共查询到20条相似文献,搜索用时 31 毫秒
1.
Let and be polynomials with real zeros satisfying An?1 = Bn?1 = 0, and let Using the recently proved validity of the van der Waerden conjecture on permanents, some results on the real zeros of H(x) are obtained. These results are related to classical results on composite polynomials. 相似文献
2.
Stanisław Lewanowicz 《Journal of Computational and Applied Mathematics》1979,5(3):193-206
In this paper we are constructing a recurrence relation of the form for integrals (called modified moments) in which Ck(λ) is the k-th Gegenbauer polynomial of order , and f is a function satisfying the differential equation of order n, where p0, p1, …, pn ? 0 are polynomials, and mk〈λ〉[p] is known for every k. We give three methods of construction of such a recurrence relation. The first of them (called Method I) is optimum in a certain sense. 相似文献
3.
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
4.
Ludwig Arnold 《Linear algebra and its applications》1976,13(3):185-199
It is proved that Wigner's semicircle law for the distribution of eigenvalues of random matrices, which is important in the statistical theory of energy levels of heavy nuclei, possesses the following completely deterministic version. Let An=(aij), 1?i, ?n, be the nth section of an infinite Hermitian matrix, {λ(n)}1?k?n its eigenvalues, and {uk(n)}1?k?n the corresponding (orthonormalized column) eigenvectors. Let , put (bookeeping function for the length of the projections of the new row v1n of An onto the eigenvectors of the preceding matrix An?1), and let finally (empirical distribution function of the eigenvalues of . Suppose (i) , (ii) limnXn(t)=Ct(0<C<∞,0?t?1). Then ,where W is absolutely continuous with (semicircle) density 相似文献
5.
If f is a positive function on (0, ∞) which is monotone of order n for every n in the sense of Löwner and if Φ1 and Φ2 are concave maps among positive definite matrices, then the following map involving tensor products: is proved to be concave. If Φ1 is affine, it is proved without use of positivity that the map is convex. These yield the concavity of the map (0<p?1) (Lieb's theorem) and the convexity of the map (0<p?1), as well as the convexity of the map .These concavity and convexity theorems are then applied to obtain unusual estimates, from above and below, for Hadamard products of positive definite matrices. 相似文献
6.
It is shown that if satisfies , where σk(A) denotes the sum of all kth order subpermanent of A, then Per[λJn+(1?λ)A] is strictly decreasing in the interval 0<λ<1. 相似文献
7.
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
8.
Scott B. Guthery 《Journal of Number Theory》1974,6(3):201-210
If f is a monotone function subject to certain restrictions, then one can associate with any real number x between zero and one a sequence {an(x)} of integers such that . In this paper properties of the function F defined by , where g is any function satisfying the same restrictions as f, are discussed. Principally, F is found to be useful in finding stationary measures on the sequences {an(x)}. 相似文献
9.
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
10.
Robert S Strichartz 《Journal of Functional Analysis》1982,49(1):91-127
The composition of two Calderón-Zygmund singular integral operators is given explicitly in terms of the kernels of the operators. For φ?L1(Rn) and ε = 0 or 1 and ∝ φ = 0 if ε = 0, let Ker(φ) be the unique function on Rn + 1 homogeneous of degree ?n ? 1 of parity ε that equals φ on the hypersurface x0 = 1. Let Sing(φ, ε) denote the singular integral operator , which exists under suitable growth conditions on ? and φ. Then Sing(φ, ε1) Sing(ψ, ε2)f = ?2π2(∝ φ)(∝ ψ)f + Sing(A, ε1, + ε2)f, where (with notation ). This result is used to show that the mapping ψ → A is a classical pseudo-differential operator of order zero if φ is smooth, with top-order symbol , where θ(ξ) is a cut-off function. These results are generalized to singular integrals with mixed homogeneity. 相似文献
11.
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
12.
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
13.
Dudley Paul Johnson 《Stochastic Processes and their Applications》1985,19(1):183-187
We show that under mild conditions the joint densities Px1,…,xn) of the general discrete time stochastic process Xn on can be computed via where ? is in a Hilbert space , and T (x), x ? are linear operators on . We then show how the Central Limit Theorem can easily be derived from such representations. 相似文献
14.
Let x?Sn, the symmetric group on n symbols. Let θ? Aut(Sn) and let the automorphim order of x with respect to θ be defined by where xθ is the image of x under θ. Let αg? Aut(Sn) denote conjugation by the element g?Sn. Let where s and k are positive integers and denotes a divides b. Further h(s, k : n) ≡ b(1; s, k : n), where 1 denotes the identity automorphim. If g?Sn let c = f(g, s) denote the number of symbols in g which are in cycles of length not dividing the integer s, and let gs denote the product of all cycles in g whose lengths do not divide s. Then gs moves c symbols. The main results proved are: (1) recursion: if n ? c + 1 and t = n ? c ? 1 then (2) reduction: b(g; s, 1 : c)h(s, 1 : i) = b(g; s, 1 : i + c); (3) distribution: let D(θ, n) ≡ {(k, b) : k?Z+ and b = b(θ; 1, k : n) ≠ 0}; then D(θ, m) = D(φ, m) ∨ m ? N = N(θ, φ) iff θ is conjugate to φ; (4) evaluation: the number of cycles in gss of any given length is smaller than the smallest prime dividing s iff b(gs; s, 1 : c) = 1. If g = (12 … pm)t and then . 相似文献
15.
Hans J Bentz 《Journal of Number Theory》1982,15(2):252-274
Chebyshev has noticed a certain predominance of primes of the form 4n + 3 over those of the form 4n + 1. He asserted that . This was unproven until today. G. H. Hardy, J. E. Littlewood and E. Landau have shown its equivalence with an analogue to the famous Riemann hypothesis, namely, L(s, χ1mod 4) ≠ 0, . S. Knapowski and P. Turán have given some similar (unproven) relations, e.g., , which are also equivalent to the above. Using Explixit Formulas the author shows that holds without any conjecture. (In addition, the order of magnitude of divergence is calculated.) It turns out that (1) is only a special case (in several respects). At first, it may be enlarged into Then, it can be generalised to a wider class of progressions. For example, the same is true if one sums over the primes in the classes 3n + 2 and 3n + 1, with a “?” and a “+” sign, respectively. All results of this type depend on the location of the first nontrivial zero of the corresponding L-series. D. Shanks has given some arguments for the predominance of primes in residue classes of nonquadratic type. He conjectured “If m1 mod k is a quadratic residue and m2 mod k a non-residue, then there are “more” primes congruent m2 than congruent m1 mod k.” This indeed turns out to be true in the sense of (1), not only for k = 3, 4, but for some higher moduli as well. Finally, numerical calculations were made to investigate the behaviour of Δ3(X) ? π(X, 2 mod 3) ? π(X, 1 mod 3) in the interval 2 ≤ X ≤ 18, 633, 261. No zero was found in this range. In the analogue case of Δ4(X) ? π(X, 3 mod 4) ? π(X, 1 mod 4) the first sign change occurs at X = 26, 861. 相似文献
16.
Let Ms, be the number of solutions of the equation in the finite field GF(p). For a prime p ≡ 1(mod 3), , , and . Here d is uniquely determined by . 相似文献
17.
Allen J. Schwenk 《Discrete Mathematics》1977,18(1):71-78
Let denote the polynomial obtained from the cycle index of the symmetric group Z(Sn) by replacing each variable si by f(x1). Let f(x) have a Taylor series with radius of convergence ? of the form f(x)=xk + ak+1xk+1 + ak+2xk+2+? with every a1?0. Finally, let 0<x<1 and let x??. We prove that This limit is used to estimate the probability (for n and p both large) that a point chosen at random from a random p-point tree has degree n + 1. These limiting probabilities are independent of p and decrease geometrically in n, contrasting with the labeled limiting probabilities of .In order to prove the main theorem, an appealing generalization of the principle of inclusion and exclusion is presented. 相似文献
18.
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
19.
Generating functions are obtained for certain types of permutations analogous to up-down and down-up permutations. In each case the generating function is a quotient of entire functions; the denominator in each case is φ02(x) ? φ1(x)φ3(x), where 相似文献