Physical Sciences and Mathematics Commons^™

Injective Tensor Products Of Tree Spaces, Milena Venkova, Christopher Boyd, Costas Poulios

Articles

We study tensor products on tree spaces; in particular, we give necessary and sufficient conditions for the n-fold injective tensor product of tree spaces to contain a copy of l_1.

Multivariate Statistical Methodologies Applied In Biomedical Raman Spectroscopy: Assessing The Validity Of Partial Least Squares Regression Using Simulated Model Datasets., Mark Keating

Articles

Raman spectroscopy is fast becoming a valuable analytical tool in a number of biomedical scenarios, most notably disease diagnostics. Importantly, the technique has also shown increasing promise in the assessment of drug interactions on a cellular and subcellular level, particularly when coupled with multivariate statistical analysis. However, an important consideration, both with Raman spectroscopy and the associated statistical methodologies, is the accuracy of these techniques and more specifically the sensitivities which can be achieved and ultimately the limits of detection of the various methods. The purpose of this study is thus the construction of a model simulated data set with …

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

Articles

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.

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

Articles

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.

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.

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.

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.

Abstract Criteria For Multiple Solutions To Nonlinear Coupled Equations Involving Magnetic Schrodinger Operators, Mattias Enstedt, Michael Melgaard

Articles

We consider a system of nonlinear coupled equations involving magnetic Schrodinger

operators and general potentials. We provide a criteria for the existence of multiple

solutions to these equations. As special cases we get the classical results on

existence of innitely many distinct solutions within Hartree and Hartree-Fock

theory of atoms and molecules subject to an external magnetic fields. We also

extend recent results within this theory, including Coulomb system with a constant

magnetic field, a decreasing magnetic field and a "physically measurable" magnetic field.

Solutions To Quasi-Relativistic Multi-Configurative Hartree-Fock Equations In Quantum Chemistry, Carlos Argáez García, Michael Melgaard

Articles

We establish existence of infinitely many distinct solutions to the multi-configurative Hartree-Fock type equations for N-electron Coulomb systems with quasi-relativistic kinetic energy for the n th electron. Finitely many of the solutions are interpreted as excited states of the molecule. Moreover, we prove existence of a ground state. The results are valid under the hypotheses that the total charge Z of K nuclei is greater than N-1 and that Z is smaller than a critical charge. The proofs are based on a new application of the Lions-Fang-Ghoussoub critical point approach to nonminimal solutions on a complete analytic Hilbert-Riemann manifold.

Holomorphic Liftings From Infinite Dimensional Spaces, Sean Dineen, Milena Venkova

Articles

We obtain a number of positive solutions to a holomorphic lifting problem on a domain in a locally convex space.

Existence Of A Solution To Hartree-Fock Equations With Decreasing Magnetic Field, Mattias Enstedt, Michael Melgaard

Articles

In the presence of an external magnetic field, we prove existence of a ground state within the Hartree–Fock theory of atoms and molecules. The ground state exists provided the magnetic field decreases at infinity and the total charge of nuclei exceeds , where is the number of electrons. In the opposite direction, no ground state exists if .