Hamiltonian collapsing of irrational Lagrangian submanifolds with small first Betti number

Among the main symplectic invariants of a closed Lagrange submanifoldL of the cotangent of ⁿ is the tubular radiusR(L) defined as the smallest tube D(r) × ^n–1 of ⁿ T^{* n} in whichL can be pushed by an Hamiltonian diffeotopy of ⁿ. We show here, using pseudo-holomorphic techniques, that such a submanifold cannot collapse if the first Betti number ofL is smaller than 3 and if the Maslov class ofL does not vanish; in other words,R(L) is then strictly positive and one can actually give an explicit lower bound in terms of the Liouville and Maslov classes ofL.Partially supported by Research Grants NSERC OGP0092913 and FCAR EQ3518     

Kac-Moody symmetry of generalized non-linear Schrödinger equations

The classical non-linear Schrödinger equation associated with a symmetric Lie algebra =km is known to possess a class of conserved quantities which from a realization of the algebrak []. The construction is now extended to provide a realization of the Kac-Moody algebrak[, ^–1] (with central extension). One can then define auxiliary quantities to obtain the full algebra [, ^–1]. This leads to the formal linearization of the system.     

On the isomorphism of a ‘quantum logic’ with the logic of the projections in a hilbert space

A topology is introduced in a logic using the set of pure states of . It is shown that , equipped with this topology, under suitable conditions, determines the division ring , or 2e. With the continuity of the antiautomorphism of the division ring added, it is shown that these conditions are necessary and sufficient for the projective logic to be isomorphic with the projective logic of the projections in a Hilbert space over , or 2e.P. Cotta-Ramusino gratefully acknowledges a fellowship of the Consiglio Nazionale delle Ricerche.     

Theory of a Pair of Quantum Bits

In this paper we present a thorough study of the theory of a pair of qubits, whose Hilbert space can be identified with ^{2 2}. Given an hermitian operator of trace 1 in ^{2 2} we focus on the following Problems: Problem 1: Find conditions that guarantee that is a state, that is, positive semidefinite. Problem 2: Find conditions that guarantee that a given state is separable, or that is a convex combination of products of one-particle states. The language we develop for our investigation makes use of the observation that ^{2 2} carries representations of the special unitary group SU(2) in two dimensions and of the direct product of this group by itself. We introduce a new type of observable called Bell observable (section 5) and a new measure of entanglement called concurrence, which is closely related to the concurrence introduced by Wootters (Physical Review Letters (1998) 80, 2245–2248) (section 8). The work has been inspired by the works of Wootters (Physical Review Letters (1997) 78, 5022–5025; Physical Review Letters (1998) 80, 2245–2248) and members of the Horodecki family (cf Horodecki and Horodecki, Physical Review A (1996) 54, 1838–1843; Horodecki et al., Physics Letters A (1996a) 223, 1–8; Physics Letters A (1996b) 222, 21–25) and reproduces some of their results.     

Influence of relative humidity on surface conductivity of germanium

A study was made of the influence of the relative humidity (from 0.5–90%) on the surface conductivity of bothp- andn-type germanium having differing resistivity. It was. found that the range of changes in the surface potential did not depend on the volume properties of germanium. An analysis of the experimental results shows that the surface potential has values approximately in the range from _s =–3kT/q to _s =+5kT/q for a change in humidity from 0.5 to 90%. Assuming that the Fermi level changes on the outer side of the oxide layer by approximately the same value as on the germaniumgermanium oxide boundary as a result of the increase and decrease of slow states, it follows that the levels of the slow centres are considerably distant from the Fermi level during the whole humidity range. It is also shown that on the first monomolecular layer of adsorbed water there is an increment of (7.5±0.8) x 10¹⁰ donor levels per cm². This increase in levels either decreases with the number of adsorbed monomolecular layers of water and for 90% humidity has approximately the value 2×10¹⁰ cm^–2 or remains almost unchanged with the number of layers, if it is assumed that for _s =6.0kT/q the Fermi level passes through the centres of adsorbed water.

---

0,5 90% p n. . , _s =– 3kT/q _s =+ 5kT/q 0,5 90%. , - , , . , (7,5 ±0,8). 10¹⁰ cm². 90% . 2. 10¹⁰ cm^–2 , , , _s =6,0kT/q .

---

---

In conclusion the author thanks Dr. J. Taue for instigating this work and S. Koc, C. Sc., for remarks and the interest with which he followed the work. He also thanks A.Müller for measuring the Hall constant in the samples.     

Projective quantum spaces

Associated to the standard SU _q (n) R-matrices, we introduce quantum spheresS _q ^2n-1 , projective quantum spaces _q ^n-1 , and quantum Grassmann manifoldsG _k( _q ⁿ ). These algebras are shown to be homogeneous spaces of standard quantum groups and are also quantum principle bundles in the sense of T. Brzeziski and S. Majid.     

A borel-weil-bott approach to representations of sl q (2, ℂ)

We use a quite concrete and simple realization of sl _q (2, ) involving finite difference operators. We interpret them as derivations (in the noncommutative sense) on a suitable graded algebra, which gives rise to the noncommutative scheme ¹ II ^1* as the counterpart of the standard ¹ = Sl(2, )/B.     

Noncommutative Geometry and the Regularization Problem of 4D Quantum Field Theory

We give a noncommutative version of the complex projective space ² and show that scalar QFT on this space is free of UV divergencies. The tools necessary to investigate quantum fields on this fuzzy ² are developed and several possibilities to introduce spinors and Dirac operators are discussed.     

Eine verbesserte Schleifvorrichtung zur Herstellung von Kugeln aus harten Materialien

Ohne Zusammenfassung

---

     

Finite dimensional representations of the quantum Lorentz group

All finite dimensional irreducible representations of the quantum Lorentz group SL _q (2,) are described explicitly and it is proved all finite dimensional representations of SL _q (2,) are completely reducible. The conjecture of Podle and Woronowicz will be answered affirmatively.     

The Lie algebra of the sl(2, ℂ)-valued automorphic functions on a torus

It is shown that the Lie algebra of the automorphic, meromorphic sl(2, )-valued functions on a torus is a geometric realization of a certain infinite-dimensional finitely generated Lie algebra. In the trigonometric limit, when the modular parameter of the torus goes to zero, the former Lie algebra goes over into the sl(2, )-valued loop algebra, while the latter goes into the Lie algebra (A ₁ ⁽¹⁾ )/(centre).     

Period doubling bifurcations for families of maps on ℝ n

Infinite sequences of period doubling bifurcations in one-parameter families (1-pf) of maps enjoy very strong universality properties: This is known numerically in a multitude of cases and has been shown rigorously for certain 1-pf of maps on the interval. These bifurcations occur in 1-pf of analytic maps at values of the parameter tending to a limit with the asymptotically geometric ratio 1 /4.6692 ....In this paper we indicate the main steps of a proof that the same is true for 1-pf of analytic maps from ⁿ to ⁿ , whose restriction to ⁿ is real.Work supported by the Fonds National Suisse, and by the National Science Foundation under Grant PHY-79-16812.     

Geometry and quantization

Gravity is treated geometrically in terms of nonlinear realizations ofGL(4, ) with particular reference to almost complex structures. This approach is used to carry out a Bargmann-Segal type quantization of space-time via the vector and spinor structures of the tangent space that formulates the theory of measurement as a quantum theory quantized in terms of a basic unit of length that appears in a new uncertainty relation. The theory is also used to discuss the gauge conditions for quantum gravity and the Kostant theory of quantization applied using a line bundle with structure groupGL(2, )/SL(2, ).     

ℂP 2 as a gravitational Instanton

We compare some of the properties of P ² with those of the SU(2) Yang-Mills Instanton and conclude that P ² may be regarded as a gravitational pseudoparticle surrounded by an event horizon.     

Direct evaluation of iwasawa and triangle decompositions for the real forms of lie algebrasgl(n + 1, ℂ)

The Iwasawa and triangle decompositions for any real form of Lie algebrasgl(n + 1, ) are given. Construction of these decompositions is based on the explicit calculation of the Cartna automorphisms with the help of which the real forms of Lie algebrasgl(n + 1, ) are defined.     

Quantum Riemann surfaces: II. The discrete series

We continue our study of noncommutative deformations of two-dimensional hyperbolic manifolds which we initiated in Part I. We construct a sequence of ^*-algebras which are quantizations of a compact Riemann surface of genus g corresponding to special values of the Planck constant. These algebras are direct integrals of finite-dimensional ^*-algebras.Supported by DOE under Grant DE-FG02-88ER25065.     

Geodesic Incompleteness in the ℂP 1 Model on a Compact Riemann Surface

It is proved that the moduli space of static solutions of the P ¹ model on spacetime ×, where is any compact Riemann surface, is geodesically incomplete with respect to the metric induced by the kinetic energy functional. The geodesic approximation predicts, therefore, that lumps can collapse and form singularities in finite time in these models.     

Common tangent space ℝ 1 4 fromU(2) charges

It is discussed how a common space-time can be constructed from a proposed hiddenU(2) world. Schrödinger's idea to obtain discrete eigenvalues by solving the Maxwell equations for the fieldF on compact spaces without boundaries is modified by orthogonality and identification concepts for the potentialsA. Using residue classes with respect to the metric (Clifford algebra), a common spinor space ⁴=RL and a common Minkowski tangent space ₁ ⁴ are bilinearly constructed from tangent spaces ofU(2) individuals [U(2) manifolds with orthogonal potentials]. The space constructed has the following properties. (1) There are algebraic elements for the identification ofU(2) individuals from ₁ ⁴ as spinors and vectorsA. (2) The transfer of the potentials fromU(2) via ⁴ to ₁ ⁴ is linear. (3) The hiddenU(2) content of the left- and right-handed spaces (L, R) is quite different. The potentials on U(2) individuals are transformed into complex wave functions on the spinor space and into 1-formsA on ₁ ⁴ that can be enlarged to gauge potentials. The construction is discussed from an old point of view of Einstein's, starting with the electric charge as the primary concept for quantum theory. The construction of the tangent space ₁ ⁴ does not depend on a preceding introduction of any points (uncertainty). The identity problem of the interpretation of the quantum theory is discussed in some detail. It is indicated how the algebraic, partiallyad hoc constructions can give a rigid frame for further analytical work.     

Nontrivial extensions of a representation of the Poincaré group with mass and helicity zero by its tensorial product

Let U(a, ) be a representation of the Poincaré group with mass and helicity zero, realized in the space of C -functions with compact support on ³, without the origin. Let U ⁽²⁾(a, ) denote the tensorial product of U(a, ) by itself. We explicitly determine the cocycles of extension of U(a, ) by U ⁽²⁾(a, ) and we prove that the nontrivial cohomology is indexed by (u(),),u()D ¹ _]0,3\,.     

Are there critical points on the boundaries of singular domains?

Generalizing a result of E. Ghys, we prove a general theorem that implies that if a rational functionf of the Riemann sphere of degree 2 leaves invariant a singular domainC (a disk or a ring) on which the rotation number off satisfies a diophantine condition, provided that on f is injective, then each boundary component ofC contains critical point off. The injectivity condition is always satisfied for singular disks associated to linearizable periodic elliptic points off(z)=z ⁿ +a, withn,n2 anda. We also show that the singular disks, associated to periodic elliptic points off(z)=e ^az that satisfy a diophantine condition, are unbounded in . In the end of the paper, we give a survey of the theory of iteration of entire functions of .