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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The Mathematics of Tetris

I am a big fan of the oldschool games and I once noticed that there is a sort parity associated to one and only one Tetris piece, the $\color{purple}{\text{T}}$ piece. This parity is found with no ...
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Rotation invariant tensors

It is often claimed that the only tensors invariant under the orthogonal transformations (rotations) are the Kronecker delta $\delta_{ij}$, the Levi-Civita epsilon $\epsilon_{ijk}$ and various ...
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Invariants of a matrix

I'm teaching a course in physics, and I need a simple and intuitive proof that a matrix ($3\times3$, but it doesn't matter) has exactly 1 invariant which is linear in its entries, 2 that are ...
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186 views

Using the invariance principle: how to solve $n+d(n)+d(d(n))=m$?

Let $d(n)$ be the digital sum of $n$. How to solve $n+d(n)+d(d(n))=m$, where $n$ and $m$ are natural?
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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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86 views

Numbers in a sequence

The number sequence $1, 9, 8, 2...$ satisfies the following rule: each element of the sequence starting from the fifth, is equal to the last digit of the sum of the previous four members. Will we ...
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1answer
77 views

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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114 views

Inertial Frames of Refereence

I am told that in Newtonian mechanics, no coordinate system is "superior" to any other. Also, all inertial frames are in a state of constant, rectilinear motion with respect to one another. So am I ...
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notation for invariation

Let $\Lambda = \{T \in \operatorname{Her}_2(\mathcal{O}) ; T \ge 0\})$ and $\mathcal{O}$ the maximal order of some quadratic imaginary number field. I write $T[U] := U^* \cdot T \cdot U$ where $U$ is ...
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697 views

The Invariance Principle

I had come across a problem practicing to get better at approaching different types of problems from different field topics and this one had got me kind of stuck in what direction to go. Not so ...
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1answer
113 views

Connection between the Tutte and characteristic polynomials?

Both the Tutte polynomial $T_G(x,y)$ and the characteristic polynomial $\phi_G(x)$ encode a great amount of structure of the input graph $G$. I've read somewhere that the Tutte polynomial has a kind ...
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30 views

Prove that n is divisible by 4 in a cylic sum with variables which have only two possible values

It is known that $a_1, a_2, a_3, ... , a_n \in \left\{-1, 1 \right\}$ and $S = a_1a_2a_3a_4 + a_2a_3a_4a_5 + ... + a_na_1a_2a_3 = 0$ Prove that $n \equiv 0\space(mod\space 4)$ I know this problem ...
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60 views

Constructing the sequence: $0\rightarrow (x-y)^{S_2} \stackrel{f}{\rightarrow} k[x+y,xy]\stackrel{g}{\rightarrow} k[y]$

Let $S_2$, a group of two elements, act on $k[x,y]$ by permuting $x$ and $y$. It is clear that $$ 0\rightarrow (x-y) \rightarrow k[x,y]\rightarrow \dfrac{k[x,y]}{(x-y)}\cong k[y] \rightarrow 0 $$ ...
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1answer
35 views

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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1answer
160 views

Identifying $k[x_1,x_2,y_1,y_2]^{\epsilon}$ with $k[x,y]\wedge k[x,y]$

Suppose the symmetric group $S_2$ of order 2 acts on $k^4=Spec \;k[x_1, x_2, y_1, y_2]$ by the following: for $\sigma\not=e$, $$\sigma\circ(x_1, x_2, y_1, y_2)=(x_2,x_1,y_2,y_1).$$ That is, the ...
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1answer
98 views

Is there any invariance under the inversion mapping?

In geometry, there is a transformation called the inversion mapping which maps nonorthogonal circles into nonorthogonal lines and vice versa.(If I make a mistake, inform me, since I am not very ...
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30 views

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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30 views

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. ...
2
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1answer
95 views

A Unique Invariant subspace for a set of matrices

Im wondering if anyone can give me a good reference or answer this question which may have already be solved. For a set of generic $n\times n$ matrices $A_1,A_2,...,A_k$, such that they share only ...
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40 views

Rotation invariance of higher than 2 dimensions

According to this $f_2(x_1,x_2) = x_1^2 + x_2^2$ is invariant under rotation. I wanted to ask if a function $f_n(x_1,x_2,...,x_n) = x_1^2 + x_2^2 + ...+ x_n^2$ is also rotation invariant. In other ...
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1answer
99 views

Does a constant of motion always imply hamiltonian?

If a dynamical system has a constant of motion, and is not already evidently Hamiltonian, is it always possible to use a change of variables and obtain a Hamiltonian system? Edit: the constant of ...
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1answer
36 views

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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55 views

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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1answer
121 views

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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1answer
41 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) ...
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1answer
34 views

Prove that the toral endomorphism $T_A: \mathbb{R}^d / \mathbb{Z}^d \rightarrow \mathbb{R}^d / \mathbb{Z}^d$ is not ergodic.

Prove that the toral endomorphism $T_A: \mathbb{R}^d / \mathbb{Z}^d \rightarrow \mathbb{R}^d / \mathbb{Z}^d$ is not ergodic. A is an integer matrix such that A has an eigenvalue which is a ...
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1answer
56 views

Expectation and Variance

I was asked to give a simple example to show that E[XY] = E[X]E[Y] is not necessarily true if and when X and Y are not independent. I can't seem to find a simple example without ending with a proof by ...
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53 views

If $M$ is invariant under $L$ is it a vector space

I'm reading Linear Algebra by Peterson, and the excercise on p.31 reads as follows: Let $L: W \to V$ be a linear operator and $V$ a vector space over $\mathbb{F}$. Show that if $M \subset V$ is ...
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121 views

Prove that the existence of a bridge is an invariant

An invariant is a property $P$ that is shared by all isomorphic graphs. In other words, a property $P$ is an invariant provided that whenever $G_1$ and $G_2$ are isomorphic graphs, if $G_1$ satisfies ...
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1answer
109 views

Vertex of a pentagon-Does the algorithm always stop?

To each vertex of a pentagon,we assign an integer $x_{i}$ with sum $s=\sum x_{i}>0$. If x,y,z are the numbers assigned to 3 successive vertices and if $y<0$,then we replace $(x,y,z)$ by ...
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747 views

Time-invariant IVP

How does one know that a system (of differential equation and initial value constraints) is time-invariant (perhaps by inspection...)? What are the implications of a system with this property (esp. ...
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1answer
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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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1answer
35 views

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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308 views

How to prove that the Kronecker delta is the unique isotropic tensor of order 2?

Is there a way to prove that the Kronecker delta $\delta_{ij}$ is indeed the only isotropic second order tensor (i.e. invariant under rotation), i.e. so we can write $T_{ij} = \lambda \delta_{ij}$ ...
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75 views

Finding an invariant polynomial under a matrix action

I asked this question as a mathematica question: http://mathematica.stackexchange.com/questions/41689/finding-a-certain-invariant-polynomial-using-matrix-coordinates but maybe it will get more ...
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For general non-symmetric square matrices is there a matrix norm that is invariant under similarity transformations?

I think that there is no similarity-invariant matrix norm for general matrices. But are there similarity invariant norms for special types of matrices (e.g. for matrices whose eigevalues are different ...
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357 views

Time-invariant Differential Equation

I was wondering if it is safe to assume that $$ (1/t)y'(t) + (1/t)y(t) = 0 $$ is a time-varying differential equation. I would say these (1/t) factors cancel out and make the eq. time invariant. ...
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39 views

A set is partitioned into $k$ non-empty sets, and the difference between elements of each subset is given, reconstruct each subset

We have a set $A = \{1, 2, 3, \dots, n\}$. This set is partitioned into $k$ non-empty subsets: $$A_1 = \{a_1, a_2, \dots, a_{m_1}\}$$ $$A_2 = \{a_{m_1 + 1}, a_{{m_1} + 2}, \dots, a_{m_2}\}$$ $$.$$ ...
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finding invariant subgroups under all automorphisms

I'm trying to find group G s.t every subgroup of G is invariant under all automorphisms, or conditions for G. For example; cyclic groups and simple groups have this condition.
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How to show that the Harris corner detector is rotation invariant

I understand what it means for the Harris corner detector to be rotation invariant. But what steps do you need to take to show that it is rotation invariant?
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Calculate the variance from a function of normal random variable

I am new to the topic that I found difficulty for the question: Given the function $g(x) = e^{-X}$, $X \sim N(0,1)$, calculate the variance of $g(x)$. I know the answer is $e(e-1)$. But I don't ...
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Prove a transformation is a variational symmetry for J

The following problem is from The Calculus of Variations by B.von Brunt (page 215, Exercise 9.2.1) Let $$ J(y)=\int_a^b xy'^2\mathrm{d}x. $$ Show that the transformation $$ X=x+\epsilon2x\ ...
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217 views

Maximal Positive Invariant Set — Some fine print

I would like to share something I noticed on the definition of Maximal Positively Invariant Sets. Definition 1. For a discrete-time system of the form $x_{k+1}=f(x_{k})$ (and $x_{k}\in ...
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$S_k$ action on $A/I$

Let $S_2$ be a finite group of order $2$ and let $S_2$ act on $k[x,y]$ by interchanging $x$ and $y$, where $k=\overline{k}$. Then since $$ R = \left( \dfrac{k[x,y]}{(x+y)} \right)^{S_2} = ...
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184 views

Invariant set: sufficient condition

Given a system $\dot x = f(x)$, $x \in \mathbb{R}^n$ with a smooth $f(x)$. Let $D$ be a set in $\mathbb{R}^n$ with a smooth boundary $\partial D$ such that $\left.\langle f(x), n(x) \rangle ...
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$[$Is connected$]$ & $[$Has a simple cycle of length $m$$]$ : An Analysis on Invariance

How are the statements $$^{\dagger_1}\text{Is connected.}~~~~~^{\dagger_2}\text{Has a simple cycle of length $m$.}$$ an invariant?
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64 views

Prove that an algorithm cannot reach a given goal

We are given an algorithm that, in each step takes a set $\left\{a, b, c\right\}$ It takes any two variables $a, b$ at random and changes them to $0.6 + 0.8b$ and $0.8a - 0.6b$. The initial value of ...
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1answer
34 views

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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1answer
46 views

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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2answers
39 views

Is this proposition posible? [duplicate]

In a board, you have $13$ White round pieces, $15$ Black round pieces, and $17$ Red round pieces. In each round you can choose two different color pieces and change them with two other pieces of ...