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Questions tagged [geometric-invariant]

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1answer
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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2answers
63 views

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 ...
6
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2answers
472 views

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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0answers
4k views

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 ...
6
votes
1answer
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 ) $ ...
2
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1answer
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 ...
2
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2answers
31 views

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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1answer
54 views

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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0answers
91 views

$G$-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 $\Phi \colon M \to \mathfrak{g^*}$ ...
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Affine group, identification and multiplication law

I have a question about the group of affine transformations of $\mathbb{R^2}$. Where by that I mean the following: $Aff(\mathbb{R^2})=\{AX + b\mid A \in GL_2(\mathbb{R^2}), b \in \mathbb{R^2}\}=\...
6
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1answer
362 views

Is there a orientable surface that is topologically isomorphic to a nonorientable one?

Is there a surface that is orientable which is topologically homeomorphic to a nonorientable one, or is orientability a topological invariant.
4
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1answer
214 views

problem in solving this problem from olympiad(use of invariant)

Start with the set $\{3, 4, 12\}$. In each step you may choose two of the numbers $a$, $b$ and replace them by $0.6a − 0.8b$ and $0.8a + 0.6b$. Can you reach $\{4, 6, 12\}$ in finitely many steps: ...
2
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0answers
70 views

Introduction to Euler structures

I am looking for a basic text on Euler structures, in particular smooth Euler structures, and the relation to combinatorial Euler structures; It is known that given a combinatorial Euler structure on ...
0
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0answers
47 views

Questions about the Moduli Space of Vector Bundles of rank 2 with trivial determinant over a curve of genus $g$ [duplicate]

Let $M_0$ be the moduli space of rank 2 semi-stable vector bundles over X with trivial determinant which is a singular projective variety of dimension $3g-3$. $M_0$ is constructed as the $GIT$ ...
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1answer
157 views

How to define a “distance” from point to line in 3D projective space which is projectively invariant?

Since the concept of distance in Euclidean space is not invariant in projective space, that is , distance is invariant under Euclidean transformations but not under projective transformations, is it ...
1
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1answer
110 views

Why are invariants of Homology 3-Spheres interesting?

I see how invariants (of any kind of mathematical objects) are interesting in general, since classification is interesting, but mostly in case there is a vast variety of such objects that we do not ...