Garrett Birkhoff

Bio: Garrett Birkhoff is an academic researcher from Harvard University. The author has contributed to research in topics: Abstract algebra & Cellular algebra. The author has an hindex of 46, co-authored 160 publications receiving 14845 citations. Previous affiliations of Garrett Birkhoff include University of Massachusetts Amherst.

The Logic of Quantum Mechanics

Abstract: One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes It asserts that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and that in particular one can never predict with certainty both the position and the momentum of S, (Heisenberg’s Uncertainty Principle) It further asserts that most pairs of observations are incompatible, and cannot be made on S, simultaneously (Principle of Non-commutativity of Observations)

The Logic of Quantum Mechanics

Abstract: One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes. It asserts that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and that in particular one can never predict with certainty both the position and the momentum of S, (Heisenberg’s Uncertainty Principle). It further asserts that most pairs of observations are incompatible, and cannot be made on S, simultaneously (Principle of Non-commutativity of Observations).

On the Structure of Abstract Algebras

Abstract: The following paper is a study of abstract algebras qua abstract algebras. As no vocabulary suitable for this purpose is current, I have been forced to use a number of new terms, and extend the meaning of some accepted ones.

Ordinary Differential Equations

Abstract: First--Order of Differential Equations. Second--Order Linear Equations. Linear Equations with Constant Coefficients. Power Series Solutions. Plane Autonomous Systems. Existence and Uniqueness Theorems. Approximate Solutions. Efficient Numerical Integration. Regular Singular Points. Sturm--Liouville Systems. Expansions in Eigenfunctions. Appendices. Bibliography. Index.

The entity-relationship model: toward a unified view of data

Abstract: A data model, called the entity-relationship model, is proposed. This model incorporates some of the important semantic information in the real world. A special diagramatic technique is introduced as a tool for data base design. An example of data base design and description using the model and the diagramatic technique is given. Some implications on data integrity, information retrieval, and data manipulation are discussed.The entity-relationship model can be used as a basis for unification of different views of data: the network model, the relational model, and the entity set model. Semantic ambiguities in these models are analyzed. Possible ways to derive their views of data from the entity-relationship model are presented.

Foundations of modern probability

Abstract: * Measure Theory-Basic Notions * Measure Theory-Key Results * Processes, Distributions, and Independence * Random Sequences, Series, and Averages * Characteristic Functions and Classical Limit Theorems * Conditioning and Disintegration * Martingales and Optional Times * Markov Processes and Discrete-Time Chains * Random Walks and Renewal Theory * Stationary Processes and Ergodic Theory * Special Notions of Symmetry and Invariance * Poisson and Pure Jump-Type Markov Processes * Gaussian Processes and Brownian Motion * Skorohod Embedding and Invariance Principles * Independent Increments and Infinite Divisibility * Convergence of Random Processes, Measures, and Sets * Stochastic Integrals and Quadratic Variation * Continuous Martingales and Brownian Motion * Feller Processes and Semigroups * Ergodic Properties of Markov Processes * Stochastic Differential Equations and Martingale Problems * Local Time, Excursions, and Additive Functionals * One-Dimensional SDEs and Diffusions * Connections with PDEs and Potential Theory * Predictability, Compensation, and Excessive Functions * Semimartingales and General Stochastic Integration * Large Deviations * Appendix 1: Advanced Measure Theory * Appendix 2: Some Special Spaces * Historical and Bibliographical Notes * Bibliography * Indices

Closed-form solution of absolute orientation using unit quaternions

Abstract: Finding the relationship between two coordinate systems using pairs of measurements of the coordinates of a number of points in both systems is a classic photogrammetric task . It finds applications i n stereoph and in robotics . I present here a closed-form solution to the least-squares problem for three or more paints . Currently various empirical, graphical, and numerical iterative methods are in use . Derivation of the solution i s simplified by use of unit quaternions to represent rotation . I emphasize a symmetry property that a solution to thi s problem ought to possess . The best translational offset is the difference between the centroid of the coordinates i n one system and the rotated and scaled centroid of the coordinates in the other system . The best scale is equal to th e ratio of the root-mean-square deviations of the coordinates in the two systems from their respective centroids . These exact results are to be preferred to approximate methods based on measurements of a few selected points . The unit quaternion representing the best rotation is the eigenvector associated with the most positive eigenvalue o f a symmetric 4 X 4 matrix . The elements of this matrix are combinations of sums of products of correspondin g coordinates of the points .

The entity-relationship model: toward a unified view of data

Abstract: A data model, called the entity-relationship model, is proposed. This model incorporates some of the important semantic information about the real world. A special diagrammatic technique is introduced as a tool for database design. An example of database design and description using the model and the diagrammatic technique is given. Some implications for data integrity, information retrieval, and data manipulation are discussed.The entity-relationship model can be used as a basis for unification of different views of data: the network model, the relational model, and the entity set model. Semantic ambiguities in these models are analyzed. Possible ways to derive their views of data from the entity-relationship model are presented.

Decoherence, einselection, and the quantum origins of the classical

Abstract: as quantum engineering. In the past two decades it has become increasingly clear that many (perhaps all) of the symptoms of classicality can be induced in quantum systems by their environments. Thus decoherence is caused by the interaction in which the environment in effect monitors certain observables of the system, destroying coherence between the pointer states corresponding to their eigenvalues. This leads to environment-induced superselection or einselection, a quantum process associated with selective loss of information. Einselected pointer states are stable. They can retain correlations with the rest of the universe in spite of the environment. Einselection enforces classicality by imposing an effective ban on the vast majority of the Hilbert space, eliminating especially the flagrantly nonlocal ''Schrodinger-cat states.'' The classical structure of phase space emerges from the quantum Hilbert space in the appropriate macroscopic limit. Combination of einselection with dynamics leads to the idealizations of a point and of a classical trajectory. In measurements, einselection replaces quantum entanglement between the apparatus and the measured system with the classical correlation. Only the preferred pointer observable of the apparatus can store information that has predictive power. When the measured quantum system is microscopic and isolated, this restriction on the predictive utility of its correlations with the macroscopic apparatus results in the effective ''collapse of the wave packet.'' The existential interpretation implied by einselection regards observers as open quantum systems, distinguished only by their ability to acquire, store, and process information. Spreading of the correlations with the effectively classical pointer states throughout the environment allows one to understand ''classical reality'' as a property based on the relatively objective existence of the einselected states. Effectively classical pointer states can be ''found out'' without being re-prepared, e.g, by intercepting the information already present in the environment. The redundancy of the records of pointer states in the environment (which can be thought of as their ''fitness'' in the Darwinian sense) is a measure of their classicality. A new symmetry appears in this setting. Environment-assisted invariance or envariance sheds new light on the nature of ignorance of the state of the system due to quantum correlations with the environment and leads to Born's rules and to reduced density matrices, ultimately justifying basic principles of the program of decoherence and einselection.