Application of collocation method for solving a parabolic‐hyperbolic free boundary problem which models the growth of tumor with drug application

In this article, we want to solve a free boundary problem which models tumor growth with drug application. This problem includes five time dependent partial differential equations. The tumor considered in this model consists of three kinds of cells, proliferative cells, quiescent cells, and dead cells. Three different first‐order hyperbolic equations are given that describe the evolution of cells and other two second‐order parabolic equations describe the diffusion of nutrient and drug concentration. We solve the problem using the collocation method. Then, we prove stability and convergence of method. Also, some examples are considered to show the efficiency of method. Copyright © 2016 John Wiley & Sons, Ltd.     

Characterization of graphs with hall number 2

Hall's condition is a simple requirement that a graph G and list assignment L must satisfy if G is to have a proper L‐colouring. The Hall number of G is the smallest integer m such that whenever the lists on the vertices each has size at least m and Hall's condition is satisfied a proper L‐colouring exists. Hilton and P.D. Johnson introduced the parameter and showed that a graph has Hall number 1 if and only if every block is a clique. In this paper we give a forbidden‐induced‐subgraph characterization of graphs with Hall number 2. © 2003 Wiley Periodicals, Inc. J Graph Theory 45: 81–100, 2004     

Importance of RNA secondary structure information for yeast donor and acceptor splice site predictions by neural networks

Previously, Patterson et al. showed that mRNA structure information aids splice site prediction in human genes [Patterson, D.J., Yasuhara, K., Ruzzo, W.L., 2002. Pre-mRNA secondary structure prediction aids splice site prediction. Pac. Symp. Biocomput. 7, 223-234]. Here, we have attempted to predict splice sites in selected genes of Saccharomyces cerevisiae using the information obtained from the secondary structures of corresponding mRNAs. From Ares database, 154 genes were selected and their structures were predicted by Mfold. We selected a 20-nucleotide window around each site, each containing 4 nucleotides in the exon region. Based on whether the nucleotide is in a stem or not, the conventional four-letter nucleotide alphabet was translated into an eight-letter alphabet. Two different three-layer-based perceptron neural networks were devised to predict the 5' and 3' splice sites. In case of 5' site determination, a network with 3 neurons at the hidden layer was chosen, while in case of 3' site 20 neurons acted more efficiently. Both neural nets were trained applying Levenberg-Marquardt backpropagation method, using half of the available genes as training inputs and the other half for testing and cross-validations. Sequences with GUs and AGs non-sites were used as negative controls. The correlation coefficients in the predictions of 5' and 3' splice sites using eight-letter alphabet were 98.0% and 69.6%, respectively, while these values were 89.3% and 57.1% when four-letter alphabet is applied. Our results suggest that considering the secondary structure of mRNA molecules positively affects both donor and acceptor site predictions by increasing the capacity of neural networks in learning the patterns.     

The second kind Chebyshev–Newton–Cotes quadrature rule (open type) and its numerical improvement

The weighted Newton–Cotes quadrature rules of open type are denoted by

where w(x) is a positive function and is the step size. Various cases can be selected for the weight function of the above formula. In this paper, we consider as the main weight function and study the general formula:

The precision degree of the above formula is n + 1 for even n’s and is n for odd n’s but if one considers its upper and lower bounds as two additional variables, a nonlinear system will be derived whose solution improves the precision degree of above formula up to degree n + 2 numerically. In this way, some examples are given to show the numerical superiority of our idea.