Ersin Aslan

A numerical approach with error estimation to solve general integro-differential-difference equations using Dickson polynomials

In this paper, a matrix method based on the Dickson polynomials and collocation points is introduced for the numerical solution of linear integro-differential-difference equations with variable coefficients under the mixed conditions. In addition, in order to improve the numerical solution, an error analysis technique relating to residual functions is performed. Some linear and nonlinear numerical examples are given to illustrate the accuracy and applicability of the method. Eventually, the obtained results are discussed according to the parameter-α of Dickson polynomials and the residual error estimation.

Multi-level reranking approach for bug localization

Bug fixing has a key role in software quality evaluation. Bug fixing starts with the bug localization step, in which developers use textual bug information to find location of source codes which have the bug. Bug localization is a tedious and time consuming process. Information retrieval requires understanding the programmes goal, coding structure, programming logic and the relevant attributes of bug. Information retrieval IR based bug localization is a retrieval task, where bug reports and source files represent the queries and documents, respectively. In this paper, we propose BugCatcher, a newly developed bug localization method based on multi-level re-ranking IR technique. We evaluate BugCatcher on three open source projects with approximately 3400 bugs. Our experiments show that multi-level reranking approach to bug localization is promising. Retrieval performance and accuracy of BugCatcher are better than current bug localization tools, and BugCatcher has the best Top N, Mean Average Precision MAP and Mean Reciprocal Rank MRR values for all datasets.

The Average Lower Connectivity of Graphs

For a vertex of a graph , the lower connectivity, denoted by , is the smallest number of vertices that contains and those vertices whose deletion from produces a disconnected or a trivial graph. The average lower connectivity denoted by is the value . It is shown that this parameter can be used to measure the vulnerability of networks. This paper contains results on bounds for the average lower connectivity and obtains the average lower connectivity of some graphs.

Edge-Neighbor-Rupture Degree of Graphs

The edge-neighbor-rupture degree of a connected graph is defined to be , where is any edge-cut-strategy of , is the number of the components of , and is the maximum order of the components of . In this paper, the edge-neighbor-rupture degree of some graphs is obtained and the relations between edge-neighbor-rupture degree and other parameters are determined.

Neighbor Isolated Tenacity of Graphs

The tenacity of a graph is a measure of the vulnerability of a graph. In this paper we investigate a refinement that involves the neighbor isolated version of this parameter. The neighbor isolated tenacity of a noncomplete connected graph G is defined to be NIT(G) = min {|X |+ c(G/X) / i(G/X) , i(G/X) ≥ 1} where the minimum is taken over all X , the cut strategy of G , i (G /X )is the number of components which are isolated vertices of G /X and c (G /X ) is the maximum order of the components of G /X . Next, the relations between neighbor isolated tenacity and other parameters are determined and the neighbor isolated tenacity of some special graphs are obtained. Moreover, some results about the neighbor isolated tenacity of graphs obtained by graph operations are given.

A Numerical Approach Technique for Solving Generalized Delay Integro-Differential Equations with Functional Bounds by Means of Dickson Polynomials

In this study, we have considered the linear classes of differential-(difference), integro-differential-(difference) and integral equations by constituting a generalized form, which contains proportional delay, difference, differentiable difference or delay. To solve the generalized form numerically, we use the efficient matrix technique based on Dickson polynomials with the parameter-α along with the collocation points. We also encode the useful computer program for susceptibility of the technique. The residual error analysis is implemented by using the residual function. The consistency of the technique is analyzed. Also, the numerical results illustrated in tables and figures are compared.

The average scattering number of graphs

The scattering number of a graph is a measure of the vulnerability of a graph. In this paper we investigate a refinement that involves the average of a local version of the parameter. If v is a vertex in a connected graph G , then s c v (G ) = max { ω (G − S v ) − | S v | }, where the maximum is taken over all disconnecting sets S v of G that contain v . The average scattering number of G denoted by s c av (G ), is defined as scav (G) = Σv ∈ V(G) scv(G) / n , where n will denote the number of vertices in graph G . Like the scattering number itself, this is a measure of the vulnerability of a graph, but it is more sensitive. Next, the relations between average scattering number and other parameters are determined. The average scattering number of some graph classes are obtained. Moreover, some results about the average scattering number of graphs obtained by graph operations are given.

The average lower reinforcement number of a graph

Let G = (V (G ),E (G )) be a simple undirected graph. The reinforcement number of a graph is a vulnerability parameter of a graph. We have investigated a refinement that involves the average lower reinforcement number of this parameter. The lower reinforcement number , denoted by r e ∗ (G ), is the minimum cardinality of reinforcement set in G that contains the edge e ∗ of the complement graph G . The average lower reinforcement number of G is defined by r av (G)=1/|E(G )| ∑e* * ∈ E (G ) r e* (G ) .In this paper, we define the average lower reinforcement number of a graph and we present the exact values for some well−known graph families.

Weak-Rupture Degree of Graphs

The rupture degree of a graph is a measure of the vulnerability of a graph. In this paper we investigate a refinement that involves the weak version of the parameter. The weak-rupture degree of a connected graph G is defined to be Rω(G) = max⁡{ ω(G−S)−| S |−me(G−S) : S ⊆ V (G),ω(G−S) > 1 } where ω(G−S) is the number of the components of (G−S) and me(G−S) is the number of edges of the largest component of G−S. Like the rupture degree itself, this is a measure of the vulnerability of a graph, but it is more sensitive. This paper, the weak-rupture degree of some special graphs are obtained and some bounds of the weak-rupture degree are given. Moreover some results about the weak-rupture degree of graphs obtained by graph operations are given.