# 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.

241 questions
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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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### Confusion concerning the Euclidean Norm and unitary invariance on $\mathbb C^n$

For a unitarily invariant norm on $\mathbb C^n$, how do I show that $||x||=||x||_2||e_1||$? I can show that $||e_1||=1$ for the Euclidean norm by definition, and is therefore unitarily invariant, does ...
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### Invariants between two isomorphic vector spaces

I have a general question about isomorphisms between vector spaces. From a general point of view, there are common properties (invariants) between two isomorphic structures (e.g., properties about ...
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### Proof that $a\nabla^2 u = bu$ is the only homogenous second order 2D PDE unchanged/invariant by rotation

Looking for feedback and maybe simpler intuition for my proof of the theorem, shown below The statement of the theorem: Theorem Among all second-order homogeneous PDEs in two dimensions ...
1answer
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### What can I say about the form of an invariant function?

I have a general scalar function which has the properties: \begin{align} f(s\,a,b,c)&=s\,f(a,b,c)\\ f(s\,a,s\,b,s\,c)&=f(a,b,c) \end{align} where $s$ can be any real number, so the invariance ...
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### Calculating tangent vector of curve s(P,$\alpha$) at given point $\alpha$ = 0. http://yann.lecun.com/exdb/publis/pdf/simard-00.pdf

I am reading one chapter where tangent vector is calculated for the given curve $s(P,\alpha)$ at $\alpha=0$ by differentiating with respect to $\alpha$; $\frac{\partial s(P,\alpha)}{\partial\alpha}$. ...
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### Invariant properties of functions under coordinate transformations

I am interested in what sort of properties are preserved for a function defined on a smooth manifold $M$. Preservation in the sense of a physicist, i.e., invariant under coordinate transformations. ...
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### Definition of invariant ideal and invariant set of points?

I am reading a book about Invariant theory. It is said there the ideal $I$ is invariant action of a group of $n\times n$ matrices. Which element of $I$ are polynomials in $n$ variables. and also said ...
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### Simple example of invariants

The combinatorics textbook I'm reading introduces invariants with the following example: There are three piles with $n$ tokens each. In every step we are allowed to choose two piles, take one ...
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### A new graph invariant? The maximum number of non-equivalent colorings with $n$ colors.

Consider (proper) coloring of a finite graph $G$ with exactly $n$ colors and the following coloring transformation: choose an edge of the graph with the end nodes of colors $a$ and $b$ and swap the ...
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### Proving two sequences overlap

$A =a_1,a_2,a_3,\dots,a_{25}$, strictly increasing with each term being non-negative integer and striclty less than 50 $B=a_1+2,a_2+2\dots,a_{25}+2$ with same condition as last except that $50\equiv0$...
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### Prove interior point is a topological invariant

Let $(X,\tau)\cong(Y,\tau^*)$, $A\subseteq X$ and $p\in int(A)$. Let $f:(X,\tau)\to (Y,\tau^*)$ be a homeomorphism. To prove that the property of interior point is a topological property we shall show ...
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### QFT, Noether and Invariance, Complex fields, Equal mass

The problem statement, all variables and given/known data Question attached: Hi I am pretty stuck on part d. I've broken the fields into real and imaginary parts as asked to and tried to compare ...