New variational principles for systems of partial differential equations

The question of finding variational principles for coupled systems of first order partial differential equations is considered. Using a potential representation for solutions of the first order system a higher order system is obtained. Existence of a variational principle follows if the original system can be transformed to a self-adjoint higher order system. Existence of variational principles for all linear wave equations with constant coefficients having real dispersion relations is established. The method of adjoining some of the equations of the original system to a suitable Lagrangian function by the method of Lagrange multipliers is used to construct new variational principles for a class of linear systems. The equations used as side conditions must satisfy highly-restrictive integrability conditions. In the more difficult nonlinear case the system of two equations in two independent variables can be analyzed completely. For systems determined by two conservation laws the side condition must be a conservation law in addition to satisfying the integrability conditions.

Boundary value problems for stochastic differential equations

A theory of two-point boundary value problems analogous to the theory of initial value problems for stochastic ordinary differential equations whose solutions form Markov processes is developed. The theory of initial value problems consists of three main parts: the proof that the solution process is markovian and diffusive; the construction of the Kolmogorov or Fokker-Planck equation of the process; and the proof that the transistion probability density of the process is a unique solution of the Fokker-Planck equation.

It is assumed here that the stochastic differential equation under consideration has, as an initial value problem, a diffusive markovian solution process. When a given boundary value problem for this stochastic equation almost surely has unique solutions, we show that the solution process of the boundary value problem is also a diffusive Markov process. Since a boundary value problem, unlike an initial value problem, has no preferred direction for the parameter set, we find that there are two Fokker-Planck equations, one for each direction. It is shown that the density of the solution process of the boundary value problem is the unique simultaneous solution of this pair of Fokker-Planck equations.

This theory is then applied to the problem of a vibrating string with stochastic density.

Habitat enhancing marine structures: Creating habitat in urban waters

Although maritime regions support a large portion of the world’s human population, their value as habitat for other species is overlooked. Urban structures that are built in the marine environment are not designed or managed for the habitat they provide, and are built without considering the communities of marine organisms that could colonize them (Clynick et al., 2008). However, the urban waterfront may be capable of supporting a significant proportion of regional aquatic biodiversity (Duffy-Anderson et al., 2003). While urban shorelines will never return to their original condition, some scientists think that the habitat quality of urban waterfronts could be significantly improved through further research and some design modifications, and that many opportunities exist to make these modifications (Russel et al., 1983, Goff, 2008). Habitat enhancing marine structures (or HEMS) are a potentially promising approach to address the impact of cities on marine organisms including habitat fragmentation and degradation. HEMS are a type of habitat improvement project that are ecologically engineered to improve the habitat quality of urban marine structures such as bulkheads and docks for marine organisms. More specifically, HEMS attempt to improve or enhance the physical habitat that organisms depend on for survival in the inter- and sub-tidal waterfronts of densely populated areas. HEMS projects are targeted at areas where human-made structures cannot be significantly altered or removed. While these techniques can be used in suburban or rural areas restoration or removal is preferred in these settings, and HEMS are resorted to only if removal of the human-made structure is not an option. Recent research supports the use of HEMS projects. Researchers have examined the communities found on urban structures including docks, bulkheads, and breakwaters. Complete community shifts have been observed where the natural shoreline was sandy, silty, or muddy. There is also evidence of declines in community composition, ecosystem functioning, and increases in non-native species abundances in assemblages on urban marine structures. Researchers have identified two key differences between these substrates including the slope (seawalls are vertical; rocky shores contain multiple slopes) and microhabitat availability (seawalls have very little; rocky shores contain many different types). In response, researchers have suggested designing and building seawalls with gentler slopes or a combination of horizontal and vertical surfaces. Researchers have also suggested incorporating microhabitat, including cavities designed to retain water during low tide, crevices, and other analogous features (Chapman, 2003; Moreira et al., 2006) (PDF contains 4 pages)

Effect of an annular pupil filter on differential confocal microscopy

Combining differential confocal microscopy and an annular pupil filter, we obtained the normalized axial intensity distribution curve of an optical system. We used the sharp slopes of the axial response curve of the optical system to measure the surface profile of a reflection grating. Experimental results prove that this method can extend the axial dynamic range and improve the transverse resolution of three-dimensional profilometry by sacrificing axial resolution. (C) 2000 Optical Society of America.

Resource management of dynamic coastal environments using synoptic measures from remote sensing

In addition to providing vital ecological services, coastal areas of North Carolina provide prized areas for habitation, recreation, and commercial fisheries. However, from a management perspective, the coasts of North Carolina are highly variable and complex. In-water constituents such as nutrients, suspended sediments, and chlorophyll a concentration can vary significantly over a broad spectrum of time and space scales. Rapid growth and land-use change continue to exert pressure on coastal lands. Coastal environments are also very vulnerable to short-term (e.g., hurricanes) and long-term (e.g., sea-level rise) natural changes that can result in significant loss of life, economic loss, or changes in coastal ecosystem functioning. Hence, the dynamic nature, effects of human-induced change over time, and vulnerability of coastal areas make it difficult to effectively monitor and manage these important state and national resources using traditional data collection technologies such as discrete monitoring stations and field surveys. In general, these approaches provide only a sparse network of data over limited time and space scales and generally are expensive and labor-intensive. Products derived from spectral images obtained by remote sensing instruments provide a unique vantage point from which to examine the dynamic nature of coastal environments. A primary advantage of remote sensing is that the altitude of observation provides a large-scale synoptic view relative to traditional field measurements. Equally important, the use of remote sensing for a broad range of research and environmental applications is now common due to major advances in data availability, data transfer, and computer technologies. To facilitate the widespread use of remote sensing products in North Carolina, the UNC Coastal Studies Institute (UNC-CSI) is developing the capability to acquire, process, and analyze remotely sensed data from several remote sensing instruments. In particular, UNC-CSI is developing regional remote sensing algorithms to examine the mobilization, transport, transformation, and fate of materials between coupled terrestrial and coastal ocean systems. To illustrate this work, we present the basic principles of remote sensing of coastal waters in the context of deriving information that supports efficient and effective management of coastal resources. (PDF contains 4 pages)

The politics of participation: Evaluating stakeholder equity in co-management regimes

Within natural resource management, there is increasing criticism of the traditional model of top-down management as a method of governance, as researchers and managers alike have recognized that resources can frequently be better managed when stakeholders are directly involved in management. As a result, in recent years the concept of co-management of natural resources, in which management responsibilities are shared between the government and stakeholders, has become increasingly popular, both in the academic literature and in practice. However, while co-management has significant potential as a successful management tool, the issue of equity in co-management has rarely been addressed. It is necessary to understand the differential impacts on stakeholders of co-management processes and the degree to which diverse stakeholders are represented within co-management. Understanding the interests of various stakeholders can be a way to more effectively address the distributional and social impacts of coastal policies, which can in turn increase compliance with management measures and lead to more sustainable resource management regimes. This research seeks to take a closer look at the concepts of co-management and participation through a number of case studies of marine protected areas (MPAs) in the Caribbean. (PDF contains 4 pages)

To pave or not to pave: A social landscape analysis of land use decision-making in the towns of the lamprey watershed

Population pressure in coastal New Hampshire challenges land use decision-making and threatens the ecological health and functioning of Great Bay, an estuary designated as both a NOAA National Estuarine Research Reserve and an EPA National Estuary Program site. Regional population in the seacoast has quadrupled in four decades resulting in sprawl, increased impervious surface cover and larger lot rural development (Zankel, et.al., 2006). All of Great Bay’s contributing watersheds face these challenges, resulting in calls for strategies addressing growth, development and land use planning. The communities within the Lamprey River watershed comprise this case study. Do these towns communicate upstream and downstream when making land use decisions? Are cumulative effects considered while debating development? Do town land use groups consider the Bay or the coasts in their decision-making? This presentation, a follow-up from the TCS 2008 conference and a completed dissertation, will discuss a novel social science approach to analyze and understand the social landscape of land use decision-making in the towns of the Lamprey River watershed. The methods include semi-structured interviews with GIS based maps in a grounded theory analytical strategy. The discussion will include key findings, opportunities and challenges in moving towards a watershed approach for land use planning. This presentation reviews the results of the case study and developed methodology, which can be used in watersheds elsewhere to map out the potential for moving towns towards EBM and watershed-scaled, land use planning. (PDF contains 4 pages)

Multiscale model reduction methods for deterministic and stochastic partial differential equations

Partial differential equations (PDEs) with multiscale coefficients are very difficult to solve due to the wide range of scales in the solutions. In the thesis, we propose some efficient numerical methods for both deterministic and stochastic PDEs based on the model reduction technique.

For the deterministic PDEs, the main purpose of our method is to derive an effective equation for the multiscale problem. An essential ingredient is to decompose the harmonic coordinate into a smooth part and a highly oscillatory part of which the magnitude is small. Such a decomposition plays a key role in our construction of the effective equation. We show that the solution to the effective equation is smooth, and could be resolved on a regular coarse mesh grid. Furthermore, we provide error analysis and show that the solution to the effective equation plus a correction term is close to the original multiscale solution.

For the stochastic PDEs, we propose the model reduction based data-driven stochastic method and multilevel Monte Carlo method. In the multiquery, setting and on the assumption that the ratio of the smallest scale and largest scale is not too small, we propose the multiscale data-driven stochastic method. We construct a data-driven stochastic basis and solve the coupled deterministic PDEs to obtain the solutions. For the tougher problems, we propose the multiscale multilevel Monte Carlo method. We apply the multilevel scheme to the effective equations and assemble the stiffness matrices efficiently on each coarse mesh grid. In both methods, the $\KL$ expansion plays an important role in extracting the main parts of some stochastic quantities.

For both the deterministic and stochastic PDEs, numerical results are presented to demonstrate the accuracy and robustness of the methods. We also show the computational time cost reduction in the numerical examples.

Equivalent differential equations for nonlinear dynamical systems

A technique for obtaining approximate periodic solutions to nonlinear ordinary differential equations is investigated. The approach is based on defining an equivalent differential equation whose exact periodic solution is known. Emphasis is placed on the mathematical justification of the approach. The relationship between the differential equation error and the solution error is investigated, and, under certain conditions, bounds are obtained on the latter. The technique employed is to consider the equation governing the exact solution error as a two point boundary value problem. Among other things, the analysis indicates that if an exact periodic solution to the original system exists, it is always possible to bound the error by selecting an appropriate equivalent system.

Three equivalence criteria for minimizing the differential equation error are compared, namely, minimum mean square error, minimum mean absolute value error, and minimum maximum absolute value error. The problem is analyzed by way of example, and it is concluded that, on the average, the minimum mean square error is the most appropriate criterion to use.

A comparison is made between the use of linear and cubic auxiliary systems for obtaining approximate solutions. In the examples considered, the cubic system provides noticeable improvement over the linear system in describing periodic response.

A comparison of the present approach to some of the more classical techniques is included. It is shown that certain of the standard approaches where a solution form is assumed can yield erroneous qualitative results.

Stability of parametrically excited differential equations

Sufficient stability criteria for classes of parametrically excited differential equations are developed and applied to example problems of a dynamical nature.

Stability requirements are presented in terms of 1) the modulus of the amplitude of the parametric terms, 2) the modulus of the integral of the parametric terms and 3) the modulus of the derivative of the parametric terms.

The methods employed to show stability are Liapunov’s Direct Method and the Gronwall Lemma. The type of stability is generally referred to as asymptotic stability in the sense of Liapunov.

The results indicate that if the equation of the system with the parametric terms set equal to zero exhibits stability and possesses bounded operators, then the system will be stable under sufficiently small modulus of the parametric terms or sufficiently small modulus of the integral of the parametric terms (high frequency). On the other hand, if the equation of the system exhibits individual stability for all values that the parameter assumes in the time interval, then the actual system will be stable under sufficiently small modulus of the derivative of the parametric terms (slowly varying).

DIFFERENTIAL GEOMETRICAL METHODS IN THE STUDY OF OPTICAL-TRANSMISSION (SCALAR THEORY) .1. STATIC TRANSMISSION CASE

Static optical transmission is restudied by postulation of the optical path as the proper element in a three-dimensional Riemannian manifold (no torsion); this postulation can be applied to describe the light-medium interactive system. On the basis of the postulation, the behaviors of light transmitting through the medium with refractive index n are investigated, the investigation covering the realms of both geometrical optics and wave optics. The wave equation of light in static transmission is studied modally, the postulation being employed to derive the exact form of the optical field equation in a medium (in which the light is viewed as a single-component field). Correspondingly, the relationships concerning the conservation of optical fluid and the dynamic properties are given, and some simple applications of the theories mentioned are presented.

DIFFERENTIAL GEOMETRICAL METHODS IN THE STUDY OF OPTICAL-TRANSMISSION (SCALAR THEORY) .2. TIME-DEPENDENT TRANSMISSION THEORY

By generalization of the methods presented in Part I of the study [J. Opt. Soc. Am. A 12, 600 (1994)] to the four-dimensional (4D) Riemannian manifold case, the time-dependent behavior of light transmitting in a medium is investigated theoretically by the geodesic equation and curvature in a 4D manifold. In addition, the field equation is restudied, and the 4D conserved current of the optical fluid and its conservation equation are derived and applied to deduce the time-dependent general refractive index. On this basis the forces acting on the fluid are dynamically analyzed and the self-consistency analysis is given.

Effects of global change on the functioning of Mediterranean Rivers

138 p.