# Questions tagged [geometric-invariant]

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16 questions
84 views

### Do two exponential spirals intersect?

I have lists of complex points: orbit of complex point z under quadratic function f(z) = z*z I know that lists are: z, z^2, z^4, z^8, ... (r,t), (r^2, 2*t), .....
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### The sum of squares of distances from the vertexes of regular polygon to the any line that passes the center of it. [closed]

To prove that it is geometric invariant I need to find some others. I was thinking about proving it by the Pythagorean theorem, using the fact that in all cases the distance from the vertex to the ...
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### How can I get better at solving problems using the Invariance Principle?

I have some questions regarding the Invariance Principle commonly used in contest math. It is well known that even though invariants can make problems easier to solve, finding invariants can be really,...
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### How to find Invariant Lines and Lines of Invariant Points, without utilising Eigenvectors?

this is my first post so I do apologise regarding any formatting issues! I have a question regarding Invariant Lines and Lines of Invariant Points; from what I can gather, an Invariant Line is one of ...
309 views

### Is dot product the only rotation invariant function?

I am looking for rotation invariant scalar functions $f(x,y): x,y \in R^3$ that are not some scalar function over the dot product (or norm), i.e. $f \neq g(x\cdot y, \Vert x \Vert, \Vert y \Vert )$ ...
382 views

### Invariance of the second moment of area of a regular polygon

Consider a $n$-sided regular (convex) polygon and its circumscribed circle of radius $r$, centered in $(0,0)$. Fixing $(r,0)$ as the coordinate of the first vertex, the $n$ vertices of the polygon ...
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### Dimension of the group of all motions in $\mathbb{R}^n$ which leaves a fixed r-plane invariant.

As the title, I would like to ask the dimension of the group of all motions in $\mathbb{R}^n$ which leaves a fixed r-plane $L^0_r$ invariant. Here is my observation, but I don't know if it is useful ...
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### What is known about rational points on the ideal of relations / syzygy ideal?

What is known about rational points on the ideal of relations / syzygy ideal? Let $G$ be a finite group, with $|G|=n$. Then $G$ acts on $\mathbb{Q}[x_1,\cdots,x_n]$ through the regular representation (...
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### $G$-invariant functions on Manifolds

In a paper I saw the following statement: Let $M$ be a connected symplectic manifold and $G$ be a compact Liegroup acting symplectically and hamiltonian on $M$. Let $\Phi \colon M \to \mathfrak{g^*}$ ...
41 views