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1.
<正>Pfaffians and Representations of the Symmetric Group Alain LASCOUX Pfaffians of matrices with entries z[i,j]/(x_i+x_j),or determinants of matrices with entries z[i,j]/(x_i-x_j),where the antisymmetrical indeterminates z[i,j]satisfy the Pl_(u|¨)cker relations,can be identified with a trace in an irreducible representation of a product of two  相似文献   

2.
For an integer m ≥ 4, we define a set of 2[m/2] × 2[m/2] matrices γj (m), (j = 0, 1,..., m - 1) which satisfy γj (m)γk (m) +γk (m)γj (m) = 2ηjk (m)I[m/2], where (ηjk (m)) 0≤j,k≤m-1 is a diagonal matrix, the first diagonal element of which is 1 and the others are -1, I[m/2] is a 2[m/1] × 2[m/2] identity matrix with [m/2] being the integer part of m/2. For m = 4 and 5, the representation (m) of the Lorentz Spin group is known. For m≥ 6, we prove that (i) when m = 2n, (n ≥ 3), (m) is the group generated by the set of matrices {T|T=1/√ξ((I+k) 0 + 0 I-K) ( U 0 0 U), (ii) when m = 2n + 1 (n≥ 3), (m) is generated by the set of matrices {T|T=1/√ξ(I -k^- k I)U,U∈ (m-1),ξ=1-m-2 ∑k,j=0 ηkja^k a^j〉0, K=i[m-3 ∑j=0 a^j γj(m-2)+a^(m-2) In],K^-=i[m-3∑j=0 a^j γj(m-2)-a^(m-2) In]}  相似文献   

3.
Abstract: In this paper, we consider the Goldbach's problem for matrix rings, namely, we decompose an n ×n (n > 1) matrix over a principal ideal domain R into a sum of two matrices in Mn(R) with given determinants. We prove the following result: Let n > 1 be a natural number and A = (αij) be a matrix in Mn(R). Define d(A) := g.c.d{αij}. Suppose that p and q are two elements in R. Then (1) If n > 1 is even, then A can be written as a sum of two matrices X, Y in Mn(R) with det(X) = p and det(Y) = q if and only if d(A) |p-q; (2) If n > 1 is odd, then A can be written as a sum of two matrices X, Y in Mn(R) with det(X) = p and det(Y) = q if and only if d(A) |p + q. We apply the result to the matrices in Mn(Z) and Mn(Q[x]) and prove that if R = Z or Q[x], then any nonzero matrix A in Mn(R) can be written as a sum of two matrices in Mn(R) with prime determinants.  相似文献   

4.
The completely positive matrices are important in the study of block designs arizing incombinatorial analysis (see [11])and are related to copositive matrices,which have appli-cations in control theory and in multimobical programming (see [12]).An n×n matrix A is said to be completely positive if A can be factored as A=BB~T forsome n×m real matrix B with nonnegative entries for some m<∞.If A is a completelypositive matrix,then the smallest number of columns in a nonnegative matrix B such that  相似文献   

5.
In this article, we consider the integral representation of harmonic functions. Using a property of the modified Poisson kernel in a half plane, we prove that a harmonic function u(z) in a half plane with its positive part u^+(z) = max{u(z), 0} satisfying a slowly growing condition can be represented by its integral of a measure on the boundary of the half plan.  相似文献   

6.
Let R be a ring with 1, I be a nilpotent subring of R (there exists a natural number n, such that In = 0), and I be generated by {xj |j ∈ J} as ring. Write U = 1 + I, and it is a nilpotent group with class ≤ n - 1. Let G be the subgroup of U which is generated by {1 + xj|j ∈ J}. The group constructed in this paper indicates that the nilpotency class of G can be less than that of U.  相似文献   

7.
This paper studies framings in Banach spaces, a concept raised by Casazza, Han and Larson, which is a natural generalization of traditional frames in Hilbert spaces and unconditional bases in Banach spaces. The minimal unconditional bases and the maximal unconditional bases with respect to framings are introduced. Our main result states that, if (xi, fi) is a framing of a Banach space X, and (eimin) and (eimax) are the minimal unconditional basis and the maximal unconditional basis with respect to (xi, fi), respectively, then for any unconditional basis (ei) associated with (xi, fi), there are A,B 〉 0 such that A||i=1∑∞aieimin||≤||i=1∑∞aiei||≤B||i=1∑∞aieimax|| for all (ai) ∈ c00.
It means that for any framing, the corresponding associated unconditional bases have common upper and lower bounds.  相似文献   

8.
It is shown that an n × n matrix of continuous linear maps from a pro-C^*-algebra A to L(H), which verifies the condition of complete positivity, is of the form [V^*TijФ(·)V]^n i,where Ф is a representation of A on a Hilbert space K, V is a bounded linear operator from H to K, and j=1,[Tij]^n i,j=1 is a positive element in the C^*-algebra of all n×n matrices over the commutant of Ф(A) in L(K). This generalizes a result of C. Y.Suen in Proc. Amer. Math. Soc., 112(3), 1991, 709-712. Also, a covariant version of this construction is given.  相似文献   

9.
Let n≥2 be an integer number. In this paper, we investigate the generalized Hyers Ulam- Rassias stability in Banach spaces and also Banach modules over a Banach algebra and a C*-algebra and the stability using the alternative fixed point of an n-dimensional cubic functional equation in Banach spaces:f(2∑j=1^n-1 xj+xn)+f(2∑j=1^n-1 xj-xn)+4∑j=1^n-1f(xj)=16f(∑j=1^n-1 xj)+2∑j=1^n-1(f(xj+xn)+f(xj-xn)  相似文献   

10.
For a sparse non-singular matrix A, generally A~(-1)is a dense matrix. However, for a class of matrices,A~(-1)can be a matrix with off-diagonal decay properties, i.e., |A_(ij)~(-1)| decays fast to 0 with respect to the increase of a properly defined distance between i and j. Here we consider the off-diagonal decay properties of discretized Green's functions for Schr¨odinger type operators. We provide decay estimates for discretized Green's functions obtained from the finite difference discretization, and from a variant of the pseudo-spectral discretization. The asymptotic decay rate in our estimate is independent of the domain size and of the discretization parameter.We verify the decay estimate with numerical results for one-dimensional Schr¨odinger type operators.  相似文献   

11.
Let S={x1,…,xn} be a set of n distinct positive integers. For x,yS and y<x, we say the y is a greatest-type divisor of x in S if yx and it can be deduced that z=y from yz,zx,z<x and zS. For xS, let GS(x) denote the set of all greatest-type divisors of x in S. For any arithmetic function f, let (f(xi,xj)) denote the n×n matrix having f evaluated at the greatest common divisor (xi,xj) of xi and xj as its i,j-entry and let (f[xi,xj]) denote the n×n matrix having f evaluated at the least common multiple [xi,xj] of xi and xj as its i,j-entry. In this paper, we assume that S is a gcd-closed set and . We show that if f is a multiplicative function such that (fμ)(d)∈Z whenever and f(a)|f(b) whenever a|b and a,bS and (f(xi,xj)) is nonsingular, then the matrix (f(xi,xj)) divides the matrix (f[xi,xj]) in the ring Mn(Z) of n×n matrices over the integers. As a consequence, we show that (f(xi,xj)) divides (f[xi,xj]) in the ring Mn(Z) if (fμ)(d)∈Z whenever and f is a completely multiplicative function such that (f(xi,xj)) is nonsingular. This confirms a conjecture of Hong raised in 2004.  相似文献   

12.
Assume F={f1,. . .,fn} is a family of nonnegative functions of n−1 nonnegative variables such that, for every matrix A of order n, |aii|>fi (moduli of off-diagonal entries in row i of A) for all i implies A nonsingular. We show that there is a positive vector x, depending only on F, such that for all A=(aij), and all i, fi≥∑j|aij|{xj}/xi. This improves a theorem of Ky Fan, and yields a generalization of a nonsingularity criterion of Gudkov. Dedicated to Charles A. Micchelli, in celebration of his 60th birthday and our 30 years of friendship  相似文献   

13.
Let S={x 1,x 2,…,xn } be a naturally ordered set of distinct positive integers. S is called a k-set if k= gcd(xi ,xj ) for xi xj any in S. In this paper k-sets are characterized by certain conditions on the determinants of some matrices associated with S.  相似文献   

14.
Let λ be the upper Lyapunov exponent corresponding to a product of i.i.d. randomm×m matrices (X i) i 0/∞ over ℂ. Assume that theX i's are chosen from a finite set {D 0,D 1...,D t-1(ℂ), withP(X i=Dj)>0, and that the monoid generated byD 0, D1,…, Dq−1 contains a matrix of rank 1. We obtain an explicit formula for λ as a sum of a convergent series. We also consider the case where theX i's are chosen according to a Markov process and thus generalize a result of Lima and Rahibe [22]. Our results on λ enable us to provide an approximation for the numberN ≠0(F(x)n,r) of nonzero coefficients inF(x) n.(modr), whereF(x) ∈ ℤ[x] andr≥2. We prove the existence of and supply a formula for a constant α (<1) such thatN ≠0(F(x)n,r) ≈n α for “almost” everyn. Supported in part by FWF Project P16004-N05  相似文献   

15.
On the divisibility of power LCM matrices by power GCD matrices   总被引:3,自引:0,他引:3  
Let S = {x 1, ..., x n } be a set of n distinct positive integers and e ⩾ 1 an integer. Denote the n × n power GCD (resp. power LCM) matrix on S having the e-th power of the greatest common divisor (x i , x j ) (resp. the e-th power of the least common multiple [x i , x j ]) as the (i, j)-entry of the matrix by ((x i , x j ) e ) (resp. ([x i , x j ] e )). We call the set S an odd gcd closed (resp. odd lcm closed) set if every element in S is an odd number and (x i , x j ) ∈ S (resp. [x i , x j ] ∈ S) for all 1 ⩽ i, jn. In studying the divisibility of the power LCM and power GCD matrices, Hong conjectured in 2004 that for any integer e ⩾ 1, the n × n power GCD matrix ((x i , x j ) e ) defined on an odd-gcd-closed (resp. odd-lcm-closed) set S divides the n × n power LCM matrix ([x i , x j ] e ) defined on S in the ring M n (ℤ) of n × n matrices over integers. In this paper, we use Hong’s method developed in his previous papers [J. Algebra 218 (1999) 216–228; 281 (2004) 1–14, Acta Arith. 111 (2004), 165–177 and J. Number Theory 113 (2005), 1–9] to investigate Hong’s conjectures. We show that the conjectures of Hong are true for n ⩽ 3 but they are both not true for n ⩾ 4. Research is partially supported by Program for New Century Excellent Talents in University, by SRF for ROCS, SEM, China and by the Lady Davis Fellowship at the Technion, Israel Research is partially supported by a UGC (HK) grant 2160210 (2003/05).  相似文献   

16.
The existence and uniqueness of a surface with given geometric characteristics is one of the important topical problems of global differential geometry. By stating this problem in terms of analysis, we arrive at second-order elliptic and parabolic partial differential equations. In the present paper we consider generalized solutions of the Monge-Ampère equation ||z ij || = ϕ(x, z, p) in Λ n , wherez = z(x 1,...,z n ) is a convex function,p = (p 1,...,P n) = (∂z/∂x 1,...,ϖz/ϖx n), andz ij =ϖ 2 z/ϖx i ϖx j. We consider the Cayley-Klein model of the space Λ n and use a method based on fixed point principle for Banach spaces. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 763–768, November, 1998.  相似文献   

17.
DNA labelled graphs with DNA computing   总被引:2,自引:0,他引:2  
Let k≥2, 1≤i≤k andα≥1 be three integers. For any multiset which consists of some k-long oligonucleotides, a DNA labelled graph is defined as follows: each oligonucleotide from the multiset becomes a point; two points are connected by an arc from the first point to the second one if the i rightmost uucleotides of the first point overlap with the i leftmost nucleotides of the second one. We say that a directed graph D can be(k, i;α)-labelled if it is possible to assign a label(l_1(x),..., l_k(x))to each point x of D such that l_j(x)∈{0,...,a-1}for any j∈{1,...,k}and(x,y)∈E(D)if and only if(l_k-i 1(x),..., l_k(x))=(l_1(y),..., l_i(y)). By the biological background, a directed graph is a DNA labelled graph if there exist two integers k, i such that it is(k, i; 4)-labelled. In this paper, a detailed discussion of DNA labelled graphs is given. Firstly, we study the relationship between DNA labelled graphs and some existing directed graph classes. Secondly, it is shown that for any DNA labelled graph, there exists a positive integer i such that it is(2i, i; 4)-labelled. Furthermore, the smallest i is determined, and a polynomial-time algorithm is introduced to give a(2i, i; 4)-labelling for a given DNA labelled graph. Finally, a DNA algorithm is given to find all paths from one given point to another in a(2i, i; 4)-labelled directed graph.  相似文献   

18.
In this paper we shall first introduce the Pascal k-eliminated functional matrices Pn,k [x y z] and CPn,k [x y z]. Then, using these matrices we obtain several important combinatorial identities. Finally, using the matrix inversion of Pn,k [x y z] and CPn,k [x y z], we derive an interesting formula for Eulerian numbers [7]  相似文献   

19.
Let be an exponential polynomial over a field of zero characteristic. Assume that for each pair i,j with ij, α i j is not a root of unity. Define . We introduce a partition of into subsets (1≤im), which induces a decomposition of f into , so that, for 1≤im, , while for , the number either is transcendental or else is algebraic with not too small a height. Then we show that for all but at most solutions x∈ℤ of f(x)= 0, we have
Received: 7 August 1998  相似文献   

20.
In this paper we classify the centers localized at the origin of coordinates, the cyclicity of their Hopf bifurcation and their isochronicity for the polynomial differential systems in \mathbbR2{\mathbb{R}^2} of degree d that in complex notation z = x + i y can be written as
[(z)\dot] = (l+i) z + (z[`(z)])\fracd-52 (A z4+j[`(z)]1-j + B z3[`(z)]2 + C z2-j[`(z)]3+j+D[`(z)]5), \dot z = (\lambda+i) z + (z \overline{z})^{\frac{d-5}{2}} \left(A z^{4+j} \overline{z}^{1-j} + B z^3 \overline{z}^2 + C z^{2-j} \overline{z}^{3+j}+D \overline{z}^5\right),  相似文献   

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