### G. Durán-Meza, J. López-García and J.L. del Río-Correa

## Abstract

In this work is presented a pedagogical point of view of multifractal analysis deoxyribonucleic acid (DNA) sequences is presented. The DNA sequences are formed by 4 nucleotides (adenine, cytosine, guanine, and tymine). Following Jeffrey’s paper we associated a simple contractive function to each nucleotide, and constructed the Hutchinson’s operator *W*, which was used to build covers of different sizes of the unitary square *Q*, thus *W ^{k}*(

*Q*) is a cover of

*Q*, conformed by 4

*squares*

^{k}*Q*of size 2

_{k}*, as each*

^{−k}*Q*corresponds to a unique subsequence of nucleotides with length

_{k}*k*:

*b*

_{1}

*b*

_{2}

*...b*. Besides, it is obtained the optimal cover

_{k}*C*to the fractal

_{k}*F*generated for each DNA sequence was obtained. We made a multifractal decomposition of

*C*in terms of the sets

_{k}*J*conformed by the

_{α}*Q*’s with the same value of the Holder exponent

_{k}*α*, and determined

*f*(

*α*), the Hausdorff dimension of

*J*, using the curdling theorem.

_{α}### Riaz Ahmad, Asma Farooqi, Jiazhong Zhang and Nasir Ali

## Abstract

A steady flow of a power law fluid through an artery with a stenosis has been analyzed. The equation governing the flow is derived under the assumption of mild stenosis. An exact solution of the governing equation is obtained, which is then used to study the effects of various parameters of interest on axial velocity, resistance to flow and shear stress distribution. It is found that axial velocity increases while resistance to flow decreases when going from shear-thinning to shear-thickening fluid. Moreover, the magnitude of shear stress decreases by increasing the tapering parameter. This problem was already addressed by Nadeem et al. [14], but the results presented by them were erroneous due to a mistake in the derivation of the governing equation of the flow. This mistake is highlighted in the "Formulation of the Problem" section.

### Maximilian Voit and Hildegard Meyer-Ortmanns

## Abstract

We consider a heteroclinic network in the framework of winnerless competition, realized by generalized Lotka-Volterra equations. By an appropriate choice of predation rates we impose a structural hierarchy so that the network consists of a heteroclinic cycle of three heteroclinic cycles which connect saddles on the basic level. As we have demonstrated in previous work, the structural hierarchy can induce a hierarchy in time scales such that slow oscillations modulate fast oscillations of species concentrations. Here we derive a Poincaré map to determine analytically the number of revolutions of the trajectory within one heteroclinic cycle on the basic level, before it switches to the heteroclinic connection on the second level. This provides an understanding of which parameters control the separation of time scales and determine the decisions of the trajectory at branching points of this network.

### R. A. Mundewadi and S. Kumbinarasaiah

## Abstract

A numerical method is developed for solving the Abel′s integral equations is presented. The method is based upon Hermite wavelet approximations. Hermite wavelet method is then utilized to reduce the Abel′s integral equations into the solution of algebraic equations. Illustrative examples are included to demonstrate the validity, efficiency and applicability of the proposed technique. Algorithm provides high accuracy and compared with other existing methods.

### Shanu Goyal, Pravin Garg and Vishnu Narayan Mishra

## Abstract

In this paper, we define new graph operations *F*-composition *F* (*G*)[*H*], where *F* (*G*) be one of the symbols *S*(*G*)*,M*(*G*)*,Q*(*G*)*,T*(*G*),Λ(*G*),Λ[*G*]*,D*
_{2}(*G*)*,D*
_{2}[*G*]. Further, we give some results for the Wiener indices of the these graph operations.

### Guillermo de Anda-Jáuregui, Cristobal Fresno, Diana García-Cortés, Jesús Espinal Enríquez and Enrique Hernández-Lemus

## Abstract

Biological systems exhibit unique phenotypes as the result of the expression of a genomic program. The regulation of this program is a complex phenomenon, wherein different regulatory mechanisms are involved. The deregulation of this program is at the centre of the emergence of diseases such as breast cancer. In particular, it has been observed that coregulation patterns between physically distant genes are lost in breast cancer.

In this work, we present a systematic study of chromosome-wide gene coregulation patterns in breast cancer as inferred by information theoretical measures over large (whole-genome expression in several hundred transcriptomes) experimental data corpora. We analyzed the chromosomal distance decay of correlations and found it to be with fat-tail distribution in breast cancer while being fundamentally constant in nontumour samples.

After model discrimination analyses, we concluded that the behaviour of the breast cancer distributions belongs to an intermediate regime between power law and Weibull distributions, with distinctive contributions corresponding to different chromosomes. This behaviour may have biological implications in terms of the organization of the gene regulatory program, and the changes found in this program between health and disease.

### Veniamin Smirnov and Dimitri Volchenkov

## Abstract

Inhomogeneous density of states in a discrete model of Standard & Poor’s 500 phase space leads to inequitable predictability of market events. Most frequent events might be efficiently predicted in the long run as expected from Mean reversion theory. Stocks have different mobility in phase space. Highly mobile stocks are associated with less unsystematic risk. Less mobile stocks might be cast into disfavor almost indefinitely. Relations between information components in Standard & Poor’s 500 phase space resemble of those in unfair coin tossing.

### Leonid Bunimovich, Dallas Smith and Benjamin Webb

## Abstract

The method of isospectral network reduction allows one the ability to reduce a network while preserving the network’s spectral structure. In this paper we describe a number of recent applications of the theory of isospectral reductions. This includes finding hidden structures, specifically latent symmetries, in networks, uncovering different network hierarchies, and simultaneously determining different network cores. We also specify how such reductions can be interpreted as dynamical systems and describe the type of dynamics such systems have. Additionally, we show how the recent theory of equitable decompositions can be paired with the method of isospectral reductions to decompose networks.

### Jianzhang Wu, Jiabin Yuan and Wei Gao

## Abstract

In software definition networks, we allow transmission paths to be selected based on real-time data traffic monitoring to avoid congested channels. Correspondingly, this motivates us to study the existence of fractional factors in different settings. In this paper, we present several extend sufficient conditions for a graph admits ID-Hamiltonian fractional (*g, f* )factor. These results improve the conclusions originally published in the study by Gong et al. [2].