Questions tagged [geometric-invariant]

19 questions
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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.
PyRulez
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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, really hard. There is this problem from Arthur…
6
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Clairaut differential equations and elliptic discriminants

I was solving this math.SE question, which was asking to solve the Clairaut differential equation $y= xy' - (y')^3$. Just to have nicer signs, I then looked at the equivalent equation $$ y= xy' + (y')^3 .$$ The main trajectories of this differential…
6
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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 ) $ Do they exist ? Edit: Edited to clarify that the…
frishcor
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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: Invariant here is that $a^2+b^2$ remains constant.…
blue boy
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Insight on difference between Euler characteristics of 2 manifolds: $\chi(U)-\chi(V)$?

For the Euler characteristic, we have the inclusion-exclusion principle: $$\chi(U\cup V) = \chi(U)+\chi(V)-\chi(U \cap V),$$ and also the connected sum property: $$ \chi(U\#V) = \chi(U)+\chi(V)-\chi(S^n). $$ However, is there any relation or…
2
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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 are given by: $$P_i =…
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2 answers

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 to help solving the problem. (1) Since the group…
YC H
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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 a manifold, one can construct a smooth one by…
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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 possible to define a distance from point to lines…
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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 understand well (for example knots or…
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Role of finite generation of the ring of invariants in the existence of a categorical quotient

From the Geometric Invariant Theory book [Mumford - Fogarty - Kirwan], we have the following theorem ([MFK,Theorem 1.1) Let $X$ be an affine scheme over a field $k$, let $G$ be a reductive algebraic group over $k$, let $\sigma:G\times X \to X$ be…
Conjecture
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Polygon / Any shape invariant for comparison or fiting

For my personal curiosity, I was wondering which would be simplest algorithmic way to compare two shapes to say whether they are the same or not. After some researches, I found out that there are many graphical/visual tools that rely on either…
Charaf
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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 (it permutes the $x_i$). Let…
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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 which a point on said line will map to another…
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