On the Number of Cyclic Projective Planes

AbstractAn explicit formula for the number of finite cyclic projective planes (or planar difference sets) is derived by applying Ramanujan sums (Von Sterneck numbers) and Möbius inversion over the set partition lattice to counting one-to-one solution vectors of multivariable linear congruences

Best Approximations in Preduals of Von Neumann Algebras

This paper characterises the semi-Chebychev subspaces of preduals of von Neumann algebras. As an application it generalises the theorem of Doob, that says that H01 has unique best approximations in L1(T), to a large class of preannihilators of natural non-selfadjoint operator algebras including the nest algebras. Then it studies the semi-Chebychev subspaces of the trace class operators and shows that the only Chebychev *-diagrams are ‘triangular

The Cohomology of the Steendrod Algebra and Representations of the General Linear Groups

Let Tr_k be the algebraic transfer that maps from the coinvariants of certain GL_k-representation to the cohomology of the Steenrod algebra. This transfer was defined by W. Singer as an algebraic version of the geometrical transfer tr_k : pi_*^S((B[doublestrike V]_k)_+) --\u3e pi_*^S(S^0). It has been shown that the algebraic transfer is highly nontrivial, more precisely, that Tr_k is an isomorphism for k = 1, 2, 3 and that T_r = ⊕_k(Tr_k) is a homomorphism of algebras. In this paper, we first recognize the phenomenon that if we start from any degree d, and apply Sq^0 repeatedly at most (k- 2) times, then we get into the region, in which all the iterated squaring operations are isomorphisms on the coinvariants of the GL_k-representation. As a consequence, every finite Sq^0-family in the coinvariants has at most (k - 2) non zero elements. Two applications are exploited. The first main theorem is that Tr_k is not an isomorphism for k gte 5. Furthermore, Tr_k is not an isomorphism in infinitely many degrees for each k \u3e 5. We also show that if Tr_ell detects a nonzero element in certain degrees of Ker(Sq^0), then it is not a monomorphism and further, Tr_k is not a monomorphism in infinitely many degrees for each k \u3e ell. The second main theorem is that the elements of any Sq^0-family in the cohomology of the Steenrod algebra, except at most its first (k - 2) elements, are either all detected or all not detected by Tr_k, for every k. Applications of this study to the cases k = 4 and 5 show that Tr_4 does not detect the three families g, D_3, p\u27 and Tr_5 does not detect the family {h_(n+1)g_n|n gte 1}

How to solve an operator equation

This article summarizes a series of lectures delivered at the Mathernatics Departrnent of the University of Leipzig, Germany in April 1991, which were to overview techniques for solving operator equations en C*-algebras connected with methods developed in a Spanish-Gerrnan research project on "Structure and Applications of C*-Algebras of Quotients" (SACQ) . One of the researchers in this project was Professor Pere Menal until his unexpected death this April. To his mernory this paper shall be dedicate

Characterization of the best approximations by classic cubic splines

This study deals specifically with classical cubic splines. Based on a lemma of John Rice, best approximation in the uniform norm by cubic splines is explored. The purpose of this study is to characterize the best approximation to a given continuous function f(x) by a cubic spline with fixed knots by counting alternating extreme points of its error function E(t)