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 work, we perform univariate approximation with rates, basic and fractional, of continuous functions that take values into an arbitrary Banach space domain on a closed interval or all reals, by quasi-interpolation neural network operators. These approximations are achieved deriving Jackson-type inequalities via the first modulus continuity hand function its abstract integer derivative Ca...

In the paper, authors find series expansions and identities for positive integer powers of inverse (hyperbolic) sine tangent, composite incomplete gamma function with hyperbolic sine, in terms first kind Stirling numbers, apply a newly established expansion to derive closed-form formula specific partial Bell polynomials representation generalized logsine function, deduce combinatorial involving...

The paper presents in detail a relatively simple implementation method of the hyperbolic tangent function, particularly targeted for FPGAs. research goal proposed was to examine usage approximation ordinary or Chebyshev polynomials function. Several miscellaneous versions have been considered. They differ polynomial degree, number intervals which domain function is divided, etc. Both floating-p...

Artificial neural networks (ANNs) are a widely used machine learning approach for estimating low-cycle fatigue parameters. ANN structure has its parameters such as hidden layers, neurons, activation functions, training and so forth, these have significant influence over the results. Three layer combinations, neurons ranging from 1 to 25, different functions like hyperbolic tangent sigmoid (tans...

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 ...

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...

In this paper we present a modified version of the Hyperbolic Tangent Activation Function as a learning unit generator for neural networks. The function uses an integer calibration constant as an approximation to the Euler number, e, based on a quadratic Real Number Formula (RNF) algorithm and an adaptive normalization constraint on the input activations to avoid the vanishing gradient. We demo...