We give a short proof of a formula of de Shalit, expressing the cup product of two vector valued one forms of the second kind on a Mumford curve in terms of Coleman integrals and residues. The proof uses the notion of double indices on curves and their reciprocity laws.

In this paper, we introduce the idea of stochastic integrals with respect to an increasing process in the G-framework and extend G-Itô’s formula. Moreover, we study the solvability of the scalar valued stochastic differential equations driven by G-Brownian motion with reflecting boundary conditions (RGSDEs).

In this paper, we shall discuss a newly introduced concept of center-radius total-ordered relations between two intervals. Here, address the Hermite–Hadamard-, Fejér- and Pachpatte-type inequalities by considering interval-valued Riemann–Liouville fractional integrals. Interval-valued for new class preinvexity, i.e., cr-h-preinvexity, are estimated. The operator is used first time to prove such...

Using order relations on the space of fuzzy numbers, we define the levels of fuzzy number valued mappings and we present some topologicals and geometrics properties. We use this concept to locate points and the set of points that solve a problem of fuzzy optimization. Consequently we obtain a necessary condition, for means of fuzzy integrals, for fuzzy global optimality.

We consider nearest–neighbor random walks on free products of finitely many copies of the integers with amalgamation over nontrivial subgroups. When all the subgroups have index two, we find the Green function of the random walks in terms of complete elliptic integrals. Our technique is to apply Voiculescu’s operator–valued R–transform.

Variational integrals whose energy densities are represented by Nfunctions h of at least quadratic growth are considered. Under rather general conditions on h, almost everywhere regularity of vector-valued local minimizers is established, and it is possible to include the case of higher order variational problems without essential changes in the arguments. §

We construct and analyze Gauss-type quadrature rules with complex-valued nodes and weights to approximate oscillatory integrals with stationary points of high order. The method is based on substituting the original interval of integration by a set of contours in the complex plane, corresponding to the paths of steepest descent. Each of these line integrals shows an exponentially decaying behavi...