In this article we prove two new criteria for the existence of rational general integral of an algebraic differential equation (cf. [4]) on the complex projective plane. These results are stated in terms of divisors and zero-cycles of a projective variety and are built by using intersection numbers of projective curves and multiplicities of singularities. We also prove a new result giving new s...

Nonlinear quadratic functional integral equations are often applicable for instance in the theory of radiative transfer, kinetic theory of gases, in the theory of neutron transport, in traffic theory and in numerous branches of mathematical physics (cf. [3, 7, 10, 11]). Additionally, some problems (motivated by some practical interests in plasma physics) was investigated in [28]. It is worthwhi...

in this paper, we present an efficient method for determining the solution of the stochastic second kind volterra integral equations (svie) by using the taylor expansion method. this method transforms the svie to a linear stochastic ordinary differential equation which needs specified boundary conditions. for determining boundary conditions, we use the integration technique. this technique give...

We show the relation between Appell's remark on the central projection in the dynamic of a particle and T.-Y. Thomas and Nijenhuis studies of the polynomial first integrals of the geodesic flow on a space of constant curvature. Among the consequences: the space of leading terms of a quadratic first integral of an equation¨q = f (q), q ∈ IR n , is isomorphic to the space of quadrilin-ear forms o...

We present an a-posteriori method for computing rigorous upper and lower bounds of the J-integral in two dimensional linear elasticity. The J-integral, which is typically expressed as a contour integral, is recast as a surface integral which yields a quadratic continuous functional of the displacement. By expanding the quadratic output about an approximate finite element solution, the output is...

Generally, in the financial literature, the notion of quadratic VaR is implicitly confused with the Delta-Gamma VaR, because more authors dealt with portfolios that contained derivatives instruments. In this paper, we postpone to estimate the Value-at-Risk of a quadratic portfolio of securities (i.e equities) without the Delta and Gamma greeks, when the joint underlying log-returns changes with...

Let n 4, and let Q ∈ Z[X1,. .. , Xn] be a non-singular quadratic form. When Q is indefinite we provide new upper bounds for the least non-trivial integral solution to the equation Q = 0, and when Q is positive definite we provide improved upper bounds for the least positive integer k for which the equation Q = k is insoluble in integers, despite being soluble modulo every prime power.

An algorithm is given for determining all power integral bases in orders of totally real cyclic sextic number fields. The orders considered are in most cases the maximal orders of the fields. The corresponding index form equation is reduced to a relative Thue equation of degree 3 over the quadratic subfield and to some inhomogeneous Thue equations of degree 3 over the rationals. At the end of t...

Let n > 4, and let Q ∈ Z[X1, . . . , Xn] be a non-singular quadratic form. When Q is indefinite we provide new upper bounds for the least non-trivial integral solution to the equation Q = 0, and when Q is positive definite we provide improved upper bounds for the greatest positive integer k for which the equation Q = k is insoluble in integers, despite being soluble modulo every prime power.

Let f be an indefinite ternary integral quadratic form and let q be a nonzero integer such that −qdet(f) is not a square. Let N(T, f, q) denote the number of integral solutions of the equation f(x) = q where x lies in the ball of radius T centered at the origin. We are interested in the asymptotic behavior of N(T, f, q) as T → ∞. We deduce from the results of our joint paper with Z.Rudnick that...