scholarly journals On the eigenvalues of some class of pseudo-linear transformations

2007 ◽  
pp. 79-83
Author(s):  
Milica Andjelic
Keyword(s):  
Linear Transformation ◽  
Closed Field ◽  
Linear Transformations ◽  
Algebraically Closed Field ◽  
New Criterion

We develop a connection between the eigenvalues of a class of pseudo-linear transformation over a field K and the eigenvalues of a certain linear transformation. We give a new criterion for this class to be diagonalizable over algebraically closed field.

The Dynamical Mordell–Lang Conjecture for Skew-Linear Self-Maps. Appendix by Michael Wibmer

2018 ◽  
Vol 2020 (21) ◽  
pp. 7433-7453
Author(s):  
Dragos Ghioca ◽  
Junyi Xie
Keyword(s):  
Linear Transformation ◽  
Difference Equations ◽  
Dense Subset ◽  
Closed Field ◽  
Open Dense Subset ◽  
Irreducible Curve ◽  
Algebraically Closed Field ◽  
Set Of Points ◽  
Dense Set

Abstract Let $k$ be an algebraically closed field of characteristic $0$, let $N\in{\mathbb{N}}$, let $g:{\mathbb{P}}^1{\longrightarrow } {\mathbb{P}}^1$ be a nonconstant morphism, and let $A:{\mathbb{A}}^N{\longrightarrow } {\mathbb{A}}^N$ be a linear transformation defined over $k({\mathbb{P}}^1)$, that is, for a Zariski-open dense subset $U\subset{\mathbb{P}}^1$, we have that for $x\in U(k)$, the specialization $A(x)$ is an $N$-by-$N$ matrix with entries in $k$. We let $f:{\mathbb{P}}^1\times{\mathbb{A}}^N{\dashrightarrow } {\mathbb{P}}^1\times{\mathbb{A}}^N$ be the rational endomorphism given by $(x,y)\mapsto (\,g(x), A(x)y)$. We prove that if $g$ induces an automorphism of ${\mathbb{A}}^1\subset{\mathbb{P}}^1$, then each irreducible curve $C\subset{\mathbb{A}}^1\times{\mathbb{A}}^N$ that intersects some orbit $\mathcal{O}_f(z)$ in infinitely many points must be periodic under the action of $f$. Furthermore, in the case $g:{\mathbb{P}}^1{\longrightarrow } {\mathbb{P}}^1$ is an endomorphism of degree greater than $1$, then we prove that each irreducible subvariety $Y\subset{\mathbb{P}}^1\times{\mathbb{A}}^N$ intersecting an orbit $\mathcal{O}_f(z)$ in a Zariski dense set of points must be periodic. Our results provide the desired conclusion in the Dynamical Mordell–Lang Conjecture in a couple new instances. Moreover, our results have interesting consequences toward a conjecture of Rubel and toward a generalized Skolem–Mahler–Lech problem proposed by Wibmer in the context of difference equations. In the appendix it is shown that the results can also be used to construct Picard–Vessiot extensions in the ring of sequences.

scholarly journals A New Criterion for Affineness

ISRN Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Jing Zhang
Keyword(s):  
Characteristic Zero ◽  
Large Degree ◽  
Direct Application ◽  
Closed Field ◽  
Affine Variety ◽  
Algebraically Closed Field ◽  
New Criterion

We show that an irreducible quasiprojective variety of dimension defined over an algebraically closed field with characteristic zero is an affine variety if and only if () = 0 and () = 0 for all , , where is any hypersurface with sufficiently large degree. A direct application is that an irreducible quasiprojective variety over is a Stein variety if it satisfies the two vanishing conditions. Here, all sheaves are algebraic.

scholarly journals TWO NON-NILPOTENT LINEAR TRANSFORMATIONS THAT SATISFY THE CUBIC q-SERRE RELATIONS

2007 ◽  
Vol 06 (03) ◽  
pp. 477-503 ◽  
Author(s):  
TATSURO ITO ◽  
PAUL TERWILLIGER
Keyword(s):  
Closed Field ◽  
Linear Transformations ◽  
Algebraically Closed Field ◽  
Root Of Unity ◽  
Finite Dimensional ◽  
Tridiagonal Pairs ◽  
Nonzero Scalar

Let 𝕂 denote an algebraically closed field with characteristic 0, and let q denote a nonzero scalar in 𝕂 that is not a root of unity. Let 𝔸q denote the unital associative 𝕂-algebra defined by generators x,y and relations [Formula: see text] where [3]q = (q3 - q-3)/(q - q-1). We classify up to isomorphism the finite-dimensional irreducible 𝔸q-modules on which neither of x,y is nilpotent. We discuss how these modules are related to tridiagonal pairs.

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 The strong simple connectedness of tame algebras with separating almost cyclic coherent Auslander–Reiten components

2021 ◽  
Author(s):  
Piotr Malicki
Keyword(s):  
Closed Field ◽  
Algebraically Closed Field ◽  
Main Application ◽  
Finite Dimensional ◽  
Simple Connectedness ◽  
Tame Algebras

AbstractWe study the strong simple connectedness of finite-dimensional tame algebras over an algebraically closed field, for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. As the main application we describe all analytically rigid algebras in this class.

scholarly journals On Unramified Separable Abelian p-Extensions of Function Fields I

1959 ◽  
Vol 14 ◽  
pp. 223-234 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Galois Group ◽  
Function Field ◽  
Function Fields ◽  
Abelian Extension ◽  
Closed Field ◽  
Jacobian Variety ◽  
Algebraically Closed Field

Let k be an algebraically closed field of characteristic p>0. Let K/k be a function field of one variable and L/K be an unramified separable abelian extension of degree pr over K. The galois automorphisms ε1, …, εpr of L/K are naturally extended to automorphisms η(ε1), … , η(εpr) of the jacobian variety JL of L/k. If we take a svstem of p-adic coordinates on JL, we get a representation {Mp(η(εv))} of the galois group G(L/K) of L/K over p-adic integers.

scholarly journals CHARACTER CLUSTERS FOR LIE ALGEBRA MODULES OVER A FIELD OF NONZERO CHARACTERISTIC

2013 ◽  
Vol 89 (2) ◽  
pp. 234-242 ◽  
Author(s):  
DONALD W. BARNES
Keyword(s):  
Lie Algebra ◽  
Saturated Formation ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Nonzero Characteristic ◽  
Composition Factors

AbstractFor a Lie algebra $L$ over an algebraically closed field $F$ of nonzero characteristic, every finite dimensional $L$-module can be decomposed into a direct sum of submodules such that all composition factors of a summand have the same character. Using the concept of a character cluster, this result is generalised to fields which are not algebraically closed. Also, it is shown that if the soluble Lie algebra $L$ is in the saturated formation $\mathfrak{F}$ and if $V, W$ are irreducible $L$-modules with the same cluster and the $p$-operation vanishes on the centre of the $p$-envelope used, then $V, W$ are either both $\mathfrak{F}$-central or both $\mathfrak{F}$-eccentric. Clusters are used to generalise the construction of induced modules.

scholarly journals Classification of one-dimensional superattracting germs in positive characteristic

2014 ◽  
Vol 35 (7) ◽  
pp. 2242-2268 ◽  
Author(s):  
MATTEO RUGGIERO
Keyword(s):  
Positive Characteristic ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Dimensional Version

We give a classification of superattracting germs in dimension $1$ over a complete normed algebraically closed field $\mathbb{K}$ of positive characteristic up to conjugacy. In particular, we show that formal and analytic classifications coincide for these germs. We also give a higher-dimensional version of some of these results.

scholarly journals Geometric Weil representation in characteristic two

2011 ◽  
Vol 11 (2) ◽  
pp. 221-271 ◽  
Author(s):  
Alain Genestier ◽  
Sergey Lysenko
Keyword(s):  
Weil Representation ◽  
Closed Field ◽  
Triangulated Category ◽  
Algebraically Closed Field ◽  
Witt Vectors ◽  
Suitable Sense ◽  
Characteristic Two ◽  
Greenberg Realization ◽  
Geometric Analogue

AbstractLet k be an algebraically closed field of characteristic two. Let R be the ring of Witt vectors of length two over k. We construct a group stack Ĝ over k, the metaplectic extension of the Greenberg realization of $\operatorname{\mathbb{S}p}_{2n}(R)$. We also construct a geometric analogue of the Weil representation of Ĝ, this is a triangulated category on which Ĝ acts by functors. This triangulated category and the action are geometric in a suitable sense.

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