The baker's map with a convex hole

We consider the baker's map $B$ on the unit square $X$ and an open convex set $H\subset X$ which we regard as a hole. The survivor set $\mathcal J(H)$ is defined as the set of all points in $X$ whose $B$-trajectories are disjoint from $H$. The main purpose of this paper is to study holes $H$ for which $\dim_H \mathcal J(H)=0$ (dimension traps) as well as those for which any periodic trajectory of $B$ intersects $\overline H$ (cycle traps). We show that any $H$ which lies in the interior of $X$ is not a dimension trap. This means that, unlike the doubling map and other one-dimensional examples, we can have $\dim_H \mathcal J(H)>0$ for $H$ whose Lebesgue measure is arbitrarily close to one. Also, we describe holes which are dimension or cycle traps, critical in the sense that if we consider a strictly convex subset, then the corresponding property in question no longer holds. We also determine $\delta>0$ such that $\dim_H \mathcal J(H)>0$ for all convex $H$ whose Lebesgue measure is less than $\delta$. This paper may be seen as a first extension of our work begun in [3, 4, 6, 7, 13] to higher dimensions.Comment: 31 pages, 10 figure

Magnetic ionic liquids in bioanalytical chemistry

A prerequisite to the analysis of complex biological samples is the isolation, enrichment, and purification of target biomolecules. Nucleic acids, proteins, and intact cells have been established as important biomarkers for diagnostic, prognostic, molecular biology, and food safety applications. However, the accuracy and reproducibility of experimental measurements involving these analytes are dependent upon the purity of the sample that is subjected to instrumental analysis. Traditional extraction and purification approaches are limited by tedious and time-consuming sample handling and manipulation steps that severely limit sample throughput. Given the urgency of providing rapid results in virtually every field of science, new methods that improve the speed, sensitivity, and selectivity of bioanalytical methods are highly desirable. Magnetic ionic liquids (MILs) are a class of molten salts with melting points at or below 100 ðC that exhibit susceptibility to external magnetic fields. As a subclass of ionic liquids (ILs), MILs are comprised entirely of organic/inorganic cations and anions that can be functionalized to adopt myriad properties including negligible vapor pressure at ambient conditions, tunable viscosity, hydrophobicity, and unique solvation interactions. Depending on the identity of the paramagnetic component within the MIL structure, it is also possible to enhance the effective magnetic moment of the resulting MIL solvent. Transition metal and rare earth metal complexes are the most common components employed for imparting magnetic susceptibility to MILs. The unique physicochemical properties and paramagnetic nature of MILs have driven intense interest in their application as extraction/purification solvents in bioanalytical applications. The extraction of DNA from biological samples often involves numerous, lengthy centrifugation steps to isolate the nucleic acid from cellular debris and interfering agents. Three tetrahaloferrate(III)-based MILs were synthesized and investigated for the rapid extraction and preconcentration of DNA from aqueous solution and bacterial cell lysate. After MILs were dispersed in aqueous DNA samples, application of an external magnetic field permitted precise control of the DNA-enriched MIL solvent. The extracted DNA was then recovered in sufficient quantity and quality for polymerase chain reaction (PCR) amplification. In order to further reduce analysis times and facilitate method automation, the direct amplification of DNA from the MIL extraction phase was investigated. MILs were initially discovered to inhibit PCR amplification due to both the heavily alkylated cations and metal-containing anions in the MIL structure. Furthermore, hydrolysis of the iron(III) center in the MIL anion resulted in a decrease in solution pH that contributed to the diminished activity of the DNA polymerase. By developing a PCR mixture containing albumin, additional MgCl2 cofactor, metal chelators, and a higher buffer capacity, amplification was restored for samples containing the MIL solvent. By coupling rapid extraction with downstream DNA amplification directly from the MIL solvent, the MIL-based method was circumvented time-consuming DNA recovery procedures and was capable of extracting DNA from bacterial cell lysate for immediate amplification by PCR. Since many clinically relevant nucleic acid targets exist at extremely low concentrations in biological samples, an enrichment or preconcentration step is often required prior to detection in order to minimize signals from untargeted biomolecules in diagnostic assays. Background DNA represents a particularly challenging interference due to its chemical similarity to target nucleic acids. Sequence-specific DNA capture can be accomplished by capitalizing on the innate ability of nucleic acid sequences to recognize their complements via base-pairing. Ion-tagged oligonucleotides (ITOs) were developed in order to impart sequence selectivity to the MIL solvent. ITOs were synthesized by coupling an imidazolium-based ion tag with a thiolated DNA sequence using thiol-ene click chemistry. Longer alkyl chains in the ion tag structure resulted in ITOs with higher affinities for a manganese(II) hexafluoroacetylacetonate-based hydrophobic MIL support. After hybridization of the ITO with a target DNA sequence, addition of the MIL solvent support to the solution enabled sequence-specific DNA extraction. The target sequence was then recovered using elevated temperature for real-time quantitative PCR (qPCR). Compared to a commercially available magnetic bead-based method, the ITO-MIL approach yielded ten-fold greater quantity of target DNA from a sample containing interfering genomic DNA. By avoiding challenges associated with magnetic bead aggregation, the particle-free ITO-MIL method represents a promising alternative for targeted DNA extraction. In many cases, DNA is isolated from a sample and subsequently stored for a period of time prior to analysis. During this timeframe that may last days or weeks, the nucleic acid is susceptible to chemical degradation or hydrolysis by enzymes such as deoxyribonuclease I (DNase I). In order to develop a workflow that would permit the extraction and storage of DNA within the same medium, MILs were also studied as solvents for the preservation of DNA. Plasmid DNA (pDNA) treated with DNase I and stored in MIL solvent was stable for up to 72 h at room temperature, whereas aqueous DNA samples were completely degraded under similar conditions. Partitioning of DNase I to the MIL phase was dependent upon the cation structure of the MIL, with a smaller amount of DNase I partitioning to an ammonium-based MIL relative to a phosphonium-based MIL. Importantly, PCR amplification of pDNA treated with DNase I and stored in aqueous buffer at −20 ðC for 1 week was unsuccessful, while pDNA stored in MIL under the same conditions provided an amplicon after PCR. The analysis of DNA plays a key role in the detection of pathogenic bacteria in foods. Although DNA analysis can provide information about bacteria species or serotype, the isolation of viable bacteria is essential for microbiological cultures that are used to unambiguously determine whether a microorganism is live or dead. Traditional bacteria enrichment methods that involve overnight cultures or immunoaffinity capture are limited by lengthy incubation times and complicated/expensive substrates (e.g., immunosorbent beads), respectively. In order to address the shortcomings of conventional bacteria enrichment approaches, MILs were investigated as solvents for the preconcentration of viable bacteria. By dispersing MIL in an aqueous suspension of Escherichia coli K12, bacteria were rapidly extracted (approximately 30 s) and isolated by application of a magnetic field. Viable E. coli were recovered from the MIL phase using a nutrient broth and were subjected to detection by either microbiological culture or qPCR amplification. Detection limits as low as 100 CFU mL−1 were achieved using the MIL-based method that required just 5 min to complete. The MIL-based method was used for successful extraction and qPCR-based detection of E. coli from milk samples

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Convert Client Training to eLearning

eBay Enterprise Inc. (EE) provides end to end e-Commerce solutions for large clients such as Toys R Us, Sony, Ralph Lauren, and Dick’s Sporting Goods. EE must train their clients how to use proprietary software to manage their website. To date this training has been 100% instructor-led. An eLearning experience will improve client education and reduce costs for EE and its clients alike. The final deliverable for this project is a single eLearning course made up of several smaller modules that focus on the EE Manager software application. The course will be accessible to clients via the EE LMS 24 hours a day. The course will be created using Articulate Storyline and will include interactive video as well as audio for each module. Each of the modules will have assessments to gauge the learner’s knowledge of a given topic. Signatures will be required by the project manager and SME before each module is considered complete. Once all modules have been signed off on, a final course approval will need to be obtained from the Project Manager and SME before the project is considered complete