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.

learn more… | top users | synonyms

7
votes
2answers
126 views

Prove that if fewer than $n$ students in class are initially infected, the whole class will never be completely infected.

During 6.042, the students are sitting in an $n$ × $n$ grid. A sudden outbreak of beaver flu (a rare variant of bird flu that lasts forever; symptoms include yearning for problem sets and craving for ...
1
vote
0answers
32 views

Theorems of euclidean geometry as invariable properties of geometric configurations

Is there some book, or systematic theory, that proves theorems of euclidean geometry by viewing them as invariable properties of certain geometric configurations ? So that from an easy special case, ...
1
vote
1answer
30 views

Invariants/monovariants: numbers on a board

The numbers from $1$ through $2008$ are written on a blackboard. Every second, Dr. Math erases four numbers of the form $a, b, c, a+b+c$, and replaces them with the numbers $a+b, b+c, c+a$. Prove ...
3
votes
1answer
39 views

Similarity classes of matrices

Let $M_n(K)$ be the set of all $n\times n$ matrices over a field $K$. If $\mathcal{R}$ is the equivalence relation defined by matrix similarity, what does the quotient $M_n(K)/\mathcal{R}$ looks like? ...
1
vote
1answer
80 views

Is the mapping “positive stochastic matrix onto its Perron-projection” continuous?

I am dealing with a topological question concerning the mapping that maps a positive stochastic matrix onto its invariant distribution. I am asking myself if such a mapping is continuous (or ...
1
vote
3answers
45 views

Does a repeated eigenvalue always mean that there is an eigenplane under the transformation matrix?

If you have a 3x3 matrix, if you find that it has repeated eigenvalues, does this mean that there is an invariant plane (or plane of invariant points if eigenvalue=1)? I always thought that there was ...
-1
votes
2answers
16 views

Matrices for action wrt basis

Consider the permutation representation where $G=S_3$ on $\mathbb{C^3}$ with the action: $\pi(g)e_i=e_{g(i)}$ $W=\{ \lambda_i e_i ; \sum \lambda_i=0 \}$ is an invariant subsoace of vector space $V$ ...
0
votes
2answers
61 views

Invariant subspaces

Let $V$ is a finite dimensional vector space over $\mathbb{C}$ and $T$ be a linear operator on $V$ . How to prove $T$ has an invariant subspace of dimension $k$ for each $k = 1,2, \ldots ,\text{dim}V$ ...
0
votes
1answer
42 views

Invariant subspaces and dimentions

Let V is a finite dimensional vector space over C and T be a linear operator on V . How to prove T has an invariant subspace of dimension k for each k = 1,2, . . . ,dimV .
2
votes
0answers
41 views

Isospectral transformation of ODE

Is there a transformation of coefficients of differential operators w/periodic coefficients on the real line $a_n(x+2\pi)=a_n(x)$, that preserves the eigenvalues of their monodromy matrices? $$Df(x)=\...
2
votes
2answers
57 views

Invariant subspaces

Let T be a linear operator on a finite dimensional vector space V over a field F such that every subspace of V is invariant under T then how to prove T is digonalizable ? Is the converse true ?
-1
votes
1answer
41 views

Compactness and positive invariance of set under flow of ODEs

Given a system of ODEs, $$x'=y$$ $$y'=x-x^3-y$$ $$x(0)=x_0$$ $$y(0)=y_0,$$ also given a set $S=\{(x,y):V(x,y)\le k, x>0\}$, $V(x,y)=-\frac{x^2}{2}+\frac{x^4}{4}+\frac{y^2}{2}$, where $-\frac{1}{4}&...
0
votes
3answers
43 views

Positive invariance of a set under a system of ODEs

Given the system of ODEs, $$x'=x(1-x-y)$$ $$y'=y(x-1),$$ $Q=\{(x,y):x\ge 0, y\ge 0\}$, and $S=(x,y)\in Q:x+y\le k$, $k>1$, I need to show that $S$ is invariant under this system of ODEs. Attempted ...
0
votes
1answer
38 views

Invariance of sets for systems of ODEs

Given the system of ODEs $$x' = x(1-y)$$ $$y'=y(x-1),$$ let the set $Q=\{(x,y):x\ge 0, y\ge 0\}$. Explain why $Q$ is invariant for this system of ODEs. My explanation: If $x > 1$ and $y<1$ then ...
4
votes
6answers
216 views

Proof there is no way to chose signs to make sequential sum $1+2+3+\cdots+10$ even [closed]

I've figured that for the sum $$1+2+3+4+5+6+7+8+9+10=55$$ There is no way to chose the signs of the numbers to get an even sum. I'm really struggling to prove this and would appreciate some ...
0
votes
2answers
16 views

Which complex polynomials in 3 variables are $GL_3(\Bbb{C})$ invariant?

A polynomial $p(x,y,z) \in \Bbb{C}[x,y,z]$ is $GL_3(\Bbb{C})$-invariant if $$ \forall \sigma \in GL_3(\Bbb{C}): p(\sigma(x,y,z)) = p(x,y,z).$$ How to characterize the set of $GL_3(\Bbb{C})$ invariant ...
0
votes
1answer
38 views

Jumping on the Coordinate lattice grid

Mr. Fat moves around on the lattice points according to the following rules: From point (x, y) he may move to any of the points $(y, x), (3x, −2y), (−2x, 3y), (x+1, y+4)$ and $(x − 1, y − 4).$ Show ...
0
votes
0answers
7 views

Closed Under Linear Transformation vs Invariance - Formally Naming a property of a set

I have a set of points in $\mathbb{R}^2$ given as $G(t;z_0,\theta_0)$ where $z_0 \in \mathbb{R}^2$ and $\theta_0$ represents a specific orientation. I have proved that, $G(t;z_0,\theta_0) = z_0 + \...
0
votes
1answer
60 views

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 $ ...
1
vote
2answers
62 views

$T$ 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 = \mathrm{im}...
1
vote
1answer
47 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 ...
2
votes
0answers
31 views

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 $\...
0
votes
1answer
35 views

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 ...
4
votes
2answers
63 views

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 ...
1
vote
1answer
44 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 ...
0
votes
0answers
22 views

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 ...
0
votes
0answers
112 views

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 ...
3
votes
2answers
45 views

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 ...
0
votes
1answer
47 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?
2
votes
2answers
74 views

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. ...
0
votes
0answers
32 views

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 $\arg\...
2
votes
1answer
46 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 ...
0
votes
1answer
22 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 ...
1
vote
1answer
49 views

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 ...
0
votes
1answer
22 views

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. ...
1
vote
1answer
47 views

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 ...
0
votes
1answer
46 views

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 $...
0
votes
0answers
91 views

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 ...
0
votes
1answer
71 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 ...
0
votes
0answers
47 views

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 ...
0
votes
1answer
20 views

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 ...
0
votes
0answers
16 views

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. Re-...
0
votes
1answer
49 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 = ...
2
votes
0answers
32 views

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 $N\in\...
2
votes
3answers
102 views

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 ...
2
votes
2answers
116 views

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 ...
1
vote
1answer
32 views

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 $<ap^1,bp^2>$ ...
1
vote
1answer
58 views

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 ...
1
vote
1answer
367 views
2
votes
1answer
20 views

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 $$...