In this study, we investigate the long-time behavior of global solution anisotropic quasi-geostrophic equation, denoted by $$\theta $$ , where belongs to space $$C_b({\mathbb {R}}^+,H^s({\mathbb {R}}^2))$$ . Our results demonstrate that decays zero as time approaches infinity in $$L^p({\mathbb {R}}^2)$$ norm, with $$p\ge 2$$ Additionally, establish limit $$\Vert \theta (t)\Vert _{H^s}$$ t tends...

We describe a new series of identities, which hold for certain general theta series, in two completely independent variables. We provide explicit examples of these identities involving the Dedekind eta function, Jacobi theta functions, and various theta functions of Ramanujan. Introduction Let z ∈ H = {x+ yi : x, y ∈ R, y > 0} and for each x ∈ R set q = exp(2πixz) and e(x) = exp(2πix). The Dede...

Abstract Given $$E_0, E_1, F_0, F_1, E$$ E 0 , 1 F rearrangement invariant function spaces, $$a_0$$ a , $$a_1$$ $$\mathrm {b}_0$$ b {b}_1$$ {b}$$ slowly varying f...

0. Introduction. Let X be a non-singular, connected, projective curve defined over an algebraically closed field k. Let U denote the set of isomorphism classes of stable vector bundles on X with given degree d and rank r. In the sixties, C.S. Seshadri and D. Mumford ([14], [17] and [18]) supplied U with a natural structure of quasi-projective variety, together with a natural compactification U ...

The problem of indexing multidimensional objects is considered. First, a classification of existing methods is given along with a discussion of the major issues involved in multidimensional data indexing. Second, a variation to Guttman’s R-trees (R+-trees) that avoids overlapping rectangles in intermediate nodes of the tree is introduced. Algorithms for searching, updating, initial packing and ...

Previous studies indicate that 11,12,15-trihydroxyeicosatrienoic acid (11,12,15-THETA), an endothelium-derived hyperpolarizing factor in the rabbit aorta, mediates a portion of the relaxation response to acetylcholine by sequential metabolism of arachidonic acid by 15-lipoxygenase, hydroperoxide isomerase, and epoxide hydrolase. To determine the stereochemical configuration of the endothelial 1...

It is shown that for a metric space (M,d) the following are equivalent: (i) M is a complete R-tree; (ii) M is hyperconvex and has unique metric segments.