Superstring theoryをめぐって(放談室)

Real and Complex Analysis

General Topology-I

Tables of integrals

This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

A first course in noncommutative rings, by T. Y. Lam. Pp. 385. £37 (pb), £62.50 (hb). 2001. ISBN 0 387 95325 6 (pb), 0 387 95183 0 (hb) (Springer-Verlag).

Octahedral norms and convex combination of slices in Banach spaces

Contents1. Introduction.2. Basicnotionsandresultsontransitivityofthenorm.Transitivity.Almost-transitivity.Transitivityversusalmost-transitivity.Convextransitivity.Maximalityofnorm.Transitivityandisomorphicconditions.3. Isometricone-dimensionalperturbationsoftheidentity.4. MultiplicativecharacterizationofHilbertspaces.5. TransitivityofBanachspaceshavingaJordanstructure.6. Thegeometryofconvex-transitiveBanachspaces.1. IntroductionInthispaperwedealwiththoseconceptsandresultsgeneratedaroundtheBanach-Mazur\rotation"problemifeverytransitiveseparableBanachspaceis a Hilbert space. Basic references in this ﬂeld are the book of S. Rolewicz[65]andthesurveypaperofF.Cabello[23]. WerecallthataBanachspaceiscalledtransitivewhenever,giventwopointsinitsunitsphere,thereexistsasurjectivelinearisometryonthespacemappingoneofsuchpointsintotheother. Asfarasweknow,theBanach-Mazurrotationproblemremainsopentodate. However,theliteraturedealingwithtransitivityof(possiblynonsep-1

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Transitivity of the Norm on Banach Spaces

Using a quantum version of the Arnold transformation of classical mechanics, all quantum dynamical systems whose classical equations of motion are non-homogeneous linear second-order ordinary differential equations, including systems with friction linear in velocity, can be related to the quantum free-particle dynamical system. This transformation provides a basic (Heisenberg?Weyl) algebra of quantum operators, along with well-defined Hermitian operators which can be chosen as evolution-like observables and complete the entire Schr?dinger algebra. It also proves to be very helpful in performing certain computations quickly, to obtain, for example, wavefunctions and closed analytic expressions for time-evolution operators.

The quantum Arnold transformation

We prove that, given a real JB*-triple X, there exists a nonempty relatively weakly open subset of the closed unit ball of X with diameter less than 2 (if and) only if the Banach space of X is isomorphic to a Hilbert space. Moreover we give the structure of real JB*-triples whose Banach spaces are isomorphic to Hilbert spaces. Such real JB*-triples are also characterized in two different purely algebraic ways.

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Relatively weakly open sets in closed balls of Banach spaces, and real JB ∗ -triples of finite rank

Different families of states, which are solutions of the time-dependent free Schrodinger equation, are imported from the harmonic oscillator using the Quantum Arnold Transformation introduced in a previous paper. Among them, infinite series of states are given that are normalizable, expand the whole space of solutions, are spatially multi-localized and are eigenstates of a suitably defined number operator. Associated with these states new sets of coherent and squeezed states for the free particle are defined representing traveling, squeezed, multi-localized wave packets. These states are also constructed in higher dimensions, leading to the quantum mechanical version of the Hermite-Gauss and Laguerre-Gauss states of paraxial wave optics. Some applications of these new families of states and procedures to experimentally realize and manipulate them are outlined.

Julio Becerra Guerrero

Papers

Octahedral norms and convex combination of slices in Banach spaces

Transitivity of the Norm on Banach Spaces

The quantum Arnold transformation

Relatively weakly open sets in closed balls of Banach spaces, and real JB ∗ -triples of finite rank

Harmonic states for the free particle