This special issue is based on scientific research works presented at the Workshop "Mathematical Physics: Symmetries" (Zacatecas, México, December 1-2, 2005). Among the authors the reader will find the well-known names of Profs. N. V. Mitskievich, M. Socolovsky, M. Napsuciale, A. Gutiérrez-Rodríguez, Z. Oziewicz, J. Keller, R. Alvarado Flores, J.A. Pérez, J. Martínez Ortiz, H. Rosu and J.P. Treviño, M. Agüero Granados, A. Enciso Muñoz. Several participants (A.E. Chubykalo, V.V. Dvoeglazov, A. Espinoza Garrido, V. Kravchenko, H. Oviedo, A.I. Nesterov) had their presentations published with other Journals before the Workshop. This Workshop continued the series of the small Workshops organized in 2005 by various Mexican Universities on the subjects of SUSY, neutrinos and flavor in the particle theory. The Workshop being announced as national, in fact, it was truly international, because many of its participants came from Eastern Europe to México as visiting scientists in the 90s. The topic "Symmetries" is of interest to several faculty members of the Zacatecas Physics School, while other events (held in 1999 and 2000) were also related to. We invited experts in group theory, non-associative algebras, category theory, differential geometry, non-linear theories, classical field theory, gauge theories, spin, etc in order to provide further impetus to these research areas at the Zacatecas University and in M\'exico altogether (the attention paid to them here is not yet sufficient).

Let us review briefly the papers included in this volume. The Mitskievich paper begins with a statement that "it seems that everybody knows what the electromagnetic field is". On the other hand, in the text we find the following statements, "gravitation does automatically regularize the global characteristics so that in general relativity local singularities do not anymore lead to an integral energy divergence" (remember the well-known Einstein problems of constructing the unified field theory?); "in 2 + 1 [dimension] field constructed from the vector (rank 1) potential ... essentially has properties of a fluid and not of electromagnetic field..."; "a complete elimination of the pure null type field is 'possible' only asymptotically"; "in the impure case it is impossible to transform away the counterparts of ... fields... However it is then possible to find a frame(s) in which these vectors are mutually parallel, so that the Poynting vector vanishes" (the discussion of these issues can be found in [1]); "when we speak about 'propagation of information', we do not speak strictly about propagation of electromagnetic field in the general sense: the question about the carrier of information ..still remains open"; "there is ... a widespread prejudice that non-inertial frames cannot be used in or they contradict to the special theory of relativity, but this is nothing more than a prejudice" etc. Are many experts familiar with them? Hopefully, people would read his recently published book [2] where these questions from the still open field of Relativistic Physics in Arbitrary Reference Frame have been discussed in detail. Apart, a careful analysis of the Liénard-Wiechert solution has been given in the works by Prof. Mitskievich.

The next paper is by M. Socolovsky from the Universidad Nacional Autónoma de México. He proves the existence of the charge conjugation operator (and, hence, the antiparticle waves) in the nonrelativistic quantum mechanics. The presentation of the basics of the Galilean group, and the corresponding Lagrangian formulation for the Pauli equation are very useful to students. I would like to mention a fact that the interest in the Galilean symmetries has grown recently [3]. The didactic presentation by M. Napsuciale and M.G. Carillo-Ruiz continues the volume. It is dedicated to the study of the Poincare group in d =3+1 and d = 2 + 1 dimensions, as well as the inversion operators C,P and T. In this connection, I recommend the classical books [4], and some of the papers of mine [5].

A theoretical investigation of the neutrino processes is a very important task, including the analysis within extensions of the Standard Model. Prof. Gutiérrez with pupils discusses it in the LRSM (the left-right symmetric model). We decided to include two of his papers, which serve, in fact, as basis for the Theses of his pupils F. Ramí rez-Sánchez and F. Péerez Vargas. The significant result is such that "the spectrum of energy of the electrons for [the beta-decay of tritium] is not affected by the left-right asymmetry parameter". It should be noted that such calculations are necessary in view of the fact that the neutrino oscillations [6] are not described in the Standard Model (the double beta decay, which has been recently claimed to be discovered, as well [7]). The second paper is very useful, indeed, because it presents the basic formalism of the Left-Right Symmetric Model and asserts that the number of visible species of leptons is three.

The Categorical Relativity of Prof. Oziewicz's is the result of his long-standing arguments, "Two Einstein's postulates have nothing to do with the Lorentz group invariance"...Ridiculous? Some people allege that this is true. More rarely in one's opinion would be the author's allegation that "the electromagnetic field is observer-independent" and that the proper time is not unique. The next paper is the one by Prof. Keller. He continues with his research based on the postulates that are given in the book [8], namely "the laws of physics are invariant under the 5D-Poincaré group which includes 5D-Lorentz transformations". The theories in five dimensions have been considered by many authors since the original works by Kaluca, Klein, Dirac (see ref. [9]), Rumer, Kadyshevsky, Greiner etc.

The Hamiltonization of arbitrary ordinary differential equations is an important issue. I prefer, of course, a set of the relativistic equations [10]. R. Alvarado Flores generalizes the Hojman framework which is much more general than that proposed earlier (including assessment from the mathematical viewpoint). The Pérez paper comes up next in our special issue. It is fully mathematical. It is a survey on the meaning of symmetry for a topologist. The J. Martńez Ortiz paper is also a mathematical one. It is related to his PhD Thesis defended recently in St. Petersburg (Russia). Multidisciplinary papers by H. Rosu, J.P. Treviño et al. and A. Enciso Muñoz finish up the special issue. I only wanted to mention that the wavelet transform theory (used in the Rosu papers) found its applications in many areas, including the electromagnetic theory [11]. I am sorry to say that the latter papers are rather of technical interest than of educational one, while the aim of the Workshop was to attract young students to work in mathematical physics.

Finally, I want to thank all participants (particularly outside of Zacatecas) for having come here and presented their thoughtful pieces of research. Next, I apologise for not being able to cite all relevant papers of these (and other) auhors in this Editorial Introduction.

The Workshop has been supported by the PROMEP (SEP, México), Secretarí a Académica de la UAZ (Dr. H. Rene Vega; I believe that participants will remember his inauguration talk - it was published in the UAZ periodicals), CIP UAZ (the Coordinador Mtro. Édgar Hurtado Hernández) and UAF UAZ (the Director Fis. A. Beltrán Mendosa; Cuerpo Académico de Partí culas, Campos y Astrofí sica, Drs. A.A. Espinoza Garrido, A.E. Chubykalo, A. Gutierrez, J. Martinez). We extend our special thanks to our Publisher, our authors and our friends.

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