A Parseval-Goldstein type theorem on the widder potential transform and its applications

In this paper a Parseval-Goldstein type theorem involving the Widder potential transform and a Laplace type integral transform is given. The theorem is then shown to yield a relationship between the -transform and the Laplace type integral transform. The theorem yields some simple algorithms for evaluating infinite integrals. Using the theorem and its results, a number of new infinite integrals of elementary and special functions are presented. Some illustrative examples are also given

The ongoing pursuit of neuroprotective therapies in Parkinson disease

Many agents developed for neuroprotective treatment of Parkinson disease (PD) have shown great promise in the laboratory, but none have translated to positive results in patients with PD. Potential neuroprotective drugs, such as ubiquinone, creatine and PYM50028, have failed to show any clinical benefits in recent high-profile clinical trials. This 'failure to translate' is likely to be related primarily to our incomplete understanding of the pathogenic mechanisms underlying PD, and excessive reliance on data from toxin-based animal models to judge which agents should be selected for clinical trials. Restricted resources inevitably mean that difficult compromises must be made in terms of trial design, and reliable estimation of efficacy is further hampered by the absence of validated biomarkers of disease progression. Drug development in PD dementia has been mostly unsuccessful; however, emerging biochemical, genetic and pathological evidence suggests a link between tau and amyloid-β deposition and cognitive decline in PD, potentially opening up new possibilities for therapeutic intervention. This Review discusses the most important 'druggable' disease mechanisms in PD, as well as the most-promising drugs that are being evaluated for their potential efficiency in treatment of motor and cognitive impairments in PD

Identities on fractional integrals and various integral transforms

In this work we will introduce theorems relating the Riemann-Liouville fractional integral and the Weyl fractional integral to some well-known integral transforms including Laplace transforms, Stieltjes transforms, generalized Stieltes transforms, Hankel transforms, and K-transforms. As applications of the theorems and their results, a number of infinite integrals of elementary functions and special functions are evaluated and some illustrative examples are presented. © 2006 Elsevier Inc. All rights reserved

Undergraduate research in mathematics as a curricular option

In this paper, a model is outlined for integrating research activities with undergraduates within the mathematics curriculum. Introducing a sequence of courses designed to engage students in research projects has brought about a change in the mathematical culture of students. The history and challenges associated with the creation of this program are discussed, indicating the positive outcomes it has had on student learning. Also discussed is the shift in departmental thoughts on student capabilities. Specific examples of student work are cited

A Generalization of the Krätzel Function and Its Applications

In this paper, we introduce new functions Yρ,rν(x) as a generalization of the Krätzel function. We investigate recurrence relations, Mellin transform, fractional derivatives, and integral of the function Yρ,rν(x). We show that the function Yρ,rν(x) is the solution of differential equations of fractional order

Parseval-Goldstein type identities involving the L4-transform and the P4-transform and their applications

In the present article the authors introduce several new integral transforms including the L4-transform and the P4-transform as generalizations of the classical Laplace transform and the classical Stieltjes transform, respectively. It is shown that the second iterate of the L4-transform is essentially the P4-transform. Using this relationship, a number of new Parseval-Goldstein type identities are obtained for these and many other well-known integral transforms. The identities proven in this article give rise to useful corollaries for evaluating infinite integrals of special functions. Some examples are also given as illustrations of the results presented here