Some applications of Tristram-Levine signatures and relation to Vassiliev invariants

Tristram and Levine introduced a continuous family of signature invariants for knots. We show that any possible value of such an invariant is realized by a knot with given Vassiliev invariants of bounded degree. We also show that one can make a knot prime preserving Alexander polynomial and Vassiliev invariants of bounded degree. Finally, the Tristram-Levine signatures are applied to obtain a condition on (signed) unknotting number.     

Dynamic network reliability modeling under nonhomogeneous Poisson processes

In this paper, we consider a two-state (up and down) network consisting of n links. We study the D-spectrum based dynamic reliability of the network under the assumption that the links are subject to failure according to a nonhomogeneous Poisson process. Several mixture representations are provided for the reliability function of residual lifetime of used networks, under different conditions on the status of the network or its links. These representations enable us to explore the residual reliability of operating networks in terms of the reliability functions of residual lifetimes of upper record values. The distribution function of inactivity time of a network is examined under the condition that the network has failed by inspection time t. Stochastic ordering properties of the residual lifetimes of networks under conditional D-spectra are investigated. Several examples and graphs are also provided to illustrate the established results.     

Immuno‐capture as ultimate sample cleanup in LC‐MS/MS determination of the early stage biomarker ProGRP

This paper presents a selective and efficient sample preparation procedure for NLLGLIEAK, signature peptide for the small cell lung cancer (SCLC) biomarker ProGRP, in human serum. The procedure is based on immuno‐capture of ProGRP in 96‐wells microtiter plates coated with the mAb E146. After immuno‐capture and thorough rinse, trypsin was added for in‐well digestion. Subsequently the signature peptide was enriched by SPE and determined by LC‐MS/MS. Various steps in the procedure were optimized to achieve a low LOD such as dilution of sample, tryptic digestion, and SPE cleanup and peptide enrichment conditions. A single quadropole MS was used during optimization of the method. A triple quadropole MS was used in the method evaluation in order to improve sensitivity. The evaluation showed good repeatability (RSD, 11.9–17.5%), accuracy (3.0–6.6%), and linearity (r² = 0.995) in the tested range (0.5–50 ng/mL). LOD and LOQ were in the pg/mL area (0.20 and 0.33 ng/mL, respectively), enabling the determination of clinically relevant concentrations. The method was applied to two patient samples and showed good agreement with an established immunological reference method. The final method was compared to a previous published LC‐MS method for the determination of ProGRP in serum based on protein precipitation and online sample cleanup. Both showed acceptable method performance, however, the immuno‐capture LC‐MS method was superior with respect to sensitivity. This illustrates the large potential of immuno‐capture sample preparation prior to LC‐MS in protein biomarker quantification.     

Using a signature function to determine resonant and attenuant 2-cycles in the Smith–Slatkin population model

We study the responses of discretely reproducing populations to periodic fluctuations in three parameters: the carrying capacity and two demographic characteristics of the species. We prove that small 2-periodic fluctuations of the three parameters generate 2-cyclic oscillations of the population. We develop a signature function for predicting the responses of populations to 2-periodic fluctuations. Our signature function is the sign of a weighted sum of the relative strengths of the oscillations of the three parameters. Periodic environments are deleterious for populations when the signature function is negative, while positive signature functions signal favorable environments. We compute the signature function for the Smith–Slatkin model, and use it to determine regions in parameter space that are either favorable or detrimental to the species.     

Lifting the 3-dimensional invariant of 2-plane fields on 3-manifolds

Homotopy classes of plane fields on 3-manifolds have been classified using a 2-dimensional invariant Γ and a 3-dimensional invariant θ by R. Gompf. Under regular covering maps, Γ lifts in the natural way. The lifting property of θ remained unresolved. In this paper, we present the lifting property of θ together with applications to Lens spaces. The applications help in specifying the liftings of the contact structures of the Lens space L(p,1) when lifted to S³.     

Global unit squares and local unit squares

For units of any Galois number field, we study the relations between the units being global squares and those being local squares.     

Generation of Dynamic Infrared Outer Space Scene

This paper proposes a new method to generate the dynamic infrared outer space scene in real time. In the paper, we first model the dynamic infrared background of star sky based on the infrared star catalogue and temporal and spacial coordinate transformations. We also present a new concept called infrared star magnitude for given infrared receiving wavebands to display the infrared star sky effectively. Then we put forward an infrared image synthesis model for outer space targets based on the method of divided elements and energy equilibrium. To generate the dynamic infrared outer space scene, several simplification and accelerating techniques such as Billboard, Levels of Details and Occlusion Culling are hired to accelerate the generating speed of the scene. Finally infrared images of targets, background of outer space scene at different infrared receiving wavebands are drawn and the dynamic infrared outer space scene is successfully carried out in real time. * Corresponding author     

Upper bound for the alternation number of a torus knot

We give an upper bound for the alternation number of a torus knot which is of either 3-, 4-, or 5-braid or of other special types. Using the inequality relating the alternation number, signature, and Rasmussen s-invariant, discovered by Abe, we determine the alternation numbers of the torus knots T(3,l), , and T(4,5). Also, for any positive integer k we construct infinitely many 3-braid knots with alternation number k.     

Multiple endonucleases improve MALDI-MS signature digestion product detection of bacterial transfer RNAs

Individual transfer ribonucleic acids (tRNAs) in a complex mixture can be identified by the matrix-assisted laser desorption/ionization mass spectrometry (MALDI-MS) detection of their signature digestion products. Signature digestion products are endonuclease digestion products whose mass-to-charge value is unique thus corresponding to only a single tRNA. To improve the effectiveness of this approach, we have expanded the applicable endonucleases and examined the use of multiple endonucleases for tRNA identification. The purine specific endonucleases RNase T1 and RNase TA generate the largest number of predicted signature digestion products. Experimentally, MALDI-MS analysis of endonuclease digests from Escherichia coli and Bacillus subtilis finds that any two endonucleases used in combination increases tRNA identification by about 25% over the number identified with a single endonuclease. Using three endonucleases, RNase T1, RNase A, and RNase TA, further improves the number of tRNAs identified by 10–15% over those found with two endonucleases. Limitations in the MALDI-MS approach for complex mixtures were revealed in this study, suggesting that the direct MALDI-MS analysis of signature digestion products is more effective for organisms having 30 or less unique tRNAs. Figure Signature digestion products for tRNA^Cys