Odd subgraphs and matchings

Let G be a graph and f : V(G)→{1,3,5,…}. Then a subgraph H of G is called a (1,f)-odd subgraph if deg_H(x){1,3,…,f(x)} for all xV(H). If f(x)=1 for all xV(G), then a (1,f)-odd subgraph is nothing but a matching. A (1,f)-odd subgraph H of G is said to be maximum if G has no (1,f)-odd subgraph K such that |K|>|H|. We show that (1,f)-odd subgraphs have some properties similar to those of matchings, in particular, we give a formula for the order of a maximum (1,f)-odd subgraph, which is similar to that for the order of a maximum matching.     

Statistical inference of 2-type critical Galton–Watson processes with immigration

In this paper the asymptotic behavior of the conditional least squares estimators of the offspring mean matrix for a 2-type critical positively regular Galton–Watson branching process with immigration is described. We also study this question for a natural estimator of the spectral radius of the offspring mean matrix, which we call criticality parameter. We discuss the subcritical case as well.     

Polycapillary‐optics‐based micro‐XANES and micro‐EXAFS at a third‐generation bending‐magnet beamline

A focusing system based on a polycapillary half‐lens optic has been successfully tested for transmission and fluorescence µ‐X‐ray absorption spectroscopy at a third‐generation bending‐magnet beamline equipped with a non‐fixed‐exit Si(111) monochromator. The vertical positional variations of the X‐ray beam owing to the use of a non‐fixed‐exit monochromator were shown to pose only a limited problem by using the polycapillary optic. The expected height variation for an EXAFS scan around the Fe K‐edge is approximately 200 µm on the lens input side and this was reduced to ～1 µm for the focused beam. Beam sizes (FWHM) of 12–16 µm, transmission efficiencies of 25–45% and intensity gain factors, compared with the non‐focused beam, of about 2000 were obtained in the 7–14 keV energy range for an incoming beam of 0.5 × 2 mm (vertical × horizontal). As a practical application, an As K‐edge µ‐XANES study of cucumber root and hypocotyl was performed to determine the As oxidation state in the different plant parts and to identify a possible metabolic conversion by the plant.     

On the exterior algebra method applied to restricted set addition

In 1994 Dias da Silva and Hamidoune solved a long-standing open problem of Erd?s and Heilbronn using the structure of cyclic spaces for derivatives on Grassmannians and the representation theory of symmetric groups. They proved that for any subset A$A$ of the p$p$-element group Z/pZ$\mathbb{Z}/p\mathbb{Z}$ (where p$p$ is a prime), at least min{p,m|A|−m²+1}$min\{p,m\left|A\right|-m^2+1\}$ different elements of the group can be written as the sum of m$m$ different elements of A$A$. In this note we present an easily accessible simplified version of their proof for the case m=2$m=2$, and explain how the method can be applied to obtain the corresponding inverse theorem.     

Tree metrics and edge-disjoint S-paths

Given an undirected graph \(G=(V,E)\) with a terminal set \(S \subseteq V\) , a weight function on terminal pairs, and an edge-cost \(a: E \rightarrow \mathbf{Z}_+\) , the \(\mu \) -weighted minimum-cost edge-disjoint \(S\) -paths problem ( \(\mu \) -CEDP) is to maximize \(\sum \nolimits _{P \in \mathcal{P}} \mu (s_P,t_P) - a(P)\) over all edge-disjoint sets \(\mathcal{P}\) of \(S\) -paths, where \(s_P,t_P\) denote the ends of \(P\) and \(a(P)\) is the sum of edge-cost \(a(e)\) over edges \(e\) in \(P\) . Our main result is a complete characterization of terminal weights \(\mu \) for which \(\mu \) -CEDP is tractable and admits a combinatorial min–max theorem. We prove that if \(\mu \) is a tree metric, then \(\mu \) -CEDP is solvable in polynomial time and has a combinatorial min–max formula, which extends Mader’s edge-disjoint \(S\) -paths theorem and its minimum-cost generalization by Karzanov. Our min–max theorem includes the dual half-integrality, which was earlier conjectured by Karzanov for a special case. We also prove that \(\mu \) -EDP, which is \(\mu \) -CEDP with \(a = 0\) , is NP-hard if \(\mu \) is not a truncated tree metric, where a truncated tree metric is a weight function represented as pairwise distances between balls in a tree. On the other hand, \(\mu \) -CEDP for a truncated tree metric \(\mu \) reduces to \(\mu '\) -CEDP for a tree metric \(\mu '\) . Thus our result is best possible unless P = NP. As an application, we obtain a good approximation algorithm for \(\mu \) -EDP with “near” tree metric \(\mu \) by utilizing results from the theory of low-distortion embedding.     

Lithiation of 2-(chloroaryl)-2-aryl-1,3-dioxolanes with butyllithium activated by N,N,N′,N″,N″-pentamethyldiethylenetriamine

Intramolecular competition of variously substituted phenyl rings of benzophenone ketals in lithiation reactions proved to be a useful tool to study both ortho-directing ability and long-range effects of the substituents. The regioselectivities observed in the reaction of benzophenone ketals having one or two chloro substituents in one of the aryl rings with butyllithium complexed to N,N,N′,N″,N″-pentamethyldiethylenetriamine demonstrate the significance of both ortho- and meta-acidifying effect of the chloro substituents. The lithio species thus generated were carboxylated resulting in new polysubstituted benzoic acids.     

Synthesis and stereochemistry of 3‐acetyl‐5‐aminospiro[1,3,4‐thiadiazoline‐2,4′‐thioflavans]

Under acetylating conditions racemic thioflavanone thiosemicarbazones cyclize into racemic 3‐acetyl‐spiro[1,3,4‐thiadiazoline‐2,4′‐thioflavans] and a racemic 3‐acetylspiro[1,3,4‐oxadiazoline‐2,4′‐thioflavan] with trans O(1) or S(1) and Ph(2′_eq). Hindered rotation of the endocyclic N(3) acetyl group spirothia‐diazolines caused the formation of isomers separable by HPLC. X‐ray diffraction analyses, ¹H‐, ¹³C‐, and ¹⁵N NMR measurements as well as MOPAC QM calculations were performed to reveal the structures of these isomers.     

The effect of electron correlation on the conformational space of melatonin

The pineal gland hormone melatonin regulates several physiological processes including circadian rhythm and also alleviates oxidative stress‐induced degenerative diseases. In spite of its important biological roles, no high level ab initio conformational study has been conducted to reveal its structural features. In this work, the conformational flexibility of melatonin was investigated using correlated ab initio calculations. Conformers, obtained previously at the Hartree‐Fock level (HF/6‐31G*), were fully optimized using second order Møller‐Plesset perturbation theory applying the frozen core approximation (MP2(FC)/6‐31G*). Furthermore, single‐point MP4(SDQ,FC)/6‐31G*//MP2(FC)/6‐31G* computations were performed to investigate the effect of higher order perturbation terms. The HF and MP2 conformational spaces are considerably different: the initial 128 structures converged into 102 different local minima as confirmed by frequency calculations; 28 new minima appeared and 26 previous HF local minima disappeared; no “all‐trans” C3 side chain conformations are seen at the MP2(FC) level. The MP2 global minimum conformation is stabilized by an aromatic‐side chain interaction. © 2008 Wiley Periodicals, Inc. J Comput Chem, 2008     

pH transients during salt removal in isoelectric trapping separations: a curse revisited

The pH transients that occur during isoelectric trapping separations as a result of the removal of nonampholytic ionic components have been re-examined. Salts containing strong electrolyte anions and cations, both with equal and dissimilar mobilities, have been studied using anodic and cathodic buffering membranes whose pH values were both equidistant and nonequidistant from pH 7. The direction and magnitude of the pH transient (acidic or basic) was found to depend on both the mobilities of the anion and cation (mu(anion)/mu(cation)) and the pH difference between pH 7 and the pH of the buffering membranes (|pH(memb) (anodic) - 7|/|7 - pH(memb) (cathodic)|). When |pH(memb) (anodic) - 7|/|7 - pH(memb) (cathodic)| = 1, mu(anion)/mu(cation)<1 leads to an acidic pH transient, mu(anion)/mu(cation) = 1 eliminates the pH transient and mu(anion)/mu(cation)>1 leads to a basic pH transient. When mu(anion)/mu(cation) = 1, |pH(memb) (anodic) - 7|/|7 - pH(memb) (cathodic)|<1 leads to a basic pH transient, |pH(memb) (anodic) - 7|/|7 - pH(memb) (cathodic)| = 1 eliminates the pH transient and |pH(memb) (anodic) - 7|/|7 - pH(memb) (cathodic)|>1 leads to an acidic pH transient. By selecting appropriate anodic and cathodic buffering membranes to adjust the |pH(memb) (anodic) - 7|/|7 - pH(memb) (cathodic)| value, pH transients caused by dissimilar anion and cation mobilities can be avoided.     

The Mitsunobu Inversion Reaction of Sterically Hindered 17-Hydroxy Steroids

The Mitsunobu inversion reaction of 3-methoxyestra-1,3,5(10)-trien-17-ol is dramatically influenced by the acidic component. There appears to be a relationship between the dissociation constant of the electron-withdrawing substituent on the aryl acid and the overall effectiveness of the reaction, with more acidic species generally providing a higher yield of inverted product.