If C is a q-ary code of length n and a and b are two codewords, then c is called a descendant of a and b if ci 2 fai; big for i = 1; : : : ; n. We are interested in codes C with the property that, given any descendant c, one can always identify at least one of the `parent' codewords in C. We study bounds on F (n; q), the maximal cardinality of a code C with this property, which we call the iden...

For each countable group Q we produce a short exact sequence 1 → N → G → Q → 1 where G is f.g. and has a graphical 1 6 presentation and N is f.g. and satisfies property T . As a consequence we produce a group N with property T such that Out(N) is infinite. Using the tools developed we are also able to produce examples of nonHopfian and non-coHopfian groups with property T . One of our main tool...

If C is a q-ary code of length n and a and b are two codewords, then c is called a descendant of a and b if ci 2 fai; big for i = 1; : : : ; n. We are interested in codes C with the property that, given any descendant c, one can always identify at least one of the `parent' codewords in C. We study bounds on F (n; q), the maximal cardinality of a code C with this property, which we call the iden...

We demonstrate analytically and numerically the possibility that the fractal property of a scale-free network cannot be characterized by a unique fractal dimension and the network takes a multifractal structure. It is found that the mass exponents τ(q) for several deterministic, stochastic, and real-world fractal scale-free networks are nonlinear functions of q, which implies that structural me...

A set X ⊆ 2 has property (s) (Marczewski (Szpilrajn)) iff for every perfect set P ⊆ 2 there exists a perfect set Q ⊆ P such that Q ⊆ X or Q∩X = ∅. Suppose U is a nonprincipal ultrafilter on ω. It is not difficult to see that if U is preserved by Sacks forcing, i.e., it generates an ultrafilter in the generic extension after forcing with the partial order of perfect sets, then U has property (s)...

A set X ⊆ 2 has property (s) (Marczewski (Szpilrajn)) iff for every perfect set P ⊆ 2 there exists a perfect set Q ⊆ P such that Q ⊆ X or Q∩X = ∅. Suppose U is a nonprincipal ultrafilter on ω. It is not difficult to see that if U is preserved by Sacks forcing, i.e., it generates an ultrafilter in the generic extension after forcing with the partial order of perfect sets, then U has property (s)...

To complement the property of Q-order of convergence we introduce the notions of Q-superorder and Q-suborder of convergence. A new definition of exact Q-order of convergence given in this note generalizes one given by Potra. The definitions of exact Q-superorder and exact Q-suborder of convergence are also introduced. These concepts allow the characterization of any sequence converging with Q-o...

We prove the following conjecture due to Bryant Mathews (2008). Let Qi be the orthogonal grassmannian of totally isotropic i-planes of a non-degenerate quadratic form q over an arbitrary field (where i is an integer satisfying 1 ≤ i ≤ m := [(dim q)/2]). Assume that for a given i, the form q has the following property (possessed by the generic quadratic form): the degree of each closed point on ...

Consider an operator dQ(f) = d dxk (Q(x)f(x)) where Q(x) is some fixed polynomial of degree k. One can easily see that dQ has exactly one polynomial eigenfunction pn(x) in each degree n ≥ 0 and its eigenvalue λn,k equals (n+k)! n! . A more intriguing fact is that all zeros of pn(x) lie in the convex hull of the set of zeros to Q(x). In particular, if Q(x) has only real zeros then each pn(x) enj...

A word w over a *nite alphabet is said to be n-collapsing if for an arbitrary *nite automaton A = 〈Q; −·−〉, the inequality |Q ·w|6 |Q| − n holds provided that |Q · u|6 |Q| − n for some word u (depending on A). We give an algorithm to test whether a word is 2-collapsing. To this aim we associate to every word w a *nite family of *nitely generated subgroups in *nitely generated free groups and pr...