We give sufficient cohomological criteria for the classes of given varieties over a field k to be algebraically independent in the Grothendieck ring of varieties over k and construct some examples.

The graph copy function when restricted to the set of trees or of rooted trees is called the tree copy function or the rooted tree copy function, respectively. We show that both the set of tree copy functions and the set of rooted tree copy functions are strongly linearly independent. We also show that the set of all tree copy functions are algebraically independent but the set of all rooted tr...

In this note we investigate arithmetic properties of values of the Tschakaloff function Tq(z) = P∞ ν=0 q −ν(ν+1)/2zν , |q| > 1. One of the open problems is proving linear independence of the values of Tq(z) with different q. The only result obtained in this direction in [1] is very restrictive. We refer an interested reader to the survey [2] for an account of known linear and algebraic independ...

An expression in terms of (cyclotomic) harmonic sums can be simplified by the quasi-shuffle algebra in terms of the so-called basis sums. By construction, these sums are algebraically independent within the quasi-shuffle algebra. In this article we show that the basis sums can be represented within a tower of difference ring extensions where the constants remain unchanged. This property enables...

At the time of Independence India, challenge was to create an image a progressive nation through mode architectural expression celebrate nation’s independence. Though it couldn’t happen right away as majority Indian architects at independence were under influence architecture Britishers or working in firms established by British architects. So, change gradual with its inception pre- era 1930s a...

We show that the classical kernel and domain functions associated to an n-connected domain in the plane are all given by rational combinations of three or fewer holomorphic functions of one complex variable. We characterize those domains for which the classical functions are given by rational combinations of only two or fewer functions of one complex variable. Such domains turn out to have the ...