Let R be a commutative ring with identity and M an R-module. The Scalar-Product Graph of M is defined as the graph GR(M) with the vertex set M and two distinct vertices x and y are adjacent if and only if there exist r or s belong to R such that x = ry or y = sx. In this paper , we discuss connectivity and planarity of these graphs and computing diameter and girth of GR(M). Also we show some of...

Warped products provide a rich class of physically significant geometric objects. Warped product construction is an important method to produce a new metric with a base manifold and a fibre. We construct compact base manifolds with a positive scalar curvature which do not admit any non-trivial quasi-Einstein warped product, and non compact complete base manifolds which do not admit any non-triv...

‎Let $p$ be a prime with $pgeq 7$ and $q=2(p-1)$‎. ‎In this paper‎ ‎we prove the existence of a nontrivial product of‎ ‎filtration $s+4$ in the stable homotopy groups of spheres‎. ‎This nontrivial‎ ‎product is shown to be represented up to a nonzero scalar by‎ ‎the product element $widetilde{gamma}_{s}b_{n-1}g_{0}in‎ ‎{Ext}_{mathcal{A}}^{s+4,(p^n+sp^2+sp+s)q+s-3}(mathbb{Z}/p,mathbb{Z}/p)$‎ ‎in ...

‎let $p$ be a prime with $pgeq 7$ and $q=2(p-1)$‎. ‎in this paper‎ ‎we prove the existence of a nontrivial product of‎ ‎filtration $s+4$ in the stable homotopy groups of spheres‎. ‎this nontrivial‎ ‎product is shown to be represented up to a nonzero scalar by‎ ‎the product element $widetilde{gamma}_{s}b_{n-1}g_{0}in‎ ‎{ext}_{mathcal{a}}^{s+4,(p^n+sp^2+sp+s)q+s-3}(mathbb{z}/p,mathbb{z}/p)$‎ ‎in ...

In 1953 Peter Roquette gave an arithmetic proof of the Riemann hypothesis for algebraic function fields over a finite constants field, which was proved by André Weil in 1940. The construction of Weil’s scalar product is essential in Roquette’s theory. In this paper a scalar product for algebraic function fields over a number field is constructed which is the analogue of Weil’s scalar product.

Privacy-preserving machine learning enables the training of models on decentralized datasets without need to reveal data, both horizontal and vertically partitioned data. However, it relies specialized techniques algorithms perform necessary computations. The privacy preserving scalar product protocol, which dot vectors revealing them, is one popular example for its versatility. Unfortunately, ...