Our main goal is to give proofs of all results announced by Oleg Izhboldin in [13]. In particular, we establish Izhboldin’s criterion for stable equivalence of 9-dimensional forms. Several other related results, some of them due to the author, are also included. All the fields we work with are those of characteristic different from 2. In these notes we consider the following problem: for a give...

The Arnold stability criterion suggests that a stationary ow of an ideal incompressible uid is stable if a certain quadratic form is deenite. We show that, in three or more dimensions, this quadratic form is never deenite. Typically the form is indeenite, and the spectrum of the associated Hermitian operator ranges from ?1 to 1. The exceptional case is where the velocity eld is harmonic (soleno...

Minimizing a nonlinear functional is presented as a way of obtaining a planar mapping that matches two similar images. A smoothing term is added to the nonlinear functional to penalize discontinuous and irregular solutions. One option for the smoothing term is a quadratic form generated by a linear differential operator. The functional is then minimized using the Fourier representation of the p...

In this research paper, the problem of optimization of quadratic forms associated with the dynamics of Hopfield-Amari neural network is considered. An elegant (and short) proof of the states at which local/global minima of quadratic form are attained is provided. A theorem associated with local/global minimization of quadratic energy function using the HopfieldAmari neural network is discussed....

A continuous quadratic form on a real Banach space X is called decomposable if it is the difference of two nonnegative (i.e., positively semidefinite) continuous quadratic forms. We prove that if X belongs to a certain class of superreflexive Banach spaces, including all Lp(μ) spaces with 2 ≤ p < ∞, then each continuous quadratic form on X is decomposable. On the other hand, on each infinite-di...

In this note we study an analogy between a positive definite quadratic form for elliptic curves over finite fields and a positive definite quadratic form for elliptic curves over the rational number field. A question is posed of which an affirmative answer would imply the analogue of the Riemann hypothesis for elliptic curves over the rational number field.

We prove that the representations numbers of a ternary definite integral quadratic form defined over Fq[t], where Fq is a finite field of odd characteristic, determine its integral equivalence class when q is large enough with respect to its successive minima. Equivalently, such a quadratic form is determined up to integral isometry by its theta series.

In this paper, we first discuss the bentness of a large class of quadratic Boolean functions in polynomial form f(x) = ∑n 2 −1 i=1 Tr n 1 (cix 1+2i) + Tr n/2 1 (cn/2x 1+2n/2), where ci ∈ GF (2) for 1 ≤ i ≤ n 2 − 1 and cn/2 ∈ GF (2). The bentness of these functions can be connected with linearized permutation polynomials. Hence, methods for constructing quadratic bent functions are given. Furthe...

In this paper, we prove compactness for the full set of solutions to the Yamabe Problem if n ≤ 24. After proving sharp pointwise estimates at a blowup point, we prove the Weyl Vanishing Theorem in those dimensions, and reduce the compactness question to showing positivity of a quadratic form. We also show that this quadratic form has negative eigenvalues if n ≥ 25.