This book contains a group of lectures illustrating–at an undergraduate level– the mathematical structure of Celestial and Terrestrial Dynamics (the original fields of physical investigations of Classical Dynamics). On the one hand, these fields will provide the physical contents of the mathematical model described in [1],[2]. On the other hand, the problems of celestial and terrestrial dynamics, once framed into the above model, will be able to be given a sound conceptual foundation and a thorough mathematical discussion. From the philosophical point of view, celestial and terrestrial dynamics are no longer disjoint areas of knowledge since Newtonian synthesis, which recognised gravitation as their common core.

This book contains a group of lectures illustrating - at an undergraduate level - the mathematical structure of Classical Dynamics, both in its historical (Newtonian-d'Alembertian) formulation and in its analytical (Lagrangian-Hamiltonian) version. From the empirical point of view, dynamics basically deals with the problem (chapter 1) of ‘predicting’ all the motions, with respect to a non-influential observer, that are possible for a constrained particle system under the action of a given force field. In this connection, the first step is to establish a ‘time-evolution law’ characterising the dynamically possible motions of the given mechanical system, and the second step is that of carrying out a discussion of the above law in order to obtain a qualitative picture and/or a quantitative determination of the dynamically possible motions. From the mathematical point of view, 1 the time-evolution law established in Newtonian-d’Alembertian dynamics (chapter 2) will be shown to result in an implicit differential equation–d’Alembert equation– expressed and elementarily discussed in the geometrical (i.e. coordinate-free) formalism of the Euclidean affine space perceived by and associated with the observer. d’Alembert equation will be obtained from the historical Newton equation for unconstrained systems by taking the possible dynamical effects of the constraints into due consideration, and will then be specialised in the classical Newton and Euler’s equations for rigid systems.

The aim of this introduction to the geometry of Classical Dynamics is to lead postgraduate students of Mathematics or Physics from the historical Newtonian-d'Alembertian dynamics up to the border with the modern (geo metrical) Lagrangian-Hamiltonian dynamics, without making any direct use of the traditional (analytical) formulation of the latter. The purpose is to cover the big gap existing in literature between the (empirical) elementary presentation of historical dynamics and the (abstract) differential-geometric formulation of analytical dynamics. The method is the pedagogical principle of working from bottom up, in order to highlight roots, motivations and inductive proceedings of the modern differential-geometrical developments in such an ancient physical area.

In order to analyze the solution behavior of general substitution ciphers for cryptanalysis, for the first time this book utilizes the methods of set optimization. The book investigates how to proceed optimally in a specific case. Linguistic data structures, such as the frequencies of the different letters, bigrams and trigrams, are divided into sets and described by order relations appropriate to the German and English languages. In the context of cryptanalysis this allows new solutions of existing models within the scope of set-valued order relations. This approach leads to an optimal strategy.

This book investigates compensation payments to property owners for aircraft noise in urban conflict situations in the region surrounding the airport for the first time on the basis of a set-valued conjugate duality. This optimal perturbation approach serves as justification for the realization of variable results. This "dual" socio-economy indicates the action strategy of different interest groups. Linear point sets as a new optimum set varying according to socio-economic properties and a payment function are therefore well substantiated.

"... The present book narrates this mathematical journey by providing a complete and self-contained treatment of the eigenstructure of the Laplacian on an equilateral triangle. The historical context and practical significance of the problem is carefully traced. The separate cases of Dirichlet, Neumann, radiation, absorbing and impedance boundary conditions are individually and exhaustively treated with the Dirichlet and Neumann cases also extended from the continuous to the discrete Laplacian. Corresponding results for the Sturm-Liouville boundary value problem under an impedance boundary condition are reviewed and applied to the parallel plate waveguide. Polygons with trigonometric eigenfunctions receive comprehensive study. Application to modal degeneracy in equilateral triangular waveguides has also been included."

"... Contributor of a substantial addition to the literature of triangle geometry, Brian J. McCartin of Kettering University, a prize-winning author, has fallen in love with the equilateral triangle. Even as mathematician Lewis Carroll had his Snark pursued with “forks and hope,” McCartin has pursued the equilateral triangle with a fine-tooth comb supported by an eagle eye. He has tracked down the multiple personality of the equilateral triangle as it appears in history, design; in theorems of plane geometry; in applications, games, recreational mathematics, competitions (e.g., the Olympiads); and in popular culture."