Derivatives of tangent function and tangent numbers

In the paper, by induction, the Fa\`a di Bruno formula, and some techniques in the theory of complex functions, the author finds explicit formulas for higher order derivatives of the tangent and cotangent functions as well as powers of the sine and cosine functions, obtains explicit formulas for two Bell polynomials of the second kind for successive derivatives of sine and cosine functions, presents curious identities for the sine function, discovers explicit formulas and recurrence relations for the tangent numbers, the Bernoulli numbers, the Genocchi numbers, special values of the Euler polynomials at zero, and special values of the Riemann zeta function at even numbers, and comments on five different forms of higher order derivatives for the tangent function and on derivative polynomials of the tangent, cotangent, secant, cosecant, hyperbolic tangent, and hyperbolic cotangent functions.Comment: 17 page

Derivative Polynomials and Closed-Form Higher Derivative Formulae

In a recent paper, Adamchik [V.S. Adamchik, On the Hurwitz function for rational arguments, Appl. Math. Comp. 187 (2007) 3--12] expressed in a closed form symbolic derivatives of four functions belonging to the class of functions whose derivatives are polynomials in terms of the same functions. In this sequel, simple closed-form higher derivative formulae which involve the Carlitz-Scoville higher order tangent and secant numbers are derived for eight trigonometric and hyperbolic functions.Comment: 7 page

Functional programming framework for GRworkbench

The software tool GRworkbench is an ongoing project in visual, numerical General Relativity at The Australian National University. Recently, the numerical differential geometric engine of GRworkbench has been rewritten using functional programming techniques. By allowing functions to be directly represented as program variables in C++ code, the functional framework enables the mathematical formalism of Differential Geometry to be more closely reflected in GRworkbench . The powerful technique of `automatic differentiation' has replaced numerical differentiation of the metric components, resulting in more accurate derivatives and an order-of-magnitude performance increase for operations relying on differentiation

Extension of information geometry for modelling non-statistical systems

In this dissertation, an abstract formalism extending information geometry is introduced. This framework encompasses a broad range of modelling problems, including possible applications in machine learning and in the information theoretical foundations of quantum theory. Its purely geometrical foundations make no use of probability theory and very little assumptions about the data or the models are made. Starting only from a divergence function, a Riemannian geometrical structure consisting of a metric tensor and an affine connection is constructed and its properties are investigated. Also the relation to information geometry and in particular the geometry of exponential families of probability distributions is elucidated. It turns out this geometrical framework offers a straightforward way to determine whether or not a parametrised family of distributions can be written in exponential form. Apart from the main theoretical chapter, the dissertation also contains a chapter of examples illustrating the application of the formalism and its geometric properties, a brief introduction to differential geometry and a historical overview of the development of information geometry.Comment: PhD thesis, University of Antwerp, Advisors: Prof. dr. Jan Naudts and Prof. dr. Jacques Tempere, December 2014, 108 page