, LNCS 10388 1 13 2017 [2] J.A. Aledo, L.G. Diaz, S. Martinez, J.C. Valverde, On the periods of parallel dynamical systems, Complexity 2017 (2017), Article ID 7209762, 6 pages. 10.1155/2017/7209762 . Aledo J.A. Diaz L.G. Martinez S. Valverde J.C. On the periods of parallel dynamical systems Complexity 2017 2017 Article ID 7209762 6 10.1155/2017/7209762 [3] J.A. Aledo, L.G. Diaz, S. Martinez, J.C. Valverde, On periods and equilibria of sequential dynamical systems, Inf. Sci. 409–410 (2017) 27–34. 10.1016/j.ins.2017.05.002 . Aledo J.A. Diaz L.G. Martinez S

### Juan A. Aledo, Luis G. Diaz, Silvia Martinez and Jose C. Valverde

### S.O. Pyskunov, Yu.V. Maksimyk and V.V. Valer

strains: ( Δ σ i j ) m = C i j k l ( { Δ ε k l } m − { ( Δ ε k l ) T } m ) . $$\begin{array}{} \displaystyle [{\left( {\Delta {\sigma _{ij}}} \right)_m} = {C_{ijkl}}\left( {{{\left\{ {\Delta {\mkern 1mu} {\varepsilon _{kl}}} \right\}}_m} - {{\left\{ {{{\left( {\Delta {\mkern 1mu} {\varepsilon _{kl}}} \right)}^T}} \right\}}_m}} \right). \end{array}$$ (10) Creep problem solution considering damage accumulation is being executed by means of step-by-step algorithm on the parameter of time. When starting each iteration n of a step m , stress values σ ij are

### Arturo Álvarez-Arenas, Juan Belmonte-Beitia and Gabriel F. Calvo

biological literature, there is a vast range of values for the diffusion and proliferation coefficients. To carry out the estimations, we resort to the following value for the proliferation ρ = 0.2 day −1 , which is in the range [0.01–0.5] day −1 , taken from [ 28 , 54 ] and D = 0.05 mm 2 /day (which is in the range [0.0004–0.1] mm 2 /day) [ 39 ]. Finally, we take α = 1/10 day −1 , L = 85 mm, x 0 = 10 mm, c = c min = 2 ( 1 − β ) , $\begin{array}{} c=c_{\text{min}}=2\sqrt{(1-\beta)}, \end{array} $ M = 0.3, b = 0.005, a = ( c − c 2 − 4 ( 1 − β − V

### Bo Zhao and Hualong Wu

structures (See Farahani et al. [ 1 ], Jamil et al. [ 2 ], Gao et al. [ 3 – 7 ] and Gao and Wang [ 8 – 10 ] for more details). The notation and terminology that were used but undefined in this paper can be found in [ 11 ]. Now, we present some important indices which will be computed in the next section. The Shultz polynomial is denoted as S c ( G , x ) = Σ { u , v } ⊆ V ( G ) ( d ( u ) + d ( v ) ) x d ( u , v ) . $$\begin{array}{} \displaystyle Sc(G,x) = \mathop \Sigma \limits_{\{ u,v\} \subseteq V(G)} (d(u) + d(v)){x^{d(u,v)}}. \end{array}$$ The additively

### Muhammad Kamran Jamil, Mohammad Reza Farahani, Muhammad Imran and Mehar Ali Malik

: ∏ 1 ( G ) = ∏ ν ∈ V ( G ) d ( ν ) 2 ∏ 2 ( G ) = ∏ u ν ∈ E ( G ) d ( u ) ⋅ d ( ν ) $$\begin{array}{} \displaystyle \begin{array}{*{20}{c}} {{\prod _1}(G) = {\prod _{\nu \in V(G)}}d{{(\nu )}^2}} \hfill \\ {{\prod _2}(G) = {\prod _{u\nu \in E(G)}}d(u) \cdot d(\nu )} \hfill \\ \end{array} \end{array}$$ More history and results on Zagreb and multiplicative Zagreb indices can be found in [ 4 ]– [ 10 ]. Recently, Ghorbani and Hosseinzadeh [ 11 ] defined the eccentric versions of Zagreb indices. These are named as third and fourth Zagreb indices and defined

### Mohammad Reza Farahani, Muhammad Kamran Jamil and Muhammad Imran

-dimensional graph of the Titania nanotube, TiO 2 [ m , n ], is shown in Figure 1 , where m and n denotes the number of octagons in a column and the number of octagons in a row of the Titania nanotube. This graph has 2(3n+2)(m+1) vertices and 10 mn + 6 m + 8 n + 4 edges. Fig. 1 2-dimensional model to Titania nanotube TiO 2 [m,n] . By using the orthogonal cuts and cut method of the Titania nanotubes, we can determine all edge cuts of the Titania nanotubes. The edge cut C(e) is an orthogonal cut, such that the set of all edges f ∈ E

### J. Llibre

partial results about the center–focus problem for degenerate centers, see for instance [ 10 ]. The following result about the C ∞ first integrals of all centers is due to Mazzi and Sabatini [ 18 ]. Theorem 2 Any analytic differential system having a center has a local C ∞ first integral defined in its neighborhood. About the analytic integrals of all centers there is the next result prove in [ 15 ]. Theorem 3 Any analytic differential system having a center at the point p has a local analytic first integral defined in a punctured neighborhood of p

### Harishchandra S. Ramane and Raju B. Jummannaver

corresponding to a drug structure with vertex (atom) set V ( G ) and edge (bond) set E ( G ). The edge joining the vertices u and ν is denoted by uν . Thus, if uv ∊ E ( G ) then u and ν are adjacent in G . The degree of a vertex u , denoted by d ( u ), is the number of edges incident to u . Several topological indices such as Estrada index [ 1 ], Zagreb index [ 8 ], PI index [ 10 ], eccentric index [ 11 ], and Wiener index [ 12 ] have been introduced in the literature to study the chemical and pharmacological properties of molecules. Furtula and Gutman [ 3

### Mehdi Alaeiyan, Mohammad Reza Farahani and Muhammad Kamran Jamil

based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges J. Math. Chem 46 1369 1376 10.1007/s10910-009-9520-x [11] M.R. Farahani. (2012), Computing some connectivity indices of nanotubes, Advances in Materials and Corrosion, 1, 57-60. Farahani M.R. 2012 Computing some connectivity indices of nanotubes Advances in Materials and Corrosion 1 57 60 [12] M.R. Farahani. (2013), Fifth geometric-arithmetic index of TURC 4 C 8 ( S ) nanotubes, Journal of Chemica Acta, 2(1), 62-64. Farahani M.R. 2013 Fifth geometric-arithmetic index of TURC

### Agnieszka Podstawczyńska and Scott D. Chambers

inventory for applied atmospheric studies. Atmos. Environ., 43 (8), 1536–1539. 12. Griffiths, A. D., Zahorowski, W., Element, A., & Werczynski, S. (2010). A map of radon flux at the Australian land surface. Atmos. Chem. Phys ., 10 , 8969–8982. 13. Karstens, U., Schwingshackl, C., Schmithusen, D., & Levin, I. (2015). A process-based 222 radon flux map for Europe and its comparison to long-term observations. Atmos. Chem. Phys ., 15 , 12845–12865. DOI: 10.5194/acp-15-12845-2015. 14. Chambers, S. D., Williams, A. G., Zahorowski, W., Griffiths, A