Multivariate splines and polynomial interpolation

Akopyan, A. A.; Saakyan, A. A.

doi:10.1070/rm1993v048n05abeh001067

Cited by 6 publications

(5 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The space P(X) below is the central space of the zonotope Z(X), while the space D(X) is the central space of the hyperplane arrangement H(X, 0). The theory of this pair of polynomial spaces was developed in the 80s and 90s in [1,2,3,8,22,23,24,25,19,14,15,17,18,13,33,34]. We have discussed some historical aspects of this theory in the Introduction, and will discuss the history of specific results in more detail towards the end of this section.…”

Section: Resultsmentioning

confidence: 99%

“…In the unimodular case, however, the discrete version coincides with the continuous version, which is the version studied here. 3 One needs also to choose correctly the basis for IR n in the definition of the edge set.…”

Section: Generalmentioning

confidence: 99%

“…Group representations are connected with a discrete version of zonotopal algebras, that are not discussed in this paper. In the unimodular case, however, the discrete version coincides with the continuous version, which is the version studied here 3. One needs also to choose correctly the basis for IR n in the definition of the edge set.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Zonotopal algebra

Holtz

Ron

2011

Advances in Mathematics

114

View full text Add to dashboard Cite

A wealth of geometric and combinatorial properties of a given linear endomorphism $X$ of $\R^N$ is captured in the study of its associated zonotope $Z(X)$, and, by duality, its associated hyperplane arrangement ${\cal H}(X)$. This well-known line of study is particularly interesting in case $n\eqbd\rank X \ll N$. We enhance this study to an algebraic level, and associate $X$ with three algebraic structures, referred herein as {\it external, central, and internal.} Each algebraic structure is given in terms of a pair of homogeneous polynomial ideals in $n$ variables that are dual to each other: one encodes properties of the arrangement ${\cal H}(X)$, while the other encodes by duality properties of the zonotope $Z(X)$. The algebraic structures are defined purely in terms of the combinatorial structure of $X$, but are subsequently proved to be equally obtainable by applying suitable algebro-analytic operations to either of $Z(X)$ or ${\cal H}(X)$. The theory is universal in the sense that it requires no assumptions on the map $X$ (the only exception being that the algebro-analytic operations on $Z(X)$ yield sought-for results only in case $X$ is unimodular), and provides new tools that can be used in enumerative combinatorics, graph theory, representation theory, polytope geometry, and approximation theory.Comment: 44 pages; updated to reflect referees' remarks and the developments in the area since the article first appeared on the arXi

show abstract

Section: Resultsmentioning

confidence: 99%

Section: Generalmentioning

confidence: 99%

See 1 more Smart Citation

Zonotopal algebra

Holtz

Ron

2011

Advances in Mathematics

114

View full text Add to dashboard Cite

show abstract

“…It is known that the reaction of halobutenes and -butadienes with sulfur is used for the synthesis of the corresponding halogenothiophenes. 58 In the case of nitrodiene 17, the reaction follows a different route: the heterocyclisation involves the nitro group and is accompanied by the evolution of sulfur dioxide. The yield of isothiazole 18 reaches 52%.…”

Section: A Synthesis Of Isothiazoles From Compounds Containing the N7...mentioning

confidence: 99%

Isothiazoles (1,2-thiazoles): synthesis, properties and applications

Kaberdin¹,

Поткин²

2002

Russ. Chem. Rev.

View full text Add to dashboard Cite

The most recent achievements in the chemistry of The most recent achievements in the chemistry of isothiazoles are generalised and systematised. The main applica-isothiazoles are generalised and systematised. The main applications of isothiazole derivatives are surveyed. The bibliography tions of isothiazole derivatives are surveyed. The bibliography includes 293 references includes 293 references. .

show abstract

“…We have also considered an alternative computational approach to this problem, described briefly below at the end of section 4; but a direct computation with fat points turns out to be more efficient in the cases we are considering. For a survey of this problem from the point of view of approximation and interpolation theory, the article [3] and the more recent [21] are excellent. For a survey of the problem from an algebraic geometry viewpoint, and its connections with other geometric and algebraic problems, as the Waring problem for forms, see [6], [23] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Bivariate Hermite Interpolation and Linear Systems of Plane Curves With Base Fat Points

Ciliberto¹,

Cioffi

Miranda

et al. 2003

Computer Mathematics

View full text Add to dashboard Cite

It is still an open question to determine in general the dimension of the vector space of bivariate polynomials of degree at most d which have all partial derivatives up through order m i − 1 vanish at each point p i (i = 1,. .. , n), for some fixed integer m i called multiplicity at p i. When the multiplicities are all equal, to m say, this problem has been attacked by a number of authors (Lorentz and Lorentz, Ciliberto and Miranda, Hirschowitz) and there are a number of good conjectures (Hirschowitz, Ciliberto and Miranda) on the dimension of these interpolating spaces. The determination of the dimension has been already solved for m ≤ 12 and all d and n by a degeneration technique and some ad hoc geometric arguments. Here this technique is applied up through m = 20; since it fails in some cases, we resort (in these exceptional cases) to the bivariete Hermite interpolation with the support of a simple idea suggested by Gröbner bases computation. In summary we are able to prove that the dimension of the vector space is the expected one for 13 ≤ m ≤ 20.

show abstract

Multivariate splines and polynomial interpolation

Abstract: The mixed mercury halide lasers HgBr2iHgCI2 and HgBr,iHgl, have each been operated with wavelength-selective cavities and have each demonstrated their ability to provide high-Dower tunable radiation in two different parts of the visible spectrum.

Cited by 6 publications

References 41 publications

Zonotopal algebra

Zonotopal algebra

Isothiazoles (1,2-thiazoles): synthesis, properties and applications

Bivariate Hermite Interpolation and Linear Systems of Plane Curves With Base Fat Points

Contact Info

Product

Resources

About