Results Of Singer Research Articles

Derivations of Murray-von Neumann Algebras

A Murray-von Neumann algebra is the algebra of operators affiliated with a finite von Neumann algebra. In this article, we study derivations of Murray-von Neumann algebras and their properties. We show that the "extended derivations" of a Murray-von Neumann algebra, those that map the associated finite von Neumann algebra into itself, are inner. In particular, we prove that the only derivation that maps a Murray-von Neumann algebra associated with a von Neumann algebra of type ${\rm II}_1$ into that von Neumann algebra is 0. This result is an extension, in two ways, of Singer's seminal result answering a question of Kaplansky, as applied to von Neumann algebras: the algebra may be non-commutative and contain unbounded elements. In another sense, as we indicate in the introduction, all the derivation results including ours extend what Singer's result says, that the derivation is the 0-mapping, numerically in our main theorem and cohomologically in our theorem on extended derivations. The cohomology in this case is the Hochschild cohomology for associative algebras.

THE SPECTRUM OF RANDOM KERNEL MATRICES: UNIVERSALITY RESULTS FOR ROUGH AND VARYING KERNELS

We consider random matrices whose entries are [Formula: see text] or f(‖Xi – Xj‖2) for iid vectors Xi ∈ ℝp with normalized distribution. Assuming that f is sufficiently smooth and the distribution of Xi's is sufficiently nice, El Karoui [The spectrum of Kernel random matrices, Ann. Statist.38(1) (2010) 1–50, MR 2589315 (2011a.62187)] showed that the spectral distributions of these matrices behave as if f is linear in the Marčhenko–Pastur limit. When Xi's are Gaussian vectors, variants of this phenomenon were recently proved for varying kernels, i.e. when f may depend on p, by Cheng–Singer [The spectrum of random inner-product Kernel matrices, preprint (2012), arXiv:1202.3155 [math.PR]]. Two results are shown in this paper: first, it is shown that for a large class of distributions the regularity assumptions on f in El Karoui's results can be reduced to minimal; and second, it is shown that the Gaussian assumptions in Cheng–Singer's result can be removed, answering a question posed in [The spectrum of random inner-product Kernel matrices, preprint (2012), arXiv:1202.3155 [math.PR]] about the universality of the limiting spectral distribution.

Module structures and the derived functors of iterated loop functors on unstable modules over the Steenrod algebra

The calculation of the iterated loop functors and their left derived functors on the category of unstable modules over the Steenrod algebra is a non-trivial problem; Singer constructed an explicit and functorial chain complex to calculate these functors. The results of Singer are analysed to give information on the behaviour of these functors with respect to the nilpotent filtration of the category of unstable modules. We show that, if an unstable module M supports an action of an unstable algebra K , then the derived functors of the iterated loop functors applied to M support actions of iterated doubles of K . This allows the finiteness results of Henn on unstable modules which support actions of unstable algebras to be applied to deduce structural results on the derived functors of iterated loops on such modules.

FURTHER TOPOLOGICAL PROOFS OF GRIBOV AMBIGUITIES

We show the existence of Gribov ambiguity for gauge group SU (N) and large classes of space-time manifolds with dimension less than or equal to four, working in the continuous category, extending previous results of Singer. In lower dimensions, and in four dimensions with gauge group SU (N), N≥3, we require only that the manifold be compact and orientable. In four dimensions with gauge group SU (2) there is a slight complification due to the fact that π4( SU (2)) does not vanish, though we are still able to state a useful result for that case. Some discussion of motivation is presented, in particular as regards to recent gauge-fixing proposals which arise in work that attempts to relate the Gribov ambiguity to confinement.

Strengthening of Singer's result on absence of global gauge fixing

Strengthening of Singer's result on absence of global gauge fixing

Inference from censored Markov chains with applications to multiwave panel data

We consider some problems of inference from censored discrete-time Markov chains. The censoring mechanism is general enough to be useful in applications and maintains the conditional multinomial sampling model for successive transitions. Procedures are given for nonparametric estimation of transition probabilities and tests of the intrinsic-time and time-homogeneity hypotheses. These procedures apply to multiwave panel data when continuous-time Markov chains are only periodically monitored. To illustrate this we build on a result of Singer and Cohen (1979) by showing that a nonhomogeneous Markov chain without an intrinsic-time is a reasonable model for transition data from a WHO malaria survey.

Periodic Linear Transformations of Affine and Projective Geometries

Introduction. In a paper called “A Theorem in Finite Projective Geometry and some Applications to Number Theory” [Trans. Amer. Math. Soc, vol. 43 (1938), 377-385], J. Singer proved that the finite projective geometry PG(s — 1,pn), that is the projective geometry of dimension s — 1 whose coordinate field is the Galois field GF(pn), admits a collineation L of period q = (psn — 1)/ (pn — 1). Since this q is the number of points of PG(s — 1, pn), Singer's result states that the points of PG(s — 1, pn) are cyclically arranged. Singer's construction of L uses the notion of a “primitive irreducible polynomial of degree 5 belonging to a field GF(pn) which defines a PG(s — 1, pn).”

THE INFLUENCE OF VITAMIN EDEFICIENCY ON THE THYROID AND BASAL METABOLISM OF MALE RATS1

SINGER (1) AND BARRIE (2) have presented data which indicated that vitamin-Edeficiency has an influence on the histology of the thyroid gland. The former described a hypoplasia of the thyroid of 12 to 18-month E-deficient female rats, and the latter a hypoplasia of the thyroid of both E-low mothers and young born to them. The results of Singer and Barrie have been questioned by Telford, Emerson and Evans (3), who reported that they could find no evidence of hypothyroidism in E-low female rates or in young born to them. We have previously reported (4) that the weight of the thyroids of male rats deficient in vitamin E for 6 months or more was approximately twice that of normal controls, whereas the weight of thyroids from female E-deficient rats was not significantly different from that of normals in any of the age groups studied