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Mathematics

Utah State University

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Articles 1 - 12 of 12

Full-Text Articles in Physical Sciences and Mathematics

Foundations Of Wave Phenomena, Charles G. Torre Dec 2016

Foundations Of Wave Phenomena, Charles G. Torre

Charles G. Torre

This is an undergraduate text on the mathematical foundations of wave phenomena. Version 8.2.

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Geometrization Conditions For Perfect Fluids, Scalar Fields, And Electromagnetic Fields, Charles G. Torre, Dionisios Krongos Jul 2015

Geometrization Conditions For Perfect Fluids, Scalar Fields, And Electromagnetic Fields, Charles G. Torre, Dionisios Krongos

Charles G. Torre

Rainich-type conditions giving a spacetime “geometrization” of matter fields in general relativity are reviewed and extended. Three types of matter are considered: perfect fluids, scalar fields, and electromagnetic fields. Necessary and sufficient conditions on a spacetime metric for it to be part of a perfect fluid solution of the Einstein equations are given. Formulas for constructing the fluid from the metric are obtained. All fluid results hold for any spacetime dimension. Geometric conditions on a metric which are necessary and sufficient for it to define a solution of the Einstein-scalar field equations and formulas for constructing the scalar field from …

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Perihelion Precession In General Relativity, Charles G. Torre Apr 2014

Perihelion Precession In General Relativity, Charles G. Torre

Charles G. Torre

This is a Maple worksheet providing a relatively quick and informal sketch of a demonstration that general relativistic corrections to the bound Kepler orbits introduce a perihelion precession. Any decent textbook will derive this result. My analysis aligns with that found in the old text "Introduction to General Relativity", by Adler, Bazin and Schiffer. The plan of the analysis is as follows. * Model the planetary orbits as geodesics in the (exterior) Schwarzschild spacetime. * Compute the geodesic equations. * Simplify them using symmetries and first integrals. * Isolate the differential equation expressing the radial coordinate as a function of …

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The Spacetime Geometry Of A Null Electromagnetic Field, Charles G. Torre Feb 2014

The Spacetime Geometry Of A Null Electromagnetic Field, Charles G. Torre

Charles G. Torre

We give a set of local geometric conditions on a spacetime metric which are necessary and sufficient for it to be a null electrovacuum, that is, the metric is part of a solution to the Einstein-Maxwell equations with a null electromagnetic field. These conditions are restrictions on a null congruence canonically constructed from the spacetime metric, and can involve up to five derivatives of the metric. The null electrovacuum conditions are counterparts of the Rainich conditions, which geometrically characterize non-null electrovacua. Given a spacetime satisfying the conditions for a null electrovacuum, a straightforward procedure builds the null electromagnetic field from …

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Multi-Scale Analysis Of Seed Dispersal Contributes Tothe Resolution Of Reid’S Paradox, James A. Powell, N. E. Zimmermann Jan 2004

Multi-Scale Analysis Of Seed Dispersal Contributes Tothe Resolution Of Reid’S Paradox, James A. Powell, N. E. Zimmermann

James A. Powell

‘‘Reid’s paradox’’ is the mismatch between theoretical estimates of invasion rates for plants and ‘‘observed’’ rates of plant migration, particularly in the Holocene postglacial migration northwards. While Reid couched his paradox in terms of the migration of oaks in Great Britain, observers have documented the same problem in a wide variety of species. In almost all cases, these authors suggest that occasional, long-distance events, probably mitigated by active dispersal factors (ants, birds, rodents) are responsible. Clark and co-workers have shown that order statistics can bridge the gap between theory and predictions, essentially using ‘‘fat-tailed’’ dispersal kernels raised to high powers …

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Assessing The Impacts Of Global Climate Changeon Forest Pests, J. A. Logan, J. Reniere, James A. Powell Jan 2003

Assessing The Impacts Of Global Climate Changeon Forest Pests, J. A. Logan, J. Reniere, James A. Powell

James A. Powell

No abstract provided.

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Ghost Forests, Global Warming And The Mountain Pine Beetle, J. A. Logan, James A. Powell Jan 2001

Ghost Forests, Global Warming And The Mountain Pine Beetle, J. A. Logan, James A. Powell

James A. Powell

No abstract provided.

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Games To Teach Mathematical Modelling, James A. Powell, J. Cangelosi, A. M. Harris Jan 1998

Games To Teach Mathematical Modelling, James A. Powell, J. Cangelosi, A. M. Harris

James A. Powell

We discuss the use of in-class games to create realistic situations for mathematical modelling. Two games are presented which are appropriate for use in post-calculus settings. The first game reproduces predator{prey oscillations and the second game simulates disease propagation in a mixing population. When used creatively these games encourage students to model realistic data and apply mathematical concepts to understanding the data.

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The Inverse Problem Of The Calculus Of Variations For Scala Fourth Order Ordinary Differential Equations, Mark E. Fels Jan 1996

The Inverse Problem Of The Calculus Of Variations For Scala Fourth Order Ordinary Differential Equations, Mark E. Fels

Mark Eric Fels

A simple invariant characterization of the scalar fourth-order ordinary differential equations which admit a variational multiplier is given. The necessary and sufficient conditions for the existence of a multiplier are expressed in terms of the vanishing of two relative invariants which can be associated with any fourth-order equation through the application of Cartan's equivalence method. The solution to the inverse problem for fourth-order scalar equations provides the solution to an equivalence problem for second-order Lagrangians, as well as the precise relationship between the symmetry algebra of a variational equation and the divergence symmetry algebra of the associated Lagrangian.

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Beam Collapse As An Explanation Foranomalous Ocular Damage, James A. Powell, J. V. Moloney, A. C. Newell, R. A. Albanese Jan 1993

Beam Collapse As An Explanation Foranomalous Ocular Damage, James A. Powell, J. V. Moloney, A. C. Newell, R. A. Albanese

James A. Powell

The basic mathematical phenomena relevant to ocular damage caused by ultrashort laser pulses are discussed with the use of mathematical results and numerical modeling. The primary effects of nonlinear self-focusing and beam collapse are examined in the ocular safety context. Finite-time material response and group-velocity dispersion are discussed as possible mitigating factors. An argument is presented that indicates that the initial stages of beam collapse are essentially two-dimensional. Experiments are suggested that might help distinguish the most important contributing factors in the damage regime. The numerical methodology is detailed in an appendix.

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Competition Between Generic And Nongeneric Fronts Inenvelope Equations, James A. Powell, A. C. Newell, C. K. R. T. Jones Jan 1991

Competition Between Generic And Nongeneric Fronts Inenvelope Equations, James A. Powell, A. C. Newell, C. K. R. T. Jones

James A. Powell

Arguments are presented for understanding the selection of the speed and the nature of the fronts that join stable and unstable states on the supercritical side of first-order phase transitions. It is suggested that from compact support, nonpositive-definite initial conditions, observable front behavior occurs only when the asymptotic spatial structure of a trajectory in the Galilean ordinary differential equation (ODE) corresponds to the most unstable temporal mode in the governing partial differential equation (PDE). This selection criterion distinguishes between a "nonlinear" front, which has its origin in the first-order nature of the bifurcation, and a "linear" front. The nonlinear front …

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Natural Variational Principles On Riemannian Structures, Ian M. Anderson Jan 1984

Natural Variational Principles On Riemannian Structures, Ian M. Anderson

Ian M. Anderson

No abstract provided.

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