Unconditional basis recursively generated in L~p on compact sets

We construct an unconditional basis in the Banach space Lp(Ω, ρ) for p > 1 by using the refinement equation and the basic operation of translation and scale, where Ω is a compact subset in Rn. We also give an algorithm of how to construct an unconditional basis in Lp(Ω,ρ). At the end of this paper, we give the characterization of the functions in Lp (Ω,ρ) by using the wavelet coefficients.     

The transition matrix between PBW basis and semicanonical basis of U +(\mathfrak{s}\mathfrak{l}_n (ℂ))

We prove that the transition matrix between a special Poincaré-Birkhoff-Witt(PBW)basis and the semicanonical basis of U+(sln(C))is upper triangular and unipotent under any order which is compatible with the partial order deg.     

Anti-commutative Grbner-Shirshov basis of a free Lie algebra

The concept of Hall words was first introduced by P. Hall in 1933 in his investigation on groups of prime power order. Then M. Hall in 1950 showed that the Hall words form a basis of a free Lie algebra by using direct construction, that is, first he started with a linear space spanned by Hall words, then defined the Lie product of Hall words and finally checked that the product yields the Lie identities. In this paper, we give a Grbner-Shirshov basis for a free Lie algebra. As an application, by using the ...     

Block sequences in nuclear Fréchet spaces with basis

It is shown that a block sequence in a nuclear Fréchet space with a basis has a block extension if and only if the subspace it generates is complemented. In addition, a short proof is given of the following result of Dubinsky and Robinson: a nuclear Fréchet space is isomorphic to = R^N, N = {1,2,...} if it has a basis such that any block sequence with blocks of length 2 of any permutation of this basis has a block extension. It is shown that a similar result holds without considering permutations of the basis if the length of the blocks is arbitrary.Translated from Matematicheskie Zametki, Vol. 17, No. 6, pp. 899–908, June, 1975.The author wishes to thank B. S. Mityagin for calling his attention to this problem and for his valuable suggestions.     

W-Gröbner basis and monomial ideals under polynomial composition

The notion of weakly relatively prime and W-Gröbner basis in K[x ₁, x ₂, …, x _n ] are given. The following results are obtained: for polynomials f ₁, f ₂, …, f _m , \(\{ f_1^{\lambda _1 } ,f_2^{\lambda _2 } ,...,f_m^{\lambda _m } \} \) is a Gröbner basis if and only if f ₁, f ₂, …, f _m are pairwise weakly relatively prime with λ ₁, λ ₂, …, λ _m arbitrary non-negative integers; polynomial composition by Θ = (θ ₁, θ ₂, …, θ _n ) commutes with monomial-Gröbner bases computation if and only if θ ₁, θ ₂, …, θ _m are pairwise weakly relatively prime.     

Anti-commutative Gröbner-Shirshov basis of a free Lie algebra

The concept of Hall words was first introduced by P. Hall in 1933 in his investigation on groups of prime power order. Then M. Hall in 1950 showed that the Hall words form a basis of a free Lie algebra by using direct construction, that is, first he started with a linear space spanned by Hall words, then defined the Lie product of Hall words and finally checked that the product yields the Lie identities. In this paper, we give a Gröbner-Shirshov basis for a free Lie algebra. As an application, by using the Composition-Diamond lemma established by Shirshov in 1962 for free anti-commutative (non-associative) algebras, we provide another method different from that of M. Hall to construct a basis of a free Lie algebra.     

Construction of ‘Wachspress type’ rational basis functions over rectangles

In the present paper, we have constructed rational basis functions ofC ⁰ class over rectangular elements with wider choice of denominator function. This construction yields additional number of interior nodes. Hence, extra nodal points and the flexibility of denominator function suggest better approximation.     

Geometric properties of Fréchet spaces and selection of basis sequences

For an arbitrary nuclear Fréchet spaceE possessing the well-known geometric propertiesD ₁ (DN) and Ω, certain sequences of functionals and elements with power-type estimates of the norms are selected. With the help of these objects, two isomorphic spacesK ₁ andK ₂ of power series of infinite type and continuous mapsJ ₂: K₂→E andJ ₁: E→K₁ are defined. Under some auxiliary conditions, it is proved that the_i are isomorphisms and the spaceE possesses a basis. Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 102–111, July, 1999.     

Computing polynomial univariate representations of zero-dimensional ideals by Grbner basis

Rational Univariate Representation(RUR) of zero-dimensional ideals is used to describe the zeros of zero-dimensional ideals and RUR has been studied extensively.In 1999,Roullier proposed an efficient algorithm to compute RUR of zero-dimensional ideals.In this paper,we will present a new algorithm to compute Polynomial Univariate Representation(PUR) of zero-dimensional ideals.The new algorithm is based on some interesting properties of Grbner basis.The new algorithm also provides a method for testing separating elements.     

Convergence rates in ℓ1-regularization when the basis is not smooth enough

Sparsity promoting regularization is an important technique for signal reconstruction and several other ill-posed problems. Theoretical investigation typically bases on the assumption that the unknown solution has a sparse representation with respect to a fixed basis. We drop this sparsity assumption and provide error estimates for nonsparse solutions. After discussing a result in this direction published earlier by one of the authors and co-authors, we prove a similar error estimate under weaker assumptions. Two examples illustrate that this set of weaker assumptions indeed covers additional situations which appear in applications.     

The existence of optimal control on the basis of Weierstrass’s theorem

The problem of existence of an optimal control is solved on the basis of Weierstrass’s classical theorem if the set of admissible controls belongs to the class of piecewise continuous functions. In the process of describing admissible controls, the main assumption is that the number of switchings (points of discontinuity) is uniformly bounded and not just finite, as in the main problem of optimal control theory. On the one hand, this assumption does not restrict the spectrum of optimal control applications. On the other hand, it fits the Weierstrass’s theorem owing to the convenience in characterizing the sequential compactness. The formulation of Weierstrass’s theorem, which asserts the existence of continuous function extrema on sequentially compact sets, is customary, and its proof complies with the traditional scheme, whereas the concepts (convergent sequences and some others) are adapted to the peculiarity of optimal problems.     

Extremes of projections of functional time series on data–driven basis systems

The paper is concerned with the extreme behavior of projections of time series of functions onto data-driven basis systems, for example, on the estimated functional principal components. The coefficients of these projections, called the scores, encode the shapes of the curves. Within the framework of functional data analysis, the extreme shapes are those corresponding to multivariate extremes of the scores. The scores are not directly observable, and must be computed from the data. Even for iid Gaussian functions, they form a triangular array of dependent non–Gaussian vectors. Thus, even though the extreme behavior of the population scores of Gaussian functions follows from well–known results, it is not clear what the extreme behavior of their approximations computed from the data is. We clarify these issues for Gaussian functions and for more general functional time series whose projections are in the Gumbel domain of attraction.     

Existence of relative integral basis over quadratic fields and Pólya property

Acta Mathematica Hungarica - For $$L/K$$ a finite extension of algebraic number fields, L may or may not have a relative integral basis over K. We show the existence of relative integral basis of a...     

Skew-commutator relations and Gröbner-Shirshov basis of quantum group of type F 4

We give a Grobner-Shirshov basis of quantum group of type F4 by using the Ringel-Hall algebra approach. We compute all skew-commutator relations between the isoclasses of indecomposable representations of Ringel- Hall algebras of type F4 by using an ＇inductive＇ method. Precisely, we do not use the traditional way of computing the skew-commutative relations, that is first compute all Hall polynomials then compute the corresponding skew- commutator relations; instead, we compute the ＇easier＇ skew-commutator relations which correspond to those exact sequences with middle term indecomposable or the split exact sequences first, then ＇deduce＇ others from these ＇easier＇ ones and this in turn gives Hall polynomials as a byproduct. Then using the composition-diamond lemma prove that the set of these relations constitute a minimal CrSbner-Shirshov basis of the positive part of the quantum group of type F4. Dually, we get a Grobner-Shirshov basis of the negative part of the quantum group of type F4. And finally, we give a Gr6bner-Shirshov basis for the whole quantum group of type F4.     

A basis for the type 〈n〉 hyperidentities of the medial variety of semigroups

Summary The medical varietyMV of semigroups is the variety defined by the medial identityxyzw = xzyw. This variety is known to satisfy the medial hyperidentitiesF(G(x ₁₁ ,, x _1n ),, G(x _n1 ,, x _nn )) = G(F(x ₁₁ ,, x _n1 ),, F(x _1n ,, x _nn )), forn 1. Taylor has observed in [2] thatMV also satisfies some other hyperidentities, which are not consequences of the medial ones. In [4] the author introduced a countably infinite family of binary hyperidentities called transposition hyperidentities, which are natural generalizations of then = 2 medial hyperidentity. It was shown that this family is irredundant, and that no finite basis is possible for theMV hyperidentities with one binary operation symbol.In this paper, we generalize the concept of a transposition hyperidentity, and extend it to cover arbitrary arityn 2. We show that theMV hyperidentities with onen-ary operation symbol have no finite basis, but do have a countably infinite basis consisting of these transposition hyperidentities.Research supported by NSERC of Canada.