共查询到20条相似文献,搜索用时 31 毫秒
1.
J.F. Colombeau 《Journal of Mathematical Analysis and Applications》1983,94(1):96-115
If Ω denotes an open subset of n (n = 1, 2,…), we define an algebra (Ω) which contains the space ′(Ω) of all distributions on Ω and such that is a subalgebra of (Ω). The elements of (Ω) may be considered as “generalized functions” on Ω and they admit partial derivatives at any order that generalize exactly the derivation of distributions. The multiplication in (Ω) gives therefore a natural meaning to any product of distributions, and we explain how these results agree with remarks of Schwartz on difficulties concerning a multiplication of distributions. More generally if q = 1, 2,…, and —a classical Schwartz notation—for any G1,…,Gq∈G(σ), we define naturally an element . These results are applied to some differential equations and extended to the vector valued case, which allows the multiplication of vector valued distributions of physics. 相似文献
2.
Michel Talagrand 《Comptes Rendus Mathematique》2003,337(7):477-480
Consider a random Hamiltonian for We assume that the family is jointly Gaussian centered and that for =ξ(N?1∑i?Nσ1iσ2i) for a certain function ξ on . F. Guerra proved the remarkable fact that the free energy of the system with Hamiltonian is bounded below by the free energy of the Parisi solution provided that ξ is convex on . We prove that this fact remains (asymptotically) true when the function ξ is only assumed to be convex on . This covers in particular the case of the p-spin interaction model for any p. To cite this article: M. Talagrand, C. R. Acad. Sci. Paris, Ser. I 337 (2003). 相似文献
3.
Robert L McFarland 《Journal of Combinatorial Theory, Series A》1973,15(1):1-10
A construction is given for difference sets in certain non-cyclic groups with the parameters , , , n = q2s for every prime power q and every positive integer s. If qs is odd, the construction yields at least inequivalent difference sets in the same group. For q = 5, s = 2 a difference set is obtained with the parameters (v, k, λ, n) = (4000, 775, 150, 625), which has minus one as a multiplier. 相似文献
4.
Here it is proved that a cyclic (n, k) code over GF(q) is equidistant if and only if its parity check polynomial is irreducible and has exponent where a divides q ? 1 and (a, k) = 1. The length n may be any multiple of e. The proof of this theorem also shows that if a cyclic (n,k) code over GF(q) is not a repetition of a shorter code and the average weight of its nonzero code words is integral, then its parity check polynomial is irreducible over GF(q) with exponent where a divides q ? 1. 相似文献
5.
6.
Tosio Kato 《Journal of Functional Analysis》1973,12(4):415-417
It is shown that the method of Chernoff developed in the preceding paper can be modified to prove the essential self-adjointness on C0∞(Rm) of all positive powers of the Schrödinger operator T = ? Δ + q if q real and in C∞(Rm) and if . 相似文献
7.
Yasuhiro Takeuchi Norihiko Adachi 《Journal of Mathematical Analysis and Applications》1981,79(1):141-162
This paper presents sufficient conditions for the existence of a nonnegative and stable equilibrium point of a dynamical system of Volterra type, (1) , for every q = (q1,…, qn)T?Rn. Results of a nonlinear complementarity problem are applied to obtain the conditions. System (1) has a nonnegative and stable equilibrium point if (i) f(x) = (f1(x),…,fn(x))T is a continuous and differentiable M-function and it satisfies a certain surjectivity property, or (ii), f(x) is continuous and strongly monotone on R+0n. 相似文献
8.
Malcolm R. Adams 《Journal of Functional Analysis》1983,52(3):420-441
Let Q be a self-adjoint, classical, zeroth order pseudodifferential operator on a compact manifold X with a fixed smooth measure dx. We use microlocal techniques to study the spectrum and spectral family, {ES}S∈ as a bounded operator on L2(X, dx).Using theorems of Weyl (Rend. Circ. Mat. Palermo, 27 (1909), 373–392) and Kato (“Perturbation Theory for Linear Operators,” Springer-Verlag, 1976) on spectra of perturbed operators we observe that the essential spectrum and the absolutely continuous spectrum of Q are determined by a finite number of terms in the symbol expansion. In particular SpecESSQ = range(q(x, ξ)) where q is the principal symbol of Q. Turning the attention to the spectral family {ES}S∈, it is shown that if is considered as a distribution on ×X×X it is in fact a Lagrangian distribution near the set where (s, x, y, σ, ξ,η) are coordinates on T1(×X×X) induced by the coordinates (s, x, y) on ×X×X. This leads to an easy proof that is a pseudodifferential operator if ?∈C∞() and to some results on the microlocal character of Es. Finally, a look at the wavefront set of leads to a conjecture about the existence of absolutely continuous spectrum in terms of a condition on q(x, ξ). 相似文献
9.
Real constant coefficient nth order elliptic operators, Q, which generate strongly continuous semigroups on L2(k) are analyzed in terms of the elementary generator, , for n even. Integral operators are defined using the fundamental solutions pn(x, t) to ut = Au and using real polynomials ql,…, qk on m by the formula, for q = (ql,…, qk), m. It is determined when, strongly on L2(k), . If n = 2 or k = 1, this can always be done. Otherwise the symbol of Q must have a special form. 相似文献
10.
M.P Heble 《Journal of Mathematical Analysis and Applications》1983,93(2):363-384
Given a cocycle a(t) of a unitary group {U1}, ?∞ < t < ∞, on a Hilbert space , such that a(t) is of bounded variation on [O, T] for every T > O, a(t) is decomposed as a(t) = f;t0Usxds + β(t) for a unique x ? , β(t) yielding a vector measure singular with respect to Lebesgue measure. The variance is defined as if existing. For a stationary diffusion process on 1, with Ω1, the space of paths which are natural extensions backwards in time, of paths confined to one nonsingular interval J of positive recurrent type, an information function I(ω) is defined on , based on the paths restricted to the time interval [0, 1]. It is shown that is continuous and bounded on . The shift τt, defines a unitary representation {Ut}. Assuming , dm being the stationary measure defined by the transition probabilities and the invariant measure on J, has a C∞ spectral density function f;. It is then shown that σ2({Ut}, I) = f;(O). 相似文献
11.
A.M Fink 《Journal of Mathematical Analysis and Applications》1977,61(2):404-408
We show how inequalities of the type when F(0) = 0 can be used to find lower bounds of the first eigenvalue of the integral equation F(z) = λ ∝0ak(s, z)F(s) ds. 相似文献
12.
Ian Knowles 《Journal of Mathematical Analysis and Applications》1978,66(3):574-585
A sufficient condition is given for the operator T0: C∞0(Rm) → L2(Rm) given by to be essentially self-adjoint. This condition is sufficiently general to admit certain potentials q having unbounded oscillations in a neighborhood of ∞. 相似文献
13.
Noboru Ito 《Journal of Combinatorial Theory, Series A》1980,29(2):251-253
A lower bound is given for the minimum weight of the symmetry code C(q) over GF(3), which is introduced by Pless [3]. 相似文献
14.
Nonlinear partial differential operators having the form G(u) = g(u, D1u,…, DNu), with g?C(R × RN), are here shown to be precisely those operators which are local, (locally) uniformly continuous on, , and (roughly speaking) translation invariant. It is also shown that all such partial differential operators are necessarily bounded and continuous with respect to the norm topologies of . 相似文献
15.
On , n?1 and n≠2, we prove the existence of a sharp constant for Sobolev inequalities with higher fractional derivatives. Let s be a positive real number. For n>2s and any function satisfies where the operator (?Δ)s in Fourier spaces is defined by . To cite this article: A. Cotsiolis, N.C. Tavoularis, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 801–804. 相似文献
16.
Let and denote respectively the space of n×n complex matrices and the real space of n×n hermitian matrices. Let p,q,n be positive integers such that p?q?n. For , the (p,q)-numerical range of A is the set , where Cp(X) is the pth compound matrix of X, and Jq is the matrix Iq?On-q. Let denote n or . The problem of determining all linear operators T: → such that is treated in this paper. 相似文献
17.
F. Götze 《Journal of multivariate analysis》1981,11(2):260-272
Let P(Θ, τ) 6 , θ ∈ Θ ? , τ ∈ T ? p denote a family of probability measures, where τ denotes the vector of nuisance parameters. Starting from randomized asymptotic maximum likelihood (as. m. l.) estimators for (θ, τ) we construct randomized estimators which are asymptotically median unbiased up to resp. test procedures which are as. similar of level (for testing θ = θ0, τ ∈ T against one sided alternatives). The estimation procedures are second-order efficient in the class of estimators which are median unbiased up to and the test procedures are second-order efficient in the class of tests which are as. of level . These results hold without any continuity condition on the family of probability measures. 相似文献
18.
Hermann König 《Journal of Functional Analysis》1977,24(1):32-51
For an open set Ω ? N, 1 ? p ? ∞ and λ ∈ +, let denote the Sobolev-Slobodetzkij space obtained by completing in the usual Sobolev-Slobodetzkij norm (cf. A. Pietsch, “r-nukleare Sobol. Einbett. Oper., Ellipt. Dgln. II,” Akademie-Verlag, Berlin, 1971, pp. 203–215). Choose a Banach ideal of operators , 1 ? p, q ? ∞ and a quasibounded domain Ω ? N. Theorem 1 of the note gives sufficient conditions on λ such that the Sobolev-imbedding map exists and belongs to the given Banach ideal : Assume the quasibounded domain fulfills condition Ckl for some l > 0 and 1 ? k ? N. Roughly this means that the distance of any to the boundary ?Ω tends to zero as for , and that the boundary consists of sufficiently smooth ?(N ? k)-dimensional manifolds. Take, furthermore, 1 ? p, q ? ∞, p > k. Then, if μ, ν are real positive numbers with λ = μ + v ∈ , μ > λ S(; p,q:N) and v > N/l · λD(;p,q), one has that belongs to the Banach ideal . Here λD(;p,q;N)∈+ and λS(;p,q;N)∈+ are the D-limit order and S-limit order of the ideal , introduced by Pietsch in the above mentioned paper. These limit orders may be computed by estimating the ideal norms of the identity mappings lpn → lqn for n → ∞. Theorem 1 in this way generalizes results of R. A. Adams and C. Clark for the ideals of compact resp. Hilbert-Schmidt operators (p = q = 2) as well as results on imbeddings over bounded domains.Similar results over general unbounded domains are indicated for weighted Sobolev spaces.As an application, in Theorem 2 an estimate is given for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in , where Ω fulfills condition C1l.For an open set Ω in N, let denote the Sobolev-Slobodetzkij space obtained by completing in the usual Sobolev-Slobodetzkij norm, see below. Taking a fixed Banach ideal of operators and 1 ? p, q ? ∞, we consider quasibounded domains Ω in N and give sufficient conditions on λ such that the Sobolev imbedding operator exists and belongs to the Banach ideal. This generalizes results of C. Clark and R. A. Adams for compact, respectively, Hilbert-Schmidt operators (p = q = 2) to general Banach ideals of operators, as well as results on imbeddings over bounded domains. Similar results over general unbounded domains may be proved for weighted Sobolev spaces. As an application, we give an estimate for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in , where Ω is a quasibounded open set in N. 相似文献
19.
Lyle Ramshaw 《Journal of Number Theory》1981,13(2):138-175
Let θ be an irrational number, and consider sequences of the form ωθ = 〈kθ〉k≥0 of points in the circle R/Z. By employing symmetry, we can show that the discrepancy DN(ωθ) of the finite sequence 〈kθ〉0≤k<N is determined by its behavior on the N arcs whose endpoints are iθ and (N ? 1 ? i)θ for 0 ≤ i < N. We then use continued fraction methods to analyze its behavior on these arcs. The resulting expression for DN(ωθ has several consequences. First, we show that the discrepancies DN(ωσ) and DN(ωτ) are closely related if σ and τ are equivalent irrationals; in particular, we prove the equality . Finally, we compute a tight asymptotic bound on DN(ωθ) when θ has the special form for some positive integer m by showing that 相似文献
20.
Let (m?n) denote the linear space of all m × n complex or real matrices according as = or . Let c=(c1,…,cm)≠0 be such that c1???cm?0. The c-spectral norm of a matrix A?m×n is the quantity . where σ1(A)???σm(A) are the singular values of A. Let d=(d1,…,dm)≠0, where d1???dm?0. We consider the linear isometries between the normed spaces and , and prove that they are dual transformations of the linear operators which map (d) onto (c), where . 相似文献