On the expected number of generators of a submodule of a free module over a finite principal ideal ring

This work concerns the expected minimal cardinality of a generating set of an arbitrary submodule of a finitely generated free module over a finite (commutative)principal ideal ring (PIR). Our primary objective is to obtain ring- and module theoretic generalization of the work that had been done earlier in the finite field vector space situation by Dobbs and Lancaster.In Chapter I, we provide background material, conventions, and some of the definitions needed for the work in Chapters II-IV.Chapter II addresses finding the expected minimal cardinality of a generating set of an arbitrary submodule of a finitely generated free module over a finite special principal ideal ring (SPIR). We generalize the various “weighted” limiting expected values found by Dobbs in the field-vector space situation to an SPIR-module setting which Dobbs had explored in a special case. As a byproduct, we find a formula for the number of submodules of a given finitely generated free module (over a finiteSPIR) which require a prescribed number of generators.In Chapter III, using a basic structure theorem of PIRs and results from Chapter II, we obtain limiting expected values for the PIR-module situation. Herewe investigate the various “weighting” contexts for limiting expected values which were introduced by Dobbs and we also address some other “weighting” contexts that do not arise naturally for vector spaces.In contrast to the PIR settings in Chapters II and III, Chapter IV investigates some ring-module situations which have limiting expected values that behave in qualitatively different ways from the answers found in the (S)PIR settings

Closed k-stop distance in graphs

Discussions Mathematicae Graph Theory 31 (2011) 533 – 545 bibte

γ-labelings of complete bipartite graphs

Explicit formulae for the γ-min and γ-max labeling values of complete bipartite graphs are given, along with γ-labelings which achieve these extremes. A recursive formula for the γ-min labeling value of any complete multipartite is also presented

Insecticidal Toxic Proteins Produced by Photorhabdus luminescens akhurstii, a Symbiont of Heterorhabditis indica

We describe the isolation and characterization of an insect pathogenic bacterium from the entomopathogenic nematode Heterorhabditis indica (Karnataka strain), an isolate from the southern regions of India. The strain has been identified and characterized by phenotypic, biochemical tests and PCR-RFLP analysis of the 16S rRNA gene as Photorhabdus luminescens subsp. akhurstii. The insecticidal toxin complex produced by this bacterium has been purified through a series of steps including ultrafiltration, anion exchange chromatography, and gel filtration chromatography. The toxin consists of two protein complexes of approximately 1,000 kD and was active against the larvae of Spodoptera litura and Galleria mellonella