Decreasing Regularly Varying Solutions of Sublinearly Perturbed Superlinear Thomas–Fermi Equation

We consider the perturbed Thomas–Fermi equation $$\begin{array}{ll} x^{\prime \prime}\, =\, p(t)|x|^{\gamma-1}x\, +\, q(t)|x|^{\delta-1}x, \qquad \qquad \qquad (A) \end{array}$$ where δ and γ are positive constants with \({\delta < 1 < \gamma}\) and p(t) and q(t) are positive continuous functions on \({[a,\infty), a > 0}\) . Our aim here is to establish criteria for the existence of positive solutions of (A) decreasing to zero as \({t \to \infty}\) in the case where p(t) and q(t) are regularly varying functions (in the sense of Karamata). Generalization of the obtained results to equations of the form $$\begin{array}{ll} \left(r(t)x^{\prime}\right)^{\prime}\, =\, p(t)|x|^{\gamma-1}x \,+ \,q(t)|x|^{\delta-1}x, \qquad \qquad \qquad (B) \end{array}$$ is also given.     

Zigzag‐Helix Transformation of Expanded Polyvaline Induced by Racemization

Biological macromolecules are essentially homochiral. For example, proteins mostly consist of l ‐amino acids. What happens when a chiral molecule meets itself in a mirror? For expanded polyvaline, zigzag‐helix transformation occurs. In this study, expanded polyvalines containing bis(pyridine)silver(I) moieties were synthesized and isolated as single crystals. The molecular structures were determined by X‐ray analysis, which revealed that chiral expanded poly(l ‐valine) and poly(d ‐valine) form zigzag chains. However, racemic mixture of these molecules form left‐ and right‐handed 4₁ helices that retain the original sequences. These secondary structures can be transformed by only flipping the C‐terminal amide plane for each unit, which is reminiscent of the relationship between an α‐helix and a β‐strand. Such expanded polypeptides can be built up into expanded protein, forming a tailor‐made three‐dimensional structure, which will lead to new functions.     

Direct Detection of Hemi-methylated DNA by SRA-fused Luciferase Based on Bioluminescence Resonance Energy Transfer

Hemi-methylated DNA is double stranded (ds)DNA where only one of the complementary strands is methylated. Hemi-methylated DNA levels on a tandem repeat NBL2 and Sat2 are increased in ovarian carcinomas. To quantify hemi-methylation levels, a bisulfite-based method is widely used, but this approach requires multiple steps. Therefore, a simple hemi-methylation assay is required for cancer diagnosis and to elucidate the functions of hemi-methylated DNA. In this study, we aimed to develop a homogeneous assay for the detection of the hemi-methylated DNA level using SET- and RING-associated (SRA) domain-fused luciferase (SRA-luciferase), which recognizes hemi-methylated DNA. In the assay, the hemi-methylation level was quantified by measuring the bioluminescence resonance energy transfer signal between the fusion protein and the DNA intercalating dye. The fusion protein specifically recognizes hemi-methylated DNA to excite the dye by luciferase luminescence. Our results demonstrated that the hemi-methylation level was easily quantified within 35?min with an R² value of 0.99.     

Plasma lipid analysis by hydrophilic interaction liquid chromatography coupled with electrospray ionization tandem mass spectrometry

A novel method for the analysis of endogenous lipids and related compounds was developed employing hydrophilic interaction liquid chromatography with electrospray ionization tandem mass spectrometry. A hydrophilic interaction liquid chromatography with carbamoyl stationary phase achieved clear separation of phosphatidylcholine, lysophosphatidylcholine, sphingomyelin, ceramide, and mono‐hexsosyl ceramide groups with good peak area repeatability (RSD% < 10) and linearity (R² > 0.99). The established method was applied to human plasma assays and a total of 117 endogenous lipids were successfully detected and reproducibly identified. In addition, we investigated the simultaneous detection of small polar metabolites such as amino and organic acids co‐existing in the same biological samples processed in a single analytical run with lipids. Our results show that hydrophilic interaction liquid chromatography is a useful tool for human plasma lipidome analysis and offers more comprehensive metabolome coverage.     

Positive Solutions of Fourth Order Thomas-Fermi Type Differential Equations in the Framework of Regular Variation

The fourth order nonlinear differential equations A with regularly varying coefficient q(t) are studied in the framework of regular variation. It is shown that thorough and complete information can be acquired about the existence of all possible regularly varying solutions of (A) and their accurate asymptotic behavior at infinity.     

A study of the physico-chemical properties of 1,3,5-trihydroxy-1,3,5-triazin-2,4,6[1H,3H,5H]-trione

Acetyl (Ia) and pivaloyl (Ib) triesters of the 1N,3N,5N-trihydroxy-1,3,5-triazin-2,4,6[1H,3H,5H]-trione (I) were synthesised. The spectrophotometric and potentiometric investigation of I revealed a weak acidic properties of triprotonic acid (pK_a1=5.23, pK_a2=6.32, and pK_a3=7.93). The MS and TGA analyses of I indicated on hydroxyisocyanate as possible degradation product. The chelating ability of I with Fe(III)-ion was preliminary explored. IR measurements of aqueous solutions of I in the presence of Fe(III) ion showed the possible chelating ability of all hydroxamic moieties. The chemical structures and properties of investigated compounds were derived from the results of IR, ¹H and ¹³C NMR, UV and MS spectrometric data, as well as thermogravimetric and potentiometric analysis.     

Nonoscillatory half-linear differential equations and generalized Karamata functions

We introduce a natural generalization of the concept of regularly varying functions in the sense of Karamata, and show that the class of generalized Karamata functions is a well-suited framework for the study of the asymptotic behavior of nonoscillatory solutions of the half-linear differential equation of the type

equation(A)

(p(t)|y^′|^α-1y^′)^′+q(t)|y|^α-1y=0.$(p(t)|y^{\prime }{|}^{\alpha -1}{y}^{\prime }{)}^{\prime }+q(t)|y{|}^{\alpha -1}y=0\text{.}$