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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Show that $\mu (\left[a, b\right)) = \ln \frac{1 + b}{1+a}$ (a measure on $X = [0,1]$) is invariant wrt. $x \rightarrow \left\{\frac{1}{x}\right\}$

Show that $\mu (\left[a, b\right)) = \ln \frac{1 + b}{1+a}$ (a measure on $X = [0,1]$) is invariant wrt. $x \rightarrow \left\{\frac{1}{x}\right\}$ My question here is: What does it mean, that a ...
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When will the algorithm stop. While $a>0$, do if $a<b$ then $(a,b)\rightarrow (2a,b-a)$ else $(a,b)\rightarrow (a-b,2b)$

I came to this question in the Problem Solving Strategies. We start with the state $(a,b)$ where $a,b$ are positive integers. To this initial sate we apply the following algoritm While $a>0$, do if ...
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How to prove that there is no 5-ary operation in a clone satisfying certain conditions

Question: How to prove that there is no 5-ary operation $f \in \mathcal{C}$ satisfying $f(2, 1, 3, 4, 3) = 1$ and $f(2, 1, 1, 4, 3) = 2$? The $\mathcal{C} = Clo(\textbf{A})$ is a clone of $\textbf{A}$,...
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Is there a name for this property of functions on groups?

Let $G$ be a group and $F:G^n \to G$ with the following property: If $x_1,…,x_n,h \in G$, then $F(hx_1,…,hx_n)=hF(x_1,…,x_n)$. Is there a name for this type of function property? It is something I’ve ...
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Is Haar measure reflection invariant?

Let $G$ be a locally compact group and $\mu$ be a Haar measure on $G.$ Is $\mu$ necessarily reflection invariant i.e. can we always say that $\mu (E) = \mu (E^{-1})\ $? where $E \subseteq G$ is a ...
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A question about the definition of equivariant map

If $S$ is a set of functions from $X$ to $Y$ then I can consider the action of a group $G$ on $S$ via its action on $X$ and $Y$ by the formula $$(g \cdot f)(x) = g \cdot f(g^{-1} \cdot x),$$ So we are ...
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Why pawns starting at chessboard squares $(1,1)$ and $(8,8)$ that move orthogonally at each step will never swap positions?

Let's say we have a chessboard (i.e an $8×8$ grid). Let's assume each cell is identified by two coordinates (integer numbers) ranging from $1$ to $8$. Assume to have a red pawn in position $(1, 1)$ ...
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How does the chain rule imply that for a solution to the diffusion equation, we have an invariance that $u(x,t)=u(1-x,t)$?

Consider the diffusion equation $u_{t}=u_{x x}$ in $\{0<x<1,0<t<\infty\}$ with $u(0, t)=u(1, t)=0$ and $u(x, 0)=4 x(1-x) .$ $\text {Show that } u(x, t)=u(1-x, t) \text { for all } t \geq 0 ...
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LaSalle's invariance principle with largest positive invariant set?

I have read many versions of LaSalle's invariance principle. One of them is the following: For $D\subset\mathbb{R}^{n}$ open let $f:D\to\mathbb{R}^{n}$ be $C^{1}(D;\mathbb{R}^{n})$. Consider $\dot{x}=...
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A question about definition of equivariance for a function.

Equivariance in deep learning, it is actually defined as a property of some function $f$ to permute the outputs according to permutation of inputs, i.e. $$f(PAP^T) = Pf(A)P^T$$ for some permutation ...
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How to derive the Euclidean metric from translation and rotation invariance?

I want to know which assumptions are sufficient to derive the Euclidean metric. Suppose $d(\mathbf{x}, \mathbf{y})$ is a metric on $\mathbb{R}^n$. Also suppose that: $d$ is translation invariant, i.e....
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Combinatorial Invariant Problem Solving Question

Consider the following problem. Take an $nxn$ table $A$ with entries $\pm 1$. A permitted move is multiplying any row or column by -1. How many $nxn$ tables exist that can be transformed to a table ...
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Understanding Invariant Simplification Steps

I'm trying to make sense of the steps of a "simplification" given in a text book I'm reading. The whole thing seems a little elaborate to explain why $p—c = (p+1) —(c+1)$. I can follow the ...
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How to check invariance using scalar products induced by standard norm?

At my university, we were to solve this exercise during representation theory lectures. We consider the space $E$ of traceless Hermitian matrices as a real vector space of dimension 3: $$\begin{...
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When is a foliation invariant under a diffeomorphism?

Let $\mathcal{F}$ be a Foliation on a Manifold $M$ and $g:M\longrightarrow M$ a Diffeomorphism. We say that the foliation $\mathcal{F}$ in invariant under the diffeomorphism $g$ if the diffeomorphism $...
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Convergence to the mean under the invariant measure of an Itô diffusion

Let $X:=\{X_t\}_{t\in[0,T]}$ be the unique strong solution to the Itô diffusion $$ dX_t = a(t,X_t)dt + b(t,X_t)dW_t, $$ where $a,b$ are such that the conditions for the existence of the invariant ...
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What is the list of all the regular planar graphs with equivalent vertices?

Platonic solids, Archimedean solids, prisms and anti prism all have planar graphs where all the vertices are equivalent, or in other words, for any 2 vertices v1 and v2 there is an automorphism that ...
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Translation Invariance of MaxPooling

I am currently reading Chapter 15 of the book "Deep Learning Architectures, A Mathematical Approach" by Ovidiu Calin (see https://www.springer.com/de/book/9783030367206) and I am having ...
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Result of replacing $1$ to $n$ in pairs by the sum?

I have the following problem: Alice writes the numbers $1, 2, 3, 4, 5, 6, \ldots, n$ on a blackboard. Bob selects two of these numbers, erases both of them, and writes down their sum on the ...
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Prove in this case that one can always find among the triangles two that are congruent.

Four congruent right triangles are given. Adriana can cut one of them along the altitude and repeat the operation several times with the newly obtained triangles. Prove that no matter how Adriana ...
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What are the invariants of this field's equation of motion?

Background Let's say I have a field/fluid which occupies a volume element $dV$ then equation of motion is given by: $$ \left( \frac{1}{c} \frac{\partial V}{\partial t} \right)^2 = \phi_{xy}^2 + \phi_{...
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Mother wavelet and Lorentz invariance

Can we choose a mother wavelet that is a Lorentz variant in practical applications of wavelet transform in Minkowski space?
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Example of a risk measure that is not law-invariant

In some theorems about risk measures, the property of law invariance is required. Let $\mathcal{Z} = \mathcal{L}(\Omega, \mathcal{F}, P)$. A risk measure $\rho\colon \mathcal{Z}\to \mathbb{R}$ is law ...
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Problem with piecewise functions, translational invariance (CNN)

I don't understand how to do this question: Compute the convolution $(f*g)(x)$ for the functions $f(x)$ and $g(x)$ defined as, $f(x)$=\begin{cases} 0&\text{if}\, x\ < a\\ \exp(-x)&\text{...
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Deriving general conic equations using coordinate invariance

The general equation of a parabola is $$\left(\frac{ax+by+c}{\sqrt{a^2+b^2}} \right) ^2=(Latus Rectum).\frac{bx-ay+c'}{\sqrt{a^2+b^2}}$$ To get to this from $y^2=4ax$, and similarly for other conics, ...
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Dodecahedron faces are buttons, vertices have counters that track the use of the buttons

As a follow up to this question, I'm trying to teach invariants by creating a game. The idea is to start with a dodecahedron where each of the 20 vertices has a counter on it and each of the 12 faces ...
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Are there mathematically interesting symmetric matrices with the other symmetries of the square?

Symmetric matrices are square arrays of integers with a symmetry with respect to the main diagonal. However, there are several other symmetries of the square (symmetry with respect to the other ...
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proving an equation is invariant under a given transformation

I'm trying to prepare for an upcoming test and I was going through my textbook and came across the the question below. I've tried for a while but was unable to solve it. Any tips/hints would be ...
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Lie Algebra - Theory of prolongations, Criterion of invariance and splitting of defining equations. Could someone explain these concepts to me?

We're having lectures about Lie Algebras right now, and those were the topics the last time. Our prof is not doing a good job at explaining, he basically reads down his notes without trying to explain ...
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Invariance of rank under diffeomorphism

So I am kind of confused by this: We have $U \subset \mathrm{R}^m$ and $V \subset \mathrm{R}^n$ as open sets. If $f: U \rightarrow V$ is a diffeomorphism then we essentially have $m=n$. Same holds ...
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Correlations estimator is invariant by space and location

I need to check that Is invariant by space and location. Do not even know where or how to start exactly.
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Prove an algorithm is finite ,removing intersecting lines between dots and adding non intersecting lines

We have 2n dots in 2D space and each 2 dots are connected randomly (for each dot in the space we have a straight line to another dot ) .because these dots are connected randomly we will have some ...
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Forming circuit with straight moves

I am reading about a problem where we have a rectangular board, and we have to show that it is impossible to complete a circuit of the board if both sides have odd length. Circuit is a sequence if ...
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Olver "Application of Lie Groups to Differential equations" Proposition 2.18.

I have trouble with proving (and even intuitively understanding) the following proposition in Olver's book "Application of Lie Groups to Differential equations". Proposition 2.18. Let $G$ ...
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To solve a combinatorics question of a system of $n$ friends using invariance

Q. There are $n$ people at a party. Prove that there are two people such that, of the remaining $n-2$ people, there are at least $\lfloor{\frac{n}{2}}\rfloor-1$ of them, each of whom either knows both ...
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Showing subspace of linear operator is an invariant subspace

I am stuck on the proof of the statement written below. How would we go about proving this fact? It seems intuitively correct. Any ideas on how to proceed? This is the assertion I'm interested in ...
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Invariant problem-Chess board

The Question There is an integer in each square of an $8\times8$ chessboard. In one move, you may choose any $4\times4$ or $3\times3$ square and add $1$ to each integer of the chosen square. Can you ...
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Is there a link between two magic squares with the same constant?

For instance we consider the magic squares of order $3$ with the constant $15$. We can find : \begin{array}{ | l | c | r | } \hline 8 & 3 & 4 \\ \hline 1 & 5 & 9 \\ \...
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Tiling $3k\times (3k-2)$ board with $L$- Trominoes

Consider a $3k\times (3k-2)$ board. For which values of $k$ can we cover the board with $L$- trominoes? For this, It was clear that for $k$ even, we are done, because then we can divide the board ...
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Show that $P$ lies inside an even number of triangles with vertices among $A_1 , . . . , A _{2m}$

Assume a convex $2m$-gon $A_1 , . . . , A _{2m}$ . In its interior we choose a point $P$ , which does not lie on any diagonal. Show that $P$ lies inside an even number of triangles with vertices among ...
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For the following class of matrices, are the determinants invariant under permutations?

I want to ask a question regarding the invariance of determinants under permutation. The following matrix is the one I want to discuss here. (It's just a symmetric matrix with non-zero elements at the ...
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Arthur Engel - Invariant for cyclic difference

Just started solving and came across this example:- Consider any four integers $a,b,c,d $ where not all are equal. It is allowed to change this sequence to $a-b,b-c,c-d,d-a$. Prove that at least one ...
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Overpopulated apartment

$119$ people live in a building with $120$ apartments. An apartment is called overpopulated if there are more than $15$ people living in it. Each day the tenants of some overpopulated apartment have a ...
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Coloring a $10×10$ grid

Given a $10×10$ grid with $9$ red blocks and $91$ white blocks, in each step we color one red block black and after that one white block red until there are $91$ black blocks and $9$ red blocks. Prove ...
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Given a finite number of stones. We place every stone on an integer. Prove that, given different movements, we can only make finite number of moves

Given a finite number of stones. We place every stone on an integer (a number $x$ where $x\in Z$) and maybe multiple stones on an integer. On each move we can make one of the following movements: ...
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Prove that there exists a four colored intersection in a four colored $100×100$ grid [duplicate]

A $100×100$ grid is colored with four colors. There are exactly 25 blocks of each color in every row and column. Prove that there exists an intersection between two rows and columns such that all four ...
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How to show that a projection matrix is invariant w.r.t the choice of a g-inverse

Let X be an $(nxk)$-matrix (not necessarily of full rank) and $P = X(X'X)^-X'$, i should show that P is invariant with respect to the choice of the g-inverse of $X'X$. As a hint I should first verify ...
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Why is the derivative of a quadratic invariant zero if the matrix is symmetric?

I was shown a proof in class, where it was said that all Runge-Kutta methods preserve the quadratic invariant $$I(y) = y^TCy \quad ,$$ where $C$ is a quadratic symmetric matrix, under the condition $$...
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Partition 100 people, 4 from each country into 4 groups with conditions

This is a problem from the $2005$ All-Russian Olympiad. Problem is as follows: $100$ people from $25$ countries, four from each country, sit in a circle. Prove that one may partition them onto $4$ ...
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Invariance of generalized eigenspace

I have a lemma saying that each of the generalized eigenspaces of a linear operator $T$ is invariant under $T$. This means that if $E_j$ is a generalized eigenspace then $T:E_j \rightarrow E_j.$ The ...

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