共查询到20条相似文献,搜索用时 31 毫秒
1.
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
2.
Zhang Xianke 《Journal of Number Theory》1984,18(3):350-355
Let k = (√u) (u ≠ 1 squarefree), K any possible cyclic quartic field containing k. A close relation is established between K and the genus group of k. In particular: (1) Each K can be written uniquely as K = (√vwη), where η is fixed in k and satisfies η ? 1, (η) = 2√u, |2| = |(√u)|, (v, u) = 1, v ∈ is squarefree, w|u, 0 < w < √u. Thus if u ≠ a2 + b2, there is no K ? k. If u = a2 + b2 then for each fixed v there are 2g ? 1K ? k, where g is the number of prime divisors of u. (2) has a relative integral basis (RIB) (i.e., OK is free over Ok) iff N(ε0) = ?1 and w = 1, where ε0 is the fundamental unit of k, (or, equivalently, iff K = (√vε0√u), (v, u) = 1). (3) A RIB is constructed explicitly whenever it exists. (4) disc(K) is given. In particular, the following results are special cases of (2): (i) Narkiewicz showed in 1974 that has a RIB if u is a prime; (ii) Edgar and Peterson (J. Number Theory12 (1980), 77–83) showed that for u composite there is at least one K ? k having no RIB. Besides, it follows from (4) that the classification and integral basis of K given by Albert (Ann. of Math.31 (1930), 381–418) are wrong. 相似文献
3.
George Dassios 《Journal of Differential Equations》1983,49(1):124-141
We show that if is a star-shaped body in 3 and u is a solution of the equation of linearized homogeneous isotropic elasticity in the exterior of which vanishes on ? and has initially compact support, then the rate of decay of the total elastic energy in any sphere of finite radius is bounded by t? for large values of the time t. 相似文献
4.
《Nonlinear Analysis: Theory, Methods & Applications》2004,57(2):153-172
We study the Cauchy problem for the equation ∂tuε−Δuε=−βε(uε) in (0,∞)×Rn as , where the nonlinearity βε is assumed to converge to a measure concentrated at . In this paper we allow for sign changes of βε and uε. The solutions are uniformly Lipschitz continuous in space and Hölder continuous in time. We show that each limit of uε is a solution of the free boundary problem ∂tu−Δu=0 in {u>0}∩(0,∞)×Rn,|∇u+|2−|∇u−|2=g on (∂{u>0}∪∂{u<0})∩((0,∞)×Rn) in the sense of domain variations. Depending on the structure of the nonlinearity the function g in the condition on the free boundary need not be a constant. 相似文献
5.
David M. Mason 《Stochastic Processes and their Applications》1983,15(1):99-109
Let Gn denote the empirical distribution based on n independent uniform (0, 1) random variables. The asymptotic distribution of the supremum of weighted discrepancies between Gn(u) and u of the forms 6wv(u)Dn(u)6 and 6wv(Gn(u))Dn(u)6, where Dn(u) = Gn(u)?u, wv(u) = (u(1?u))?1+v and 0 ? v < is obtained. Goodness-of-fit tests based on these statistics are shown to be asymptotically sensitive only in the extreme tails of a distribution, which is exactly where such statistics that use a weight function wv with ? v ? 1 are insensitive. For this reason weighted discrepancies which use the weight function wv with 0 ? v < are potentially applicable in the construction of confidence contours for the extreme tails of a distribution. 相似文献
6.
7.
R.S. Singh 《Journal of multivariate analysis》1976,6(2):338-342
Let Xj = (X1j ,…, Xpj), j = 1,…, n be n independent random vectors. For x = (x1 ,…, xp) in Rp and for α in [0, 1], let Fj(x) = αI(X1j < x1 ,…, Xpj < xp) + (1 ? α) I(X1j ≤ x1 ,…, Xpj ≤ xp), where I(A) is the indicator random variable of the event A. Let Fj(x) = E(Fj(x)) and Dn = supx, α max1 ≤ N ≤ n |Σ0n(Fj(x) ? Fj(x))|. It is shown that P[Dn ≥ L] < 4pL exp{?2(L2n?1 ? 1)} for each positive integer n and for all L2 ≥ n; and, as n → ∞, with probability one. 相似文献
8.
Results on partition of energy and on energy decay are derived for solutions of the Cauchy problem . Here the Aj's are constant, k × k Hermitian matrices, x = (x1,…, xn), t represents time, and u = u(t, x) is a k-vector. It is shown that the energy of Mu approaches a limit , where M is an arbitrary matrix; that there exists a sufficiently large subspace of data ?, which is invariant under the solution group U0(t) and such that depending on ? and that the local energy of nonstatic solutions decays as . More refined results on energy decay are also given and the existence of wave operators is established, considering a perturbed equation at infinity. 相似文献
9.
Bent Fuglede 《Journal of Functional Analysis》1974,16(1):101-121
In Rn let Ω denote a Nikodym region (= a connected open set on which every distribution of finite Dirichlet integral is itself in . The existence of n commuting self-adjoint operators such that each Hj is a restriction of (acting in the distribution sense) is shown to be equivalent to the existence of a set Λ ?Rn such that the restrictions to Ω of the functions exp i ∑ λjxj form a total orthogonal family in . If it is required, in addition, that the unitary groups generated by H1,…, Hn act multiplicatively on , then this is shown to correspond to the requirement that Λ can be chosen as a subgroup of the additive group Rn. The measurable sets Ω ?Rn (of finite Lebesgue measure) for which there exists a subgroup Λ ?Rn as stated are precisely those measurable sets which (after a correction by a null set) form a system of representatives for the quotient of Rn by some subgroup Γ (essentially the dual of Λ). 相似文献
10.
M.M. Robertson 《Journal of Number Theory》1977,9(2):258-270
For 1 ≦ l ≦ j, let l = ?h=1q(l){alh + Mv: v = 0, 1, 2,…}, where j, M, q(l) and the alh are positive integers such that j > 1, al1 < … < alq(2) ≦ M, and let l′ = l ∪ {0}. Let p(n : ) be the number of partitions of n = (n1,…,nj) where, for 1 ≦ l ≦ j, the lth component of each part belongs to l and let p1(n : ) be the number of partitions of n into different parts where again the lth component of each part belongs to l. Asymptotic formulas are obtained for p(n : ), p1(n : ) where all but one nl tend to infinity much more rapidly than that nl, and asymptotic formulas are also obtained for p(n : ′), p1(n ; ′), where one nl tends to infinity much more rapidly than every other nl. These formulas contrast with those of a recent paper (Robertson and Spencer, Trans. Amer. Math. Soc., to appear) in which all the nl tend to infinity at approximately the same rate. 相似文献
11.
《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
12.
In this Note we study the Schrödinger equation i?tu+Δu+V0u+V1u=0 on with initial condition , where V0 is a coulombian potential, singular at finite distance and V1 is an electric potential, possibly unbounded. Both of them may depend on space and time variables. We prove that this problem is well-posed and that the regularity of the initial data is conserved for the solution. The detailed proof will be given elsewhere (Baudouin et al., in press). To cite this article: L. Baudouin et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003). 相似文献
13.
The term condition considered here is the property of an operation ? that holds iff ? and all of its variants obtained by permuting the variables satisfy (for all x,y,u1,…v1,…). Clones consisting entirely of operations satisfying this term condition are called TC clones; algebras whose clone of term operations is a TC clone are called TC algebras; varieties such that every algebra in the variety is a TC algebra are called TC varieties. The paper is a systematic study of these notions, giving primary attention to operations and algebras on finite base sets, and to varieties generated by finite algebras. It is proved, among other results, that the number of n-ary TC operations on a k-element set is logarithmically asymptotic to k(k?1)n when n increases without bound and k is held fixed; that there exist only countably many TC clones on any finite set; that the maximal TC clones on a finite set are finite in number (for each set). Some necessary conditions for an algebra to generate a TC variety are given, also some sufficient conditions. 相似文献
14.
Motivated by problems occurring in the empirical identification and modelling of a n-dimensional ARMA time series X(t) we study the possibility of obtaining a factorization (I + a1B + … + apBp) X(t) = [Πi=1p (I ? αiB)] X(t), where B is the backward shift operator. Using a result in [3] we conclude that as in the univariate case such a factorization always exists, but unlike the univariate case in general the factorization is not unique for given a1, a2,…, ap. In fact the number of possibilities is limited upwards by , there being cases, however, where this maximum is not reached. Implications for the existence and possible use of transformations which removes nonstationarity (or almost nonstationarity) of X(t) are mentioned. 相似文献
15.
We show that for every , there exists a bounded variation function such that u=ei? a.e. on and |?|BV?2|u|BV. The constant 2 is optimal in dimension n>1. To cite this article: J. Dávila, R. Ignat, C. R. Acad. Sci. Paris, Ser. I 337 (2003). 相似文献
16.
For a dense Gδ of pairs (λ, α) in R2, we prove that the operator (Hu)(n) = u(n + 1) + u(n ?1) + λ cos(2παn + θ) u(n) has a nowhere dense spectrum. Along the way we prove several interesting results about the case of which we mention: (a) If qθ is not an integral multiple of π, then all gaps are open, and (b) If q is even and θ is chosen suitably, then the middle gap is closed for all λ. 相似文献
17.
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
18.
Héctor J Sussmann 《Journal of Differential Equations》1982,43(3):399-418
We study the trajectories of systems , where X is a continuous “extendably piecewise analytic” vector field, i.e., a continuous vector field X such that the domain of ? admits a locally finite partition into sets such that for each A ∈ there is a vector field XA which is analytic on a neighborhood of the closure of A and whose restriction to A coincides with that of X. We prove that the trajectories are piecewise analytic, with a priori bounds on the number of switchings for all trajectories that stay in a fixed compact set and whose duration does not exceed a fixed number T. This result implies the existence of a regular synthesis for optimal control problems with a strictly convex Lagrangian, and a linear dynamics with polyhedral constraints on the controls. 相似文献
19.
20.
Yu.V. Egorov V.A. Galaktionov V.A. Kondratiev S.I. Pohozaev 《Comptes Rendus Mathematique》2002,335(10):805-810
We study the asymptotic behaviour of global bounded solutions of the Cauchy problem for the semilinear 2mth order parabolic equation ut=?(?Δ)mu+|u|p in RN×R+, where m>1, p>1, with bounded integrable initial data u0. We prove that in the supercritical Fujita range p>pF=1+2m/N any small global solution with nonnegative initial mass, , exhibits as t→∞ the asymptotic behaviour given by the fundamental solution of the linear parabolic operator (unlike the case where solutions can blow-up for any arbitrarily small initial data). A discrete spectrum of other possible asymptotic patterns and the corresponding monotone sequence of critical exponents , where p0=pF, are discussed. To cite this article: Yu.V. Egorov et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 805–810. 相似文献