Nonreduced extended affine root systems of nullity 3

In 1985 K. Saito [S] introduced the concept of an extended affine root system (EARS). His study of these root systems was motivated by his interest in singularities. Later in [AABGP], this notion played an important role in the study of extended affine Lie algebras. Saito classified the EARS's of nullity ≤ 2 which have the further property that the quotient modulo a "marking" is reduced. In [AABGP], a construction was given of all EARS's, and this was used to give a classification of EARS's of reduced type. However, when the EARS was not reduced, the isomorphism problem for the construction is quite difficult and the classification was only done for EARS'S with nullity ≤ 2. The present paper extends this classification to nullity 3.     

Kennzeichnung Verallgemeinerter Euklidischer Ebenen Durch Ähnlichkeitsrelationen

We show that generalized euclidean planes (or metric affine planes) can be characterized by a similarity relation on the set of all non-collinear point triples or of the set of all point triples of an affine plane. In the second case most of the properties of the relation can be formulated independent of the geometrical structure.     

Specializations of nonsymmetric Macdonald–Koornwinder polynomials

The purpose of this article is to work out the details of the Ram–Yip formula for nonsymmetric Macdonald–Koornwinder polynomials for the double affine Hecke algebras of not-necessarily reduced affine root systems. It is shown that the \(t\rightarrow 0\) equal-parameter specialization of nonsymmetric Macdonald polynomials admits an explicit combinatorial formula in terms of quantum alcove paths, generalizing the formula of Lenart in the untwisted case. In particular, our formula yields a definition of quantum Bruhat graph for all affine root systems. For mixed type, the proof requires the Ram–Yip formula for the nonsymmetric Koornwinder polynomials. A quantum alcove path formula is also given at \(t\rightarrow \infty \). As a consequence, we establish the positivity of the coefficients of nonsymmetric Macdonald polynomials under this limit, as conjectured by Cherednik and the first author. Finally, an explicit formula is given at \(q\rightarrow \infty \), which yields the p-adic Iwahori–Whittaker functions of Brubaker, Bump, and Licata.     

Torsion de groupes munis d'une donnée radicielle

We consider groups endowed with root data associated with non-necessarily finite root systems. We generalise to these groups the twisting methods of Chevalley groups initiated by Steinberg and Ree. The resulting theorem (proved in 1988) can be applied to Kac–Moody groups: see for instance two papers published by J. Ramagge in 1995 [J. Ramagge, On certain fixed point subgroups of affine Kac–Moody groups, J. Algebra 171 (2) (1995) 473–514; J. Ramagge, A realization of certain affine Kac–Moody groups of types II and III, J. Algebra 171 (3) (1995) 713–806].     

Jordan centroids

Central simple triples are important for the classification of prime Jordan triples of Clifford type in arbitrary characterstics. For triples and pairs (or even for unital Jordan algebras of characteristic 2), there is no workable notion of center, and the concept of “central simple” system must be understood as “centroid-simple”. The centroid of a Jordan system (algebra, triple, or pair) consists of the “natural” scalars for that system: the largest unital, commutative ring Γ such that the system can be considered as a quadratic Jordan system over Γ. In this paper we will characterize the centroids of the basic simple Jordan algebras, triples, and pairs. (Consideration of the tangled ample outer ideals in Jordan algebras of quadratic forms will be left to a separate paper.) A powerful tool is the Eigenvalue Lemma, that a centroidal transformation on a prime system over φ which has an eigenvalue α in φ must actually be scalar multiplication by α. An important consequence is that a prime system over φ with reduced elements P_xJ = φx (or which grows reduced elements under controlled conditions) must already be central, Γ = φ.     

Difference Fourier transforms for nonreduced root systems

In the first part of the paper kernels are constructed which meromorphically extend the Macdonald-Koornwinder polynomials in their degrees. In the second part the kernels associated with rank one root systems are used to define nonsymmetric variants of the spherical Fourier transform on the quantum SU(1,1) group. Related Plancherel and inversion formulas are derived using double affine Hecke algebra techniques.     

Bispectral quantum Knizhnik�CZamolodchikov equations for arbitrary root systems

The bispectral quantum Knizhnik–Zamolodchikov (BqKZ) equation corresponding to the affine Hecke algebra H of type A _N-1 is a consistent system of q-difference equations which in some sense contains two families of Cherednik’s quantum affine Knizhnik–Zamolodchikov equations for meromorphic functions with values in principal series representations of H. In this paper, we extend this construction of BqKZ to the case where H is the affine Hecke algebra associated with an arbitrary irreducible reduced root system. We construct explicit solutions of BqKZ and describe its correspondence to a bispectral problem involving Macdonald’s q-difference operators.     

中约化子空间上的仿射与伪仿射对偶小波标架

本文讨论中约化子空间上的仿射(伪仿射)对偶小波标架.我们建立了仿射系与伪仿射系之间的一个标架 和对偶标架保持定理,并且在没有任何衰减性假设的条件下获得了仿射(伪仿射)对偶小波标架在傅立叶域上的一个刻画.进一步, 我们也给出了仿射Parseval标架在傅立叶域上的刻画.       

A presentation for reduced extended affine weyl groups

In 1985 K. Saito [Sal] introduced the concept of an extended affine Weyl group (EAWG), the Weyl group of an extended affine root system (EARS). In [A2, Section 5J, we gave a presentation called “a presentation by conjugation” for the class of EAWGs of index zero, a subclass of EAWGs. In this paper we will give a presentation wh.ich we call a “generalized present.ation by conjugation” for the class of reduced EAWGs. If the extended affine Weyl group is of index zero this presentation reduces to “a presentation by conjugation”. Our main result states that when the nullity of the EARS is 2, these two presentations coincide that is, EAWGs of nullity 2 have “a presentation by conjugation”. In [ST] another presentation for EAWGs of nullity 2 is given.     

Computation of Affine Covariants of Quadratic Bivariate Differential Systems

This paper deals with affine covariants of autonomous differential systems. The main result is the construction of a minimal system of generators of the algebra of affine covariants of quadratic bivariate differential systems which is helpful in qualitative and numerical study. To this end, we establish a theorem (true for general systems of dimension n and degree m) which provides a procedure of construction of systems of generators for affine covariants from those of center-affine invariants. After applying this theorem to the case n=m=2 we give the expansions of the obtained affine covariants in terms of center-affine covariants. All algorithms constitute the package SIB.     

Classification of partitions of all triples on ten points into copies of Fano and affine planes

We continue our study of partitions of the full set of triples chosen from a v-set into copies of the Fano plane PG(2,2) (Fano partitions) or copies of the affine plane AG(2,3) (affine partitions) or into copies of both of these planes (mixed partitions). The smallest cases for which such partitions can occur are v=8 where Fano partitions exist, v=9 where affine partitions exist, and v=10 where both affine and mixed partitions exist. The Fano partitions for v=8 and the affine partitions for v=9 and 10 have been fully classified, into 11, two and 77 isomorphism classes, respectively. Here we classify (1) the sets of i pairwise disjoint affine planes for i=1,…,7, and (2) the mixed partitions for v=10 into their 22 isomorphism classes. We consider the ways in which these partitions relate to the large sets of AG(2,3).     

Mathematical modelling and theoretical analysis of nonholonomic kinematic systems with a class of rheonomous affine constraints

In this paper, we deal with kinematic control systems subject to a class of rheonomous affine constraints. We first define A-rheonomous affine constraints and explain a geometric representation method for them. Next, we derive a necessary and sufficient condition for complete nonholonomicity of the A-rheonomous affine constraints. Then, a mathematical model of nonholonomic kinematic systems with A-rheonomous affine constraints (NKSARAC), which is included in the class of nonlinear affine control systems, is introduced. Theoretical analysis on linearly-approximated systems and accessibility for the NKSARAC is also shown. Finally, we apply the results to some physical examples in order to confirm the effectiveness of them.     

Root subsystems of loop extensions

We completely classify the real root subsystems of root systems of loop algebras of Kac–Moody Lie algebras. This classification involves new notions of “admissible subgroups” of the coweight lattice of a root system Ψ, and “scaling functions” on Ψ. Our results generalise and simplify earlier work on subsystems of real affine root systems.     

Direct Unions of Lie Tori (Realization of Locally Extended Affine Lie Algebras)

In this work, we consider realizations of locally extended affine Lie algebras, in the level of core modulo center. We provide a framework similar to the one for extended affine Lie algebras by “direct unions.” Our approach suggests that the direct union of existing realizations of extended affine Lie algebras, in a rigorous mathematical sense, would lead to a complete realization of locally extended affine Lie algebras, in the level of core modulo center. As an application of our results, we realize centerless cores of locally extended affine Lie algebras with specific root systems of types A₁, B, C, and BC.     

Extended affine Weyl groups and Frobenius manifolds

We define certain extensions of affine Weyl groups (distinct from these considered by K. Saito [S1] in the theory of extended affine root systems), prove an analogue of Chevalley Theorem for their invariants, and construct a Frobenius structure on their orbit spaces. This produces solutions F(t₁, ..., t_n) of WDVV equations of associativity polynomial in t₁, ..., t_n-1, exp t_n.     

Weak(quasi-)affine bi-frames for reducing subspaces of L~2(R~d)

Since a frame for a Hilbert space must be a Bessel sequence, many results on(quasi-)affine bi-frame are established under the premise that the corresponding(quasi-)affine systems are Bessel sequences. However,it is very technical to construct a(quasi-)affine Bessel sequence. Motivated by this observation, in this paper we introduce the notion of weak(quasi-)affine bi-frame(W(Q)ABF) in a general reducing subspace for which the Bessel sequence hypothesis is not needed. We obtain a characterization of WABF, and prove the equivalence between WABF and WQABF under a mild condition. This characterization is used to recover some related known results in the literature.     

A new algorithm for Chebyshev minimum-error multiplication of reduced affine forms

Reduced affine arithmetic (RAA) eliminates the main deficiency of the standard affine arithmetic (AA), i.e. a gradual increase of the number of noise symbols, which makes AA inefficient in a long computation chain. To further reduce overestimation in RAA computation, a new algorithm for the Chebyshev minimum-error multiplication of reduced affine forms is proposed. The algorithm yields the minimum Chebyshev-type bounds and works in linear time, which is asymptotically optimal. We also propose a simplified \(\mathcal {O}(n\log n)\) version of the algorithm, which performs better for low dimensional problems. Illustrative examples show that the presented approach significantly improves solutions of many numerical problems, such as the problem of solving parametric interval linear systems or parametric linear programming, and also improves the efficiency of interval global optimisation.     

Simple left-symmetric algebras with solvable Lie algebra

Left-symmetric algebras (LSAs) are Lie admissible algebras arising from geometry. The leftinvariant affine structures on a Lie groupG correspond bijectively to LSA-structures on its Lie algebra. Moreover if a Lie group acts simply transitively as affine transformations on a vector space, then its Lie algebra admits a complete LSA-structure. In this paper we studysimple LSAs having only trivial two-sided ideals. Some natural examples and deformations are presented. We classify simple LSAs in low dimensions and prove results about the Lie algebra of simple LSAs using a canonical root space decomposition. A special class of complete LSAs is studied.     

不确定广义系统的仿射二次稳定

本文讨论了具有仿射型不确定性的广义系统稳定问题，给出了不确定广义系统的仿射二次稳定定义及仿射二次H∞性能指标定义，利用线性矩阵不等式给出了不确定广义系统的仿射二次稳定的充分条件和系统具有仿射二次H∞性能指标的充分条件。     

On the Lax Pairs of the Symmetric Painlevé Equations

The symmetric forms of the Painlevé equations are a sequence of nonlinear dynamical systems in N + 1 variables that admit the action of an extended affine Weyl group of type     , as shown by Noumi and Yamada. They are equivalent to the periodic dressing chains studied by Veselov and Shabat, and by Adler. In this paper, a direct derivation of the symmetries of a corresponding sequence of ( N + 1) × ( N + 1) matrix linear systems (Lax pairs) is given. The action of the generators of the extended affine Weyl group of type     on the associated Lax pairs is realized through a set of transformations of the eigenfunctions, and this extends to an action of the whole group.