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1.
In this paper, we establish the formulas of the maximal and minimal ranks of the Hermitian matrix expression ${C_{5}-A_{3}X_{1}A_{3}*-A_{4}X_{2}A_{4}*}$ where X 1 and X 2 are Hermitian solutions to two systems of matrix equations A 1 X 1 = C 1,X 1 B 1 = C 2 and A 2 X 2 = C 3,X 2 B 2 = C 4, respectively. Using this result and matrix rank method, we give necessary and sufficient conditions for the existence of Hermitian solutions to a system of five matrix equations ${A_{1}X_{1}=C_{1},X_{1}B_{1}=C_{2},A_{2}X_{2}=C_{3},X_{2}B_{2}=C_{4} ,A_{3}X_{1}A_{3}*+A_{4}X_{2}A_{4}*=C_{5}}$ by rank equalities. The general expressions and extreme ranks of the Hermitian solutions X 1 and X 2 to the system mentioned above are also presented. 相似文献
2.
Kh. D. Ikramov 《Journal of Mathematical Sciences》2004,121(4):2458-2464
A matrix
is said to be accretive-dissipative if, in its Hermitian decomposition
, both matrices B and C are positive definite. Further, if B= I
n, then A is called a Buckley matrix. The following extension of the classical Fischer inequality for Hermitian positive-definite matrices is proved. Let
be an accretive-dissipative matrix, k and l be the orders of A
11 and A
22, respectively, and let m = min{k,l}. Then
For Buckley matrices, the stronger bound
is obtained. Bibliography: 5 titles. 相似文献
3.
For fixed generalized reflection matrix P, i.e. P
T
= P, P
2 = I, then matrix X is said to be generalized bisymmetric, if X = X
T
= PXP. In this paper, an iterative method is constructed to find the generalized bisymmetric solutions of the matrix equation A
1
X
1
B
1 + A
2
X
2
B
2 + ⋯ + A
l
X
l
B
l
= C where [X
1,X
2, ⋯ ,X
l
] is real matrices group. By this iterative method, the solvability of the matrix equation can be judged automatically. When
the matrix equation is consistent, for any initial generalized bisymmetric matrix group , a generalized bisymmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors,
and the least norm generalized bisymmetric solution group can be obtained by choosing a special kind of initial generalized
bisymmetric matrix group. In addition, the optimal approximation generalized bisymmetric solution group to a given generalized
bisymmetric matrix group in Frobenius norm can be obtained by finding the least norm generalized bisymmetric solution group of the new matrix equation
, where . Given numerical examples show that the algorithm is efficient.
Research supported by: (1) the National Natural Science Foundation of China (10571047) and (10771058), (2) Natural Science
Foundation of Hunan Province (06JJ2053), (3) Scientific Research Fund of Hunan Provincial Education Department(06A017). 相似文献
4.
Alain Grigis 《Geometriae Dedicata》2009,143(1):81-88
We present a construction of triangulations of the torus
\mathbbTn{\mathbb{T}^n} with 2
n+1−1 vertices, starting from the Delaunay triangulation of a generic lattices. The typical example is the Kühnel–Lassmann triangulation,
with help of the lattice A*n{A^{*}_{n}} . Concerning this triangulation, we also prove a result of unicity for some values of n. 相似文献
5.
Igor V. Protasov 《Algebra Universalis》2009,62(4):339-343
Let ${\mathbb{A}}Let
\mathbbA{\mathbb{A}} be a universal algebra of signature Ω, and let I{\mathcal{I}} be an ideal in the Boolean algebra
P\mathbbA{\mathcal{P}_{\mathbb{A}}} of all subsets of
\mathbbA{\mathbb{A}} . We say that I{\mathcal{I}} is an Ω-ideal if I{\mathcal{I}} contains all finite subsets of
\mathbbA{\mathbb{A}} and f(An) ? I{f(A^{n}) \in \mathcal{I}} for every n-ary operation f ? W{f \in \Omega} and every A ? I{A \in \mathcal{I}} . We prove that there are 22à0{2^{2^{\aleph_0}}} Ω-ideals in
P\mathbbA{\mathcal{P}_{\mathbb{A}}} provided that
\mathbbA{\mathbb{A}} is countably infinite and Ω is countable. 相似文献
6.
Yuexu Zhao 《Bulletin of the Brazilian Mathematical Society》2006,37(3):377-391
Let X1, X2, ... be i.i.d. random variables with EX1 = 0 and positive, finite variance σ2, and set Sn = X1 + ... + Xn. For any α > −1, β > −1/2 and for κn(ε) a function of ε and n such that κn(ε) log log n → λ as n ↑ ∞ and
, we prove that
*Supported by the Natural Science Foundation of Department of Education of Zhejiang Province (Grant No. 20060237 and 20050494). 相似文献
7.
Zheng Yan LIN Sung Chul LEE 《数学学报(英文版)》2006,22(2):535-544
Let {Xn,n ≥ 0} be an AR(1) process. Let Q(n) be the rescaled range statistic, or the R/S statistic for {Xn} which is given by (max1≤k≤n(∑j=1^k(Xj - ^-Xn)) - min 1≤k≤n(∑j=1^k( Xj - ^Xn ))) /(n ^-1∑j=1^n(Xj -^-Xn)^2)^1/2 where ^-Xn = n^-1 ∑j=1^nXj. In this paper we show a law of iterated logarithm for rescaled range statistics Q(n) for AR(1) model. 相似文献
8.
The solvability in anisotropic spaces
, σ ∈ ℝ+, p, q ∈ (1, ∞), of the heat equation ut − Δu = f in ΩT ≡ (0, T) × Ω is studied under the boundary and initial conditions u = g on ST, u|t=0 = u0 in Ω, where S is the boundary of a bounded domain Ω ⊂ ℝn. The existence of a unique solution
of the above problem is proved under the assumptions that
and under some additional conditions on the data. The existence is proved by the technique of regularizers. For this purpose
the local-in-space solvability near the boundary and near an interior point of Ω is needed. To show the local-in-space existence,
the definition of Besov spaces by the dyadic decomposition of a partition of unity is used. This enables us to get an appropriate
estimate in a new and promising way without applying either the potential technique or the resolvent estimates or the interpolation.
Bibliography: 26 titles.
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 348, 2007, pp. 40–97. 相似文献
9.
Aimé Lachal 《Journal of Theoretical Probability》2006,19(4):757-771
Let (B
t
)
t≥ 0 be standard Brownian motion starting at y and set X
t
= for , with V(y) = y
γ if y≥ 0, V(y) = −K(−y)γ if y≤ 0, where γ and K are some given positive constants. Set . In this paper, we provide some formulas for the probability distribution of the random variable as well as for the probability (or b)}. The formulas corresponding to the particular cases x = a or b are explicitly expressed by means of hypergeometric functions.
相似文献
10.
The iterative method of the generalized coupled Sylvester-conjugate matrix equations \(\sum\limits _{j=1}^{l}\left (A_{ij}X_{j}B_{ij}+C_{ij}\overline {X}_{j}D_{ij}\right )=E_{i} (i=1,2,\cdots ,s)\) over Hermitian and generalized skew Hamiltonian solution is presented. When these systems of matrix equations are consistent, for arbitrary initial Hermitian and generalized skew Hamiltonian matrices X j (1), j = 1,2,? , l, the exact solutions can be obtained by iterative algorithm within finite iterative steps in the absence of round-off errors. Furthermore, we provide a method for choosing the initial matrices to obtain the least Frobenius norm Hermitian and generalized skew Hamiltonian solution of the problem. Finally, numerical examples are presented to demonstrate the proposed algorithm is efficient. 相似文献
11.
Vladislav Kargin 《Probability Theory and Related Fields》2007,139(3-4):397-413
Let X
i
denote free identically-distributed random variables. This paper investigates how the norm of products behaves as n approaches infinity. In addition, for positive X
i
it studies the asymptotic behavior of the norm of where denotes the symmetric product of two positive operators: . It is proved that if EX
i
= 1, then is between and c
2
n for certain constant c
1 and c
2. For it is proved that the limit of exists and equals Finally, if π is a cyclic representation of the algebra generated by X
i
, and if ξ is a cyclic vector, then for all n. These results are significantly different from analogous results for commuting random variables. 相似文献
12.
We prove that a (globally) subanalytic function ${f : X \subset {\bf Q}^{n}_{p} \rightarrow {\bf Q}_{p}}We prove that a (globally) subanalytic function f : X ì Qnp ? Qp{f : X \subset {\bf Q}^{n}_{p} \rightarrow {\bf Q}_{p}} which is locally Lipschitz continuous with some constant C is piecewise (globally on each piece) Lipschitz continuous with possibly some other constant, where the pieces can be taken
to be subanalytic. We also prove the analogous result for a subanalytic family of functions fy : Xy ì Qnp ? Qp{f_{y} : X_{y} \subset {\bf Q}^{n}_{p} \rightarrow {\bf Q}_{p}} depending on p−adic parameters. The statements also hold in a semi-algebraic set-up and also in a finite field extension of Q
p
. These results are p−adic analogues of results of K. Kurdyka over the real numbers. To encompass the total disconnectedness of p−adic fields, we need to introduce new methods adapted to the p−adic situation. 相似文献
13.
Markus Haase 《Integral Equations and Operator Theory》2006,56(2):197-228
We generalize a Hilbert space result by Auscher, McIntosh and Nahmod to arbitrary Banach spaces X and to not densely defined injective sectorial operators A. A convenient tool proves to be a certain universal extrapolation space associated with A. We characterize the real interpolation space
( X,D( Aa ) ?R( Aa ) )q,p{\left( {X,\mathcal{D}{\left( {A^{\alpha } } \right)} \cap \mathcal{R}{\left( {A^{\alpha } } \right)}} \right)}_{{\theta ,p}}
as
{ x ? X|t - q\textRea y1 ( tA )x, t - q\textRea y2 ( tA )x ? L*p ( ( 0,¥ );X ) } {\left\{ {x\, \in \,X|t^{{ - \theta {\text{Re}}\alpha }} \psi _{1} {\left( {tA} \right)}x,\,t^{{ - \theta {\text{Re}}\alpha }} \psi _{2} {\left( {tA} \right)}x \in L_{*}^{p} {\left( {{\left( {0,\infty } \right)};X} \right)}} \right\}} 相似文献
14.
Rumi Shindo 《Mediterranean Journal of Mathematics》2011,8(1):81-95
Let A, B be uniform algebras. Suppose that A
0, B
0 are subgroups of A
−1, B
−1 that contain exp A, exp B respectively. Let α be a non-zero complex number. Suppose that m, n are non-zero integers and d is the greatest common divisor of m and n. If T : A
0 → B
0 is a surjection with ||T(f)mT(g)n - a||¥ = ||fmgn - a||¥{\|T(f)^{m}T(g)^{n} - \alpha\|_{\infty} = \|f^{m}g^{n} - \alpha\|_{\infty}} for all f,g ? A0{f,g \in A_0}, then there exists a real-algebra isomorphism [(T)\tilde] : A ? B{\tilde{T} : A \rightarrow B} such that [(T)\tilde](f)d = (T(f)/T(1))d{\tilde{T}(f)^d = (T(f)/T(1))^d} for every f ? A0{f \in A_0}. This result leads to the following assertion: Suppose that S
A
, S
B
are subsets of A, B that contain A
−1, B
−1 respectively. If m, n > 0 and a surjection T : S
A
→ S
B
satisfies ||T(f)mT(g)n - a||¥ = ||fmgn - a||¥{\|T(f)^{m}T(g)^{n} - \alpha\|_{\infty} = \|f^{m}g^{n} - \alpha\|_{\infty}} for all f, g ? SA{f, g \in S_A}, then there exists a real-algebra isomorphism [(T)\tilde] : A ? B{\tilde{T} : A \rightarrow B} such that [(T)\tilde](f)d = (T(f)/T(1))d{\tilde{T}(f)^d = (T(f)/T(1))^d} for every f ? SA{f \in S_A}. Note that in these results and elsewhere in this paper we do not assume that T(exp A) = exp B. 相似文献
15.
Let {A, B} and {C, D} be diagonalizable pairs of order n, i.e., there exist invertible matrices P, Q and X, Ysuchthat A = P∧Q, B = PΩQ, C =XГY, D= X△Y, where
∧ = diag(α1, α2, …, αn), Ω= diag(βl, β2, …βn), Г=diag(γ1,γ2,…,γn), △=diag(δl,δ2,…,δn). Let ρ((α,β), (γ,δ))=|αδ-βγ|/√|α|^2+|β|^2√|γ|^2+|δ|^2.In this paper, it will be proved that there is a permutation τ of {1,2,... ,n} such that n∑i=1[ρ((αi,βi),(γτ(i),δτ(i)))]^2≤n[1-1/κ^2(Y)κ^2(Q)(1-d2F(Z,W)/n)], where κ(Y) = ||Y||2||Y^-1||2,Z= (A,B),W= (C, D) and dF(Z,W) = 1/√2||Pz* -Pw*||F. 相似文献 16.
Let be a field of characteristic and S
1 the unit circle. We prove that the shc-structure on a cochain algebra (A,d
A
) induces an associative product on the negative cyclic homology HC
*−
A. When the cochain algebra (A,d
A
) is the algebra of normalized cochains of the simply connected topological space X with coefficients in , then HC
*−
A is isomorphic as a graded algebra to the S
1-equivariant cohomology algebra of LX, the free loop space of X. We use the notion of shc-formality introduced in Topology 41, 85–106 (2002) to compute the S
1-equivariant cohomology algebras of the free loop space of the complex projective space when n + 1 = 0 [p] and of the even spheres S
2n
when p = 2.
相似文献
17.
Let {X
t
: 0 ≦ t ≦ 1} be a centered stationary Gaussian process, with correlation function satisfying the condition ρ(t) = 1 − t
β
L(t), 0 < β < 2, and let L be a slowly varying function at zero. Observing the process at points i/N, i = 0,1,..., N and considering |X
i/N
− X
(i-1)/N
|
p
with p > 0, we study the properties of the Donsker line associated with p-th order variations
. We also study the relationship between the number of crossings of a regularization of the initial process and the local
time of the initial process. The results depend on the values of β.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
18.
In this paper, we consider the Schrödinger type operator ${H = (-\Delta _{\mathbb {H}}^n)^2 +V ^{2}}
19.
Let A
0, ... , A
n−1 be operators on a separable complex Hilbert space , and let α0,..., α
n−1 be positive real numbers such that 1. We prove that for every unitarily invariant norm,
20.
Let {X
n
; n ≥ 1} be a sequence of independent and identically distributed U[0,1]-distributed random variables. Define the uniform empirical process $F_n (t) = n^{ - \tfrac{1}
{2}} \sum\nolimits_{i = 1}^n {(I_{\{ X_i \leqslant t\} } - t),0} \leqslant t \leqslant 1,\left\| {F_n } \right\| = \sup _{0 \leqslant t \leqslant 1} \left| {F_n (t)} \right|
$F_n (t) = n^{ - \tfrac{1}
{2}} \sum\nolimits_{i = 1}^n {(I_{\{ X_i \leqslant t\} } - t),0} \leqslant t \leqslant 1,\left\| {F_n } \right\| = \sup _{0 \leqslant t \leqslant 1} \left| {F_n (t)} \right|
. In this paper, the exact convergence rates of a general law of weighted infinite series of E {‖F
n
‖ − ɛg
s
(n)}+ are obtained. 相似文献
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