Let (ei) be a dictionary for a separable Banach space X . We consider the problem of approximation by linear combinations of dictionary elements with quantized coefficients drawn usually from a ‘finite alphabet’. We investigate several approximation properties of this type and connect them to the Banach space geometry of X . The existence of a total minimal system with one of these properties, ...

are classes of starlike and strongly starlike functions of order β (0 < β ≤ 1), respectively. Note that S∗(β)⊂ S∗ for 0< β< 1 and S∗(1)= S∗ [5]. Kanas [2] introduced the subclass R̄δ(β) of function f ∈ S as the following. Definition 1.1. For δ ≥ 0, β ∈ (0,1], a function f normalized by (1.1) belongs to R̄δ(β) if, for z ∈D−{0} and Dδf(z)≠ 0, the following holds: ∣∣∣arg z ( Dδf(z) )′ Dδf(z) ∣∣∣≤ βπ...

Denote by x a random infinite path in the graph of Pascal’s triangle (left and right turns are selected independently with fixed probabilities) and by dn(x) the binomial coefficient at the n’th level along the path x. Then for a dense Gδ set of θ in the unit interval, {dn(x)θ} is almost surely dense but not uniformly distributed modulo 1.

Since its introduction in the year 1998 by Watts and Strogatz, the clustering coefficient has become a frequently used tool for analyzing graphs. In 2002 the transitivity was proposed by Newman, Watts and Strogatz as an alternative to the clustering coefficient. As many networks considered in complex systems are huge, the efficient computation of such network parameters is crucial. Several algo...

We define and study certain moduli stacks of modules equipped with a Frobenius semi-linear endomorphism. These stacks can be thought of as parametrizing the coefficients of a variable Galois representation and are global variants of the spaces of Kisin–Breuil Φ-modules used by Kisin in his study of deformation spaces of local Galois representations. A version of a rigid analytic period map is d...

We present a new type of probabilistic model which we call DISsimilarity COefficient Networks (DISCO Nets). DISCO Nets allow us to efficiently sample from a posterior distribution parametrised by a neural network. During training, DISCO Nets are learned by minimising the dissimilarity coefficient between the true distribution and the estimated distribution. This allows us to tailor the training...

number of bacteria ingested at the close of the experlment(instead of the maximum number of bacteria present, which, one is led to infer, was larger than A even at the higher temperatures) and x equal, as usual, to the number of bacteria ingested in time, T. Considering their method of analysis it is not surprising that they found it impossible to calculate the temperature coefficient of phago•...