共查询到20条相似文献,搜索用时 31 毫秒
1.
Let Mn(F) be the algebra of n×n matrices over a field F, and let A∈Mn(F) have characteristic polynomial c(x)=p1(x)p2(x)?pr(x) where p1(x),…,pr(x) are distinct and irreducible in F[x]. Let X be a subalgebra of Mn(F) containing A. Under a mild hypothesis on the pi(x), we find a necessary and sufficient condition for X to be Mn(F). 相似文献
2.
Li Qiu 《Linear algebra and its applications》2007,422(1):304-307
Let A1, … , Ak be positive semidefinite matrices and B1, … , Bk arbitrary complex matrices of order n. We show that
span{(A1x)°(A2x)°?°(Akx)|x∈Cn}=range(A1°A2°?°Ak) 相似文献
3.
Krishnaswami Alladi 《Journal of Number Theory》1982,14(1):86-98
Let p(n) denote the smallest prime factor of an integer n>1 and let p(1)=∞. We study the asymptotic behavior of the sum M(x,y)=Σ1≤n≤x,p(n)>yμ(n) and use this to estimate the size of A(x)=max|f|≤1|Σ2≤n<xμ(n)f(p(n))|, where μ(n) is the Moebius function. Applications of bounds for A(x), M(x,y) and similar quantities are discussed. 相似文献
4.
R.C. Baker 《Journal of Number Theory》2010,130(10):2119-2146
Let F(x1,…,xn) be a nonsingular indefinite quadratic form, n=3 or 4. For n=4, suppose the determinant of F is a square. Results are obtained on the number of solutions of
F(x1,…,xn)=0 相似文献
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If M is a mean on and M(f(x1),f(x2),…,f(xn))=f(M(x1,x2,…,xn)) then we say that M is invariant under f. The problem is to find a class of functions that by invariance determines a mean uniquely. We focus on the geometric mean, which can be transformed to obtain results for other means. 相似文献
7.
We show that every injective Jordan semi-triple map on the algebra Mn(F) of all n × n matrices with entries in a field F (i.e. a map Φ:Mn(F)→Mn(F) satisfying
Φ(ABA)=Φ(A)Φ(B)Φ(A) 相似文献
8.
Norihide Tokushige 《Discrete Mathematics》2010,310(3):453-460
Let m(n,k,r,t) be the maximum size of satisfying |F1∩?∩Fr|≥t for all F1,…,Fr∈F. We prove that for every p∈(0,1) there is some r0 such that, for all r>r0 and all t with 1≤t≤⌊(p1−r−p)/(1−p)⌋−r, there exists n0 so that if n>n0 and p=k/n, then . The upper bound for t is tight for fixed p and r. 相似文献
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10.
We propose a new characterization of dual bases in finite fields. Let A=(α1,…,αn) be a basis of F over Fq and its dual basis B=(β1,…,βn) with the transition matrix C∈GLn(Fq) such that (β1,…,βn)=(α1,…,αn)C. We show that holds for all 1?k?n, where Tk∈Mn(Fq) satisfies αk(α1,…,αn)=(α1,…,αn)Tk. Conversely, suppose F=Fq(αk′) and for some 1?k′?n and G∈GLn(Fq), then B is equivalent to (α1,…,αn)G. As applications, we can construct the dual basis of a given basis A or determine whether the dual basis of A satisfies the desired conditions from Tk. This generalizes the results obtained by Liao and Sun for normal bases. Furthermore, we give a simple proof of the theorem of Gollmann, Wang and Blake for polynomial bases. 相似文献
11.
In this paper, we have found upper and lower bounds for the spectral norms of r-circulant matrices in the forms A = Cr(F0, F1, …, Fn−1), B = Cr(L0, L1, …, Ln−1), and we have obtained some bounds for the spectral norms of Kronecker and Hadamard products of A and B matrices. 相似文献
12.
Let Mn(F) denote the algebra of n×n matrices over the field F of complex, or real, numbers. Given a self-adjoint involution J∈Mn(C), that is, J=J*,J2=I, let us consider Cn endowed with the indefinite inner product [,] induced by J and defined by [x,y]?〈Jx,y〉,x,y∈Cn. Assuming that (r,n-r), 0?r?n, is the inertia of J, without loss of generality we may assume J=diag(j1,?,jn)=Ir⊕-In-r. For T=(|tik|2)∈Mn(R), the matrices of the form T=(|tik|2jijk), with all line sums equal to 1, are called J-doubly stochastic matrices. In the particular case r∈{0,n}, these matrices reduce to doubly stochastic matrices, that is, non-negative real matrices with all line sums equal to 1. A generalization of Birkhoff’s theorem on doubly stochastic matrices is obtained for J-doubly stochastic matrices and an application to determinants is presented. 相似文献
13.
Chun-Gil Park 《Journal of Mathematical Analysis and Applications》2005,307(2):753-762
It is shown that every almost linear bijection of a unital C∗-algebra A onto a unital C∗-algebra B is a C∗-algebra isomorphism when h(n2uy)=h(n2u)h(y) for all unitaries u∈A, all y∈A, and n=0,1,2,…, and that almost linear continuous bijection of a unital C∗-algebra A of real rank zero onto a unital C∗-algebra B is a C∗-algebra isomorphism when h(n2uy)=h(n2u)h(y) for all , all y∈A, and n=0,1,2,…. Assume that X and Y are left normed modules over a unital C∗-algebra A. It is shown that every surjective isometry , satisfying T(0)=0 and T(ux)=uT(x) for all x∈X and all unitaries u∈A, is an A-linear isomorphism. This is applied to investigate C∗-algebra isomorphisms between unital C∗-algebras. 相似文献
14.
The Kalmár function K(n) counts the factorizations n=x1x2…xr with xi?2(1?i?r). Its Dirichlet series is where ζ(s) denotes the Riemann ζ function. Let ρ=1.728… be the root greater than 1 of the equation ζ(s)=2. Improving on preceding results of Kalmár, Hille, Erd?s, Evans, and Klazar and Luca, we show that there exist two constants C5 and C6 such that, for all n, holds, while, for infinitely many n's, .An integer N is called a K-champion number if M<N⇒K(M)<K(N). Several properties of K-champion numbers are given, mainly about the size of the exponents and the number of prime factors in the standard factorization into primes of a large enough K-champion number.The proof of these results is based on the asymptotic formula of K(n) given by Evans, and on the solution of a problem of optimization. 相似文献
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Aleksander Grytczuk 《Journal of Number Theory》2010,130(7):1480-1487
Let Fn be a binary form with integral coefficients of degree n?2, let d denote the greatest common divisor of all non-zero coefficients of Fn, and let h?2 be an integer. We prove that if d=1 then the Thue equation (T) Fn(x,y)=h has relatively few solutions: if A is a subset of the set T(Fn,h) of all solutions to (T), with r:=card(A)?n+1, then
- (#)
- h divides the numberΔ(A):=∏1?k<l?rδ(ξk,ξl),
17.
Let [n] denote the set of positive integers {1,2,…,n}. An r-partial permutation of [n] is a pair (A,f) where A⊆[n], |A|=r and f:A→[n] is an injective map. A set A of r-partial permutations is intersecting if for any (A,f), (B,g)∈A, there exists x∈A∩B such that f(x)=g(x). We prove that for any intersecting family A of r-partial permutations, we have .It seems rather hard to characterize the case of equality. For 8?r?n-3, we show that equality holds if and only if there exist x0 and ε0 such that A consists of all (A,f) for which x0∈A and f(x0)=ε0. 相似文献
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Sukumar Das Adhikari 《Journal of Combinatorial Theory, Series A》2008,115(1):178-184
Let G be a finite abelian group of order n and let A⊆Z be non-empty. Generalizing a well-known constant, we define the Davenport constant of G with weight A, denoted by DA(G), to be the least natural number k such that for any sequence (x1,…,xk) with xi∈G, there exists a non-empty subsequence (xj1,…,xjl) and a1,…,al∈A such that . Similarly, for any such set A, EA(G) is defined to be the least t∈N such that for all sequences (x1,…,xt) with xi∈G, there exist indices j1,…,jn∈N,1?j1<?<jn?t, and ?1,…,?n∈A with . In the present paper, we establish a relation between the constants DA(G) and EA(G) under certain conditions. Our definitions are compatible with the previous generalizations for the particular group G=Z/nZ and the relation we establish had been conjectured in that particular case. 相似文献
20.
Let F be a field with ∣F∣ > 2 and Tn(F) be the set of all n × n upper triangular matrices, where n ? 2. Let k ? 2 be a given integer. A k-tuple of matrices A1, …, Ak ∈ Tn(F) is called rank reverse permutable if rank(A1 A2 ? Ak) = rank(Ak Ak−1 ? A1). We characterize the linear maps on Tn(F) that strongly preserve the set of rank reverse permutable matrix k-tuples. 相似文献