Cubature formulas are obtained which are optimal or asymptotically optimal on given sets of functions. These formulas consist of line integrals which may be evaluated by optimal or asymptotically optimal quadrature formulas. The advantage of these formulas over the optimal and asymptotically optimal cubature formulas with rectangular-lattices of knots is shown.

where 1 2 1 2 , , , r r θ θ are any real numbers, and n is any positive integer. We can obtain the infinite series forms of these double integrals using Taylor series expansions and integration term by term theorem; these are the major results of this paper (i.e., Theorems 1-3). Adams et al. [1], Nyblom [2], and Oster [3] provided some techniques to solve the integral problems. Yu [4-29], Yu an...

Background: A decreased rate of successful helicobacter pylori (H.pylori) infection treatment has revealed serious demand for more effective regimens to eradicate infection. Therefore, probiotics have recently been considered to increase the rate of antibiotic regimens efficacy in H. pylori infections. In current randomized controlled trial, we evaluated the effect of double strain probiotic co...

Let S⊂L2⊂S∗ be the Gel'fand triple over Bernoulli space, where elements of S∗ are called generalized functionals. In this paper, we define integrals functionals with respect to a spectral measure (projection operator-valued measure) in framework S⊂L2⊂S∗, and examine their fundamental properties. New notions introduced, several results obtained examples also shown.

‎in this paper, we formulate the fourth order sturm-liouville problem (fslp) as a lie group matrix differential equation. by solving this ma- trix differential equation by lie group magnus expansion, we compute the eigenvalues of the fslp. the magnus expansion is an infinite series of multiple integrals of lie brackets. the approximation is, in fact, the truncation of magnus expansion and a gauss...

We obtain the double factorization of braided bialgebras or braided Hopf algebras, give relation among integrals and semisimplicity of braided Hopf algebra and its factors.

High-level ab initio quantum-mechanical methods have been used to study the HFS molecule. The most advanced method involves a triple-zeta plus double polarization plusf functions (TZ2P +f, basis set in conjunction with the singleand double-excitation coupled-cluster method perturbatively including connected triple excitations, i.e., CCSD (T) . Two of the three fundamental vibrational frequencie...

We obtain special values results for the triple product Lfunction attached to a Hilbert modular cuspidal eigenform over a totally real quadratic number field and an elliptic modular cuspidal eigenform, both of level one and even weight. Replacing the elliptic modular cusp form by a specified Eisenstein series, we renormalize the integral defining the triple product L-function in order to obtain...

CONTEXT When clinicians assess the validity of randomized controlled trials (RCTs), they commonly evaluate the blinding status of individuals in the RCT. The terminology authors often use to convey blinding status (single, double, and triple blinding) may be open to various interpretations. OBJECTIVE To determine physician interpretations and textbook definitions of RCT blinding terms. DESI...

The purpose of this appendix is to review the basics of vector calculus in the two dimensions. Most, if not all, this material should be familiar to the student who has taken a basic course in multivariable calculus, but it is worth collecting together the necessary results. We will assume you are familiar with the basics of partial derivatives, including the equality of mixed partical (assumin...