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 permutation representation is reducible, by finding G-invariant subspace [duplicate]

$(\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 $ ...
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$T$ Invariant subspace

Please help me to write a step by step solution to this problem: Let $T$ be a linear operator defined on a finite dimensional vector space $V$. If $W$ is a $T$ invariant subspace with $V = ...
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33 views

Solving a Chessboard problem using the Invariance principle

Problem Statement There is an integer in each square of an 8 x 8 chessboard. In one move, you may choose any 4 x 4 or ...
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Invariant equation

If I have the equation: $x=-$tan$x$ I can send $x \rightarrow -x$ and the equation doesn't change. If I define $x>0$ one of the possible ranges $x$ can take without solving the equation is ...
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Applying invariance principle on a problem on sequence of positive integers

The problem statement: Start with the positive integers 1,...,4n-1. In one move you may replace any two integers by their difference. Prove that an even integer ...
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Understading the integral form of a conservation law

When I think of a conservation law I think of a continuity equation like the following $$\partial_t \rho = -\nabla \cdot \vec j$$ But now I'm reading a book on electrodynamics (that's honestly a bit ...
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37 views

What is an invariant Simplex

Let $$\Delta=\{x_i\in\mathbb{R}^k_+: \sum_{i=1}^kx_i=1\}$$ where k is a positive integer. I've read in a book the following: The simplex $\Delta \subset \mathbb{R}^k$ can be shown to be invariant ...
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Invariance of stationary wavelet transform

Suppose we are given 64 points $x_1,\ldots ,x_{64}$ and divide them into two groups $x_1,\ldots, x_{32}$ and $x_{33},\ldots , x_{64}$. Then we apply stationary wavelet transform to both these groups ...
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Left invariant Vector Field on $S^2$

How intuitively look like all left invariant vector fields on this manifold: the 2 dimensional unit sphere $S^2$ with the smooth structure inherited from $\mathbb R^3$? Why all left invariant vector ...
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Are Bezier curves invariant under conformal mapping?

I've spent quite a bit of time on google trying to find information on whether or not Bezier curves are invariant under conformal mapping (i.e. a conformal mapping of all points on the curve is the ...
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38 views

$(Y_i - \hat{Y}_i)(\hat{Y}_i - \bar{Y}_i) = 0$

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

The following is an algorithm which finds the maximum value in a list of integers, and I want to prove that it is correct by using a loop invariant. ...
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arg min-invariance for norm of vectorfield under linear transformation

Given a vectorfield $\vec{F}(\vec{c}) \in \mathbb{R}^n$ which is a function of some parameters $\vec{c}$, what constraints must you have on a matrix such that when you act on the vectorfield the ...
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40 views

find an invariant

I've been reading about the use of invariants in contest math. I saw the following problem (in my own words): There are $N = 2n$ numbers placed on a circle. Then we increase two any consecutive ...
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19 views

Invariance Dealing with Infected Squares

Twelve 1x1 cells of a 10x10 square are infected. Two cells are called neighbors if they share at least one vertex (thus an inner cell has 8 neighbors). In one unit time, the cells with at least four ...
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Numbers written on a board

The numbers $1,2,...,n$ are written on a board ($n\in\mathbb N$). In each step we take any two numbers $a,b$, remove them, and write either $a-b$ or $a+b$ on the board. After $n-1$ steps there will be ...
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Invariance problem dealing with the sums of units digits

We may write all the digits from 1 to 9 in a row in any order we like, and then we write plus signs between some digits (as many plus signs as we like). Finally, we evaluate the obtained expression. ...
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Lyapunov invariant set for affine systems

Given a linear system $\dot{x}=Ax$ such that the real part of every eigenvalue of $A$ is less than $0$, Lyapunov's equation $A^T P + P A = -Q$ with $Q$ being any suitably sized positive definite ...
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Are similar complex matrices again similar when each is expressed as a real matrix?

We know that, relative to this ordered basis {$(1,0),(i,0),(0,1), (0,i)$}, we can express a 2x2 complex matrix mapping $C^2 -> C^2$ as a $4x4$ real matrix (representing the same transformation of ...
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Proving span of a complex eigenvectors is an invariant subspace

Following these notes:http://www.math.uwaterloo.ca/~jmckinno/Math225/Week11/Lecture3r.pdf We wish to prove validity of the following in bold: Theorem 9.4.2 Suppose that $λ = a + bi$, $b \neq 0$, is ...
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70 views

State machine scenario: finding invariant

Alice, Bob, and Charles want to evenly distribute a dozen doughnuts. Initially, Alice has 5, Bob has 3, and Charles has 4. However, they want to do it according to the following rules: 1) Bob may ...
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Invariant question: Fifteen Puzzle

The Fifteen Puzzle consists of sliding square tiles numbered $1...15$ held in a $4\times4$ frame with one empty square. Any tile adjacent to the empty square can slide into it. The standard initial ...
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Finding invariants

If in a given ecosystem there are 30 chameleons living on an island: 15 red, 7 blue, 8 green. When two of a different color meet, they both change into a third color (ie. if a red and blue meet, they ...
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Verification about group actions and “uniformity” of an action

I spent some time revisiting group actions this week. I was hoping to get someone to verify a seemingly straightforward claim. I also had a thought on how "uniform" an action is on a space. ...
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43 views

Specific matrix has no 2-dimensional invariant subspaces

I have the endomorphism $$ M = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix} $$ of a real vector space $V$. Note that this matrix is nilpotent (with $M^3 = ...
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Attractor $A$ with neighborhood $V$ such that $f^N(V)\subset V$ and $A=\bigcap_{n\in\mathbb{N}}f^n(V)$, then $\omega(x)\subset A$ for all $x\in V$?

Let $f\colon X\to X$ be a continuous map on a (compact) topological space $X$. Let $A$ be an attractor for $f$, i.e. there is a neighborhood $V$ of $A$ such that $f^N(V)\subset V$ for some ...
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Fibonacci Loop Invariants

I've taking an Algorithms course. This is non-graded homework. The concept of loop invariants are new to me and it's taking some time to sink in. This was my first attempt at a proof of correctness ...
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show that the product of two delta functions δ(x)δ(y) is invariant under rotation around the origin.

Show that the product of two delta functions $\delta{(x)}$$\delta{(y)}$ is invariant under rotation around the origin. This is a problem from Zee's textbook on Gravity on page 51. The book was ...
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Given a vector $\vec{p}$ = $<p^1,p^2>$ show $<ap^1,bp^2>$ for $a \ne b$ is not a vector.

In the textbook Einstein Gravity in a nutshell by Zee, on page 43, it claims that given a vector $\vec{p}$ = $<p^1,p^2>$ then for any 2 real numbers $a\ne b$ then the object $<ap^1,bp^2>$ ...
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Scale invariant ODE. Is this general method correct?

Recently, a question I asked had the differential equation $y''=xyy'$. A trick to solving this quickly is to notice that scaling $y$ by $a$ and $x$ by $b$ shows that $a=1/b^2$ is the condition that ...
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Necessary condition for a finite cyclic sum of length $4$ made of $1$ and $-1$ to be $0$

This is something I observed when I was reading the classic Problem-Solving Strategies by Arthur Engel. I liked the way he solved the following problem: Let $a_1,\ldots,a_n\in\{-1,1\}$ such that ...
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Can we obtain the pair $(1,50)$ with these following operations?

It's a problem from some russian competition: We're given a card with two positive integers $(a,b)$ and we have tree machines which generate another card from the one we insert on it(I assume we ...
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31 views

Why is $U$ $T$-invariant?

Let $V$ a finite dimensional vector space and two sub-spaces, $U, W$ such that $V = U \oplus W$. Let's assume $T$ is a linear operator such that $W$ is $T$-invariant. Why is it true that $U$ is also ...
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Application of invariant method

On the board is written the number $18$. Every minute the number is replaced by the product with 2 or 3,or by the quotient of the division with 2 or 3. Show that after $60$ minutes the number cannot ...
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Is there a Fourier invariant basis?

There are some functions which are invariant under Fourier transformation up to scaling factors, eg. sech(pi*x), Gaussian function etc.. Is there a set of basis functions, which form an invariant ...
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Is seven-dimensional cross product rotationally invariant?

For three-dimensional cross product, the following property holds true: \begin{equation} (R\mathbf x) \times (R \mathbf y)=R(\mathbf x \times \mathbf y) \end{equation} where $R\in SO(3)$. Is the ...
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Proving a $k$-multilinear symmetric map is invariant iff a condition is satisfied

In Huybrecht's book on complex geometry, he states the following lemma on page 193: Lemma 4.4.2: The $k$-multilinear symmetric map $P$ is invariant if and only if for all $B,B_1,\ldots, B_k \in ...
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What should you call a property, like an invariant, but that is reversed instead of preserved?

Suppose $P$ is some property of some objects and $f$ is a function on those objects. If $Px$ implies $Pf(x)$ and $\lnot Px$ implies $\lnot Pf(x)$, then we might say that "$P$ is invariant under $f$". ...
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Gaussian Distribution Under Orthogonal Transformation

Let $\mathbf{H}\in \mathbb{R}^{n\times n}$ be a random matrix whose every element has a Gaussian distribution with mean $m_{ij}$ and variance $\sigma^2$ i.e. ...
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three grasshoppers jumping on a plane.

The problem is dead simple: Three grasshoppers sit on a plane not in a line. Every second just one of the grasshoppers hops symmetrically over one of the others. Can they return to the initial ...
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Proofs involving Subspace Invariance

Let $ V $ be a finite dimensional vector space over a Field $ F $ and let $ T \in End(V) $. i) If $ S \in End(V) $ is such that $ ST = TS $, show that $ Im(S) $ is a $T$-Invariant Subspace of $ V $ ...
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42 views

Matrix properties invariant under scalar multiplication

Given a square real matrix $A\in M_n(\Bbb R)$, what are ALL the properties invariant under scalar multiplication? In other words: which are the properties shared by all the $\lambda A$'s when ...
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Write $V=P_2(\mathbb{R})$ as a direct sum of $V=W_1\oplus W_2 \oplus W_3$

So, if I let $T:P_2(\mathbb{R}) \rightarrow P_2(\mathbb{R})$ and is a linear endomorphism given by $T(f(x))=f(x)-f(2x-1)$. Then I have to write$V=P_2(\mathbb{R})$ as a direct sum of $V=W_1\oplus W_2 ...
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Boundedness of RHS implies existance of invariant cube

Consider a system of ODEs of the form $$\dot x_1 = f_1(x_1,x_2)-g_1(x_1,x_2)x_1 \\ \dot x_2 = f_2(x_1,x_2)-g_2(x_1,x_2)x_2,$$where $f,g$ are bounded, Lipschitz-continuous functions. (Then by the ...
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Inequality involving absolute moment and variance

Suppose $X\in\Omega$ is a random variable and $f:\Omega\rightarrow [0,1]$. Is the following true: $$E[|f(X)-E[f(X)]|]^2\leq \operatorname{Var}[f(X)]?$$ This was stated without proof in a research ...
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Creating all strongly connected graphs with given in-degree with equal probability

I am looking for a way to sample uniformly from the space of all strongly connected directed graphs (without self-loops) of $n$ nodes and in-degree $k=(k_1,...,k_n)$ with $1 \leq k_i \leq n-1$. In ...
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Invariant dimension property and a ring epimorphism

In Hungerford's Algebra, p. 186, the Proposition 2.11 says Let $f:R\to S$ be a nonzero epimorphism of rings with identity. If $S$ has the invariant dimension property (IDP), then so does $R$. It is ...
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What does rotational invariance mean in statistics?

What does rotational invariance mean in statistics? The property that the normal distribution satisfies for independent normal distributed $X_i$, $\Sigma_i X_i$ is also normal with variance $\Sigma_i ...
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51 views

How to show that the cross-ratio is invariant under Moebius transformations without using generator matrices of $\text{GL}(2, \mathbb{C})$

I am given the following problem set: Show that the cross-ratio is invariant under Moebius transformation, meaning that $$D \left(L_g(z_1),L_g(z_1),L_g(z_1),L_g(z_1)\right) = D(z_1,z_2,z_3,z_4) ...