Wilson loops provide a convenient method of obtaining gauge invariant observables from lattices to strings. On the other hand the product integral formalism was developed in connection with matrix valued differential equations, having built in the feature of order of the matrices. Consequently these come quite handy when dealing with path ordered quantities. As the theory of product integrals i...

We consider binomial and inverse binomial sums at infinity and rewrite them in terms of a small set of constants, such as powers of π or log(2). In order to perform these simplifications, we view the series as specializations of generating series. For these generating series, we derive integral representations in terms of root-valued iterated integrals. Using substitutions, we express the inter...

In ordinary tomography the objective is to reconstruct a scalar function f from all its line integrals. The question dealt with in this paper is if it is possible to do the same for a vector valued function, when we know the line integrals of f dl, where l is the directional vectors of the lines. The answer is that it is only possible to regain r f. Some simulations and experiments done in asso...

— In this paper, we consider the problem of deriving new eigenvalue distributions of real-valued Wishart matrices that arises in many scientific and engineering applications. The distributions are derived using the tools from the theory of skew symmetric matrices. In particular, we relate the multiple integrals of a determinant, which arises while finding the eigenvalue distributions, in terms ...

We develop a technique for obtaining the fourth moment bound on normal approximation of F, where F is an Rd-valued random vector whose components are functionals Gaussian fields. This study transcends case vectors multiple stochastic integrals, which has been subject research so far. perform this task by investigating relationship between expectations two operators ? and ?*. Here, operator was ...

In this paper, we study fuzzy calculus in two main branches differential and integral.  Some rules for finding limit and $gH$-derivative of $gH$-difference, constant multiple of two fuzzy-valued functions are obtained and we also present fuzzy chain rule for calculating  $gH$-derivative of a composite function.  Two techniques namely,  Leibniz's rule and integration by parts are introduced for ...

In this article, we study analogues of the van der Corput lemmas [19] involving Bessel functions. harmonic analysis, one most important estimates is lemma, which an estimate oscillatory integrals. This was first obtained by Dutch mathematician Johannes Gaultherus Corput. Van interested in behavior for large positive λ integral R b a e iλφ(x)ψ(x)dx, where φ real-valued smooth function (the phase...

where s is a “type parameter”, usually s = −i or s = 1, s = −i standing for “oscillatory”, s = 1 for “probabilistic type”. Φ is a real-valued function, in the probabilistic case assumed to be lower bounded; it is called “phase function”; f is a complex-valued function, called “amplitude function”; γ is thought to be a member of some space Γ “on which the integration extends”; dγ is a heuristic ...

In this contribution we define a new class of non-linear integrals based on decomposition integrals. These integrals are motivated by greediness of many real-life situations. Another view on this new class of integrals is that it is a generalization of both the Shilkret and PAN integrals. Moreover, it can be seen as an iterated Shilkret integral. Also, an example in time-series analysis is prov...