Tight Heffter Arrays Exist for all Possible Values

A tight Heffter array is an matrix with nonzero entries from such that (i) the sum of the elements in each row and each column is 0, and (ii) no element from appears twice. We prove that exist if and only if both m and n are at least 3. If H has the property that all entries are integers of magnitude at most , every row and column sum is 0 over the integers, and H also satisfies ), we call H an integer Heffter array. We show integer Heffter arrays exist if and only if . Finally, an integer Heffter array is shiftable if each row and column contains the same number of positive and negative integers. We show that shiftable integer arrays exists exactly when both are even.     

Triple Systems are Eulerian

An Euler tour of a hypergraph (also called a rank‐2 universal cycle or 1‐overlap cycle in the context of designs) is a closed walk that traverses every edge exactly once. In this paper, using a graph‐theoretic approach, we prove that every triple system with at least two triples is eulerian, that is, it admits an Euler tour. Horan and Hurlbert have previously shown that for every admissible order >3, there exists a Steiner triple system with an Euler tour, while Dewar and Stevens have proved that every cyclic Steiner triple system of order >3 and every cyclic twofold triple system admits an Euler tour.     

d‐Fructose‐Decorated Poly(ethylene imine) for Human Breast Cancer Cell Targeting

The high affinity of GLUT5 transporter for d ‐fructose in breast cancer cells has been discussed intensely. In this contribution, high molar mass linear poly(ethylene imine) (LPEI) is functionalized with d ‐fructose moieties to combine the selectivity for the GLUT5 transporter with the delivery potential of PEI for genetic material. The four‐step synthesis of a thiol‐group bearing d ‐fructose enables the decoration of a cationic polymer backbone with d ‐fructose via thiol‐ene photoaddition. The functionalization of LPEI is confirmed by 2D NMR techniques, elemental analysis, and size exclusion chromatography. Importantly, a d ‐fructose decoration of 16% renders the polymers water‐soluble and eliminates the cytotoxicity of PEI in noncancer L929 cells, accompanied by a reduced unspecific cellular uptake of the genetic material. In contrast, the cytotoxicity as well as the cell specific uptake is increased for triple negative MDA‐MB‐231 breast cancer cells. Therefore, the introduction of d ‐fructose shows superior potential for cell targeting, which can be assumed to be GLUT5 dependent.

     

Group Classification of Equations of the Form y″ = f(x,y)

The problem of classification of ordinary differential equations of the form y = f(x,y) by admissible local Lie groups of transformations is solved. Standard equations are listed on the basis of the equivalence concept. The classes of equations admitting a oneparameter group and obtained from the standard equations by invariant extension are described.     

Absolute quantification of superoxide dismutase in cytosol and mitochondria of mice hepatic cells exposed to mercury by a novel metallomic approach

In the last years, the development of new methods for analyzing accurate and precise individual metalloproteins is of increasing importance, since numerous metalloproteins are excellent biomarkers of oxidative stress and diseases. In that way, methods based on the use of post column isotopic dilution analysis (IDA) or enriched protein standards are required to obtain a sufficient degree of accuracy, precision and high limits of detection. This paper reports the identification and absolute quantification of Cu,Zn-superoxide dismutase (Cu,Zn-SOD) in cytosol and mitochondria from mice hepatic cells using a innovative column switching analytical approach. The method consisted of orthogonal chromatographic systems coupled to inductively coupling plasma-mass spectrometry equipped with a octopole reaction systems (ICP-ORS-MS) and UV detectors: size exclusion fractionation (SEC) of the cytosolic and mitochondrial extracts followed by online anion exchange chromatographic (AEC) separation of Cu/Zn containing species. After purification, Cu,Zn-SOD was identified after tryptic digestion by molecular mass spectrometry (MS). The MS/MS spectrum of a doubly charged peptide was used to obtain the sequence of the protein using the MASCOT searching engine. This optimized methodology reduces the time of analysis and avoids the use of sample preconcentration and clean-up procedures, such as cut-off centrifuged filters, solid phase extraction (SPE), precipitation procedures, off-line fractions insolates, etc. In this sense, the method is robust, reliable and fast with typical chromatographic run time less than 20 min. Precision in terms of relative standard deviation (n = 5) is of 3–5% and detection limits is 0.21 ng Cu g⁻¹.     

二次李三系的分解定理

有限维李三系称为二次的,如果它容许一个非退化的不变对称双线性型.无论是李三系分解为不可分解理想的直和,还是二次李三系分解为不可分解非退化理想的正交直和,证明了这两类分解在同构意义下都是唯一的.