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1 Symmetry in General.- 1-1. Introduction.- 1-2. Definition of Symmetry.- 1-3. Symmetry in Our Culture.- 1-4. Symmetry in Nature.- 1-5. Symmetry in Science.- 1-6. References.- 2 Symmetry in Chemistry.- 2-1. Introduction.- 2-2. Symmetry Elements.- 2-3. Point Groups.- 2-4. Rules for Classification of Molecules into Point Groups.- 2-5. Problems.- 2-6. References.- 2-7. Bibliography.- 3 Elementary Mathematics.- 3-1. Definition of a Group.- 3-2. Finite and Infinite Groups.- 3-3. Subgroups.- 3-4. Classes.- 3-5. Definition of a Matrix.- 3-6. Multiplication of Matrices.- 3-7. Transpose of a Matrix.- 3-8. Representation of Groups.- 3-9. Problems.- 3-10. References.- 4 The Character Table.- 4-1. Introduction.- 4-2. Types of Representations.- 4-3. Character Tables for the More Common Point Groups.- 4-4. Problem.- 5 Derivation of Selection Rules.- 5-1. Derivation of Selection Rules for Nonlinear Molecules of the Most Common Point Groups.- 5-2. The Td Point Group.- 5-3. The D4h Point Group.- 5-4. The Oh Point Group.- 5-5. The D3h Point Group.- 5-6. The C3v Point Group.- 5-7. The C2v Point Group.- 5-8. Derivation of Selection Rules for Linear Molecules.- 5-9. Problem.- 5-10. References.- 6 Potential Force Fields.- 6-1. The General Quadratic Potential Function.- 6-2. The Central Force Field.- 6-3. The Valence Force Field.- 6-4. The Generalized Valence Force Field.- 6-5. The Urey-Bradley Function.- 6-6. References.- 7 The Normal Coordinate Treatment for Molecules with C2v, C3v, Td, and Oh Symmetry.- 7-1. Procedure Necessary in the NCT Method.- 7-2. Normal Coordinate Treatment of H2O (C2v Symmetry).- 7-3. Normal Coordinate Treatment of NH3(C3v Symmetry).- 7-4. Normal Coordinate Treatment of CH4(Td Symmetry).- 7-5. Normal Coordinate Treatment of UF6(Oh Symmetry).- 7-6. Some Results of NCT of Molecules.- 7-7. The Product Rule.- 7-8. The Sum Rule.- 7-9. Summary.- 7-10. Problems.- 7-11. References.- 8 Applications of Group Theory.- 8-1. Introduction.- 8-2. Procedure Used in Determing the Structure of a Molecule.- 8-3. Examples Illustrating the Use of Group Theory in Determining Molecular Structure.- 8-4. Practice Problem.- 8-5. Selection of Rules for A3 to A8 Molecules.- 8-6. Forbidden Vibrations for Several Point Groups.- 8-7. Problems.- 8-8. References.- Appendices.- 1. Conversion Table for Wavelength and Wave Number Units.- 3. Diagrams of Normal Vibrations for Common Point Groups.- 4. Calculation of Thermodynamic Functions for Vibrational-Rotational Spectra.- 6. General Method of Obtaining Molecular Symmetry Coordinates.- 7. Conversion Chart for Various Forms of Energy and Velocities.

1 Introduction.- 2 Rational matrices and vector spaces.- 3 Representations of linear time-invariant systems.- 4 Minimality and transformation groups.- 5 Realization in minimal first-order form.- 6 Structural invariants.- 7 Conclusions.

Blowup of small data solutions for a class of quasilinear wave equations in two dimensions: an outline of the proof.- Concentration effects in critical nonlinear wave equation and scattering theory.- Lower semicontinuity of weighted path length in BV.- Time decay of Lp norms for solutions of the wave equation on exterior domains.- Sobolev embeddings in Weyl-Hörmander calculus.- About the Cauchy problem for a system of conservation laws.- Global existence of the solutions and formation of singularities for a class of hyperbolic systems.- A class of solvable operators.- On the uniqueness of the Cauchy problem under partial analyticity assumptions.- Nonlinear wave diffraction.- Caustics for dissipative semilinear oscillations.- Geometric optics and the bottom of the spectrum.- Hypoellipticity for a class of infinitely degenerate elliptic operators.- Regularity of solutions to characteristic boundary value problem for symmetric systems.

Process Level Large Deviations for a Class of Piecewise Homogeneous Random Walks.- Mutual Singularity of Genealogical Structures of Fleming-Viot and Continuous Branching Processes.- Regular Conditional Expectations and the Continuum Hypothesis.- Necessary and Sufficient Conditions for Weak Convergence of One-Dimensional Markov Processes.- Inverse Subordination of Excessive Functions.- Jumping Branching Measure-Valued Processes.- The Boundedness of Branching Markov Processes.- A Limit Theorem for Weighted Branching Process Trees.- Loop Condensation Effects in the Behavior of Random Walks.- On the Stability of Solutions of Stochastic Evolution Equations.- Harmonic Functions on Riemannian Manifolds: A Probabilistic Approach.- On a Problem Suggested by A.D. Wentzell.- Regularity Properties of a Supercritical Superprocess.- A Lemma on Super-Brownian Motion with Some Applications.- Sequential Screening of Significant Variables of an Additive Model.- Brownian Bandits.- Lyapunov Exponents and Distributions of Magnetic Fields in Dynamo Models.- The Strong Markov Property of the Support of Super-Brownian Motion.- Some Results on Random Walks on Groups.- Diffusions as Integral Curves, or Stratonovich without Itô.- Convex Solutions to Variational Inequalities and Multidimensional Singular Control.- Regularity of Self-Diffusion Coefficient.- Representation Results for Stopping Times in Jump-with-Drift Processes.