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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 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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1answer
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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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15 views

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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2answers
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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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217 views

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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1answer
53 views

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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1answer
42 views

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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1answer
88 views

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
41 views

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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1answer
75 views

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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15 views

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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1answer
19 views

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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1answer
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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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1answer
55 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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3answers
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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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1answer
172 views

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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1answer
68 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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1answer
115 views

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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0answers
37 views

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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1answer
54 views

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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0answers
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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 ...
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1answer
53 views

Lebesgue measure - swap invariance

Lebesgue measure has the property of Translation invariance, and my question is whether it is invariant under swaps. In particular, let $A\subseteq \mathbb R^n$ and let $A^{i\leftrightarrow j}$ denote ...
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1answer
57 views

Invariance of subgroups using the mapping of an isotypical decomposition in representation theory

I was presented with this question as a study problem in my representation theory course in college and have spent hours trying to solve it or find something similar. Any help or direction on this ...
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1answer
43 views

Projective invariants of quadrilaterals/group of quadrilaterals

I am looking for projective invariant properties of quadrilaterals or even a group of quadrilaterals. Example: In Multiple View Geometry in Computer Vision by Hartley and Zisserman I read that ...
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1answer
453 views

Rigorous proof to show that the $15$-Puzzle problem is unsolvable

So this is supposedly a very popular puzzle by Sam Loyd. (I don't want answerers to provide solutions directly from some website etc. I mean, an ingenious solution is more welcome please.) ...
3
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1answer
60 views

Two Player Strategy Game

I have recently been struggling on a problem involving a modified game of Nim. I have tried finding an invariant or monovariant, but to no avail. "In a game, Players X and Y take turns taking chips ...
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0answers
38 views

Poles and Zeros of a Linear Transform

Suppose I have some linear transform $T: V \rightarrow W$ where $V \in \mathbb{C}^{N}$ and $W \in \mathbb{C}^{M}$ representing some Linear Time Invariant (LTI) system. Since the space is discrete, I ...
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2answers
51 views

Writing a short but rigorous proof

The problem is There are $a$ white, $b$ black, and $c$ red chips on a table. In one step, you may choose two chips of different colors and replace them by a chip of the third color. Prove that a ...
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1answer
188 views

Diffeomorphism invariance, Lie derivative

There is written in the Hamilton's Ricci flow book about Lie Derivative this: ...
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1answer
30 views

system delay: $x(2t-t_o) \,\,or \,\, x(2t-2t_o)$?

I have a system as this: $x(t)--->system-->y(t) = x(2t)$ If i delay the output y(t), what do i get $y(t) = x(2t-t_o)\,\, or\,\, y(t) = x(2t-2t_o)$? What is the correct expression and why?
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Matrix permutation-similarity invariants

https://en.wikipedia.org/wiki/Matrix_similarity https://en.wikipedia.org/wiki/Permutation_matrix The determinant and trace (and characteristic polynomial coefficients) are well-known similarity ...