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1.
A special class n of n×n matrices is described, which has tensor rank n over the real field. A tensor base for general symmetric, persymmetric, both symmetric and persymmetric matrices and Toeplitz symmetric matrices can be defined in terms of the tensor bases of l for some different values of l. It is proved that both symmetric and persymmetric n×n matrices and Toeplitz symmetric n×n matrices have tensor rank and 2n?2, respectively, in the real field. 相似文献
2.
《Nonlinear Analysis: Theory, Methods & Applications》2004,57(3):349-362
We show that in a smooth bounded domain , n⩾2, all global nonnegative solutions of ut−Δum=up with zero boundary data are uniformly bounded in by a constant depending on and τ but not on u0, provided that 1<m<p<[(n+1)/(n−1)]m. Furthermore, we prove an a priori bound in depending on under the optimal condition 1<m<p<[(n+2)/(n−2)]m. 相似文献
3.
B. Roth 《Journal of Functional Analysis》1975,18(4):329-337
Let [(Ω)]p be the Cartesian product of the space of real-valued infinitely differentiable functions on a connected open set Ω in n with itself p-times. The finitely generated submodules of [(Ω)]p are of the form im(F) where F: [(Ω)]q → [(Ω)]p is a p × q matrix of infinitely differentiable functions on Ω. Let . The main results of the present paper are that for Ω ? n, if the finitely generated submodule im(F) is closed in [(Ω)]p, then for every x?ω with rank(F(x)) < r there exists an r × r sub-matrix A of F such that x is a zero of finite order of det(A), and for Ω ? 1 the converse also holds. 相似文献
4.
Let T be a linear transformation on the set of m × n matrices with entries in an algebraically closed field. If T maps the set of all matrices whose rank is k into itself, and if, then the rank of A is the rank of T(A) for every m × n matrix. 相似文献
5.
An n × n matrix A is called involutory iff A2=In, where In is the n × n identity matrix. This paper is concerned with involutory matrices over an arbitrary finite commutative ring R with identity and with the similarity relation among such matrices. In particular the authors seek a canonical set with respect to similarity for the n × n involutory matrices over R—i.e., a set of n × n involutory matrices over R with the property that each n × n involutory matrix over R is similar to exactly on matrix in . Because of the structure of finite commutative rings and because of previous research, they are able to restrict their attention to finite local rings of characteristic a power of 2, and although their main result does not completely specify a canonical set for such a ring, it does solve the problem for a special class of rings and shows that a solution to the general case necessarily contains a solution to the classically unsolved problem of simultaneously bringing a sequence A1,…,Av of (not necessarily involutory) matrices over a finite field of characteristic 2 to canonical form (using the same similarity transformation on each Ai). (More generally, the authors observe that a theory of similarity fot matrices over an arbitrary local ring, such as the well-known rational canonical theory for matrices over a field, necessarily implies a solution to the simultaneous canonical form problem for matrices over a field.) In a final section they apply their results to find a canonical set for the involutory matrices over the ring of integers modulo 2m and using this canonical set they are able to obtain a formula for the number of n × n involutory matrices over this ring. 相似文献
6.
《Nonlinear Analysis: Theory, Methods & Applications》2004,56(2):185-199
In this paper we are concerned with positive solutions of the doubly nonlinear parabolic equation ut=div(um−1|∇u|p−2∇u)+Vum+p−2 in a cylinder , with initial condition u(·,0)=u0(·)⩾0 and vanishing on the parabolic boundary . Here (resp. ) is a bounded domain with smooth boundary, , , 1<p<N and m+p−2>0. The critical exponents are found and the nonexistence results are proved for . 相似文献
7.
Peter D Johnson 《Discrete Mathematics》1984,48(1):83-85
If A is a set colored with m colors, and B is colored with n colors, the coloring of A × B obtained by coloring (a, b) with the pair (color of a, color of b) will be called an m × n simple product coloring (SPC) of A × B. SPC's of Cartesian products of three or more sets are defined analogously. It is shown that there are 2 × 2, and 2 × 2 × 2 SPC's of 2 and 3 which forbid the distance one; that there is no 2k SPC of k forbidding the distance one, for k > 3; and that there is no 2 × 2 SPC of × (√15), and thus none of 2, forbidding the distance 1. 相似文献
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10.
Analogues of characterizations of rank-preserving operators on field-valued matrices are determined for matrices witheentries in certain structures contained in the nonnegative reals. For example, if is the set of nonnegative members of a real unique factorization domain (e.g. the nonnegative reals or the nonnegative integers), M is the set of m×n matrices with entries in , and min(m,n)?4, then a “linear” operator on M preserves the “rank” of each matrix in M if and only if it preserves the ranks of those matrices in M of ranks 1, 2, and 4. Notions of rank and linearity are defined analogously to the field-valued concepts. Other characterizations of rank-preserving operators for matrices over these and other structures are also given. 相似文献
11.
Mohammed Aassila 《Comptes Rendus Mathematique》2002,334(11):961-966
We study the semilinear wave equation utt?Δu=p?k|u|m in , where p is a conformal factor approaching 0 at infinity. We prove that the solutions blow-up in finite time for small powers m, while having an arbitrarily long life-span for large m. Furthermore, we study the finite time blow-up of solutions for the class of quasilinear wave equations utt?Δu=p?k|Lu|m in . To cite this article: M. Aassila, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 961–966. 相似文献
12.
Antoine Brunel 《Comptes Rendus Mathematique》2002,334(3):205-207
We consider positive linear operators on Lp-spaces (1<p<∞), (A(Lp+)?Lp+), satisfying the inequality Am+n<Am+An for all We describe the structure of these operators (Theorem 1). As a consequence we obtain for all f∈Lp,Anf converges a.e. The last statement contains the theorem of a.e. convergence of Cesaro averages for positive mean bounded operators. To cite this article: A. Brunel, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 205–207. 相似文献
13.
《Journal of Pure and Applied Algebra》1986,42(3):237-243
Let I be an ideal, and let f = {Kn|n ≥ 0 } be a filtration of the Noetherian ring R, such that In ⊆ Kn for all n ≥ 0. We study when the Rees ring (f) is either finite or integral over the Rees ring (I), for two types of filtrations f which have recently drawn interest. If I and J are ideals in R, and if m(n) is the least power of J such that (In : Jm(n) + 1), we show that the function m(n) is eventually non-decreasing. For J regular, we characterize when it is eventually constant. 相似文献
14.
Lindsay N. Childs Gerald Garfinkel Morris Orzech 《Journal of Pure and Applied Algebra》1975,6(2):111-123
Let B(R) be the Brauer group of the integrally closed noetherian domain R with quotient field K. We reexamine the proof that B(R) → R(K) is monic for R regular from the point of view of factoriality of R and its extensions. For R local with maximal ideal m, henselization Rh and divisor class group Cl(R) we embed ker into . This is applied to obtain examples of non-regular geometric local domains R for which B(R) → B(K) is monic. 相似文献
15.
Beth Allen 《Stochastic Processes and their Applications》1984,16(3):275-285
Tangencies and level crossings of a random field X:m+×Ω→n (which is not necessarily Gaussian) are studied under the assumption that almost every sample path is continuously differentiable. If n=m and if the random field has uniformly bounded sample derivatives and uniformly bounded densities for the distributions of the Xl, then for a compact K?m+ and any fixed level, the restriction to K of almost every sample path has no tangencies to the level and at most finitely many crossings. The case of n≠m is also examined. Some generic properties, which hold for a residual set of random fields, are analyzed. Proofs involve the concepts of regularity and transversality from differential topology. 相似文献
16.
James G. Hooton 《Journal of Mathematical Analysis and Applications》1981,83(2):570-581
We provide conditions on a finite measure μ on n which insure that the imbeddings Wk, p(ndμ)?Lp(ndμ) are compact, where 1 ? p < ∞ and k is a positive integer. The conditions involve uniform decay of the measure μ for large ¦x¦ and are satisfied, for example, by . 相似文献
17.
David Gurarie 《Journal of Mathematical Analysis and Applications》1985,108(1):223-229
For elliptic operators on Rn and certain of their singular perturbations relative compactness of B with respect to A is established. This result applies to the study of Lp-spectra of elliptic operators for different p. 相似文献
18.
The Schur product of two n×n complex matrices A=(aij), B=(bij) is defined by A°B=(aijbij. By a result of Schur [2], the algebra of n×n matrices with Schur product and the usual addition is a commutative Banach algebra under the operator norm (the norm of the operator defined on n by the matrix). For a fixed matrix A, the norm of the operator B?A°B on this Banach algebra is called the Schur multiplier norm of A, and is denoted by ∥A∥m. It is proved here that for all unitary U (where ∥·∥ denotes the operator norm) iff A is a scalar multiple of a unitary matrix; and that ∥A∥m=∥A∥ iff there exist two permutations P, Q, a p×p (1?p?n) unitary U, an (n?p)×(n?p)1 contraction C, and a nonnegative number λ such that and this is so iff , where ā is the matrix obtained by taking entrywise conjugates of A. 相似文献
19.
The asymptotic behaviour as t tends to +∞ of the solution of in N × +, p > 1, was studied. It was proved that the behaviour depends strongly on the sign of and also on the rate of decay of the admissible initial data u(0, x) as tends to +∞. 相似文献
20.
Let A be a real symmetric n × n matrix of rank k, and suppose that A = BB′ for some real n × m matrix B with nonnegative entries (for some m). (Such an A is called completely positive.) It is shown that such a B exists with , where 2N is the maximal number of (off-diagonal) entries which equal zero in a nonsingular principal submatrix of A. An example is given where the least m which works is (k odd), (k even). 相似文献