共查询到20条相似文献,搜索用时 36 毫秒
1.
Jorge L.C Sanz Thomas S Huang 《Journal of Mathematical Analysis and Applications》1984,104(1):302-308
In this paper, the problem of phase reconstruction from magnitude of multidimensional band-limited functions is considered. It is shown that any irreducible band-limited function f(z1…,zn), zi ? , i=1, …, n, is uniquely determined from the magnitude of f(x1…,xn): | f(x1…,xn)|, xi ? , i=1,…, n, except for (1) linear shifts: i(α1z1+…+αn2n+β), β, αi?, i=1,…, n; and (2) conjugation: . 相似文献
2.
Si studia, in un cilindro, il problema di Dirichlet per l'equazione ellittica del II ordine: , dove è l'operatore a coefficienti discontinui sull'asse x3 già introdotto da N. Ural'tseva per mostrare che l'equazione considerata può non avere soluzione nello spazio di Sobolev W2,p(p > 2) per qualche f?Lp. In questo lavoro si danno limitazioni a priori e teoremi di esistenza e unicità in W2,p quando p varia in un intervallo (p1(α), p2(α)), dipendente dalla costante di ellitticità α. Se p = p2(α) le limitazioni a priori cadono: l'esempio è quello di Ural'tseva. 相似文献
3.
J.H Michael 《Journal of Mathematical Analysis and Applications》1981,79(1):203-217
We consider the mixed boundary value problem , where Ω is a bounded open subset of n whose boundary Γ is divided into disjoint open subsets Γ+ and Γ? by an (n ? 2)-dimensional manifold ω in Γ. We assume A is a properly elliptic second order partial differential operator on and Bj, for j = 0, 1, is a normal jth order boundary operator satisfying the complementing condition with respect to A on . The coefficients of the operators and Γ+, Γ? and ω are all assumed arbitrarily smooth. As announced in [Bull. Amer. Math. Soc.83 (1977), 391–393] we obtain necessary and sufficient conditions in terms of the coefficients of the operators for the mixed boundary value problem to be well posed in Sobolev spaces. In fact, we construct an open subset of the reals such that, if then for is a Fredholm operator if and only if s ∈ . Moreover, = ?xewx, where the sets x are determined algebraically by the coefficients of the operators at x. If n = 2, x is the set of all reals not congruent (modulo 1) to some exceptional value; if n = 3, x is either an open interval of length 1 or is empty; and finally, if n ? 4, x is an open interval of length 1. 相似文献
4.
Aimo Tietäväinen 《Journal of Number Theory》1975,7(3):353-356
Let θ(k, p) be the least s such that the congruence has a nontrivial solution. Let θ(k) = {max θ(k, p)| p > 1 + 2k}. The purpose of this note is to prove the following conjecture of S. Chowla: . 相似文献
5.
The system is investigated, where x and y are scalar functions of time (t ? 0), and n space variables , and F and G are nonlinear functions. Under certain hypotheses on F and G it is proved that there exists a unique spherically symmetric solution , which is bounded for r ? 0 and satisfies x(0) >x0, y(0) > y0, x′(0) = 0, y′(0) = 0, and x′ < 0, y′ > 0, ?r > 0. Thus, (x(r), y(r)) represents a time independent equilibrium solution of the system. Further, the linearization of the system restricted to spherically symmetric solutions, around (x(r), y(r)), has a unique positive eigenvalue. This is in contrast to the case n = 1 (i.e., one space dimension) in which zero is an eigenvalue. The uniqueness of the positive eigenvalue is used in the proof that the spherically symmetric solution described is unique. 相似文献
6.
Stanley J Benkoski 《Journal of Number Theory》1976,8(2):218-223
If r, k are positive integers, then denotes the number of k-tuples of positive integers (x1, x2, …, xk) with 1 ≤ xi ≤ n and (x1, x2, …, xk)r = 1. An explicit formula for is derived and it is shown that .If S = {p1, p2, …, pa} is a finite set of primes, then 〈S〉 = {p1a1p2a2…psas; pi ∈ S and ai ≥ 0 for all i} and denotes the number of k-tuples (x1, x3, …, xk) with 1 ≤ xi ≤ n and (x1, x2, …, xk)r ∈ 〈S〉. Asymptotic formulas for are derived and it is shown that . 相似文献
7.
A technique for the numerical approximation of matrix-valued Riemann product integrals is developed. For a ? x < y ? b, Im(x, y) denotes , and Am(x, y) denotes an approximation of Im(x, y) of the form , where ak and yik are fixed numbers for i = 1, 2,…, m and k = 1, 2,…, N and xik = x + (y ? x)yik. The following result is established. If p is a positive integer, F is a function from the real numbers to the set of w × w matrices with real elements and F(1) exists and is continuous on [a, b], then there exists a bounded interval function H such that, if n, r, and s are positive integers, , then Further, if F(j) exists and is continuous on [a, b] for j = 1, 2,…, p + 1 and A is exact for polynomials of degree less than p + 1 ? j for j = 1, 2,…, p, then the preceding result remains valid when Aj is substituted for Ij. 相似文献
8.
Daniel J. Madden 《Journal of Number Theory》1978,10(3):303-323
If k is a perfect field of characteristic p ≠ 0 and k(x) is the rational function field over k, it is possible to construct cyclic extensions Kn over k(x) such that [K : k(x)] = pn using the concept of Witt vectors. This is accomplished in the following way; if [β1, β2,…, βn] is a Witt vector over k(x) = K0, then the Witt equation generates a tower of extensions through where . In this paper, it is shown that there exists an alternate method of generating this tower which lends itself better for further constructions in Kn. This alternate generation has the form Ki = Ki?1(yi); yip ? yi = Bi, where, as a divisor in Ki?1, Bi has the form . In this form q is prime to Πpjλj and each λj is positive and prime to p. As an application of this, the alternate generation is used to construct a lower-triangular form of the Hasse-Witt matrix of such a field Kn over an algebraically closed field of constants. 相似文献
9.
The existence, uniqueness, and construction of unitary n × n matrix valued functions in Wiener-like algebras on the circle with prescribed matrix Fourier coefficients for j ? 0 are studied. In particular, if , then such an ? exists with if and only if ∥Γ0∥ ? 1, where Γv, denotes the infinite block Hankel matrix (γj + k + v), j, k = 0, 1,…, acting in the sequence space ln2. One of the main results is that the nonnegative factorization indices of every such ? are uniquely determined by the given data in terms of the dimensions of the kernels of , whereas the negative factorization indices are arbitrary. It is also shown that there is a unique such ? if and only if the data forces all the factorization indices to be nonnegative and simple conditions for that and a formula for ? in terms of certain Schmidt pairs of Γ0 are given. The results depend upon a fine analysis of the structure of the kernels of and of the one step extension problem of Adamjan, Arov, and Krein (Funct. Anal. Appl.2 (1968), 1–18). Isometric interpolants for the nonsquare case are also considered. 相似文献
10.
Herbert E. Salzer 《Journal of Computational and Applied Mathematics》1976,2(4):241-248
Gauss's (2n+1)-point trigonometric interpolation formula, based upon f(xi), i = 1(1)2n+1, gives a trigonometric sum of the nth order, S2n+1(x = a0 + ∑jn = 1(ajcos jx + bjsin jx), which may be integrated to provide formulas for either direct quadrature or stepwise integration of differential equations having periodic (or near-periodic) solutions. An “orthogonal” trigonometric sum S2r+1(x) is one that satisfies and two other arbitrarily imposable conditions needed to make S2r1(x) unique. Two proofs are given of a fundamental factor theorem for any S2n+1(x) (somewhat different from that for polynomials) from which we derive 2r-point Gaussian-type quadrature formulas, r = [n/2] + 1, which are exact for any S4r?1(x). We have where the nodes xj, j = 1(1)2r, are the zeros of the orthogonal S2r+1(x). It is proven that Aj > 0 and that 2r-1 of the nodes must lie within the interval [a,b], and the remaining node (which may or may not be in [a,b]) must be real. Unlike Legendre polynomials, any [a′,b′] other than a translation of [a,b], requires different and unrelated sets of nodes and weights. Gaussian-type quadrature formulas are applicable to the numerical integration of the Gauss (2n+1)-point interpolation formulas, with extra efficiency when the latter are expressed in barycentric form. S2r+1(x), xjandAj, j = 1(1)2r, were calculated for [a,b] = [0, π/4], 2r = 2 and 4, to single-precision accuracy. 相似文献
11.
Allan R Sampson 《Journal of multivariate analysis》1983,13(2):375-381
Let X1, …, Xp have p.d.f. g(x12 + … + xp2). It is shown that (a) X1, …, Xp are positively lower orthant dependent or positively upper orthant dependent if, and only if, X1,…, Xp are i.i.d. N(0, σ2); and (b) the p.d.f. of |X1|,…, |Xp| is TP2 in pairs if, and only if, In g(u) is convex. Let X1, X2 have p.d.f. . Necessary and sufficient conditions are given for f(x1, x2) to be TP2 for fixed correlation ?. It is shown that if f is TP2 for all ? >0. then (X1, X2)′ ~ N(0, Σ). Related positive dependence results and applications are also considered. 相似文献
12.
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
13.
J.E Nymann 《Journal of Number Theory》1975,7(4):406-412
Given a set S of positive integers let denote the number of k-tuples 〈m1, …, mk〉 for which and (m1, …, mk) = 1. Also let denote the probability that k integers, chosen at random from , are relatively prime. It is shown that if P = {p1, …, pr} is a finite set of primes and S = {m : (m, p1 … pr) = 1}, then if k ≥ 3 and where d(S) denotes the natural density of S. From this result it follows immediately that as n → ∞. This result generalizes an earlier result of the author's where and S is then the whole set of positive integers. It is also shown that if S = {p1x1 … prxr : xi = 0, 1, 2,…}, then as n → ∞. 相似文献
14.
D.C Doss 《Journal of multivariate analysis》1979,9(3):460-464
The probability generating function (pgf) of an n-variate negative binomial distribution is defined to be [β(s1,…,sn)]?k where β is a polynomial of degree n being linear in each si and k > 0. This definition gives rise to two characterizations of negative binomial distributions. An n-variate linear exponential distribution with the probability function h(x1,…,xn) is negative binomial if and only if its univariate marginals are negative binomial. Let St, t = 1,…, m, be subsets of {s1,…, sn} with empty ∩t=1mSt. Then an n-variate pgf is of a negative binomial if and only if for all s in St being fixed the function is of the form of the pgf of a negative binomial in other s's and this is true for all t. 相似文献
15.
J.L. Brenner 《Journal of Mathematical Analysis and Applications》1985,106(2):427-442
Sharp inequalities are derived for certain (polynomial-like) functions of the real variables pi (i = 1(1)σ) by interpreting pi as the probabilities that various switches be thrown in certain directions. Parameters mv in the inequalities are at first taken to be integers; later the inequalities are established when mv are arbitrary real numbers. The side condition ∑pi = 1 occurs throughout analysis, so there are many corollaries. Examples of the inequalities established are valid ifm>1 valid if m > 1, n > r + 1, 0 < p, s, p + s ? 1, and also valid if . (1.03) 相似文献
16.
J Bustoz 《Journal of Mathematical Analysis and Applications》1981,79(1):71-79
It is known that the classical orthogonal polynomials satisfy inequalities of the form Un2(x) ? Un + 1(x) Un ? 1(x) > 0 when x lies in the spectral interval. These are called Turan inequalities. In this paper we will prove a generalized Turan inequality for ultraspherical and Laguerre polynomials. Specifically if Pnλ(x) and Lnα(x) are the ultraspherical and Laguerre polynomials and . We also prove the inequality is a positive constant depending on α and β. 相似文献
17.
Elliptic boundary value problems for systems of nonlinear partial differential equations of the form , i = 1(1)N, j, k = 1(1)n, pi ? 0, ? being a small parameter, with Dirichlet boundary conditions are considered. It is supposed that a formal approximation Z is given which satisfies the boundary conditions and the differential equations upto the order χ(?) = o(1) in some norm. Then, using the theory of differential inequalities, it is shown that under certain conditions the difference between the exact solution u of the boundary value problem and the formal approximation Z, taken in the sense of a suitable norm, can be made small. 相似文献
18.
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
19.
A set {b1,b2,…,bi} ? {1,2,…,N} is said to be a difference intersector set if {a1,a2,…,as} ? {1,2,…,N}, j > ?N imply the solvability of the equation ax ? ay = b′; the notion of sum intersector set is defined similarly. The authors prove two general theorems saying that if a set {b1,b2,…,bi} is well distributed simultaneously among and within all residue classes of small moduli then it must be both difference and sum intersector set. They apply these theorems to investigate the solvability of the equations (, , , (where () denotes the Legendre symbol) and to show that “almost all” sets form both difference and sum intersector sets. 相似文献
20.
David S Jerison 《Journal of Functional Analysis》1981,43(1):97-142
For (x,y,t)∈n × n × , denote and . When α = n ? 2q, a represents the action of the Kohn Laplacian □b on q-forms on the Heisenberg group. For ?n < α < n, we construct a parametrix for the Dirichlet problem in smooth domains D near non-characteristic points of ?D. A point w of ?D is non-characteristic if one of X1,…, Xn, Y1,…, Yn is transverse to ?D at w. This yields sharp local estimates in the Dirichlet problem in the appropriate non-isotropic Lipschitz classes. The main new tool is a “convolution calculus” of pseudo-differential operators that can be applied to the relevant layer potentials, for which the usual asymptotic composition formula is false. Characteristic points are treated in Part II. 相似文献