Some dimensional results for a class of special homogeneous Moran sets

We construct a class of special homogeneous Moran sets, called {m_k}-quasi homogeneous Cantor sets, and discuss their Hausdorff dimensions. By adjusting the value of {m_k}_k?1, we constructively prove the intermediate value theorem for the homogeneous Moran set. Moreover, we obtain a sufficient condition for the Hausdorff dimension of ho- mogeneous Moran sets to assume the minimum value, which expands earlier works.     

Dimensional results for Cartesian products of homogeneous Moran sets

M(J, {m _s * n _s }, {c _s }) be the collection of Cartesian products of two homogenous Moran sets with the same ratios {c _s } where J = [0, 1]×[0, 1]. Then the maximal and minimal values of the Hausdorff dimensions for the elements in M are obtained without any restriction on {m _s n _s } or {c _s }.     

Traces of functions with majorizable derivatives

We study the limiting values (y→+0) of functionsf (x, y), x ε R_n, y > 0, for which ¦?f/?y¦≤M?(y), ¦?f/?x_k¦≤Mψk(y), M=M [f], in the case of arbitrary weight functions. It is shown that the space of traces can be described as the set of all functionsf (x, 0) which satisfy a Lipschitz condition in some metricω(x, x) associated with the weights.     

Dimensions of some fractals defined via the semigroup generated by 2 and 3

We compute the Hausdorff and Minkowski dimension of subsets of the symbolic space Σ _m ={0, ...,m?1}^? that are invariant under multiplication by integers. The results apply to the sets {x∈Σ _m :? k, x _k x _2k ... x _nk =0}, where n ≥ 3. We prove that for such sets, the Hausdorff and Minkowski dimensions typically differ.     

Quasisymmetrically minimal uniform Cantor sets

The uniform Cantor set E(n,c) of Hausdorff dimension 1, defined by a bounded sequence n of positive integers and a gap sequence c, is shown to be minimal for 1-dimensional quasisymmetric maps.     

Hyperbolically Convex Functions,Dimension and Capacity

We study a conformal map ? of the unit disk D onto a hyperbolically convex set in D, in particular the behaviour of ? on the preimage T _? = {z ? ?D:|?(z)| = 1} of ?D. The main problem is how much the Hausdorff dimension can increase for sets on T _?. The case that ?(D) is bounded by full circles is treated in more detail. In this case ? can be written as a composition sequence of mappings onto halfplanes.     

There are no de bruijn sequences of span n with complexity 2n − 1 + n + 1

If s = (s₀, s₁,…, s_2ⁿ?1) is a binary de Bruijn sequence of span n, then it has been shown that the least length of a linear recursion that generates s, called the complexity of s and denoted by c(s), is bounded for n ? 3 by 2^{n ? 1} + n ? c(s) ? 2ⁿ ?1. A numerical study of the allowable values of c(s) for 3 ? n ? 6 found that all values in this range occurred except for 2^n?1 + n + 1. It is proven in this note that there are no de Bruijn sequences of complexity 2^n?1 + n + 1 for all n ? 3.     

Relative probabilities of majority winners under partial information

Suppose the only observable characteristic of each of n random variables that is uniformly distributed on the six rankings of objects in a three-element set is its first-ranked object. Let ?(n₁,n₂,n₃) be the probability that one of the three objects has majorities over the other two within the rankings when n_j of the n rankings have the jth object in first place. It is assumed that n is odd, so that ?(n₁,n₂,n₃)=1 only if n_j≥(n+1)/2 for some j.It is shown that $\text{?(a+1,b,c)<?(a,b+1,c) if a <b,a ≤ c ≤ b+1}$ and max {b,c}≤(n?1)/2. It follows from this that ? is minimized for fixed n if and only if n_j?n_k≤1 for all j,k? {1,2,3}. However, ? does not necessarily increase when two of its arguments get farther apart. For example, ?(b,b,3)>?(b?1,b+1,3) for b≥28, and ?(b,b,2b?1)>?(b?1,b+1,2b?1) for b≥12.     

Multiparameter acoustic imaging in the born approximation

Samples of biological tissue are modelled as inhomogeneous fluids with density ?(X) and sound speed c(x) at point x. The samples are contained in the sphere |x| ? δ and it is assumed that ?(x) ? ?₀ = 1 and c(x) ? c₀ = 1 for |x| ? δ, and |γ_n(x)| ? 1 and |?γ?(x)| ? 1 where γ?(x) = ?(x) ? 1 and γ_n(x) = c^?2(x) ? 1. The samples are insonified by plane pulses s(x · θ₀ – t) where x = |θ₀| = 1 and the scattered pulse is shown to have the form |x|^?1 e_s(|x| – t, θ, θ₀) in the far field, where x = |x| θ. The response e_s(τ, θ, θ₀) is measurable. The goal of the work is to construct the sample parameters γn and γ? from e_s(τ, θ, θ₀) for suitable choiches of s, θ and θ₀. In the limiting case of constant density: γ?(x)_? 0 it is shown that Where δ represents the Dirac δ and S² is the unit sphere |θ| = 1. Analogous formulas, based on two sets of measurements, are derived for the case of variable c(x) and ?(x).     

Thue-Morse at multiples of an integer

Let t=(tn)_n?0 be the classical Thue-Morse sequence defined by , where s₂ is the sum of the bits in the binary representation of n. It is well known that for any integer k?1 the frequency of the letter “1” in the subsequence t₀,tk,t2k,… is asymptotically 1/2. Here we prove that for any k there is an n?k+4 such that tkn=1. Moreover, we show that n can be chosen to have Hamming weight ?3. This is best in a twofold sense. First, there are infinitely many k such that tkn=1 implies that n has Hamming weight ?3. Second, we characterize all k where the minimal n equals k, k+1, k+2, k+3, or k+4. Finally, we present some results and conjectures for the generalized problem, where s₂ is replaced by sb for an arbitrary base b?2.