Abstract:
The asymmetric unit of the title compound {systematic name: cis-dichloridobis[(3,7-dimethylbicyclo[3.3.1]non-1-ylmethyl)amine-κN]platinum(II) N,N-dimethylformamide solvate}, [PtCl2(C11H19N)2]·C3H7NO, consists of two metrically similar Pt complexes and two dimethylformamide solvent molecules. Each PtII center is coordinated by the amine groups of two (1-adamantylmethyl)amine ligands and two Cl atoms in a cis-square-planar arrangement. The PtII centers lie slightly outside [0.031 (4) and 0.038 (4) ] the coordination planes. The N—Pt—N and Cl—Pt—Cl angles [92.1 (4)–92.30 (11)°] are slightly more open than the N—Pt—Cl angles [87.3 (3)–88.3 (3)°]. N—H...O and N—H...Cl intermolecular hydrogen bonds are observed, forming two discrete pairs of complexes and solvent molecules.

Abstract:
Let X be a smooth compact manifold with boundary. For smooth foliations on the boundary of X admitting a `resolution' in terms of a fibration, we construct a pseudodifferential calculus generalizing the fibred cusp calculus of Mazzeo and Melrose. In particular, we introduce certain symbols leading to a simple description of the Fredholm operators inside the calculus. When the leaves of the fibration `resolving' the foliation are compact, we also obtain an index formula for Fredholm perturbations of Dirac-type operators. Along the way, we obtain a formula for the adiabatic limit of the eta invariant for invertible perturbations of Dirac-type operators, a result of independent interest generalizing the well-known formula of Bismut and Cheeger.

Abstract:
In this article, we introduce the adapted inverse iteration method to generate bicomplex Julia sets associated to the polynomial map $w^2+c$. The result is based on a full characterization of bicomplex Julia sets as the boundary of a particular bicomplex cartesian set and the study of the fixed points of $w^2+c$. The inverse iteration method is used in particular to generate and display in the usual 3-dimensional space bicomplex Julia sets that are dendrites.

Abstract:
We show that the polyhomogeneity at infinity of an asymptotically complex hyperbolic metric is preserved along the Ricci-DeTurck flow. Moreover, if the initial metric is `smooth up to the boundary', this will be preserved by the Ricci-DeTurck flow and the normalized Ricci flow. When the initial metric is K\"ahler, sharper results are obtained in terms of a potential.

Abstract:
Generalizations of the complex number system underlying the mathematical formulation of quantum mechanics have been known for some time, but the use of the commutative ring of bicomplex numbers for that purpose is relatively new. This paper provides an analytical solution of the quantum Coulomb potential problem formulated in terms of bicomplex numbers. We define the problem by introducing a bicomplex hamiltonian operator and extending the canonical commutation relations to the form [X_i,P_k] = i_1 hbar xi delta_{ik}, where xi is a bicomplex number. Following Pauli's algebraic method, we find the eigenvalues of the bicomplex hamiltonian. These eigenvalues are also obtained, along with appropriate eigenfunctions, by solving the extension of Schrodinger's time-independent differential equation. Examples of solutions are displayed. There is an orthonormal system of solutions that belongs to a bicomplex Hilbert space.

Abstract:
In this article, we introduce the concept of normal families of bicomplex holomorphic functions to obtain a bicomplex Montel theorem. Moreover, we give a general definition of Fatou and Julia sets for bicomplex polynomials and we obtain a characterization of bicomplex Fatou and Julia sets in terms of Fatou set, Julia set and filled-in Julia set of one complex variable. Some 3D visual examples of bicomplex Julia sets are also given for the specific slice $\bold{j}=0$.

Abstract:
The infinite matrix `Schwartz' group $G^{-\infty}$ is a classifying group for odd K-theory and carries Chern classes in each odd dimension, generating the cohomology. These classes are closely related to the Fredholm determinant on $G^{-\infty}.$ We show that while the higher (even, Schwartz) loop groups of $G^{-\infty},$ again classifying for odd K-theory, do \emph{not} carry multiplicative determinants generating the first Chern class, `dressed' extensions, corresponding to a star product, do carry such functions. We use these to discuss Bott periodicity for the determinant bundle and the eta invariant. In so doing we relate two distinct extensions of the eta invariant, to self-adjoint elliptic operators and to elliptic invertible suspended families and show that the corresponding $\tau$ invariant is a determinant in this sense.

Abstract:
Let $A(t)$ be an elliptic, product-type suspended (which is to say parameter-dependant in a symbolic way) family of pseudodifferential operators on the fibres of a fibration $\phi$ with base $Y.$ The standard example is $A+it$ where $A$ is a family, in the usual sense, of first order, self-adjoint and elliptic pseudodifferential operators and $t\in\bbR$ is the `suspending' parameter. Let $\pi_{\cA}:\cA(\phi)\longrightarrow Y$ be the infinite-dimensional bundle with fibre at $y\in Y$ consisting of the Schwartz-smoothing perturbations, $q,$ making $A_y(t)+q(t)$ invertible for all $t\in\bbR.$ The total eta form, $\eta_{\cA},$ as described here, is an even form on $\cA(\phi)$ which has basic differential which is an explicit representative of the odd Chern character of the index of the family: % d\eta_{\cA}=\pi_{\cA}^*\gamma_A, \Ch(\ind(A))=[\gamma_{A}]\in H^{\odd}(Y). \tag{*}\label{efatoi.5} % The 1-form part of this identity may be interpreted in terms of the $\tau$ invariant (exponentiated eta invariant) as the determinant of the family. The 2-form part of the eta form may be interpreted as a B-field on the K-theory gerbe for the family $A$ with \eqref{efatoi.5} giving the `curving' as the 3-form part of the Chern character of the index. We also give `universal' versions of these constructions over a classifying space for odd K-theory. For Dirac-type operators, we relate $\eta_{\cA}$ with the Bismut-Cheeger eta form.

Abstract:
Let X be a quasiprojective manifold given by the complement of a divisor $\bD$ with normal crossings in a smooth projective manifold $\bX$. Using a natural compactification of $X$ by a manifold with corners $\tX$, we describe the full asymptotic behavior at infinity of certain complete Kahler metrics of finite volume on X. When these metrics evolve according to the Ricci flow, we prove that such asymptotic behaviors persist at later time by showing the associated potential function is smooth up to the boundary on the compactification $\tX$. However, when the divisor $\bD$ is smooth with $K_{\bX}+[\bD]>0$ and the Ricci flow converges to a Kahler-Einstein metric, we show that this Kahler-Einstein metric has a rather different asymptotic behavior at infinity, since its associated potential function is polyhomogeneous with in general some logarithmic terms occurring in its expansion at the boundary.

Abstract:
On a smoothly stratified space, we identify intersection cohomology of any given perversity with an associated weighted $L^2$ cohomology for iterated fibred cusp metrics on the smooth stratum. In particular given a Witt space, we identify the $L^2$ cohomology of iterated fibred cusp metrics with the middle perversity intersection cohomology of the corresponding stratified space.