Caratheodory's Theorem in Depth

Test sets of the knapsack problem and simultaneous diophantine approximation

Absact This paper deals with the study of test sets of the knapsack problem and simultaneous diophantine approximation. The Graver test set of the knapsack problem can be derived from minimal integral solutions of linear diophantine equations. We present best possible inequalities that must be satisfied by all minimal integral solutions of a linear diophantine equation and prove that for the co...

A From Sylvester-Gallai Configurations to Rank Bounds: Improved Blackbox Identity Test for Depth-3 Circuits

We study the problem of identity testing for depth-3 circuits of top fanin k and degree d. We give a new structure theorem for such identities that improves the known deterministic d O(k) -time blackbox identity test over rationals (Kayal & Saraf, FOCS 2009) to one that takes dO(k 2)-time. Our structure theorem essentially says that the number of independent variables in a real depth-3 identity...

On intermediate value theorem in ordered Banach spaces for noncompact and discontinuous mappings

In this paper, a vector version of the intermediate value theorem is established. The main theorem of this article can be considered as an improvement of the main results have been appeared in [textit{On fixed point theorems for monotone increasing vector valued mappings via scalarizing}, Positivity, 19 (2) (2015) 333-340] with containing the uniqueness, convergent of each iteration to the fixe...

Fan-KKM Theorem in Minimal Vector Spaces and its Applications

In this paper, after reviewing some results in minimal space, some new results in this setting are given. We prove a generalized form of the Fan-KKM typetheorem in minimal vector spaces. As some applications, the open type of matching theorem and generalized form of the classical KKM theorem in minimal vector spaces are given.

A Fixed Point Theorem in Generalized D-metric Spaces