Showing papers in "Serdica. Mathematical Journal in 2011"

Double complexes and vanishing of Novikov cohomology

Abstract: We consider a non-standard totalisation functor to produce a cochain complex from a given double complex D∗,∗: instead of sums or products, totalisation is defined via truncated products of modules. We give an elementary proof of the fact that a double complex with exact rows (resp., columns) yields an acyclic cochain complex under totalisation using right (resp., left) truncated products. As an application we consider the algebraic mapping torus T (h) of a self map h of a cochain complex C. We show that if C consists of finitely presented modules then T (h) has trivial negative Novikov cohomology; if in addition h is a quasi-isomorphism, then T (h) has trivial positive Novikov cohomology as well. As a consequence we obtain a new proof that a finitely dominated cochain complex over a Laurent polynomial ring has trivial Novikov cohomology. Finiteness conditions for chain complexes of modules play an important role in both algebra and topology. For example, given a group G one might ask whether the trivial G-module Z admits a resolution by finitely generated projective Z[G]-modules; existence of such resolutions is relevant for the study of group cohomology of G, and has applications in the theory of duality groups [B75]. For topologists, finite domination of chain complexes is related, among other things, to questions about finiteness of CW complexes, the topology of ends of manifolds, and obstructions for the existence of nonsingular closed 1-forms [Ran95, S06]. A cochain complex C of R[z, z−1]-modules is called finitely dominated if it is homotopy equivalent, as a complex of R-modules, to a bounded complex of finitely generated projective R-modules. Finite domination of C can be characterised in various ways; Brown considered compatibility of the functors M 7→ H∗(C;M) and M 7→ H∗(C;M) with products and direct limits, respectively [B75, Theorem 1], while Ranicki showed that C is finitely dominated if and only if the Novikov cohomology of C is trivial [Ran95, Theorem 2] (see also Definition 2.3 and Corollary 2.7 below). Our approach to Novikov cohomology involves a non-standard totalisation functor. The key fact is that certain double complexes are converted into acyclic cochain complexes (Proposition 1.2) which is proved by an elementary calculation. As an application we obtain a new result for vanishing of Novikov cohomology of algebraic mapping tori (Theorem 2.5), and recover the “only-if” part of Ranicki’s criterion for finite domination over Date: 08.09.2011. 2000 Mathematics Subject Classification. Primary 18G35; Secondary 55U15. This work was supported by the Engineering and Physical Sciences Research Council [grant number EP/H018743/1]. 1

On realizability of p-groups as Galois groups

Abstract: In this article we survey and examine the realizability of $p$-groups as Galois groups over arbitrary fields. In particular we consider various cohomological criteria that lead to necessary and sufficient conditions for the realizability of such a group as a Galois group, the embedding problem (i.e., realizability over a given subextension), descriptions of such extensions, automatic realizations among $p$-groups, and related topics.

Zolotarev's proof of Gauss reciprocity and Jacobi symbols

Abstract: Egor Zolotarev (Nouvelle demonstration de la loi de reciprocite de Legendre, Nouv. Ann. Math (2), 11 (1872), p. 354-362) noticed that sign of permutation x 7−→ ax mod p of the set of mod p residues is the Legendre symbol ( a p ) and has applied this observation to his proof of Gauss Reciprocity. Actually, the sign of permutation x 7−→ ax modn of the set of modn residues is the Jacobi symbol ( a n ) for composite n. We prove Reciprocity Law for Jacobi symbols directly, using a variant of Zolotarev reasoning.

Optimal investment under stochastic volatility and power type utility function

Abstract: In this work we will study a problem of optimal investment in financial markets with stochastic volatility with small parameter. We used the averaging method of Bogoliubov for limited development for the optimal strategies when the small parameter of the model tends to zero and the limit for the optimal strategy and demonstrated the convergence of these optimal strategies.

Bayesian and Frequentist Two-Sample Predictions of the Inverse Weibull Model Based on Generalized Order Statistics

Abstract: This paper is concerned with the problem of deriving Bayesian prediction bounds for the future observations (two-sample prediction) from the inverse Weibull distribution based on generalized order statistics (GOS). Study the two side interval Bayesian prediction, point prediction under symmetric and asymmetric loss functions and the maximum likelihood (ML) prediction using “plug-in” procedure for future observations from the inverse Weibull distribution based on GOS. Study the problem of predicting future records based on observed progressive type II censored data and observed order statistics from the inverse Weibull distribution. Finally, a numerical example using real data are used to illustrate the procedure.