# Tagged Questions

Questions about symmetry, in group theory, geometry or elsewhere in mathematics. See https://en.wikipedia.org/wiki/Symmetry

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### Rotational Symmetry for Rangoli Designs

I am viewing these Rangoli Patterns And it says Which one has no line symmetry but matches four times as it turns around? But to me, they all have line symmetry, i.e. you fold it in half, ...
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### Nontrivial Symmetry of an Integral Depending on a Parameter

We have defined the following function \chi(\gamma)=\frac{1}{2\pi}\int{\rm d}^2{\bf x}_2\frac{{\bf x}_{10}^2}{{\bf x}_{20}^2{\bf x}_{21}^2}\left[\left(\frac{{\bf x}_{21}^2}{{\bf x}_{...
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### $O(d,d)$ vector

The $O(d,d)$ matrix $M$ ($M\in O(d,d)$) is defined as the matrix satisfying $$M^T \eta M=\eta,\quad \eta=\left(\begin{matrix}0 & 1\\1 & 0 \end{matrix}\right).$$ So if we are given some ...
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### Killing fields and symmetries

In my answer to Why are Killing fields relevant in physics? I wrote: The flow of a Killing field can drag the whole manifold. Since Killing fields are divergence free and thus their flows have no ...
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### Invariants of a PDE by Lie Symmetries

I have a little Problem in understanding how to derive invariants form Lie Symmetries (or theire infinitesimals). As one can show the heat equation $u_t=u_{xx}$ has the following symmetries (...
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### Symmetry of Predicate and Universal Quantifiers

$$\forall x \forall y P(x,y) \implies \forall x \forall y P(x,y) \land P(y,x)$$ I guess the above statement is valid but no idea how to formally prove it, any idea?
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### Solving Lie's invariant condtion for first order ODE

I write down the general equation $Y_{x}+(Y_{y}-X_{x})F-X_{y}F^{2}= XF_{x}+YF_{y}$ and assume that X=a(x) and Y=b(x)y, after that I can't see anyway to solve it for a and b. How can I get solution ...
### Isotropic left invariant Riemannian metric on $GL_n^+$?
I am trying to see if it's possible to construct a left invariant isotropic Riemannian metric on $GL_n^+$. (the group of $n \times n$ invertible real matrices with positive determinant) (When by ...