e article is a tribute to Hermann Minkowski leading from his geometry of numbers to an attempt at using Finsler geometry for a break of Lorentz invariance.

Here we have discussed on Fermi-Dirac statistics, in particular, on its brief historical progress, derivation, consequences, applications, etc. Importance of Fermi-Dirac statistics has been discussed even in connection with the current progress in science. This article is mainly aimed to the undergraduate and graduate students.

Personal recollections about Alexandre Grothendieck and early days of his theory of motives

What can we discover? As the discussion in this paper is limited to ontological considerations, it does not deal with the discovery of new concepts. It raises the following question: What are the entities or existents that we can discover? There are two kinds of such entities: (1) actual entities, and (2) possible entities, which are pure possibilities. The paper explains why the first kind of discovery depends primarily on the second kind. The paper illustrates the discoveries of individual pure possibilities by presenting examples such as the Higgs particle, Dirac’s positron, and Pauli-Fermi’s neutrino.

In the physics literature «ergodicity» is taken to mean that a system, including a macroscopic one, visits all microscopic states in a relatively short time. We show that this is an impossibility even if that time is billions of years. We also suggest that this feature does not contradict most physical considerations since those considerations deal with correlations of only a few particles.

It is proposed that the recent controversy over «time-symmetric quantum counterfactuals» (TSQCs), based on the Aharonov-Bergmann-Lebowitz Rule for measurements of pre- and post-selected systems, can be clarified by taking TSQCs to be counterfactuals with a specific type of compound antecedent. In that case, inconsistency proofs such as that of Sharp and Shanks (1993) are not applicable, and the main issue becomes not whether such statements are true, but whether they are nontrivial. The latter question is addressed and answered in the negative. Thus it is concluded that TSQCs, understood as counterfactuals with a compound antecedent, are true but only trivially so, and provide no new contingent information about specific quantum systems (except in special cases already identified in the literature).

We review and strengthen the arguments given by Einstein to derive his first gravitational field equation for static fields and show that, although it was ultimately rejected, it follows from General Relativity (GR) for negligible pressure. Using this equation and considerations folowing directly from the equivalence principle (EP), we show how Schwarzschild metric and other vacum metrics can be obtained immediately. With this results and some basic principles, we obtain the metric in the general spherically symmetric case and the corresponding hydrostatic equilibrium equation. For this metrics we obtain the motion equations in a simple and exact manner that clearly shows the three sources of difference (implied by various aspects of the EP) with respect to the Newtonian case and use them to study the classical tests of GR.

Classical tensors, the familiar mathematical objects denoted by symbols such as t i , t ij and t ij k , are usually interpreted either as ‘coordinatizable objects’ with coordinates changing in a specific way under a change of coordinate system or as elements of tensor spaces of the form V ⊗n ⊗(V ∗ ) ⊗m . An alternative interpretation of classical tensors as linear tensor maps of the form V ⊗m →V ⊗n is presented here. In this interpretation, tensor multiplication is seen as generalized function composition. Representations of classical tensors by means of arrays are also considered.

This paper contains a discussion of a library of formalized mathematics for the proof assistant Coq which the author worked on in 2011-13.

We study Saccheri’s three hypotheses on a two right-angled isosceles quadrilateral, with a rectilinear summit side. We claim that in the Hilbert`s foundation of geometry the euclidean parallelism is a theorem and as that it can be used, in the hyperbolic geometry