Lower bounds for eigenvalues of a quadratic form relative to a positive quadratic form

1968 ◽  
Vol 27 (5) ◽  
pp. 398-406 ◽  
Author(s):  
Norman W. Bazley ◽  
Wolfgang Börsch-Supan ◽  
David W. Fox
Keyword(s):  
Quadratic Form ◽  
Lower Bounds ◽  
Positive Quadratic Form

The Distribution of Zeros of Quadratic Forms over Finite Fields

2015 ◽  
Vol 7 (2) ◽  
pp. 18
Author(s):  
Ali H. Hakami
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Finite Field ◽  
Finite Fields ◽  
Lower Bounds ◽  
Quadratic Forms ◽  
Distribution Of Zeros ◽  
Linearly Independent ◽  
Set Of Points

Let $m$ be a positive integer with $m < p/2$ and $p$ is a prime. Let $\mathbb{F}_q$ be the finite field in $q = p^f$ elements, $Q({\mathbf{x}})$ be a nonsinqular quadratic form over $\mathbb{F}_q$ with $q$ odd, $V$ be the set of points in $\mathbb{F}_q^n$ satisfying the equation $Q({\mathbf{x}}) = 0$ in which the variables are restricted to a box of points of the type\[\mathcal{B}(m) = \left\{ {{\mathbf{x}} \in \mathbb{F}_q^n \left| {x_i  = \sum\limits_{j = 1}^f {x_{ij} \xi _j } ,\;\left| {x_{ij} } \right| < m,\;1 \leqslant i \leqslant n,\;1 \leqslant j \leqslant f} \right.} \right\},\]where $\xi _1 , \ldots ,\xi _f$ is a basis for $\mathbb{F}_q$ over $\mathbb{F}_p$ and $n > 2$ even. Set $\Delta  = \det Q$ such that $\chi \left( {( - 1)^{n/2} \Delta } \right) = 1.$ We shall motivate work of (Cochrane, 1986) to obtain lower bounds on $m,$ size of the box $\mathcal{B},$ so that $\mathcal{B} \cap V$ is nonempty. For this we show that the box $\mathcal{B}(m)$ contains a zero of $Q({\mathbf{x}})$ provided that $m \geqslant p^{1/2}.$ We also show that the box $\mathcal{B}(m)$ contains $n$ linearly independent zeros of $Q({\mathbf{x}})$ provided that $m \geqslant 2^{n/2} p^{1/2} .$

scholarly journals Addendum to ‘Sharp inequalities that generalize the divergence theorem: an extension of the notion of quasi-convexity’

2015 ◽  
Vol 471 (2176) ◽  
pp. 20140886 ◽  
Author(s):  
Graeme W. Milton
Keyword(s):  
Quadratic Form ◽  
Boundary Conditions ◽  
Lower Bounds ◽  
Wide Class ◽  
Quadratic Functions ◽  
General Boundary ◽  
Divergence Theorem ◽  
General Boundary Conditions ◽  
Lagrange Equations ◽  
Quasi Convexity

The paper ‘Sharp inequalities that generalize the divergence theorem: an extension of the notion of quasi-convexity’ published in Proc. R. Soc. A 2013, 469, 20130075 ( doi:10.1098/rspa.2013.0075 ) is clarified. Notably, much more general boundary conditions are given under which sharp lower bounds on the integrals of certain quadratic functions of the fields can be obtained. More precisely, if the quadratic form is Q *-convex then any solution of the Euler–Lagrange equations will necessarily minimize the integral. As a consequence, strict Q *-convexity is found to be an appropriate condition to ensure uniqueness of the solutions of a wide class of linear Euler–Lagrange equations in a given domain Ω with appropriate boundary conditions.

scholarly journals Criteria for extreme forms

1959 ◽  
Vol 1 (1) ◽  
pp. 17-20 ◽  
Author(s):  
E. S. Barnes
Keyword(s):  
Quadratic Form ◽  
Local Maximum ◽  
Polyhedral Cone ◽  
Fundamental Region ◽  
Positive Quadratic Form ◽  
Complete Knowledge ◽  
Coefficient Space

A positive quadratic form, of determinantand minimum M for integral, is said to be extreme if the ratiois a (local) maximum for small variations in the coefficients.Minkowski [3] has given a criterion for extreme forms in terms of a fundamental region (polyhedral cone) in the coefficient space. This criterion, however, involves a complete knowledge of the edges of the region and is therefore of only theoretical value.

The Construction of Perfect and Extreme Forms

1966 ◽  
Vol 18 ◽  
pp. 147-158 ◽  
Author(s):  
P. R. Scott
Keyword(s):  
Quadratic Form ◽  
Finite Number ◽  
Positive Quadratic Form ◽  
Image Position ◽  
Minimal Vectors

Letbe a positive quadratic form of determinantD,and letMbe the minimum off(x) for integral x ≠ 0. Thenf(x) assumes the valueMfor a finite number of integral vectors x =±mk(k= 1 , … ,s)called theminimal vectors.

scholarly journals Eigenvalue Bounds for the Signless $p$-Laplacian

10.37236/6683 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Elizandro Max Borba ◽  
Uwe Schwerdtfeger
Keyword(s):  
Quadratic Form ◽  
Lower Bounds ◽  
Unit Sphere ◽  
Laplacian Matrix ◽  
Upper And Lower Bounds ◽  
Eigenvalue Bounds ◽  
Smallest Eigenvalue ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian ◽  
Largest Eigenvalues

We consider the signless $p$-Laplacian $Q_p$ of a graph, a generalisation of the quadratic form of the signless Laplacian matrix (the case $p=2$). In analogy to Rayleigh's principle the minimum and maximum of $Q_p$ on the $p$-norm unit sphere are called its smallest and largest eigenvalues, respectively. We show a Perron-Frobenius property and basic inequalites for the largest eigenvalue and provide upper and lower bounds for the smallest eigenvalue in terms of a graph parameter related to the bipartiteness. The latter result generalises bounds by Desai and Rao and, interestingly, at $p=1$ upper and lower bounds coincide.

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