Asymptotic Results for Semigroups of Order-Preserving Partial Transformations

Let 𝒫  _n be the semigroup of all decreasing and order-preserving partial transformations of an n-element chain, and let E(𝒫  _n ) be its set of idempotents. Among other results, asymptotic formulae for |𝒫  _n | and |E(𝒫  _n )|/|𝒫  _n | are obtained. Similar results for 𝒫  _n the (larger) semigroup of all order-preserving partial transformations of an n-element chain are also obtained.     

Simple Formulae for the Number of Quadrics and Symmetric Forms of Modules Over Local Rings

Simple formulae of similar type for the number of quadrics and symmetric forms of modules over local rings are found analytically. This simplification of known formulae of Levchuk and Starikova is achieved using integral representation method of combinatorial summation and leads to number-theoretical identities of new type. The problem of algebraic interpretation of new formulae is posed.     

Quantification of hard X-ray photoelectron spectroscopy: Calculating relative sensitivity factors for 1.5- to 10-keV photons in any instrument geometry

A method for the rapid determination of theoretical relative sensitivity factors (RSFs) for hard X-ray photoelectron spectroscopy (HAXPES) instruments of any type and photon energy has been developed. We develop empirical functions to describe discrete theoretically calculated values for photoemission cross sections and asymmetry parameters across the photon energy range from 1.5 to 10 keV for all elements from lithium to californium. The formulae describing these parameters, in conjunction with similar practical estimates for inelastic mean free paths, allow the calculation of a full set of theoretical sensitivity factors for a given X-ray photon energy, X-ray polarisation and instrument geometry. We show that the anticipated errors on these RSFs are less than the typical errors generated by extracting X-ray photoelectron spectroscopy (XPS) intensities from the spectra and thus enable adequate quantification for any XPS/HAXPES experiment up to 10 keV. A spreadsheet implementation of this method is provided in the supporting information, along with example RSFs for existing commercial instruments.     

Effect of cocrystallization due to bile acid on molecular-ion sensitivity of biological phospholipids in Bi cluster time-of-flight secondary ion mass spectrometry measurements

Bi cluster time-of-flight secondary ion mass spectrometry (TOF-SIMS) is a useful method for evaluating organic surfaces. However, its ability to detect large molecules is limited. One of the problems is that the sensitivities of macromolecules are lower than those of small molecules because larger molecules tend to exhibit lower ionization efficiencies and/or higher probabilities of fragmentation. Matrix-enhanced (ME)-SIMS is a sensitivity enhancement technique for intact molecular ions. The crystal structure of a mixed substance composed of an analyte and a matrix is known to affect the sensitivity of the analysis target. In this study, the effect of cocrystallization, which occurs due to the presence of bile acid, on the molecular-ion sensitivity was investigated using Bi cluster TOF-SIMS. Biological phospholipids and bile acids, which exhibit surfactant behaviors, were selected as the evaluated molecules and additives, respectively. The mass spectra indicated that the secondary-ion yields of phospholipids with bile acid were substantially greater than those of the pristine lipid. Specifically, samples with an analyte/bile acid ratio of 1:100 achieved approximately 60–100-fold sensitivity enhancement of [M + H]⁺ and [2M + H]⁺ molecular ions than the sensitivity achieved with the pristine samples. In the evaluation of molecular distribution, higher signal counts of intact ions were obtained from the cocrystallization area, although less-fragmented ions were emitted from these regions. Consequently, the results indicate that the cocrystallization due to the presence of bile acid provides an effective crystal structure for facilitating emission of larger molecules.     

Coomassie staining provides routine (sub)femtomole in‐gel detection of intact proteoforms: Expanding opportunities for genuine Top‐down Proteomics

Modified colloidal Coomassie Brilliant Blue (cCBB) staining utilising a novel destain protocol and near‐infrared fluorescence detection (nIRFD) rivals the in‐gel protein detection sensitivity (DS) of SYPRO Ruby. However, established DS estimates are likely inaccurate in terms of 2DE‐resolved proteoform ‘spots’ since DS is routinely measured from comparatively diffuse protein ‘bands’ following wide‐well 1DE. Here, cCBB DS for 2DE‐based proteomics was more accurately determined using narrow‐well 1DE. As precise estimates of protein standard monomer concentrations are essential for accurate quantitation, coupling UV absorbance with gel‐based purity assessments is described. Further, as cCBB is compatible with both nIRFD and densitometry, the impacts of imaging method (and image resolution) on DS were assessed. Narrow‐well 1DE enabled more accurate quantitation of cCBB DS for 2DE, achieving (sub)femtomole DS with either nIRFD or densitometry. While densitometry offers comparative simplicity and affordability, nIRFD has the unique potential for enhanced DS with Deep Imaging. Higher‐resolution nIRFD also improved analysis of a 2DE‐resolved proteome, surpassing the DS of standard nIRFD and densitometry, with nIRFD Deep Imaging further maximising proteome coverage. cCBB DS for intact proteins rivals that of mass spectrometry (MS) for peptides in complex mixtures, reaffirming that 2DE‐MS currently provides the most routine, broadly applicable, robust, and information‐rich Top‐down approach to Discovery Proteomics.     

Metallated porphyrin noncovalent interaction with reduced graphene oxide‐modified electrode for amperometric detection of environmental pollutant hydrazine

A reduced graphene oxide/platinum(II) tetraphenylporphyrin nanocomposite (RGO/Pt‐TPP)‐modified glassy carbon electrode was developed for the selective detection of hydrazine. The RGO/Pt‐TPP nanocomposite was successfully prepared via noncovalent π–π stacking interaction. The prepared nanocomposite was characterized using nuclear magnetic resonance, electrochemical impedance, ultraviolet–visible and Raman spectroscopies, scanning electron microscopy and X‐ray diffraction. The electrochemical detection of hydrazine was performed via cyclic voltammetry and amperometry. The RGO/Pt‐TPP nanocomposite exhibited good electrocatalytic activity towards detection of hydrazine with low overpotential and high oxidation peak current. The fabricated sensor exhibited a wide linear range from 13 nM to 232 μM and a detection limit of 5 nM. In addition, the fabricated sensor selectively detected hydrazine even in the presence of 500‐fold excess of common interfering ions. The fabricated electrode exhibited good sensitivity, stability, repeatability and reproducibility. In addition, the practical applicability of the sensor was evaluated in various water samples with acceptable recoveries.     

(1,k)-configurations and facets for packing problems

A new class of facets for knapsack polytopes is obtained. This class of inequalities is shown to define a polytope with zero–one vertices only. A combinatorial inequality is obtained from Fulkerson's max—max inequality.     

On a viscous Hamilton-Jacobi equation with an unbounded potential term

We present comparison, uniqueness and existence results for unbounded solutions of a viscous Hamilton-Jacobi or eikonal equation. The equation includes an unbounded potential term V(x) subject to a quadratic upper bound. The results are obtained through a tailor-made change of variables in combination with the Hopf-Cole transformation. An integral representation formula for the solution of the Cauchy problem is derived in the case where V(x)=ω²|x|²/2.     

Banach空间非线性脉冲积分方程解的存在性

利用Monch不动点定理和分段估计方法,本文研究Banach空间非线性脉冲积分方程解的存在性,但是我们不使用脉冲项的紧型条件和非紧型测度估计的限制性条件.作为一个应用,我们讨论Banach空间一阶非线性脉冲微分方程终值问题解的存在性.