Let k be an algebraically closed field, char k 6= 2, 3, and let X ⊂ P be an elliptic curve with defining polynomial f . We show that any non-trivial torsion point of order r, determines up to equivalence, a unique minimal matrix Φr of size 3r×3r with linear polynomial entries such that det Φr = f . We also show that the identity, thought of as the trivial torsion point of order r, determines up...

A fundamental question in the study of integral quadratic forms is the representation problem which asks for an effective determination of the set of integers represented by a given quadratic form. A related and equally interesting problem is the representation of integers by inhomogeneous quadratic polynomials. An inhomogeneous quadratic polynomial is a sum of a quadratic form and a linear for...

We find a lower bound on the number of imaginary quadratic extensions of the function field Fq(T ) whose class groups have an element of a fixed order. More precisely, let q ≥ 5 be a power of an odd prime and let g be a fixed positive integer ≥ 3. There are q 1 2+ 1 g ) polynomials D ∈ Fq[T ] with deg(D) ≤ ` such that the class groups of the quadratic extensions Fq(T, √ D) have an element of or...

We study the functional codes of second order defined by G. Lachaud on X ⊂ P(Fq) a quadric of rank(X )=3,4,5 or a non-degenerate hermitian variety. We give some bounds for the number of points of quadratic sections of X , which are the best possible and show that codes defined on non-degenerate quadrics are better than those defined on degenerate quadrics. We also show the geometric structure o...

Consider the elliptic curves given by En,θ : y 2 = x + 2snx − (r − s)nx where 0 < θ < π, cos(θ) = s/r is rational with 0 ≤ |s| < r and gcd(r, s) = 1. These elliptic curves are related to the θ-congruent number problem as a generalization of the congruent number problem. For xed θ this family corresponds to the quadratic twist by n of the curve Eθ : y 2 = x +2sx − (r − s)x. We study two special ...

The specific force equation has many applications in open channel flow problems. Quantifying of the hydraulic jump phenomenon is an important application of this equation. This equation has a direct solution only for the rectangular channels. The trial and error procedure as well as the graphical solution are the existing methods of solving hydraulic jump equations. No direct solutions are avai...

we propose a method of estimation of the derivatives of probability density based on wavelets methods for a sequence of random variables with a common one-dimensional probability density function and obtain an upper bound on lp-losses for such estimators. we suppose that the process is strongly mixing and we show that the rate of convergence essentially depends on the behavior of a special quad...

The morphology of glacial valley can be described in terms of power law or quadratic equations fitted to valley cross-profiles. These two models are used to study the cross-profiles of 86 valley cross-profiles in the Zardkuh Mountain in order to understand the evolutional patterns of valleys. Assessment of using the power law function indicates that b values for both valley sides range from 1.0...

This paper addresses the problems caused by an inappropriate selection of sliding surface parameters in fuzzy sliding mode controllers via an optimization approach. In particular, the proposed method employs the parallel distributed compensator scheme to design the state feedback based control law. The controller gains are determined in offline mode via a linear quadratic regular. The particle ...