Hilbert’s proof, apart from the determination of the best possible constant π csc(π/p), was published by Weyl [7]. The calculation of the constant, and the integral analogue of Hilbert’s double series theorem (for p = 2) are due to Schur [6]. The generalizations to other p′s of both the discrete and integral versions of this result were discovered later on by Hardy and Riesz and published by Ha...

Let X be a finite connected CW -complex. Suppose that its fundamental group π is residually finite, i.e., there is a nested sequence . . . ⊂ Γm+1 ⊂ Γm ⊂ . . . ⊂ π of in π normal subgroups of finite index whose intersection is trivial. Then we show that the p-th L2-Betti number of X is the limit of the sequence bp(Xm)/[π : Γm] where bp(Xm) is the (ordinary) p-th Betti number of the finite coveri...

In this article λ-method of Mitrinović-Vasić [1] is applied to improve the upper bound for the arc sin function of L. Zhu [4]. 1. Inequalities of Shafer-Fink’s type D. S. Mitrinović in [1] considered the lower bound of the arc sin function, which belongs to R. E. Shafer. Namely, the following statement is true. Theorem 1.1 For 0 ≤ x ≤ 1 the following inequalities are true: 3x 2 + √ 1− x2 ≤ 6( √...

Experiments have been carried out in a 0.1 m internal diameter, 1.88 m height vertical Plexiglas column with an antenna type modified air sparger in the gas-liquid distributor section. This arrangement provides uniform mixing of the fluids, ensures the gas entering the fluidizing section as fine bubbles and reduces the pressure drop encountered through a conventional distributor used for the pu...

In the absence of governing equations, dimensional analysis is a robust technique for extracting insights and finding symmetries in physical systems. Given measurement variables parameters, Buckingham Pi theorem provides procedure set dimensionless groups that spans solution space, although this not unique. We propose an automated approach using symmetric self-similar structure available data t...

In physics and engineering, data is represented as attribute-value pairs with corresponding measurement units. Due to the functional model building process in the natural sciences, any functional relationship with measurement units based on a dimensions concept can be written in terms of dimensionless groups. This is guaranteed by the Pi-Theorem of Buckingham which is based on the condition of ...

We investigate generalizations of pebbling numbers and of Graham’s pebbling conjecture that π(G × H) ≤ π(G)π(H), where π(G) is the pebbling number of the graph G. We develop new machinery to attack the conjecture, which is now twenty years old. We show that certain conjectures imply others that initially appear stronger. We also find counterexamples that show that Sjöstrand’s theorem on cover p...

In the field of case-based reasoning in artificial intelligence, the general derivation of so-called similarity measures is still an unresolved open question. In this work the theoretical framework of dimensional analysis is used to derive appropriate similarity measures for a case-based reasoning technique. For the subclass of all case descriptions in engineering and physics consisting of real...

Consider a given abstract root system Δ and some simple root system Π ⊂ Δ. We may represent β ∈ Δ as a linear combination of elements of Π with integer coefficients of the same sign. By definition, we know that the elements of Π are linearly independent. Thus, such a representation of β is unique and exactly one of ±β is a positive root associated to Π. Δ+ is simply the set of all positive root...

This paper is a survey of some recent work of mine done jointly with V. Kumar Murty, N. Saradha, S. Gun and P. Rath. In all of these works, the motivating question is the following. Given an automorphic representation π , we consider the L-series L(s, π) attached to π according to the Langlands formalism. We are interested in the possible transcendence of special values L(k, π) when k is a posi...