On the neutrix convolution product ofx − r Inx − andx + −3

The existence of the neutrix convolution product of distributionx _? ^r Inx _? andx ₊ ^?3 is proved and some convolution products are evaluated.     

The Contraction of S2p−1 to Hp−1

 A contraction of the sphere , considered as the homogeneous space , to the Heisenberg group is defined. The infinite dimensional irreducible unitary representations of Heisenberg group are then shown to be the limits of the irreducible representations of which are class-1 with respect to . Our results generalise the earlier results of Fulvio Ricci. (Received 1 July 1998; in revised form 3 November 1998)     

Gorenstein Right Derived Functors of − ⊗ −with Respect to Semidualizing Modules

We study Gorenstein right derived functors of ? ? ?with respect to semidualizing modules. As applications, some new criteria for a semidualizing module to be dualizing are given provided that R is a ring with a dualizing complex.     

On the Diophantine Equations (2 n − 1)(6 n − 1) = x 2 and (a n − 1)(a kn − 1) = x 2

In this paper we prove that the equation (2 ⁿ – 1)(6 ⁿ – 1) = x ² has no solutions in positive integers n and x. Furthermore, the equation (a ⁿ – 1) (a ^kn – 1) = x ² in positive integers a > 1, n, k > 1 (kn > 2) and x is also considered. We show that this equation has the only solutions (a,n,k,x) = (2,3,2,21), (3,1,5,22) and (7,1,4,120).     

Steiner systems andn −1-isomorphisms

In this work we introduce the concept of n ^–1-isomorphism between Steiner systems (this coincides with the concept of isomorphism whenever n=1).Precisely two Steiner systems S₁ and S₂ are said to be n^–1-isomorphic if there exist n partial systems S _i ⁽¹⁾ ,...,S _i ⁽ⁿ⁾ contained in Si, i.{1,2},such that S ₁ ^(k) and S ₂ ^(k) are isomorphic for each k{1,..., n}.The n^–1-isomorphisms are also used to study nets replacements, see Ostrom [8], and to study the transformation methods of designs and other incidence structures introduced in [9] and generalized in [1] and [10].Work done under the auspicies of G.N.S.A.G.A. supported by 40% grants of M.U.R.S.T.     

The solution of the matrix equations AXB−CXD=E AND (YA−DZ,YC−BZ)=(E,F)

The conditions for the existence of a unique solution of the matrix equation AXB−CXD=E are proved to be that (i) the pencils A - λC and D - λB are regular, and (ii) the spectra of the pencils have an empty intersection. A numerical algorithm for solving the equation is proposed. The possibility of a least-squares-type solution is briefly discussed. The set of equations (YA−DZ,YC−BZ)=(E,F) is proved to be equivalent to the aforementioned equation, and its solution is also investigated. A numerical algorithm is proposed.     

On the Diophantine equation x 2 − kxy + y 2 − 2 n = 0

In this study, we determine when the Diophantine equation x ²?kxy+y ²?2 ⁿ = 0 has an infinite number of positive integer solutions x and y for 0 ? n ? 10. Moreover, we give all positive integer solutions of the same equation for 0 ? n ? 10 in terms of generalized Fibonacci sequence. Lastly, we formulate a conjecture related to the Diophantine equation x ² ? kxy + y ² ? 2 ⁿ = 0.     

Quadratic differentials (A(z − a)(z − b)/(z − c)2) dz2 and algebraic Cauchy transform

We discuss the representability almost everywhere (a.e.) in C of an irreducible algebraic function as the Cauchy transform of a signed measure supported on a finite number of compact semi-analytic curves and a finite number of isolated points. This brings us to the study of trajectories of the particular family of quadratic differentials A(z ? a)(z ? b)×(z ? c)^?2 dz². More precisely, we give a necessary and sufficient condition on the complex numbers a and b for these quadratic differentials to have finite critical trajectories. We also discuss all possible configurations of critical graphs.