An efficient numerical approximation for the linear class of Fredholm integro-differential equations based on Cattani’s method

This paper provides with a generalization of the work by Cattani (Math. Probl. Eng. (2008) 1–24), who has introduced the connection coefficients of the Shannon wavelets. We apply the Shannon wavelets approximation based on Cattani’s connection coefficients together the collocation points for solving the linear Fredholm integro-differential equations. Finally, numerical results are included to demonstrate the validity and applicability of the method and some comparisons are made with existing results.     

An extension of Babbage’s criterion for primality

Let n > 1 and k > 1 be positive integers. We show that if $$\left( {\begin{array}{*{20}c} {n + m} \\ n \\ \end{array} } \right) \equiv 1 (\bmod k)$$ for each integer m with 0 ≤ m ≤ n ? 1, then k is a prime and n is a power of this prime. In particular, this assertion under the hypothesis that n = k implies that n is a prime. This was proved by Babbage, and thus our result may be considered as a generalization of this criterion for primality.     

Evolutionary methods for unsupervised feature selection using Sammon’s stress function

In this paper, four methods are proposed for feature selection in an unsupervised manner by using genetic algorithms. The proposed methods do not use the class label information but select a set of features using a task independent criterion that can preserve the geometric structure (topology) of the original data in the reduced feature space. One of the components of the fitness function is Sammon’s stress function which tries to preserve the topology of the high dimensional data when reduced into the lower dimensional one. In this context, in addition to using a fitness criterion, we also explore the utility of unfitness criterion to select chromosomes for genetic operations. This ensures higher diversity in the population and helps unfit chromosomes to become more fit. We use four different ways for evaluation of the quality of the features selected: Sammon error, correlation between the inter-point distances in the two spaces, a measure of preservation of cluster structure found in the original and reduced spaces and a classifier performance. The proposed methods are tested on six real data sets with dimensionality varying between 9 and 60. The selected features are found to be excellent in terms of preservation topology (inter-point geometry), cluster structure and classifier performance. We do not compare our methods with other methods because, unlike other methods, using four different ways we check the quality of the selected features by finding how well the selected features preserve the “structure” of the original data.     

Feasible smooth method based on Barzilai–Borwein method for stochastic linear complementarity problem

In this paper, we propose a feasible smooth method based on Barzilai–Borwein (BB) for stochastic linear complementarity problem. It is based on the expected residual minimization (ERM) formulation for the stochastic linear complementarity problem. Numerical experiments show that the method is efficient.     

A variant of Newton’s method based on Simpson’s three-eighths rule for nonlinear equations

We propose a new variant of Newton’s method based on Simpson’s three-eighth rule. It can be shown that the new method is cubically convergent.     

Lyapunov’s direct method for linear systems of functional-differential equations in Sobolev space

We establish a criterion for exponential stability in the H¹-topology in terms of operator inequalities for a linear FDE system of retarded type by Lyapunov’s direct method. As a corollary, some sufficient condition of exponential stability in terms of the matrix specifying the Stieltjes integral is obtained in the autonomous case. A few examples illustrating the results are exhibited.     

A new infeasible interior-point method based on Darvay’s technique for symmetric optimization

We present a full Nesterov and Todd step primal-dual infeasible interior-point algorithm for symmetric optimization based on Darvay’s technique by using Euclidean Jordan algebras. The search directions are obtained by an equivalent algebraic transformation of the centering equation. The algorithm decreases the duality gap and the feasibility residuals at the same rate. During this algorithm we construct strictly feasible iterates for a sequence of perturbations of the given problem and its dual problem. Each main iteration of the algorithm consists of a feasibility step and some centering steps. The starting point in the first iteration of the algorithm depends on a positive number ξ and it is strictly feasible for a perturbed pair. The feasibility steps find strictly feasible iterates for the next perturbed pair. By using centering steps for the new perturbed pair, we obtain strictly feasible iterates close to the central path of the new perturbed pair. The algorithm finds an ?-optimal solution or detects infeasibility of the given problem. Moreover, we derive the currently best known iteration bound for infeasible interior-point methods.     

An extension of one direction in Marty’s normality criterion

We prove the following extension of one direction in Marty’s theorem: If $k$ is a natural number, $\alpha >1$ and $\mathcal{F }$ is a family of functions meromorphic on a domain $D$ all of whose poles have multiplicity at least $\frac{k}{\alpha -1}$ , then the normality of $\mathcal{F }$ implies that the family $$\begin{aligned} \left\{ \frac{|f^{(k)}|}{1+|f|^\alpha }\,:\, f\in \mathcal{F }\right\} \end{aligned}$$ is locally uniformly bounded.     

An approximation algorithm for quadratic dynamic systems based on N. Chomsky’s grammar for Taylor’s formula

Single-step methods for the approximate solution of the Cauchy problem for dynamic systems are discussed. It is shown that a numerical integration algorithm with a high degree of accuracy based on Taylor’s formula can be proposed in the case of quadratic systems. An explicit estimate is given for the remainder. The algorithm is based on N. Chomsky’s generative grammar for the language of terms of Taylor’s formula.     

An extension of benveniste-matsuki’s method on 6-canonical maps for threefolds 1

The aim of this paper is to show that 6-canonical maps of nonsingular minimal 3 folds of general type are birational for p _g (X)≥2.     

Global convergence of Newton’s method on an interval

The solution of an equation f(x)= given by an increasing function f on an interval I and right-hand side , can be approximated by a sequence calculated according to Newtons method. In this article, global convergence of the method is considered in the strong sense of convergence for any initial value in I and any feasible right-hand side. The class of functions for which the method converges globally is characterized. This class contains all increasing convex and increasing concave functions as well as sums of such functions on the given interval. The characterization is applied to Keplers equation and to calculation of the internal rate of return of an investment project.An earlier version was presented at the Joint National Meeting of TIMS and ORSA, Las Vegas, May 7–9, 1990. Financial support from Økonomisk Forskningsfond, Bodø, Norway, is gratefully acknowledged. The author thanks an anonymous referee for helpful comments and suggestions.     

Improving discrimination in data envelopment analysis without losing information based on Renyi’s entropy

Data envelopment analysis (DEA) heavily depends on the dimensionality of the variables, and previous studies address the problem by decreasing the dimensionality with a minimal loss of information. Since the lost information can also have the impact on the evaluation performance, this paper accordingly proposes an approach to improve the discriminatory power of DEA without losing any variables information and without requiring any additional preferential information. Furthermore, an accelerating approach based on the concept of parallel computing is introduced to solve the multi-subsets problem. Listing all the possible variables subsets as the nodes, then the DEA efficiencies under each node are calculated, and the corresponding purity of information can be scientifically generated based on Renyi’s entropy. Subsequently, the important degrees of nodes are obtained by normalizing the purities of information, and the comprehensive efficiency scores can be finally generated by the weighted sum between the important degrees and the efficiencies under the corresponding node. Two specific examples are provided to evaluate the performance.     

Rothe’s method for an integrodifferential equation with integral conditions

This work deals with an integrodifferential equation with initial, Newman and integral conditions. The existence and uniqueness of the weak solution in an appropriate sense is proved by the method of Rothe.     

Numerical method for an approximate solution to Bulgakov’s problem

A great deal of the attention devoted to controlled system dynamics optimization in the literature is given to Bulgakov’s familiar problem of the accumulation of perturbation in a linear system. A more general case of this problem is considered in this work. An approximate method for calculating the value characterizing maximum dispersion is discussed.     

An efficient algorithm for solving rank one perturbed linear Diophantine systems using Rosser��s approach

Recently, we described a generalization of Rosser’s algorithm for a single linear Diophantine equation to an algorithm for solving systems of linear Diophantine equations. Here, we make use of the new formulation to present a new algorithm for solving rank one perturbed linear Diophantine systems, based on using Rosser’s approach. Finally, we compare the efficiency and effectiveness of our proposed algorithm with the algorithm proposed by Amini and Mahdavi-Amiri (Optim Methods Softw 21:819–831, 2006).     

An acceleration-scale model of IING’s diffusion based on force analysis

ABSTRACT

The diffusion of Internet-based Intangible Network Goods (IINGs) shows new characteristics completely different from that of traditional material products. This paper aims to establish new models to describe and predict IING’s diffusion at the aggregate level. Firstly, we transform the key factors affecting IING’s diffusion into driving forces, resistant forces, and variable forces. Secondly, we analyse the dynamic changes of these forces in different diffusion stages and obtain the acceleration model of IING’s diffusion. Then, since acceleration is the second derivative of scale, we further establish the scale model of IING’s diffusion. As the scale model can predict the number of IING’s adopters at a particular time and the acceleration model can explain the dynamic changes of scale, we combine them as the acceleration-scale model to describe IING’s diffusion. Finally, we make comparisons between the acceleration-scale model and the Bass model based on three cases. Different from the previous studies, we found that IING’s diffusion rate is asymmetric. The diffusion rate of successful IING is right skewed while the diffusion rate of failed IING is left skewed. The results also shows that the acceleration-scale model has a better predictive performance than the Bass model, no matter the diffusion is successful or failed