共查询到20条相似文献,搜索用时 15 毫秒
1.
Let π = (a1, a2, …, an), ? = (b1, b2, …, bn) be two permutations of . A rise of π is pair ai, ai+1 with ai < ai+1; a fall is a pair ai, ai+1 with ai > ai+1. Thus, for i = 1, 2, …, n ? 1, the two pairs ai, ai+1; bi, bi+1 are either both rises, both falls, the first a rise and the second a fall or the first a fall and the second a rise. These possibilities are denoted by RR, FF, RF, FR. The paper is concerned with the enumeration of pairs π, p with a given number of RR, FF, RF, FR. In particular if ωn denotes the number of pairs with RR forbidden, it is proved that , . More precisely if ω(n, k) denotes the number of pairs π, p with exactly k occurences of RR(or FF, RF, FR) then . 相似文献
2.
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
3.
R.C. Griffiths 《Journal of multivariate analysis》1975,5(2):271-277
Orthogonal polynomials on the multivariate negative binomial distribution, where α > 0, Θ1 > 0, x = ΣΘi, x0, x1, …, xp = 0,1, … are constructed and their properties studied. 相似文献
4.
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
5.
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
6.
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
7.
Charles J Monlezun 《Journal of Mathematical Analysis and Applications》1974,47(1):133-152
A theory of scattering for the time dependent evolution equations (1) is developed. The wave operators are defined in terms of the evolution operators Uj(t, s), which govern (1). The scattering operator remains unitary. Sufficient conditions for existence and completeness of the wave operators are obtained; these are the main results. General properties, such as the chain rule and various intertwining relations, are also established. Applications include potential scattering (H0(t) = ?Δ, Δ denoting the Laplacian, and H1(t) = ?Δ + q(t, ·)) and scattering for second-order differential operators with coefficients constant in the spatial variable (). 相似文献
8.
Allen Devinatz 《Journal of Functional Analysis》1979,32(3):312-335
Let Ω be a domain in Rn and T = ∑j,k = 1n(?j ? ibj(x)) ajk(x)(?k ? ibk(x)), where the ajk and the bj are real valued functions in , and the matrix (ajk(x)) is symmetric and positive definite for every . If T0 is the same as T but with bj = 0, j = 1,…, n, and if u and Tu are in , then T. Kato has established the distributional inequality ū) Tu]. He then used this result to obtain selfadjointness results for perturbed operators of the form T ? q on Rn. In this paper we shall obtain Kato's inequality for degenerate-elliptic operators with real coefficients. We then use this to get selfadjointness results for second order degenerate-elliptic operators on Rn. 相似文献
9.
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
10.
D Zwick 《Journal of Mathematical Analysis and Applications》1984,104(2):435-436
For a(1) ? a(2) ? ··· ? a(n) ? 0, b(1) ? b(2) ? ··· ? b(n) ? 0, the ordered values of ai, bi, i = 1, 2,…, n, m fixed, m ? n, and p ? 1 it is shown that where is the integer such that and . The inequality is shown to be sharp. When p < 1 and a(i)'s are in increasing order then the inequality is reversed. 相似文献
11.
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
12.
For a sequence A = {Ak} of finite subsets of N we introduce: , , where A(m) is the number of subsets Ak ? {1, 2, …, m}.The collection of all subsets of {1, …, n} together with the operation constitutes a finite semi-group N∪ (semi-group N∩) (group ). For N∪, N∩ we prove analogues of the Erdös-Landau theorem: δ(A+B) ? δ(A)(1+(2λ)?1(1?δ(A>))), where B is a base of N of the average order λ. We prove for analogues of Schnirelmann's theorem (that δ(A) + δ(B) > 1 implies δ(A + B) = 1) and the inequalities λ ? 2h, where h is the order of the base.We introduce the concept of divisibility of subsets: a|b if b is a continuation of a. We prove an analog of the Davenport-Erdös theorem: if d(A) > 0, then there exists an infinite sequence {Akr}, where Akr | Akr+1 for r = 1, 2, …. In Section 6 we consider for analogues of Rohrbach inequality: , where g(n) = min k over the subsets {a1 < … < ak} ? {0, 1, 2, …, n}, such that every m? {0, 1, 2, …, n} can be expressed as m = ai + aj.Pour une série A = {Ak} de sous-ensembles finis de N on introduit les densités: , où A(m) est le nombre d'ensembles Ak ? {1, 2, …, m}. L'ensemble de toutes les parties de {1, 2, …, n} devient, pour les opérations , un semi-groupe fini N∪, N∩ ou un groupe N1 respectivement. Pour N∪, N∩ on démontre l'analogue du théorème de Erdös-Landau: δ(A + B) ? δ(A)(1 + (2λ)?1(1?δ(A))), où B est une base de N d'ordre moyen λ. On démontre pour l'analogue du théorème de Schnirelmann (si δ(A) + δ(B) > 1, alors δ(A + B) = 1) et les inégalités λ ? 2h, où h est l'ordre de base. On introduit le rapport de divisibilité des enembles: a|b, si b est une continuation de a. On démontre l'analogue du théorème de Davenport-Erdös: si d(A) > 0, alors il existe une sous-série infinie {Akr}, où Akr|Akr+1, pour r = 1, 2, … . Dans le Paragraphe 6 on envisage pour N∪, les analogues de l'inégalité de Rohrbach: , où g(n) = min k pour les ensembles {a1 < … < ak} ? {0, 1, 2, …, n} tels que pour tout m? {0, 1, 2, …, n} on a m = ai + aj. 相似文献
13.
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
14.
Christer Borell 《Journal of Mathematical Analysis and Applications》1973,41(2):300-312
We study certain functionals and obtain an inverse Hölder inequality for n functions f1a1,…,fnan (fk concave, 1 dimension).We also prove a multidimensional inverse Hölder inequality for n functions f1,…,fn, where .Finally we give an inverse Minkowski inequality for concave functions. 相似文献
15.
Tor Helleseth 《Discrete Mathematics》1978,23(3):301-307
Let {aj}and {adj} be two binary maximal length linear sequences of period 2n?1. The cross-correlation function is defined as for t = 0, 1,…, 2n ? 2. We find the values and the number of occurrences of each value of Cd(t) when (mod 8). 相似文献
16.
R.J Cook 《Journal of Number Theory》1983,17(1):80-92
Let k be an odd positive integer. Davenport and Lewis have shown that the equations with integer coefficients, have a nontrivial solution in integers x1,…, xN provided that Here it is shown that for any ? > 0 and k > k0(?) the equations have a nontrivial solution provided that 相似文献
17.
Some quadratic identities associated with positive definite Hermitian matrices are derived by use of the theory of reproducing kernels. For example, the following identity is obtained: Let{Aj}mj=1 be N × N positive definite Hermitian matrices. Then, for any complex vector x ∈ N, we have the identity . The minimum is taken here over all the decompositions x =∑mj=1xj. This identity gives, in a sense, a precise converse for an inequality which was derived by T. Ando. Moreover, this paper shows that the sum of two reproducing kernels is naturally related to the harmonic-arithmetic-mean inequality for matrices and also that the geometric-arithmetic-mean inequality for matrices can be naturally interpreted in terms of tensor-product spaces. 相似文献
18.
19.
Let Ω be a simply connected domain in the complex plane, and , the space of functions which are defined and analytic on , if K is the operator on elements defined in terms of the kernels ki(t, s, a1, …, an) in by is the identity operator on , then the operator I ? K may be factored in the form (I ? K)(M ? W) = (I ? ΠK)(M ? ΠW). Here, W is an operator on defined in terms of a kernel w(t, s, a1, …, an) in by Wu = ∝antw(t, s, a1, …, an) u(s, a1, …, an) ds. ΠW is the operator; ΠWu = ∝an ? 1w(t, s, a1, …, an) u(s, a1, …, an) ds. ΠK is the operator; ΠKu = ∑i = 1n ? 1 ∝aitki(t, s, a1, …, an) ds + ∝an ? 1tkn(t, s, a1, …, an) u(s, a1, …, an) ds. The operator M is of the form m(t, a1, …, an)I, where and maps elements of into itself by multiplication. The function m is uniquely derived from K in the following manner. The operator K defines an operator on functions u in , by . A determinant of the operator is defined as an element of . This is mapped into by setting an + 1 = t to give m(t, a1, …, an). The operator I ? ΠK may be factored in similar fashion, giving rise to a chain factorization of I ? K. In some cases all the matrix kernels ki defining K are separable in the sense that ki(t, s, a1, …, an) = Pi(t, a1, …, an) Qi(s, a1, …, an), where Pi is a 1 × pi matrix and Qi is a pi × 1 matrix, each with elements in , explicit formulas are given for the kernels of the factors W. The various results are stated in a form allowing immediate extension to the vector-matrix case. 相似文献
20.
Gideon Nettler 《Journal of Number Theory》1981,13(4):456-462
In a previous paper it was proven that given the continued fractions where the a's and b's are positive integers, then A, B, A ± B, and AB are irrational numbers if for all n sufficiently large, and transcendental numbers if for all n sufficiently large. Using a more direct approach it is proven in this paper that A, B, A ± B, and AB are transcendental numbers if an > bn > an?1(n?1)2 for all n sufficiently large. 相似文献