The Inverse Iteration Method for Julia Sets in the 3-Dimensional Space

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.Comment: 16 pages, 4 figure

Finite-Dimensional Bicomplex Hilbert Spaces

This paper is a detailed study of finite-dimensional modules defined on bicomplex numbers. A number of results are proved on bicomplex square matrices, linear operators, orthogonal bases, self-adjoint operators and Hilbert spaces, including the spectral decomposition theorem. Applications to concepts relevant to quantum mechanics, like the evolution operator, are pointed out.Comment: 21 page

The bicomplex quantum Coulomb potential problem

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.Comment: Clarifications; some figures removed; version to appear in Can. J. Phy

On the Klein-Gordon equation and hyperbolic pseudoanalytic function theory

Elliptic pseudoanalytic function theory was considered independently by Bers and Vekua decades ago. In this paper we develop a hyperbolic analogue of pseudoanalytic function theory using the algebra of hyperbolic numbers. We consider the Klein-Gordon equation with a potential. With the aid of one particular solution we factorize the Klein-Gordon operator in terms of two Vekua-type operators. We show that real parts of the solutions of one of these Vekua-type operators are solutions of the considered Klein-Gordon equation. Using hyperbolic pseudoanalytic function theory, we then obtain explicit construction of infinite systems of solutions of the Klein-Gordon equation with potential. Finally, we give some examples of application of the proposed procedure

cis-Bis[(1-adamantylmeth­yl)amine-κN]­dichloridoplatinum(II) N,N-dimethyl­formamide solvate

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) Å] 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

The Marmara Sea Gateway since ~16 ky BP: non-catastrophic causes of paleoceanographic events in the Black Sea at 8.4 and 7.15 ky BP

The Late Quaternary history of connection of the Black Sea to the Eastern Mediterranean has been intensely debated. Ryan, Pitman and coworkers advocate two pulses of outflow from the Black Sea to the world ocean at ~16–14.7 ky BP and ~11–10 ky BP. From ~14.7–11 ky BP and from ~10–8.4 ky BP, they suggest that the level of the Black Sea fell to ~ -100 m. At 8.4 ky BP, they further claim that a catastrophic flood occurred in a geological instant, refilling the Black Sea with saline waters from the Mediterranean. In contrast, we continue to gather evidence from seismic profiles and dated cores in the Marmara Sea which demonstrate conclusively that the proposed flood did not occur. Instead, the Black Sea has been at or above the Bosphorus sill depth and flowing into the world ocean unabated since ~10.5 ky BP. This conclusion is based on continuous Holocene water-column stratification (leading to sapropel deposition in the Marmara Sea and the Aegean Sea), proxy indicators of sea-surface salinity, and migration of endemic species across the Bosphorus in both directions whenever appropriate hydrographic conditions existed in the strait. The two pulses of outflow documented by Ryan, Pitman and coworkers find support in our data, and we have modified our earlier interpretations so that these pulses now coincide with the development of mid-shelf deltas: \Delta 2 (16–14.7 ky BP) and \Delta 1 (10.5–9 ky BP) at the southern end of the Bosphorus Strait. However, continued Black Sea outflow after 9 ky BP prevented the northward advection of Mediterranean water and the entry of open-marine species into the Black Sea for more than 1000 years. Sufficient Mediterranean water to change the Sr-isotopic composition of slope and shelf water masses was not available until ~8.4 ky BP (along with the first arrival of many varieties of marine fauna and flora), whereas euryhaline molluscs did not successfully populate the Black Sea shelves until ~7.15 ky BP. Instead of relying on catastrophic events, we recognize a slow, progressive reconnection of the Black Sea to the world ocean, accompanied by significant time lags

Development of an improved model estimating the nutrient content of the bole for four boreal tree species

An improved model for estimating nutrient contents of the commercial portion of tree boles was developed for four boreal tree species (Populus tremuloides Michx., Betula papyrifera Marsh., Picea glauca (Moench) Voss, and Abies balsamea (L.) Mill.). This model considers the spatial pattern of variation of nutrient concentrations inside the bole and its relationships with tree size. For all species-nutrient combinations, no significant pattern was found for vertical variations in nutrient concentrations, while two types of nonlinear models, using distance from the tree periphery as the independent variable, fit the pattern of horizontal (or radial) variations. These patterns of variability were used to estimate the global nutrient concentration of the bole by using mathematical integration. The values obtained with this method were generally lower, especially for large stems, than values obtained with traditional methods that do not consider the variability of nutrient concentrations inside the bole. This improved model would permit better estimates of the mounts of nutrients lost in biomass upon forest harvesting, as well as internal cycling of nutrients within the bole