# Tagged Questions

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

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### Proving that 1- and 2-d simple symmetric random walks return to the origin with probability 1

How does one prove that a simple (steps of length $1$ in directions parallel to the axes) symmetric (each possible direction is equally likely) random walk in $1$ or $2$ dimensions returns to the ...
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### Why does Group Theory not come in here?

Here is a list of questions that I find quite similar, for the one and only reason that they all show much "symmetry". Which is at the same time my problem, because I don't have a very precise notion ...
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### What is the difference between using $PAP^{-1}$ and $PAP^{T}$ to diagonalize a matrix?

What is the difference between using $PAP^{-1}$ and $PAP^{T}$ to diagonalize a matrix? Can both methods be used to diagonalize a diagonalizable matrix $A$? Also does $A$ been symmetric or not ...
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### Interesting $\sum_{i=0}^{m-1}(-1)^{i}(m-i)^{2} = \sum_{i=1}^{m}i$

I found that for m $\in N$ $$\sum_{i=0}^{m-1}(-1)^{i}(m-i)^{2} = \sum_{i=1}^{m}i.$$ I found it after doing an exercise. For example: $$5^{2}-4^{2}+3^{2}-2^{2}+1^{2} = 1 + 2 + 3 + 4 + 5 = 15.$$ For ...
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### Number of reflection symmetries of a basketball

Excerpt from John Horton Conway, The Symmetries of Things, pg. 12. Basketballs have two planes of reflective symmetry, as do tennis balls. I read this sentence and it immediately struck me as ...
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### Seeking Better (Symmetry Exploiting) Solution and Generalization of An Inequality

Given positive variable $(x,y,a,b)$ where $x+y=1$, how does one "slickly" prove the following inequality? $$f(x,y) := \frac{xa+yb}{\sqrt{xa^2+yb^2}}\ge \frac{2\sqrt{ab}}{a+b}.$$ or simply f(x,y) := ...
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### What is the dimension of a certain vector subspace of invariant matrices?

$\newcommand{\Sig}{\Sigma}$ Let $\Sig$ be a diagonal matrix with strictly positive entries on the diagonal. Define $V=\{B \in M_n\mid B\Sig +\Sig B^T=\Sig B +B^T \Sig \}$ (where $M_n$ is the vector ...
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### Combining symbols with symmetry

So this question has probably been answered already, but I can't find a good answer through searching google or this site. Basically, if I have n symbols, how many n-length combinations of the ...
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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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### Lie Algebra: Optimal system of one-dimensional sub-algebras of the heat equation

This is a follow up question to Invariants of a PDE by Lie Symmetries, as I tried to follow the reasoning from the book Applications of Lie Groups to Differential Equations (Peter J. Olver, Example 3....