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Parametric Möbius inversion formulas

Beifang Chen
1997 Discrete Mathematics  
A M6bius inversion formula on the product of a locally finite poset and an arbitrary set is presented. The M6bius inversion of Number Theory is modified to several inversion formulas.  ...  The idea of introducing parameters should be useful in the applications of some combinatorial inversion formulas and inverse transformations.  ...  The multi-dimensional versions of the inversion formulas (7)-(12) are straightforward. For instance, (7) and (8) Example 2.  ... 
doi:10.1016/s0012-365x(96)00085-4 fatcat:bbwx754tyfbn5biynu5evyd26u

Modelling relational statistics with Bayes Nets

Oliver Schulte, Hassan Khosravi, Arthur E. Kirkpatrick, Tianxiang Gao, Yuke Zhu
2013 Machine Learning  
We render the computation of these empirical frequencies tractable in the presence of negated relations by the inverse Möbius transform.  ...  We represent class statistics using Parametrized Bayes Nets (PBNs), a first-order logic extension of Bayes nets.  ...  Acknowledgements Lise Getoor's work on Statistical Relational Models inspired us to consider class-level modelling with Parametrized Bayes nets; we thank her for helpful comments and encouragement.  ... 
doi:10.1007/s10994-013-5362-7 fatcat:yj7obxtsebc7bdqvrtsbthfcva

Möbius invariant metrics on the space of knots [article]

Jun O'Hara
2020 arXiv   pre-print
function to produce a M\"obius invariant weighted inner product on the tangent space of the space of knots, and show that some kind of M\"obius invariant knot energies can produce M\"obius invariant and parametrization  ...  G V 3 e (f ) = (V (f )) −3 G e (f ) (f ∈ K • ) is a Möbius invariant and parametrization independent gradient with respect to · , · V 3 .  ...  Let us look for a weighted inner product that is compatible with Möbius transformations and that is independent of the parametrization of knots.  ... 
arXiv:1905.06098v3 fatcat:svkh66l7lrap7ffhm3mprddgle

The Möbius Curvature of Bezier Curves

Filiz ERTEM KAYA
2021 European Journal of Science and Technology  
The aim of this study is to observe the Möbius curvature is computed by me as using curvature of Bezier curve is therefore proportional to the differentials of the curvature also correspond to a such as  ...  The Möbius curvature of Bezier curve has different value according to the control points. Also when the different cases may ocur, it has different values according to the angle is constant or not.  ...  If take a parametrization-invariant Möbius invariant known as the inversive Bezier curve or Möbius curvature of Bezier curve at 0  t point. Proof.  ... 
doi:10.31590/ejosat.992818 fatcat:7xa77bi5ifdird3vprnko457xe

Rational Bézier Formulas with Quaternion and Clifford Algebra Weights [chapter]

Rimvydas Krasauskas, Severinas Zubė
2014 Geometry and Computing  
We consider Bézier-like formulas with weights in quaternion and geometric (Clifford) algebra for parametrizing rational curves and surfaces.  ...  Such formulas reproduce well known biquadratic parametrizations of principal Dupin cyclide patches, and are characterized in general as special Darboux cyclide patches.  ...  Möbius Transformations in R 3 Möbius (M) transformations in space are generated by inversions in R 3 with respect to spheres.  ... 
doi:10.1007/978-3-319-08635-4_8 fatcat:ijaix3n5prbzze4de2gf4miady

Ptolemy spaces with strong inversions

A. Smirnov
2014 St. Petersburg Mathematical Journal  
It is proved that a compact Ptolemy space with many strong inversions that contains a Ptolemy circle is Möbius equivalent to an extended Euclidean space.  ...  Since X is not Möbius equivalent to p R, there is x ∈ X \ σ. Let c : R → X ω be a unit speed parametrization of the Ptolemy line = σ \ ω such that the horosphere H of through c(0) contains x .  ...  We define a strong space inversion, or s-inversion for brevity, with respect to distinct ω, ω ∈ X and a metric sphere S ⊂ X between ω, ω as a Möbius automorphism ϕ = ϕ ω,ω ,S : X → X such that (1) ϕ is  ... 
doi:10.1090/s1061-0022-2014-01327-1 fatcat:bj2326u2pbbpxjtzkrzacl7oui

Cauchy integrals and Möbius geometry of curves

David E. Barrett, Michael Bolt
2007 Asian Journal of Mathematics  
Inversive arc-length: From (5) we see that | Im Sγ * (t)| dt defines a parametrization-independent Möbius-invariant 1-form on γ.  ...  Our arc can be reconstructed up to post-composition with a Möbius transformation from the inversive curvature (viewed as a function of inversive arc-length) by solving the equation Sγ * = ±i + κ inv [Leh  ... 
doi:10.4310/ajm.2007.v11.n1.a6 fatcat:bdp6hcvmofco3my7padjo52urm

A stochastic interpretation of the Riemann zeta function [chapter]

Kenneth S. Alexander, Kenneth Baclawski, Gian-Carlo Rota
2003 Gian-Carlo Rota on Analysis and Probability  
Mobius Inversion on an Infinite Lattice We will use Mobius inversion on L, so it is useful to formalize the basic convergence result we need. THEOREM 1. Let P be a locally finite poset.  ...  Therefore Absolute convergence clearly also holds. i We now give an example of the use of Mobius inversion in the setting of Section 2. THEOREM 2.  ... 
doi:10.1007/978-1-4612-2070-1_41 fatcat:ylhf4bgsujbhpa23bcvy6y2tua

A stochastic interpretation of the Riemann zeta function

K. S. Alexander, K. Baclawski, G. C. Rota
1993 Proceedings of the National Academy of Sciences of the United States of America  
Mobius Inversion on an Infinite Lattice We will use Mobius inversion on L, so it is useful to formalize the basic convergence result we need. THEOREM 1. Let P be a locally finite poset.  ...  Therefore Absolute convergence clearly also holds. i We now give an example of the use of Mobius inversion in the setting of Section 2. THEOREM 2.  ... 
doi:10.1073/pnas.90.2.697 pmid:11607353 pmcid:PMC45731 fatcat:4n6dbln5ezg63ac3qfbfkibtya

Generalized KP hierarchy: Möbius symmetry, symmetry constraints and Calogero–Moser system

L.V. Bogdanov, B.G. Konopelchenko
2001 Physica D : Non-linear phenomena  
A more general class of multicomponent Möbius-type symmetries is studied.  ...  It is demonstrated that symmetry constraints of KP hierarchy defined using multicomponent Möbius-type symmetries give rise to Calogero-Moser system.  ...  this is nothing more then one-parametric subgroup of the Möbius group.  ... 
doi:10.1016/s0167-2789(01)00161-0 fatcat:h7h43ddarbfn3csefbexwrjbty

Page 7317 of Mathematical Reviews Vol. , Issue 97M [page]

1997 Mathematical Reviews  
Summary: “A Mobius inversion formula on the product of a lo- cally finite poset and an arbitrary set is presented. The MObius in- version of number theory is modified to several inversion formulas.  ...  {For the entire collection see MR 97k:68005. } 97m:05007 0S5A15 Chen, Beifang (HK-HKST; Kowloon) Parametric Mobius inversion formulas. (English summary) Discrete Math. 169 (1997), no. 1-3, 211-215.  ... 

The Radon Transform on SL(2,R)/SO(2,R)

D. I. Wallace, Ryuji Yamaguchi
1986 Transactions of the American Mathematical Society  
Technically, the inversion depends heavily on Fourier analysis on R2.  ...  For R2, the solution to this problem was the inversion of the original Radon-John transform.  ...  In §4, we discuss an inversion formula. We work with four different parametrizations of P.  ... 
doi:10.2307/2000470 fatcat:xqfre7tvnngvbhmzxyzdqi3e44

Generalized KP hierarchy: Möbius Symmetry, Symmetry Constraints and Calogero-Moser System [article]

L.V. Bogdanov, B.G. Konopelchenko
1999 arXiv   pre-print
this is nothing more then one-parametric subgroup of the Möbius group.  ...  Generic Möbius transformation can be represented as composition of translation, scaling and inversion.  ... 
arXiv:solv-int/9912005v1 fatcat:fxepgerzfjbzjcezlmt3gmuv4u

The Radon transform on ${\rm SL}(2,{\bf R})/{\rm SO}(2,{\bf R})$

D. I. Wallace, Ryuji Yamaguchi
1986 Transactions of the American Mathematical Society  
Technically, the inversion depends heavily on Fourier analysis on R2.  ...  For R2, the solution to this problem was the inversion of the original Radon-John transform.  ...  In §4, we discuss an inversion formula. We work with four different parametrizations of P.  ... 
doi:10.1090/s0002-9947-1986-0849481-7 fatcat:sc3wrxoikvepdctydtjdwhxxwy

Numerical harmonic analysis on the hyperbolic plane

Buma Fridman, Peter Kuchment, Kirk Lancaster, Serguei Lissianoi, Mila Mogilevsky, Daowei Ma, Igor Ponomarev, Vassilis Papanicolaou
2000 Applicable Analysis  
The study is motivated by the hyperbolic geometry approach to the linearized inverse conductivity problem, suggested by C. A. Berenstein and E. Casadio Tarabusi.  ...  Results are reported of a numerical implementation of the hyperbolic Fourier transform and the geodesic and horocyclic Radon transforms on the hyperbolic plane, and of their inverses.  ...  Inversion of Fourier and Radon transforms on the hyperbolic plane A Fourier inversion formula was obtained by Helgason [15] : f (z) = 1 4π R ∂D f (λ, b)e (iλ+1) z,b λ th πλ 2 dλdb. (2) Together with formula  ... 
doi:10.1080/00036810008840889 fatcat:tu6bk7y36jfnjbxsihhz7v4f5q
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