In this paper we define a Poincaré-Reidemeister scalar product on the determinant line of the cohomology of any flat vector bundle over a closed orientable odd-dimensional manifold. It is a combinatorial “torsion-type” invariant which refines the PR-metric introduced in [Fa] and contains an additional sign or phase information. We compute the PR-scalar product in terms of the torsions of Euler ...

Given a smooth compact manifold with boundary, we study variational properties of the volume functional and area restricted to space Riemannian metrics prescribed curvature. We obtain sufficient necessary condition for metric be critical point. As by-product, very natural analogue V-statics is obtained. In second part, using Yamabe invariant in boundary setting, solve Kazdan–Warner–Kobayashi pr...

The symmetric eigenvalue decomposition and the singular value decomposition (SVD) are fundamental matrix decompositions with many applications. Conventional algorithms for computing these decompositions are suboptimal in view of recent trends in computer architectures, which require minimizing communication together with arithmetic costs. Spectral divideand-conquer algorithms, which recursively...

In this note we comment recent paper by A.V. Minkevich 'Regular inflationary cosmology and gauge theories of gravity' on the 'bouncing' character of flat-type inflationary cosmolog-ical models within gauge theories of gravity. Recently [1] it was argued that inflationary cosmological solutions in the case of flat models within the framework of gauge theories of gravity have 'bouncing' character...

We study the weak-field limit of the static spherically symmetric solution of the locally conformally invariant theory advocated in the recent past by Mannheim and Kazanas as an alternative to Einstein’s General Relativity. In contrast with the previous works, we consider the physically relevant case where the scalar field that breaks conformal symmetry and generates fermion masses is nonzero. ...

A new matrix function corresponding to the scalar unwinding number of Corless, Hare, and Jeffrey is introduced. This matrix unwinding function, U , is shown to be a valuable tool for deriving identities involving the matrix logarithm and fractional matrix powers, revealing, for example, the precise relation between logAα and α logA. The unwinding function is also shown to be closely connected w...

Abstract In this paper, by applying a nonlinear alternative principle of Leray–Schauder and Guo–Krasnosel’skii fixed point theorem on compression expansion cones, together with truncation technique, we study the existence multiplicity noncollision periodic solutions to third-order singular dynamical systems. By combining analysis sign Green’s function for linear equation, consider systems where...