In this paper, the well-known nonconforming Morley element for biharmonic equations in two spatial dimensions is extended to any higher dimensions in a canonical fashion. The general -dimensional Morley element consists of all quadratic polynomials defined on each -simplex with degrees of freedom given by the integral average of the normal derivative on each -subsimplex and the integral average...

An integral symmetric matrix S = (sij) ∈ M sym m (Z) with sii ∈ 2Z gives rise to an integral quadratic form q(x) = 12 xSx on Z. If S is positive definite, the number r(q, t) of solutions x ∈ Z of the equation q(x) = t is finite, and it is one of the classical tasks of number theory to study the qualitative question which numbers t are represented by q or the quantitative problem to determine th...

the study of stability problems of functional equations was motivated by a question of s.m. ulam asked in 1940. the first result giving answer to this question is due to d.h. hyers. subsequently, his result was extended and generalized in several ways.we prove some hyperstability results for the equation g(ax+by)+g(cx+dy)=ag(x)+bg(y)on restricted domain. namely, we show, under some weak natural...

Let Sr be the blow-up of P 2 in r general points, i.e., a smooth Del Pezzo surface of degree 9 − r. For r ≤ 7, we determine the quadratic equations defining its Cox ring explicitly. The ideal of the relations in Cox(S8) is calculated up to radical. As conjectured by Batyrev and Popov, all the generating relations are quadratic.

The type II1 unprojection is, by definition, the generic complete intersection type II unprojection, in the sense of [P] Section 3.1, for the parameter value k = 1, and depends on a parameter n ≥ 2. Our main results are the explicit calculation of the linear relations of the type II1 unprojection for any value n ≥ 2 (Theorem 3.16) and the explicit calculation of the quadratic equation for the c...

This paper is concerned with an efficient dual boundary element method for 2d crack problems under antiplane shear loading. The dual equations are the displacement and the traction boundary integral equations. When the displacement equation is applied on the outer boundary and the traction equation on one of the crack surfaces, general crack problems with anti-plane shear loading can be solved ...

In this paper, we use a combination of Legendre and Block-Pulse functionson the interval [0; 1] to solve the nonlinear integral equation of the second kind.The nonlinear part of the integral equation is approximated by Hybrid Legen-dre Block-Pulse functions, and the nonlinear integral equation is reduced to asystem of nonlinear equations. We give some numerical examples. To showapplicability of...

It is shown that the set of all paraperspective images with arbitrary reference point and the set of aane images of a 3-D object are identical. Consequently, all uncalibrated paraper-spective images of an object can be constructed from a 3-D model of the object by applying an aane transformation to the model and every aane image of the object represents some uncalibrated paraperspective image o...

The Hyers–Ulam–Rassias stability of the conditional quadratic functional equation of Pexider type f (x+y)+f (x−y) = 2g(x)+2h(y), x ⊥ y is proved where ⊥ is the orthogonality in the sense of Rätz.

First, the quadratic functional equation of Pexider type will be solved. By applying this result, we will also solve some functional equations of Pexider type which are closely associated with the quadratic equation.