for certain explicitly computable constants ai. The difficult part of extending Ingham’s result to include the lower-order terms is asymptotically evaluating the off-diagonal terms. The family of all primitive Dirichlet L-functions of modulus q is similar in some ways to the Riemann zeta function in t-aspect, but is more difficult to analyze. In 1981, Heath-Brown obtained an asymptotic formula ...

A diagonal Latin square is a Latin square whose main diagonal and back diagonal are both transversals. In this paper we give some constructions of pairwise orthogonal diagonal Latin squares. As an application of such constructions we obtain some new infinite classes of pairwise orthogonal diagonal Latin squares which are useful in the study of pairwise orthogonal diagonal Latin squares.

For an invertible diagonal matrix D , the convergence of the power scaled matrix sequence N N D A  is investigated. As a special case, necessary and sufficient conditions are given for the convergence of N N D T  , where T is triangular. These conditions involve both the spectrum as well as the diagraph of the matrix T . The results are then used to privide a new proof for the convergence of ...

This article discusses two theorems of Georg Cantor: Cantor’s Little Theorem and Cantor’s Diagonal Theorem.These results are obtained by generalizing the method of proof ofthe well known Cantor’s theorem about the cardinalities of a setand its power set. As an application of these, Gödel’s firstincompleteness theorem is proved. Hints are given as to how to derive other deeper results including ...

We use the grid consisting of bits of 3n to motivate the definition of 2-adic numbers. Specifically, we exhibit diagonal stripes in the bits of 32 n , which turn out to be the first in an infinite sequence of such structures. Our observations are explained by a 2-adic power series, providing some regularity among the disorder in the bits of powers of 3. Generally, the base-p representation of k...

We define the following mappings: the characteristic function of a subset of a set, the inclusion function (injection or embedding), the projections from a Cartesian product onto its arguments and diagonal function (inclusion of a set into its Cartesian square). Some operations on functions are also defined: the products of two functions (the complex function and the more general product-functi...

Lipshitz showed that the diagonal of a D-finite power series is still D-finite, but his proof seems hard to implement. This paper may be regarded as the first step towards an efficient algorithm realizing Lipshitz’s theory. We show that the idea of a reduced form may be a big saving for computing the D-finite functional equation. For the residue in one variable of a rational function, we develo...

This paper studies power allocation in coordinated multi-point (CoMP) transmission of 3GPP LTE-Advanced system with remote radio units(RRUs) power constraints. We apply block diagonal (BD) precoding to downlink transmission, and assume perfect knowledge of downlink channels and transmit messages at each transmit point. We propose a modified water-filling power (MWF) allocation algorithm in orde...

We obtain several results and examples concerning the general question “When must a space with a small diagonal have a Gδ-diagonal?”. In particular, we show (1) every compact metrizably fibered space with a small diagonal is metrizable; (2) there are consistent examples of regular Lindelöf (even hereditarily Lindelöf) spaces with a small diagonal but no Gδ-diagonal; (3) every first-countable he...

In this paper a new and efficient method for the reliable load flow solution of power network branches controlled by flexible AC transmission systems (FACTS) is presented. In the various methods proposed in past, the formulations was derived in which the Jacobian matrix is in UDU form, where U is a constant upper triangular matrix depending solely on system topology and D is a block diagonal ma...