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q-Blossoming: A new approach to algorithms and identities for q-Bernstein bases and q-Bézier curves

2012
*
Journal of Approximation Theory
*

We introduce a new variant of

doi:10.1016/j.jat.2011.09.006
fatcat:newbrv2f6vbtfb4755rlkqiupy
*the**blossom*,*the*q-*blossom*, by altering*the*diagonal property of*the*standard*blossom*. ... new,*affine**invariant*, recursive evaluation algorithms. Using two of these new recursive evaluation algorithms, we construct a recursive subdivision algorithm for q-Bézier curves. ... Thus in*the*first algorithm*the*intermediate values are*affine**invariant*, but in*the*second algorithm only*the*final result is*affine**invariant*. ...##
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q-Blossoming for analytic functions

2018
*
Numerical Algorithms
*

We construct a q-analogue of

doi:10.1007/s11075-018-0595-y
fatcat:j727hd2yhzgixo524hzul4ofyy
*the**blossom*for*analytic*functions,*the**analytic*q-*blossom*. This q-analogue also extends*the*notion of q-*blossoming*from polynomials to*analytic*functions. ...*The*goal of this paper is to generalize this*analytic**blossom*to an*analytic*q-*blossom*, where once again*the*classical derivative is replaced by a quantum derivative, and then to apply this*analytic*q-*blossom*... Since*the*q-Poisson basis functions are non-negative on [0, 1] and form a partition of unity, q-Poisson curves are*affine**invariant*and lie in*the*convex hull of their control points. ...##
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Manifold splines

2006
*
Graphical Models
*

We study

doi:10.1016/j.gmod.2006.03.004
fatcat:vlkc4cify5ffjf7dflyxy36cwa
*the**affine*structure of domain manifolds in depth and prove that*the*existence of manifold splines is equivalent to*the*existence of a manifold's*affine*atlas. ... As a result, our new spline surface defined over any manifold is a piecewise polynomial surface with high parametric continuity without*the*need for any patching and/or trimming operations. ... Acknowledgments*The*authors thank*the*reviewers for their careful review and helpful suggestions. This work was partially supported by*the*NSF CAREER Award CCF-0448339 to X. ...##
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Manifold splines

2005
*
Proceedings of the 2005 ACM symposium on Solid and physical modeling - SPM '05
*

We study

doi:10.1145/1060244.1060249
dblp:conf/sma/GuHQ05
fatcat:inyekbp2gvf3tlerj6qoiv236i
*the**affine*structure of domain manifolds in depth and prove that*the*existence of manifold splines is equivalent to*the*existence of a manifold's*affine*atlas. ... As a result, our new spline surface defined over any manifold is a piecewise polynomial surface with high parametric continuity without*the*need for any patching and/or trimming operations. ... Acknowledgments*The*authors thank*the*reviewers for their careful review and helpful suggestions. This work was partially supported by*the*NSF CAREER Award CCF-0448339 to X. ...##
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Inflection points and singularities on C-curves

2004
*
Computer Aided Geometric Design
*

We show that all so-called C-curves are

doi:10.1016/j.cagd.2003.11.002
fatcat:5uesosprezbxjfotireyioh5za
*affine*images of trochoids or sine curves and use this relation to investigate*the*occurrence of inflection points, cusps, and loops. ...*The*results are summarized in a shape diagram of C-Bézier curves, which is useful when using C-Bézier curves for curve and surface modeling. ... Acknowledgements This work was partly supported by*the*National Natural Science ...##
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A Blossoming Development of Splines

2006
*
Synthesis Lectures on Computer Graphics and Animation
*

*The*primary analysis tool used in this lecture is

*blossoming*, which gives an elegant labeling of

*the*control points that allows us to analyze their properties geometrically. ... Bézier/B-splines represent polynomials and piecewise polynomials in a geometric manner using sets of control points that define

*the*shape of

*the*surface. ...

*invariance*: T( i P i B n i (u)) = i T(P i )B n i (u) . ...

##
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Gerhard Hochschild (1915/2010) A Mathematician of the XXth Century
[article]

2011
*
arXiv
*
pre-print

Gerhard Hochschild's contribution to

arXiv:1104.0335v1
fatcat:d4ylcde575grzm47junz6qfceq
*the*development of mathematics in*the*XX century is succinctly surveyed. ... Indeed, if we have an*affine*algebraic group H acting on an*affine*variety with ring of polynomials R, as one expects that*the**invariant*polynomials R H separate*the*geometric orbits -at least generically-to ... His collaboration with Mostow*blossomed*with*the*stream of papers written on Representative functions (see in [14] ,*the*articles by A. Magid and D. ...##
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Page 4693 of Mathematical Reviews Vol. , Issue 86j
[page]

1986
*
Mathematical Reviews
*

In this paper,

*the*author shows that*the*Pythagoras*blossoms*may be obtained as*the*Julia set of a pair of randomly iterated*affine*maps. ...*The*result is*the*Pythagoras tree.*The*limit points of this construction are*the**blossoms*on*the*tree. ...##
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Approximate Reachability Computation for Polynomial Systems
[chapter]

2006
*
Lecture Notes in Computer Science
*

We propose a method to do so using

doi:10.1007/11730637_13
fatcat:hzvovfkdovc5xgw2ufvqqkhipy
*the*Bézier control net of*the*polynomial map and*the**blossoming*technique to compute this control net. We also prove that our overall method is of order 2. ... In order to use this scheme to compute all*the*states reachable by*the*system starting from some initial set, we then consider*the*problem of computing*the*image of a set by a multivariate polynomial. ...*invariants*[29] ). ...##
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Integrable Mappings from a Unified Perspective
[article]

2019
*
arXiv
*
pre-print

for

arXiv:1901.08174v1
fatcat:cif7lktanfeorbd5aegwaiew3a
*the*associated probability distributions. ...*The*two models are integrable and our analysis uncovers*the*geometric sources of this integrability and uses this to conceptually explain*the*rigorous existence and structure of elegant closed form expressions ... Hence our map determines an algebro-geometric addition law on each of*the**affinely*smooth level sets of*the**invariant*. ...##
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Page 5168 of Mathematical Reviews Vol. , Issue 2002G
[page]

2002
*
Mathematical Reviews
*

Milbrandt, A geo- metrically motivated

*affine**invariant*norm (267-280); A. Nawotki, Exploiting wavelet coefficients for modifying functions (281-292); M. Robinson, M. I. G. Bloor and M. J. ... Goldman,*Blossoming*and divided difference (155-184); S. Hahmann, G.-P. Bonneau and R. Taleb, Localizing*the*4-split method for G' free- form surface fitting (185-198); B. Heckel, A. E. Uva, B. ...##
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Political Culture and Covalent Bonding. A Conceptual Model of Political Culture Change

2015
*
Social Science Research Network
*

Project"s goals are defined with respect to

doi:10.2139/ssrn.2556161
fatcat:pbc725xmonbjrpfh6pbnzcuvgy
*the*specificity of*the*Eastern European geopolitical space:*the*impact of*the*political culture legacy of*the*Eastern European communist regimes on*the*political ... Acknowledgements*The*ongoing research reported here represents a research project initiated and developed by*the*Author as*the*Founding President of*the*European Research Group on Political Attitudes and ...*The*approaches are based on rather numerical combinatorial techniques, using intensively empirical data to*analytical*purposes. ...##
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s-power series: an alternative to Poisson expansions for representing analytic functions

2005
*
Computer Aided Geometric Design
*

Morin and Goldman [Computer Aided Geometric Design 17 (2000) 813] have recently presented a remarkable new framework, based on employing Poisson series, for describing

doi:10.1016/j.cagd.2004.09.003
fatcat:lj4hwky3bjf6fbnrw5gyw2tmsi
*analytic*functions in CAD. ... We compare this Poisson formulation with s-power series, modified Newton series that can be regarded as*the*two-point analogue of Taylor expansions. ... Acknowledgement This work is supported by*the*Spanish Ministerio de Ciencia y Tecnología, under research grant DPI2003-02598. ...##
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:{unav)

2012
*
Advances in Computational Mathematics
*

Several features of toric patches are considered:

doi:10.1023/a:1015289823859
fatcat:fxwvxeh7knc5rleqgzkr3imqie
*affine**invariance*, convex hull property, boundary curves, implicit degree and singular points. ...*The*method of subdivision into tensor product surfaces is introduced. Fundamentals of a multidimensional variant of this theory are also developed. ... Goldman for very helpful comments, and to referees for pointing out some errors in*the*first version of*the*paper. ...##
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First steps in symplectic and spectral theory of integrable systems
[article]

2013
*
arXiv
*
pre-print

Even though quantum integrable systems date back to

arXiv:1306.0124v1
fatcat:hmpcrwcaffg3zepjfkw6izke7u
*the*early days of quantum mechanics, such as*the*work of Bohr, Sommerfeld and Einstein,*the*theory did not*blossom*at*the*time. ...*The*paper intends to lay out*the*first steps towards constructing a unified framework to understand*the*symplectic and spectral theory of finite dimensional integrable Hamiltonian systems. ... His numerous comments prompted us to rethink parts of*the*paper and have greatly improved*the*exposition and*the*clarity. ...
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