The Moore-Penrose Pseudoinverse. A Tutorial Review of the Theory

In the last decades the Moore-Penrose pseudoinverse has found a wide range of applications in many areas of Science and became a useful tool for physicists dealing, for instance, with optimization problems, with data analysis, with the solution of linear integral equations, etc. The existence of such applications alone should attract the interest of students and researchers in the Moore-Penrose pseudoinverse and in related sub jects, like the singular values decomposition theorem for matrices. In this note we present a tutorial review of the theory of the Moore-Penrose pseudoinverse. We present the first definitions and some motivations and, after obtaining some basic results, we center our discussion on the Spectral Theorem and present an algorithmically simple expression for the computation of the Moore-Penrose pseudoinverse of a given matrix. We do not claim originality of the results. We rather intend to present a complete and self-contained tutorial review, useful for those more devoted to applications, for those more theoretically oriented and for those who already have some working knowledge of the sub ject.Comment: 23 page

New data on the pteromalid wasps (Hymenoptera: Chalcidoidea: Pteromalidae) of the fauna of Turkey

Abstract. The сhalcid wasps from the family Pteromalidae (Chalcidoidea) were studied in three Turkish provinces: Nevşehir, Muğla and Antalya. In total, 28 species from 18 genera were collected. The subfamily Trigonoderinae (Chalcidoidea: Pteromalidae), four genera (Plutothrix Förster, 1856, Stenoselma Delucchi, 1956, Trigonoderus Westwood, 1832, and Trychnosoma Graham, 1957) and 11 species are recorded for Turkey for the first time: Anisopteromalus apiovorus Rasplus, 1988, A. quinarius Gokhman et Baur, 2014, Cyrtoptyx gallicola Dzhanokmen, 1976, Homoporus gibbiscuta (Thomson, 1878), H. nypsius (Walker, 1839), Plutothrix coelius (Walker, 1839), Pteromalus speculifer Graham, 1981, P. temporalis (Graham, 1969), Stenoselma nigrum Delucchi, 1956, Trigonoderus cyanescens (Förster, 1841), and Trychnosoma punctipleura (Thomson, 1878). According to our study and previously published data the pteromalid fauna of Turkey comprises currently 176 species belonging to 59 genera of five subfamilies