Classical vs Quantum Groups as Symmetries of Quantized Systems; M. Arik, G. UEnel. Algebraic Model of an Oblate Top; R.Bijker, A. Leviatan. The Mass-Squared Operator and the Einstein-Hilbert Action for Rescaled Lorentz Metrics; E. Binz, P. Oellers. Multichannel Dynamic Symmetry; J. Cseh. Kazhdan-Lusztig Polynomials, Subsingular Vectors and Conditionally Invariant; q-Deformed Equations; V.K. Dobrev. On a Path to Nonlinear Quantum Mechanics; H.-D. Doebner, J.-D. Hennig. Quantum Mechanical Problems with q-Deformations and over the p-Adic Number Fields; I.H. Duru. A Symmetry Adapted Algebraic Approach to Molecular Spectroscopy; A. Frank, et al. Dyson Boson-Fermion Realization of Lie (Super)Algebras; D.V. Fursa, et al. Formal Languages for Quasicrystals; J.G. Escudero. On Quadratic and Nonquadratic Forms: Applications to R2m R2m-n Nonbijective Transformations; M. Kibler. Quantization of Systems with Constraints; J.R. Klauder. Automorphisms and Discrete Fiber Bundles; P. Kramer, et al. Algebraic Approach to Baryon Structure; A. Leviatan, R. Bijker. Discrete Reflection Groups and Induced Representations of Poincare Group on the Lattice; M. Lorente. 10 Additional Articles. Index.

Dynamical Symmetries of Relativistic Two - and Many Body Systems.- Algebraic Treatment of Multistep Processes in Electron-Molecule Scattering.- Symmetry, Constitutive Laws of Bounded Smoothly Deformable Media and Neumann Problems.- Imitation of Symmetries in Local Quantum Field Theory.- Representations of Quantum Groups.- Algebras and Symmetries - Quantum Mechanical Symmetry Breaking.- q-Analysis and Quantum Groups.- Orthogonal Polynomials and Coherent States.- Algebraic Model for Molecular Electronic Spectra.- Highest Weight Unitary Modules for Non-Compact Groups and Applications to Physical Problems.- Scattering Theory and the Group Representation Matrix.- Predicting "Anyons" The Origins of Fractional Statistics in Two-Dimensional Space.- The Role of Parabose-Statistics in Making Abstract Quantum Theory Concrete.- On Quantized Verma Modules.- The Role of Spectrum Generating Algebras and Dynamic Symmetries in Molecular Physics.- O(4) Symmetry and Angular Momentum Theory in Four Dimensions.- Representations of the Quantum Algebras Uq(su(2)) and Uq(su(1, 1)).- From "Quantum Groups" to "Quasi-Quantum Groups".- Symmetries of Icosahedral Quasicrystals.- Molecular Symmetry Adapted Bases in the Born-Oppenheimer Approximation.- On Certain Submodules of the Enveloping Fields.- Invariants and States Generating Symmetry of Nonstationary Systems.- Origins of Nuclear and Hadron Symmetries.- Projection Operator Method and Q-Analog of Angular Momentum Theory.- Symmetry Breaking and Fractional Quantization of Quantum Systems.- New Phases of D ? 2 Current and Diffeomorphism Algebras in Particle Physics.- Algebra and Geometry in the Theory of Mixed States.- Group Structures and the Interacting Boson Approximation for Nuclei.- Paradigms of Quantum Algebras.

Dynamical Symmetry Breaking and the Onset of Chaos in the Interacting Boson Model of Nuclei; Y. Alhassid. Generating the Spectrum of Nonlinear Hamiltonians; L.Ya. Baranov, R.D. Levine. Dynamical Symmetry and String Theory; I. Bars. U(7) Spectrum Generating Algebra for Rotations and Vibrations in Triatomic Molecules; R. Bijker, et al. Algebraic Treatment of Collective Excitations in Baryon Spectroscopy; R. Bijker, A. Leviatan. Can Iachello's Idea of a Spectral Supersymmetry be Extended into the Relativistic Domain; A. Bohm, L.C. Biedenharn. Low Lying Electric Dipole Excitations and the Interacting Boson Model; P. von Brentano, et al. Effective Charges, the Valence pn Interaction, and the IBM; R.F. Casten, A. Wolf. Dynamical Symmetries in Superdeformed Nuclei; J.A. Cizewski. Dynamical Algebras and Syperalgebras for Interacting Itinerant Many-Electron Systems; A. Danani, M. Rasetti. Number and Isospin Dependence of the IBM3 Hamiltonian; J.P. Elliott, et al. 43 additional articles. Index.

2 Elementary systems on G-manifolds. 292 3 Two elementary systems: The(GxG)-manifold and its subma- fold splitting underdiag(GxG). 293 4 Examples of internal coordinates on(GxG). 297 5 Kronecker products and two-particle state decompositions on(Gx G). 298 6 Fusion of two elementary systems on(GxG). 300 7 Elementary systems on the Poincare-manifold. ' 301 7. 1 Mackey and covariant ?elds. 301 7. 2 From covariant to Mackey ?elds. 302 7. 3 Obstruction of Poincare-manifolds ' by covariant ?elds. 303 8 Relativistic position operators and coordinates. 305 8. 1 Position operators for relativistic Mackey ?elds. 305 8. 2 Position operators for ?elds with spin. 306 9 From Dirac ?elds to Bargmann-Wigner ?elds by fusion. 307 10 Elementary systems in interaction. 309 10. 1 Euclidean invariant interactions. 309 10. 2 Interacting Dirac spinor ?elds. 310 11 Scission of an elementary system. 311 12 Conclusion. 313 13 Appendix. 313 13. 1 A: Orthogonality and completeness of unitary represen- tions. 313 13. 2 B: Parameters, cosets and multiplication rules forSl(2,C). 313 13. 3 C: Observables in the relativistic 2-body system. 314 Propagation in crossed electric and magnetic ?elds 317 T. Kramer, C. Bracher 1 Introduction 317 2 Elastic scattering and quantum sources 318 2. 1 Connection to the propagator 319 2. 2 Currents generated by quantum sources 321 2. 3 Density of States 322 2. 4 Construction of the Green function 322 3 Matter waves in crossed electric and magnetic ?elds 324 3. 1 The quantum propagator 324 3.

Invariant Theory of Matrices; H. Aslaksen, et al. Symmetries of Elementary Particles Revisited; A.O. Barut. Perturbative SU(1,1); H. Beker. A Dual Structure for the Quantal Rotation Group, SU(2); L.C. Biedenharn, M.A. Lohe. Some Points in the Quantization of Relativistic Grassmann Dependent Interaction Systems; A. Del Sol Mesa, R.P. Martinez y Romero. q-Difference Intertwining Operators for Uq(sI(4)) and q-Conformal Invariant Equations; V.K. Dobrev. A Quantum Mechanical Evolution Equation for Mixed States from Symmetry and Kinematics; H.D. Doebner, J.D. Hennig. Quantum Mechanical Motions over the Group Manifolds and Related Potentials; I.H. Duru. Quantum Violation of Weak Equivalence Principal in the Brans-Dicke Theory; Y. Fujii. Quantum Unitary and Pseudounitary Groups and Generalized Hadron Mass Relations; A.M. Gavrilik. Linear Coxeter Groups; J. Getino. Diffeomorphism Groups, Quasiinvariant Measures, and Infinite Quantum Systems; G.A. Goldin, U. Moschella. Algebraic Shells and the Interacting Boson Model of the Nucleus; B. Gruber. Recent Developments in the Application of Vector Coherent States; K.T. Hecht. Algebraic Theory of the Threebody Problem; F. Iachello. 18 additional articles. Index.

The international Symposium "Symmetries in Science X" was held at the Col legium Mehrerau in Bregenz, Austria, during the period July 13-18, 1997. Some 40 invited scientists from 13 countries participated in the meeting. The Symposium was sponsored by the Technische Universitat Graz, the Land Vorarlberg, Austria, and the Landeshauptstadt Bregenz, Vorarlberg, Austria. I wish to thank Dr. Hubert Regner of the Amt der Vorarlberger Landesregierung, tlw Biirgermeister and Landtagsprasident Dipl. -Vw. Siegfried Gasser for financial assistence for the meeting, and Frater Albin Printschler of the Cloister Mehrerau for his continued support of the Symposium series. Bruno Gruber y CONTENTS Quantization on a Lie Group: Higher-Order Polarizations 1 V. Aldaya, J. Guerrero, and G. Marmo A Comparison between Algebraic Models of Molecular Spectroscopy 37 R. Bijker, A. Frank, R. Lemus, J. M. Arias, and F. Perez-Bernal On Disc: rete Media, Their Interaction Forms and the Origin of Non-Exactness of the Virtual Work . . . . . . . . . . . . . . . . . . . . . 47 E. Binz The k-Fermions and Objects Interpolating between Fermions and Bosons 63 M. Daoud, Y. Hassouni, and M. Kibler Polynomial Realization of the Uq(sl(3)) Gel'fand-(Weyl)-Zetlin Basis and Irregular Irreps at Roots of Unity . . . . . . . . . . . . . . 79 V. K. Dobrev and R. Truini Nonstandard Deformation U (son): The Imbedding U (son) C Uq(sln) and Representations . . . . . . . . . . . . . . . . . . . . . . 121 A. M. Gavrilik, N. Z. Iorgov, and A. U. Klimyk Algebraic: Methods in Physics 135 F. Iaehello Possible Origins of Quantum Fluctuations 145 M. Kanenaga and M."

The international Symposium "Symmetries in Science X" was held at the Col legium Mehrerau in Bregenz, Austria, during the period July 13-18, 1997. Some 40 invited scientists from 13 countries participated in the meeting. The Symposium was sponsored by the Technische Universitat Graz, the Land Vorarlberg, Austria, and the Landeshauptstadt Bregenz, Vorarlberg, Austria. I wish to thank Dr. Hubert Regner of the Amt der Vorarlberger Landesregierung, tlw Biirgermeister and Landtagsprasident Dipl. -Vw. Siegfried Gasser for financial assistence for the meeting, and Frater Albin Printschler of the Cloister Mehrerau for his continued support of the Symposium series. Bruno Gruber y CONTENTS Quantization on a Lie Group: Higher-Order Polarizations 1 V. Aldaya, J. Guerrero, and G. Marmo A Comparison between Algebraic Models of Molecular Spectroscopy 37 R. Bijker, A. Frank, R. Lemus, J. M. Arias, and F. Perez-Bernal On Disc: rete Media, Their Interaction Forms and the Origin of Non-Exactness of the Virtual Work . . . . . . . . . . . . . . . . . . . . . 47 E. Binz The k-Fermions and Objects Interpolating between Fermions and Bosons 63 M. Daoud, Y. Hassouni, and M. Kibler Polynomial Realization of the Uq(sl(3)) Gel'fand-(Weyl)-Zetlin Basis and Irregular Irreps at Roots of Unity . . . . . . . . . . . . . . 79 V. K. Dobrev and R. Truini Nonstandard Deformation U (son): The Imbedding U (son) C Uq(sln) and Representations . . . . . . . . . . . . . . . . . . . . . . 121 A. M. Gavrilik, N. Z. Iorgov, and A. U. Klimyk Algebraic: Methods in Physics 135 F. Iaehello Possible Origins of Quantum Fluctuations 145 M. Kanenaga and M."

The Symposium "Symmetries in Science II" was held at Southern Illinois University, Carbondale, during the period March 24-26, 1986, following the Einstein Centennial Symposium "Symmetries in Science" after a lapse of seven years. As it was the case for the original Symposium, the 1986 Symposium was truly interdisciplinary and truly international. I wish to thank all participants who made the. effort to come to Carbondale, Illinois, from allover the world. At this point I also wish to express my sincere thanks to Dr. Albert Somit, President of Southern Illinois University at Carbondale, and Dr. John C. Guyon, Vice President for Academic Affairs and Research at Southern Illinois University at Carbondale. Their generous support and encouragement was instrumental in getting the Symposium organized. In addition I wish to thank Associate Vice President Charles B. Klasek, Dr. Russell R. Dutcher, Dean of the College of Science, John H. Yopp, Associate Dean, College of Science, Dr. Subir K. Bose, Chairman of the Physics Department, Dr. James Tyrrell, Chairman of the Chemistry Department, Dr. Jared H. Dorn, Director of International Programs and Services, Dr. Rhonda Jo Vinson, Director of International and Economic Development, Dr. Tommy T. Dunagan, Vice President of Sigma Xi at Southern Illinois University, Dr. George Garoian, Professor of Zoology, Dr. Ann Phillippi, Assistant Professor of Zoology and Dr. Linda R. Gannon, Coordinator of Women's Studies, for their support and assistance.

The international Symposium "Symmetries in Science IX" was held at the Col- Mehrerau in Bregenz, Austria, during the period August 6-10, 1996. Some 40 legium invited scientists from 13 countries participated in the meeting. The Symposium was sponsored by Southern Illinois University at Carbondale, the Land Vorarlberg, Austria, and the Landeshauptstadt Bregenz, Vorarlberg, Austria. I wish to thank Chancellor Emeritus John C. Guyon of Southern Illinois Univer- sity at Carbondale, Dr. Hubert Regner of the Amt der Vorarlberger Landesregierung, and Biirgermeister and Landtagspriisident Dipl.-Vw. Siegfried Gasser for financial as- sistence for the meeting and Frater Albin Printschler of the Cloister Mehrerau for his continued support of the Symposium series. Bruno Gruber v CONTENTS Classical vs. Quantum Groups as Symmetries of Quantized Systems 1 M. Arik and C. Vnel Algebraic Model of an Oblate Top 9 R. Bijker and A. Leviatan The Mass-Squared Operator and the Einstein-Hilbert Action for Rescaled Lorentz Metrics ...25 E. Binz and P. Oellers Multichannel Dynamic Symmetry 37 J. Cseh Kazhdan-Lusztig Polynomials, Subsingular Vectors and Conditionally Invariant (q-Deformed) Equations ...47 V. K. Dobrev On a Path to Nonlinear QuantuIll Mechanics 81 H.-D. Doebner and J.-D. Hennig Quantum Mechanical Problems with q-Deformations and over the p-Adic Number Fields ...93 .

Thf3 symposium "Symmetries in Science VIII" was held in August of 1994 at the Cloister Mehrerau in Bregenz, Austria. The symposium was supported by Southern Illinois University at Carbondale, the Land Vorarlberg, and the Landeshaupstaot Bregenz. I wish to thank Dr. John C. Guyon, President of Southern Illinois University at Carbondale; Dr. Hubert Regner, Amt der Vorarlberger Landesregierung; and Dipl. Vw. Siegfried Gasser, Buergermeister der Landeshauptstadt Bregenz and Lantagsabgeordneter, for their generous support of the symposium. Finally I wish to thank Frater Albin of the Cloister Mehrerau for his support and cooperation in this endeavor, which made for a successful meeting in a most pleasant environment. Bruno Gruber v CONTENTS On Om x Gin Highest Weight Vectors Helmer Aslaksen, Eng-Chye Tan, and Chen-bo Zhu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Invariant Theory of Matrices Helmer Aslaksen, Eng-Chye Tan, and Chen-bo Zhu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 Symmetries of Elementary Particles Revisited A. O. Barut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Perturbative SU(1,1) Haluk Seker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ., . . . . . . . . . ., . . . . . . . . . . . . . . . . . . . . . . ., . . . . . . . . . . . . . . . . . . . . 25 A Dual Structure for the Quantal Rotation Group, SU(2) L. C. Biedenharn and M. A. Lohe . . . . . . . . . . . . ., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Some Points in the Quantization of Relativistic Grassmann Dependent Interaction Systems A. Del Sol Mesa and R, P. Martinez y Romero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ., . . . . . 49 for Uq(sl(4)) and q-Conformal q-Difference Intertwining Operators I nvariant Equations V. K. Dobrev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 A Quantum Mechanical Evolution Equation for Mixed States from Symmetry and Kinematics H. -D. Doebner and J. D. Hennig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 vii Quantum Mechanical Motions over the Group Manifolds and Related Potentials I. H. Duru . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ."

The Symposium "Symmetries in Science V: Algebraic Systems: their Represen- tations, Realizations and Physical Applications" was held at the Landesbildungszen- trum Schloss Hofen, Vorarlberg, Austria, during the period July 30 - August 3, 1990. Leading scientists presented summaries, as well as recent research developments, con- cerning their own field of specialization to the assembled international group of re- searchers. These review-oriented presentations are included in this volume in order to make them available to the general scientific community. As was the case for the preceding Symposia we need to thank those who made Symposium V possible through their continued and generous support. These include, on part of the State of Vorarlbergj Dr. Martin Purtscher, Landeshauptmann Dr. Guntram Lins, Landesrat Ms. Elisabeth Gehrer, Landesratin and on part of Southern Illinois University at Carbondale; Dr. John C. Guyon, President Dr. Benjamin A. Shepherd, Vice President for Academic Affairs and Research Dr. John H. Yopp, Dean of the Graduate School Dr. Charles B. Klasek, Executive Assistant to the President for International and Economic Development Dr. Russell R. Dutcher, Dean of the College of Science Dr. Victoria J. Molfese, Director, Office of Research Development and Administration Dr. James D. Quisenberry, Acting Director of International Programs and Services Dr. Frank C. Sanders, Chairman of the Department of Physics. Finally, we need to especially thank Dr.

Southern Illinois University at Carbondale undertook to honor Albert Einstein as scientist and as humanitarian in commemo ration of his lOOth birthday during an "Albert Einstein Centennial Week," February 23 - March 2, 1979. During the course of this week two Symposia were held, entitled "Symmetries in Science" and "Einstein: Humanities Conscience," in addition to cultural and social activities honoring Einstein. This volume presents the Symposium "Symmetries in Science." It reflects the outstanding response that was given to our "Albert Einstein Centennial Week" by the international community of scientists. The motivation to have a celebration honoring Albert Einstein at Southern Illinois University at Carbondale was supplied by Dr. Paul A. Schilpp, the editor of the "Library of Living Philo sophers." Albert Einstein has contributed to this series with his autobiographical notes, a kind of autobiography of his scientific life, in a volume entitled "Einstein: Scientist-Philosopher," the most popular among all the outstanding volumes of this series. Dr. Paul A. Schilpp's presence at Southern Illinois University at Carbondale provided a natural link for an Einstein Celebration as a kind of a continuation of the contribution he made to mankind through the Einstein volume of his "Library of Living Philosophers.""

The Symposium "Symmetries in Science VII: Spectrum Generating Algebras and Dynamic Symmetries in Physics" was held at the Southern Illinois University at Carbondale in Niigata, Japan Campus, during the period August 28-31, 1992. The Symposium was held in honor of Professor Francesco lachello on the occasion of his 50th birthday. We wish to thank the colleagues and friends of Franco for their participation in the Symposium as well as for contributing articles to this volume honoring him. It was their commitment and involvement which made this Symposium a success. We also wish to thank Dr. Jared H. Dorn, the director of SIUC-N, for his support in the planning and the execution of the Symposium. Moreover we wish to thank Mayor Nobuo Kumakura of Nakajo town and Mr. Kaichi Suzuki of the school entity "The Pacific" for their friendly support. Bruno Gruber, SIUC-N Takaharu Otsuka, University of Tokyo v LAUDATIO ON THE OCCASION OF THE 50TH BIRTHDAY OF PROFESSOR FRANCESCO IACHELLO I first met Franco lachello in 1974. Driving a smart Alfa-Romeo, he came to meet me at the station at Groningen where I was to spend a summer conducting research.

The symposium, "Symmetries in Science IV: Biological and Biophysical Systems," was held at Schloss Hofen, Vorarlberg, during the period July 24-27, 1989. Its purpose was to promote interaction between scientists working in the area of biological-biophysical research with scientists working in physics, mathematical physics, and mathematics. Reviews in the field of biological-biophysical research were presented by the participants, and subsequently these presentations were analyzed by the interdisciplinary group in a workshop and by means of round table dis cussions. This volume contains the review presentations as a means of making this subject available to the general scientific community. The symposium was co-sponsored by: Southern Illinois University at Carbondale Land Vorarlberg U.S. Office of Naval Research, London We wish to thank these institutions for their support. Moreover, we wish to thank the following individuals: Dr. John C. Guyon, President Southern Illinois University at Carbondale Dr. Benjamin A. Shepherd Vice President for Academic Affairs and Research Southern Illinois University at Carbondale Dr. Martin Purtscher Landeshauptmann des Landes Vorarlberg Dr. Guntram Lins Landesrat des Landes Vorarlberg Dr. Hubert Regner Vorstand der Abt. Volksausbildung und Wissenschaft Their generous support and steady encouragement has made Symposium IV possible."

The Symposium "Symmetries in Science III" was held at Schloss Hofen, Vorarl- berg, Austria, during the period July 25-28, 1988. Some 70 invited scientists from 24 countries attended the Symposium and presented reviews, research papers and re- ports, as well as participated in workshops. The format of the Symposium was delib- erately kept international and interdisciplinary, and also provided an opportunity for presentations of symmetry applications in fields other than in the conventional disci- plines. The Symposium was co-sponsored by Southern Illinois University at Carbondale Land Vorarlberg, U.S. Office of Naval Research (Special Session), and IBM Austria. Support by these institutions is gratefully acknowledged. Several individuals have contributed significantly to the success of Sym- sium III. They are Dr. John C. Guyon, President, Southern Illinois University, at Carbondale, Dr. Martin Purtscher, Landeshauptman des Landes Vorarlberg, Dr. Guntram Lins, Landesrat des Landes Vorarlberg, and Dr. Hubert Regner, Direktor, Landesbildungszentrum Schloss Hofen. I wish to thank these individuals for their support of Symposium III, also in the name of the participating scientists. The Symposium provided these scientists with an opportunity to get together and exchange scientific information in an efficient manner, and in a very pleasant environment.

2 Elementary systems on G-manifolds. 292 3 Two elementary systems: The(GxG)-manifold and its subma- fold splitting underdiag(GxG). 293 4 Examples of internal coordinates on(GxG). 297 5 Kronecker products and two-particle state decompositions on(Gx G). 298 6 Fusion of two elementary systems on(GxG). 300 7 Elementary systems on the Poincare-manifold. ' 301 7. 1 Mackey and covariant ?elds. 301 7. 2 From covariant to Mackey ?elds. 302 7. 3 Obstruction of Poincare-manifolds ' by covariant ?elds. 303 8 Relativistic position operators and coordinates. 305 8. 1 Position operators for relativistic Mackey ?elds. 305 8. 2 Position operators for ?elds with spin. 306 9 From Dirac ?elds to Bargmann-Wigner ?elds by fusion. 307 10 Elementary systems in interaction. 309 10. 1 Euclidean invariant interactions. 309 10. 2 Interacting Dirac spinor ?elds. 310 11 Scission of an elementary system. 311 12 Conclusion. 313 13 Appendix. 313 13. 1 A: Orthogonality and completeness of unitary represen- tions. 313 13. 2 B: Parameters, cosets and multiplication rules forSl(2,C). 313 13. 3 C: Observables in the relativistic 2-body system. 314 Propagation in crossed electric and magnetic ?elds 317 T. Kramer, C. Bracher 1 Introduction 317 2 Elastic scattering and quantum sources 318 2. 1 Connection to the propagator 319 2. 2 Currents generated by quantum sources 321 2. 3 Density of States 322 2. 4 Construction of the Green function 322 3 Matter waves in crossed electric and magnetic ?elds 324 3. 1 The quantum propagator 324 3.

Proceedings of a symposium at Vorarlberg, Austria, July 1989, called to allow interaction between scientists working in areas of biological and biophysical research, and those working in physics and mathematics. The 11 papers include discussions of such topics as symmetry in synthetic and natural pe