Questions tagged [involutions]
For problems related to involutions , that is functions that are their own inverses .
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Examples of decompositions of involutions
The book "Algebra and Geometry" by Beardon introduces Theorem 6.1.4 which states that "Every isometry of $\mathbb{R}^3$ is the composition of at most four reflections", and then ...
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A certain subfield of $\mathbb{C}(x,y)$
Let $u,v \in \mathbb{C}[x,y]$, so $\mathbb{C}(u,v) \subseteq \mathbb{C}(x,y)$.
A monomial $x^iy^j$ of $\mathbb{C}[x,y]$ has one of the following forms:
$i$ is even, $j$ is even, "type 1 ...
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Involution on polynomial rings
A map $\ast$ on a ring $R$ is said to be involution if the following properties holds for each element of $a,b\in R$:
$(a+b)^\ast=a^\ast + b^\ast$
$(ab)^\ast=b^\ast a^\ast$
$(a^\ast)^\ast=a$.
I want ...
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Does duality mapping preserve cross ratio?
I'm new to projective geometry.
I learned the definition of cross ratio of 4 collinear points and that of 4 concurrent lines in $\mathit{P}\mathbb{R}^{2}$.
The question is, by duality we can map 4 ...
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Involutions in PCO
In the algebraic group $G=\operatorname {PCO}(4,K)$ where $K$ is an algebraically closed field of an odd characteristic, how many different classes of involutions are there in $G \setminus \...
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Recurrence differential equations arising from the Normal PDF
Let $a,b,c:\mathbb{R}\rightarrow\mathbb{R}$ be differentiable functions, with $a(t)\rightarrow -\infty$ and $c(t)\rightarrow -\infty$ as $t\rightarrow -\infty$, and with $a(t)\rightarrow\infty$ and $c(...
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Centralizer generators
This is a question posted on overflow but no reply has been received.
In the algebraic group $G=\operatorname {PSO}(4,K)<\operatorname {PCGO}(4,K)$ where $K$ is an algebraically closed field of an ...
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Passing an involution to a quotient algebra
This question is inspired by the discussion under this MO answer. I hope I have captured correctly what is going on in the below.
Let $A_0$ be a finitely generated universal unital complex algebra $...
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Every continuous involution on R^n has a fixed point [duplicate]
I'm looking for a reference which supports the claim that every continuous involution on $\mathbb{R}^n$ has a fixed point.
This fact is discussed here, but I'd like to see this as a standalone claim ...
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Could any "incoherently involutive" endofunctor be made "coherently involutive"?
Given a category $C$ with an endofunctor $F:C \to C$ and a natural isomorphism $\epsilon:FF \cong 1_C$, call the pair $(F, \epsilon)$ an involutive endofunctor.
Also, call the pair $(F, \epsilon)$ a ...
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Solving the Functional Equation $f(f(x))=x$ gone wrong
For a function to be its inverse (i.e. an involution), it needs to satisfy the functional equation $f(x)=f^{-1}(x)$ or $f(f(x))=x$. I expressed $f^{\circ n}(x)$ (the composition of $f(x)$ to itself $n$...
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$L^1(G)$ is a Banach *-Algebra [duplicate]
Let G be a locally compact group with left Haar measure $\mu$. In Principles of Harmonic Analysis, it's affirmed that $L^1(G)$ is a Banach *-Algebra, where the multiplication operation is convolution ...
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Does every complex involutive algebra admit at least one non-trivial C*-seminorm?
Let $A$ be a unital involutive algebra over $\mathbb{C}$. A $C^*$-seminorm is a seminorm $p$ such that $p(x^*x) = p(x)^2, \forall x \in A$.
I understand that if the spectral radius of $x \in A$ given ...
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Determine the involution when given two pairs of point on a line
I'm studying about involution on a projective line (line with point at infinity). An involution is a map from a projective line $l$ to itself that satisfied $f \circ f =$ is the identity map.
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In a finite reflection group, an involution is a product of commuting reflections
I am working through the book Reflection Groups and Coxeter Groups by Humphreys. I got stuck while trying Exercise 1.12.3:
If $w \in W$ is an involution, prove that $w$ can be written as a product of ...
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