Mathematics Commons^™

A Numerical Method For A Nonlinear Singularly Perturbed Interior Layer Problem Using An Approximate Layer Location, Jason Quinn

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A class of nonlinear singularly perturbed interior layer problems is examined in this paper. Solutions exhibit an interior layer at an a priori unknown location. A numerical method is presented that uses a piecewise uniform mesh refined around approximations to the first two terms of the asymptotic expansion of the interior layer location. The first term in the expansion is used exactly in the construction of the approximation which restricts the range of problem data considered. The method is shown to converge point-wise to the true solution with a first order convergence rate (overlooking a logarithmic factor) for sufficiently small …

Bergm: Bayesian Exponential Random Graphs In R, Alberto Caimo, Nial Friel

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In this paper we describe the main featuress of the Bergm package for the open-source R software which provides a comprehensive framework for Bayesian analysis for exponential random graph models: tools for parameter estimation, model selection and goodness-of-fit diagnostics. We illustrate the capabilities of this package describing the algorithms through a tutorial analysis of three network datasets.

New Insights On Free Energies And Saint-Venant’S Principle In Viscoelasticity, L. Diseri, G. Gentili, John Murrough Golden

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This work was conceived in 1999 and brought near completion by 2003. Giorgio Gentili was deeply involved in this research until his untimely death. He is greatly missed. Work pressures on the other authors forced a postponement of research on this topic, originally envisaged as lasting a few months but in the event it turned out to be nearly ten years. We now dedicate this work to the memory of Giorgio and to his Family.

Fully Inert Subgroups Of Abelian P-Groups, Brendan Goldsmith, Luigi Salce, Paolo Zanardo

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A subgroup H of an Abelian group G is said to be fully inert in G, if for every endomorphism ϕ of G, the factor group (H+ϕ(H))/H" role="presentation"> is finite. This notion arises in the study of the dynamical properties of endomorphisms (entropy). The principal result of this work is that fully inert subgroups of direct sums of cyclic p-groups are commensurable with fully invariant subgroups of the direct sum.

Matrix G-Strands, Darryl Holm, Rossen Ivanov

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We discuss three examples in which one may extend integrable Euler–Poincare ordinary differential equations to integrable Euler–Poincare partial differential
equations in the matrix G-Strand context. After describing matrix G-Strand examples for SO(3) and SO(4) we turn our attention to SE(3) where the matrix G-Strand equations recover the exact rod theory in the convective representation. We then find a zero curvature representation of these equations and establish the conditions under which they are completely integrable. Thus, the G-Strand equations turn out to be a rich source of integrable systems. The treatment is meant to be expository and most concepts are explained …

Hamiltonian Approach To The Modeling Of Internal Geophysical Waves With Vorticity, Alan Compelli

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We examine a simplified model of internal geophysical waves in a rotational 2-dimensional water-wave system, under the influence of Coriolis forces and with gravitationally induced waves. The system consists of a lower medium, bound underneath by an impermeable flat bed, and an upper lid. The 2 media have a free common interface. Both media have constant density and constant (non-zero) vorticity. By examining the governing equations of the system we calculate the Hamiltonian of the system in terms of its conjugate variables and perform a variable transformation to show that it has canonical Hamiltonian structure. We then linearize the system, …

Symmetry And Reductions Of Integrable Dynamical Systems: Peakon And The Toda Chain Systems, Vladimir Gerdjikov, Rossen Ivanov, Gaetano Vilasi

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We are analyzing several types of dynamical systems which are both integrable and important for physical applications. The first type are the so-called peakon systems that appear in the singular solutions of the Camassa-Holm equation describing special types of water waves. The second type are Toda chain systems, that describe molecule interactions. Their complexifications model soliton interactions in the adiabatic approximation. We analyze the algebraic aspects of the Toda chains and describe their real Hamiltonian forms.

Complex Absorbing Potential Method For Dirac Operators. Clusters Of Resonances, Jimmy Kungsman, Michael Melgaard

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For both nonrelativistic and relativistic Hamiltonians, the Complex Absorbing Potential (CAP) method has been applied extensively to calculate resonances in Physics and Chemistry. We study clusters of resonances for the perturbed Dirac operator near the real axis and, in the semiclassical limit, we establish the CAP method rigorously by showing that resonances are perturbed eigenvalues of the nonselfadjoint CAP Hamiltonian, and vice versa.

Evaporation Screening, Scattering, And Quantum Tunneling Near The Horizons Of Schwarzschild And Reissner-Nordstroem Black Holes, Emil Prodanov

Articles

For the radial motion of massive particles with large angular momenta in Schwarzschild geometry and that of massive charged particles with large
angular momenta or energy in a particular range in Reissner-Nordstroem geometry, there exist classically forbidden regions on the outside of
the respective event horizons which scatter certain in-falling geodesics or screen some of the black holes' evaporation by reflecting the emitted

particles back into the black holes. Quantum tunneling across this forbidden regions is studied.

Hamiltonian Formulation Of 2 Bounded Immiscible Media With Constant Non-Zero Vorticities And A Common Interface, Alan Compelli

Articles

We examine a 2-dimensional water-wave system, with gravitationally induced waves, consisting of a lower medium bound underneath by an impermeable flat bed and an upper medium bound above by an impermeable lid such that the 2 media have a free common interface. Both media have constant density and constant (non-zero) vorticity. By examining the governing equations of the system we calculate the Hamiltonian of the system in terms of it's conjugate variables and per- form a variable transformation to show that it has canonical Hamiltonian structure.

The Behavior Of Normality When Iteratively Finding The Normal To A Line In An Lp Geometry, David L. Farnsworth, Joshua M. Fitzhugh

Articles

The normal direction to the normal direction to a line in Minkowski geometries generally does not give the original line. We show that in lp geometries with p > 1 repeatedly finding the normal line through the origin gives sequences of lines that monotonically approach specific lines of symmetry of the unit circle. Which lines of symmetry that are approached depends upon the value of p and the slope of the initial line.

A Construction That Produces Wallis-Type Formulas, Joshua M. Fitzhugh, David L. Farnsworth

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Generalizations of the geometric construction that repeatedly attaches rectangles to a square, originally given by Myer- son, are presented. The initial square is replaced with a rectangle, and also the dimensionality of the construction is increased. By selecting values for the various parameters, such as the lengths of the sides of the original rectangle or rectangular box in dimensions more than two and their relationships to the size of the attached rectangles or rectangular boxes, some interesting formulas are found. Examples are Wallis-type infinite-product formulas for the areas of p-circles with p > 1.

Holomorphic Basis For Families Of Subspaces Of A Banach Space, Milena Venkova, Christopher Boyd

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In this article we investigate the connection between a family of com-plemented subspaces of a Banach space having a holomorphic basis, and being holomorphically complemented.

Counting The Number Of Squares Reachable In K Knight's Moves., Amanda M. Miller, David L. Farnsworth

Articles

from its initial position on an infinite chessboard are derived. The number of squares reachable in exactly k moves are 1, 8, 32, 68, and 96 for k = 0, 1, 2, 3, and 4, respectively, and 28k – 20 for k ≥ 5. The cumulative number of squares reach- able in k or fever moves are 1, 9, 41, and 109 for k = 0, 1, 2, and 3, respectively, and 14k-squared – 6k + 5 for k ≥ 4. Although these formulas are known, the proofs that are presented are new and more mathematically accessible then preceding proofs.

Pricing European And American Options In The Heston Model With Accelerated Explicit Finite Differencing Methods, Conall O'Sullivan, Stephen O'Sullivan

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No abstract provided.

On The Quadratic Bundles Related To Hermitian Symmetric Spaces, Tihomir Valchev

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We develop the direct scattering problem for quadratic bundles associated to Hermitian symmetric spaces. We adapt the dressing method for quadratic bundles which allows us to find special solutions to multicomponent derivative Schrodinger equation for instance. The latter is an infinite dimensional Hamiltonian system possessing infinite number of integrals of motion. We demonstrate how one can derive them by block diagonalization of the corresponding Lax pair.

Knight's Tours On 3 X N Chessboards With A Single Square Removed, Amanda M. Miller, David L. Farnsworth

Articles

The following theorem is proved: A knight’s tour exists on all 3 x n chessboards with one square removed unless: n is even, the removed square is (i, j) with i + j odd, n = 3 when any square other than the center square is removed, n = 5, n = 7 when any square other than square (2, 2) or (2, 6) is removed, n = 9 when square (1, 3), (3, 3), (1, 7), (3, 7), (2, 4), (2, 6), (2, 2), or (2, 8) is removed, or n = 11 when square (1, 3), (2, 4), …

G-Strands And Peakon Collisions On Diff(R), Darryl Holm, Rossen Ivanov

Articles

A G-strand is a map g : R x R --> G for a Lie group G that follows from Hamilton's principle for a certain class of G-invariant Lagrangians. Some G-strands on finite-dimensional groups satisfy 1+1 space-time evolutionary equations that admit soliton solutions as completely integrable Hamiltonian systems. For example, the SO(3)-strand equations may be regarded physically as integrable dynamics for solitons on a continuous spin chain. Previous work has shown that G-strands for diffeomorphisms on the real line possess solutions with singular support (e.g. peakons). This paper studies collisions of such singular solutions of G-strands when G = Diff( …

Mass Transfer From A Vertical Flat Plate Due To A Constant Upward Flow, David Mcdonnell, Brendan Redmond, Lawrence J. Crane

Articles

This paper examines the mass transfer from a vertical flat surface of a soluble material due to a constant upward flow. The mass transfer rate due to this upward flow is calculated and used to obtain the distance along the surface at which the boundary layer separates. For relatively large velocities no separation will occur and the solution approaches that of forced convection on a horizontal surface.

Spectral Multiplicity Of Selfadjoint Schrodinger Operators On Star-Graphs With Standard Interface Conditions, Sergey Simonov, Harald Woracek

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We analyze the singular spectrum of selfadjoint operators which arise from pasting a finite number of boundary relations with a standard interface condition. A model example for this situation is a Schroedinger operator on a star-shaped graph with continuity and Kirchhoff conditions at the interior vertex. We compute the multiplicity of the singular spectrum in terms of the spectral measures of the Weyl functions associated with the single (independently considered) boundary relations. This result is a generalization and refinement of Theorem of I.S. Kac.

Extreme Wave Events In Ireland: 14 680 Bp–2012, Laura Cooke, John M. Dudley, Frédéric Dias

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The island of Ireland is battered by waves from all sides, most ferociously on the west coast as the first port of call for waves travelling across the Atlantic Ocean. However, when discussing ocean events relevant to the nation of Ire- land, one must actually consider its significantly larger designated continental shelf, which is one of the largest seabed territories in Europe. With this expanded definition, it is not surprising that Ireland has been subject to many oceanic events which could be designated as “extreme”; in this paper we present what we believe to be the first catalogue of such …

On The Persistence Properties Of The Cross-Coupled Camassa-Holm System, David Henry, Darryl Holm, Rossen Ivanov

Articles

In this paper we examine the evolution of solutions, that initially have compact support, of a recently-derived system of cross-coupled Camassa-Holm equations. The analytical methods which we employ provide a full picture for the persistence of compact support for the momenta. For solutions of the system itself, the answer is more convoluted, and we determine when the compactness of the support is lost, replaced instead by an exponential decay rate.

Particle Trajectories In Extreme Stokes Waves Over Inifinte Depth, Tony Lyons

Articles

We investigate the velocity field of fluid particles in an extreme water wave over infinite depth. It is shown that the trajectories of the particles within the fluid and along the free surface do not form closed paths over the course of one period, but rather undergo a positive drift in the direction of wave propagation. In addition it is shown that the wave crest cannot form a stagnation point despite the velocity of the fluid being zero there.

Mathematical Modelling At Secondary School: The Macsi-Clongowes Wood College Experience, J.P.F. Charpin, S. O'Hara, Dana Mackey

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In Ireland, to encourage the study of STEM (science, technology, engineering and mathematics) subjects and particularly mathematics, the Mathematics Applications Consortium for Science and Industry (MACSI) and Clongowes Wood College (County Kildare, Ireland) organized a mathematical modelling workshop for senior cycle secondary school students. Participants developed simple mathematical models for everyday life problems with an open-ended answer. The format and content of the workshop are described and feedback from both students and participating teachers is provided. For nearly all participants, this workshop was an enjoyable experience which showed mathematics and other STEM components in a very positive way.

Weyl-Titchmarsh Type Formula For Periodic Schrodinger Operator With Wigner-Von Neumann Potential, Sergey Simonov, Pavel Kurasov

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Schroedinger operator on the half-line with periodic background potential perturbed by a certain potential of Wigner-von Neumann type is considered. The asymptotics of generalized eigenvectors for the values of the spectral parameter from the upper half-plane and on the absolutely continuous spectrum is established. Weyl-Titchmarsh type formula for this operator is proven.

Mass Transfer From A Vertical Flat Plate Due To Natural Convection With A Constant Counterflow, David Mcdonnell, Brendan Redmond, Lawrence J. Crane

Articles

This paper first examines the mass transfer from a vertical flat surface of a soluble material due to natural convection. A perturbation term is then introduced into the stream function to model the introduction of a constant counterflow. The effect this counterflow has on both the overall mass transfer and the overall velocity profile is studied in detail.

Bayesian Model Selection For Exponential Random Graph Models, Alberto Caimo, Nial Friel

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Exponential random graph models are a class of widely used exponential family models for social networks. The topological structure of an observed network is modelled by the relative prevalence of a set of local sub-graph configurations termed network statistics. One of the key tasks in the application of these models is which network statistics to include in the model. This can be thought of as statistical model selection problem. This is a very challenging problem---the posterior distribution for each model is often termed ``doubly intractable'' since computation of the likelihood is rarely available, but also, the evidence of the posterior …

Endomorphisms Of Abelian Groups With Small Algebraic Entropy, D. Dikranjan, Ketao Gong, P. Zanardo

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We study the endomorphisms ϕ of abelian groups G having a “small” algebraic entropy h (where “small” usually means ). Using essentially elementary tools from linear algebra, we show that this study can be carried out in the group , where an automorphism ϕ with must have all eigenvalues in the open circle of radius 2, centered at 0 and ϕ must leave invariant a lattice in , i.e., be essentially an automorphism of . In particular, all eigenvalues of an automorphism ϕ with must be roots of unity. This is a particular case of a more general fact known …

Eigenfunction Expansions Associated With The One-Dimensional Schrödinger Operator, Daphne Gilbert

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We consider the form of eigenfunction expansions associated with the time-independent Schrödinger operator on the line, under the assumption that the limit point case holds at both of the infinite endpoints. It is well known that in this situation the multiplicity of the operator may be one or two, depending on properties of the potential function. Moreover, for values of the spectral parameter in the upper half complex plane, there exist Weyl solutions associated with the restrictions of the operator to the negative and positive half-lines respectively, together with corresponding Titchmarsh-Weyl functions.

Dark Solitons Of The Qiao's Hierarchy, Rossen Ivanov, Tony Lyons

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We obtain a class of soliton solutions of the integrable hierarchy which has been put forward in a series of works by Z. Qiao. The soliton solutions are in the class of real functions approaching constant value fast enough at infinity, the so-called 'dark solitons'.