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1.
Yongge Tian 《Linear and Multilinear Algebra》2013,61(2):123-147
The solvability conditions of the following two linear matrix equations (i)A1X1B1 +A2X2B2 +A3X3B3 =C,(ii) A1XB1 =C1 A2XB2 =C2 are established using ranks and generalized inverses of matrices. In addition, the duality of the three types of matrix equations (iii) A 1 X 1 B 1+A 2 X 2 B 2+A 3 X 3 B 3+A 4 X 4 B 4=C, (iv) A 1 XB 1=C 1 A 2 XB 2=C 2 A 3 XB 3=C 3 A 4 XB 4=C 4, (v) AXB+CXD=E are also considered. 相似文献
2.
Summary Letf, G1 × G2 C, where G
i
(i = 1, 2) denote arbitrary groups and C denotes the set of complex numbers. The general solutions of the following functional equationsf(x
1
y
1
,x
2
y
2
) +f(x
1
y
1
,x
2
y
2
-1
) +f(x
1
y
1
-1
,x
2
y
2
) +f(x
1
y
1
-1
,x
2
y
2
-1
) =f(x
1
,x
2
)F(y
1
,y
2
) +F(x
1
,x
2
)f(y
1
,y
2
) (1) andf(x
1
y
1
,x
2
y
2
) +f(x
1
y
1
,x
2
y
2
-1
) +f(x
1
y
1
-1
,x
2
y
2
) +f(x
1
y
1
-1
,x
2
y
2
-1
) =f(x
1
,x
2
)f(y
1
,y
2
) +F(x
1
,x
2
)F(y
1
,y
2
) (2) are determined assuming thatf satisfies the conditionf(x
1y1z1, x2) = f(x1z1y1, x2), f(x1, x2y2z2) = f(x1, x2z2y2) (C) for allx
i, yi, xi Gi (i = 1, 2). The functional equations (1) and (2) are generalizations of the well known rectangular type functional equationf(x
1 + y1, x2 + y2) + f(x1 + y1, x2 – y2) + f(x1 – y1, x2 + y2) + f(x1 – y1, x2 – y2) = 4f(x1, x2) studied by J. Aczel, H. Haruki, M. A. McKiernan and G. N. Sakovic in 1968. 相似文献
3.
A. S. Dzhumadil’daev 《Journal of Mathematical Sciences》2007,144(2):3909-3925
An algebra with the identity t
1(t
2
t
3) = (t
1
t
2+t
2
t
1)t
3 is called Zinbiel. For example, ℂ[x] under the multiplication
is Zinbiel. Let a ○
q
b = a ○ b + q b ○ a be a q-commutator, where q ∈ ℂ. We prove that for any Zinbiel algebra A the corresponding algebra under the commutator A
(−1) = (A, ○−1) satisfies the identities t
1
t
2 = −t
2
t
1 and (t
1
t
2)(t
3
t
4) + (t
1
t
4)(t
3
t
2) = jac(t
1, t
2, t
3)t
4 + jac(t
1, t
4, t
3)t
2, where jac(t
1, t
2, t
3) = (t
1
t
2)t
3 + (t
2
t
3)t
1 + (t
3
t
1)t
2. We find basic identities for q-Zinbiel algebras and prove that they form varieties equivalent to the variety of Zinbiel algebras if q
2 ≠ 1.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 57–78, 2005. 相似文献
4.
Write p
1, p
2…p
m
for the permutation matrix δ
pi, j
. Let S
n
(M) be the set of n×n permutation matrices which do not contain the m×m permutation matrix M as a submatrix. In [7] Simion and Schmidt show bijectively that |S
n
(123) |=|S
n
(213) |. In [9] this was generalised to a bijection between S
n
(12 p
3…p
m
) and S
n
(21 p
3…p
m
). In the present paper we obtain a bijection between S
n
(123 p
4…p
m
) and S
n
(321 p
4…p
m
).
Revised: March 24, 1999 相似文献
5.
The matrix least squares (LS) problem minx ||AXB^T--T||F is trivial and its solution can be simply formulated in terms of the generalized inverse of A and B. Its generalized problem minx1,x2 ||A1X1B1^T + A2X2B2^T - T||F can also be regarded as the constrained LS problem minx=diag(x1,x2) ||AXB^T -T||F with A = [A1, A2] and B = [B1, B2]. The authors transform T to T such that min x1,x2 ||A1X1B1^T+A2X2B2^T -T||F is equivalent to min x=diag(x1 ,x2) ||AXB^T - T||F whose solutions are included in the solution set of unconstrained problem minx ||AXB^T - T||F. So the general solutions of min x1,x2 ||A1X1B^T + A2X2B2^T -T||F are reconstructed by selecting the parameter matrix in that of minx ||AXB^T - T||F. 相似文献
6.
Melis Minisker 《Semigroup Forum》2009,78(1):99-105
In this paper we consider certain ranks of some semigroups. These ranks are r
1(S),r
2(S),r
3(S),r
4(S) and r
5(S) as defined below. We have r
1≤r
2≤r
3≤r
4≤r
5. The semigroups are CL
n
,CL
m
×CL
n
,Z
n
and SL
n
. Here CL
n
is a chain with n elements, Z
n
is the zero semigroup on n elements and SL
n
is the free semilattice generated by n elements and having 2
n
−1 elements. We find many of the ranks for these classes of semigroups. 相似文献
7.
Abstract Consider two independent random variables x and y with means and standard deviations μ x ,μ y ,σ x , and σ y , respectively. Let F x (t) = P[(x - μ, x )/σ x ≤ t] and F y (t) = P[(y - μ y )/σ y ≤ t]. In this article we address the problem of testing the null hypothesis H 0 : F x ≡ F y , against the alternative H 1 : F x ≡ F y . A graphical tool called T 3 plot for checking normality of independently and identically distributed univariate data was proposed in an earlier article by Ghosh. In the present article we develop a two-sample T 3 plot where the basic statistic is the normalized difference between the T 3 functions for the two samples. Significant departure of this difference function from the horizontal zero line is indicative of evidence against the null hypothesis. In contrast to the one-sample problem, the common distribution function under the null hypothesis is not specified in the two-sample case. Bootstrap is used to construct the acceptance region under H 0, for the two-sample T 3 plot. 相似文献
8.
We consider Abelian p-groups (p ≥ 3) A
1 = D
1 ⊕ G
1 and A
2 = D
2 ⊕ G
2, where D
1 and D
2 are divisible and G
1 and G
2 are reduced subgroups. We prove that if the automorphism groups Aut A
1 and Aut A
2 are elementarily equivalent, then the groups D
1, D
2 and G
1, G
2 are equivalent, respectively, in the second-order logic. 相似文献
9.
Claude Marion 《代数通讯》2013,41(3):853-925
Let p 1, p 2, p 3 be primes. This is the second article in a series of three on the (p 1, p 2, p 3)-generation of the finite projective special unitary and linear groups PSU3(p n ), PSL3(p n ), where we say a noncyclic group is (p 1, p 2, p 3)-generated if it is a homomorphic image of the triangle group T p 1, p 2, p 3 . This paper is concerned with the case where p 1 = 2 and p 2 = p 3. We determine for any prime p 2 the prime powers p n such that PSU3(p n ) (respectively, PSL3(p n )) is a quotient of T = T 2, p 2, p 2 . We also derive the limit of the probability that a randomly chosen homomorphism in Hom(T, PSU3(p n )) (respectively, Hom(T, PSL3(p n ))) is surjective as p n tends to infinity. 相似文献
10.
We introduce a notion of relative efficiency for axiom systems. Given an axiom system Aβ for a theory T consistent with S12, we show that the problem of deciding whether an axiom system Aα for the same theory is more efficient than Aβ is II2-hard. Several possibilities of speed-up of proofs are examined in relation to pairs of axiom systems Aα, Aβ, with Aα ? Aβ, both in the case of Aα, Aβ having the same language, and in the case of the language of Aα extending that of Aβ: in the latter case, letting Prα, Prβ denote the theories axiomatized by Aα, Aβ, respectively, and assuming Prα to be a conservative extension of Prβ, we show that if Aα — Aβ contains no nonlogical axioms, then Aα can only be a linear speed-up of Aβ; otherwise, given any recursive function g and any Aβ, there exists a finite extension Aα of Aβ such that Aα is a speed-up of Aβ with respect to g. Mathematics Subject Classification: 03F20, 03F30. 相似文献
11.
12.
The bounded edge-connectivity λk(G) of a connected graph G with respect to is the minimum number of edges in G whose deletion from G results in a subgraph with diameter larger than k and the edge-persistence D+(G) is defined as λd(G)(G), where d(G) is the diameter of G. This paper considers the Cartesian product G1×G2, shows λk1+k2(G1×G2)≥λk1(G1)+λk2(G2) for k1≥2 and k2≥2, and determines the exact values of D+(G) for G=Cn×Pm, Cn×Cm, Qn×Pm and Qn×Cm. 相似文献
13.
A graph G is called the 2-amalgamation of subgraphs G1 and G2 if G = G1 ∪ G2 and G1 ∩ G2 = {x, y}, 2 distinct points. in this case we write G = G1∪{x, y} G2. in this paper we show that the orientable genus, γ(G), satisfies the inequalities γ(G1) + γ(G2) ? 1 ≤ γ(G1 ∪{x, y} G2) ≤ γ(G1) + γ(G2) + 1 and that this is the best possible result, i. e., the resulting three values for γ(G1 ∪{x, y} G2) which are possible can actually be realized by appropriate choices for G1 and G2. 相似文献
14.
We study the existence and shape preserving properties of a generalized Bernstein operator B
n
fixing a strictly positive function f
0, and a second function f
1 such that f
1/f
0 is strictly increasing, within the framework of extended Chebyshev spaces U
n
. The first main result gives an inductive criterion for existence: suppose there exists a Bernstein operator B
n
: C[a, b] → U
n
with strictly increasing nodes, fixing f0, f1 ? Un{f_{0}, f_{1} \in U_{n}} . If Un ì Un + 1{U_{n} \subset U_{n + 1}} and U
n+1 has a non-negative Bernstein basis, then there exists a Bernstein operator B
n+1 : C[a, b] → U
n+1 with strictly increasing nodes, fixing f
0 and f
1. In particular, if f
0, f
1, . . . , f
n
is a basis of U
n
such that the linear span of f
0, . . . , f
k
is an extended Chebyshev space over [a, b] for each k = 0, . . . , n, then there exists a Bernstein operator B
n
with increasing nodes fixing f
0 and f
1. The second main result says that under the above assumptions the following inequalities hold
Bn f 3 Bn+1 f 3 fB_{n} f \geq B_{n+1} f \geq f 相似文献
15.
We study the attractors of a finite system of planar contraction similarities S
j
(j=1,...,n) satisfying the coupling condition: for a set {x
0,...,x
n} of points and a binary vector (s
1,...,s
n
), called the signature, the mapping S
j
takes the pair {x
0,x
n} either into the pair {x
j-1,x
j
} (if s
j
=0) or into the pair {x
j
, x
j-1} (if s
j
=1). We describe the situations in which the Jordan property of such attractor implies that the attractor has bounded turning, i.e., is a quasiconformal image of an interval of the real axis. 相似文献
16.
Let Aj, Bj be complex B-spaces, j = 0, 1, Aθ and Bθ–the complex-interpolation spaces generated by the couples (A0, A1) and (B0, B1), resp., by CALDERON'S/LIONS'S method. Let T: A0 ∧ A1 → B0 → B1 be an operator satisfying some conditions such as continuity, estimates etc. in terms of the norms of Aj, Bj (j = 0, 1). We consider the question which one of these properties is inherited to T when A0 → A1 and B0 → B1 are equipped with the norm of Aθ and Bθ. 相似文献
17.
Douglas Bauer 《Journal of Graph Theory》1980,4(2):219-232
A degree sequence π = (d1, d2,…,dp), with d1 ≥ d2 ≥…≥ dp, is line graphical if it is realized by the line graph of some graph. Degree sequences with line-graphical realizations are characterized for the cases d1 = p - 1, d1 = p - 2, d1 ≤ 3, and d1 = dp. It is also shown that if a degree sequence with d1 = p-1 is line graphical, it is uniquely line graphical. It follows that with possibly one exception each line-graphical realization of an arbitrary degree sequence must have either C5, 2K1, + K2, K1 + 2K2, or 3K1, as an induced subgraph. 相似文献
18.
Let R be a local ring and let (x
1, …, x
r) be part of a system of parameters of a finitely generated R-module M, where r < dimR
M. We will show that if (y
1, …, y
r) is part of a reducing system of parameters of M with (y
1, …, y
r) M = (x
1, …, x
r) M then (x
1, …, x
r) is already reducing. Moreover, there is such a part of a reducing system of parameters of M iff for all primes P ε Supp M ∩ V
R(x
1, …, x
r) with dimR
R/P = dimR
M − r the localization M
P of M at P is an r-dimensional Cohen-Macaulay module over R
P.
Furthermore, we will show that M is a Cohen-Macaulay module iff y
d is a non zero divisor on M/(y
1, …, y
d−1) M, where (y
1, …, y
d) is a reducing system of parameters of M (d:= dimR
M). 相似文献
19.
《代数通讯》2013,41(1):379-389
Abstract Let d 1 : k[X] → k[X] and d 2 : k[Y] → k[Y] be k-derivations, where k[X] ? k[x 1,…,x n ], k[Y] ? k[y 1,…,y m ] are polynomial algebras over a field k of characteristic zero. Denote by d 1 ⊕ d 2 the unique k-derivation of k[X, Y] such that d| k[X] = d 1 and d| k[Y] = d 2. We prove that if d 1 and d 2 are positively homogeneous and if d 1 has no nontrivial Darboux polynomials, then every Darboux polynomial of d 1 ⊕ d 2 belongs to k[Y] and is a Darboux polynomial of d 2. We prove a similar fact for the algebra of constants of d 1 ⊕ d 2 and present several applications of our results. 相似文献
20.
Hongbo Zhang 《代数通讯》2013,41(4):1420-1427
An element of a ring R is called “strongly clean” if it is the sum of an idempotent and a unit that commute, and R is called “strongly clean” if every element of R is strongly clean. A module M is called “strongly clean” if its endomorphism ring End(M) is a strongly clean ring. In this article, strongly clean modules are characterized by direct sum decompositions, that is, M is a strongly clean module if and only if whenever M′⊕ B = A 1⊕ A 2 with M′? M, there are decompositions M′ = M 1⊕ M 2, B = B 1⊕ B 2, and A i = C i ⊕ D i (i = 1,2) such that M 1⊕ B 1 = C 1⊕ D 2 = M 1⊕ C 1 and M 2⊕ B 2 = D 1⊕ C 2 = M 2⊕ C 2. 相似文献
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