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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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Proof - Possibility to transform all vertex’s value to 1

A big equilateral triangle is made up of smaller equilateral triangles. The relation is for n order of the bigger triangle, the number of inner triangles are n^2. Example of such a triangle with n = ...
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Is the usual metric on $\Bbb{N}^\Bbb{N}$ left invariant on $S(\Bbb{N})$?

Let $\Bbb{N}^\Bbb{N}$ be the set of all functions $(x_n\mid n\in\Bbb{N})$ from $\Bbb{N}$ into itself (I identify sequences with their images, as usual). I know this is a metrizable space with ...
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What is the operational way of discovering scale invariance of differential equations?

Context The answer here by @Keenan Pepper gives an instance for what it means for an algebraic or trigonometric formula to be scale invariant. For quick reference, I quote his answer here but with a ...
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Show that Eigenspace is invariant under flow in gradient vector field

I'm struggling with where to start on the following question Consider the gradient vector field $$ \frac{d}{dt} \begin{pmatrix} x \\ y \end{pmatrix}= \begin{pmatrix} \frac{\partial V}{\...
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Why do we only consider homogeneous invariants?

Let $G$ be a finite matrix group, $G \subseteq GL_{n}(\mathbb{C})$. Consider the polynomial ring in $n$ variables; $\mathbb{C}[x_1,...,x_n]$. It is known that the ivnariant subring $\mathbb{C}[x_1,.....
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Why is the stabilizer of a line segment a cylinder?

There is a passage in a paper I'm reading discussing the stabilizer of an edge. For an edge (passing through the origin), its stabilizer (in $\operatorname{GL}_2(\mathbb{R})$) must fix the ...
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Arthur engel Question 22 invariance problem

Question: There is a chip on each dot. In one move, you may simultaneously move any two chips by one place in opposite directions. The goal is to get all chips into one dot. When can this goal be ...
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Dimension of the invariant subspace

Let $\Gamma \subseteq GL_{n}(\mathbb{C})$ be a finite matrix group. Let this finite matrix group act on $f(x_1,...,x_n) \in \mathbb{C}[x_1,...,x_n]$ like so: $$\Gamma \cdot f(x_1,...,x_n) = f(\Gamma \...
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Confusion concerning the Euclidean Norm and unitary invariance on $\mathbb C^n$

For a unitarily invariant norm on $\mathbb C^n$, how do I show that $||x||=||x||_2||e_1||$? I can show that $||e_1||=1$ for the Euclidean norm by definition, and is therefore unitarily invariant, does ...
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Invariants between two isomorphic vector spaces

I have a general question about isomorphisms between vector spaces. From a general point of view, there are common properties (invariants) between two isomorphic structures (e.g., properties about ...
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Proof that $a\nabla^2 u = bu$ is the only homogenous second order 2D PDE unchanged/invariant by rotation

Looking for feedback and maybe simpler intuition for my proof of the theorem, shown below The statement of the theorem: Theorem Among all second-order homogeneous PDEs in two dimensions ...
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What can I say about the form of an invariant function?

I have a general scalar function which has the properties: \begin{align} f(s\,a,b,c)&=s\,f(a,b,c)\\ f(s\,a,s\,b,s\,c)&=f(a,b,c) \end{align} where $s$ can be any real number, so the invariance ...
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Calculating tangent vector of curve s(P,$\alpha$) at given point $\alpha$ = 0. http://yann.lecun.com/exdb/publis/pdf/simard-00.pdf

I am reading one chapter where tangent vector is calculated for the given curve $s(P,\alpha)$ at $\alpha=0$ by differentiating with respect to $\alpha$; $\frac{\partial s(P,\alpha)}{\partial\alpha}$. ...
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Invariant properties of functions under coordinate transformations

I am interested in what sort of properties are preserved for a function defined on a smooth manifold $M$. Preservation in the sense of a physicist, i.e., invariant under coordinate transformations. ...
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Definition of invariant ideal and invariant set of points?

I am reading a book about Invariant theory. It is said there the ideal $I$ is invariant action of a group of $n\times n$ matrices. Which element of $I$ are polynomials in $n$ variables. and also said ...
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Simple example of invariants

The combinatorics textbook I'm reading introduces invariants with the following example: There are three piles with $n$ tokens each. In every step we are allowed to choose two piles, take one ...
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A new graph invariant? The maximum number of non-equivalent colorings with $n$ colors.

Consider (proper) coloring of a finite graph $G$ with exactly $n$ colors and the following coloring transformation: choose an edge of the graph with the end nodes of colors $a$ and $b$ and swap the ...
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Proving the existance of a one- or two dimensional subspace that is both A and B invariant.

I've just started learning about invariant subspaces and I came across this exercise: Let $V$ be a finite-dimensional space over the field $\mathbb{C}$. Prove that if endomorphism matrices $A$ and $...
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Projective-invariant differential operator

This question has been cross-posted to MathOverflow. Suppose we want a differential operator $T$ acting on functions $\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that \begin{align*} &T(g) = 0 \...
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Expression relating second invariant to eigenvalues matrix

Consider a matrix $A\in\mathbb{R}^{3\times 3}$. We know that it has the three invariants $$ I_1 = tr(A),\\ I_2 = 1/2(tr(A)^2-tr(A^2)),\\ I_3 = \det(A).$$ Also, the first and third can be expressed as ...
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Set invariant under the map $x \mapsto -x$ that is a translation of symmetric sets

Let $F \subset \mathbb{R}^n$ be an open bounded (non empty) convex set and assume that it is invariant under the map $x \mapsto -x$ (which means that $F= \{ x : x \in F \} = \{ -x : x \in F\} = -F$). ...
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When normalizing an equation, why are the non-dimensional terms expected to be order 1?

I'm familiar with the procedure of normalization, but I'm unfamiliar with some of the theory involved. For instance, using the Navier Stokes equations where the density and viscosity can be treated ...
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Noether on symmetry of a Weierstrass representation

Can someone explain which Weierstrass representation Emmy Noether refers to below? As context, Noether's second Conservation Theorem says if a Lagrangian has an infinite-dimensional Lie group of ...
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What are some other topological invariants apart from connectedness, compactness and fundamental groups?

I realized that, for any pair of non-homeomorphic topological spaces that I know of, those three invariants are usually sufficient to prove that the two spaces are not the same. So, for example: the ...
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Find invariant points under matrix transformation (degeneracy)

I have the matrix $$Q=\begin{bmatrix}-1&2\\0&1\\\end{bmatrix}$$ and want to find the invariant points. To do this, I solve the equation: $$\begin{bmatrix}-1&2\\0&1\\\end{...
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Invariance of nabla under pure rotation

The problem goes as follows: Show that, under a rotation $\vec {\nabla} = \hat{i} \frac{\partial}{\partial x}+ \hat{j} \frac{\partial}{\partial y}+ \hat{k} \frac{\partial}{\partial z} = \hat{i'} \frac{...
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How to know what combination of variables a function depends on from it's PDE?

We know that if $f(x,y)$ satisfies $\partial_xf(x,y)+\partial_yf(x,y)=0$, then $f(x,y)$ only depends on $x-y$, i.e. $f(x,y)$ is actually $f(x-y)$. Is there a systematic way of know what a function ...
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Testing if two matrices differ only by permutation? (generalized graph isomorphism problem)

In graph isomorphism problem, for which Babai's quasi polynomial algorithm is currently under review (stack), we ask if two adjacency matrices: of $\{0,1\}$ coefficients differ only by a permutation. ...
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$\sum_{i=1}^nx_i\frac{\partial}{\partial x_i}$ is invariant under all matrices

Consider the differential operator $D:=\sum_{i=1}^nx_i\frac{\partial}{\partial x_i}$. Then I claim that $D(u\circ A)(x)=D(u)(Ax)$. The proof is as follows: First note that $Du(x)=\langle x, \nabla_x u(...
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Pawn visiting every square on a chessboard exactly once and returning to its right

A pawn moves across $n\times n$ chesssboard so that in one move it can shift one square to the right, one square upward, or along a diagonal down and left. Can the pawn go through all the squares on ...
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Testing if two finite sets of points differ only by rotation (unordered, in polynomial time in size and dimension)?

Imagine we have two size $m$ sets (without order) of points $X=\{x^i\}_{i=1..m}, Y=\{y^i\}_{i=1..m} \subset \mathbb{R}^n$ and we want to answer the question if they differ only by rotation: if there ...
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Making elements of an array divisible by three

Consider the following $6$x$6$ array $$ \begin{matrix} 2 & 0 & 1 & 0 & 2 & 0 \\ 0 & 2 & 0 & 1 & 2 & 0 \\ 1 & 0 & 2 & 0 & 2 & 0 \\ 0 & ...
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Number of football fans in a town Coursera question, why will it terminate?

I've come to a problem while doing the Mathematical thinking Course on Coursera. The problem statement is: There are two football teams in a town. Each of the citizens is supporting one of the teams....
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Topological invariance of dimension

I am starting to study smooth manifolds with the book of Lee. At the beginning, he states this theorem, which is then proven later on with advanced techniques: Theorem 1.2 (Topological Invariance of ...
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1answer
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Ordering states for an integer-incrementing game.

There is a game played with a one-dimensional array of non-negative integers such as $$\underline{1}\;\underline{2}\;\underline{0}\;\underline{0}\;\underline{1}$$ Given such an array of $n$ numbers,...
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Strange text about a strange attractor

I'm reading a text about the Lorenz equations, for $r>1$ and all the other parameters positive. At one point the author says My questions are: 1) Why is $L$ a Liapunov function? There are ...
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Invariant Description of Displacements

Consider a collection of points, or particles (e.g., a crystal lattice). Is there a way to mathematically describe the Cartesian displacements of these points from their initial positions, such that ...
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If $f(T)g(T)=0$ and $V_1=g(T)(V)$, $V_2=f(T)(V)$ proof that $V=V_1\oplus V_2$

$\Bbb K$ is a field, $V$ and let $T:V\longrightarrow V$ be a linear map. $\Bbb K[x]$ is the vector space of the polynomials over $\Bbb K$. Suppose that $f,g\in \Bbb K[x]$, $f(T)g(T)=0$, $V_1=g(T)(V)$...
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“Invariants” of Exotic spheres

An exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere. Naively, I thought that there is no algebraic topological invariant ...
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1answer
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What is the name of this theorem? convex set

I have seen this theory but I am wondering what is the name of this theorem? If $~~~~~~$ 1- $\mathbb D$ is a closed convex set and real $~~~~~~$ 2- $f:\mathbb D \rightarrow\mathbb R$ is a convex ...
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60 views

Where does this proof of ergodicity fail?

I am a bit confused and was hoping the folk of MSE would help (and point out where the error is made). Suppose $(X,\mathcal{B},\mu,R)$ and $(Y,\mathcal{C},\nu,S)$ are ergodic measure preserving ...
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Invariant principle in tranformation number

In a pentagon, each vertex is assigned a real number which sum is positive. If there is a negative number $y$, use the transformation $$T:(x,y,z)\mapsto(x+y,-y,y+z)$$ where $x,y,$ and $z$ are ...
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Invariance of the probability of an event related to an urn… with a weird constraint

Introduction to the problem A set $C$ contains $c\in \mathbb{N}$ elements of three different kinds: There are $\alpha\in\mathbb{N}$ elements of kind $A$, $\beta\in\mathbb{N}$ elements of kind $B$, ...
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Suppose $A=\bigcup_i A_i$ is f-invariant. Then $f^n(A_i)\subseteq A$ for all $n\in\mathbb{N}$?

Suppose $A=\bigcup_{i\in I}A_i, A_i\subseteq X,$ for some index set $I$ and $A$ is $f$-invariant for some function $f\colon X\to X$, Does this imply that, for each $i\in I$, $$ f^n(A_i)\subseteq A~\...
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79 views

Showing measure is invariant/ergodic for a skew product.

Let $(X,\mathcal{B},\mu,T)$ be a measure preserving dynamical system for which the probability measure $\mu$ is ergodic. Suppose we have a compact group $G$ with Haar measure $\nu$ and a measurable ...
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Show that the annulus $1 \leq x^2 + y^2 \leq 3$ is positively invariant

Given the planar dynamical system $$\dot x = x - y - x^3, \\ \dot y = x + y - y^3$$ show this is positively invariant in the annulus $1 \leq x^2 + y^2 \leq 3$. Hints: $$x^4 + y^4 = (x^2 ...
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Invariants of a general 2nd order tensor

I am attempting to write out three invariants of a general (not necessarily symmetric or Cartesian), 2nd order tensor which we will call $\mathbf{T} = T^i_{\;j}\mathbf{e}_i\mathbf{e}^j \,, \;\; i,j=1,...
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Proving two sequences overlap

$A =a_1,a_2,a_3,\dots,a_{25}$, strictly increasing with each term being non-negative integer and striclty less than 50 $B=a_1+2,a_2+2\dots,a_{25}+2$ with same condition as last except that $50\equiv0$...
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Prove interior point is a topological invariant

Let $(X,\tau)\cong(Y,\tau^*)$, $A\subseteq X$ and $p\in int(A)$. Let $f:(X,\tau)\to (Y,\tau^*)$ be a homeomorphism. To prove that the property of interior point is a topological property we shall show ...
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QFT, Noether and Invariance, Complex fields, Equal mass

The problem statement, all variables and given/known data Question attached: Hi I am pretty stuck on part d. I've broken the fields into real and imaginary parts as asked to and tried to compare ...