Catalan Triangulations of the Möbius Band

A Catalan triangulation of the Möbius band is an abstract simplicial complex triangulating the Möbius band which uses no interior vertices, and has vertices labelled 1, 2, …, n in order as one traverses the boundary. We prove two results about the structure of this set, analogous to well-known results for Catalan triangulations of the disk. The first is a generating function for Catalan triangulations of M having n vertices, and the second is that any two such triangulations are connected by a sequence of diagonal-flips.     

Asymptotic analysis of surface waves at vacuum/porous medium and liquid/porous medium interfaces

Received November 1, 2001 / Published online February 4, 2002     

FMR and TEM Studies of Co and Ni Nanoparticles Implanted in the SiO<Subscript>2</Subscript> Matrix

Fused silica plates have been implanted with 40 keV Co⁺ or Ni⁺ ions to high doses in the range of (0.25–1.0) × 10¹⁷ ions/cm², and magnetic properties of the implanted samples have been studied with ferromagnetic resonance (FMR) technique supplemented by transmission electron microscopy, electron diffraction and energy dispersive X-ray spectroscopy. The high-dose implantation with 3d-ions results in the formation of cobalt and nickel metal nanoparticles in the irradiated subsurface layer of the SiO₂ matrix. Co and Ni nanocrystals with hexagonal close packing and face-centered cubic structures have a spherical shape and the sizes of 4–5 nm (for cobalt) and 6–14 nm (for nickel) in diameter. Room-temperature FMR signals from ensembles of Co and Ni nanoparticles implanted in the SiO₂ matrix exhibit an out-of-plane uniaxial magnetic anisotropy that is typical for thin magnetic films. The dose and temperature dependences of FMR spectra have been analyzed using the Kittel formalism, and the effective magnetization and g-factor values have been obtained for Co- and Ni-implanted samples. Nonsymmetric FMR line shapes have been fitted by a sum of two symmetrical curves. The dependences of the magnetic parameters of each curve on the implantation dose and temperature are presented.     

Dynamics of self-excited oscillators with neutral delay coupling

The work is devoted to analytic and numeric investigation of dynamical behavior in a system of two Van der Pol (VdP) oscillators coupled by a non-dispersive elastic rod. The model is rigorously reduced to a system of nonlinear neutral differential delay equations. For the case of relatively small coupling and moderate delay, an approximate analytic investigation can be accomplished by means of an averaging procedure. The region of synchronization in the space of parameters is established and characteristic bifurcations are revealed. A numeric study confirms the validity of the analytic approach in the synchronization region. Beyond this region, the averaging approach is no more valid. A multitude of quasiperiodic and chaotic-like orbits has been revealed. Especially interesting behavior occurs in the case of relatively large delays and corresponds to sequential quenching and excitation of the VdP oscillators. This regime is also explored analytically, by means of a large-delay approximation, which reduces the system to a perturbed discrete map.     

Sturm Sequences and Random Eigenvalue Distributions

This paper proposes that the study of Sturm sequences is invaluable in the numerical computation and theoretical derivation of eigenvalue distributions of random matrix ensembles. We first explore the use of Sturm sequences to efficiently compute histograms of eigenvalues for symmetric tridiagonal matrices and apply these ideas to random matrix ensembles such as the β-Hermite ensemble. Using our techniques, we reduce the time to compute a histogram of the eigenvalues of such a matrix from O(n ²+m) to O(mn) time where n is the dimension of the matrix and m is the number of bins (with arbitrary bin centers and widths) desired in the histogram (m is usually much smaller than n). Second, we derive analytic formulas in terms of iterated multivariate integrals for the eigenvalue distribution and the largest eigenvalue distribution for arbitrary symmetric tridiagonal random matrix models. As an example of the utility of this approach, we give a derivation of both distributions for the β-Hermite random matrix ensemble (for general β). Third, we explore the relationship between the Sturm sequence of a random matrix and its shooting eigenvectors. We show using Sturm sequences that assuming the eigenvector contains no zeros, the number of sign changes in a shooting eigenvector of parameter λ is equal to the number of eigenvalues greater than λ. Finally, we use the techniques presented in the first section to experimentally demonstrate a O(log n) growth relationship between the variance of histogram bin values and the order of the β-Hermite matrix ensemble. This research was supported by NSF Grant DMS–0411962.     

Structure and the magnetic and magneto-optical properties of Co-Sm-O nanogranular films

The magnetic properties and magneto-optical effects in nanocomposites based on Co-Sm-O films prepared through pulsed plasma sputtering of a SmCo₅ target are investigated. It is shown that, depending on the technological conditions and regimes of subsequent annealing, the films can have different structures from cobalt nanoparticles distributed in the dielectric samarium oxide matrix with a magnetic phase volume of more than 60% to a continuous polycrystalline cobalt film with embedded samarium oxide nanoparticles. The evolution of the spectra of the magneto-optical Kerr effect and the field dependences of the magnetization is studied as a function of the film structure.     

Magnetic and magneto-optical properties of ion-synthesized cobalt nanoparticles in silicon oxide

The magnetic and magneto-optical properties of ion-synthesized cobalt nanoparticles in the amorphous silicon oxide matrix are investigated as a function of the implantation dose. The analysis of the field dependences of the magnetization and the magneto-optical Faraday and Kerr effects demonstrates that, as the ion implantation dose increases, the superparamagnetic behavior of an ensemble of cobalt nanoparticles at room temperature gives way to a ferromagnetic response with the anisotropy characteristic of a thin magnetic film. The magnetization curves for the superparamagnetic and ferromagnetic ensembles of cobalt nanoparticles are simulated to determine their average sizes and the filling density in the irradiated layer of the silicon dioxide matrix. It is revealed that the spectral dependences of the Faraday and Kerr effects for ion-synthesized cobalt nanoparticles differ substantially from those for continuous cobalt films due to the localized excitations of free electrons in the nanoparticles.     

Group labelings of graphs

Given a graph Γ an abelian group G, and a labeling of the vertices of Γ with elements of G, necessary and sufficient conditions are stated for the existence of a labeling of the edges in which the label of each vertex equals the product of the labels of its incident edges. Such an edge labeling is called compatible. For vertex labelings satisfying these conditions, the set of compatible edge labelings is enumerated.     

Stochastic web as a generator of three-dimensional quasicrystal symmetry

It is shown that two coupled oscillators perturbed by periodic kicks generate a thin stochastic web in the four-dimensional phase space, which differs from the Arnold web. Under some resonance-type condition the web possesses a quasicrystal-type symmetry. In three-dimensional coordinate space, the web's symmetry corresponds to the icosahedral one and, due to that, the original four-dimensional map can be considered as a dynamical generator of the quasicrystal-type tiling of three-dimensional space.