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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What is the invariance group of $f(\mathbf{z})=(\mathbf{z}^T\mathbf{z}) (\mathbf{z}^*(\mathbf{z}^*)^T )$?

What is the invariance group of $f(\mathbf{z})=(\mathbf{z}^T\mathbf{z})(\mathbf{z}^*(\mathbf{z}^*)^T)$ ? where $f:\mathbb{C}^n\to \mathbb{R}$. We can think of $\mathbf{z}$ has a complex vector such ...
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Rotational invariance of functions

What is the difference between a) $\mathbf{f} \rightarrow \mathbf{f}(\mathbf{R}^T\cdot\mathbf{x})=\mathbf{f}(\mathbf{x}^\prime)$, (Wikipedia, the article about rotational invariance) and b) $\...
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Hint on the Fifteen (Slide Puzzle) Problem?

I've been working on The Fifteen puzzle, which is to transform the slide puzzle in configuration $A$ below into configuration $B$, but I'm unsure how to proceed. Can someone provide a small hint as to ...
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System of equations invariant under a permutation.

Is there a theory or branch of Math which considers systems of equations in which the subscript variables can be permuted to obtain the same system, ie, they are invariant under some permutations. ...
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Behavior of the intersection between of stable and unstable manifold.

Consider a differentiable map $\Psi: M \rightarrow M$ , where $M$ is a differentiable manifold. Let $x^*$ be a hyperbolic fixed point and suppose that the relative stable and unstable manifold ...
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Copula invariance examples

I am learning about copulas and have problems understanding copula invariance property. Let $X$ and $Y$ be continuously distributed random variables. I get that if $(X,Y)$ have copula $C$, then $(2X, ...
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In proving the “Invariance Under Translation of Definite Integrals” by substitution, is the continuity of the integrated function mandatory?

In the beginning, I shall apologise if my question has been answered before (I can't find such an answer), and if my question is naive (I am not of a Mathematical or even scientific background, I am ...
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Finding the invariant lines of $\left(\begin{smallmatrix} 3 & -1 \\ 6 & -2 \end{smallmatrix}\right)$

How do I find the invariant lines of: \begin{pmatrix} 3 & -1 \\ 6 & -2 \end{pmatrix} So my thoughts were to assume the invariant line has the form $y=mx$ as it will pass through the origin, ...
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Invariance of domain theorem and systems of equations

I have to deduce from invariance of domain theorem that system of $n$ equations with $n$ variables has one solution. I have no idea how to do this, I don't see any connection between open sets and ...
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Intersection of acceleration and deceleration curves using differential equations

An object is moving with given acceleration and then decelerating to certain point and velocity, while whole movement must be fastest as possible - so only the one switch between acceleration and ...
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Example of a left-invariant vector space

in order to work with group operations and flows I'm looking for (simple) examples of left-invariant vector spaces! Do you have some?
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To what is $V-E+F$ transformed to when $n>3$?

I guess that convex polyhedra can be well-defined in $\mathbb R^n$ when $n>3$ and that they are well-studied so would like to know to what does the expression $V-E+F$ transforms to when $n>3$ ...
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Properties of $\mathcal{C}^1$-diffeomorphisms which keep invariant the uniform distribution on the n-cube?

Let us consider the n-cube (n-dimensional hypercube) $H_P={]0,\,1[}^P$ and let $\psi:\,H_P\rightarrow{}H_P$ be a $\mathcal{C}^1$-diffeomorphism which keep the uniform distribution (with respect to the ...
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Invariant distribution of hidden markov model

We know that a time homogenous Markov chain which is aperiodic and irreducible converges to its unique invariant distribution. Suppose we have a hidden markov model where $X$ is the underlying Markov ...
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Definition: Invariants of Hopf modules

In my lecture notes it says: Let H be a $k$-Hopf algebra. Let M be a left H-module. The invariants of H on M are defined as the $k$-vector subspace $M^{H}$:= {m $\in$ M | h.m=$\epsilon$(h)m for ...
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How to define “number of cycles which make up a graph” as a graph invariant formally?

I'll consider a graph (quiver) $E=(E^0,E^1,s,r)$, where $E^0$ is the set of vertices, $E^1$ is the set of arrows, and $s,r\colon E^1\to E^0$ are the source and range maps. I allow loops and multiple ...
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Simpler proof for tensor invariance

I tried to prove to myself the invariance of $$ (X_{11} - X_{22})^2 + (X_{12}+X_{21})^2 $$ for a 2D tensor $X$ under rotation. I managed to do it by showing that $$ \left( [ R X R^{-1} ]_{11} - [ R X ...
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How to tell if a line of invariance goes through the origin with eigenvectors

I’m trying to figure out how to calculate the lines of invariance of a transformation using eigenvectors. I’m nearly there I just don’t understand how to work out whether it passes through the origin (...
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Conditional expectation under operations

I am a bit confused about how the conditional expectation works under operations. I've seen steps such as $(E[\theta | X])^2 = E[\theta^2 | x]$ Done in proofs without much explanation. My question ...
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Unreachable states when rotating rows and columns on a 3x3 grid

I came up with this problem when I'm randomly playing around with my poker cards: first, arrange the cards $1$ to $9$ in a $3 \times 3$ grid: $$ \begin{array}{|c|c|c|} \hline 1 & 2 & 3 \\ \...
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Looking for a proof of LaSalle's invariance principle for a dynamical system on a manifold.

I found the following version of LaSalle's theorem and it appears to be stayed differently from the original. Consider the smooth dynamical system on an $n-$manifold $M$ given by $\dot{x} = X(x)$ and ...
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Is this actually a valid proof? Multivariable Calculus

So I was asked to prove that the curl of a vector field $\mathbf F=(f_1,f_2,f_3)$ is invariant under change of basis, where the initial and final basis are both orthonormal. In particular, let the ...
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An example of a non-invariant measure on a compact Lie group

I would like to construct a non-invariant measure on a compact Lie group but I'm not sure what is allowed and what the consequences are. Take the simplest example of $SO(2)$. The unnormalized ...
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In what sense are these two invariant measures on $SU(2)$ proportional?

An element $g$ of $SU(2)$ is of the following form: $$ g=\begin{bmatrix} z_1 & z_2\\ -\bar{z}_2 & \bar{z}_1 \end{bmatrix}, $$ where $z_i$ are complex satisfying $|z_1|^2+|z_2|^2=1$. I can ...
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Prob. 19, Sec. 2.3, in Herstein's TOPICS IN ALGEBRA, 2nd ed: Bracketing any $n$-tuple of elements in a set with an associative binary operation [duplicate]

Here is Prob. 19, Sec. 2.3, in the book Topics in Algebra by I.N. Herstein, 2nd edition: If $S$ is a set closed under an associative operation, prove that no matter how you bracket $a_1a_2 \ldots ...
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Does there exist an initial arrangement of 10 black squares such that all the squares will ultimately be black?

Let there be a $12×12$ table of white squares. We draw $10$ squares in black. If a white square has $2$ black neighbours, then we draw it in black. We say that $2$ squares are neighbours if they have ...
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Invariant Measure under dilatation

We know that the Lebesgue measure is dilatation-invariant, namely $\lambda( \alpha A) = \alpha \lambda(A),$ for any $\alpha > 0$ and a Borel $A$. What are the conditions for non Lebesgue measure ...
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Rotation in-variance in d=2+1 dimensions (cherns-simons term).

this is probably a stupid question, but, does rotational invariance in $d=2+1$ mean to only rotate the spatial coordinates and not the time. I mean bascially I want to show that $ \int d^3 x \...
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Why is stability conserved during a bifurcation?

Suppose we have some quantity $J$ which is defined for a particular (1D) bifurcation in the following way: $J = \text{The number of stable equilibria - the number of unstable equilibria}$. This ...
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Invariance of the x and y axis

My question is about the invariance of the x and y-axis of the following system of differential equations: $$\begin{pmatrix} \dot{x}(t)\\ \dot{y}(t) \end{pmatrix} = \begin{pmatrix} 3x(t)\\ -2y(t)+x(...
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Prove that there are 2 numbers whose difference is divisible by 2n. [closed]

I have been trying to solve this problem using pigeon hole principle but I think it has some subtleties I might not be paying attention to : We have the natural numbers $1,2,...,2n$ and we have ...
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Prove that all the numbers in the table are even.

In each cell of a 20 x 20 table an integer is written , such that for every 7 rows and 7 columns that we consider the sum of these 49 cells (the cells where the rows and columns intersect) is an even ...
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Regarding translation invariance (equivariance) property of risk measure

While studying some literatures for stochastic programming and risk averse optimization, I read that risk measure is called as coherent, when it satisfies: Convexity Monotonicity TRANSLATIONAL ...
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Explanation of Solution of a question based on Invariance Principle .

"In the parliament of Sikinia, each member has at most three enemies. Prove that the house can be separated into two houses, so that each member has at most one enemyin his own house." This is an ...
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Need hint for this olympiad combinatorics problem

I've been thinking about this combinatorics problem for a few days but couldn't figure it out, a hint would be highly appreciated. Numbers $1,2,\dots,2019$ are written on a board. You keep doing the ...
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Combinatorics problem about odd counts

There are boys and girls studying at a school. A group of boys is called good if every girl knows at least one boy from the group. A group of girls is called good if every boy knows at least one ...
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Understanding “perimeter” invariant

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 ...
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Proving basis functions span the rotation-invariant subspace of R3

What would be the correct way to prove a set of basis functions spans the rotation-invariant subspace of R3? I have a set of function that I know spans R3, and by combining the projection results in a ...
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87 views

Smoothly homeomorphic for invariance of domain and invariance of dimension

Follow-up to this: Do homeomorphic smooth manifolds, like diffeomorphic ones, have the same dimension? Based on this question Viewing invariance of domain as a converse of invariance of dimension, ...
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Viewing invariance of domain as a converse of invariance of dimension

My book is An Introduction to Manifolds by Loring W. Tu. Corollary 8.7 is (smooth) invariance of dimension, and Theorem 22.3 is smooth invariance of domain. I view these as converses and think of ...
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Noether's theorem in SmoothLife?

Conway's Game of Life, being discretized in both space and time domains, has no locally conserved quantities. SmoothLife, however, is a generalization of the Game of Life to a continuous and spatially-...
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Change two consecutive numbers to their average

Here is a well-known beautiful problem I can’t solve. It probably has been already been asked but I couldn’t find it, so please either write a solution or write a link. Thanks. Given $2^n$ integers ...
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We have a sequence $a_0,a_1,a_2,…,a_9$ so that each member is $1$ or $-1$. Is it possible: $a_0a_1+a_1a_2+…+a_8a_9+a_9a_0=0$

We have a sequence $a_0,a_1,a_2,...,a_9$ so that each member is $1$ or $-1$. Is it possible: $$a_0a_1+a_1a_2+...+a_8a_9+a_9a_0=0$$ This problem was given on contest, but I don't know how to solve it. ...
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Classical Lagrangian invariant under transformation

Consider the Lagrangian $$L(q_1, q_2, \dot{q_1}, \dot{q_2}) = \dot{q_1}^2 - \dot{q_2}^2 + q_1 ^2 - q_2 ^2$$ (Set aside any concerns about the possibility of the kinetic energy being negative.) Show ...
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Proof - Possibility to transform all vertex’s value to 1

A big equilateral triangle is made up of smaller equilateral triangles. The relation is for n order of the bigger triangle, the number of inner triangles are n^2. Example of such a triangle with n = ...
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Is the usual metric on $\Bbb{N}^\Bbb{N}$ left invariant on $S(\Bbb{N})$?

Let $\Bbb{N}^\Bbb{N}$ be the set of all functions $(x_n\mid n\in\Bbb{N})$ from $\Bbb{N}$ into itself (I identify sequences with their images, as usual). I know this is a metrizable space with ...
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What is the operational way of discovering scale invariance of differential equations?

Context The answer here by @Keenan Pepper gives an instance for what it means for an algebraic or trigonometric formula to be scale invariant. For quick reference, I quote his answer here but with a ...
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Show that Eigenspace is invariant under flow in gradient vector field

I'm struggling with where to start on the following question Consider the gradient vector field $$ \frac{d}{dt} \begin{pmatrix} x \\ y \end{pmatrix}= \begin{pmatrix} \frac{\partial V}{\...
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Why do we only consider homogeneous invariants?

Let $G$ be a finite matrix group, $G \subseteq GL_{n}(\mathbb{C})$. Consider the polynomial ring in $n$ variables; $\mathbb{C}[x_1,...,x_n]$. It is known that the ivnariant subring $\mathbb{C}[x_1,.....
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Dimension of the invariant subspace

Let $\Gamma \subseteq GL_{n}(\mathbb{C})$ be a finite matrix group. Let this finite matrix group act on $f(x_1,...,x_n) \in \mathbb{C}[x_1,...,x_n]$ like so: $$\Gamma \cdot f(x_1,...,x_n) = f(\Gamma \...

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