The Asymptotics of Wilkinson's Iteration: Loss of Cubic Convergence

One of the most widely used methods for eigenvalue computation is the $QR$ iteration with Wilkinson's shift: here the shift $s$ is the eigenvalue of the bottom $2\times 2$ principal minor closest to the corner entry. It has been a long-standing conjecture that the rate of convergence of the algorithm is cubic. In contrast, we show that there exist matrices for which the rate of convergence is strictly quadratic. More precisely, let $T_X$ be the $3 \times 3$ matrix having only two nonzero entries $(T_X)_{12} = (T_X)_{21} = 1$ and let $T_L$ be the set of real, symmetric tridiagonal matrices with the same spectrum as $T_X$. There exists a neighborhood $U \subset T_L$ of $T_X$ which is invariant under Wilkinson's shift strategy with the following properties. For $T_0 \in U$, the sequence of iterates $(T_k)$ exhibits either strictly quadratic or strictly cubic convergence to zero of the entry $(T_k)_{23}$. In fact, quadratic convergence occurs exactly when $\lim T_k = T_X$. Let $X$ be the union of such quadratically convergent sequences $(T_k)$: the set $X$ has Hausdorff dimension 1 and is a union of disjoint arcs $X^\sigma$ meeting at $T_X$, where $\sigma$ ranges over a Cantor set.Comment: 20 pages, 8 figures. Some passages rewritten for clarit

Ribotyping of streptococcus uberis from a dairy's environment, bovine feces and milk

Streptococcus uberis is a major cause of bovine mastitis and infections commonly result from environmental exposure to the pathogen. To identify specific sources of mastitis-causing S. uberis strains, samples were collected monthly from the environment and feces of dry cows in a grazing herd. Environmental and fecal strains of S. uberis were compared to those found in milk. S. uberis was detected in 63% of 94 environmental samples, including water, soil, plant matter, bedding material, flies, and hay, in 23% of 107 fecal samples, and in 4% of 787 milk samples. Automated PvuII ribotyping revealed 48 ribotypes among 266 isolates. Per sample, up to five ribotypes were detected. The distribution of ribotypes did not differ significantly among environmental, fecal and milk samples. Specific environmental sources or strains of udder-pathogenic S. uberis were not identified. Fecal shedding was not persistent and did not differ between dry-off and calving. The proportion of fecal samples containing S. uberis was highest during the summer grazing season. S. uberis was common in farm soil (31 of 35 samples) but not in non-farm soil (0 of 11 samples). We hypothesize that fecal shedding of S. uberis may play a role in maintenance of S. uberis populations in the dairy ecosystem

Molecular Subtyping And Tracking Of Listeria Monocytogenes In Latin-style Fresh-cheese Processing Plants

Latin-style fresh cheeses, which have been linked to at least 2 human listeriosis outbreaks in the United States, are considered to be high-risk foods for Listeria monocytogenes contamination. We evaluated L. monocytogenes contamination patterns in 3 Latin-style fresh-cheese processing plants to gain a better understanding of L. monocytogenes contamination sources in the manufacture of these cheeses. Over a 6-mo period, 246 environmental samples were collected and analyzed for L. monocytogenes using both the Food and Drug Administration (FDA) method and the Biosynth L. monocytogenes detection system (LMDS). Finished cheese samples from the same plants (n = 111) were also analyzed by the FDA method, which was modified to include L. monocytogenes plating medium (LMPM) and the L. monocytogenes confirmatory plating medium (LMCM) used in the LMDS method. Listeria monocytogenes was detected in 6.3% of cheese and 11.0% of environmental samples. Crates, drains, and floor samples showed the highest contamination rates, with 55.6, 30.0, and 20.6% L. monocytogenes positive samples, respectively. Finished products and food contact surfaces were positive in only one plant. The FDA method showed a higher sensitivity than the LMDS method for detection of L. monocytogenes from environmental samples. The addition of LMPM and LMCM media did not further enhance the performance of the FDA method for L. monocytogenes detection from finished products. Molecular subtyping (PCR-based allelic analysis of the virulence genes actA and hly and automated ribotyping) was used to track contamination patterns. Ribotype DUP-1044A, which had previously been linked to a 1998 multistate human listeriosis outbreak in the United States, was the most commonly identified subtype (20/36 isolates) and was isolated from 2 plants. This ribotype was persistent and widespread in one factory, where it was also responsible for the contamination of finished products. We hypothesize that this ribotype may represent a clonal group with a specific ability to persist in food processing environments. While previous listeriosis outbreaks were linked to Latin-style fresh cheeses made from unpasteurized milk, the presence of this organism in pasteurized cheese products illustrates that persistent environmental contamination also represents an important source of finished product contamination.87928032812Outbreak of listeriosis associated with homemade Mexican-style cheese - North Carolina, October 2000 - January 2001 (2001) MMWR, 50, pp. 560-562Arimi, S.M., Ryser, E.T., Pritchard, T.J., Donnelly, C.W., Diversity of Listeria ribotypes recovered from dairy cattle, silage, and dairy processing environments (1997) J. 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Mastitis-causing streptococci are important contributors to bacterial counts in raw bulk tank milk

The objective of this study was to probe the contribution of streptococci to the microbial quality of raw milk. Over a 5-month period, bulk tank milk samples from 48 New York State dairy farms were analyzed qualitatively for bacterial ecology and quantitatively for total bacterial, streptococcal, staphylococcal, and gram-negative bacterial counts. Linear regression analysis was used to determine the contribution of differential counts to total bacterial counts. Streptococci, staphylococci, and gram-negative bacteria accounted for 69, 3, and 3% of total bacterial count variability, respectively. Randomly selected Streptococcus isolates from each bulk tank milk sample were identified to species by means of the API 20 STREP identification system. The most commonly identified streptococcal species were Streptococcus uberis, Aerococcus viridans, and Streptococcus agalactiae, which were detected in 81, 50, and 31% of 48 bulk tank samples, respectively. For five herds, S. uberis isolates from bulk tank milk and individual cows were characterized by PvuII ribotyping. A farm-specific dominant ribotype was found in each bulk tank sample, and that ribotype was isolated from at least one cow within each herd of origin. Bacteriological and strain typing data indicate that control of streptococci, specifically mastitis-causing species, is important for improvement of the microbial quality of raw milk in New York State

Development of molecular typing methods for Bacillus spp. and Paenibacillus spp. isolated from fluid milk products

Bacillus spp. and related sporeformers are important food spoilage organisms. While use of molecular subtyping methods has provided important information on the ecology and transmission of foodborne pathogens, the lack of rapid, reliable, and affordable subtyping methods for Bacillus spp. has limited our ability to understand and control their transmission throughout the food chain. We used a previously described collection of Bacillus spp. and Paenibacillus spp. isolated from dairy products to develop a DNA sequencing-based subtyping approach for these spoilage microorganisms. After optimization of polymerase chain reaction (PCR) parameters, primers targeting the rpoB housekeeping gene allowed for successful amplification in all isolates. rpoB sequencing allowed differentiation of 29 subtypes (that is, sequence types) among the 57 isolates characterized. Phylogenetic analyses of rpoB sequences revealed distinct monophyletic lineages that correlated with bacterial genera (Bacillus and Paenibacillus) as well as with species or species-like assemblages within each genus. rpoB sequencing provided improved subtype discrimination over 16S rDNA sequencing; therefore, rpoB sequencing allows for both sensitive subtype discrimination as well as for species and genus identification. Analysis of subtypes isolated over time in dairy products revealed the presence of both persistent and transient bacterial subtypes, indicating that application of these methods can improve our understanding of the ecology of these spoilage organisms and can help in identification of bacterial niches that may contribute to the persistence of these spoilage organisms in food systems

Evaluation of different methods to detect microbial hygiene indicators relevant in the dairy industry

AbstractIt is estimated that 19% of the total food loss from retail, food service, and households comes from dairy products. A portion of this loss may be attributed to premature spoilage of products due to lapses in sanitation and postpasteurization contamination at the processing level. Bacterial groups including coliforms, Enterobacteriaceae (EB), and total gram-negative organisms represent indicators of poor sanitation or postpasteurization contamination in dairy products worldwide. Although Petrifilms (3M, St. Paul, MN) and traditional selective media are commonly used for the testing of these indicator organism groups throughout the US dairy industry, new rapid methods are also being developed. This project was designed to evaluate the ability of different methods to detect coliforms, EB, and other gram-negative organisms isolated from various dairy products and dairy processing environments. Using the Food Microbe Tracker database, a collection of 211 coliform, EB, and gram-negative bacterial isolates representing 25 genera associated with dairy products was assembled for this study. We tested the selected isolates in pure culture (at levels of approximately 15 to 300 cells/test) to evaluate the ability of 3M Coliform Petrifilm to detect coliforms, 3M Enterobacteriaceae Petrifilm, violet red bile glucose agar, and an alternative flow cytometry-based method (bioMérieux D-Count, Marcy-l’Étoile, France) to detect EB, and crystal violet tetrazolium agar to detect total gram-negative organisms. Of the 211 gram-negative isolates tested, 82% (174/211) had characteristic growth on crystal violet tetrazolium agar. Within this set of 211 gram-negative organisms, 175 isolates representing 19 EB genera were screened for detection using EB selective/differential testing methods. We observed positive results for 96% (168/175), 90% (158/175), and 86% (151/175) of EB isolates when tested on EB Petrifilm, violet red bile glucose agar, and D-Count, respectively; optimization of the cut-off thresholds for the D-Count may further improve its sensitivity and specificity, but will require additional data and may vary in food matrices. Additionally, 74% (129/175) of the EB isolates tested positive as coliforms. The data obtained from this study identify differences in detection between 5 microbial hygiene indicator tests and highlight the benefits of EB and total gram-negative testing methods

Longitudinal assessment of dairy farm management practices associated with the presence of psychrotolerant Bacillales spores in bulk tank milk on 10 New York State dairy farms

The ability of certain spore-forming bacteria in the order Bacillales (e.g., Bacillus spp., Paenibacillus spp.) to survive pasteurization in spore form and grow at refrigeration temperatures results in product spoilage and limits the shelf life of high temperature, short time (HTST)-pasteurized fluid milk. To facilitate development of strategies to minimize contamination of raw milk with psychrotolerant Bacillales spores, we conducted a longitudinal study of 10 New York State dairy farms, which included yearlong monthly assessments of the frequency and levels of bulk tank raw milk psychrotolerant spore contamination, along with administration of questionnaires to identify farm management practices associated with psychrotolerant spore presence over time. Milk samples were first spore pasteurized (80°C for 12 min) and then analyzed for sporeformer counts on the initial day of spore pasteurization (SP), and after refrigerated storage (6°C) for 7, 14, and 21 d after SP. Overall, 41% of samples showed sporeformer counts of >20,000 cfu/mL at d 21, with Bacillus and Paenibacillus spp. being predominant causes of high sporeformer counts. Statistical analyses identified 3 management factors (more frequent cleaning of the bulk tank area, the use of a skid steer to scrape the housing area, and segregating problem cows during milking) that were all associated with lower probabilities of d-21 Bacillales spore detection in SP-treated bulk tank raw milk. Our data emphasize that appropriate on-farm measures to improve overall cleanliness and cow hygiene will reduce the probability of psychrotolerant Bacillales spore contamination of bulk tank raw milk, allowing for consistent production of raw milk with reduced psychrotolerant spore counts, which will facilitate production of HTST-pasteurized milk with extended refrigerated shelf life