Abe Shenitzer

A Simple Non-Euclidean Geometry and Its Physical Basis

1. What is geometry?.- 2. What is mechanics?.- I. Distance and Angle Triangles and Quadrilaterals.- 3. Distance between points and angle between lines.- 4. The triangle.- 5. Principle of duality coparallelograms and cotrapezoids.- 6. Proof s of the principle of duality.- II. Circles and Cycles.- 7. Definition of a cycle radius and curvature.- 8. Cyclic rotation diameters of a cycle.- 9. The circumcycle and incycle of a triangle.- 10. Power of a point with respect to a circle or cycle inversion.- Conclusion.- 11. Einsteins principle of relativity and Lorentz transformations.- 12. Minkowskian geometry.- 13. Galilean geometry as a limiting case of Euclidean and Minkowskian geometry.- Supplement A. Nine plane geometries.- Supplement B. Axiomatic characterization of the nine plane geometries.- Supplement C. Analytic models of the nine plane geometries.- Answers and Hints to Problems and Exercises.- Index of Names.- Index of Subjects.

The principle of maximum entropy

There is no need to stress the importance of variational problems in mathematics and its applications. The list of variational problems, of different degrees of difficulty, is very long, and it stretches from famous minimum and maximum problems of antiquity, through the variational problems of analytical mechanics and theoretical physics, all the way to the variational problems of modern opera t ions research. While maximizing or minimizing a function or a functional is a routine procedure, some special variational problems give solutions which either unify previously unconnected results or match surprisingly well the results of our experiments. Such variational problems are called variational principles. Whether or not the architecture of our world is based on variational principles is a philosophical problem. But it is a sound strategy to discover and apply variational principles in order to acquire a better understanding of a part of this architecture. In applied mathematics we get a model by taking into account some connections and, inevitably, ignoring others. One way of making a model convincing and useful is to obtain it as the solution of a variational problem. The aim of the present paper is to bring some arguments in favour of the promotion of the variational problem of entropy maximization to the rank of a variational principle.

The genesis of the abstract group concept : a contribution to the history of the origin of abstract group theory

In this book, Hans Wussing sets out to trace the process of abstraction that led finally to the axiomatic formulation of the abstract notion of group. His main thesis is that the roots of the abstract notion of group do not lie, as frequently assumed, only in the theory of algebraic equations; they are also to be found in the geometry and the theory of numbers of the end of the 18th and the first half of the 19th centuries.The book takes us from Lagrange via Cauchy, Abel, and Galois to Serret and Camille Jordan. It then turns to Cayley, to Felix Kleins Erlangen Program, and to Sophus Lie, and ends with a sketch of the state of group theory about 1920, when the axiom systems of Webber had been formalized and investigated in their own right.

The Problem of Squarable Lunes

Translators note. Hippocrates of Chios (second half of the fifth century BCE) seems to have been the first mathematician to square a figure with curved boundary. The figure in question was a lune. (For details about Hippocrates and his work see pp. 131-136 in B. L. van der Waerden, Science Awakening, P. Noordhoff, Groningen, 1954.) A lune is a figure bounded by two circular arcs with a common chord. Figure 1 shows a concave-convex lune and Figure 2 shows a convex lune.

On the Appearance of Moving Bodies

The first reaction of anyone who reads about the special theory of relativity is invariably amazement. Can the rate of the flow of time be relative? Can the dimensions of moving objects depend on the observer? Sheer nonsense! And yet thats the way things are and common sense turns out to be a poor guide. The one thing that is undoubtedly true is that the theory of relativity is fascinating and intrigues not only physicists but also scientific amateurs and enthusiasts.

The Poincare conjecture

On November 11, 2002, Grigoriĭ Yakovlevich Perelman, a geometer working in the St. Petersburg section of the Steklov Mathematical Institute at Fontanka 27, published on the internet a forty-page paper titled “An Entropy Formula for the Ricci Flow and Its Geometric Applications.” The fourth page of the dry introduction, full of technical terms, ends with the sentence: Finally, in Section 13, we give a brief sketch of a proof of the Geometrization Conjecture.