Partial Differential Equations Commons^™

Valuation Of Financial Derivatives Subject To Liquidity Risk, Yanan Jiang

UNLV Theses, Dissertations, Professional Papers, and Capstones

Valuation of financial derivatives subject to liquidity risk remains an open problem in finance. This dissertation focuses on the valuation of European-style call option under limited market liquidity through the dynamic management of a portfolio of assets. We investigate liquidity from three perspectives: market breadth, depth, and immediacy. We present a general framework of valuation based on the optimal realization of a performance index relative to the set of all feasible portfolio trajectories. Numerical examples are then presented and analyzed that show option price increases as the market transitions from liquid to less liquid state. Furthermore, buying and selling activities, …

Stiefel And Grassmann Manifolds In Quantum Chemistry, Eduardo Chiumiento, Michael Melgaard

Articles

We establish geometric properties of Stiefel and Grassmann manifolds which arise in relation to Slatertype variational spaces in many-particle Hartree-Fock theory and beyond. In particular, we prove thatthey are analytic homogeneous spaces and submanifolds of the space of bounded operators on the single-particle Hilbert space. As a by-product we obtain that they are complete Finsler manifolds. These geometric properties underpin state-of-the-art results on existence of solutions to Hartree-Fock type equations.

Systems Of Nonlinear Wave Equations With Damping And Supercritical Sources, Yanqiu Guo

Department of Mathematics: Dissertations, Theses, and Student Research

We consider the local and global well-posedness of the coupled nonlinear wave equations

u_tt – Δu + g₁(u_t) = f₁(u, v)

v_tt – Δv + g₂(v_t) = f₂(u, v);

in a bounded domain Ω subset of the real numbers (Rⁿ) with a nonlinear Robin boundary condition on u and a zero boundary conditions on v. The nonlinearities f₁(u, v) and f₂(u, v) are with supercritical exponents …

On Soliton Interactions For A Hierarchy Of Generalized Heisenberg Ferromagnetic Models On Su(3)/S(U(1) $\Times$ U(2)) Symmetric Space, Vladimir Gerdjikov, Georgi Grahovski, Alexander Mikhailov, Tihomir Valchev

Articles

We consider an integrable hierarchy of nonlinear evolution equations (NLEE) related to linear bundle Lax operator L. The Lax representation is Z2 \times Z2 reduced and is naturally associated with the symmetric space SU(3)/S(U(1) \times U(2)). The simplest nontrivial equation in the hierarchy is a generalization of Heisenberg ferromagnetic model. We construct the N-soliton solutions for an arbitrary member of the hierarchy by using the Zakharov-Shabat dressing method with an appropriately chosen dressing factor. Two types of soliton solutions: quadruplet and doublet solitons are found. The one-soliton solutions of NLEEs with even and odd dispersion laws have different properties. In …

Heterogeneous Multiscale Modeling Of Advection-Diffusion Problems, David J. Gardner, Daniel R. Reynolds

Mathematics Research

No abstract provided.

Effects Of Stochastic Freshwater Flux On The Atlantic Thermohaline Circulation, Alyssa Pampell, Alejandra Aceves

Mathematics Research

No abstract provided.

On The Stability Of A Microstructure Model, Mihhail Berezovski, Arkadi Berezovski

Publications

Abstract

The asymptotic stability of solutions of the Mindlin-type microstructure model for solids is analyzed in the paper. It is shown that short waves are asymptotically stable even in the case of a weakly non-convex free energy dependence on microdeformation.

Research highlights

The Mindlin-type microstructure model cannot describe properly short wave propagation in laminates. A modified Mindlin-type microstructure model with weakly non-convex free energy resolves this discrepancy. It is shown that the improved model with weakly non-convex free energy is asymptotically stable for short waves.

Controlling Nanoparticles Formation In Molten Metallic Bilayers By Pulsed-Laser Interference Heating, Mikhail Khenner, Sagar Yadavali, Ramki Kalyanaraman

Mathematics Faculty Publications

The impacts of the two-beam interference heating on the number of core-shell and embedded nanoparticles and on nanostructure coarsening are studied numerically based on the non-linear dynamical model for dewetting of the pulsed-laser irradiated, thin (< 20 nm) metallic bilayers. The model incorporates thermocapillary forces and disjoining pressures, and assumes dewetting from the optically transparent substrate atop of the reflective support layer, which results in the complicated dependence of light reflectivity and absorption on the thicknesses of the layers. Stabilizing thermocapillary effect is due to the local thickness-dependent, steady- state temperature profile in the liquid, which is derived based on the mean substrate temperature estimated from the elaborate thermal model of transient heating and melting/freezing. Linear stability analysis of the model equations set for Ag/Co bilayer predicts the dewetting length scales in the qualitative agreement with experiment.

Singular Solutions Of Coss-Coupled Epdiff Equations: Waltzing Peakons And Compacton Pairs, Colin Cotter, Darryl Holm, Rossen Ivanov, James Percival

Conference papers

We introduce EPDiff equations as Euler-Poincare´ equations related to Lagrangian provided by a metric, invariant under the Lie Group Diff(Rⁿ). Then we proceed with a particular form of EPDiff equations, a cross coupled two-component system of Camassa-Holm type. The system has a new type of peakon solutions, 'waltzing' peakons and compacton pairs.

The Generalised Zakharov-Shabat System And The Gauge Group Action, Georgi Grahovski

Articles

The generalized Zakharov-Shabat systems with complex-valued non-regular Cartan elements and the systems studied by Caudrey, Beals and Coifman (CBC systems) and their gauge equivalent are studied. This study includes: the properties of fundamental analytical solutions (FAS) for the gauge-equivalent to CBC systems and the minimal set of scattering data; the description of the class of nonlinear evolutionary equations, solvable by the inverse scattering method, and the recursion operator, related to such systems; the hierarchies of Hamiltonian structures. The results are illustrated on the example of the multi-component nonlinear Schrodinger (MNLS) equations and the corresponding gauge-equivalent multi-component Heisenberg ferromagnetic (MHF) type …

Integrable Models For Shallow Water With Energy Dependent Spectral Problems, Rossen Ivanov, Tony Lyons

Articles

We study the inverse problem for the so-called operators with energy depending potentials. In particular, we study spectral operators with quadratic dependence on the spectral parameter. The corresponding hierarchy of integrable equations includes the Kaup–Boussinesq equation. We formulate the inverse problem as a Riemann–Hilbert problem with a Z₂ reduction group. The soliton solutions are explicitly obtained.

Wave Propagation And Dispersion In Microstructured Solids, Arkadi Berezovski, Juri Engelbrecht, Mihhail Berezovski

Publications

A series of numerical simulations is carried on in order to understand the accuracy of dispersive wave models for microstructured solids. The computations are performed by means of the finite-volume numerical scheme, which belongs to the class of wave-propagation algorithms. The dispersion effects are analyzed in materials with different internal structures: microstructure described by micromorphic theory, regular laminates, laminates with substructures, etc., for a large range of material parameters and wavelengths.

The Generalised Zakharov-Shabat System And The Gauge Group Action, Georgi Grahovski

Articles

The generalized Zakharov–Shabat systems with complex-valued non-regular Cartan elements and the systems studied by Caudrey, Beals and Coifman (CBC systems) and their gauge equivalent are studied. This study includes: the properties of fundamental analytical solutions (FAS) for the gauge-equivalent to CBC systems and the minimal set of scattering data; the description of the class of nonlinear evolutionary equations, solvable by the inverse scattering method, and the recursion operator, related to such systems; the hierarchies of Hamiltonian structures. The results are illustrated on the example of the multi-component nonlinear Schr¨odinger (MNLS) equations and the corresponding gauge-equivalent multi-component Heisenberg ferromagnetic (MHF) type …

Formation Of Organized Nanostructures From Unstable Bilayers Of Thin Metallic Liquids, Mikhail Khenner, Sagar Yadavali, Ramki Kalyanaraman

Mathematics Faculty Publications

Dewetting of pulsed-laser irradiated, thin (< 20 nm), optically reflective metallic bilayers on an optically transparent substrate with a reflective support layer is studied within the lubrication equations model. A steady-state bilayer film thickness (h) dependent temperature profile is derived based on the mean substrate temperature estimated from the elaborate thermal model of transient heating and melting/freezing. Large thermocapillary forces are observed along the plane of the liquid-liquid and liquid-gas interfaces due to this h-dependent temperature, which, in turn, is strongly influenced by the h-dependent laser light reflection and absorption. Consequently the dewetting is a result of the competition between thermocapillary and intermolecular forces. A linear analysis of the dewetting length scales established that the non-isothermal calculations better predict the experimental results as compared to the isothermal case within the bounding Hamaker coefficients. Subsequently, a computational non-linear dynamics study of the dewetting pathway was performed for Ag/Co and Co/Ag bilayer systems to predict the morphology evolution. We found that the systems evolve towards formation of different morphologies, including core-shell, embedded, or stacked nanostructure morphologies.

A Dynamical Study Of The Evolution Of Pressure Waves Propagating Through A Semi-Infinite Region Of Homogeneous Gas Combustion Subject To A Time-Harmonic Signal At The Boundary, John Eslick

LSU New Orleans Theses and Dissertations

In this dissertation, the evolution of a pressure wave driven by a harmonic signal on the boundary during gas combustion is studied. The problem is modeled by a nonlinear, hyperbolic partial differential equation. Steady-state behavior is investigated using the perturbation method to ensure that enough time has passed for any transient effects to have dissipated. The zeroth, first and second-order perturbation solutions are obtained and their moduli are plotted against frequency. It is seen that the first and second-order corrections have unique maxima that shift to the right as the frequency decreases and to the left as the frequency increases. …

Boundary Element Method (Bem) And Method Of Fundamental Solutions (Mfs) For The Boundary Value Problems Of The 2-D Laplace's Equation, Ermes Anthony Salgado-Ibarra

UNLV Theses, Dissertations, Professional Papers, and Capstones

In this thesis we study the solution of the two dimensional Laplace equation by the boundary Element method (BEM) and the method of fundamental solutions (MFS). Both the BEM and MFS used to solve boundary value problems involving the Laplace equation 2-D settings. Both methods rely on the use of fundamental solution of the Laplace's equation (the solution of Laplace's equation in the distributional sense). We will contrast and compare the results we get using the BEM with results we get using the MFS.

The Principle Of Linearized Stability For Size-Structured Population Models, M. El-Doma

Applications and Applied Mathematics: An International Journal (AAM)

The principle of linearized stability for size-structured population dynamics models is proved giving validity to previous stability results reported in, for example, El-Doma (2008-1). In particular, we show that if all the roots of the characteristic equation lie to the left of the imaginary axis then the steady state is locally exponentially stable, and on the other hand, if there is at least one root that lies to the right of the imaginary axis then the steady state is unstable. We also point out cases when there is resonance

The Singular Perturbation In The Analysis Of Mode I Fracture Based Upon A New Multiscale Theory, Kai-Bin Fu

Applications and Applied Mathematics: An International Journal (AAM)

A theory of fracture is presented that is based upon an extension of continuum mechanics to the nanoscale fracture through the incorporation of long-range intermolecular forces which correct bulk material descriptions near interfaces. To be consistent with the literature, constant surface energies are assigned to interfaces. In the analysis of mode I fracture for quasi-brittle material, it is observed that the incorporation of these long-range intermolecular forces predicts sharp fracture tip rather than blunt and avoids stress singularity. The details of the singular perturbation employed in this analysis are also presented.

Nondestructive Electrothermal Detection Of Corrosion, Brittany Ambeau, Harris Enniss, Stefan Schnake

Mathematical Sciences Technical Reports (MSTR)

Nondestructive testing and imaging plays an important role in many industries, e.g., the monitoring and maintenance of corrosion in aircraft. The general technique is to input energy in some form into an object, observe the object’s response, and from this input-output information determine the internal structure. New techniques are always being explored, and recently there has been much interest in methods that use multiple forms of energy. In this vein, we examine a new technique for imaging corrosion or material loss in an object by combining electrical and thermal measurements on some accessible portion of the object’s outer boundary. The …

Cloaking Against Thermal Imaging, Maple So

Mathematical Sciences Technical Reports (MSTR)

There has been a lot of recent interest in cloaking and invisibility in the mathematics and science communities, and in fact physically plausible mechanisms have been proposed (some built) for cloaking an object against detection using a variety of electromagnetic methods. The ideas are very general, however, and should allow one to design cloaks that work against other forms of imaging. We examine the possibility of cloaking an object to make it invisible to an observer using thermal energy (heat) as the imaging tool. Specifically, we desire to cloak an object inside a two-dimensional disk by cutting a small hole …