Extending Itô's formula to non-smooth functions is important both in theory and applications. One of the fairly general extensions of the formula, known as Meyer–Itô, applies to one dimensional ...semimartingales and convex functions. There are also satisfactory generalizations of Itô's formula for diffusion processes where the Meyer–Itô assumptions are weakened even further. We study a version of Itô's formula for multi-dimensional finite variation Lévy processes assuming that the underlying function is continuous and admits weak derivatives. We also discuss some applications of this extension, particularly in finance.
The aim of this paper is to develop fractional calculus for vector‐valued functions using the weak Riemann integral. Also, we establish the existence of weak solutions for a class of fractional ...differential equations with fractional weak derivatives.
In this paper, we develop the weighted square integral inequalities for the difference of two smooth superharmonic functions. Then we prove the existence and integrability of the Sobolev derivative ...for superharmonic functions. The inequalities are generalized for the difference of two weak superharmonic functions. We also establish that the superharmonic approximation is indeed the better imitation of the exact unknown solution rather than the usual uniform approximation.
Sobolev of the Euler school Kutateladze, S. S.
Siberian mathematical journal,
09/2008, Letnik:
49, Številka:
5
Journal Article
Recenzirano
This is a short overview of the origins of distribution theory as well as a few signposts of the life of Sergeĭ Sobolev (1908–1989) and his contribution to the formation of the modern outlook of ...mathematics.
Solutions are in existence for segmentation and restoration of medical images. This paper goes a step forward by going in for establishing an framework which includes solving full domain and ...segmented domain demanding an unique solution. This paves way for focusing the diseased part of the organ on the image in an augmented way. Treating the full image as a domain, medical image as a separate entity and segmented region as a separate entity in the form of matrix the partial derivative equations are solved by generating the linear equations. Finding the co factored elements of this matrix to applying the gradient method to Hessian matrix forms among other matrices are part of the procedure. Use of the topological derivative is attested by the perturbed domain. Cost functional calculation is aided by this topological derivative. Perturbation is introduced by the diffusion process. Banach space introduces subspaces into the problem. With the calculated cost functional value, weak derivative helps in identifying edges highlighting the diseased part. Further the distributional derivative helps to arrive at the splayed out results for the most infected regions.
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