The aim of this short lecture course is to develop Selberg's trace formula for a compact hyperbolic surface M, and discuss some of its applications. The main motivation for our studies is quantum chaos: the Laplace-Beltrami operator −∆ on the surface M represents the quantum Hamiltonian of a particle, whose classical dynamics is governed by the (strongly chaotic) geodesic flow on the unit tange...

In the present analysis, we have modeled the governing equations of a two dimensional hyperbolic tangent fluid model. Using the assumption of long wavelength and low Reynolds number, the governing equations of hyperbolic tangent fluid for an asymmetric channel have been solved using the regular perturbation method. The expression for pressure rise has been calculated using numerical integration...

Since my graduate work and throughout my postdoctoral career, I have been interested in hyperbolic dynamical systems, more precisely Anosov flows. Roughly speaking, a smooth flow ft on a closed Riemannian manifold M is called Anosov if the tangent bundle of M splits into three continuous sub-bundles, TM = E⊕E⊕E, such that E is tangent to the flow direction, E is exponentially contracted and E i...

This paper discusses the propagation of signals in generic densely connected multilayered feedforward neural networks. It is concluded that the dense connecting combined with the hyperbolic tangent activation functions of the neurons may cause a highly random, spurious generalization, that decreases the overall performance and reliability of a neural network and can be mistaken with overfitting...

In this paper, using a hyperbolic tangent function tanhðbxÞ, b > 0, we develop a non-iterative method to estimate a root of an equation f(x) = 0. The problem of finding root is transformed to evaluating an integral, and thus we need not take account of choosing initial guess. The larger the value of b, the better the approximation to the root. Alternatively we employ the signum function sgnðxÞ ...

In this paper, a new segmentation method using hyperbolic tangent fuzzy cmeans (MHTFCM) algorithm for medical image segmentation. The proposed method uses two hyperbolic tangent functions for clustering of images. The performance of the proposed algorithm is tested on OASIS-MRI image dataset. The performance is tested in terms of score, number of iterations (NI) and execution time (TM) under di...