Globevnik [G] answered the question in the affirmative for rotation invariant families. Agranovsky and Globevnik [AG] resolved the question for families of circles when f is a rational or real-analytic functions of two real variables. Ehrenpreis [E2] independently found solution for the circles of constant radii with centers on a line and real-analytic functions. In [T] we obtain the affirmativ...

Let G be a closed group of automorphisms of a graph X. We relate geometric properties of G and X, such as amenability and unimodularity, to properties of G-invariant percolation processes on X, such as the number of infinite components, the expected degree, and the topology of the components. Our fundamental tool is a new masstransport technique that has been occasionally used elsewhere and is ...

We study invariant and bi-invariant metrics on groups focusing finite G . show that non-equivalent (bi) are in 1-1 correspondence with unitary symmetric (conjugate) partitions To every metric group ( , d ) we associate to it the symmetry weighted graph of distances. Using these objects can classify all equivalence classes for small groups. then number give an expression such terms Bell numbers,...

By a 2-group we mean a groupoid equipped with a weakened group structure. It is called split when it is equivalent to the semidirect product of a discrete 2-group and a oneobject 2-group. By a permutation 2-group we mean the 2-group Sym(G) of self-equivalences of a groupoid G and natural isomorphisms between them, with the product given by composition of self-equivalences. These generalize the ...

We investigate the pair-splitting number spair which is a variation of splitting number, pair-reaping number rpair which is a variation of reaping number and cardinal invariants of ideals on ω. We also study cardinal invariants of Fσ ideals and their upper bounds and lower bounds. As an application, we answer a question of S. Solecki by showing that the ideal of finitely chromatic graphs is not...

In this paper, we conduct a series of experiments to evaluate the translation invariant property of a set of discrete wavelet features in a face graph. Using local-area power spectrum estimation based on discrete wavelet transform, we compute a feature vector that possesses both efficient space-frequency structure and translation invariant properties.

On the level of Lie algebras, the Kontsevich integral of a knot (a graph-valued invariant) becomes the colored Jones function (a power series invariant). Rozansky conjectured and the authors proved a Rationality Conjecture for the Kontsevich integral. In this note, we explain how the Rationality Conjecture is related to Lie groups.

Let X be a connected graph. An automorphism of X is said to be parabolic if it leaves no finite subset of vertices in X invariant and fixes precisely one end of X and hyperbolic if it leaves no finite subset of vertices in X invariant and fixes precisely two ends of X. Various questions concerning dynamics of parabolic and hyperbolic automorphisms