In the work of Rostami et al., Bogomolov multiplier a Lie algebra L over field Ω is defined as particular factor subalgebra exterior product L∧L. If finite dimensional, we identify this object certain subgroup second cohomology group H2(L,Ω) by deriving Hopf-Type formula. As an application, affirmatively answer two questions posed Kunyavskiĭ regarding invariance under isoclinism algebras and ex...

For any n-dimensional compact Riemannian manifold (M, g) without boundary and another compact Riemannian manifold (N,h), we establish the uniqueness of the heat flow of harmonic maps from M to N in the class C([0, T ),W ). For the hydrodynamic flow (u, d) of nematic liquid crystals in dimensions n = 2 or 3, we show the uniqueness holds for the class of weak solutions provided either (i) for n =...

The orthoscheme complex of a graded poset is metrization its order such that the simplex each maximal chain isometric to Euclidean vertices $$0, e_1,e_1+e_2,\ldots , e_1+e_2+ \cdots + e_n$$ . This notion was introduced by Brady and McCammond in geometric group theory, has applications discrete optimization submodularity theory. We address question which can yield with CAT(0) property. modular ...

Let F be a field and let F(X1,…,Xn) the of rational functions in n variables X1,…,Xn over F. T=X1+⋯+Xn∈F(X1,…,Xn) m positive integer such that charF∤m. Is it possible to express each Xi as function X1m…,Xnm T F? It is not difficult prove this can done but another matter show how done. We answer above question affirmatively with nonconstructive proof constructive proof.

Should we operate on elderly patients? And, if them, should provide them with complete treatment of their disease? The answers to these 2 questions seem obvious, and are convinced that vast majority the cardiac surgeons would respond affirmatively both. In reality, however, problem combined surgeries in is multifaceted; not only must consider operating an aging perhaps fragile population, but a...

Ge asked the question whether $LF_{\infty}$ can be embedded into $LF_2$ as a maximal subfactor. We answer it affirmatively by three different approaches, all containing same key ingredient: existence of subgroups with infinite index. also show that point stabilizer for every faithful, 4-transitive action on an set give rise to von Neumann subalgebras. Combining this known results constructing h...