How large are the level sets of the Takagi function?

Let T be Takagi's continuous but nowhere-differentiable function. This paper considers the size of the level sets of T both from a probabilistic point of view and from the perspective of Baire category. We first give more elementary proofs of three recently published results. The first, due to Z. Buczolich, states that almost all level sets (with respect to Lebesgue measure on the range of T) are finite. The second, due to J. Lagarias and Z. Maddock, states that the average number of points in a level set is infinite. The third result, also due to Lagarias and Maddock, states that the average number of local level sets contained in a level set is 3/2. In the second part of the paper it is shown that, in contrast to the above results, the set of ordinates y with uncountably infinite level sets is residual, and a fairly explicit description of this set is given. The paper also gives a negative answer to a question of Lagarias and Maddock by showing that most level sets (in the sense of Baire category) contain infinitely many local level sets, and that a continuum of level sets even contain uncountably many local level sets. Finally, several of the main results are extended to a version of T with arbitrary signs in the summands.Comment: Added a new Section 5 with generalization of the main results; some new and corrected proofs of the old material; 29 pages, 3 figure

Helical distribution of the bacterial chemoreceptor via colocalization with the Sec protein translocation machinery

In Escherichia coli, chemoreceptor clustering at a cell pole seems critical for signal amplification and adaptation. However, little is known about the mechanism of localization itself. Here we examined whether the aspartate chemoreceptor (Tar) is inserted directly into the polar membrane by using its fusion to green fluorescent protein (GFP). After induction of Tar–GFP, fluorescent spots first appeared in lateral membrane regions, and later cell poles became predominantly fluorescent. Unexpectedly, Tar–GFP showed a helical arrangement in lateral regions, which was more apparent when a Tar–GFP derivative with two cysteine residues in the periplasmic domain was cross-linked to form higher oligomers. Moreover, similar distribution was observed even when the cytoplasmic domain of the double cysteine Tar–GFP mutant was replaced by that of the kinase EnvZ, which does not localize to a pole. Observation of GFP–SecE and a translocation-defective MalE–GFP mutant, as well as indirect immunofluorescence microscopy on SecG, suggested that the general protein translocation machinery (Sec) itself is arranged into a helical array, with which Tar is transiently associated. The Sec coil appeared distinct from the MreB coil, an actin-like cytoskeleton. These findings will shed new light on the mechanisms underlying spatial organization of membrane proteins in E. coli

Factors Associated with Revision Surgery after Internal Fixation of Hip Fractures

Background: Femoral neck fractures are associated with high rates of revision surgery after management with internal fixation. Using data from the Fixation using Alternative Implants for the Treatment of Hip fractures (FAITH) trial evaluating methods of internal fixation in patients with femoral neck fractures, we investigated associations between baseline and surgical factors and the need for revision surgery to promote healing, relieve pain, treat infection or improve function over 24 months postsurgery. Additionally, we investigated factors associated with (1) hardware removal and (2) implant exchange from cancellous screws (CS) or sliding hip screw (SHS) to total hip arthroplasty, hemiarthroplasty, or another internal fixation device. Methods: We identified 15 potential factors a priori that may be associated with revision surgery, 7 with hardware removal, and 14 with implant exchange. We used multivariable Cox proportional hazards analyses in our investigation. Results: Factors associated with increased risk of revision surgery included: female sex, [hazard ratio (HR) 1.79, 95% confidence interval (CI) 1.25-2.50; P = 0.001], higher body mass index (fo

Determination of endogenous abscisic acid levels in immature cereal embryos during in vitro culture

Levels of endogenous abscisic acid (ABA) in immature wheat (Triticum aestivum cv. Timmo) and barley (Hordeum vulgare cv. Golden Promise) embryos have been determined by enzyme-linked immunosorbent assay. Embryos of both cereal species showed an increase in ABA content during development on the parent plant. Immature embryos were excised and cultured in vitro on nutrient media that led to precocious germination or on media containing 9% (w/v) mannitol that maintained their developmental arrest. Barley and wheat embryos responded to these culture conditions in an identical manner with respect to changes in morphology, fresh weight, protein and lectin content. However, in complete contrast, the ABA content of barley embryos increased by an order of magnitude during culture on mannitol, whereas that of wheat embryos showed no significant change. The results are discussed within the context of the role of ABA in the regulation of embryo development

Localization and environmental regulation of MCP-like proteins in Rhodobacter sphaeroides

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/71531/1/j.1365-2958.1999.01226.x.pd