Combinatorial bases of modules for affine Lie algebra B 2 (1)

We construct bases of standard (i.e. integrable highest weight) modules L(Λ) for affine Lie algebra of type B ₂ ⁽¹⁾ consisting of semi-infinite monomials. The main technical ingredient is a construction of monomial bases for Feigin-Stoyanovsky type subspaces W(Λ) of L(Λ) by using simple currents and intertwining operators in vertex operator algebra theory. By coincidence W(kΛ₀) for B ₂ ⁽¹⁾ and the integrable highest weight module L(kΛ₀) for A ₁ ⁽¹⁾ have the same parametrization of combinatorial bases and the same presentation P/I.     

Gröbner-Shirshov bases for quantum enveloping algebras

We give a method for finding Gröbner-Shirshov bases for the quantum enveloping algebras of Drinfel’d and Jimbo, show how the methods can be applied to Kac-Moody algebras, and explicitly find the bases for quantum enveloping algebras of typeA _N (forq ⁸≠1).     

On Pimsner–Popa bases

In this paper, we examine bases for finite index inclusion of II₁ factors and connected inclusion of finite dimensional C ^?-algebras. These bases behave nicely with respect to basic construction towers. As applications we have studied automorphisms of the hyperfinite II₁ factor R which are ‘compatible with respect to the Jones’ tower of finite dimensional C ^?-algebras’. As a further application, in both cases we obtain a characterization, in terms of bases, of basic constructions. Finally we use these bases to describe the phenomenon of multistep basic constructions (in both the cases).     

On certain spaces of lattice diagram determinants

The aim of this work is to study some lattice diagram determinants Δ_L(X,Y) as defined in (Adv. Math. 142 (1999) 244) and to extend results of Aval et al. (J. Combin. Theory Ser. A, to appear). We recall that M_L denotes the space of all partial derivatives of Δ_L. In this paper, we want to study the space M_i,j^k(X,Y) which is defined as the sum of M_L spaces where the lattice diagrams L are obtained by removing k cells from a given partition, these cells being in the “shadow” of a given cell (i,j) in a fixed Ferrers diagram. We obtain an upper bound for the dimension of the resulting space M_i,j^k(X,Y), that we conjecture to be optimal. This dimension is a multiple of n! and thus we obtain a generalization of the n! conjecture. Moreover, these upper bounds associated to nice properties of some special symmetric differential operators (the “shift” operators) allow us to construct explicit bases in the case of one set of variables, i.e. for the subspace M_i,j^k(X) consisting of elements of 0 Y-degree.     

Compactly supported bidimensional wavelet bases with hexagonal symmetry

We study a class of subband coding schemes allowing perfect reconstruction for a bidimensional signal sampled on the hexagonal grid. From these schemes we construct biorthogonal wavelet bases ofL ²(R ²) which are compactly supported and such that the sets of generating functionsψ ₁,ψ ₂,ψ ₃ for the synthesis and \(\tilde \psi _1 , \tilde \psi _2 , \tilde \psi _3 ,\) for the analysis, as well as the scaling functions φ and \(\tilde \varphi \) , are globally invariant by a rotation of 2π/3. We focus on the particular case of linear splines and we discuss how to obtain a higher regularity. We finally present the possibilities of sharp angular frequency resolution provided by these new bases.     

Minimal Gröbner bases and the predictable leading monomial property

We focus on Gröbner bases for modules of univariate polynomial vectors over a ring. We identify a useful property, the “predictable leading monomial (PLM) property” that is shared by minimal Gröbner bases of modules in F[x]^q, no matter what positional term order is used. The PLM property is useful in a range of applications and can be seen as a strengthening of the wellknown predictable degree property (= row reducedness), a terminology introduced by Forney in the 70’s. Because of the presence of zero divisors, minimal Gröbner bases over a finite ring of the type Z_p^r (where p is a prime integer and r is an integer >1) do not necessarily have the PLM property. In this paper we show how to derive, from an ordered minimal Gröbner basis, a so-called “minimal Gröbner p-basis” that does have a PLM property. We demonstrate that minimal Gröbner p-bases lend themselves particularly well to derive minimal realization parametrizations over Z_p^r. Applications are in coding and sequences over Z_p^r.     

Matroids with few non-common bases

In [On Mills's conjecture on matroids with many common bases, Discrete Math. 240 (2001) 271-276], Lemos proved a conjecture of Mills [On matroids with many common bases, Discrete Math. 203 (1999) 195-205]: for two (k+1)-connected matroids whose symmetric difference between their collections of bases has size at most k, there is a matroid that is obtained from one of these matroids by relaxing n₁ circuit-hyperplanes and from the other by relaxing n₂ circuit-hyperplanes, where n₁ and n₂ are non-negative integers such that n₁+n₂≤k. In [Matroids with many common bases, Discrete Math. 270 (2003) 193-205], Lemos proved a similar result, where the hypothesis of the matroids being k-connected is replaced by the weaker hypothesis of being vertically k-connected. In this paper, we extend these results.     

The structurally stable linear systems on the half-line are those with exponential dichotomies

Let A be a selft-adjoint operator on the Hilbert space L²Ω, ?) = {u ε L_loc²(Ω)|∫_Ω|² ?(x)dx < + ∞} defined by means of a closed, semibounded, sesquilinear form a(·, ·). We obtain a necessary and sufficuents condition for the spectrum of A to be discrete. We apply this result to a Sturm-Liouville problem for an elliptic operator with discontinuous coefficients defined on an unbounded domain and to the study of the spectrum of a Hamiltonian defined by a pseudodifferential operator.     

Wavelet packets associated with nonuniform multiresolution analyses

A multiresolution analysis was defined by Gabardo and Nashed for which the translation set is a discrete set which is not a group. We construct the associated wavelet packets for such an MRA. Further, from the collection of dilations and translations of the wavelet packets, we characterize the subcollections which form orthonormal bases for L²(R).     

New Properties of the Lie Algebra Variety N2A

We continue to consider the properties of the almost polynomial growth variety of Lie algebras over a field of characteristic zero defined by the identity (x ₁ x ₂)(x ₃ x ₄)(x ₅ x ₆)?≡?0. Here we have constructed the bases of its multilinear parts and proved the formulas for the colength and codimension sequences of this variety.     

The local determination of Jordan bases for H-self-adjoint operators

Let H be an invertible self-adjoint operator on a finite dimensional Hilbert space $\text{X}$. A linear operator A is said to be H-self-adjoint (or self-adjoint relative to H) if HA = A¹H. Let σ(A) denote, as usual, the spectrum of A. If A is H-self-adjoint, then A is similar to A¹ and λ ∈ σ(A) implies λ&#x0304; ∈ σ (A), so that the spectrum of A issymmetric with respect to the real axis. Given spectral information for A at an eigenvalue λ₀ (≠ λ&#x0304;₀), we investigate the corresponding information at λ&#x0304;₀ and, in particular, the unique pairing of Jordan bases for the root subspaces at λ₀ and λ&#x0304;₀.     

The biweight enumerator of self-orthogonal binary codes

Let C be a binary code of length n and let J_C (a, b, c, d) be its biweight enumerator. If n is even and C is self-dual, then J_C is an element of the ring R^$\text{G}$ of absolute invariants of a certain group $\text{G}$. Under the additional assumption that all codewords of C have weight divisible by 4, a similar result holds with a different group. If n is odd and C is maximal self-orthogonal, then J_C is an element of a certain R^$\text{G}$-module. Again a similar result holds if the codewords of C have weights divisible by 4. The groups involved are related to finite groups generated by reflections. In this paper the structure of these groups is described, and polynomial bases for the rings and modules in question are obtained. This answers a question posed in The Theory of Error- correcting Codes by F.J. MacWilliams and N.J.A. Sloane.     

Arithmetical progressions formed byk different Lehmer pseudoprimes

LetU _n=(αⁿ-β²)/(α-β) forn odd andU _n=(αⁿ-βⁿ)/(α²-β²) for evenn, where α and β are distinct roots of the trinomialf(z)=z ²-√Lz+Q andL>0 andQ are rational integers.U _n is then-th Lehmer number connected withf(z). A compositen is a Lehmer pseudoprime for the bases α and β ifU _n??(n)≡0 (modn), where?(n)=(LD/n) is the Jacobi symbol. IfD=L?4Q>0, U _n denotesn-th Lehmer number,p>3 and 2p?1 are primes,p(2p-1)+(α²-β²)², (α^2p-1±β^2p-1)/(α±β) are composite then the numbers (α^2p-1+β^2p-1)/(α+β), (α^2p-β^2p)/(α²-β²), (α^2p-1-β^2p-1)/(α-β) are lehmer pseudoprimes for the bases α and β and form an arithmetical progression. IfD>0 then from hypothesisH of A. Schinzel on polynomials it follows that for every positive integerk there exists infinitely many arithmetic progressions formed fromk different Lehmer pseudoprimes for the bases α and β.     

Fractional Covers for Convolution Products

A non-zero vector-valued sequence u ∈ ?^q(X′) is a cover for a subset M of ?_P(X) if, for some 0 < α ≤ 1, ∥u * h∥ _∞ ≥ α ∥u∥_q ∥h∥_p for all h ∈ M. Covers of ?¹ = ?¹(R) are important in worst case system identification in ?¹ and in the reconstruction of elements in a normed space from corrupted functional values. We investigate the existence of covers for certain naturally occurring subspaces of ?^p(X). We show that there exist finitely supported covers for some subspaces, and obtain lower bounds for their ’lengths’. We also obtain similar results for covers associated with convolution products for spaces of measurable vector-valued functions defined on the positive real axis.     

Invariant G2V algorithm for computing SAGBI-Gröbner bases

Faugère and Rahmany have presented the invariant F5 algorithm to compute SAGBI-Grbner bases of ideals of invariant rings. This algorithm has an incremental structure, and it is based on the matrix version of F5 algorithm to use F5 criterion to remove a part of useless reductions. Although this algorithm is more efficient than the Buchberger-like algorithm, however it does not use all the existing criteria (for an incremental structure) to detect superfluous reductions. In this paper, we consider a new algorithm, namely, invariant G2V algorithm, to compute SAGBI-Grbner bases of ideals of invariant rings using more criteria. This algorithm has a new structure and it is based on the G2V algorithm; a variant of the F5 algorithm to compute Grbner bases. We have implemented our new algorithm in Maple , and we give experimental comparison, via some examples, of performance of this algorithm with the invariant F5 algorithm.     

Conjugate transforms for τT semigroups of probability distribution functions

This paper introduces the use of conjugate transforms in the study of τ_T semigroups of probability distribution functions. If Δ⁺ denotes the space of one-dimensional distribution functions concentrated on [0, ∞) and T is a t-norm, i.e., a suitable binary operation on [0, 1], then the operation τ_T is defined for F, G in Δ⁺by τ_T(F, G)(x) = sup_{u+v = x}T(F(u), G(v)) for all x. The pair (Δ⁺, τ_T) is then a semigroup. For any Archimedean t-norm T, a conjugate transform C_T is defined on (Δ⁺, τ_T). These transforms are shown to play a role similar to that played by the Laplace transform on the convolution semigroup. Thus a theory of “characteristic functions” for τ_T semigroups is developed. In addition to establishing their basic algebraic properties, we also use conjugate transforms to study the algebraic questions of the cancellation law, infinitely divisible elements, and solutions of equations in τ_T semigroups.     

On the uniqueness of symmetric bases in finite dimensional Banach spaces

Suppose{e _i} _i=1 ⁿ and{f _i} _i=1 ⁿ are symmetric bases of the Banach spacesE andF. Letd(E,F)≦C andd(E,l _n ² )≧n' for somer>0. Then there is a constantC _r=C_r(C)>0 such that for alla _i∈Ri=1,...,n $$C_r^{ - 1} \left\| {\sum\limits_{i = 1}^n {a_i e_i } } \right\| \leqq \left\| {\sum\limits_{i = 1}^n {a_i f_i } } \right\| \leqq C_r \left\| {\sum\limits_{i = 1}^n {a_i e_i } } \right\|$$ We also give a partial uniqueness of unconditional bases under more restrictive conditions.     

Calculating all elements of minimal index in the infinite parametric family of simplest quartic fields

It is a classical problem in algebraic number theory to decide if a number field is monogeneous, that is if it admits power integral bases. It is especially interesting to consider this question in an infinite parametric familiy of number fields. In this paper we consider the infinite parametric family of simplest quartic fields K generated by a root ξ of the polynomial P _t (x) = x ⁴ ? tx ³ ? 6x ² +tx+1, assuming that t > 0, t ≠ 3 and t ² +16 has no odd square factors. In addition to generators of power integral bases we also calculate the minimal index and all elements of minimal index in all fields in this family.     

Spectral Models for Orthonormal Wavelets and?Multiresolution Analysis of L 2(?)

Spectral representations of the dilation and translation operators on L ²(?) are built through appropriate bases. Orthonormal wavelets and multiresolution analysis are then described in terms of rigid operator-valued functions defined on the functional spectral spaces. The approach is useful for computational purposes.     

On the Stability of One Characterization of Stable Distributions

In 1923, G. Polya proved that if X ₁ and X ₂ are independent identically distributed random variables (i.i.d.r.v.) with finite variance, then the distributions of X ₁ and (X ₁+X ₂)/ $\sqrt 2$ are coincidental iff X ₁ has the normal distribution with zero mean. Is an analogous theorem possible for an couple of statistics X ₁ and (X ₁+X ₂)/2^1/α if α<2? P. Lévy constructed an example that denies that hypothesis. However, having supplemented the condition of coincidence of the distributions of X ₁ and (X ₁+X ₂)/2^1/α with a similar condition, namely, requiring, in addition, for the distributions of X ₁ and (X ₁+X ₂+X ₃)/3^1/α to be coincident (here X ₁,X ₂ and X ₃ are i.i.d.r.v.), P. Lévy has proved that X ₁ and X ₂ have a strictly stable distribution. The stability of this characterization in a metric λ₀ (that is defined in the class of characteristic functions by analogy with a uniform metric defined in the class of distributions) without an additional symmetry assumption as well as the stability in a Lévy metric L are analizied in this paper.