Primitive Ideals and Automorphisms of Quantum Matrices

Let be a field and q be a nonzero element of that is not a root of unity. We give a criterion for 〈0〉 to be a primitive ideal of the algebra of quantum matrices. Next, we describe all height one primes of ; these two problems are actually interlinked since it turns out that 〈0〉 is a primitive ideal of whenever has only finitely many height one primes. Finally, we compute the automorphism group of in the case where m ≠ n. In order to do this, we first study the action of this group on the prime spectrum of . Then, by using the preferred basis of and PBW bases, we prove that the automorphism group of is isomorphic to the torus when m ≠ n and (m,n) ≠ (1, 3),(3, 1). This research was supported by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme and by Leverhulme Research Interchange Grant F/00158/X.     

Approximating periodic functions in the uniform metric by Jackson type polynomials

Let C be the space of continuous 2π-periodic functions f with the norm . Let , where , be the Jackson polynomials of a function f, E _n (f) be the best approximation of f in the space C by trigonometric polynomials of order n, and let , be the function trigonometrically conjugate to the primitive of f. The paper establishes results of the following types:

Deep Matrix Algebras of Finite Type

Deep matrix algebras based on a set over a ring are introduced and studied by McCrimmon when has infinite cardinality. Here, we construct a new -module that is faithful for of any cardinality. For a field of arbitrary characteristic and , is studied in depth. The algebra is shown to possess a unique proper non-zero ideal, isomorphic to . This leads to a new and interesting simple algebra, , the quotient of by its unique ideal. Both and the quotient algebra are shown to have centers isomorphic to . Via the new faithful representation, all automorphisms of are shown to be inner in the sense of Definition 18.Presented by D. Passman.     

Beurling Zeta Functions, Generalised Primes, and Fractal Membranes

We study generalised prime systems , with tending to infinity) and the associated Beurling zeta function . Under appropriate assumptions, we establish various analytic properties of , including its analytic continuation, and we characterise the existence of a suitable generalised functional equation. In particular, we examine the relationship between a counterpart of the Prime Number Theorem (with error term) and the properties of the analytic continuation of . Further we study ‘well-behaved’ g-prime systems, namely, systems for which both the prime and integer counting function are asymptotically well-behaved. Finally, we show that there exists a natural correspondence between generalised prime systems and suitable orders on . Some of the above results are relevant to the second author’s theory of ‘fractal membranes’, whose spectral partition functions are given by Beurling-type zeta functions, as well as to joint work of that author and R. Nest on zeta functions attached to quasicrystals.The work of M. L. Lapidus was partially supported by the U. S. National Science Foundation under grant DMS-0070497.     

On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems

We consider the computation of stable approximations to the exact solution of nonlinear ill-posed inverse problems F(x) = y with nonlinear operators F : X → Y between two Hilbert spaces X and Y by the Newton type methods

Hilbert Space Representations of Cross Product Algebras II

In this paper, we study and classify Hilbert space representations of cross product -algebras of the quantized enveloping algebra with the coordinate algebras of the quantum motion group and of the complex plane, and of the quantized enveloping algebra with the coordinate algebras of the quantum group and of the quantum disc. Invariant positive functionals and the corresponding Heisenberg representations are explicitly described.Presented by S.L. Woronowicz.     

Chains and antichains in partial orderings

We study the complexity of infinite chains and antichains in computable partial orderings. We show that there is a computable partial ordering which has an infinite chain but none that is or , and also obtain the analogous result for antichains. On the other hand, we show that every computable partial ordering which has an infinite chain must have an infinite chain that is the difference of two sets. Our main result is that there is a computably axiomatizable theory K of partial orderings such that K has a computable model with arbitrarily long finite chains but no computable model with an infinite chain. We also prove the corresponding result for antichains. Finally, we prove that if a computable partial ordering has the feature that for every , there is an infinite chain or antichain that is relative to , then we have uniform dichotomy: either for all copies of , there is an infinite chain that is relative to , or for all copies of , there is an infinite antichain that is relative to .     

Three-dimensional sets with small sumset

We describe the structure of three dimensional sets of lattice points, having a small doubling property. Let be a finite subset of ℤ³ such that dim = 3. If and , then lies on three parallel lines. Moreover, for every three dimensional finite set that lies on three parallel lines, if , then is contained in three arithmetic progressions with the same common difference, having together no more than terms. These best possible results confirm a recent conjecture of Freiman and cannot be sharpened by reducing the quantity υ or by increasing the upper bounds for .     

Precise Asymptotics in the Law of the Iterated Logarithm of Moving-Average Processes

In this paper, we discuss the moving-average process Xk = ∑i=-∞ ^∞ ai＋kεi, where {εi;-∞ 〈 i 〈 ∞} is a doubly infinite sequence of identically distributed ψ-mixing or negatively associated random variables with mean zeros and finite variances, {ai;-∞ 〈 i 〈 -∞） is an absolutely solutely summable sequence of real numbers.     

Bass Numbers of Local Cohomology Modules with Respect to an Ideal

Let be a commutative Noetherian local ring and let be an ideal of R. We give some inequalities between the Bass numbers of an R–module and those of its local cohomology modules with respect to . As an application of these inequalities, we recover results of Delfino-Marley and Kawasaki by showing that for a minimax R-module M and for any non-negative integer i, the Bass numbers of the ith local cohomology module are finite if one of the following holds:

Abelian crossed products and scalar extensions of central simple algebras

Let be a central simple algebra over a field k and G be a commutative group with . It is proved that there exists a regular field extension E/k preserving indices of k-algebras such that is a crossed product with the group G. Bibliography: 11 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 356, 2008, pp. 179-188.     

Quelques Résultats sur les Algèbres Flexibles Préhilbertiennes sans Diviseurs de Zéro Vérifiant ∥a 2∥ = ∥a∥2

Résumé. Soit A une algèbre réelle. On suppose que l’espace vectoriel A est muni d’une norme ∥.∥ préhilbertienne vérifiant ∥a ²∥ = ∥a∥² pour tout . Si A est flexible, sans diviseurs de zéro et de dimension ≤ 4, alors A est isomorphe à ou , ce qui généralise un théorème d’El-Mallah [1]. Si A est flexible, sans diviseurs de zéro, contenant un idempotent central et vérifiant la propriété d’Osborn, alors A est de dimension finie et isomorphe à , ou . Enfin nous montrons qu’une algèbre normée préhilbertienne unitaire d’unité e telle que ∥e∥ = 1 est flexible et vérifie ∥a ²∥ = ∥ a∥².

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Let A be a real algebra. Assuming that a vector space A is endowed with a pre-Hilbert norm ∥.∥ satisfying ∥a ²∥ = ∥a∥² for all . If A is flexible, without divisor of zero and of a dimension ≤ 4, then A is isomorphic to or , which generalize El-Mallah’s theorem [1]. If A is flexible, without divisor of zero, containing a central idempotent and satisfying Osborn’s properties, then A is finite dimensional and isomorphic to , or . Finally we prove that a normed pre-Hilbert algebra with unit e such that ∥e∥ = 1 is flexible and satisfies ∥a ²∥ = ∥a∥².

     

On the Number of Topological Types Occurring in a Parameterized Family of Arrangements

Let be an o-minimal structure over ℝ, a closed definable set, and

Exotic smooth structures on small 4-manifolds

Let M be either or . We construct the first example of a simply-connected irreducible symplectic 4-manifold that is homeomorphic but not diffeomorphic to M. Dedicated to Ronald J. Stern on the occasion of his sixtieth birthday Mathematics Subject Classification (2000)  Primary 57R55; Secondary 57R17, 57M05     

Some remarks on systems of elliptic equations doubly critical in the whole $${\mathbb{R}^N}$$

We study the existence of different types of positive solutions to problem

Indecomposable Elements in K1 of a Smooth Projective Variety

Let X be a smooth projective variety over the complex numbers. We consider the cohomology of the sheaves and arising from Deligne–Beilinson cohomology and the Hodge filtration on the singular cohomology of X. We show that one can identify with the image of the truncated regulator map c¯2,1. In particular, this implies that is countable. Since this group is a direct summand of coker , this gives a partial answer to Voisin's conjecture that cocker() is countable. In the case of X a surface, we prove that the Albanese kernel T(X) is isomorphic to the group of global sections of if and only if pg=0.     

Positive weight quadrature on the sphere and monotonicities of Jacobi polynomials

In 2000, Reimer proved that a positive weight quadrature rule on the unit sphere has the property of quadrature regularity. Hesse and Sloan used a related property, called Property (R) in their work on estimates of quadrature error on . The constants related to Property (R) for a sequence of positive weight quadrature rules on can be estimated by using a variation on Reimer’s bounds on the sum of the quadrature weight within a spherical cap, with Jacobi polynomials of the form , in combination with the Sturm comparison theorem. A recent conjecture on monotonicities of Jacobi polynomials would, if true, provide improved estimates for these constants. The work was carried out while the author was a PhD student at the School of Mathematics, University of New South Wales.     

Another hybrid conjugate gradient algorithm for unconstrained optimization

Another hybrid conjugate gradient algorithm is subject to analysis. The parameter β _k is computed as a convex combination of (Hestenes-Stiefel) and (Dai-Yuan) algorithms, i.e. . The parameter θ _k in the convex combination is computed in such a way so that the direction corresponding to the conjugate gradient algorithm to be the Newton direction and the pair (s _k , y _k ) to satisfy the quasi-Newton equation , where and . The algorithm uses the standard Wolfe line search conditions. Numerical comparisons with conjugate gradient algorithms show that this hybrid computational scheme outperforms the Hestenes-Stiefel and the Dai-Yuan conjugate gradient algorithms as well as the hybrid conjugate gradient algorithms of Dai and Yuan. A set of 750 unconstrained optimization problems are used, some of them from the CUTE library.      

Nilpotent Bicone and Characteristic Submodule of a Reductive Lie Algebra

Let be a finite-dimensional complex reductive Lie algebra and S() its symmetric algebra. The nilpotent bicone of is the subset of elements (x, y) of whose subspace generated by x and y is contained in the nilpotent cone. The nilpotent bicone is naturally endowed with a scheme structure, as nullvariety of the augmentation ideal of the subalgebra of generated by the 2-order polarizations of invariants of . The main result of this paper is that the nilpotent bicone is a complete intersection of dimension , where and are the dimensions of Borel subalgebras and the rank of , respectively. This affirmatively answers a conjecture of Kraft and Wallach concerning the nullcone [KrW2]. In addition, we introduce and study in this paper the characteristic submodule of . The properties of the nilpotent bicone and the characteristic submodule are known to be very important for the understanding of the commuting variety and its ideal of definition. The main difficulty encountered for this work is that the nilpotent bicone is not reduced. To deal with this problem, we introduce an auxiliary reduced variety, the principal bicone. The nilpotent bicone, as well as the principal bicone, are linked to jet schemes. We study their dimensions using arguments from motivic integration. Namely, we follow methods developed by Mustaţǎ in [Mu]. Finally, we give applications of our results to invariant theory.