This paper deals with a relation between Hardy-Hilbert’s integral inequality and Mulholland’s integral inequality with a best constant factor, by using the Beta function and introducing a parameter λ. As applications, the reverse, the equivalent form and some particular results are considered.

We prove that every n-vertex graph of genus 9 and maximal degree k has an edge separator of size O( ..;gTffl). The upper bound is best possible to within a constant factor. This extends known results on planar graphs and similar results about vertex separators. We apply the edge separatorto the isoperimetric number problem, graph embeddings and lower bounds for crossing numbers.

By means of weight functions and Hermite-Hadamard's inequality, and introducing a discrete interval variable, a more accurate half-discrete Hardy-Hilbert-type inequality related to the kernel of arc tangent function and a best possible constant factor is given, which is an extension of a published result. The equivalent forms and the operator expressions are also considered.

By using the way of weight functions and technique of real analysis, a new multiple half-discrete Hilbert-type inequality with the best constant factor is given. As applications, the equivalent forms, operator expressions as well as some reverse inequalities are also considered. Mathematics subject classification (2010): 26D15, 47A07.

We study the multiple-precision addition of two positive floating-point numbers in base 2, with exact rounding, as specified in the MPFR library, i.e. where each number has its own precision. We show how the best possible complexity (up to a constant factor that depends on the implementation) can be obtain.

By introducing multi-parameters, applying the weight coefficients and Hermite-Hadamard's inequality, we give a reverse of the extended Mulholland inequality in the whole plane with the best possible constant factor. The equivalent forms and a few particular cases are also considered.

Clustering can play a critical role in increasing the performance and lifetime of wireless networks. The facility location problem is a general abstraction of the clustering problem and this paper presents the first constant-factor approximation algorithm for the facility location problem on unit disk graphs (UDGs), a commonly used model for wireless networks. In this version of the problem, co...

We consider the online search problem in which a server starting at origin of d-dimensional Euclidean space has to find an arbitrary hyperplane. The best-possible competitive ratio and length shortest curve from each point on unit sphere can be seen are within constant factor other. show that this is ?(d)?O(d3/2).