Smooth analysis of the condition number and the least singular value

Let \a be a complex random variable with mean zero and bounded variance. Let $N_{n}$ be the random matrix of size $n$ whose entries are iid copies of \a and $M$ be a fixed matrix of the same size. The goal of this paper is to give a general estimate for the condition number and least singular value of the matrix $M + N_{n}$, generalizing an earlier result of Spielman and Teng for the case when \a is gaussian. Our investigation reveals an interesting fact that the "core" matrix $M$ does play a role on tail bounds for the least singular value of $M+N_{n}$. This does not occur in Spielman-Teng studies when \a is gaussian. Consequently, our general estimate involves the norm $\|M\|$. In the special case when $\|M\|$ is relatively small, this estimate is nearly optimal and extends or refines existing results.Comment: 20 pages. An erratum to the published version has been adde

Uwarunkowania bioutylizacji odpadow tluszczowych w warunkach laboratoryjnych

Celem pracy była próba optymalizacji procesu utylizacji odpadów porafinacyjnych w postaci zaolejonej ziemi bielącej w warunkach laboratoryjnych. Zaolejona ziemia bieląca charakteryzowała się 14% zawartością tłuszczu oraz pH 4,3. W pracy uwzględniono wpływ odpowiednio dobranych szczepów bakteryjnych i drożdżowych, pH oraz dodatku węgla brunatnego i torfu na stopień rozkladu substancji tłuszczowych zawartych w odpadach. Najlepsze efekty redukcji tłuszczu uzyskano po zastosowaniu kompozytu, który był mieszaniną szczepów pochodzących z kolekcji i izolowanych z zaolejonej ziemi bielącej. Na intensywność mineralizacji odpadów tłuszczowych przez drobnoustroje lipolityczne najkorzystniejszy wpływ miał ok. 35% dodatek mieszaniny węgla brunatnego i torfu oraz neutralizacja pH z 4,3 do 8,0.The aim of study was to optimize the process of post-refinement waste (in this case the oiled bleaching soil) utilization on laboratory scale. The oiled bleaching soil was characterized by 14% aliphatic substance content and pH 4.3. The paper assesses the effect of such factors as aptly selected bacteria and yeast strains, pH, as well as addition of brown coal and peat on the efficacy of the aliphatic substance degradation process. Best results in fat reduction were obtained with the use of a composite consisting of library strains and strains isolated from oiled bleaching soil. Mineralization of fatty waste by lipolytical bacteria was most efficient when ca 35% brown coal and peat was added at pH neutralized from 4.3 to 8.0

The phytobenthos of the Hornsund fiord, SW Spitsbergen

The species composition and biomass of material collected in the summers 1981 and 1983 from 69 stations in the Hornsund fiord were analysed. 48 macroalgal taxa were found: 13 green algae, 18 brown algae and 17 red algae. 19 of these taxa were not reported from the nearby Isfjorden. The overall biomass for the littoral zone varied from 0 to 216gnT2 formalin d.w., although in J of samples the biomass was between 20 and 80 gm_z. The dominant species were Fucus distichus, Acrosiphonia duriusada and Pilayella littoralis. Using cluster analysis, an attempt was made to delimit regions with a characteristic species composition. The results obtained, however, do not permit this in the littoral zone of the Hornsund fiord