Questions tagged [invariance]

A property of an object is called invariant if, given some steps that alter the object, always remains, no matter what steps are used in what order.

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Invariant norms of tensors

I am aware that certain special quantities of tensors (e.g. trace, det) are invariants (i.e. unchanging wrt coordinate system changes). This wikipedia article says that the invariants of tensors are coefficients of the characteristic polynomial of…
user3658307
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Why is integral over a domain invariant?

I am reading Pavel Grinfeld's "introduction to tensor analysis and the calculus of moving surfaces" and have some confusion in the following pages: The author says:"The definition of the integral (14.8) involves a limiting process, in which the…
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How to determine the invariant of a set?

Consider all colorings of the set $S=\{1,2,3,4,5,7\}$ in $R$, $G$, and $B$. Let $\sigma$ be the permutation given by $$\sigma= \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 5 & 4 & 7 & 1 & 3 & 6 & 2 \\ \end{pmatrix} $$ Determine…
idknuttin
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Invariant in a sequence

Consider a sequence given by $a_0 = 1, a_1 = 0, a_2 = 1, a_3=0, a_4=1, a_5=0, a_n = a_{n-1}+a_{n-2}+a_{n-3}+a_{n-4}+a_{n-5}+a_{n-6}$ How to prove that $I\left(a_i,a_{i+1},a_{i+2},a_{i+3},a_{i+4},a_{i+5}\right) =…
Kushal Sharma
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Prove that the ellipsoid $x^T W x \leq 1$ is invariant under $f (x) = A x$

Given matrix $W \succ 0 $ and a set $\mathcal{Z} := \{z \mid z^T W z \leq 1\}$, prove that if $Az \in \mathcal{Z}$ and $z \in\mathcal{Z}$, then the following inequality holds $$ A^T W A - W \preceq 0$$
user353855
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Is the mapping "positive stochastic matrix onto its Perron-projection" continuous?

I am dealing with a topological question concerning the mapping that maps a positive stochastic matrix onto its invariant distribution. I am asking myself if such a mapping is continuous (or measurable) with respect to suitable topologies…
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Invariant subspaces

Let $V$ is a finite dimensional vector space over $\mathbb{C}$ and $T$ be a linear operator on $V$ . How to prove $T$ has an invariant subspace of dimension $k$ for each $k = 1,2, \ldots ,\text{dim}V$ . Can it be solved by induction ?
Shona
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Invariant subspaces and dimentions

Let V is a finite dimensional vector space over C and T be a linear operator on V . How to prove T has an invariant subspace of dimension k for each k = 1,2, . . . ,dimV .
Shona
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endomorphism and subspace

Let $\phi: V \to V$ be an endomorphism over a $\mathbb{C}$-field. Show: If there is a decomposition $V = V_1 \oplus V_2$ with $V_1,V_2 \neq \{0\}$ and $\phi$-invariant, then there exist $\phi$-invariant $U_1,U_2 \subset V$ with $\operatorname{dim}…
J.kolab
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Positive invariance of a set under a system of ODEs

Given the system of ODEs, $$x'=x(1-x-y)$$ $$y'=y(x-1),$$ $Q=\{(x,y):x\ge 0, y\ge 0\}$, and $S=(x,y)\in Q:x+y\le k$, $k>1$, I need to show that $S$ is invariant under this system of ODEs. Attempted solution: $x'=x(1-(x+y))\ge x(1-k) \le 0$. Suppose…
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Invariance of sets for systems of ODEs

Given the system of ODEs $$x' = x(1-y)$$ $$y'=y(x-1),$$ let the set $Q=\{(x,y):x\ge 0, y\ge 0\}$. Explain why $Q$ is invariant for this system of ODEs. My explanation: If $x > 1$ and $y<1$ then $x'$ and $y'$ are both greater than $0$, which means…
sequence
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Which complex polynomials in 3 variables are $GL_3(\Bbb{C})$ invariant?

A polynomial $p(x,y,z) \in \Bbb{C}[x,y,z]$ is $GL_3(\Bbb{C})$-invariant if $$ \forall \sigma \in GL_3(\Bbb{C}): p(\sigma(x,y,z)) = p(x,y,z).$$ How to characterize the set of $GL_3(\Bbb{C})$ invariant polynomials ?
Teddy
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Jumping on the Coordinate lattice grid

Mr. Fat moves around on the lattice points according to the following rules: From point (x, y) he may move to any of the points $(y, x), (3x, −2y), (−2x, 3y), (x+1, y+4)$ and $(x − 1, y − 4).$ Show that if he starts at $(0, 1)$ he can never get to…
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Show permutation representation is reducible, by finding G-invariant subspace

$(\pi,V) $ is the permutation representation of the symmetric group $S_5 $, $ V=C^5$ and the action of standard basis vectors of $ V$ is given by $\pi(\sigma)e_i=e_{\sigma(i)} $ for $\sigma\in S_5 $ $i=1,...,5$ Show that $(\pi , V) $ is not…
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$(Y_i - \hat{Y}_i)(\hat{Y}_i - \bar{Y}_i) = 0$

$(Y_i - \hat{Y}_i)(\hat{Y}_i - \bar{Y}) = 0$ in the image below (third and fourth line of the proof!). Why?
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