Periodic simple groups of finitary linear transformations

We study the problem of determining, for a polynomial function$f$on a vector space$V$, the linear transformations$g$of$V$such that$f\circ g=f$. When$f$is invariant under a simple algebraic group$G$acting irreducibly on$V$, we note that the subgroup of$\text{GL}(V)$stabilizing$f$often has identity component$G$, and we give applications realizing various groups, including the largest exceptional group$E_{8}$, as automorphism groups of polynomials and algebras. We show that, starting with a simple group$G$and an irreducible representation$V$, one can almost always find an$f$whose stabilizer has identity component$G$, and that no such$f$exists in the short list of excluded cases. This relies on our core technical result, the enumeration of inclusions$G<H\leqslant \text{SL}(V)$such that$V/H$has the same dimension as$V/G$. The main results of this paper are new even in the special case where$k$is the complex numbers.

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Linear Transformations, Projection Operators and Generalized Inverses; A Geometric Approach

10.21236/ada197608 ◽

1988 ◽

Author(s):

C. R. Rao

Keyword(s):

Geometric Approach ◽

Generalized Inverses ◽

Projection Operators ◽

Linear Transformations

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Nonuniform Spatial Deformation of Light Fields by Locally Linear Transformations

ACM Transactions on Graphics ◽

10.1145/3072959.2928267 ◽

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

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A Comparison of Estimates of Michaelis-Menten Kinetic Constants from Various Linear Transformations

Journal of Biological Chemistry ◽

10.1016/s0021-9258(17)45254-9 ◽

1965 ◽

Vol 240 (2) ◽

pp. 863-869 ◽

Cited By ~ 49

Author(s):

John E. Dowd ◽

Douglas S. Riggs

Keyword(s):

Kinetic Constants ◽

Linear Transformations ◽

Menten Kinetic

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

Biology ◽

10.3390/biology10010045 ◽

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.

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A diameter bound for finite simple groups of large rank

Journal of the London Mathematical Society ◽

10.1112/jlms.12018 ◽

2017 ◽

Vol 95 (2) ◽

pp. 455-474 ◽

Cited By ~ 2

Author(s):

Arindam Biswas ◽

Yilong Yang

Keyword(s):

Simple Groups ◽

Finite Simple Groups ◽

Large Rank

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Conversion Between Cubic Bezier Curves and Catmull–Rom Splines

SN Computer Science ◽

10.1007/s42979-021-00770-x ◽

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.

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