On a Semilinear Parabolic Problem with Four-Point Boundary Conditions

This paper studies a semilinear parabolic equation in 1D along with nonlocal boundary conditions. The value at each boundary point is associated with the value at an interior point of the domain, which is known as a four-point boundary condition. First, the solvability of a steady-state problem is addressed and a constructive algorithm for finding a solution is proposed. Combining this schema with the semi-discretization in time, a constructive algorithm for approximation of a solution to a transient problem is developed. The well-posedness of the problem is shown using the semigroup theory in C-spaces. Numerical experiments support the theoretical algorithms.

A Constructive Algorithm for Transforming Conics in a Circle on Imaginary Laguerre Points

The article is devoted to the consideration of the properties of Laguerre points in their constructive relationship with the solution of the problem of transforming a pair of conical curves into two circles. Geometric schemes are presented that reveal the principle of specifying a collinear transformation that establishes a correspondence between conics and their images in the form of circles, including if the Laguerre points defining this transformation are imaginary. In the solution of the problem, a geometric scheme for the formation of the main conjugations of conics is given, one of which is a circle. The conclusion is substantiated that the radical axis of the bundle of circles is a circle that has split into a pair of straight lines, one of which is infinitely distant. An algorithm for the formation of a series of elliptic curves induced by Laguerre points is presented.

A hybrid constructive algorithm incorporating teaching-learning based optimization for neural network training

In neural networks, simultaneous determination of the optimum structure and weights is a challenge. This paper proposes a combination of teaching-learning based optimization (TLBO) algorithm and a constructive algorithm (CA) to cope with the challenge. In literature, TLBO is used to choose proper weights, while CA is adopted to construct different structures in order to select the proper one. In this study, the basic TLBO algorithm along with an improved version of this algorithm for network weights selection are utilized. Meanwhile, as a constructive algorithm, a novel modification to multiple operations, using statistical tests (MOST), is applied and tested to choose the proper structure. The proposed combinatorial algorithms are applied to ten classification problems and two-time-series prediction problems, as the benchmark. The results are evaluated based on training and testing error, network complexity and mean-square error. The experimental results illustrate that the proposed hybrid method of the modified MOST constructive algorithm and the improved TLBO (MCO-ITLBO) algorithm outperform the others; moreover, they have been proven by Wilcoxon statistical tests as well. The proposed method demonstrates less average error with less complexity in the network structure.