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The award supported Peter Bosler as a postdoc in the Department of Mathematics at the University of Michigan. The project developed a novel Lagrangian particle method (LPM) for geophysical fluid flow simulations on a rotating sphere. The method is potentially relevant to Naval operations that rely on accurate and efficient modeling of atmosphere and ocean dynamics. We developed new remeshing and refinement schemes to overcome the loss of accuracy due to distortion in the particle distribution. The results were disseminated through published articles and presentations at conferences and seminars.

This young investigator award from the Air Force Office of Scientific Research to Daniel Forger played a pivotal role in his career. Many discoveries were made and are described in over 10 published papers or manuscripts in preparation. Some research highlights include: 1) discovering that timekeeping in mammals is more complex than originally thought, 2) discovering on how to optimally stimulate neuronal (and in particular the circadian) systems and 3) mathematical methods for studying biological clocks. Forger remains very grateful to the AFOSR for the opportunity to perform this work., AFOSR Young Investigator Award.

This project draws together a team of researchers to develop new grid-free tools for plasma simulations. The objectives are: (1) to develop a grid-free field solver, fluid model, and kinetic model, (2) to evaluate these tools in comparison with traditional mesh-based methods, and (3) to demonstrate the capability of the grid-free approach in an application of USAF interest. The field solver will use boundary integral methods and a recently developed tree code algorithm to compute the electrostatic force induced by a set of charged particles. The kinetic model will use tree-based statistical procedures for inter-particle collisions. Various hybrid grid-free simulations will be possible using these tools. The resulting grid-free approach will have several advantages over traditional mesh-based methods: it will eliminate the need for elaborate volume-meshing routines required to handle complex geometry, and will avoid the spurious numerical artifacts in the particle distribution that can arise in mesh-based schemes.

This grant provided support for a postdoc at the University of Michigan to assist in developing a grid-free particle method for electrostatic plasma simulations. The aim of the work is to substantially improve the accuracy and efficiency of these simulations. The proposed method is an alternative to traditional mesh-based methods such as particle-in-cell (PlC). In the new approach, the standard Eulerian formulation of the Vlasov-Poisson equation is replaced by a Lagrangian formulation in which the charge flow map is the key unknown quantity. Discretizing the Lagrangian formulation leads to a grid-free particle method. The investigators made progress in formulating and testing this approach. Numerical results are presented for the cold one-stream and two-stream instabilities. The method is especially well suited for tracking charge transport and resolving fine structures in phase space.

An examination of the relationship between the scattering matrix (the Fourier transform of the scattering operator) and the integral equations used in the Singularity Expansion Method (SEM) established that only the complex poles off the axis are intrinsically associated with the scatterer. While it has been known in special cases that those on the axis do not contribute to the field, this appears to be the first time this relationship has been clearly exhibited. Since the scattering matrix can be shown to be analytic in a half-plane containing the axis, any integral equation should exhibit the same properties for this region. The relationship between the eigenvalues of the integral equations of SEM and the complex eigenvalues of the associated partial differential equations -- whether scalar or vector. In particular, the integral equations of SEM have at most two eigenvalues + or - and these are functions of the at-most-denumerable number of complex eigenvalues of the associated differential equations. The theory of non-self-adjoint operators in Hilbert space could have significant implications for that part of the electrical engineering formalism known as the Eigenmode Expansion Method (EEM) particularly in relevance to the use of this formalism in developing equivalent circuits. (jhd)

A wake is modelled by a vortex sheet carrying positive and negative circulation. The sheet's evolution is computed by the vortex-blob method. Initial conditions and circulation density for the vortex sheet are chosen to simulate some of the wake patterns observed in the soap-film experiments of Couder et.al.(2,3).

The report summarizes progress made in several, areas, including: estimates for finite element methods, numerical simulation of nonlinear dispersive waves, numerical models for plasticity, numerical methods for the transport equation, and computation of fluid-flow problems. (Author)

Eigenfunction expansions for nonselfadjoint operators are important for scalar and electromagnetic wave scattering. Two methods: the Singularity Expansion Method (SEM), and the Eigenmode Expansion Method (EEM) which had been developed separately were studied. Criteria for their validity were established; moreover, the poles of the Green's function of the SEM are in 1-1 correspondence with the zeros of the eigenvalues of the EEM. A constructive numerical process for determining the poles of the Green's function was developed. Among several other results was the establishment of variational principles for the spectrum of compact nonselfadjoint operators. Another research area was the singularity behavior of eigenfunction expansions of various Green's functions in electromagnetic theory. The principal result shows that in an eigenfunction of a typical Green's function the point singularity of a Green's function is represented by a layer of surface singularity. This characteristic is analogous to the Gibbs' phenomenon where the representation of a discontinuous function by an orthogonal expansion creates the spikes which are absent in the original function. (Author)

Several problems relevant in the application of mechanics to real-world problems were treated. The work included theoretical analysis of solutions of partial differential equations, detailed studies of numerical techniques for approximately solving differential equations arising in mechanics, as well as actual numerical computations using such numerical schemes to model experimental phenomena. Application areas impacted by this research include heat conduction and fluid mechanics. Also closely related are problems arising in the study of controlled fusion. In addition, the work has added significantly to theoretical understanding of numerical methods for solving various partial differential equations. (Author)

A new finite element method is proposed for the numerical solution of a class of initial-boundary value problems for first order hyperbolic systems in one space dimension. An application of our procedure to a system modeling gas flow in a pipe is discussed. Asymptotic error estimates are derived in the L sq norm in space. (Author)

This paper gives a theory of spectral approximation for closed operators in Banach spaces. The perturbation theory developed in this paper is applied to the study of a finite element procedure for approximating the spectral properties of a differential system modeling a fluid in a rotating basin. (Author)

During this period the nine scientists associated with the project produced 76 papers of which 35 have appeared in journals at this time. Many invited lectures, editorships and a chair in mathematics attest to the ability of the principal investigators. In the area of entrainment of frequency in the large involving oscillations of physical systems at resonance and across point of resonance 14 papers were produced. Pareto optimization theorems in optimal control theory account for three papers. Six papers on optical resonance of laser beams in crystals and 11 papers on evolution equations and related problems in shock wave theory, semiconductors, diffusion, combustion kinetics and ecological systems give an idea of the diversity of areas addressed by the functional analytic techniques of Professors Ceasari, Smoller Suryanarayana, Kannan and associates. (Author)

This report includes a list of papers completed and published, a description of the activities of each of the investigators, and a discussion of some of the most significant results obtained.

The various theoretical results outlined are concerned with such matters as the structure of graphs having a maximum and minimum number of 'cliques' as used in clustering and classification schemes, with the enumeration of acyclic configurations of generalizations of mathematical models of the biological growth cells and paving problems in statistical mechanics, with properties of 'anticritical' graphs such as point and line-connectivity, with duals of tree graphs, with enumeration of random walks in graphs in terms of the moments of the eigenvalues of their adjacency matrices, and with determining the minimum number of prints of a digraph which must be removed so that the result is acyclic, i.e., finding the minimum feedback vertex set of a digraph. The report also provides a listing of the publications in which the detailed accomplishments are presented., See also report dated 30 Aug 74, AD/A-003 983.

This report is a brief summary of research accomplishments by the author and his associates during the period 1 April 1973 through 30 Mar 1974. The areas discussed include graphical enumeration, e.g., counting the number of spanning trees of a graph, graph topology, generalized Ramsey theory, integral characteristic polynomials of adjacency matrices, and communication problems. A list of publications which contain the detailed research findings is provided.

Research on both internal and free-surface flows related to experimental situations was continued. Part of this research was based on using a low-order code developed previously, jointly by Pritchard and Scott. Funds for the programmer allowed two substantial new programming projects to be initiated during this time. One concerned investigation of techniques for implementing a quartic finite element method for viscous, incompressible, two-dimensional flows. The other involved a streamline-diffusion finite element method for inviscid flows.

An effort has been made to present the various topics in the theory of graphs in a logical order, to indicate the historical background, and to clarify the exposition by including figures to illustrate concepts and results. In addition, there are three appendices which provide diagrams of graphs, directed graphs, and trees. The emphasis throughout is on theorems rather than algorithms or applications, which however are occasionally mentioned., Availability: Paper copy available from Addison-Wesley Publishing Co., Reading, MA.

Contents: The four color conjecture and other graphical diseases; Several proofs of the number of labeled 2-dimensional trees; On the chromatic number of permutation graphs; The expanding unicurse; Problems and results in chromatic graph theory; Forbidden subgraphs; A proof technique in graph theory; Independence and covering numbers of line graphs and total graphs; The decline and fall of Zarankiewicz's Theorem; On the intersection number of a graph; On endomorphisms of graphs and their homomorphic images; Counterexamples in the theory of well-quasi-ordered sets; On the existence of certain minimal regular n-systems with given girth; Reconstruction of unicyclic graphs; The group of a graph whose adiacency matrix has all distinct eigenvalues; Extremal nonseparable graphs of diameter 2; A class of strongly regular graphs; The exponentiation group as the automorphism group of A graph; Remarks on the Heawood conjecture; Indifference graphs; Enumeration of Euler graphs; A graph-theoretical model for periodic discrete structures; Even and odd 4-colorings; A theorem on Tait colorings with an application to the generalized Petersen graphs; The Mobius function in combinatorial analysis and chromatic graph theory; and Key-word indexed bibliography of graph theory., Proceedings of the Ann Arbor Graph Theory Conference (2nd), Feb 68.

The opening six chapters present a coherent body of graph theoretic concepts. The remaining eight chapters report lectures presented by various seminar participants. Topics presented include: Complete Bipartite Graphs, Extremal Problems in Graph Theory, Applications of Probabilistic Methods to Graph Theory, The Minimal Regular Graph Containing a Given Graph, Various Proofs of Cayley's Formula for Counting Trees, On Well-Quasi-Ordering Trees, Universal Graphs and Graphs and Composite Games. (Author)

Laser scintillation was measured for a horizontal optical path 500 m long and 1 m high for various conditions of horizontally homogeneous turbulence. Wind direction, average vertical distributions of wind speed and temperature, and, in some cases, turbulent fluctuations of wind velocity were measured simultaneously. The results of the measurements were analyzed in relation to a set of theoretical relationships derived by Tatarski (1961) for electromagnetic wave propagation in turbulent flow. Tatarski's derivation of the scintillation frequency spectrum (at a point in a plane perpendicular to the optical path) in relation to the three-dimensional spectral density of index of refraction inhomogeneities is summarized and interpreted in relation to its validity for various conditions of turbulence. Analysis of the spectral data showed that their characteristics were similar to Tatarski's theoretical spectrum if the divergence of the laser beam, the size of the receiver aperture, the intensity of scintillation and turbulence spectra were considered. (Author)

Contents: On the history of the theory of graphs; On graphs with a constant link; Some extremal problems on r-graphs; New directions in hamiltonian graph theory; On the use of graphs in group theory; The adjacency matrix and the group of a graph; Unexplored and semi-explored territories in graph theory; Graphical enumeration methods; Isomorphisms between hypergraphs; Counting labeled acyclic digraphs; Almost all trees are cospectral; What is a map., Library of Congress catalog card no. 72-77368. Proceedings of the Ann Arbor Conference on Graph Theory (3rd), 21-23 Oct 71.

The book is a definitive treatise on the enumeration of graphs and related structural configurations, providing historical results as well as recent developments having potential application in the fields of physics, chemistry, computer science, and electrical engineering. Its ten chapters cover the subjects of labeled enumeration, Polya's Theorem, trees, graphs, diagraphs, power group enumeration, superposition, blocks, asymptotics, and as a special feature, a new comprehensive list of unsolved graphical enumeration problems. It includes an extensive bibliography and numerous exercises ranging in difficulty from routine to intractable. (Author), Prepared in cooperation with Michigan State Univ., East Lansing. Library of Congress catalog card no. 72-82653.

The report is a brief summary of the results obtained during a five year basic research effort in the theory of graphs and its application to Air Force systems design and analysis. It includes an identification of the twelve research topics which received major attention, an identification of the scientific and engineering problem areas in which theoretical results have been applied, and a listing of six books and twenty-two technical papers resulting from the grant which describe in detail the theory and methodology developed. (Author)

A new method, called the QZ algorithm, is presented for the solution of the matrix eigenvalue problem Ax = lambda Bx with the general square matrices A and B. Particular attention is paid to the degeneracies which result when B is singular. No inversions of B or its submatrices are used. The algorithm is a generalization of the QR algorithm, and reduces to it when B=I. Problems involving powers of lambda are also mentioned. (Author)

Approximations that result from the natural matching of two stable dissipative difference schemes across a coordinate line are shown to be stable. The basic ideas is to reformulate the matching scheme consistent to an equivalent initial boundary value problem and verify the algebraic conditions for stability of such systems. An interesting comparison to the above result is the case of redefinition of a scheme at a single point. In particular, the author shows that some unstable perturbations do not upset the stability of the Lax-Wendroff scheme. (Author)

The report consists of the abstracts of four Ph. D. theses. The specific topics covered are numerical solution of elliptic partial differential equations, approximation theory, and matrix computations. (Author)

The efficiency of conventional FORTRAN programs for matrix computations can often be improved by reversing the order of nested loops. Such modifications produce modest savings in many common situations and very significant savings for large problems run under an operating system which uses paging. (Author), Prepared in cooperation with Standford Univ., Calif. Dept. of Computer Science, Rept. no. STAN-CS-71-196. Revision of report dated Jun 70.

An observational study was made during August, September and October 1968 to investigate the influence of water waves on the adjacent airflow. Simultaneous measurements of wind and temperature fluctuations and waves were made from a tower on Lake Michigan. 11 periods were analyzed, and 8 included simultaneous measurements of velocity and temperature fluctuations at two levels above the mean water level. Hot-wire anemometers with X-probe sensors were used to obtain both the along-wind and vertical components of the fluctuating wind. High response resistance wires were used to measure temperature fluctuations and a capacitance gauge was used to measure the waves. Data were analyzed to obtain variance and covariance spectra over a frequency range from .01 to 15 Hz and are presented over a range from .03 to 10 Hz. (Author)

Simultaneous measurements of wind velocity components with hot-film anemometers, air temperature fluctuations, and wave heights were made about 200 miles east northeast of Barbados during BOMEX. A total of 54 hours of measurements were made during the last two weeks in May, 1969. Of these, 23 had simultaneous recordings of wind velocity components and temperature fluctuations at two heights. Measurement heights were 2, 3, 6 and 8 meters above mean water level. Three different computer facilities have been employed in processing the data to obtain probability distribution functions, joint probability distribution functions, spectrum functions and cross-spectrum functions for velocity components, temperature fluctuations and wave heights. Examples of the calculation results are given and briefly described. They show specific effects of waves on the velocity components. (Author)

Contents: Development of a dependable 'Smoke producing' rocket to a height limit of 3000 to 4000 ft; Development of photographic techniques for sequentically recording the smoke plumes in three-dimensions, and, the techniques employed in the determination of wind speeds and directions vs. heights from these readings; Development of a night lighting system for the use of 'Smoke rockets' at night; and Development and testing of a smoke rocket system to extend the useful height range to at least 8000 ft above ground.

The hydrodynamic stability of divergent jets is studied under the condition of small but nonzero Rossby number. Small disturbances to the basic state consist of fast waves of inertio-gravity type and slow waves with speed comparable to that of the basic current. It is shown that the fast waves are stable and do not amplify. It is shown also that, for a velocity profile resembling that of the Gulf Stream, the slow waves are destabilized by the presence of nonzero Rossby number. For very shallow jets it is found that the complex wave velocity is small and that, in general, disturbances will be stable unless the jet has a countercurrent. (Author), See also AD-690 555.

During the reporting the principal investigator and some of his collaborators completed seventeen research papers. This work is described. The major effort was devoted to problems of optimal control. Existence theorems were proved for lumped parameter control systems by a new type of analysis. A number of related conditions were analyzed with examples and applications. Existence theorems were proved for distributed parameter control systems by the use of Sobolev spaces technique and for partial differential equations written in the rather general form proposed by Dieudonne and Rashevski. A form of Pontryagin's principle were proved for these systems which is remarkably similar to a K. A. Lurie problem of magnetohydrodynamic channel flow. Further work concerns: measure theory, integration over a surface area, periodic solutions of nonlinear hyperbolic partial differential equations, in particular, nonlinear wave equations. (Author)

The report contains a course given in Bressanone (Italy), June 10-18, 1966. The course concerns existence theorems for problems of optimal control and the calculus of variations. These theorems extend Filippov's existence theorem in optimal control theory, and the Tonelli-Nagumo theorem in the calculus of variations. Complete proofs are given for the case of lumped parameters. For the case of distributed parameters the theorems are only stated. (Author), Presented at Centro Internazionale Matematico Estivo. Course given in Bressanoe (Italy), 10-18 Jun 66.

Research was conducted in the theory of singular integral equations which arise in certain initial value and boundary value problems that come from elliptic and hyperbolic equations. In particular, those related to axially symmetric boundary value problems were studied. Results were obtained on a singular boundary value problem for the Euler Darboux equation, an integral equation in diffraction theory, diffraction by a disc and a boundary value problem associated with the Tricomi equation. (Author)

A hot-wire anemometer system, a fast response resistance wire thermometer and a thermocouple circuit were used in an attempt to measure the structure of turbulence, the turbulent fluxes of heat and momentum and the mean vertical temperature difference over ocean waves. To study data reliability (1) the recorded information was converted to voltage analogs and recorded on a paper chart and (2) spectral analyses were made for all channels of information for a representative six minute interval. It was found that a high noise level was present in all recordings and that it could be accounted for by voltage and frequency fluctuations in the power supplied for the magnetic tape recorder. The noise level masked the temperature measurement outputs which, apparently, were very small because of the nearly neutral thermal stratification during the experiments. On the basis of the preliminary analyses it appears that meaningful velocity spectra and cross spectra for frequencies less than 1.5 Hz may be computed from the recordings; higher frequency information is probably obscured by the high noise level. (Author)

The document gives various papers done under contract on optimal control and plasma physics.