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1.
We show that the fixed elements for the natural GL m-action on the universal division algebra UD( m, n) of m generic n× n-matrices form a division subalgebra of degree n, assuming n3 and 2 mn2−2. This allows us to describe the asymptotic behavior of the dimension of the space of SL m-invariant homogeneous central polynomials p( X1,…, Xm) for n× n-matrices. Here the base field is assumed to be of characteristic zero. 相似文献
2.
Let be a finite or infinite sequence of 2×2 matrices with entries in an integral domain. We show that, except in a very special case, is (simultaneously) triangularizable if and only if all pairs ( Aj, Ak) are triangularizable, for 1 j, k∞. We also provide a simple numerical criterion for triangularization.Using constructive methods in invariant theory, we define a map (with the minimal number of invariants) that distinguishes simultaneous similarity classes for non-commutative sequences over a field of characteristic ≠2. We also describe canonical forms for sequences of 2×2 matrices over algebraically closed fields, and give a method for finding sequences with a given set of invariants. 相似文献
3.
The Toeplitz pencil conjecture stated in [W. Schmale, P.K. Sharma, Problem 30-3: singularity of a toeplitz matrix, IMAGE 30 (2003); W. Schmale, P.K. Sharma, Cyclizable matrix pairs over and a conjecture on toeplitz pencils, Linear Algebra Appl. 389 (2004) 33-42] is equivalent to a conjecture for n× n Hankel pencils of the form Hn( x)=( ci+j-n+1), where c0= x is an indeterminate, cl=0 for l<0, and for l1. In this paper it is shown to be implied by another conjecture, which we call the root conjecture. The root conjecture asserts a strong relationship between the roots of certain submaximal minors of Hn( x) specialized to have c1= c2=1. We give explicit formulae in the ci for these minors and prove the root conjecture for minors mnn, mn-1,n of degree 6. This implies the Hankel Pencil conjecture for matrices up to size 8×8. The main tools involved are a partial parametrization of the set of solutions of systems of polynomial equations that are both homogeneous and index sum homogeneous, and use of the Sylvester identity for matrices. 相似文献
4.
Suppose that G is a graph with n vertices and m edges, and let μ be the spectral radius of its adjacency matrix.Recently we showed that if G has no 4-cycle, then μ2- μn-1, with equality if and only if G is the friendship graph.Here we prove that if m9 and G has no 4-cycle, then μ2m, with equality if G is a star. For 4 m8 this assertion fails. 相似文献
5.
We develop a general context for the computation of the determinant of a Hankel matrix Hn = ( αi+j)0i,jn, assuming some suitable conditions for the exponential (or ordinary) generating function of the sequence ( αn) n0. Several well-known particular cases are thus derived in a unified way. 相似文献
6.
New pointwise inversion formulae are obtained for the d-dimensional totally geodesic Radon transform on the n-dimensional real hyperbolic space, 1 dn−1, in terms of polynomials of the Laplace–Beltrami operator and intertwining fractional integrals. Similar results are established for hyperbolic cosine and sine transforms. 相似文献
7.
A finite group G is called an ah-group if any two distinct conjugacy classes of G have distinct cardinality. We show that if G is an ah-group, then the non-abelian socle of G is isomorphic to one of the following: 1. , for 1a5, a≠2. 2. A8. 3. PSL(3,4)e, for 1e10. 4. A5×PSL(3,4)e, for 1e10. Based on this result, we virtually show that if G is an ah-group with π( G) 2,3,5,7 , then F( G)≠1, or equivalently, that G has an abelian normal subgroup.In addition, we show that if G is an ah-group of minimal size which is not isomorphic to S3, then the non-abelian socle of G is either trivial or isomorphic to one of the following: 1. , for 3a5. 2. PSL(3,4)e, for 1e10. Our research lead us to interesting results related to transitivity and homogeneousity in permutation groups, and to subgroups of wreath products of form Z2Sn. These results are of independent interest and are located in appendices for greater autonomy. 相似文献
8.
Suppose μ and ν are integer partitions of n, and N> n. It is well known that the Ferrers boards associated to μ and ν are rook-equivalent iff the multisets [ μi+ i:1 iN] and [ νi+ i:1 iN] are equal. We use the Garsia–Milne involution principle to produce a bijective proof of this theorem in which non-attacking rook placements for μ are explicitly matched with corresponding placements for ν. One byproduct is a direct combinatorial proof that the matrix of Stirling numbers of the first kind is the inverse of the matrix of Stirling numbers of the second kind. We also prove q-analogues and p, q-analogues of these results. We also use the Garsia–Milne involution principle to show that for any two rook boards B and B′, if B and B′ are bijectively rook-equivalent, then B and B′ are bijectively hit-equivalent. 相似文献
9.
The problem of approximating a given function by Dirichlet series with nonnegative coefficients is associated with the discrete spectral representation of the relaxation modulus in rheology. The main result of this paper is that if a function can be approximated arbitrarily closely by Dirichlet series with nonnegative coefficients in supremum norm or L p-norm, 1 p<∞, then it must be completely monotonic. 相似文献
11.
We extend the direct algorithm for computing the derivatives of the compactly supported Daubechies N-vanishing-moment basis functions. The method yields exact values at dyadic rationals for the nth derivative (0 n N − 1) of the basis functions, when it exists. Example results are shown for the first derivatives of the basis functions from the Daubechies N-vanishing-moment extremal phase orthonormal family (for N = 3, 4, and 5), and the CDF(2, N) spline-based biorthogonal family (for N = 6, 8, and 10). 相似文献
12.
The cone of Completely Positive (CP) matrices can be used to exactly formulate a variety of NP-Hard optimization problems. A tractable relaxation for CP matrices is provided by the cone of Doubly Nonnegative (DNN) matrices; that is, matrices that are both positive semidefinite and componentwise nonnegative. A natural problem in the optimization setting is then to separate a given DNN but non-CP matrix from the cone of CP matrices. We describe two different constructions for such a separation that apply to 5 × 5 matrices that are DNN but non-CP. We also describe a generalization that applies to larger DNN but non-CP matrices having block structure. Computational results illustrate the applicability of these separation procedures to generate improved bounds on difficult problems. 相似文献
13.
We discuss the existence of a diffeomorphism such that | where are closed differential forms and 2kn. Our main results (the case k=n having been handled by Moser [J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965) 286–294] and Dacorogna and Moser [B. Dacorogna, J. Moser, On a partial differential equation involving the Jacobian determinant, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990) 1–26]) are that- – when n is even and k=2, under some natural non-degeneracy condition, we can prove the existence of such diffeomorphism satisfying Dirichlet data on the boundary of a bounded open set and the natural Hölder regularity; at the same time we get Darboux theorem with optimal regularity;
- – we are also able to handle the degenerate cases when k=2 (in particular when n is odd), k=n−1 and some cases where 3kn−2.
Résumé
Nous montrons l'existence d'un difféomorphisme
satisfaisant
où
sont des formes différentielles fermées et 2
kn. Nos résultats principaux (le cas
k=
n a été discuté notamment dans Moser [J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965) 286–294] et Dacorogna et Moser [B. Dacorogna, J. Moser, On a partial differential equation involving the Jacobian determinant, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990) 1–26]) sont les suivants.
- – Si n est pair, k=2 et sous des conditions naturelles de non dégénérescence, nous montrons l'existence et la régularité dans les espaces de Hölder d'un tel difféomorphisme satisfaisant de plus une condition de Dirichlet. On obtient aussi le théorème de Darboux avec la régularité optimale.
- – Par ailleurs quand k=2 et n est impair ou k=n−1, ainsi que quelques cas particuliers où 3kn−2, nous montrons l'existence locale d'un tel difféomorphisme satisfaisant, en outre, des conditions de Cauchy.
Keywords: Darboux theorem; Symplectic forms; Pullback; Hölder regularity
相似文献
14.
We consider boolean circuits
C over the basis
Ω={,} with inputs
x1,
x2,…,
xn for which arrival times
are given. For 1
in we define the delay of
xi in
C as the sum of
ti and the number of gates on a longest directed path in
C starting at
xi. The delay of
C is defined as the maximum delay of an input.Given a function of the form
f(x1,x2,…,xn)=gn−1(gn−2(…g3(g2(g1(x1,x2),x3),x4)…,xn−1),xn)
where
gjΩ for 1
jn−1 and arrival times for
x1,
x2,…,
xn, we describe a cubic-time algorithm that determines a circuit for
f over
Ω that is of linear size and whose delay is at most 1.44 times the optimum delay plus some small constant.
相似文献
15.
Let μ be a real measure on the line such that its Poisson integral
M(
z) converges and satisfies|
M(
x+
iy)|
Ae−cyα,
y→+∞,for some constants
A,
c>0 and 0<α1. We show that for 1/2<α1 the measure μ must have many sign changes on both positive and negative rays. For 0<α1/2 this is true for at least one of the rays, and not always true for both rays. Asymptotical bounds for the number of sign changes are given which are sharp in some sense.
相似文献
16.
Fluid particle trajectories for the Rayleigh–Bénard problem in a cube with perfectly conducting lateral walls have been investigated. The velocity and temperature fields of the stationary flow solutions have been obtained by means of a parameter continuation procedure based on a Galerkin spectral method. The rich dynamics of the resulting fluid particle paths has been studied for three branches of stationary solutions and different values of the Rayleigh number within the range10
4Ra1.5×10
5 at a Prandtl number equal to 130. The stability properties and bifurcations of fixed points, which play a key role in the global dynamics, have been analyzed. Main periodic orbits and their stability character have also been determined. Poincaré maps reveal that regions of chaotic motion and regions of regular motion coexist inside the cavity. The boundaries of these three-dimensional regions have been determined. The metric entropy gives an indication of the mixing properties of the large chaotic zone.
相似文献
18.
In this paper we consider the classical Erdős–Rényi model of random graphs
Gn,p. We show that for
p=
p(
n)
n−3/4−δ, for any fixed
δ>0, the chromatic number
χ(
Gn,p) is a.a.s.
ℓ,
ℓ+1, or
ℓ+2, where
ℓ is the maximum integer satisfying 2(
ℓ−1)log(
ℓ−1)
p(
n−1).
相似文献
19.
There are basic equivalent assertions known for operator monotone functions and operator convex functions in two papers by Hansen and Pedersen. In this note we consider their results as correlation problem between two sequences of matrix
n-monotone functions and matrix
n-convex functions, and we focus the following three assertions at each label
n among them:
- (i) f(0)0 and f is n-convex in [0,α),
- (ii) For each matrix a with its spectrum in [0,α) and a contraction c in the matrix algebra Mn,
- (iii) The function is n-monotone in (0,α).
We show that for any
nN two conditions (ii) and (iii) are equivalent. The assertion that
f is
n-convex with
f(0)0 implies that
g(
t) is (
n-1)-monotone holds. The implication from (iii) to (i) does not hold even for
n=1. We also show in a limited case that the condition (i) implies (ii).
相似文献
20.
Let
S[
i,
j] denote the
q-Stirling numbers of the second kind. We show that the determinant of the matrix (
S[
s+
i+
j,
s+
j])
0i,jn is given by the product
. We give two proofs of this result, one bijective and one based upon factoring the matrix. We also prove an identity due to Cigler that expresses the Hankel determinant of
q-exponential polynomials as a product. Lastly, a two variable version of a theorem of Sylvester and an application are presented.
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