) =(2,unit disk) or (n, Ω) =(3,unit ball), we use the following result [1]. Let X1, X2, X3, Y1, Y2, Y3, Z1, Z2, Z3 be independent normally distributed random variables with mean 0 and variance 1/2. Let W1, W2, W3 be exponential random variables, independent of the

) =(2,unit disk) or (n, Ω) =(3,unit ball), we use the following result [1]. Let X1, X2, X3, Y1, Y2, Y3, Z1, Z2, Z3 be independent normally distributed random variables with mean 0 and variance 1/2. Let W1, W2, W3 be exponential random variables, independent of the

Hardy [1] relates the following anecdote. “I remember going to see him [Ramanujan] when he was lying ill at Putney. I had ridden in taxi–cab No. 1729, and remarked that the number (7 × 13 × 19) seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. “No, ” he replied, “it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways.”” Indeed, 1729 = 93 + 103 = 123 + 13. But there is another way in which this example is special. We know, since Euler, that the sum of two positive cubes is never a cube. But the above example shows that the sum of two positive cubes can do the next best thing – and that is, to miss a cube by as little as 1. Indeed, Ramanujan left for us infinitely many examples of just that phenomenon. In his so-called “Lost Notebook ” [4], he stated a result equivalent to the following.

In view of the current interest in d x 2 −y 2 + idxy superconductors some of their thermodynamic properties have been studied to obtain relevant information for experimental verification. The temperature dependence of the specific heat and superfluid density show marked differences in d x 2 −y 2

Abstract. In this paper, we introduce a modification of the Szász-Mirakjan type opera-tors of two variables which preserve f0 (x, y) = 1 and f3 (x, y) = x 2 + y2. We prove that this type of operators enables a better error estimation on the interval [0,∞) × [0,∞) than the classical Szász

which primes p can be written as x 2 + y 2 D for integers x, y and D.

Abstract. The paper Pythagorean triples of Polygonal Numbers [3] called our attention to search on another parametric representation for Diophantine equation of x2+y2+z2 = t2 of n-polygonal numbers. We got benefit from the identity that gives all the solutions of the equation x2+ y2+ z2 = t2

error analysis of an augmented precision algorithm for the square root, and introduce two slightly different augmented precision algorithms for the 2D-norm p x2 + y2. Then we give tight lower bounds on the minimum distance (in ulps) between p x2 + y2 and a midpoint when p x2 + y2 is not itself a

Abstract. In this paper we investigate mixed sums of squares and triangular numbers for the first time. We prove that any natural number n can be written as x 2 + y 2 + Tz with x, y, z ∈ Z and Tz = z(z + 1)/2. Also, we can express n in any of the following forms:

nearest neighbor sites. PACS numbers: 75.40.Mg, 74.20.-z, 75.10.Jm, 74.65.+n Typeset Using REVTEXSome time ago it has been proposed that the exchange of spin fluctuations could lead to superconductivity in the singlet d x 2 −y 2 channel [1]. Such mechanisms were recently discussed in connection