-dimensional eigenspace of {$A,B$}. The corresponding classical theorems on the Hermitian matrices can be derived from the results of this paper.     

Structural stability of (C, A)-marked subspaces

Given an observable pair of matrices (C, A) we consider the manifold of (C, A)-invariant subspaces having a fixed Brunovsky-Kronecker structure. Using Arnold techniques we obtain the explicit form of a miniversal deformation of a marked (C, A)-invariant subspace with respect to the usual equivalence relation. As an application, we obtain the dimension of the orbit and we characterize the structurally stable subspaces (those with open orbit).     

Feedback invariants of pairs of matrices and restrictions

If (A,B) εF^n×n×F^×mis a given pair and S is an (A,B)-invariant subspace we investigate the relationship between the feedback invariants of (A, B) and those of its restrictions

to S.     

On the Existence of Optimal Unions of Subspaces for Data Modeling and Clustering

Given a set of vectors F={f ₁,…,f _m } in a Hilbert space H\mathcal {H}, and given a family C\mathcal {C} of closed subspaces of H\mathcal {H}, the subspace clustering problem consists in finding a union of subspaces in C\mathcal {C} that best approximates (is nearest to) the data F. This problem has applications to and connections with many areas of mathematics, computer science and engineering, such as Generalized Principal Component Analysis (GPCA), learning theory, compressed sensing, and sampling with finite rate of innovation. In this paper, we characterize families of subspaces C\mathcal {C} for which such a best approximation exists. In finite dimensions the characterization is in terms of the convex hull of an augmented set C⁺\mathcal {C}^{+}. In infinite dimensions, however, the characterization is in terms of a new but related notion; that of contact half-spaces. As an application, the existence of best approximations from π(G)-invariant families C\mathcal {C} of unitary representations of Abelian groups is derived.     

$L^{\Phi}(I,X)$中的最佳同时逼近

Let X be a Banach space and Ф be an Orlicz function. Denote by L^Ф（I,X） the space of X-valued （I）-integrable functions on the unit interval I equipped with the Luxemburg norm. For f1,f2,... ,fm ∈ L^Ф（I,X）, a distance formula distv（f1,f2,... ,fm,L^Ф（I,G）） is presented, where G is a close subspace of X. Moreover, some existence and characterization results concerning the best simultaneous approximation of L^Ф （I, G） in L^Ф （I, X） axe given.     

Shorted算子的几何结构

使用算子分块矩阵的技巧,研究了shorted算子,揭示了任意一个正算子和它的shorted算子之间的几何结构关系.此外,对由一个自伴算子A和一个闭子空间S组成的元素对(A,S)的兼容性(compatibility)进行了研究.特别地,当A是正算子时得出了集合∏(A,S)={Q∈∏:R(Q)=S⊥,AQ=Q*A}非空的充要条件;并且对集合∏(A,S)进行了详细的刻化,这里∏和S⊥分别表示一个复Hilbert空间上的所有幂等算子构成的集合和子空间S的正交补空间.     

EXTREME POINTS AND SUPPORT POINTS OF A CLASS OF ANALYTIC FUNCTIONS

1IntroductionByAwedenotethespaceoffunctionsanalyticintheunitdisk.ThetopologyofAisdefinedtobethetoPologyofuniformconvergenceoncompartsubsetsoftheunitdisk.SupposethatFisacomPactsubsetofA,feF.IfthereexistsacontinuouslinearfunctionalJonA,satisfyingthatffeJisnon-constantonF,suchthatthenjiscalledasupportpointofr.ThesetofallsupportpointsofFisdenotedbysUppF.SupposethatUisasubsetofA'fEUiscalledanextremepointofUiffcannotbeexpressedasaproperconvexcombinationoftwodistinctelementsofU.Thesetbfalle…     

Feedback invariants of pairs of matrices and restrictions

If (A,B) εF ^n×n ×F ^×m is a given pair and S is an (A,B)-invariant subspace we investigate the relationship between the feedback invariants of (A, B) and those of its restrictions

to S.     

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.     

A Sufficient Condition for Lipschitz Stability of Controlled Invariant Subspaces

Given a pair of matrices (A, B) we study the Lipschitz stability of its controlled invariant subspaces. A sufficient condition is derived from the geometry of the set formed by the quadruples (A, B, F, S) where S is an (A, B)-invariant subspace and F a corresponding feedback.