Equilibrium and NMR studies on GdIII, YIII, CuII and ZnII complexes of various DTPA-N,N'-bis(amide) ligands. Kinetic stabilities of the gadolinium(III) complexes

Three DTPA-derivative ligands, the non-substituted DTPA-bis(amide) (L(0)), the mono-substituted DTPA-bis(n-butylamide) (L(1)) and the di-substituted DTPA-bis[bis(n-butylamide)] (L(2)) were synthesized. The stability constants of their Gd3+ complexes (GdL) have been determined by pH-potentiometry with the use of EDTA or DTPA as competing ligands. The endogenous Cu2+ and Zn2+ ions form ML, MHL and M(2)L species. For the complexes CuL(0) and CuL(1) the dissociation of the amide hydrogens (CuLH(-1)) has also been detected. The stability constants of complexes formed with Gd3+, Cu2+ and Zn2+ increase with an increase in the number of butyl substituents in the order ML(0) < ML(1) < ML(2). NMR studies of the diamagnetic YL(0) show the presence of four diastereomers formed by changing the chirality of the terminal nitrogens of their enantiomers. At 323 K, the enantiomerization process, involving the racemization of central nitrogen, falls into the fast exchange range. By the assignment and interpretation of 1H and 13C NMR spectra, the fractions of the diastereomers were found to be equal at pH = 5.8 for YL(0). The kinetic stabilities of GdL(0), GdL(1) and GdL(2) have been characterized by the rates of the exchange reactions occurring between the complexes and Eu3+, Cu2+ or Zn2+. The rates of reaction with Eu3+ are independent of the [Eu3+] and increase with increasing [H+], indicating the rate determining role of the proton assisted dissociation of complexes. The rates of reaction with Cu2+ and Zn2+ increase with rising metal ion concentration, which shows that the exchange can take place with direct attack of Cu2+ or Zn2+ on the complex, via the formation of a dinuclear intermediate. The rates of the proton, Cu2+ and Zn2+ assisted dissociation of Gd3+ complexes decrease with increasing number of the n-butyl substituents, which is presumably the result of steric hindrance hampering the formation or dissociation of the intermediates. The kinetic stabilities of GdL(0) and GdL(1) at pH = 7.4, [Cu2+] = 1 x 10(-6) M and [Zn(2+)] = 1 x 10(-5) M are similar to that of Gd(DTPA)2-, while the complex GdL2 possesses a much higher kinetic stability.     

Uncatalyzed reactions in the classical Belousov-Zhabotinsky system. 2. The malonic acid-bromate reaction in acidic media

The title reaction was studied with various techniques in 1 M sulfuric acid, a usual medium for the oscillatory Belousov-Zhabotinsky (BZ) reaction. It was found to be a more complex process than the bromomalonic acid (BrMA)-BrO3- reaction studied previously in the first part of this work. Malonic acid (MA) can react with acidic bromate by two parallel mechanisms. The main aim of the present research was to determine the mechanisms, the rate laws, and the rate constants for these parallel channels. In one reaction channel the first molecular products are glyoxalic acid (GOA) and CO2 while in the other channel mesoxalic acid (MOA) is the first molecular intermediate, that is, no CO2 is formed in this step. To prove these two independent routes specific colorimetric techniques were developed to determine GOA and MOA selectively. The rate of the GOA channel was determined by following the rate of the carbon dioxide evolution characteristic for this reaction route. In this step, regarding it as an overall process, one MA is oxidized to GOA and CO2 and one BrO3- is reduced to HOBr, which forms BrMA with another MA. The initial rate of the GOA channel is a bilinear function of the initial MA and BrO3- concentrations with a second-order rate constant k(GOA)= 2.4 x 10(-7) M(-1) s(-1). The rate of the other channel was calculated from the rate of the BrO3- consumption measured in separate experiments, assuming that the measured depletion is a sum of two separate terms reflecting the consumptions due to the two independent channels. In the MOA channel one MA is oxidized to MOA and one BrO3- is consumed while another MA is brominated as in the GOA channel. It was found that the initial rate of the MOA channel is also a bilinear function of the MA and BrO3- concentrations with a second-order rate constant k(MOA)= 2.46 x 10(-6) M(-1) s(-1). Separate chemical mechanisms are suggested for both channels. In all of the various bromate-substrate reactions of these mechanisms oxygen atom transfer from the bromate to the substrate occurs generating bromous acid intermediate. This can be of high importance in BZ systems as bromous acid is the autocatalytic intermediate there. GOA and MOA also can be oxidized by acidic bromate but a study of these reactions will be published later.     

Stability in the Erdős–Gallai Theorem on cycles and paths,II

The Erd?s–Gallai Theorem states that for $k\ge 3$, any $n$-vertex graph with no cycle of length at least $k$ has at most $\frac{1}{2}(k?1)(n?1)$ edges. A stronger version of the Erd?s–Gallai Theorem was given by Kopylov: If $G$ is a 2-connected $n$-vertex graph with no cycle of length at least $k$, then $e\left(G\right)\le max\{h(n,k,2),h(n,k,?\frac{k?1}{2}?)\}$, where $h(n,k,a)?\left(\genfrac{}{}{0ex}{}{k?a}{2}\right)+a(n?k+a)$. Furthermore, Kopylov presented the two possible extremal graphs, one with $h(n,k,2)$ edges and one with $h(n,k,?\frac{k?1}{2}?)$ edges.In this paper, we complete a stability theorem which strengthens Kopylov’s result. In particular, we show that for $k\ge 3$ odd and all $n\ge k$, every $n$-vertex 2-connected graph $G$ with no cycle of length at least $k$ is a subgraph of one of the two extremal graphs or $e\left(G\right)\le max\{h(n,k,3),h(n,k,\frac{k?3}{2})\}$. The upper bound for $e\left(G\right)$ here is tight.     

A Radon–Nikodym type theorem for forms

The Lebesgue decomposition theorem and the Radon–Nikodym theorem are the cornerstones of the classical measure theory. These theorems were generalized in several settings and several ways. Hassi, Sebestyén, and de Snoo recently proved a Lebesgue type decomposition theorem for nonnegative sesquilinear forms defined on complex linear spaces. The main purpose of this paper is to formulate and prove also a Radon–Nikodym type result in this setting. As an application, we present a Lebesgue type decomposition theorem and solve a special case of the infimum problem for densely defined (not necessarily bounded) positive operators.     

Complement of forms

The notions of the parallel sum, the parallel difference, and the complement of two nonnegative sesquilinear forms were introduced and studied by Hassi, Sebestyé and de Snoo in Hassi et al. (Oper Theory Adv Appl 198:211–227, 2010) and Hassi et al. (J Funct Anal 257(12):3858–3894, 2009). In this paper we continue these investigations. The Galois correspondence induced by the map ${\mathfrak{m} \mapsto \mathfrak{m}_\mathfrak{t}}$ (where ${\mathfrak{m}_\mathfrak{t}}$ denotes the ${\mathfrak{t}}$ -complement of ${\mathfrak{m}}$ ) is also studied. Inspired by the work of Eriksson and Leutwiler Eriksson and Leutwiler (Math Ann 274:301–317, 1986), we introduce the notion of quasi-unit for nonnegative sesquilinear forms. The quasi-units are characterized by means of the complement and the disjoint part. It is also shown that the ${{\mathfrak{t}}}$ -quasi-units coincide with the extreme points of the convex set ${\mathfrak{z}: 0 \leq \mathfrak{z} \leq \mathfrak{t}\}}$ .     

Sprays metrizable by Finsler functions of constant flag curvature

In this paper we characterize sprays that are metrizable by Finsler functions of constant flag curvature. By solving a particular case of the Finsler metrizability problem, we provide the necessary and sufficient conditions that can be used to decide whether or not a given homogeneous system of second order ordinary differential equations represents the geodesic equations of a Finsler function of constant flag curvature. The conditions we provide are tensorial equations on the Jacobi endomorphism. We identify the class of homogeneous SODE where the Finsler metrizability is equivalent with the metrizability by a Finsler function of constant flag curvature.     

Cycle‐Saturated Graphs with Minimum Number of Edges

A graph G is called H‐saturated if it does not contain any copy of H, but for any edge e in the complement of G, the graph contains some H. The minimum size of an n‐vertex H‐saturated graph is denoted by . We prove holds for all , where is a cycle with length k. A graph G is H‐semisaturated if contains more copies of H than G does for . Let be the minimum size of an n‐vertex H‐semisaturated graph. We have We conjecture that our constructions are optimal for . © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 203–215, 2013     

Isometric embeddings into Heisenberg groups

We study isometric embeddings of a Euclidean space or a Heisenberg group into a higher dimensional Heisenberg group, where both the source and target space are equipped with an arbitrary left-invariant homogeneous distance that is not necessarily sub-Riemannian. We show that if all infinite geodesics in the target are straight lines, then such an embedding must be a homogeneous homomorphism. We discuss a necessary and certain sufficient conditions for the target space to have this ‘geodesic linearity property’, and we provide various examples.