Z-Graded Lie Superalgebras with Depth One over Fields of Prime Characteristic

Let F be a field of characteristic p > 3, and L = ⊕ _i≥minus;1 L _i the finite-dimensional transitive Z-graded simple Lie superalgebral over F. In this paper L is determined if the Z-graded Lie superalgebra L ₋₁ L ₀ is isomorphic to the L ₋₁⊕pl (L ₋₁) or L ₋₁⊕spl (L ₋₁). Received November 10, 1999, Accepted April 7, 2000     

Leray’s inequality in general multi-connected domains in {\mathbb{R}^n}

Consider the stationary Navier–Stokes equations in a bounded domain whose boundary consists of L + 1 smooth (n − 1)-dimensional closed hypersurfaces Γ₀, Γ₁, . . . , Γ _L , where Γ₁, . . . , Γ _L lie inside of Γ₀ and outside of one another. The Leray inequality of the given boundary data β on plays an important role for the existence of solutions. It is known that if the flux on Γ _i (ν: the unit outer normal to Γ _i ) is zero for each i = 0, 1, . . . , L, then the Leray inequality holds. We prove that if there exists a sphere S in Ω separating in such a way that Γ₁, . . . , Γ _k (1 ≦ k ≦ L) are contained inside of S and that the others Γ _k+1, . . . , Γ _L are outside of S, then the Leray inequality necessarily implies that γ ₁ + · · · +  γ _k =  0. In particular, suppose that there are L spheres S ₁, . . . , S _L in Ω lying outside of one another such that Γ _i lies inside of S _i for all i = 1, . . . , L. Then the Leray inequality holds if and only if γ ₀ = γ ₁ = · · · = γ _L = 0.     

On the Generalized Enveloping Algebra of a Color Lie Algebra

Let G be an abelian group, ε an anti-bicharacter of G and L a G-graded ε Lie algebra (color Lie algebra) over a field of characteristic zero. We prove that for all G-graded, positively filtered A such that the associated graded algebra is isomorphic to the G-graded ε-symmetric algebra S(L), there is a G- graded ε-Lie algebra L and a G-graded scalar two cocycle , such that A is isomorphic to U _ω (L) the generalized enveloping algebra of L associated with ω. We also prove there is an isomorphism of graded spaces between the Hochschild cohomology of the generalized universal enveloping algebra U(L) and the generalized cohomology of the color Lie algebra L. Supported by the EC project Liegrits MCRTN 505078.     

Minimax risk over quadratically convex sets

We consider the problem of estimating a vector θ = (θ₁, θ₂,…) ∈ Θ ⊂ l ₂ from observations y _i = θ _i + σ _i x _i , i = 1, 2,…, where the random values x _i are N(0, 1), independent, and identically distributed, the parametric set Θ is compact, orthosymmetric, convex, and quadratically convex. We show that in that case, the minimax risk is not very different from sup?_L( P) \sup {\Re_L}\left( \Pi \right) , where ?_L( P) {\Re_L}\left( \Pi \right) is the minimax linear risk in the same problem with parametric set Π, and sup is taken over all the hyperrectangles Π ⊂ Θ. Donoho, Liu, and McGibbon (1990) have obtained this result for the case of equal σ _i , i = 1, 2,…. Bibliography: 4 titles.     

On comparing two chains of numerical semigroups and detecting arf semigroups

If T is a numerical semigroup with maximal ideal N , define associated semigroups B(T):=(N-N) and L(T)= \cup { (hN-hN) \colon h \geq 1 } . If S is a numerical semigroup, define strictly increasing finite sequences { B _i (S) \colon 0 ≤ i ≤β (S) } and { L _i (S) \colon 0 ≤ i ≤λ (S) } of semigroups by B ₀ (S):=S=:L ₀ (S) , B _{β (S)} (S):= \Bbb N =: L _{λ (S)} (S) , B _i+1 (S):=B(B _i (S)) for 0<i< β (S) , L _i+1 (S):=L(L _i (S)) for 0<i< λ (S) . It is shown, contrary to recent claims and conjectures, that B ₂ (S) need not be a subset of L ₂ (S) and that β (S) - λ (S) can be any preassigned integer. On the other hand, B ₂ (S) \subseteq L ₂ (S) in each of the following cases: S is symmetric;S has maximal embedding dimension;S has embedding dimension e(S) ≤ 3 . Moreover, if either e(S)=2 or S is pseudo-symmetric of maximal embedding dimension, then B _i (S) \subseteq L _i (S) for each i , 0 ≤ i ≤λ (S) . For each integer n \geq 2 , an example is given of a (necessarily non-Arf) semigroup S such that β (S) = λ (S)=n , B _i (S) = L _i (S) for all 0 ≤ i ≤ n-2 , and B _n-1 (S) \subsetneqq L _n-1 (S) . April 4, 2000     

Higher integrability for minimizers of variational integrals with nonstandard growth

We consider a (possibly) vector-valued function u: Ω→R ^N, Ω⊂R ⁿ, minimizing the integral , whereD _iu=∂u/∂x _i, or some more general functional retaining the same behaviour; we prove higher integrability forDu:D ₁u,…,D_n−1u∈L^q, under suitable assumptions ona _i(x).

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Sunto Consideriamo una funzione u: Ω→R ^N, Ω⊂R ⁿ che minimizzi l'integrale , doveD _iu=∂u/∂x_i, o un funzionale con un comportamento simile; sotto opportune ipotesi sua _i(x), dimostriamo la seguente maggiore integrabilità perDu:D ₁u,…,D_n−1uεL^q.

     

Highest weight representations of a Lie algebra of Block type

For a field F of characteristic zero and an additive subgroup G of F, a Lie algebra B(G) of the Block type is defined with the basis {Lα,i, c|α∈G, -1≤i∈Z} and the relations [Lα,i,Lβ,j] = ((i 1)β- (j 1)α)Lα β,i j αδα,-βδi j,-2c,[c, Lα,i] = 0. Given a total order (?) on G compatible with its group structure, and anyα∈B(G)0*, a Verma B(G)-module M(A, (?)) is defined, and the irreducibility of M(A,(?)) is completely determined. Furthermore, it is proved that an irreducible highest weight B(Z )-module is quasifinite if and only if it is a proper quotient of a Verma module.     

Depth Generated Simple Lie Algebras

A Lie module algebra for a Lie algebra L is an algebra and L-module A such that L acts on A by derivations. The depth Lie algebra of a Lie algebra L with Lie module algebra A acts on a corresponding depth Lie module algebra . This determines a depth functor from the category of Lie module algebra pairs to itself. Remarkably, this functor preserves central simplicity. It follows that the Lie algebras corresponding to faithful central simple Lie module algebra pairs (A,L) with A commutative are simple. Upon iteration at such (A,L), the Lie algebras are simple for all i ∈ ω. In particular, the (i ∈ ω) corresponding to central simple Jordan Lie algops (A,L) are simple Lie algebras. Presented by Don Passman.     

Construction of positive definite cubature formulae and approximation of functions via Voronoi tessellations

Let Ω ⊂ ℝ ^d be a compact convex set of positive measure. In a recent paper, we established a definiteness theory for cubature formulae of order two on Ω. Here we study extremal properties of those positive definite formulae that can be generated by a centroidal Voronoi tessellation of Ω. In this connection we come across a class of operators of the form L_n[f](x): = ?_i=1ⁿ f_i(x)(f(y_i) + á?f(y_i), x-y_i?)L_n[f](\boldsymbol{x}):= \sum_{i=1}^n \phi_i(\boldsymbol{x})(f(\boldsymbol{y}_i) + \langle\nabla f(\boldsymbol{y}_i), \boldsymbol{x}-\boldsymbol{y}_i\rangle), where y₁,..., y_n\boldsymbol{y}_1,\dots, \boldsymbol{y}_n are distinct points in Ω and {ϕ ₁, ..., ϕ _n } is a partition of unity on Ω. We present best possible pointwise error estimates and describe operators L _n with a smallest constant in an L ^p error estimate for 1 ≤ p < ∞ . For a generalization, we introduce a new type of Voronoi tessellation in terms of a twice continuously differentiable and strictly convex function f. It allows us to describe a best operator L _n for approximating f by L _n [f] with respect to the L ^p norm.     

Hochschild （Co）homology of Galois Coverings of Grassmann Algebras

Let ∧ be the Z2-Galois covering of the Grassmann algebra A over a field k of characteristic not equal to 2. In this paper, the dimensional formulae of Hochschild homology and cohomology groups of ∧ are calculated explicitly. And the cyclic homology of∧ can also be calculated when the underlying field is of characteristic zero. As a result, we prove that there is an isomorphism from i≥1 HH^i（∧） to i≥1 HH^i（∧）.     

Highest weight representations of a family of Lie algebras of Block type

For an additive subgroup G of a field F of characteristic zero, a Lie algebra B（G） of Block type is defined with basis {Lα,i｜ α∈G, i∈Z＋} and relations [Lα,i, Lβ,j] = （β-α）Lα＋β,i＋j＋（αj-βi）Lα＋β,Lα＋β,i＋j-1.It is proved that an irreducible highest weight B（Z）-module is quasifinite if and only if it is a proper quotient of a Verma module. Furthermore, for a total order λ on G and any ∧∈B（G）0^＊（the dual space of B（G）0 = span{L0,i｜i∈Z＋}）, a Verma B（G）-module M（∧,λ） is defined, and the irreducibility of M（A,λ） is completely determined.     

Analysis of Alternative Digit Sets for Nonadjacent Representations

It is known that every positive integer n can be represented as a finite sum of the form ∑_ia_i2ⁱ, where a_i ∈ {0, 1,−1} and no two consecutive a_i’s are non-zero (“nonadjacent form”, NAF). Recently, Muir and Stinson [14, 15] investigated other digit sets of the form {0, 1, x}, such that each integer has a nonadjacent representation (such a number x is called admissible). The present paper continues this line of research. The topics covered include transducers that translate the standard binary representation into such a NAF and a careful topological study of the (exceptional) set (which is of fractal nature) of those numbers where no finite look-ahead is sufficient to construct the NAF from left-to-right, counting the number of digits 1 (resp. x) in a (random) representation, and the non-optimality of the representations if x is different from 3 or −1. This paper was written while the first author was a visitor at the John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand, Johannesburg. He thanks the centre for its hospitality. He was also supported by the grant S8307-MAT of the Austrian Science Fund. This author is supported by the grant NRF 2053748 of the South African National Research Foundation. The research of this author was done while he was with the John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand, Johannesburg.     

Cayley’s decomposition and pólya’sW-property of ordinary linear differential equations

A proof is given for the equivalence of Pólya’sW- property of a linear differential equationL _n (D) y=0 to the possibility of decomposingL _n (D) ≡ Π _n ¹ [D+λ_i(x)] in a given interval. In this case a set ofn independent solutions form a Chebyshev system in the interval. An application determines intervals of non-oscillation for solutions of linear equations of the second order. Research supported by the National Science Foundation Grant No. GP-3897 with the University of Maryland.     

Cycles de congruences stratifiables dans un espace projectif de dimension impaire

Résumé Dans un espace projectif S_2n−i(n≥3) à 2n−1 dimensions, on étudie un système composé de n congruences non-paraboliques de droites L₁, L₂, ..., L_n qui jouissent de la propriété que chaque couple de congruences L_i,L _i+1(i=1, 2, ..., n; L_n+i=L_i) est stratifiable dans le sens de L_i vers L_i+1. à M. Enrico Bompiani pour son Jubilé scientifique.     

Modular Lie powers and the Solomon descent algebra

Let V be an r-dimensional vector space over an infinite field F of prime characteristic p, and let L_n(V) denote the nth homogeneous component of the free Lie algebra on V. We study the structure of L_n(V) as a module for the general linear group GL_r(F) when n=pk and k is not divisible by p and where r≥n. Our main result is an explicit 1-1 correspondence, multiplicity-preserving, between the indecomposable direct summands of L_k(V) and the indecomposable direct summands of L_n(V) which are not isomorphic to direct summands of V^⊗n. Our approach uses idempotents of the Solomon descent algebras, and in addition a correspondence theorem for permutation modules of symmetric groups. Second author supported by Deutsche Forschungsgemeinschaft (DFG-Scho 799).     

Smooth convex bodies with proportional projection functions

For a convex body K ⊂ ℝⁿ and i ∈ {1, …, n − 1}, the function assigning to any i-dimensional subspace L of ℝⁿ, the i-dimensional volume of the orthogonal projection of K to L, is called the i-th projection function of K. Let K, K ₀ ⊂ ℝⁿ be smooth convex bodies with boundaries of class C ² and positive Gauss-Kronecker curvature and assume K ₀ is centrally symmetric. Excluding two exceptional cases, (i, j) = (1, n − 1) and (i, j) = (n − 2, n − 1), we prove that K and K ₀ are homothetic if their i-th and j-th projection functions are proportional. When K ₀ is a Euclidean ball this shows that a convex body with C ² boundary and positive Gauss-Kronecker with constant i-th and j-th projection functions is a Euclidean ball. The second author was supported in part by the European Network PHD, FP6 Marie Curie Actions, RTN, Contract MCRN-511953.     

Information cells and information cell mixture models for concept modelling

By combining the prototype theory and random set theory interpretations of vague concepts, a novel structure named information cell and a combined structure named information cell mixture model are proposed to represent the semantics of vague concepts. An information cell L _i on the domain Ω has a transparent cognitive structure ‘L _i =about P _i ’ which is mathematically formalized by a 3-tuple 〈P _i ,d _i ,δ _i 〉 comprising a prototype set P _i (⊆Ω), a distance function d _i on Ω and a density function δ _i on [0,+∞). An information cell mixture model on domain Ω is actually a set of weighted information cells L _i s. A positive neighborhood function of the information cell mixture model is introduced in this paper to reflect the belief distribution of positive neighbors of the underlying concept. An information cellularization algorithm is also proposed to learn the information cell mixture model from a training data set, which is a direct application of the k-means and EM algorithms. Information cell mixture models provide some tools for information coarsening and concept modelling, and have potential applications in uncertain reasoning and classification.     

Classification of gradient space of dimension 8 by a reducible <Emphasis Type="Italic">s</Emphasis>ℓ(2, C) action

This paper deals with a reducible sℓ(2,C) action on the formal power series ring. The purpose of this paper is to confirm a special case of the Yau conjecture: Suppose that sℓ(2,C) acts on the formal power series ring via (1.1). Then I(f) = (ℓ _i1) ⊕ (ℓ _i2) ⊕... ⊕ (ℓ _is ) modulo some one dimensional sℓ(2,C) representations where (ℓ _i ) is an irreducible sℓ(2,C) representation of ℓ _i dimension and {ℓ _i1 ℓ _i2,...,ℓ _is } ⊆ {ℓ ₁ , ℓ ₂...,ℓ _r }. Unlike classical invariant theory which deals only with irreducible action and 1-dimensional representations, we treat the reducible action and higher dimensional representations successively.     

Approximation of Functions of Few Variables in?High?Dimensions

Let f be a continuous function defined on Ω:=[0,1] ^N which depends on only ℓ coordinate variables, f(x₁,?,x_N)=g(x_i1,?,x_il)f(x_{1},\ldots,x_{N})=g(x_{i_{1}},\ldots,x_{i_{\ell}}). We assume that we are given m and are allowed to ask for the values of f at m points in Ω. If g is in Lip1 and the coordinates i ₁,…,i _ℓ are known to us, then by asking for the values of f at m=L ^ℓ uniformly spaced points, we could recover f to the accuracy |g|_Lip1 L ⁻¹ in the norm of C(Ω). This paper studies whether we can obtain similar results when the coordinates i ₁,…,i _ℓ are not known to us. A prototypical result of this paper is that by asking for C(ℓ)L ^ℓ (log ₂ N) adaptively chosen point values of f, we can recover f in the uniform norm to accuracy |g|_Lip1 L ⁻¹ when g∈Lip1. Similar results are proven for more general smoothness conditions on g. Results are also proven under the assumption that f can be approximated to some tolerance ε (which is not known) by functions of ℓ variables.