I. Introduction. The solutions of the quadratic equation ax2 + bx+c = 0, where a, b, and c are real numbers, are given in exact form by the quadratic formula. Moreover, if the roots are real and irrational, Newton's method or continued fractions may be used to approximate the solutions.

version. In the abstract theory, if V is a K vector space of dimension n, there there exists a basis B = {e1, . . . , en} for V . That is, every v ∈ V is represented uniquely as a linear combination v = c1e1 + · · ·+ cnen for some coefficients cj ∈ K. Then the coordinate vector for v (relative to that basis) is Cv = (c1, . . . , cn) T ∈ Kn. If B′ = {e1, . . . , en} is another basis for V , the ...

The level N and squarefree character q of a positive definite ternary quadratic form are defined so that its associated modular form has level N and character Xg ■ We define ä collection of correspondences between classes of quadratic forms having the same level and different discriminants. This makes practical a method for finding representatives of all classes of ternary forms having a given ...

In this paper, we study the stochastic integration with respect to local times of fractional Brownian motion B with Hurst index 1/2 < H < 1. As a related problem we define the weighted quadratic covariation [f(B), B ] ) of f(B) and B as follows n−1

We prove that there are finitely many perfect powers in elliptic divisibility sequences generated by a non-integral point on curves of the form y 2 =x(x +b), where b is any positive integer. achieve this using modularity over real quadratic number fields.

The aim of the present study was the estimation of (co) variance components and genetic parameters for body weight of Moghani sheep, using random regression models based on B-Splines functions. The data set included 9165 body weight records from 60 to 360 days of age from 2811 Moghani sheep, collected between 1994 to 2013 from Jafar-Abad Animal Research and Breeding Institute, Ardabil province,...

Noncentral indefinite quadratic expressions in possibly non- singular normal vectors are represented in terms of the difference of two positive definite quadratic forms and an independently distributed linear combination of standard normal random variables. This result also ap- plies to quadratic forms in singular normal vectors for which no general representation is currently available. The ...

We attempt to generalize the familiar covariantly conserved Bel-Robinson tensor Bμναβ ∼ RR of GR and its recent topologically massive third derivative order counterpart B ∼ RDR, to quadratic curvature actions. Two very different models of current interest are examined: fourth order D = 3 “new massive”, and second order D > 4 Lanczos-Lovelock, gravity. On dimensional grounds, the candidates here...

This paper presents two results: a density estimator and an estimator of regression error density. We first propose a density estimator constructed by model selection, which is adaptive for the quadratic risk at a given point. Then we apply this result to estimate the error density in an homoscedastic regression framework Yi = b(Xi) + ǫi, from which we observe a sample (Xi, Yi). Given an adapti...

Let p ≡ 1 (mod 4) be a prime. Let a, b ∈ Z with p a(a2 + b2). In the paper we mainly determine ( b+ √ a2+b2 2 ) p−1 2 (mod p) by assuming p = c2 + d2 or p = Ax2 + 2Bxy + Cy2 with AC − B2 = a2 + b2. As an application we obtain simple criteria for εD to be a quadratic residue (mod p), where D > 1 is a squarefree integer such that D is a quadratic residue of p, εD is the fundamental unit of the qu...