Horizontal Gradient of Polynomial Functions for the Standard Engel Structure on ℝ4

2014 ◽  
Vol 22 (1) ◽  
pp. 15-34
Author(s):  
Si Tiep Dinh ◽  
Krzysztof Kurdyka
Keyword(s):  
Horizontal Gradient ◽  
Polynomial Functions ◽  
Engel Structure

ε-InSe single crystals grown by a horizontal gradient freeze method

CrystEngComm ◽  
2020 ◽  
Vol 22 (45) ◽  
pp. 7864-7869
Author(s):  
Maojun Sun ◽  
Wei Wang ◽  
Qinghua Zhao ◽  
Xuetao Gan ◽  
Yuanhui Sun ◽  
...  
Keyword(s):  
Single Crystals ◽  
Optoelectronic Device ◽  
Indium Selenide ◽  
Horizontal Gradient ◽  
Device Applications

Indium selenide (InSe) single crystals have been considered as promising candidates for future optical, electrical, and optoelectronic device applications.

scholarly journals On the group of unit-valued polynomial functions

2021 ◽  
Author(s):  
Amr Ali Al-Maktry
Keyword(s):  
Commutative Ring ◽  
Semidirect Product ◽  
Polynomial Functions ◽  
Dual Numbers ◽  
Ring Of Integers ◽  
Group Of Units ◽  
Canonical Representations ◽  
Finite Commutative Ring

AbstractLet R be a finite commutative ring. The set $${{\mathcal{F}}}(R)$$ F ( R ) of polynomial functions on R is a finite commutative ring with pointwise operations. Its group of units $${{\mathcal{F}}}(R)^\times $$ F ( R ) × is just the set of all unit-valued polynomial functions. We investigate polynomial permutations on $$R[x]/(x^2)=R[\alpha ]$$ R [ x ] / ( x 2 ) = R [ α ] , the ring of dual numbers over R, and show that the group $${\mathcal{P}}_{R}(R[\alpha ])$$ P R ( R [ α ] ) , consisting of those polynomial permutations of $$R[\alpha ]$$ R [ α ] represented by polynomials in R[x], is embedded in a semidirect product of $${{\mathcal{F}}}(R)^\times $$ F ( R ) × by the group $${\mathcal{P}}(R)$$ P ( R ) of polynomial permutations on R. In particular, when $$R={\mathbb{F}}_q$$ R = F q , we prove that $${\mathcal{P}}_{{\mathbb{F}}_q}({\mathbb{F}}_q[\alpha ])\cong {\mathcal{P}}({\mathbb{F}}_q) \ltimes _\theta {{\mathcal{F}}}({\mathbb{F}}_q)^\times $$ P F q ( F q [ α ] ) ≅ P ( F q ) ⋉ θ F ( F q ) × . Furthermore, we count unit-valued polynomial functions on the ring of integers modulo $${p^n}$$ p n and obtain canonical representations for these functions.

Level curves of open polynomial functions on the real plane

1994 ◽  
Vol 22 (14) ◽  
pp. 5973-5981
Author(s):  
J. Ferrera ◽  
M.J. de la Puente
Keyword(s):  
Real Plane ◽  
Polynomial Functions ◽  
The Real ◽  
Level Curves

A proof of a theorem by Fried and MacRae and applications to the composition of polynomial functions

2011 ◽  
Vol 97 (2) ◽  
pp. 115-124 ◽  
Author(s):  
Erhard Aichinger ◽  
Stefan Steinerberger
Keyword(s):  
Polynomial Functions

Properties of reversible polynomial functions

1997 ◽  
Vol 33 (5) ◽  
pp. 647-651
Author(s):  
A. A. Mishchenko
Keyword(s):  
Polynomial Functions
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