In this paper we apply social epistemology to mathematical proofs and their role in mathematical knowledge. The most famous modern collaborative mathematical proof effort is the Classification of Finite Simple Groups. The history and sociology of this proof have been well-documented by Alma Steingart (2012), who highlights a number of surprising and unusual features of this collaborative endeavour that set it apart from smaller-scale pieces of mathematics. These features raise a number of interesting philosophical issues, but have received very little attention. In this paper, we will consider the philosophical tensions that Steingart uncovers, and use them to argue that the best account of the epistemic status of the Classification Theorem will be essentially and ineliminably social. This forms part of the broader argument that in order to understand mathematical proofs, we must appreciate their social aspects.

On the basis of extensive studies of synthetic perovskite-structured compounds it is possible to derive a hierarchy of hettotype structures which are derivatives of the arisotypic cubic perovskite structure (ABX3), exemplified by SrTiO3 (tausonite) or KMgF3 (parascandolaite) by: (1) tilting and distortion of the BX6 octahedra; (2) ordering of A- and B-site cations; (3) formation of A-, B- or X-site vacancies. This hierarchical scheme can be applied to some naturally-occurring oxides, fluorides,hydroxides, chlorides, arsenides, intermetallic compounds and silicates which adopt such derivative crystal structures. Application of this hierarchical scheme to naturally-occurring minerals results in the recognition of a perovskite supergroup which is divided into stoichiometric and non-stoichiometricperovskite groups, with both groups further divided into single ABX3 or double A2BB'X6 perovskites. Subgroups, and potential subgroups, of stoichiometric perovskites include: (1) silicate single perovskites of the bridgmanite subgroup;(2) oxide single perovskites of the perovskite subgroup (tausonite, perovskite, loparite, lueshite, isolueshite, lakargiite, megawite); (3) oxide single perovskites of the macedonite subgroup which exhibit second order Jahn-Teller distortions (macedonite, barioperovskite); (4) fluoride singleperovskites of the neighborite subgroup (neighborite, parascandolaite); (5) chloride single perovskites of the chlorocalcite subgroup; (6) B-site cation ordered double fluoride perovskites of the cryolite subgroup (cryolite, elpasolite, simmonsite); (7) B-site cation orderedoxide double perovskites of the vapnikite subgroup [vapnikite, (?) latrappite]. Non-stoichiometric perovskites include: (1) A-site vacant double hydroxides, or hydroxide perovskites, belonging to the söhngeite, schoenfliesite and stottite subgroups; (2) Anion-deficient perovskitesof the brownmillerite subgroup (srebrodolskite, shulamitite); (3) A-site vacant quadruple perovskites (skutterudite subgroup); (4) B-site vacant single perovskites of the oskarssonite subgroup [oskarssonite]; (5) B-site vacant inverse single perovskites of the coheniteand auricupride subgroups; (6) B-site vacant double perovskites of the diaboleite subgroup; (7) anion-deficient partly-inverse B-site quadruple perovskites of the hematophanite subgroup.

Every finite group $G$ acts on some nonorientable unbordered surfaces. The minimal topological genus of those surfaces is called the symmetric crosscap number of $G$. It is known that 3 is not the symmetric crosscap number of any group but it remains unknown whether there are other such values, called gaps. In this paper we obtain group presentations which allow one to find the actions realizing the symmetric crosscap number of groups of each group of order less than or equal to 63.

In recent years, much work in descriptive set theory has been focused on the Borel complexity of naturally occurring classification problems, in particular, the study of countable Borel equivalence relations and their structure under the quasi-order of Borel reducibility. Following the approach of Louveau and Rosendal for the study of analytic equivalence relations, we study countable Borel quasi-orders.

In this paper we are concerned with universal countable Borel quasi-orders, i.e., countable Borel quasi-orders above all other countable Borel quasi-orders with regard to Borel reducibility. We first establish that there is a universal countable Borel quasi-order, and then establish that several countable Borel quasi-orders are universal. An important example is an embeddability relation on descriptive set theoretic trees.

Our main result states that embeddability of finitely generated groups is a universal countable Borel quasi-order, answering a question of Louveau and Rosendal. This immediately implies that biembeddability of finitely generated groups is a universal countable Borel equivalence relation. The same techniques are also used to show that embeddability of countable groups is a universal analytic quasi-order.

Finally, we show that, up to Borel bireducibility, there are ${2^{{\aleph _0}}}$ distinct countable Borel quasi-orders, which symmetrize to a universal countable Borel equivalence relation.

In this paper, we present a new method of enumeration of parallel manipulators with one end-effector. The method consists of enumerating all the manipulators possible with one end-effector that a single kinematic chain can originate. A very useful simplification for kinematic chain, mechanism and manipulator enumeration is their representation through graphs. The method is based on group theory where abstract structures are used to capture the internal symmetry of a structure in the form of automorphisms of a group. The concept used is orbits of the group of automorphisms of a colored vertex graph. The theory and some examples are presented to illustrate the method.

We provide a wedge decomposition of the homotopy type of the p-subgroup complex in the case of a finite solvable group G. In particular, this includes a new proof of the result of Quillen which says that this complex is contractible if and only if there is a non-trivial normal p-subgroup in G. We also provide reduction formulas for the G-module structure of the homology groups. Our results are obtained with diagram-methods by gluing the p-subgroup complex of G along the p-subgroup complex of = G/N for a normal p′-subgroup of G.

This paper provides explicit formulas for the Wall polynomials which arise in Wall's work on conjugacy classes in the unitary, symplectic and orthogonal groups. From these explicit formulas are easily derived six interesting limiting idetities incuding the two that arise in the Luszing-Macdonald Wall conjectures.