Journal of Advances in Applied Mathematics

6

Authors: Jochen Merker

In supervised machine learning, a support vector machine (SVM) constructs from binary classified training data a linear classifier by solving a linearly constrained convex optimization problem. Depending on the number N of training data and the dimension D of the feature space, it either is advantageous to solve the primal problem or the dual problem. In this article, the case D >> N is discussed where D is so large that even a calculation of the dot product of fully occupied vectors in dimension D is too slow for the desired (e.g. real-time) application. Then a way to speed up the classification is to use an SVM which constructs a sparse linear classifier by solving an optimization problem involving the 1-norm, i.e. many components of the classifying vector are zero so that much less than D multiplications are neccessary to calculate the dot product. For a soft-margin SVM, in this article a theorem on the number of non-zero components is shown.

12

Authors: Farhana Alam, M. H. Rashid, M. A. Alom

Let X and Y be Banach spaces. We study a Regula-falsi-type method for solving the generalized equations 0 ∈ f(x) + g(x) + F(x), where f : X → Y is Fréchet differentiable in a neighborhood of a solution x¯, g : X → Y is Fréchet differentiable at x¯ and F : X ⇒ 2 Y is a set valued mapping with closed graph. Under some suitable assumptions, we prove the existence of any sequence generated by the Regula-falsi-type method and establish the local convergence results of the sequence generated by this method. Indeed, we will show that the sequence generated by this method converges linearly and super-linearly

16

Authors: Lingying Teng, Xiaohu Wang

The main aim of this paper is to develop some basic theories of stochastic functional differential equations (SFDEs) with non-Lipschitz coefficients. Firstly, we show that Peano’s theorem holds for SFDEs, that is, the continuity alone is sufficient to prove the local existence of the initial value problem of SFDEs. Secondly, some new uniqueness theorems are established by the comparison methods proposed by Xu et al. And then, continuation theorems and global existence theorems for SFDEs with non-Lipschitz coefficients are obtained. Finally, an example is given to illustrate the efficiency of the obtained results.