This is a short overview of the origins of distribution theory as well as the life of Sergĕı Sobolev (1908–1989) and his contribution to the formation of the modern outlook of mathematics. Sergĕı Lvovich Sobolev belongs to the Russian mathematical school and ranks among the scientists whose creativity has produced the major treasures of the world culture. Mathematics studies the forms of reason...

The purpose of these notes is to review the basics of vector calculus in the two dimensions. We will assume you are familiar with the basics of partial derivatives, including the equality of mixed partials (assuming they are continuous), the chain rule, implicit differentiation. In addition, some familiarity with multiple integrals is assumed, although we will review the highlights. Proofs and ...

Exploring the possible links between the mathematical field of fractional calculus and the electromagnetic theory has been one of the topics of our research interests in recent years. We have studied the possibility of bringing the tools of fractional calculus and electromagnetic theory together, and have explored and developed the topic of fractional paradigm in electromagnetic theory (see e.g...

We discuss new applications of advanced tools of variational analysis and generalized differentiation to a number of important problems in optimization theory, equilibria, optimal control, and feedback control design. The presented results are largely based on the recent work by the author and his collaborators. Among the main topics considered and briefly surveyed in this paper are new calculu...

BACKGROUND AND PURPOSE Imaging-based tumor grading is highly desirable but faces challenges in sensitivity, specificity, and diagnostic accuracy. A recently proposed diffusion imaging method by using a fractional order calculus model offers a set of new parameters to probe not only the diffusion process itself but also intravoxel tissue structures, providing new opportunities for noninvasive tu...

This is a short overview of the origins of distribution theory as well as the life of Sergĕı Sobolev (1908–1989) and his contribution to the formation of the modern outlook of mathematics. Sergĕı Lvovich Sobolev belongs to the Russian mathematical school and ranks among the scientists whose creativity has produced the major treasures of the world culture. Mathematics studies the forms of reason...