This short note gives an introduction to the Riemann-Stieltjes integral on R and Rn. Some natural and important applications in probability theory are discussed. The reason for discussing the Riemann-Stieltjes integral instead of the more general Lebesgue and LebesgueStieltjes integrals are that most applications in elementary probability theory are satisfactorily covered by the Riemann-Stieltj...

‎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 show how to find all the integrating factors and corresponding first integrals for any system of Ordinary Differential Equations (ODEs). Integrating factors are shown to be all solutions of both the adjoint system of the linearised system of ordinary differential equations and a system that represents an extra adjoint-invariance condition. We present an explicit construction formula to find ...

We consider a real-valued path; it is possible to associate a tree to this path, and we explore the relations between the tree, the properties of p-variation of the path, and integration with respect to the path. In particular, the fractal dimension of the tree is estimated from the variations of the path, and Young integrals with respect to the path, as well as integrals from the rough paths t...

All three-loop on-shell HQET propagator integrals with a loop of a massive quark can be reduced to a set of master integrals, using integration by parts [1]. Some master integrals are known exactly, for other ones a few terms of ε expansions have been calculated [1]. In particular, for the integral I2, the 1/ε and O(1) terms are known. However, in some applications more terms of its ε expansion...

We consider a real-valued path; it is possible to associate a tree to this path, and we explore the relations between the tree, the properties of p-variation of the path, and integration with respect to the path. In particular, the fractal dimension of the tree is estimated from the variations of the path, and Young integrals with respect to the path, as well as integrals from the rough paths t...

We derive a t-J model with electron-phonon coupling from the three-band model, considering modulation of both hopping and Coulomb integrals by phonons. While the modulation of the hopping integrals dominates, the modulation of the Coulomb integrals cannot be neglected. The model explains the experimentally observed anomalous softening of the half-breathing mode upon doping and a weaker softenin...

In a recent work by the author an extrapolation method, the W-transformation, was developed, by which a large class of oscillatory infinite integrals can be computed very efficiently. The results of this work are extended to a class of divergent oscillatory infinite integrals in the present paper. It is shown in particular that these divergent integrals exist in the sense of Abel summability an...

The correlation functions of the spin-1/2 XXZ chain in the ground state were ex-pressed in the form of multiple integrals for −1 < ∆ ≤ 1 and 1 < ∆. In particular,adjacent four-point correlation functions were given as certain four-dimensional inte-grals. We show that these integrals can be reduced to polynomials with respect tospecific one-dimensional integrals. The results ...

Three types of integral representations for the cumulative distribution functions of convolutions of Γp(αk,Σk,∆k)–distributions with non–centrality matrices ∆k are given by integration of products of simple complex functions over the p–cube (−π, π]. In particular, the joint distribution of the diagonal elements of a generalized quadratic form XAX ′ with n independentNp(μk,Σ)–distributed columns...