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 .

     

Nonisomorphic graphs with identical atomic counts of self-returning walks: Isocodal graphs

On the basis of endospectral graphs, we present a graphical method for obtaining pairs of nonisomorphic graphs with identical atomic counts of self-returning walks.     

Thermotropic liquid crystalline polymers with low thermal transitions. I. Low melting thermotropic liquid crystalline homo- and co-polycarbonates

Due to the particular rheological properties of thermotropic liquid crystalline polymers (TLCP), their application for imaging technology has been investigated. The first class of polymers investigated in this study are the thermotropic liquid crystalline polycarbonates prepared from the solution polycondensation of p,p-dihydroxybiphenyl with bischloroformate as the aliphatic flexible spacer. From the variety of bischloroformates employed, smectic TLCP's were generally obtained with the possible presence of a nematic mesostate. The introduction of comonomers such as substituted hydroquinones or bisphenols were found to lower both the melting transitions and mesophasic range of the TLCP. From rheological characterization, the amount of nonmesogenic moieties present in the copolycarbonates were found to correlate with the increase in the melt viscosity of TLCP.     

Norm estimates of almost Mathieu operators

We estimate the norm of the almost Mathieu operator , regarded as an element in the rotation C^*-algebra . In the process, we prove for every λ∈R and the inequality

     

The Correlations of Farey Fractions

It is proved that all correlations of the sequence of Fareyfractions exist and explicit formulas are provided for the correlationmeasures.     

On Ramanujan's continued fraction for

The continued fraction in the title is perhaps the deepest of Ramanujan's -continued fractions. We give a new proof of this continued fraction, more elementary and shorter than the only known proof by Andrews, Berndt, Jacobsen, and Lamphere. On page 45 in his lost notebook, Ramanujan states an asymptotic formula for a continued fraction generalizing that in the title. The second main goal of this paper is to prove this asymptotic formula.

     

Singular integrals on symmetric spaces, II

We extend some of our earlier results on boundedness of singular integrals on symmetric spaces of real rank one to arbitrary noncompact symmetric spaces. Our main theorem is a transference principle for operators defined by -bi-invariant kernels with certain large scale cancellation properties. As an application we prove boundedness of operators defined by Fourier multipliers that satisfy singular differential inequalities of the Hörmander-Michlin type.

     

On the Parity of the Number of Partitions in Square Free Parts

Let p~(n) be the number of partitions of a positive integer n in square free parts. We prove that for large N,(a) The number of n N such that p~(n) is odd is log N,(b) The number of n N such that p~(n) is even is N/log N.