scholarly journals Linear transformations that are tridiagonal with respect to both eigenbases of a Leonard pair

2007 ◽  
Vol 420 (1) ◽  
pp. 198-207 ◽  
Author(s):  
Kazumasa Nomura ◽  
Paul Terwilliger
Keyword(s):  
Linear Transformations ◽  
Leonard Pair

scholarly journals LEONARD PAIRS AND THE ASKEY–WILSON RELATIONS

2004 ◽  
Vol 03 (04) ◽  
pp. 411-426 ◽  
Author(s):  
PAUL TERWILLIGER ◽  
RAIMUNDAS VIDUNAS
Keyword(s):  
Vector Space ◽  
Linear Transformations ◽  
Leonard Pairs ◽  
Positive Dimension ◽  
The Matrix ◽  
Leonard Pair

Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider an ordered pair of linear transformations A:V→V and A*:V→V which satisfy the following two properties: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A* is diagonal. (ii) There exists a basis for V with respect to which the matrix representing A* is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonard pair on V. Referring to the above Leonard pair, we show there exists a sequence of scalars β,γ,γ*,ϱ,ϱ*,ω,η,η* taken from K such that both [Formula: see text] The sequence is uniquely determined by the Leonard pair provided the dimension of V is at least 4. The equations above are called the Askey–Wilson relations.

Nonuniform Spatial Deformation of Light Fields by Locally Linear Transformations

2017 ◽  
Vol 36 (4) ◽  
pp. 1
Author(s):  
Clemens Birklbauer ◽  
David C. Schedl ◽  
Oliver Bimber
Keyword(s):  
Light Fields ◽  
Linear Transformations ◽  
Spatial Deformation ◽  
Locally Linear

scholarly journals Interconversion of Plasma Free Thyroxine Values from Assay Platforms with Different Reference Intervals Using Linear Transformation Methods

Biology ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 45
Author(s):  
Fanwen Meng ◽  
Jacqueline Jonklaas ◽  
Melvin Khee-Shing Leow
Keyword(s):  
Linear Transformation ◽  
Piecewise Linear ◽  
Equilibrium Dialysis ◽  
Accurate Method ◽  
Reference Intervals ◽  
Thyroid Stimulating Hormone ◽  
Free Thyroxine ◽  
Thyroid Function Tests ◽  
Linear Transformations ◽  
Transformation Methods

Clinicians often encounter thyroid function tests (TFT) comprising serum/plasma free thyroxine (FT4) and thyroid stimulating hormone (TSH) measured using different assay platforms during the course of follow-up evaluations which complicates reliable comparison and interpretation of TFT changes. Although interconversion between concentration units is straightforward, the validity of interconversion of FT4/TSH values from one assay platform to another with different reference intervals remains questionable. This study aims to establish an accurate and reliable methodology of interconverting FT4 by any laboratory to an equivalent FT4 value scaled to a reference range of interest via linear transformation methods. As a proof-of-concept, FT4 was simultaneously assayed by direct analog immunoassay, tandem mass spectrometry and equilibrium dialysis. Both linear and piecewise linear transformations proved relatively accurate for FT4 inter-scale conversion. Linear transformation performs better when FT4 are converted from a more accurate to a less accurate assay platform. The converse is true, whereby piecewise linear transformation is superior to linear transformation when converting values from a less accurate method to a more robust assay platform. Such transformations can potentially apply to other biochemical analytes scale conversions, including TSH. This aids interpretation of TFT trends while monitoring the treatment of patients with thyroid disorders.

scholarly journals Conversion Between Cubic Bezier Curves and Catmull–Rom Splines

2021 ◽  
Vol 2 (5) ◽  
Author(s):  
Soroosh Tayebi Arasteh ◽  
Adam Kalisz
Keyword(s):  
Computer Graphics ◽  
Geometric Design ◽  
The Other ◽  
Control Points ◽  
Complex Surfaces ◽  
Linear Transformations ◽  
Computer Aided Geometric Design ◽  
Cubic Curves ◽  
Original Curve ◽  
Computer Aided

AbstractSplines are one of the main methods of mathematically representing complicated shapes, which have become the primary technique in the fields of Computer Graphics (CG) and Computer-Aided Geometric Design (CAGD) for modeling complex surfaces. Among all, Bézier and Catmull–Rom splines are the most common in the sub-fields of engineering. In this paper, we focus on conversion between cubic Bézier and Catmull–Rom curve segments, rather than going through their properties. By deriving the conversion equations, we aim at converting the original set of the control points of either of the Catmull–Rom or Bézier cubic curves to a new set of control points, which corresponds to approximately the same shape as the original curve, when considered as the set of the control points of the other curve. Due to providing simple linear transformations of control points, the method is very simple, efficient, and easy to implement, which is further validated in this paper using some numerical and visual examples.

Linear transformations extract parameters of ionic currents and action potential features from pseudo-ECG in cardiomyocyte models

2020 ◽  
Author(s):  
Salim Baigildin ◽  
Konstantin Ushenin ◽  
Aigul Fabarisova ◽  
Marat Bogdanov ◽  
Olga Solovyeva
Keyword(s):  
Action Potential ◽  
Ionic Currents ◽  
Linear Transformations

scholarly journals Zero-diminishing linear transformations

1980 ◽  
Vol 80 (4) ◽  
pp. 544-544 ◽  
Author(s):  
Thomas Craven ◽  
George Csordas
Keyword(s):  
Linear Transformations

THE NATURAL PARTIAL ORDER ON LINEAR SEMIGROUPS WITH NULLITY AND CO-RANK BOUNDED BELOW

2014 ◽  
Vol 91 (1) ◽  
pp. 104-115 ◽  
Author(s):  
SUREEPORN CHAOPRAKNOI ◽  
TEERAPHONG PHONGPATTANACHAROEN ◽  
PONGSAN PRAKITSRI
Keyword(s):  
Semigroup Forum ◽  
Partial Order ◽  
Lower Bounds ◽  
Partial Orders ◽  
Natural Partial Order ◽  
Linear Transformations ◽  
Bounded Below

AbstractHiggins [‘The Mitsch order on a semigroup’, Semigroup Forum 49 (1994), 261–266] showed that the natural partial orders on a semigroup and its regular subsemigroups coincide. This is why we are interested in the study of the natural partial order on nonregular semigroups. Of particular interest are the nonregular semigroups of linear transformations with lower bounds on the nullity or the co-rank. In this paper, we determine when they exist, characterise the natural partial order on these nonregular semigroups and consider questions of compatibility, minimality and maximality. In addition, we provide many examples associated with our results.

scholarly journals Unisoft Hypervector Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Abdullah M. Al-Roqi
Keyword(s):  
Linear Transformations

The notion of unisoft subfields, unisoft algebras over unisoft subfields, and unisoft hypervector spaces are introduced, and their properties and characterizations are considered. In connection with linear transformations, unisoft hypervector spaces are discussed.

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