Classification of the relative positions between a circular hyperboloid of one sheet and a sphere

We characterize all possible relative positions between a circular hyperboloid of one sheet and a sphere through the roots of a characteristic polynomial associated to these quadrics. As an application, this provides a method to detect contact between the 2 surfaces by a simple calculation in many real world applications.     

The relative commutant of separable C-algebras of real rank zero

We answer a question of E. Kirchberg (personal communication): does the relative commutant of a separable C^∗-algebra in its ultrapower depend on the choice of the ultrafilter?     

Testing the relative positions of several points on a circle

This note introduces a programming technique that serves to facilitate the examination of the relative positions of several points on a circle, or—more generally—the relative positions of several entities that belong to a cyclic structure. The basic idea is to employ a function of the numerical representations of the individual positions which is invariant under rotation with regard to sign. The nature of the numerical representation makes it natural to introduce the term: cut-point-independent test procedure.     

Inequalities between rank moments of overpartitions

F.G. Garvan proved an inequality between crank moments and rank moments of partitions which utilizes an inequality for symmetrized rank and crank moments. In this paper, we study two symmetrized rank moments of overpartitions and prove an inequality between two rank moments of overpartitions.     

Distinct gaps between fractional parts of sequences

Let be a real number, a positive integer and a subset of . We give an upper bound for the number of distinct lengths of gaps between the fractional parts .

     

A new characterization of gaps between two subspaces

In this note, a new characterization of gaps between two subspaces of a Hilbert space is established.

     

On gaps between zeros of the Riemann zeta-function

Assuming the Riemann Hypothesis, we show that infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at least 2.7327 times the average spacing and infinitely often they differ by at most 0.5154 times the average spacing.     

Large gaps between consecutive zeros of the Riemann zeta-function

Combining the amplifiers, we exhibit other choices of coefficients that improve the results on large gaps between the zeros of the Riemann zeta-function. Precisely, assuming the Generalized Riemann Hypothesis (GRH), we show that there exist infinitely many consecutive gaps greater than 3.033 times the average spacing.     

Large gaps between the zeros of the Riemann zeta function

We show that the generalized Riemann hypothesis implies that there are infinitely many consecutive zeros of the zeta function whose spacing is three times larger than the average spacing. This is deduced from the calculation of the second moment of the Riemann zeta function multiplied by a Dirichlet polynomial averaged over the zeros of the zeta function.     

Some relations between rank of a graph and its complement

Let G be a graph of order n and rank(G) denotes the rank of its adjacency matrix. Clearly, . In this paper we characterize all graphs G such that or n + 2. Also for every integer n ? 5 and any k, 0 ? k ? n, we construct a graph G of order n, such that .     

On the equality between rank and trace of an idempotent matrix

The paper was inspired by the question whether it is possible to derive the equality between the rank and trace of an idempotent matrix by using only the idempotency property, without referring to any further features of the matrix. It is shown that such a proof can be obtained by exploiting a general characteristic of the rank of any matrix. An original proof of this characteristic is provided, which utilizes a formula for the Moore-Penrose inverse of a partitioned matrix. Further consequences of the rank property are discussed, in particular, several additional facts are established with considerably simpler proofs than those available. Moreover, a collection of new results referring to the coincidence between rank and trace of an idempotent matrix are derived as well.     

Some relations between rank, chromatic number and energy of graphs

The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G. Let G be a graph of order n and be the rank of the adjacency matrix of G. In this paper we characterize all graphs with . Among other results we show that apart from a few families of graphs, , where n is the number of vertices of G, and χ(G) are the complement and the chromatic number of G, respectively. Moreover some new lower bounds for E(G) in terms of are given.