How to Exhibit Toroidal Maps in Space

Steinitz's theorem states that a graph is the 1-skeleton of a convex polyhedron if and only if it is 3-connected and planar. The polyhedron is called a geometric realization of the embedded graph. Its faces are bounded by convex polygons whose points are coplanar. A map on the torus does not necessarily have such a geometric realization. In this paper we relax the condition that faces are the convex hull of coplanar points. We require instead that the convex hull of the points on a face can be projected onto a plane so that the boundary of the convex hull of the projected points is the image of the boundary of the face. We also require that the interiors of the convex hulls of different faces do not intersect. Call this an exhibition of the map. A map is polyhedral if the intersection of any two closed faces is simply connected. Our main result is that every polyhedral toroidal map can be exhibited. As a corollary, every toroidal triangulation has a geometric realization.     

Heteroclinic solutions for a class of the second order Hamiltonian systems

We shall be concerned with the existence of heteroclinic orbits for the second order Hamiltonian system , where q∈Rn and V∈C¹(R×Rn,R), V?0. We will assume that V and a certain subset M⊂Rn satisfy the following conditions. M is a set of isolated points and #M?2. For every sufficiently small ε>0 there exists δ>0 such that for all (t,z)∈R×Rn, if d(z,M)?ε then −V(t,z)?δ. The integrals , z∈M, are equi-bounded and −V(t,z)→∞, as |t|→∞, uniformly on compact subsets of Rn?M. Our result states that each point in M is joined to another point in M by a solution of our system.     

Switching and Swapping of Quantum Information: Entropy and Entanglement Level

Information switching and swapping seem to be fundamental elements of quantum communication protocols. Another crucial issue is the presence of entanglement and its level in inspected quantum systems. In this article, a formal definition of the operation of the swapping local quantum information and its existence proof, together with some elementary properties analysed through the prism of the concept of the entropy, are presented. As an example of the local information swapping usage, we demonstrate a certain realisation of the quantum switch. Entanglement levels, during the work of the switch, are calculated with the Negativity measure and a separability criterion based on the von Neumann entropy, spectral decomposition and Schmidt decomposition. Results of numerical experiments, during which the entanglement levels are estimated for systems under consideration with and without distortions, are presented. The noise is generated by the Dzyaloshinskii-Moriya interaction and the intrinsic decoherence is modelled by the Milburn equation. This work contains a switch realisation in a circuit form—built out of elementary quantum gates, and a scheme of the circuit which estimates levels of entanglement during the switch’s operating.     

S‐Adenosyl Methionine Cofactor Modifications Enhance the Biocatalytic Repertoire of Small Molecule C‐Alkylation

A tandem enzymatic strategy to enhance the scope of C‐alkylation of small molecules via the in situ formation of S‐adenosyl methionine (SAM) cofactor analogues is described. A solvent‐exposed channel present in the SAM‐forming enzyme SalL tolerates 5′‐chloro‐5′‐deoxyadenosine (ClDA) analogues modified at the 2‐position of the adenine nucleobase. Coupling SalL‐catalyzed cofactor production with C‐(m)ethyl transfer to coumarin substrates catalyzed by the methyltransferase (MTase) NovO forms C‐(m)ethylated coumarins in superior yield and greater substrate scope relative to that obtained using cofactors lacking nucleobase modifications. Establishing the molecular determinants that influence C‐alkylation provides the basis to develop a late‐stage enzymatic platform for the preparation of high value small molecules.     

How to Obtain the Maximum Properties Flexibility of 3D Printed Ketoprofen Tablets Using Only One Drug-Loaded Filament?

The flexibility of dose and dosage forms makes 3D printing a very interesting tool for personalized medicine, with fused deposition modeling being the most promising and intensively developed method. In our research, we analyzed how various types of disintegrants and drug loading in poly(vinyl alcohol)-based filaments affect their mechanical properties and printability. We also assessed the effect of drug dosage and tablet spatial structure on the dissolution profiles. Given that the development of a method that allows the production of dosage forms with different properties from a single drug-loaded filament is desirable, we developed a method of printing ketoprofen tablets with different dose and dissolution profiles from a single feedstock filament. We optimized the filament preparation by hot-melt extrusion and characterized them. Then, we printed single, bi-, and tri-layer tablets varying with dose, infill density, internal structure, and composition. We analyzed the reproducibility of a spatial structure, phase, and degree of molecular order of ketoprofen in the tablets, and the dissolution profiles. We have printed tablets with immediate- and sustained-release characteristics using one drug-loaded filament, which demonstrates that a single filament can serve as a versatile source for the manufacturing of tablets exhibiting various release characteristics.     

Structural Insights into the Interactions of Digoxin and Na+/K+-ATPase and Other Targets for the Inhibition of Cancer Cell Proliferation

Digoxin is a cardiac glycoside long used to treat congestive heart failure and found recently to show antitumor potential. The hydroxy groups connected at the C-12, C-14, and C-3′a positions; the C-17 unsaturated lactone unit; the conformation of the steroid core; and the C-3 saccharide moiety have been demonstrated as being important for digoxin’s cytotoxicity and interactions with Na⁺/K⁺-ATPase. The docking profiles for digoxin and several derivatives and Na⁺/K⁺-ATPase were investigated; an additional small Asn130 side pocket was revealed, which could be useful in the design of novel digoxin-like antitumor agents. In addition, the docking scores for digoxin and its derivatives were found to correlate with their cytotoxicity, indicating a potential use of these values in the prediction of the cancer cell cytotoxicity of other cardiac glycosides. Moreover, in these docking studies, digoxin was found to bind to FIH-1 and NF-κB but not HDAC, IAP, and PI3K, suggesting that this cardiac glycoside directly targets FIH-1, Na⁺/K⁺-ATPase, and NF-κB to mediate its antitumor potential. Differentially, digoxigenin, the aglycon of digoxin, binds to HDAC and PI3K, but not FIH-1, IAP, Na⁺/K⁺-ATPase, and NF-κB, indicating that this compound may target tumor autophagy and metabolism to mediate its antitumor propensity.     

Binding and chemical fractionation of heavy metals in typical peat matter

Laboratory batch studies were conducted to evaluate the binding capacity and the mobility of metal species bound to typical humus peat matter. The identification of phase composition of mineral fractions and functional groups in the organic matter was assessed. The results showed generally high, but different retention capacity and binding strength, suggesting distinct diversity in binding mechanisms, phases and chemical nature of binding sites, depending on the metal species and their input concentrations. In general, the binding capacity of peat for the metal ions studied follows the order: Cr(3+) > Cu(2+) > Zn(2+) > Cd(2+) and results in the decrease of pH in the same order, due to displacement of H(3)O(+) from the peat by metal ions. The highest metal enrichment occurs in fractions F1(EXC), F2(CARB), F4(MRO) and F5(OM) of different binding strength adequate to exchangeable, carbonatic, moderately reducible amorphous Fe-oxide and organic/ sulphidic fractions in soils and sediments. In relation to species distribution in peats, the prevailing part of Cr(3+) is strongly bound in oxidizable organic substrate, while Cu(2+) is highly enriched in the moderately reducible F4(MRO) and the most labile F2(EXC) fractions. Cd(2+) and Zn(2+) are predominantly bound in the labile F1(EXC) and F2(CARB) fractions. Diversity of the predominant binding phases for the studied metals suggests rather weak competition for binding sites between chromium and copper ions; the strongest competition between the sorbed metal ions is anticipated for F1(EXC) and F2(CARB) fractions.     

Precision and control in polymer synthesis why it's important and some recent examples of how to do it

This paper discusses some of the reasons why precision and control in polymer synthesis is of importance. By way of illustration it describes in outline recent results from the authors' laboratories in three areas. Namely; the controlled syntheses of poly(arylene vinylene)s and the influence of cis/trans vinylene content on luminescence in such polymers; the living polymerisation of highly functionalised polymers in water and the regulation of the crystallisation of calcium carbonate from water by the resultant well-defined water soluble polymers; and a simple route to hyperbranched polymers and the influence of the structure and topology of the products on solution properties. In each case the influence of control of architecture on properties will be discussed.     

Membranes with a plasma deposited titanium isopropoxide layer

Porous polypropylene membranes were coated with plasma polymerized titanium isopropoxide in a 75 kHz plasma reactor. It was noted that the presence of air in the plasma chamber increased the amount of deposited polymer. Selection of the process parameters enabled obtaining membranes with up to 300 εg cm^?2 of polymerized titanium isopropoxide. Deposition of the titanium oxide layer resulted in the reduction of permeate flux but it significantly improved the membrane photocleaning ability. The recovery index reached the level of 95 % for membranes with the highest amount of the titanium oxide deposit.