Questions tagged [geometric-invariant]
22 questions
10
votes
1 answer
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.
Christopher King
- 10,773
9
votes
2 answers
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
votes
2 answers
Mismatching Euler characteristic of the Torus
Why is it that when I try to compute the Euler characteristic for the Torus using a drawing like the following , then the number that I get is not the number that the Torus should have? Which is $0$?
I mean, if I understand correctly, given any…
Tutusaus
- 667
6
votes
1 answer
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…
Luca Ghidelli
- 1,337
6
votes
1 answer
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
- 61
- 2
4
votes
1 answer
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.…
Bluey
- 2,124
2
votes
2 answers
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…
TribalChief
- 189
2
votes
1 answer
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 =…
anderstood
- 3,554
2
votes
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
- 31
2
votes
0 answers
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…
Pandora
- 6,874
1
vote
1 answer
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…
LCFactorization
- 2,102
- 18
- 23
1
vote
1 answer
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…
Pandora
- 6,874
1
vote
2 answers
Why are the linear factors of $g.p$ are given by $(a_i, b_i)g^{-1}$?
I am reading a part of the paper below that computes the semistable locus in case of $\operatorname{Sym^3}(\mathbb{C}^2).$ Here is the part of the paper I do not understand:
Specifically, I do not understand the following:
Why are the linear…
weird
- 69
1
vote
1 answer
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
- 3,389
1
vote
1 answer
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
- 123