Morita equivalence of smooth noncommutative tori

We show that in the generic case the smooth noncommutative tori associated with two n × n real skew-symmetric matrices are Morita equivalent if and only if the matrices are in the same orbit of the natural SO(n, n∣Z) action. This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada, held by the first named author.     

Relative Regular Modules. Applications to von Neumann Regular Rings

We use the concept of a regular object with respect to another object in an arbitrary category, in order to obtain the transfer of regularity in the sense of Zelmanowitz between the categories R −mod and S −mod, when S is an excellent extension of the ring R. Consequently, if S is an excellent extension of the ring R, then S is von Neumann regular ring if and only if R is also von Neumann regular ring. In the second part, using relative regular modules, we give a new proof of a classical result: the von Neumann regular property of a ring is Morita invariant.     

Cohomogeneity One Manifolds and Self-Maps of Nontrivial Degree

We construct natural self-maps of compact cohomogeneity one manifolds and compute their degrees and Lefschetz numbers. On manifolds with simple cohomology rings this yields relations between the order of the Weyl group and the Euler characteristic of a principal orbit. As examples we determine all cohomogeneity one actions on irreducible Riemannian symmetric spaces of compact type that lead to self-maps of degree ≠ −1; 0; 1. We derive explicit formulas for new coordinate polynomial self-maps of the compact matrix groups SU(3), SU(4), and SO(2n). For SU(3) we determine precisely which integers can be realized as degrees of self-maps. Supported by a DFG Heisenberg scholarship and DFG priority program SPP 1154.     

Erasure Decoding for Gabidulin Codes

We present a new approach of the decoding algorithm for Gabidulin Codes. In the same way as efficient erasure decoding for Generalized Reed Solomon codes by using the structure of the inverse of the VanderMonde matrices, we show that, the erasure(t erasures mean that t components of a code vector are erased) decoding Gabidulin code can be seen as a computation of three matrice and an affine permutation, instead of computing an inverse from the generator or parity check matrix. This significantly reduces the decoding complexity compared to others algorithms. For t erasures with t ≤ r, where r = n − k, the erasure algorithm decoding for Gab _{n, k} (g) Gabidulin code compute the t symbols by simple multiplication of three matrices. That requires rt + r(k − 1) Galois field multiplications, t(r − 1) + (t + r)k field additions, r ² + r(k + 1) field negations and t(k + 1) field inversions.     

Weyl’s theorem for algebraically wF(p, r, q) operators with p, r?>?0 and q?≥?1

If T (or T*) is an algebraically wF(p, r, q) operator with p, r > 0 and q ≥ 1 acting in an infinite-dimensional separable Hilbert space, then we prove that Weyl’s theorem holds for f(T) for any f ∈ Hol(σ(T)), where Hol(σ(T)) is the set of all analytic functions in an open neighborhood of σ(T). Moreover, if T* is a wF(p, r, q) operator with p, r > 0 and q ≥ 1, then the a-Weyl’s theorem holds for f(T). In addition, if T (or T*) is an algebraically wF(p, r, q) operator with p, r > 0 and q ≥ 1, then we establish the spectral mapping theorems for the Weyl spectrum and for the essential approximate point spectrum of T for any f ∈ Hol(σ(T)), respectively. Finally, we examine the stability of Weyl’s theorem and the a-Weyl’s theorem under commutative perturbations by finite-rank operators.     

Noetherian Skew Inverse Power Series Rings

We study skew inverse power series extensions R[[y ^− 1; τ, δ]], where R is a noetherian ring equipped with an automorphism τ and a τ-derivation δ. We find that these extensions share many of the well known features of commutative power series rings. As an application of our analysis, we see that the iterated skew inverse power series rings corresponding to nth Weyl algebras are complete, local, noetherian, Auslander regular domains whose right Krull dimension, global dimension, and classical Krull dimension are all equal to 2n.     

Morita Invariance of the Filter Dimension and of the Inequality of Bernstein

It is proved that the filter dimension is Morita invariant. A direct consequence of this fact is the Morita invariance of the inequality of Bernstein: if an algebra A is Morita equivalent to the ring of differential operators on a smooth irreducible affine algebraic variety X of dimension n ≥ 1 over a field of characteristic zero then the Gelfand–Kirillov dimension for all nonzero finitely generated A-modules M. In fact, a stronger result is proved, namely, a Morita invariance of the holonomic number for finitely generated algebra. A direct consequence of this fact is that an analogue of the inequality of Bernstein holds for the (simple) rational Cherednik algebras H _c for integral c: for all nonzero finitely generated H _c -modules M. For these class of algebras, it gives an affirmative answer to a question of Ken Brown about symplectic reflection algebras. Presented by Alain Verschoren.     

Schur–Weyl duality for higher levels

We extend Schur–Weyl duality to an arbitrary level l ≥ 1, level one recovering the classical duality between the symmetric and general linear groups. In general, the symmetric group is replaced by the degenerate cyclotomic Hecke algebra over parametrized by a dominant weight of level l for the root system of type A_∞. As an application, we prove that the degenerate analogue of the quasi-hereditary cover of the cyclotomic Hecke algebra constructed by Dipper, James and Mathas is Morita equivalent to certain blocks of parabolic category for the general linear Lie algebra.      

Ideals in Morita Rings and Morita Semigroups

We characterize the lattice of all ideals of a Morita ring （semigroup） when the corresponding pair of rings （semigroups） in the Morita context are Morita equivalent s-unital （like-unitv） rings （semigroups）.     

Locally Closed Semirings

 We call a semiring S locally closed if for all a ∈ S there is some integer k such that 1 + a + ⋯ + a ^k  =1 + a + ⋯ + a ^k + 1 . In any locally closed semiring we may define a star operation a ↦ a ^*, where a ^* is the above finite sum. We prove that when S is locally closed and commutative, then S is an iteration semiring.     

Partitioning Posets

Given a poset P = (X, ≺ ), a partition X ₁, ..., X _k of X is called an ordered partition of P if, whenever x ∈ X _i and y ∈ X _j with x ≺ y, then i ≤ j. In this paper, we show that for every poset P = (X, ≺ ) and every integer k ≥ 2, there exists an ordered partition of P into k parts such that the total number of comparable pairs within the parts is at most (m − 1)/k, where m ≥ 1 is the total number of edges in the comparability graph of P. We show that this bound is best possible for k = 2, but we give an improved bound, , for k ≥ 3, where c(k) is a constant depending only on k. We also show that, given a poset P = (X, ≺ ) and an integer 2 ≤ k ≤ |X|, we can find an ordered partition of P into k parts that minimises the total number of comparable pairs within parts in time polynomial in the size of P. We prove more general, weighted versions of these results. Supported by an EPSRC doctoral training grant.     

Approximation properties of the Generalized Finite Element Method

In this paper, we have obtained an approximation result in the Generalized Finite Element Method (GFEM) that reflects the global approximation property of the Partition of Unity (PU) as well as the approximability of the local approximation spaces. We have considered a GFEM, where the underlying PU functions reproduce polynomials of degree l. With the space of polynomials of degree k serving as the local approximation spaces of the GFEM, we have shown, in particular, that the energy norm of the GFEM approximation error of a smooth function is O(h ^l + k ). This result cannot be obtained from the classical approximation result of GFEM, which does not reflect the global approximation property of the PU.     

Application of a Bernstein type inequality to rational interpolation in the Dirichlet space

We prove a Bernstein type inequality involving the Bergman and Hardy norms for rational functions in the unit disk \mathbb D {\mathbb D} that have at most n poles all of which are outside the disk \frac1r \mathbb D \frac{1}{r} {\mathbb D} , 0 < r < 1. The asymptotic sharpness of this inequality is shown as n → ∞ and r → 1—. We apply our Bernstein type inequality to an efficient Nevanlinna–Pick interpolation problem in the standard Dirichlet space constrained by the H2-nom. Bibliography: 14 titles.     

Submanifolds of Weyl Flat Manifolds

 We consider compact Weyl submanifolds of Weyl flat manifolds with special attention on compact Einstein-Weyl hypersurfaces. In particular, in the last part of the paper, we study Weyl submanifolds of special noncompact manifolds, called PC-manifolds.     

Locally Closed Semirings

 We call a semiring S locally closed if for all a ∈ S there is some integer k such that 1 + a + ⋯ + a ^k  =1 + a + ⋯ + a ^k + 1 . In any locally closed semiring we may define a star operation a ↦ a ^*, where a ^* is the above finite sum. We prove that when S is locally closed and commutative, then S is an iteration semiring. Partially supported by grant no. T30511 from the National Foundation of Hungary for Scientific Research and the Austrian–Hungarian Bilateral Research and Development Fund, no. A-4/1999, and by the Austrian–Hungarian Action Foundation. Partially supported by the Austrian–Hungarian Bilateral Research and Development Fund, no. A-4/1999, and by the Austrian–Hungarian Action Foundation. Received March 16, 2001     

à propos de l’indice de composition des nombres

 We define the index of composition λ(n) of an integer n ⩾ 2 as λ(n) = log n/log γ(n), where γ(n) stands for the product of the primes dividing n, and first establish that λ and 1/λ both have asymptotic mean value 1. We then establish that, given any ɛ > 0 and any integer k ⩾ 2, there exist infinitely many positive integers n such that . Considering the distribution function F(z,x) := #{n < x : λ(n) > z}, we prove that, given 1 < z < 2 and ɛ > 0, then, if x is sufficiently large,

À propos de l’indice de composition des nombres

 We define the index of composition λ(n) of an integer n ⩾ 2 as λ(n) = log n/log γ(n), where γ(n) stands for the product of the primes dividing n, and first establish that λ and 1/λ both have asymptotic mean value 1. We then establish that, given any ɛ > 0 and any integer k ⩾ 2, there exist infinitely many positive integers n such that . Considering the distribution function F(z,x) := #{n < x : λ(n) > z}, we prove that, given 1 < z < 2 and ɛ > 0, then, if x is sufficiently large,

On some imbedding relations between certain sequence spaces

We introduce a space of sequences l_p^l \ell_p^\lambda of nonabsolute type, which is a p-normed space and a BK-space in the cases of 0 < p < 1 and 1 ≤ p < ∞; respectively. Further, we deduce some imbedding relations and construct a basis for the space l_p^l \ell_p^\lambda , where 1 ≤ p < ∞.     

The Stochastic Wave Equation with Multiplicative Fractional Noise: A Malliavin Calculus Approach

We consider the stochastic wave equation with multiplicative noise, which is fractional in time with index H > 1/2, and has a homogeneous spatial covariance structure given by the Riesz kernel of order α. The solution is interpreted using the Skorohod integral. We show that the sufficient condition for the existence of the solution is α > d − 2, which coincides with the condition obtained in Dalang (Electr J Probab 4(6):29, 1999), when the noise is white in time. Under this condition, we obtain estimates for the p-th moments of the solution, we deduce its H?lder continuity, and we show that the solution is Malliavin differentiable of any order. When d ≤ 2, we prove that the first-order Malliavin derivative of the solution satisfies a certain integral equation.     

Spectral estimates for Schrödinger operators with sparse potentials on graphs

A construction of “sparse potentials,” suggested by the authors for the lattice \mathbbZ^d {\mathbb{Z}^d} , d > 2, is extended to a large class of combinatorial and metric graphs whose global dimension is a number D > 2. For the Schr?dinger operator − Δ − αV on such graphs, with a sparse potential V, we study the behavior (as α → ∞) of the number N_(−Δ − αV) of negative eigenvalues of − Δ − αV. We show that by means of sparse potentials one can realize any prescribed asymptotic behavior of N_(−Δ − αV) under very mild regularity assumptions. A similar construction works also for the lattice \mathbbZ² {\mathbb{Z}^2} , where D = 2. Bibliography: 13 titles.