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.     

Dimensions of subsets of Moran fractals related to frequencies of their codings

The Moran fractal considered in this paper is an extension of the self-similar sets satisfying the open set condition. We consider those subsets of the Moran fractal that are the union of an uncountable number of sets each of which consists of the points with their location codes having prescribed mixed group frequencies. It is proved that the Hausdorff and packing dimensions of each of these subsets coincide and are equal to the supremum of the Hausdorff (or packing) dimensions of the sets in the union. An approach is given to calculate their Hausdorff and packing dimensions. The main advantage of our approach is that we treat these subsets in a unified manner. Another advantage of this approach is that the values of the Hausdorff and packing dimensions do not need to be guessed a priori.     

Fractals related to long DNA sequences and complete genomes

In visualizing very long DNA sequences, including the complete genomes of several bacteria, yeast and segments of human genes, we encounter fractal-like patterns underlying these biological objects of prominent importance. The method used here to visualize genomes of organisms may well be used as a convenient tool to trace, e.g., evolutionary relatedness of species. We describe the method and explain the origin of the observed fractal-like patterns.     

Dimensions of fractals in the large

Fractals in the large can be generated as the invariant set of an expansive, iterated function system. A number of dimensions have been introduced and studied for such fractals. In this note we show that these dimensions coincide for large fractals generated by functions with arithmetic expansion factors, and that this common dimension is equal to the dimension of the (small) fractal generated by the inverse functions.     

On languages with non-homogeneous strings of quantifiers

It is shown that every linear string of quantifiers can be replaced by a well-ordered sequence of quantifiers. The results of this paper was part of the author’s Master Thesis, which was submitted to the Hebrew University in August, 1967. The work was done under the guidance of Professor H. Gaifman, whom I thank for his kind guidance, and his help to clarify the exposition in this article.     

Dimensions and singular traces for spectral triples, with applications to fractals

Given a spectral triple , the functionals on of the form a?τ_ω(a|D|^−α) are studied, where τ_ω is a singular trace, and ω is a generalised limit. When τ_ω is the Dixmier trace, the unique exponent d giving rise possibly to a non-trivial functional is called Hausdorff dimension, and the corresponding functional the (d-dimensional) Hausdorff functional.It is shown that the Hausdorff dimension d coincides with the abscissa of convergence of the zeta function of |D|⁻¹, and that the set of α's for which there exists a singular trace τ_ω giving rise to a non trivial functional is an interval containing d. Moreover, the endpoints of such traceability interval have a dimensional interpretation. The functionals corresponding to points in the traceability interval are called Hausdorff-Besicovitch functionals.These definitions are tested on fractals in , by computing the mentioned quantities and showing in many cases their correspondence with classical objects. In particular, for self-similar fractals the traceability interval consists only of the Hausdorff dimension, and the corresponding Hausdorff-Besicovitch functional gives rise to the Hausdorff measure. More generally, for any limit fractal, the described functionals do not depend on the generalized limit ω.     

The study of the fourier series of functions defined on moran fractals

LetE be a Moran fractal andH ^s (E) denote thes-dimensional Hausdorff measure ofE. In this paper, we define a orthonormal and complete system of functions in the Hilbert spaceL ²(E,H ^s ) and prove that partial sums of the Fourier series, with respect to , of each functionf(x)L ¹(E,H ^s ) converge tof(x) atH ^s -a.e.xE. Moreover, the Fourier series off, forfL ^p (E,H ^s ),p1, converges off inL ^p -norm. When Moran fractals degenerate into self-similar fractals, our results well agree with M. Reyes's results.This work is supported in part by the National Natural Science Foundation of China.     

Dimensions of level sets related to tangential dimensions

For level sets related to the tangential dimensions of Bernoulli measures, the Hausdorff and packing dimensions are determined. Our results extend some classical work of Besicovitch and Eggleston.     

The strong Lefschetz property for complete intersections defined by products of linear forms

We prove the strong Lefschetz property for certain complete intersections defined by products of linear forms, using a characterization of the strong Lefschetz property in terms of central simple modules.

     

Open problems and conjectures,optimal rearrangement problems related to discrete loaded strings

In this section we present some open problems and conjectures about some interesting types of difference equations. Please submit your problems and conjectures with all relevant information to G. Ladas.     

Subordination results for classes of analytic functions related to conic domains defined by a fractional operator

Let a fractional operator (n∈N₀={0,1,2,…}, 0?α<1, λ?0) be defined by

     

Property of right units of complete semigroups of binary relations defined by finite XI-semilattices of unions

We study properties of right units of complete semigroups of binary relations defined by finite XI-semilattices of unions.     

Approximation of infinite dimensional fractals generated by integral equations

We present some results concerning fractals generated by an iterated function system in the infinite dimensional space of continuous functions on a compact interval. Namely, we approximate the fractal via a finite approximant set and project this approximant set in two dimensions, in order to “draw” a picture of it.     

Rational approximation to surfaces defined by polynomials in one variable

For a Tychonoff space X, we denote by C _p (X) the space of all real-valued continuous functions on X with the topology of pointwise convergence.

In this paper we prove that:

• If every finite power of X is Lindelöf then C _p (X) is strongly sequentially separable iff X is \({\gamma}\)-set.
• \({B_{\alpha}(X)}\) (= functions of Baire class \({\alpha}\) (\({1 < \alpha \leq \omega_1}\)) on a Tychonoff space X with the pointwise topology) is sequentially separable iff there exists a Baire isomorphism class \({\alpha}\) from a space X onto a \({\sigma}\)-set.
• \({B_{\alpha}(X)}\) is strongly sequentially separable iff \({iw(X)=\aleph_0}\) and X is a \({Z^{\alpha}}\)-cover \({\gamma}\)-set for \({0 < \alpha \leq \omega_1}\).
• There is a consistent example of a set of reals X such that C _p (X) is strongly sequentially separable but B₁(X) is not strongly sequentially separable.
• B(X) is sequentially separable but is not strongly sequentially separable for a \({\mathfrak{b}}\)-Sierpiński set X.

     

Compression of character strings by an adaptive dictionary

A new technique for compression of character strings is presented. The technique is based on the use of a dictionary forest which is built simultaneously with the encoding and decoding. Codes representing substrings are addresses in the dictionary forest. Experimental results show that the length of the text can be reduced more than 50% with no a priori knowledge of the nature of the text.     

On the length of arcs in labyrinth fractals

Monatshefte für Mathematik - Labyrinth fractals are self-similar dendrites in the unit square that are defined with the help of a labyrinth set or a labyrinth pattern. In the case when the...