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This is a special volume of Communications in Mathematical Analysis in honor of the Fields Medalist Stephen Smale in recognition of his many significant and various contributions to mathematics, and in celebration of his 80th birthday. The special volume will encompass original research papers and expository articles from all fields and subfields of mathematics within the scope of Communications in Mathematical Analysis as well as areas in which Professor Smale has played a significant role, including but are not limited to, mathematical analysis, topology, dynamical systems, economics, mechanics, and computation.
The stationary Oseen equations are studied in $\mathbb R^3$ in its general form, that is, with a non-constant divergenceless function on the convective term. We prove existence, uniqueness and regularity results in weighted Sobolev spaces. From this new approach, we also state existence, uniqueness and regularity results for the generalized Oseen model which describes the rotating flows. The proofs are based on Laplace, Stokes and Oseen theories.
It is admitted in the literature on special relativity that, being velocity dependent, relativistic mass is a wild notion in the sense that it does not conform with the Minkowskian four-vector formalism. The resulting lack of clear consensus on the basic role of relativistic mass in special relativity has some influence in diminishing its use in modern books. Fortunately, relativistic mechanics is regulated by the hyperbolic geometry of Bolyai and Lobachevsky just as classical mechanics is regulated by Euclidean geometry. Guided by analogies that Euclidean geometry and classical mechanics share with hyperbolic geometry and relativistic mechanics, the objective of this article is to tame the relativistic mass by placing it under the umbrella of the Minkowskian formalism, and to present interesting consequences.
KEYWORDS: Integral bordisms in PDE's, Existence of local and global solutions in PDE's, Conservation laws, crystallographic groups, Singular PDE's, PDE's on complex manifolds category, PDE's on quantum supermanifolds category, 55N22, 58J32, 57R20, 58C50, 58J42, 20H15, 32Q55, 32S20
Exotic heat equations that allow to prove the Poincaré conjecture, some related problems and suitable generalizations too are considered. The methodology used is the PDE's algebraic topology, introduced by A. Prástaro in the geometry of PDE's, in order to characterize global solutions.
Some recent Jensen's type inequalities for log-convex functions of selfadjoint operators in Hilbert spaces under suitable assumptions for the involved operators are surveyed. Applications in relation with some celebrated results due to Hölder-McCarthy and Ky Fan are provided as well.