Theta-point exponent for polymer chain in random media

Using field-theoretic arguments for self-avoiding walks on dilute lattices with site occupation concentrationp, we show that the-point size exponent _p ⁰ of polymer chains remains unchanged for small disorder concentration (p>p _c ). At the percolation thresholdp=p _c , using a Flory-type approximation, we conjecture that _pc ⁰ =5/(d _B +7), whered _B is the percolation backbone dimension. It shows that the upper critical dimensionality for the-point transition atp=p _c shifts to a dimensiond _c >3. We also propose that the-point varies practically linearly withp for 1>pp _c .     

Fractal structure in base-catalyzed silica aerogels examined by TEM, SAXS and porosimetry

A fractal analysis of three base catalyzed silica aerogels was performed using different experimental techniques: image analysis of electron micrographs, SAXS and study of pore size distribution determined from nitrogen adsorption isotherms. The aerogels appeared to exhibit self-similar properties over the range from 3–10 to 50–130 nm. The values of mass fractal dimension varied from 1.75 to 2.05 depending on the reactants concentration (TEOS, H₂O) and were found to be similar irrespective of the method applied.     

Contractive Markov System with Constant Probabilities

In this paper, we give a simple proof that a contractive Markov system Ref. 7 with constant probabilities and a compact state space has a unique stationary initial distribution in an irreducible case and an exponential rate of convergence to the stationary initial distribution in an aperiodic case.     

Heterochromatin and Complexity: A Theoretical Approach

Heterochromatin represents 30% of eukaryotic genome in Drosophila and 15% in humans. Despite extensive research spanning many decades, its evolutionary significance, as well as the forces that guarantee its maintenance, are still elusive. Many theoretical and experimental approaches have led researchers to propose several conceptual frameworks to elucidate the nature of this huge mysterious genetic material and its spreading in all eukaryotic genomes. Junk DNA as well as selfish genetic material are two examples of such attempts, but several lines of evidence suggest that such explanations are incomplete. In fact, if the selfish DNA hypothesis does not explain the mapping of genetic functions in heterochromatin, then the junk DNA hypothesis is incomplete in describing both emergence of genetic functions and their maintenance in the eukaryotic heterochromatin. Recent developments in the physics of complex systems and mathematical concepts such as fractals provide new conceptual clues to answer several basic questions concerning the emergence of heterochromatin in eukaryotic genomes, its evolutionary significance, the forces that guarantee its maintenance, and its peculiar behavior in the eukaryotic cell. The aim of this paper is to provide a new theoretical framework for the heterochromatin, considering such genetic material in physical terms as a complex adaptive system. We apply some computer calculations to demonstrate the nonlinearity of the flux of genetic information along the phylogenic tree. Fractal dimensions of representative heterochromatic sequences are provided. A theory is proposed in which heterochromatin is considered a system that evolves in a self-organized manner at the edge of cellular and environmental chaos.     

Fractal Dimensions for Some Increments of the Uniform Empirical Process

In this paper, we investigate the limiting behavior of increments of the uniform empirical process. More precisely, we are concerned by sets of exceptional oscillation points related to large and small increments. We prove that these sets are random fractals and evaluate their Hausdorff dimensions. This work is a complement to the previous investigations carried out by Deheuvels and Mason⁽⁶⁾ where Csörg–Révész–Stute-type increments are studied.     

Analyte-receptor binding kinetics for biosensor applications

The diffusion-limited binding kinetics of antigen (analyte), in solution with antibody (receptor) immobilized on a biosensor surface, is analyzed within a fractal framework. Most of the data presented is adequately described by a single-fractal analysis. This was indicated by the regression analysis provided by Sigmaplot. A single example of a dual-fractal analysis is also presented. It is of interest to note that the binding-rate coefficient (k) and the fractal dimension (D_f) both exhibit changes in the same and in the reverse direction for the antigen-antibody systems analyzed. Binding-rate coefficient expressions, as a function of the D_f developed for the antigen-antibody binding systems, indicate the high sensitivity of thek on the D_f when both a single- and a dual-fractal analysis are used. For example, for a single-fractal analysis, and for the binding of antibody Mab 0.5β in solution to gpl20 peptide immobilized on a BIAcore biosensor, the order of dependence on the D_f was 4.0926. For a dual-fractal analysis, and for the binding of 25-100 ng/mL TRITC-LPS (lipopolysaccharide) in solution with polymyxin B immobilized on a fiberoptic biosensor, the order of dependence of the binding-rate coefficients, k₁ and k₂ on the fractal dimensions, D_f1 and D_f2, were 7.6335 and-11.55, respectively. The fractional order of dependence of thek(s) on the D_f(s) further reinforces the fractal nature of the system. Thek(s) expressions developed as a function of the D_f(s) are of particular value, since they provide a means to better control biosensor performance, by linking it to the heterogeneity on the surface, and further emphasize, in a quantitative sense, the importance of the nature of the surface in biosensor performance.     

The upper and lower quantization coefficient for Markov‐type measures

We study the asymptotic quantization error for Markov‐type measures μ on a class of ratio‐specified graph directed fractals E . Assuming a separation condition for E , we show that the quantization dimension for μ of order r exists and determine its exact value in terms of spectral radius of a related matrix. We prove that the ‐dimensional lower quantization coefficient for μ is always positive. Moreover, we establish a necessary and sufficient condition for the ‐dimensional upper quantization coefficient for μ to be finite.     

Designing Fractal Nanostructured Biointerfaces for Biomedical Applications

Fractal structures in nature offer a unique “fractal contact mode” that guarantees the efficient working of an organism with an optimized style. Fractal nanostructured biointerfaces have shown great potential for the ultrasensitive detection of disease‐relevant biomarkers from small biomolecules on the nanoscale to cancer cells on the microscale. This review will present the advantages of fractal nanostructures, the basic concept of designing fractal nanostructured biointerfaces, and their biomedical applications for the ultrasensitive detection of various disease‐relevant biomarkers, such microRNA, cancer antigen 125, and breast cancer cells, from unpurified cell lysates and the blood of patients.     

On singularity and fine spectral structure of random continued fractions

We study properties of the distribution of a random variable of the continued fraction form where are independent and not necessarily identically distributed random variables. We prove the singularity of and study the fine spectral structure of such measures.     

求解非线性方程组的混沌分形方法

混沌分形是动力系统普遍出现的一种现象,牛顿-拉夫森NR(Newton-Raphson)方法是重要的一维及多维迭代技术,其迭代本身对初始点非常敏感,该敏感区是牛顿-拉夫森法所构成的非线性离散动力系统Julia集,在Julia集中迭代函数会呈现出混沌分形现象,提出了一种寻找牛顿-拉夫森函数的Julia点的求解方法,利用非线性离散动力系统在其Julia集出现混沌分形现象的特点,提出了一种基于牛顿-拉夫森法的非线性方程组求解的新方法,计算实例表明了该方法的有效性和正确性.