Naturalising Austin

In this paper I will try to defend a quasi-naturalistic interpretation of J.L. Austin’s work. I will rely on P. Kitcher’s 1992 paper “The Naturalists Return” to compile four general criteria by which a philosopher can be called a naturalist. Then I will turn to Austin’s work and examine whether he meets these criteria. I will try to claim that versions of such naturalistic elements can be found in his work.     

Partition identities arising from theta function identities

The authors show that certain theta function identities of Schroeter and Ramanujan imply elegant partition identities.     

Polynomial identities

This article contains two surveys: (A) A historical survey of the early literature of 1920–1950 in Polynomial Identities which follows the roots and sources of PI-rings. (B) A survey of the methods of constructing identities for matrix rings, since the identity of Wagner until the central identities of Formanek and Razmyslov and the rational identity of Bergman.     

Abel's identities

We generalize Abel's identities (cf. H. Weber, Lehrbuch der Algebra,2, 205–216 (1908)) for elliptic curves.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 121, pp. 58–61, 1983.     

Universal curvature identities

We study scalar and symmetric 2-form valued universal curvature identities. We use this to establish the Gauss–Bonnet theorem using heat equation methods, to give a new proof of a result of Kuz?mina and Labbi concerning the Euler–Lagrange equations of the Gauss–Bonnet integral, and to give a new derivation of the Euh–Park–Sekigawa identity.     

Triangular extension identities

It is shown that if the - -bimodule M generates a category of - -bimodules, then the ideal of identities of the triangular extension of the direct sum of algebras and by means of the bimodule M is equal to the product of ideals of identities of the algebras and .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 132, pp. 5–11, 1983.     

On bianchi identities

Résumé  Après un bref bilan concernant preuves et applications des identités de Bianchi, nous complétons l’approche récentevia la naturalité, due à Kazdan, en l’étendant des connexions riemanniennes aux connexions linéaires; puis nous proposons pour ces connexions une nouvelle approche basée seulement sur l’identitéd ²=0 appliquée aux 1-formes.

---

---

Chargé de recherches, UMR 6621 du CNRS.     

Some combinatorial identities

The paper introduces a series of new combinatorial identities which allow one to find the value of the partial sums of some multiple power series. Examples are given of the application of these identities to the summation of infinite series and to the computation of the quantiles of discrete probability distributions.Translated from Matematicheskie Zametki, Vol. 11, No. 1, pp. 121–127, January, 1972.     

Homogeneous polynomial identities

PI-algebras are studied by attaching invariants to the homogeneous identities analogous to the invariants of the multilinear identities studied by Regev. Also, it is shown that every finitely generated PI-algebra is polynomially bounded.     

Degree three identities

Out approach differs significantly from the approaches used by other mathematicians because we introduce a canonical form for identities. It is straightforward to place an identity in canoni-cal form, and, once the identities are in canonical form, it is easy to determine if they are equivalent, quasi-equivalent, or if one implies the other. This approach is used to identify the dis-tinct minimal degree three identities under the equivalence rela-tion of quasi-equivalence. We determine the exact conditions when almost left alternativity implies almost alternativity and when almost left alternativity implies power associativity. We give a method to decide when an algehra satisfying degree three identi-ties is almost alternati.     

On Cantor identities

In this paper, some special varieties which generalize Jónsson?CTarski algebras are considered. We prove that every nontrivial algebra from such a variety is term infinite and contains infinitely many distinct proper diagonal term operations of every arity.     

Differential commutator identities

I.N. Herstein proved that if R is a prime ring satisfying a differential identity , with d a nonzero derivation of R, then R embeds isomorphically in M₂(F) for F a field. We consider a natural generalization of this result for the class of polynomials E_n(X)=[E_n-1(x₁,…,x_n-1),x_n]. Using matrix computations, we prove that if R satisfies a differential identity , or with some restrictions, then R must embed in M₂(F), but that differential identities using [[E_n,E_m],E_s] with m,n,s>1 need not force R to embed in M₂(F). These results hold if the expressions are identities for a noncommutative Lie ideal of R, rather than for R itself.     

Self-dual lattice identities

Given a finitely based self-dual variety of lattices, is it definable, modulo lattice theory, by a single self-dual lattice identity? There are infinitely many examples with “yes” as the answer and infinitely many with “no.”. Received December 20, 1999; accepted in final form July 11, 2002.     

Multigraded Jacobi identities

We construct a basis of the kernel of the Lie bracketing map restricted to the submodule of the free associative algebra generated by all words having exactly one occurrence of both of two fixed letters.