Rare Earth Complexes in Ionic Liquids - Structures, Electrochemical and Optical Properties

The thesis presented here deals with solvent-solute interactions of rare-earth ions in bis(trifluoromethanesulfonyl)amide ([Tf2N)- and trifluoromethanesulfonate ([OTf])-based ionic liquids (IL). In order to elucidate the process of solvatation, rare earth iodides, triflates and bis(trifluoromethanesulfonyl)amides are synthesized as precursors and reacted with [Tf2N]- and [OTf]-based ionic liquids, containing substituted pyrrolidinium or imidazolium heterocycles as cations. Interaction of NdI3 with the room temperature ionic liquid (RTIL) 1-butyl-1-methylpyrrolidinium [Tf2N] ([bmpyr][Tf2N]) leads to the formation of the sparingly soluble compound [bmpyr]4[NdI6][Tf2N], the anion of the RTIL being incorporated in the structure in a non-coordinating fashion. By displacing the butyl chain in [bmpyr]+ by a methyl group, the compound [mppyr]3[NdI6], a [Tf2N]-free compound is obtained. The solvatation of the other trivalent iodides of the rare-earth elements leads to similar compounds with the composition [bmpyr]4[LnI6][Tf2N] (Ln = La, Pr, Sm, Dy, Er) in which the metals are coordinated octahedrally by iodine atoms. A structural change is observed between Pr and Nd, which is manifested in the arrangement of the [LnI6]-ocahedra with respect to each other. The coordination mode of the [Tf2N]-anion was determined by reaction of Ln(Tf2N)3 with [bmpyr][Tf2N]. X-Ray structure determinations reveal that [bmpyr]2[Ln(Tf2N)5] is formed for the larger lanthanides (La-Tb), whereas the smaller ones (Dy-Lu) are coordinated by four ligands in [bmpyr][Ln(Tf2N)4]. The RTIL 1-butyl-1-methylimidazolium [Tf2N] ([bmim][Tf2N]) reacts with Eu(Tf2N)3 to give a lanthanide-based RTIL [bmim][Eu(Tf2N)5] (glass-transition point: -50°C). Ligand exchange examinations reveal that [OTf]-anions are able to displace [Tf2N]-ligands. For example the compound [bmpyr]4[Yb(OTf)6][Tf2N] can be crystallized from Yb(Tf2N)3 and [bmpyr][OTf]. Simple halides of the salt-like divalent rare-earth dissolve in [Tf2N]-based ILs under complete ligand exchange as can be shown by the formation of [mppyr]2[Yb(Tf2N)4] from YbI2 and [mppyr][Tf2N]. In addition, [mppyr]2[AE(Tf2N)4] (AE = Ca, Sr) and [mppyr][Ba(Tf2N)3] were synthesized as model compounds for the larger divalent lanthanides (Eu, Sm)

Asymptotic Delsarte cliques in distance-regular graphs

We give a new bound on the parameter $\lambda$ (number of common neighbors of a pair of adjacent vertices) in a distance-regular graph $G$, improving and generalizing bounds for strongly regular graphs by Spielman (1996) and Pyber (2014). The new bound is one of the ingredients of recent progress on the complexity of testing isomorphism of strongly regular graphs (Babai, Chen, Sun, Teng, Wilmes 2013). The proof is based on a clique geometry found by Metsch (1991) under certain constraints on the parameters. We also give a simplified proof of the following asymptotic consequence of Metsch's result: if $k\mu = o(\lambda^2)$ then each edge of $G$ belongs to a unique maximal clique of size asymptotically equal to $\lambda$, and all other cliques have size $o(\lambda)$. Here $k$ denotes the degree and $\mu$ the number of common neighbors of a pair of vertices at distance 2. We point out that Metsch's cliques are "asymptotically Delsarte" when $k\mu = o(\lambda^2)$, so families of distance-regular graphs with parameters satisfying $k\mu = o(\lambda^2)$ are "asymptotically Delsarte-geometric."Comment: 10 page

Most primitive groups are full automorphism groups of edge-transitive hypergraphs

We prove that, for a primitive permutation group G acting on a set of size n, other than the alternating group, the probability that Aut(X,Y^G) = G for a random subset Y of X, tends to 1 as n tends to infinity. So the property of the title holds for all primitive groups except the alternating groups and finitely many others. This answers a question of M. Klin. Moreover, we give an upper bound n^{1/2+\epsilon} for the minimum size of the edges in such a hypergraph. This is essentially best possible.Comment: To appear in special issue of Journal of Algebra in memory of Akos Seres

Capillary crystallization for crystal assessment and fibre-optic devices

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Evasiveness and the Distribution of Prime Numbers

We confirm the eventual evasiveness of several classes of monotone graph properties under widely accepted number theoretic hypotheses. In particular we show that Chowla's conjecture on Dirichlet primes implies that (a) for any graph $H$, "forbidden subgraph $H$" is eventually evasive and (b) all nontrivial monotone properties of graphs with $\le n^{3/2-\epsilon}$ edges are eventually evasive. ($n$ is the number of vertices.) While Chowla's conjecture is not known to follow from the Extended Riemann Hypothesis (ERH, the Riemann Hypothesis for Dirichlet's $L$ functions), we show (b) with the bound $O(n^{5/4-\epsilon})$ under ERH. We also prove unconditional results: (a$'$) for any graph $H$, the query complexity of "forbidden subgraph $H$" is $\binom{n}{2} - O(1)$; (b$'$) for some constant $c>0$, all nontrivial monotone properties of graphs with $\le cn\log n+O(1)$ edges are eventually evasive. Even these weaker, unconditional results rely on deep results from number theory such as Vinogradov's theorem on the Goldbach conjecture. Our technical contribution consists in connecting the topological framework of Kahn, Saks, and Sturtevant (1984), as further developed by Chakrabarti, Khot, and Shi (2002), with a deeper analysis of the orbital structure of permutation groups and their connection to the distribution of prime numbers. Our unconditional results include stronger versions and generalizations of some result of Chakrabarti et al.Comment: 12 pages (conference version for STACS 2010

High pressure water jet cutting and stripping

High pressure water cutting techniques have a wide range of applications to the American space effort. Hydroblasting techniques are commonly used during the refurbishment of the reusable solid rocket motors. The process can be controlled to strip a thermal protective ablator without incurring any damage to the painted surface underneath by using a variation of possible parameters. Hydroblasting is a technique which is easily automated. Automation removes personnel from the hostile environment of the high pressure water. Computer controlled robots can perform the same task in a fraction of the time that would be required by manual operation

On the automorphism groups of strongly regular graphs II

We derive strong constraints on the automorphism groups of strongly regular (SR) graphs, resolving old problems motivated by Peter Cameron's 1981 description of large primitive groups.Trivial SR graphs are the disjoint unions of cliques of equal size and their complements. Graphic SR graphs are the line-graphs of cliques and of regular bipartite cliques (complete bipartite graphs with equal parts) and their complements.We conjecture that the order of the automorphism group of a non-trivial, non-graphic SR graph is quasi-polynomially bounded, i.e., it is at most exp((logn)C) for some constant C, where n is the number of vertices.While the conjecture remains open, we find surprisingly strong bounds on important parameters of the automorphism group. In particular, we show that the order of every automorphism is O(n8), and in fact O(n) if we exclude the line-graphs of certain geometries. We prove the conjecture for the case when the automorphism group is primitive; in this case we obtain a nearly tight n1+log2n bound.We obtain these bounds by bounding the fixicity of the automorphism group, i.e., the maximum number of fixed points of non-identity automorphisms, in terms of the second largest (in magnitude) eigenvalue and the maximum number of pairwise common neighbors of a regular graph. We connect the order of the automorphisms to the fixicity through an old lemma by Ákos Seress and the author.We propose to extend these investigations to primitive coherent configurations and offer problems and conjectures in this direction. Part of the motivation comes from the complexity of the Graph Isomorphism problem