A growth estimate on the Fourier coefficients along geodesics for eigenfunctions of the Laplacian is given on compact hyperbolic manifolds. Along the way, a new summation formula is proved.

We give results on the following questions about a topologically tame hyperbolic 3-manifold M : 1. Does M have nonzero L-harmonic 1-forms? 2. Does zero lie in the spectrum of the Laplacian acting on Λ(M)/Ker(d)?

The goal of this handout is to discuss models of Hyperbolic and Euclidean Geometry, and the consistency of Hyperbolic Geometry. 1. Some concepts from Euclidean Geometry We will use circles in Euclidean Geometry to build up models for Hyperbolic Geometry. So we will review some basic facts about circles. (Mostly without proofs). In this section assume that we are working in Euclidean Geometry. D...

In this paper, we introduce two approaches to compute the matrix hyperbolic tangent. While one of them is based on its own definition and uses exponential, other focused expansion Taylor series. For second approximation, analyse different alternatives evaluate corresponding polynomials. This resulted in three stable accurate codes, which implemented MATLAB numerically computationally compared b...

Utilizing echoic intension and distribution from different organs and local details of human body, ultrasonic image can catch important medical pathological changes, which unfortunately may be affected by ultrasonic speckle noise. A feature preserving ultrasonic image denoising and edge enhancement scheme is put forth, which includes two terms: anisotropic diffusion and edge enhancement, contro...

Rectifying neurons are more biologically plausible than logistic sigmoid neurons, which are themselves more biologically plausible than hyperbolic tangent neurons. However, the latter work better for training multi-layer neural networks than logistic sigmoid neurons. This paper shows that networks of rectifying neurons yield equal or better performance than hyperbolic tangent networks in spite ...

In this paper, we have exploited the first-order sliding mode control method to track the ECG data of the human heart by three different nonlinear control laws. In order to lessen the intrinsic chattering of the classic sliding mode control system, smooth function approximations of the control input, by means of the hyperbolic tangent and the saturation function, were used. The fast Fourier tra...

We apply topological methods to study eigenvalues of the Laplacian on closed hyperbolic surfaces. For any closed hyperbolic surface S of genus g , we get a geometric lower bound on 2g 2.S/ W 2g 2.S/ > 1=4C 0.S/ > 0; where 0.S/ is an explicit constant which depends only on the systole of S . Mathematics Subject Classification (2010). Primary 30F; Secondary 35P05, 35P15.