The Ramifications of the Centres: Quantised Function Algebras at Roots of Unity

This paper continues the study of quantised function algebrasO[G] of a semisimple group G at an lth root of unity . Thesealgebras were introduced by De Concini and Lyubashenko in 1994,and studied further by De Concini and Procesi and by Gordon,amongst others. Our main purpose here is to increase understandingof the finite-dimensional factor algebras O[G](g), for g G.We determine the representation type and block structure ofthese factors, and (for many g) describe them up to isomorphism.A series of parallel results is obtained for the quantised Borelalgebras and . 2000 Mathematical Subject Classification: 16W35,17B37.     

Complex Hyperspherical Equations, Nodal-Partitioning, and First-Excited-State Theorems in ℝ n

Three problems related to the spherical quantum billiard in are considered. In the first, a compact form of the hyperspherical equations leads to their complex contracted representation. Employing these contracted equations, a proof is given of Courant's nodal-symmetry intersection theorem for diagonal eigenstates of spherical-like quantum billiards in . The second topic addresses the first-excited-state theorem for the spherical quantum billiard in . Wavefunctions for this system are given by the product form, ( )Z _q+()Y ⁽ⁿ⁾ , where is dimensionless displacement, is angular-momentum number, qis an integer function of dimension, Z() is either a spherical Bessel function (nodd) or a Bessel function of the first kind (neven) and represents (n– 1) independent angular components. Generalized spherical harmonics are written . It is found that the first excited state (i.e., the second eigenstate of the Laplacian) for the spherical quantum billiard in is n-fold degenerate and a first excited state for this quantum billiard exists which contains a nodal bisecting hypersurface of mirror symmetry. These findings establish the first-excited-state theorem for the spherical quantum billiard in . In a third study, an expression is derived for the dimension of the th irreducible representation (irrep) of the rotation group O(n) in by enumerating independent degenerate product eigenstates of the Laplacian.     

Properties of Quantum Languages

The class of quantum languages Q() over an alphabet is the class of languages accepted by quantum automata. We study properties of Q() and compare Q() with the class of regular languages R(). It is shown that Q() is closed under union, intersection, and reversal but is not closed under complementation, concatenation, or Kleene star. It is also shown that Q() and R() are incomparable. Finally, we prove that L Q() if and only if L admits a transition amplitude function satisfying a certain property and a similar characterization is given for R().     

Kontsevich and Takhtajan Construction of Star Product on the Poisson–Lie Group GL(2)

Comparing the star product defined by Takhtajan on the Poisson–Lie group GL(2) and any star product calculated from the Kontsevich's graphs (any K-star product) on the same group, we show, by direct computation, that the Takhtajan star product on GL(2) can't be written as a K-star product.     

The World According to Quantum Mechanics (Or the 18 Errors of Henry P. Stapp)

Several errors in Stapp's interpretation of quantum mechanics and its application to mental causation (Henry P. Stapp, Quantum theory and the role of mind in nature, Foundations of Physics 31, 1465–1499 (2001)) are pointed out. An interpretation of (standard) quantum mechanics that avoids these errors is presented.     

Local Primitive Causality and the Common Cause Principle in Quantum Field Theory

If (V) is a net of local von Neumann algebras satisfying standard axioms of algebraic relativistic quantum field theory and V ₁ and V ₂ are spacelike separated spacetime regions, then the system ( (V ₁ ), (V ₂ ), ) is said to satisfy the Weak Reichenbach's Common Cause Principle iff for every pair of projections A (V ₁ ), B (V ₂ ) correlated in the normal state there exists a projection C belonging to a von Neumann algebra associated with a spacetime region V contained in the union of the backward light cones of V ₁ and V ₂ and disjoint from both V ₁ and V ₂ , a projection having the properties of a Reichenbachian common cause of the correlation between A and B. It is shown that if the net has the local primitive causality property then every local system ( (V ₁ ), (V ₂ ), ) with a locally normal and locally faithful state and suitable bounded V ₁ and V ₂ satisfies the Weak Reichenbach's Common Cause Principle.     

On Fuzzy Probability Theory

Our main aim from this work is to see which theorems in classical probability theory are still valid in fuzzy probability theory. Following Gudder's approach [Demonestratio Mathematica 31(3), 1998, 235–254; Foundations of Physics, 30, 1663–1678] to fuzzy probability theory, the basic concepts of the theory, that is of fuzzy probability measures and fuzzy random variables (observables), are presented. We show that fuzzy random variables extend the usual ones. Moreover, we prove that for any separable metrizable space, the crisp observables coincide with random variables. Then we prove the existence of a joint observable for any collection of observables, and we prove the weak law of large numbers and the central limit theorem in the fuzzy context. We construct a new definition of almost everywhere convergence. After proving that Gudder's definition implies ours and presenting an example that indicates that the converse is not true, we prove the strong law of large numbers according to this definition.     

Photoelectric Properties of the In/CuIn3Se5 and In/CuGa3Se5 Structures

Using the method of oriented crystallization of a melt (a vertical variant), we grew homogeneous single crystals of the ternary compounds CuIn₃Se₅ and CuGa₃Se₅. Photosensitive structures based on them were created for the first time, the spectral dependences of the relative quantum efficiency of the phototransformation of structures were investigated, and the energy gap width of the ternary compounds was determined. It is established that the CuIn₃Se₅ and CuGa₃Se₅ compounds are materials with direct interband transitions. We show that the structures created can be used as wideband photoconverters of natural radiation.