We consider the exit measure of super-Brownian motion with a stable branching mechanism of a smooth domain D of R d. We derive lower bounds for the hitting probability of small balls for the exit measure and upper bounds in the critical dimension. This completes the results given by Sheu 20] and generalizes the results of Abraham and Le Gall 2]. We give also the Hausdorr dimension of the exit m...

In the study of modern complex analysis, the quasisymmetric condition on a map is an important topic and in the study of modern dynamical systems, an invariant measure is an important topic. In this paper, we combine these two topics together developing a new interesting topic, symmetric invariant measure. 1. Uniformly symmetric circle endomorphisms Let T = {z ∈ C | |z| = 1} be the unit circle ...

For a measurable space (E,E ), we denote by E+ the set of functions E → [0,∞] that are E → B[0,∞] measurable. It can be proved that if I : E+ → [0,∞] is a function such that (i) f = 0 implies that I(f) = 0, (ii) if f, g ∈ E+ and a, b ≥ 0 then I(af + bg) = aI(f) + bI(g), and (iii) if fn is a sequence in E+ that increases pointwise to an element f of E+ then I(fn) increases to I(f), then there a ...

In the present paper we shall propose a new measure of the nonclassical distance [1]. The proposed modification is based on the following considerations. If ρ1 and ρ2 are density operators, and F (ρ1, ρ2) is the corresponding fidelity, then from the inequalities [2] 2(1 − √ F (ρ1, ρ2)) ≤ ||ρ1 − ρ2||1 ≤ 2[1− (F (ρ1, ρ2))] 1 2 it is evident that the quantity φ(ρ) = sup ρcl F (ρcl, ρ) can be used ...

Proof of Theorem 1. We first establish the following Lemma. Lemma 1. Consider random variables X ∈ R and Y ∈ R. Let f Y |X be the Radon-Nikodym derivatives of probability measure P θ Y |X with respect to arbitrary measures QY provided that P θ Y |X QY . θ ∈ R is a parameter. f Y is the Radon-Nikodym derivatives of probability measure P θ Y with respect to QY provided that P θ Y QY . Note that i...

Let X be a space and F a family of 0, 1-valued functions on X. Vapnik and Chervonenkis showed that if F is “simple” (finite VC dimension), then for every probability measure μ on X and ε > 0 there is a finite set S such that for all f ∈ F , ∑ x∈S f(x)/|S| = [ ∫ f(x)dμ(x)]± ε. Think of S as a “universal ε-approximator” for integration in F . S can actually be obtained w.h.p. just by sampling a f...

Let the invariant probability measures for an orientation-reversing weakly expanding map of the interval [0, 1] be partially ordered by majorization. The minimal elements of the resulting poset are shown to be convex combinations of Dirac measures supported on two adjacent fixed points. A consequence is that if f : [0, 1] → R is strictly convex, then either its minimizing measure is unique and ...

We analyze a class of weakly differentiable vector fields F : R → R with the property that F ∈ L∞ and div F is a Radon measure. These fields are called bounded divergencemeasure fields. The primary focus of our investigation is to introduce a suitable notion of the normal trace of any divergence-measure field F over the boundary of an arbitrary set of finite perimeter, which ensures the validit...

In this note, I will discuss a possible relation between the Mahler measure of the colored Jones polynomial and the volume conjecture. In particular , I will study the colored Jones polynomial of the figure-eight knot on the unit circle. I will also propose a method to prove the volume conjecture for satellites of the figure-eight knot. 1. Mahler measure Let f (t) be a (non-zero) Laurent polyno...