594 research outputs found
A note on a curious formula for Euler's constant
In this short note we will use the residue theorem to establish a formula for
Euler's constant. In particular, we offer a slightly generalized version of an
interesting infinite series due to Flajolet, Gourdon, and Dumas.Comment: 4 page
A study of inverse trigonometric integrals associated with three-variable Mahler measures, and some related identities
We prove several identities relating three-variable Mahler measures to
integrals of inverse trigonometric functions. After deriving closed forms for
most of these integrals, we obtain ten explicit formulas for three-variable
Mahler measures. Several of these results generalize formulas due to Condon and
Lal\'in. As a corollary, we also obtain three -series expansions for the
dilogarithm
Local Boxicity, Local Dimension, and Maximum Degree
In this paper, we focus on two recently introduced parameters in the
literature, namely `local boxicity' (a parameter on graphs) and `local
dimension' (a parameter on partially ordered sets). We give an `almost linear'
upper bound for both the parameters in terms of the maximum degree of a graph
(for local dimension we consider the comparability graph of a poset). Further,
we give an time deterministic algorithm to compute a local box
representation of dimension at most for a claw-free graph, where
and denote the number of vertices and the maximum degree,
respectively, of the graph under consideration. We also prove two other upper
bounds for the local boxicity of a graph, one in terms of the number of
vertices and the other in terms of the number of edges. Finally, we show that
the local boxicity of a graph is upper bounded by its `product dimension'.Comment: 11 page
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