Let A ⊂ R, d ≥ 2, be a compact convex set and let μ = ̺0 dx be a probability measure on A equivalent to the restriction of Lebesgue measure. Let ν = ̺1 dx be a probability measure on Br := {x : |x| ≤ r} equivalent to the restriction of Lebesgue measure. We prove that there exists a mapping T such that ν = μ◦T−1 and T = φ ·n, where φ : A → [0, r] is a continuous potential with convex sub-level set...

Given a subvariety V of the complex algebraic torus G n m defined by polynomials of total degree at most d and a power map φ : G n m → G n m , the points x whose forward orbits O φ (x) belong to V form its stable subvariety S(V, φ). The main result of the paper provides an upper bound T = T (n, d, φ) for the number of iterations of the power map φ required to " cut off " the points of V that do...

Let BN be the unit ball in the N-dimensional complex space, for ψ, a holomorphic function in BN , and φ, a holomorphic map from BN into itself, the weighted composition operator on the weighted Hardy space H(β,BN ) is given by (Cψ,φ) f = ψ(z) f (φ(z)), where f ∈H2(β,BN ). This paper discusses the spectrum of Cψ,φ when it is compact on a certain class of weighted Hardy spaces and when the compos...

We show in this paper that the module structure and the orthogonality structure of a Hilbert C∗-module determine its inner product structure. Let A be a C∗-algebra, and E and F be Hilbert A-modules. Assume Φ : E → F is an A-module map satisfying 〈Φ(x),Φ(y)〉A = 0 whenever 〈x, y〉A = 0. Then Φ is automatically bounded. In case Φ is bijective, E is isomorphic to F . More precisely, let JE be the cl...

We consider the Schrödinger map initial-value problem { ∂tφ = φ×∆φ on R × R; φ(0) = φ0, where φ : R × R → S →֒ R is a smooth function. In all dimensions d ≥ 2, we prove that the Schrödinger map initial-value problem admits a unique global smooth solution φ ∈ C(R : H∞ Q ), Q ∈ S, provided that the data φ0 ∈ H∞ Q is smooth and satisfies the smallness condition ‖φ0−Q‖Ḣd/2 ≪ 1. We prove also that th...

Let (X, T ) and (Y, S) be two subshifts so that Y is a factor of X. For any asymptotically sub-additive potential Φ on X and a = (a, b) ∈ R with a > 0, b ≥ 0, we introduce the notions of a-weighted topological pressure and a-weighted equilibrium state of Φ. We setup the weighted variational principle. In the case that X, Y are full shifts with one-block factor map, we prove the uniqueness and G...

A map φ : K → R2 of a graph K is approximable by embeddings, if for each ε > 0 there is an ε-close to φ embedding f : K → R2. Analogous notions were studied in computer science under the names of cluster planarity and weak simplicity. In this survey we present criteria for approximability by embeddings (P. Minc, 1997, M. Skopenkov, 2003) and their algorithmic corollaries. We introduce the van K...

Let (X, T) and (Y, S) be two subshifts so that Y is a factor of X. For any asymptotically sub-additive potential Φ on X and a = (a, b) ∈ R 2 with a > 0, b ≥ 0, we introduce the notions of a-weighted topological pressure and a-weighted equilibrium state of Φ. We setup the weighted variational principle. In the case that X, Y are full shifts with one-block factor map, we prove the uniqueness and ...

Let X be a compact metric space and let f : X → X be continuous. It is well known that a given discrete dynamical system (X, f) induces a map Φ on the space F(X) of fuzzy sets. In this paper we study relations between sizes of topological entropies of a given crisp dynamical system (X, f) and its fuzzy counterpart (F(X),Φ). Among another things, we present a constructive proof of the fact that ...

Let Mφ denote the set of Borel probability measures invariant under a topological action φ on a compact metrizable space X. For a continuous function f : X → R, a measure μ ∈ Mφ is called f -maximizing if ∫ f dμ = sup{ ∫ f dm : m ∈Mφ}. It is shown that if μ is any ergodic measure in Mφ, then there exists a continuous function whose unique maximizing measure is μ. More generally, if E is a non-e...