In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers and a a particular Monte Carlo method numerically computes the Riemann integral. Whereas other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher-dimensional integral...

ai−d πd ∫ dk1 d k2 [(k1 − k)2]a1 [(k2 − k)2]a2 [(k1 − k2)]3(k 2) a4(k2 1) a5 (1.1) in d-dimensional Euclidean momentum space. It has a long and interesting history. For many years, most of the information we had about perturbative quantum field theory was coming (directly or indirectly) from this integral. All massless three-loop self-energy integrals (with integer indices) reduce to 6 master i...

We evaluate the coefficients of the leading poles of the complete two-loop quark self-energy Σ(p) in the Coulomb gauge. Working in the framework of split dimensional regularization, with complex regulating parameters σ and n/2−σ for the energy and space components of the loop momentum, respectively, we find that split dimensional regularization leads to well-defined two-loop integrals, and that...

A general Gauss divergence theorem with applications to convolution integrals of the form ∫ f(x̄)h(|x̄− ā|)dVn, where the integration extends over an n-dimensional polyhedral domain, is presented. The kernel h(|x̄ − ā|) may be singular, but the given integral must remain integrable. As a result of the Gauss theorem, the given integral is reduced to an integral over the boundary of the n-dimensiona...

where ki, i = 1, . . . , h, are loop momenta and the denominators Er are either quadratic of linear with respect to ki and external momenta q1, . . . , qN . By default, the integrals are dimensionally regularized with d = 4− 2ǫ. If the number of Feynman integrals needed for a given calculation is small or/and they are simple, one evaluates, by some methods, every scalar Feynman integral of the ...

New algebraic approach to analytical calculations of D-dimensional integrals for multi-loop Feynman diagrams is proposed. We show that the known analytical methods of evaluation of multi-loop Feynman integrals, such as integration by parts and star-triangle relation methods, can be drastically simplified by using this algebraic approach. To demonstrate the advantages of the algebraic method of ...

A line integral is defined as the integral of two-dimensional data along a (onedimensional, straight) line of given length and orientation. Line integrals are used in various forms of edge and line detectors in images and in the computation of the Radon transform. We present a recursive algorithm which enables approximation of discretized line integrals at all lengths, orientations, and locatio...

We obtain several new closed-form expressions for the evaluation of a family of infinite-domain integrals of the Whittaker functions W , x and M , x and the modified Bessel functions I x and K x with respect to the index . The new family of definite integrals is useful in a variety of contexts in mathematical physics. In particular, the integral involving K x represents a new example of the Kon...

Ordinary differential equations(ODEs) with stochastic processes in their vector field, have lots of applications in science and engineering. The main purpose of this article is to investigate the numerical methods for ODEs with Wiener and Compound Poisson processes in more than one dimension. Ordinary differential equations with Ito diffusion which is a solution of an Ito stochastic differentia...

The fuzzy integrals are a kind of fuzzy measures acting on fuzzy sets. They can be viewed as an average membershipvalue of fuzzy sets. The value of the fuzzy integral in a decision making environment where uncertainty is presenthas been well established. Most of the integral inequalities studied in the fuzzy integration context normally considerconditions such as monotonicity or comonotonicity....