An affine invariant point on the class of convex bodies Kn in R, endowed with the Hausdorff metric, is a continuous map from Kn to R which is invariant under one-to-one affine transformations A on R, that is, p ` A(K) ́ = A ` p(K) ́ . We define here the new notion of dual affine point q of an affine invariant point p by the formula q(K) = p(K) for every K ∈ Kn, where K denotes the polar of K with...

The thermodynamic association of RNA polymerase (RNAP) with DNA is sensitive to salt concentration in vitro. Paradoxically, previous studies of changes in osmolarity during steady-state cell growth found no dependence between the association of RNAP to DNA and K(+) concentration in Escherichia coli. We reevaluated this issue by following the interaction of RNAP and genomic DNA in time-course ex...

Let K be a locally compact hypergroup. In this paper we initiate the concept of fundamental domain in locally compact hypergroups and then we introduce the Borel section mapping. In fact, a fundamental domain is a subset of a hypergroup K including a unique element from each cosets, and the Borel section mapping is a function which corresponds to any coset, the related unique element in the fun...

We give a classification of dual functions, they are m -al functions. We call a function m-al with respect to an operator if the operator lives any function unchanged after m times of using the operator. And 2 ≤ m ≤ k. Functions with different m have very different properties. We give theoretical results for clones of self-dual (m = 2) and selfk -al (m = k) functions in k -valued logic at k ≤ 3...

Here R(n, k) denotes the minimum possible size of a completely separating system C on [n] with |A| = k for each A ∈ C. Values of R(n, k) are determined for ( k−1 2 ) ≤ n < (k 2 ) or 11 ≤ n ≤ 12. Using the dual interpretation of completely separating systems as antichains, this paper provides corresponding results for dual k-regular antichains.

R(n,k) denotes the minimum possible size of a completely separating system C on {1,2,…,n} with |A|=k for each set A of C. Values of R(n,k) are determined for k=11 or k=12 and several other results are mentioned. Using the dual interpretation of completely separating systems as antichains, it provides corresponding results for dual k-regular antichains.