Let G be a simple connected graph with degree sequence (d1; d2; : : :; dn) where
= d1  d2 


 dn =  > 0 and let  1   2 


  n1 >  n = 0 be the Laplacian eigenvalues of G. Let Kf(G) = n Pn1 i=1 1  i and  (G) = Qn1 i=1  i denote the Kirchho index and the number of spanning trees of G, respectively. In this paper we establish several lower bounds for Kf(G) in terms of  (G), the order, the size and maximum degree of G.

Let R be a commutative ring with identity. We give a new generalization to prime ideals called ,-prime ideal. A proper ideal P of R is called an ,-prime ideal if for all a,b in R with ab 2 P, then a 2 P or , (b) 2 P where ,2 End(R). We study some properties of ,-prime ideals analogous to prime ideals. We give some characterizations for such generalization and we prove that the intersection of all ,-primes in a ring R is the set of all ,-nilpotent elements in R. Finally, we give new versions of some famous theorems about prime ideals including ,-integral domains and ,-, elds.

In this paper, we solve the quadratic -functional equations 2f(x) + 2f(y) = f(x + y) +  􀀀 2f( (x 􀀀 y)); (0. 1) where  is a  xed non-Archimedean number with  􀀀 2 6= 3. Using the  xed point method and the direct method, we prove the Hyers-Ulam stability of the functional equation (0. 1) in non-Archimedean Banach spaces.

This study investigates the performance of the  rst reinforcement layer depth for sand subbase of a road or construction by plate load laboratory tests. Unreinforced and reinforced experiments on di erent reinforcement types were made by changing depth ratio of the  rst reinforcement layer. One type of geotextile and two di erent geogrid specimens were used in the research. 'Load-settlement curves' and 'Bearing Ratios' were studied by measuring the results of di erent settlement ratios. Finally, laboratory measurements of unreinforced and reinforced soils using geotextile reinforcement were compared with Finite-Element Model (FEM) analyses modeled in similar conditions. The results demonstrated the e ects of di erent types of reinforcements for di erent  rst reinforcement layer locations. The number of reinforcement layers was another parameter that a ected the bearing ratio along with the  rst reinforcement layer depths. It was also observed that the bearing ratio (BR) and load-settlement behavior changed signi cantly with the  rst reinforcement depth and settlements. E ects on failure modes for unreinforced and reinforced sand soils were compared for each test.

In this research paper, a numerical method is presented for solving second-order hybrid fuzzy differential equations by using fuzzy Taylor expansion under generalized Hukuhara di erentiability and also with convergence theorem. Also, the method is illustrated by solving several numerical examples. The  nal results showed that the solution of the second-order hybrid fuzzy di erential equations.

Background: Hypoxia caused by exercise condition may alter PGC-1 and HIF-1 expression in cardiac cells. Both factors are transcriptional factors which play a role in conserving energy during metabolic process. However, there is less information about the effects of long term moderate intensity exercise such as swimming on PGC-1 and HIF-1 expression in cardiac cells. Objectives: Aim of this study is to observe the effect of force swimming exercise on the expression PGC-1 and HIF-1 gene expression in cardiac muscle of mice. Methods: Male BALBc mice were divided into two groups (n = 10): control (C) and swimming (ST). ST group was subjected to moderate intensity swimming for 4 weeks, 30 minutes/day for 5 days. Hearts of animal models were preserved for cardiac morphology study using HE staining, PGC-1 and HIF-1 mRNA expression study. Results: We observed that cardiac PGC-1 expression in ST was significantly increased 1. 5 fold compared with control group (0. 64 vs 0. 44; CI = 95%,-0. 378-0. 019; P = 0. 035), but expression of HIF-1 was significantly decreased 0. 4 times lower than control group (0. 35 vs 0. 52; CI = 95%, 0. 27-0. 31; P = 0, 025). Conclusions: Taken together, ST group increased the expression of PGC-1 but decreased the expression of HIF-1 in mice cardiac muscle in response to chronic hypoxia condition.