scholarly journals Various Proofs of the Sylvester Criterion for Quadratic Forms

2017 ◽  
Vol 9 (6) ◽  
pp. 55
Author(s):  
Giorgi Giorgio
Keyword(s):  
Quadratic Forms ◽  
Homogeneous System ◽  
Algebraic Proof ◽  
Linear Homogeneous System

In the first part of the paper we present several proofs of the so-called Sylvester criterion for quadratic forms; some of the said proofs are short and easy. In the second part of the paper we give an algebraic proof of the Sylvester criterion for quadratic forms subject to a linear homogeneous system.

scholarly journals A Note on Sparse Supersaturation and Extremal Results for Linear Homogeneous Systems

10.37236/6730 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Christoph Spiegel
Keyword(s):  
Homogeneous System ◽  
Random Sets ◽  
Type Result ◽  
Linear Homogeneous System ◽  
Homogeneous Systems ◽  
Class Of Matrices

We study the thresholds for the property of containing a solution to a linear homogeneous system in random sets. We expand a previous sparse Szémeredi-type result of Schacht to the broadest class of matrices possible. We also provide a shorter proof of a sparse Rado result of Friedgut, Rödl, Ruciński and Schacht based on a hypergraph container approach due to Nenadov and Steger. Lastly we further extend these results to include some solutions with repeated entries using a notion of non-trivial solutions due to Rúzsa as well as Rué et al.

scholarly journals Lyapunov-Type Inequality for a Class of Discrete Systems with Antiperiodic Boundary Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Xin-Ge Liu ◽  
Mei-Lan Tang
Keyword(s):  
Boundary Conditions ◽  
Positive Solution ◽  
Type Inequality ◽  
Discrete Systems ◽  
Homogeneous System ◽  
Higher Order ◽  
Linear Homogeneous System ◽  
3 Dimensional ◽  
Lyapunov Type Inequality

A class of higher-order 3-dimensional discrete systems with antiperiodic boundary conditions is investigated. Based on the existence of the positive solution of linear homogeneous system, several new Lyapunov-type inequalities are established.

Existence

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Structure Theorem ◽  
Quadratic Extension ◽  
Quadratic Space ◽  
Division Algebras ◽  
Separable Extension ◽  
Valued Fields ◽  
Quaternion Division Algebra

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

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