# 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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### What is the invariance group of $f(\mathbf{z})=(\mathbf{z}^T\mathbf{z}) (\mathbf{z}^*(\mathbf{z}^*)^T )$?

What is the invariance group of $f(\mathbf{z})=(\mathbf{z}^T\mathbf{z})(\mathbf{z}^*(\mathbf{z}^*)^T)$ ? where $f:\mathbb{C}^n\to \mathbb{R}$. We can think of $\mathbf{z}$ has a complex vector such ...
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### In proving the “Invariance Under Translation of Definite Integrals” by substitution, is the continuity of the integrated function mandatory?

In the beginning, I shall apologise if my question has been answered before (I can't find such an answer), and if my question is naive (I am not of a Mathematical or even scientific background, I am ...
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### Finding the invariant lines of $\left(\begin{smallmatrix} 3 & -1 \\ 6 & -2 \end{smallmatrix}\right)$

How do I find the invariant lines of: \begin{pmatrix} 3 & -1 \\ 6 & -2 \end{pmatrix} So my thoughts were to assume the invariant line has the form $y=mx$ as it will pass through the origin, ...
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### Invariance of domain theorem and systems of equations

I have to deduce from invariance of domain theorem that system of $n$ equations with $n$ variables has one solution. I have no idea how to do this, I don't see any connection between open sets and ...
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### Intersection of acceleration and deceleration curves using differential equations

An object is moving with given acceleration and then decelerating to certain point and velocity, while whole movement must be fastest as possible - so only the one switch between acceleration and ...
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### Example of a left-invariant vector space

in order to work with group operations and flows I'm looking for (simple) examples of left-invariant vector spaces! Do you have some?
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### To what is $V-E+F$ transformed to when $n>3$?

I guess that convex polyhedra can be well-defined in $\mathbb R^n$ when $n>3$ and that they are well-studied so would like to know to what does the expression $V-E+F$ transforms to when $n>3$ ...
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### Properties of $\mathcal{C}^1$-diffeomorphisms which keep invariant the uniform distribution on the n-cube?

Let us consider the n-cube (n-dimensional hypercube) $H_P={]0,\,1[}^P$ and let $\psi:\,H_P\rightarrow{}H_P$ be a $\mathcal{C}^1$-diffeomorphism which keep the uniform distribution (with respect to the ...
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### Invariant distribution of hidden markov model

We know that a time homogenous Markov chain which is aperiodic and irreducible converges to its unique invariant distribution. Suppose we have a hidden markov model where $X$ is the underlying Markov ...
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### Definition: Invariants of Hopf modules

In my lecture notes it says: Let H be a $k$-Hopf algebra. Let M be a left H-module. The invariants of H on M are defined as the $k$-vector subspace $M^{H}$:= {m $\in$ M | h.m=$\epsilon$(h)m for ...
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### How to define “number of cycles which make up a graph” as a graph invariant formally?

I'll consider a graph (quiver) $E=(E^0,E^1,s,r)$, where $E^0$ is the set of vertices, $E^1$ is the set of arrows, and $s,r\colon E^1\to E^0$ are the source and range maps. I allow loops and multiple ...
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### Looking for a proof of LaSalle's invariance principle for a dynamical system on a manifold.

I found the following version of LaSalle's theorem and it appears to be stayed differently from the original. Consider the smooth dynamical system on an $n-$manifold $M$ given by $\dot{x} = X(x)$ and ...
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### Is this actually a valid proof? Multivariable Calculus

So I was asked to prove that the curl of a vector field $\mathbf F=(f_1,f_2,f_3)$ is invariant under change of basis, where the initial and final basis are both orthonormal. In particular, let the ...
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### An example of a non-invariant measure on a compact Lie group

I would like to construct a non-invariant measure on a compact Lie group but I'm not sure what is allowed and what the consequences are. Take the simplest example of $SO(2)$. The unnormalized ...
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### In what sense are these two invariant measures on $SU(2)$ proportional?

An element $g$ of $SU(2)$ is of the following form: $$g=\begin{bmatrix} z_1 & z_2\\ -\bar{z}_2 & \bar{z}_1 \end{bmatrix},$$ where $z_i$ are complex satisfying $|z_1|^2+|z_2|^2=1$. I can ...
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### Why is stability conserved during a bifurcation?

Suppose we have some quantity $J$ which is defined for a particular (1D) bifurcation in the following way: $J = \text{The number of stable equilibria - the number of unstable equilibria}$. This ...