Divergence bounded computable real numbers

AbstractA real x is called h-bounded computable, for some function h:N→N, if there is a computable sequence (xs) of rational numbers which converges to x such that, for any n∈N, at most h(n) non-overlapping pairs of its members are separated by a distance larger than 2-n. In this paper we discuss properties of h-bounded computable reals for various functions h. We will show a simple sufficient condition for a class of functions h such that the corresponding h-bounded computable reals form an algebraic field. A hierarchy theorem for h-bounded computable reals is also shown. Besides we compare semi-computability and weak computability with the h-bounded computability for special functions h

Amplifying Sine Unit: An Oscillatory Activation Function for Deep Neural Networks to Recover Nonlinear Oscillations Efficiently

Many industrial and real life problems exhibit highly nonlinear periodic behaviors and the conventional methods may fall short of finding their analytical or closed form solutions. Such problems demand some cutting edge computational tools with increased functionality and reduced cost. Recently, deep neural networks have gained massive research interest due to their ability to handle large data and universality to learn complex functions. In this work, we put forward a methodology based on deep neural networks with responsive layers structure to deal nonlinear oscillations in microelectromechanical systems. We incorporated some oscillatory and non oscillatory activation functions such as growing cosine unit known as GCU, Sine, Mish and Tanh in our designed network to have a comprehensive analysis on their performance for highly nonlinear and vibrational problems. Integrating oscillatory activation functions with deep neural networks definitely outperform in predicting the periodic patterns of underlying systems. To support oscillatory actuation for nonlinear systems, we have proposed a novel oscillatory activation function called Amplifying Sine Unit denoted as ASU which is more efficient than GCU for complex vibratory systems such as microelectromechanical systems. Experimental results show that the designed network with our proposed activation function ASU is more reliable and robust to handle the challenges posed by nonlinearity and oscillations. To validate the proposed methodology, outputs of our networks are being compared with the results from Livermore solver for ordinary differential equation called LSODA. Further, graphical illustrations of incurred errors are also being presented in the work.Comment: 16 Pages and 16 figure

ASU-CNN: An Efficient Deep Architecture for Image Classification and Feature Visualizations

Activation functions play a decisive role in determining the capacity of Deep Neural Networks as they enable neural networks to capture inherent nonlinearities present in data fed to them. The prior research on activation functions primarily focused on the utility of monotonic or non-oscillatory functions, until Growing Cosine Unit broke the taboo for a number of applications. In this paper, a Convolutional Neural Network model named as ASU-CNN is proposed which utilizes recently designed activation function ASU across its layers. The effect of this non-monotonic and oscillatory function is inspected through feature map visualizations from different convolutional layers. The optimization of proposed network is offered by Adam with a fine-tuned adjustment of learning rate. The network achieved promising results on both training and testing data for the classification of CIFAR-10. The experimental results affirm the computational feasibility and efficacy of the proposed model for performing tasks related to the field of computer vision.Comment: 11 pages , 8 figure

Adaptive Control for Modified Projective Synchronization of Four-dimensional Qi Chaotic System with Dispersive Term *

Abstract: This paper is concerned with the modified projective synchronization problem for the four-dimensional Qi chaotic system with uncertain parameters .The designed controller ensures that the state variables of the state variables of the master systems respectively. The results are validated using numerical simulation

Application of the Homotopy Analysis Method for Solving the Variable Coefficient KdV-Burgers Equation

The homotopy analysis method is applied to solve the variable coefficient KdV-Burgers equation. With the aid of generalized elliptic method and Fourier’s transform method, the approximate solutions of double periodic form are obtained. These solutions may be degenerated into the approximate solutions of hyperbolic function form and the approximate solutions of trigonometric function form in the limit cases. The results indicate that this method is efficient for the nonlinear models with the dissipative terms and variable coefficients

Traveling Wave Solutions of Space-Time Fractional Generalized Fifth-Order KdV Equation

The Korteweg-de Vries (KdV) equation, especially the fractional higher order one, provides a relatively accurate description of motions of long waves in shallow water under gravity and wave propagation in one-dimensional nonlinear lattice. In this article, the generalized exp⁡(-Φ(ξ))-expansion method is proposed to construct exact solutions of space-time fractional generalized fifth-order KdV equation with Jumarie’s modified Riemann-Liouville derivatives. At the end, three types of exact traveling wave solutions are obtained which indicate that the method is very practical and suitable for solving nonlinear fractional partial differential equations

MHD boundary layer flow of Carreau fluid over a convectively heated bidirectional sheet with non-fourier heat flux and variable thermal conductivity

© 2019 by the authors. In the present exploration, instead of the more customary parabolic Fourier law, we have adopted the hyperbolic Cattaneo-Christov (C-C) heat flux model to jump over the major hurdle of parabolic energy equation . The more realistic three-dimensional Carreau fluid flow analysis is conducted in attendance of temperature-dependent thermal conductivity. The other salient impacts affecting the considered model are the homogeneous-heterogeneous (h-h) reactions and magnetohydrodynamic (MHD). The boundary conditions supporting the problem are convective heat and of h-h reactions. The considered boundary layer problem is addressed via similarity transformations to obtain the system of coupled differential equations. The numerical solutions are attained by undertaking the MATLAB built-in function bvp4c. To comprehend the consequences of assorted parameters on involved distributions, different graphs are plotted and are accompanied by requisite discussions in the light of their physical significance. To substantiate the presented results, a comparison to the already conducted problem is also given. It is envisaged that there is a close correlation between the two results. This shows that dependable results are being submitted. It is noticed that h-h reactions depict an opposite behavior versus concentration profile. Moreover, the temperature of the fluid augments for higher values of thermal conductivity parameters

Structure of New Solitary Solutions for The Schwarzian Korteweg De Vries Equation And (2+1)-Ablowitz-Kaup-Newell-Segur Equation

In this research, we introduce and represent the modified Khater method on two basic models in the optical fiber. These two models describe the dynamics of the wave movement in the optical fiber.  It is a new modification of new recent method which developed by Mostafa M. A. Khater in 2017. We implement this new modified technique on Schwarzian Korteweg de Vries equation and (2+1)-Ablowitz-Kaup-Newell-Segur equation. This modification of Khater method produces more closed solutions than many other methods. Schwarzian Korteweg de Vries (SKdV) equation has a closed relationship with (2+1)-Ablowitz- Kaup-Newell-Segur equation. Schwarzian Korteweg de Vries equation prescribes the location in a micro-segment of space and motion of the isolated waves in varied fields which localized in a tiny portion of space. It is a great and basic system in fluid mechanics, nonlinear optics, plasma physics, and quantum field theory