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1.
We have implemented in Matlab a Gauss-like cubature formula over arbitrary bivariate domains with a piecewise regular boundary, which is tracked by splines of maximum degree p (spline curvilinear polygons). The formula is exact for polynomials of degree at most 2n−1 using Ncmn2 nodes, 1≤cp, m being the total number of points given on the boundary. It does not need any decomposition of the domain, but relies directly on univariate Gauss-Legendre quadrature via Green’s integral formula. Several numerical tests are presented, including computation of standard as well as orthogonal moments over a nonstandard planar region.  相似文献   

2.
The extremal problem of minimizing the error of approximate evaluation of a line integral of the first kind is considered for certain classes of functions and spatial curves determined by moduli of continuity.It is proved that if the endpoints of the interval [0, L] (where L is the length of the curve along which the integration is performed) are not included in the set of nodes of a quadrature formula for evaluating the line integral of the first kind, then the best quadrature formula for the classes m(p) ρ of functions and \({H^{{\omega _1}, \ldots ,{\omega _m}}}\) of curves is the midpoint rectangle formula. If the extreme points x = 0 and x = L of the interval are included in the set of nodes of a quadrature formula for approximately evaluating the line integral (such formulas are said to be Markov-type), then, for these classes, the best formula is the trapezoidal rule. Sharp error estimates for all considered classed of functions and curves are calculated and a generalization to more general classes is given.  相似文献   

3.
It is well known that the linear stability of elliptic Lagrangian solutions depends on the mass parameter β = 27(m 1 m 2 + m 2 m 3 + m 3 m 1)/(m 1 + m 2 + m 3)2 ∈ [0, 9] and the eccentricity e ∈ [0, 1). Based on new techniques for evaluating the hyperbolicity and the recently developed trace formula for Hamiltonian systems [9], we identify regions for (β, e) such that elliptic Lagrangian solutions are hyperbolic. Consequently, we have proven that the elliptic relative equilibrium of square central configurations is hyperbolic with any eccentricity.  相似文献   

4.
Scattering of electromagnetic (EM) waves by one and many small (ka?1) impedance particles D m of an arbitrary shape, embedded in a homogeneous medium, is studied. Analytic formula for the field, scattered by one particle, is derived. The scattered field is of the order O(a 2?κ ), where κ∈[0,1) is a number. This field is much larger than in the Rayleigh-type scattering. An equation is derived for the effective EM field scattered by many small impedance particles distributed in a bounded domain. Novel physical effects in this domain are described and discussed.  相似文献   

5.
An explicit rule is given for the product of the degree two class with an arbitrary Schubert class in the torus-equivariant homology of the affine Grassmannian. In addition a Pieri rule (the Schubert expansion of the product of a special Schubert class with an arbitrary one) is established for the equivariant homology of the affine Grassmannians of SL n and a similar formula is conjectured for Sp 2n and SO 2n+1. For SL n the formula is explicit and positive. By a theorem of Peterson these compute certain products of Schubert classes in the torus-equivariant quantum cohomology of flag varieties. The SL n Pieri rule is used in our recent definition of k-double Schur functions and affine double Schur functions.  相似文献   

6.
Ordinary N-term integral quadratures require the evaluation of the entire integrand at N points. However, m-by-n product type quadratures involve the evaluation of one factor of the integrand at m points and the reamaining factor at n points. The principal results of this paper include the generalization of the product-type quadrature concept to arbitrary weight functions and to infinite as well as finite intervals, the calculation of the mn coefficients of this product quadrature formula from the LU decoposition of one n-byn, and the extension of the precision of the formula. Nuerical examples are included to illustrate the application of Gaussian product-type quadratures and to compare them with the ordinary Gaussian quadratures.  相似文献   

7.
A formula relating the Radon transform of functions of spherical symmetries to the Radon–Kipriyanov transform Kγ for a naturalmulti-index γ is given. For an arbitrary multi-index γ, formulas for the representation of the Kγ-transform of a radial function as fractional integrals of Erdelyi–Kober integral type and of Riemann–Liouville integral type are proved. The corresponding inversion formulas are obtained. These results are used to study the inversion of the Radon–Kipriyanov transform of the generalized (generated by a generalized shift) spherical mean values of functions that belong to a weighted Lebesgue space and are even with respect to each of the weight variables.  相似文献   

8.
We show that the denominator formula for the strange series of affine superalgebras, conjectured by Kac and Wakimoto and proved by Zagier, follows from a classical determinant evaluation of Frobenius. As a limit case, we obtain exact formulas for the number of representations of an arbitrary number as a sum of 4m2/d triangles, whenever d|2m, and 4m(m+1)/d triangles, when d|2m or d|2m+2. This extends recent results of Getz and Mahlburg, Milne, and Zagier.  相似文献   

9.
Abstract

By using the white noise theory for a fractional Brownian sheet, we derive an Itô formula for the generalized functionals for the fractional Brownian sheet with arbitrary Hurst parameters H 1, H 2 ∈ (0,1). As an application, we give the integral representations for two versions of local times of a fractional Brownian sheet, respectively.  相似文献   

10.
Euler's well-known nonlinear relation for Bernoulli numbers, which can be written in symbolic notation as n(B0+B0)=−nBn−1−(n−1)Bn, is extended to n(Bk1+?+Bkm) for m?2 and arbitrary fixed integers k1,…,km?0. In the general case we prove an existence theorem for Euler-type formulas, and for m=3 we obtain explicit expressions. This extends the authors' previous work for m=2.  相似文献   

11.
A graph is called integral if all its eigenvalues (of the adjacency matrix) are integers. In this paper, the graphs K1,rKn, rKn, K1,rKm,n, rKm,n and the tree K1,sT(q,r,m,t) are defined. We determine the characteristic polynomials of these graphs and also obtain sufficient and necessary conditions for these graphs to be integral. Some sufficient conditions are found by using the number theory and computer search. All these classes are infinite. Some new results which treat interrelations between integral trees of various diameters are also found. The discovery of these integral graphs is a new contribution to the search of such graphs.  相似文献   

12.
We consider the Banach algebra $\mathfrak{A}$ of singular integral operators with matrix piecewise continuous coefficients in the reflexive Orlicz spaceL M n (Γ). We assume that Γ belongs to a certain wide subclass of the class of Carleson curves; this subclass includes curves with cusps, as well as curves of the logarithmic spiral type. We obtain an index formula for an arbitrary operator from the algebra $L_M^n (\Gamma )$ in terms of the symbol of this operator.  相似文献   

13.
We consider exact and approximate multivariate interpolation of a function f(x 1?,?.?.?.?,?x d ) by a rational function p n,m /q n,m (x 1?,?.?.?.?,?x d ) and develop an error formula for the difference f???p n,m /q n,m . The similarity with a well-known univariate formula for the error in rational interpolation is striking. Exact interpolation is through point values for f and approximate interpolation is through intervals bounding f. The latter allows for some measurement error on the function values, which is controlled and limited by the nature of the interval data. To achieve this result we make use of an error formula obtained for multivariate polynomial interpolation, which we first present in a more general form. The practical usefulness of the error formula in multivariate rational interpolation is illustrated by means of a 4-dimensional example, which is only one of the several problems we tested it on.  相似文献   

14.
By [4], a semigroupS is called an (n, m)-commutative semigroup (n, m ∈ ?+, the set of all positive integers) if $$x_1 x_2 \cdot \cdot \cdot x_n y_1 y_2 \cdot \cdot \cdot y_m = y_1 y_2 \cdot \cdot \cdot y_m x_1 x_2 \cdot \cdot \cdot x_n $$ holds for allx 1,...,x n ,y 1,...,y m S It is evident that ifS is an (n, m)-commutative semigroup then it is (n′,m′)-commutative for alln′n andm′m. In this paper, for an arbitrary semigroupS, we determine all pairs (n, m) of positive integersn andm for which the semigroupS is (n, m)-commutative. In our investigation a special type of function mapping ?+ into itself plays an important role. These functions which are defined and discussed here will be called permutation functions.  相似文献   

15.
Given complex numbers m1,l1 and nonnegative integers m2,l2, such that m1+m2=l1+l2, for any a,b=0,…,min(m2,l2) we define an l2-dimensional Barnes type q-hypergeometric integral Ia,b(z,μ;m1,m2,l1,l2) and an l2-dimensional hypergeometric integral Ja,b(z,μ;m1,m2,l1,l2). The integrals depend on complex parameters z and μ. We show that Ia,b(z,μ;m1,m2,l1,l2) equals Ja,b(eμ,z;l1,l2,m1,m2) up to an explicit factor, thus establishing an equality of l2-dimensional q-hypergeometric and m2-dimensional hypergeometric integrals. The identity is based on the duality for the qKZ and dynamical difference equations.  相似文献   

16.
The vanishing ideal I of a subspace arrangement V1V2∪?∪VmV is an intersection I1I2∩?∩Im of linear ideals. We give a formula for the Hilbert polynomial of I if the subspaces meet transversally. We also give a formula for the Hilbert series of the product ideal J=I1I2?Im without any assumptions about the subspace arrangement. It turns out that the Hilbert series of J is a combinatorial invariant of the subspace arrangement: it only depends on the intersection lattice and the dimension function. The graded Betti numbers of J are determined by the Hilbert series, so they are combinatorial invariants as well. We will also apply our results to generalized principal component analysis (GPCA), a tool that is useful for computer vision and image processing.  相似文献   

17.
In the geometric function theory (GFT) much attention is paid to various linear integral operators mapping the class S of the univalent functions and its subclasses into themselves. In [12] and [13] Hohlov obtained sufficient conditions that guarantee such mappings for the operator defined by means of Hadamard product with the Gauss hypergeometric function. In our earlier papers as [20], [19], [17] and [18], etc., we extended his method to the operators of the generalized fractional calculus (GFC, [16]). These operators have product functions of the forms m+1Fm and m+1Ψm and integral representations by means of the Meijer G- and Fox H-functions. Here we propose sufficient conditions that guarantee mapping of the univalent, respectively of the convex functions, into univalent functions in the case of the celebrated Dziok-Srivastava operator ([8] : J. Dziok, H.M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput.103, No 1 (1999), pp. 1-13) defined as a Hadamard product with an arbitrary generalized hypergeometric function pFq. Similar conditions are suggested also for its extension involving the Wright pΨq-function and called the Srivastava-Wright operator (Srivastava, [36]). Since the discussed operators include the above-mentioned GFC operators and many their particular cases (operators of the classical FC), from the results proposed here one can derive univalence criteria for many named operators in the GFT, as the operators of Hohlov, Carlson and Shaffer, Saigo, Libera, Bernardi, Erdélyi-Kober, etc., by giving particular values to the orders p ? q + 1 of the generalized hypergeometric functions and to their parameters.  相似文献   

18.
19.
An integral formula is derived, relating the six irreducible components of the intrinsic torsion of an SpnSp1 structure on a compact 4n-dimensional manifold with the Riemann curvature tensor. Some consequences of the formula are studied.  相似文献   

20.
TheH p corona problem is the following: Letg 1, ...,g m be bounded holomorphic functions with 0<δ≤Σ‖g i ‖. Can we, for anyH p function ?, findH p functionsu 1, ...,u m such that Σg i u i =?? It is known that the answer is affirmative in the polydisc, and the aim of this paper is to prove that it is in non-degenerate analytic polyhedra. To prove this, we construct a solution using a certain integral representation formula. TheH p estimate for the solution is then obtained by localization and some harmonic analysis results in the polydisc.  相似文献   

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