Relativistic Chasles' theorem and the conjugacy classes of the inhomogeneous Lorentz group

This work is devoted to the relativistic generalization of Chasles' theorem, namely to the proof that every proper orthochronous isometry of Minkowski spacetime, which sends some point to its chronological future, is generated through the frame displacement of an observer which moves with constant acceleration and constant angular velocity. The acceleration and angular velocity can be chosen either aligned or perpendicular, and in the latter case the angular velocity can be chosen equal or smaller than than the acceleration. We start reviewing the classical Euler's and Chasles' theorems both in the Lie algebra and group versions. We recall the relativistic generalization of Euler's theorem and observe that every (infinitesimal) transformation can be recovered from information of algebraic and geometric type, the former being identified with the conjugacy class and the latter with some additional geometric ingredients (the screw axis in the usual non-relativistic version). Then the proper orthochronous inhomogeneous Lorentz Lie group is studied in detail. We prove its exponentiality and identify a causal semigroup and the corresponding Lie cone. Through the identification of new Ad-invariants we classify the conjugacy classes, and show that those which admit a causal representative have special physical significance. These results imply a classification of the inequivalent Killing vector fields of Minkowski spacetime which we express through simple representatives. Finally, we arrive at the mentioned generalization of Chasles' theorem.Comment: Latex2e, 49 pages. v2: few typos correcte

Iterated harmonic numbers

The harmonic numbers are the sequence 1, 1+1/2, 1+1/2+1/3, ... Their asymptotic difference from the sequence of the natural logarithm of the positive integers is Euler's constant gamma. We define a family of natural generalizations of the harmonic numbers. The jth iterated harmonic numbers are a sequence of rational numbers that nests the previous sequences and relates in a similar way to the sequence of the jth iterate of the natural logarithm of positive integers. The analogues of several well-known properties of the harmonic numbers also hold for the iterated harmonic numbers, including a generalization of Euler's constant. We reproduce the proof that only the first harmonic number is an integer and, providing some numeric evidence for the cases j = 2 and j = 3, conjecture that the same result holds for all iterated harmonic numbers. We also review another proposed generalization of harmonic numbers.Comment: 13 pages, 2 figure

Exactly Conservative Integrators

Traditional numerical discretizations of conservative systems generically yield an artificial secular drift of any nonlinear invariants. In this work we present an explicit nontraditional algorithm that exactly conserves these invariants. We illustrate the general method by applying it to the three-wave truncation of the Euler equations, the Lotka--Volterra predator--prey model, and the Kepler problem. This method is discussed in the context of symplectic (phase space conserving) integration methods as well as nonsymplectic conservative methods. We comment on the application of our method to general conservative systems.Comment: 30 pages, postscript (1.3MB). Submitted to SIAM J. Sci. Comput

A visual tour via the Definite Integration ∫ab1xdx\int_{a}^{b}\frac{1}{x}dx

Geometrically, $\int_{a}^{b}\frac{1}{x}dx$ means the area under the curve $\frac{1}{x}$ from $a$ to $b$, where $0<a<b$, and this area gives a positive number. Using this area argument, in this expository note, we present some visual representations of some classical results. For examples, we demonstrate an area argument on a generalization of Euler's limit $\left(\lim\limits_{n\to\infty}\left(\frac{(n+1)}{n}\right)^{n}=e\right)$. Also, in this note, we provide an area argument of the inequality $b^a < a^b$, where $e \leq a< b$, as well as we provide a visual representation of an infinite geometric progression. Moreover, we prove that the Euler's constant $\gamma\in [\frac{1}{2}, 1)$ and the value of $e$ is near to $2.7$. Some parts of this expository article has been accepted for publication in Resonance - Journal of Science Education, The Mathematical Gazette, and International Journal of Mathematical Education in Science and Technology.Comment: 10 pages, 15 figure

A Note on the Unsteady Cavity Flow in a Tunnel

The unsteady internal cavitating flow such as the one observed in a pump or a turbine is studied for a simple two-dimensional model of a base-cavitating wedge in an infinite tunnel and it is shown how the cavitation compliance can be calculated using the linearized free streamline theory. Numerical values are obtained for the limiting case of a free jet. Two important features are: First, the cavitation compliance is found to be of complex form, having additional resistive and reactive terms beyond the purely inertial oscillation of the whole channel in "slug flow." Second, the compliance has a strong dependence on frequency