# Tagged Questions

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

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### How linear map transform the unit ball?

Let $f:\mathbb{R}^n \to \mathbb{R^n}$ be a linear application, we suppose that $f$ is symmetric ($\langle f(x),y\rangle=\langle x, f(y)\rangle$), without using spectral theorem how we can see that $f$ ...
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### Functional equation $P(X)=P(1-X)$ for polynomials

I have encountered the following problem : Find all polynomials $P$ such as $P(X)=P(1-X)$ on $\mathbb{C}$ and then $\mathbb{R}$. I have found that on $\mathbb{C}$ such polynomials have an even ...
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### Testing for symmetry about a curve/line

In High School Algebra , after studying how to plot a graph of $f(x)$ (rather called $y$) against $x$ in Cartesian coordinates, we studied several tests to determine the symmetry of the plotted graphs ...
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### Efficient assignment of tetrahedron's chirality

Suppose we have a regular tetrahedron delimited by four points $A_{1}, A_{2}, A_{3}, A_{4}$. There are 24 permutations of vertices, but there are only two distinct terahedra that cannot be ...
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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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### Eigenvalues of a rotationally symmetric matrix

I have a rotationally symmetric matrix of arbitrary size, for example, A = \begin{pmatrix} a & b & c & b & a \\ b & d & e & d & b \\ c & e &...
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### Dihedral groups: Relationship between symmetries and rigid motions

In Dummit & Foote, $D_{2n},\ n \geqslant 3$ is the set of symmetries of a regular $n$-gon, where a symmetry is a rigid motion of the $n$-gon which can be effected by taking a copy of the $n$-gon, ...
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### Solution of wave equation

I posted this question for the first time on Physics Stack Exchange more than one year ago. The question was closed as off topic. Even if I reworked the question no one considered the possibility ro ...
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### Is the space of rigid Riemannian metrics convex?

Let $M$ be a smooth manifold. Let $g_1,g_2$ be two rigid Riemannian metrics on it. (i.e with no isometries except the identity). Is it true that every convex combination of $g_1,g_2$ is also rigid? ...
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### Spherical functions which are invariant under a finite rotation group

Is there a nice, clean reference which lists a basis in terms of (linear combinations of) spherical harmonics for the $L^2$ space of functions defined on the sphere $\mathbb{S}^2$ which are invariant ...
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### Symmetry properties conserved after integration?

I have an integrand consisting of the variables $a,b,w,x,y,z$. Now I integrate over the variables $w,x,y,z$ ($w$ and $x$ from $0$ to $1$ and $y$ and $z$ from $0$ to infinity). I know that the ...
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### General structure of a 3 by 3 persymmetric matrix with zero eigenvalues

Here is an interesting problem that might be extended to higher dimensions, I am looking for different simple ways of describing a 3 by 3 matrix with one or two zero eigenvalues. This persymmetric ...
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### Difference between symmetry algebra and symmetry group

What is the difference between symmetry algebra and a symmetry group? I just wanted to know if my understanding is right. Lets say we have a system of differential equations. Then the symmetry group ...
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### a conjugate of a glide reflection by any isometry of the plane is again a glide reflection

Q : Prove that a conjugate of a glide reflection by any isometry of the plane is again a glide reflection, and show that the glide vectors have the same length. I know that: the conjugate of ...
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### How to “rotate” a function? Or, how to write a function which has a known, rotational symmetry with respect to another function?

EDIT 2: I've posted my "real" question here: http://mathematica.stackexchange.com/questions/115766/finding-closed-form-eigenvalues-of-a-particular-matrix I have posed my question formally in LaTeX ...
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### What is a “unique” mirror line of symmetry?

What is a "unique" mirror line of symmetry? For example why does an equilateral triangle have three mirror lines but only one "unique"mirror line of symmetry?
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### Construction of a triangle using symmetry

I need to construct a triangle $\Delta \textrm{ABC}$ knowing that $t_a = AS$, $|AS| = 6\, cm$, $|\measuredangle \textrm{BCA}| = 30°$ and $|AB| = 5.5 \,cm$. I've been told that it's possible to do it ...
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### Symmetry of Rank Two Tensor with Mixed Components

I understand that a rank two tensor (t) is classified as symmetric if $t^{ij} = t^{ji}$ or $t_{ij} = t_{ji}$. Later in my reading, I came across the following quote: It is not useful to speak of ...
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### Property / Feature of a symmetric or skew symmetric matrix for comparison

I have a symmetric matrix of size $n \times n$, where $n$ may range from $10$ to $50$. Let us call this matrix as $M_1$. If from $M_1$, I delete an $m$th row and an $m$th column, all other elements ...
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### Lie Algebra: Dimension of a one-parameter group (dimension of orbit)?

I am reading a book about Applications of Lie Groups to Differential Equations by Peter Olver. Lets say we have a PDE with $p$ independent variables and $q$ dependent variables In Chapter 3.1 (page ...
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### determine whether group is additive

I am taking a stab at group theory and in some of the questions I am working on they don't explicitly state whether a group is additive or multiplicative. $\mathbb{Z}_4$ is additive (and that makes ...
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### Why is the orthogonal group $O_2$ generated by rotations and reflection only?

Consider the following isometries in the Euclidean group of distance preserving maps of the plane $\mbox{Iso}(\mathbb R^2)$ which is generated by the following: Rotations $\rho_\theta$ about the ...
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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....
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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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### Matrix representations of particular generators of the full octahedral group

I want to find matrix representations of the generators $a, b, c$ of the full octahedral group in the presentation $$\{a,b,c \mid a^2=1,b^3=1,(ab)^4=1,ac=ca,bc=cb\}.$$ Is there a recipe to write the ...
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If I have a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ with a Lie group G as a symmetry, $f(Ax)=f(x),\quad A\in G$ how might I go about obtaining a reduced function $\tilde{f}$ on $\mathbb{R}^... 2answers 58 views ### Solving a PDE to Yield Determining Equations I'm going through an example in Peter Hydon's book "Symmetry Methods for Differential Equations" which finds the basis for the Lie Algebra of the point symmetry generators for Burgers' equations. ... 3answers 63 views ### If$A$and$B$are arbitrary$n \times n$matrices, prove that$(A^TB^TBA)$is symmetric My attempt:$(A^TB^TBA)^T$=$(A^T)^T(B^T)^TB^TA^T$=$(AB)B^TA^T\ne(A^TB^TBA)$therefore$(A^TB^TBA)$is not symmetric. 2answers 47 views ### What is the dimension of a certain vector subspace of invariant matrices?$\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$is the vector ... 1answer 83 views ### Find a basis of the space V of all symmetric 3x3 matricies, and thus determine the dimension of V I need help finding the general element or matrix of$V$. Do I need to find the basis of the nullspace and basis of the image to solve this problem? 0answers 20 views ### Identifying Symmetric Matricies If$A$and$B$are arbitrary$nxn$matrices, which of the matrices must be symmetric?$something^T$means the transpose of "something".$A^T$$A A-A^T A^TB^TBA 2answers 329 views ### Why is this definite integral antisymmetric in s\mapsto s^{-1}? I recently happened into the following integral identity, valid for positive s>0:$$\int_0^1 \log\left[x^s+(1-x)^{s}\right]\frac{dx}{x}=-\frac{\pi^2}{12}\left(s-\frac{1}{s}\right).$$The ... 2answers 257 views ### Reflection with respect to a parabola I know how to find a reflection with respect to one of the axis or with respect to the origin, but let's say I want to find the reflection with respect to a parabola, how do I do it? Let's say we have ... 1answer 22 views ### Does Linear Transformation transforms orthonormal bases of symmetric matrix into orthogonal vectors? How would you show it? I went all over the book: looked at Linear Transformation definition again, looked at orthogonality of bases for symmetric matrices. I got to the point: Since L is linear ... 1answer 21 views ### Combination of even and odd functions Can someone help me how to show that any function f(x) defined on a symmetrically placed interval can be written as a sum of an even and a odd function? What is the special role played by "... 1answer 47 views ### What is the group of rotations of a volleyball(pyritohedron)? Practice test for Abstract algebra final, very stuck on this particular question. 1answer 38 views ### Showing a function is odd I have this equation:$$ f(x) = \frac{2x^2+3}{x-2} $$and I have to prove it has half-turn symmetry around the point (2,8). I know that for a function to have half-turn symmetry, it needs to have ... 0answers 13 views ### Symmetric and reflexive closure on positive integers Finding the symmetric and reflexive closure of the relation R = \{(a, b) | a > b\} on the set of positive integers. For symmetric closure I have: R = \{(a, b) | a > b\} U \{(b, a) | a > b\... 0answers 12 views ### Determining if Poset based on Domain and Comparison Operator? Can someone help me with how to think about the below problem? I know that a poset is a relation which is reflexive, antisymmetric, and transitive, but unless I'm dealing with finite sets I have a lot ... 0answers 27 views ### Fixed-point subspace of O(2)^-, a subgroup of O(3) O(2)^- is generated by the SO(2) of rotations about the z-axis and a reflection through a vertical plane. The space V_l is generated by spherial harmonics, i.e., Cartan decomposition$$V_l=...
Consider a system of objects labeled by the index $I$, each object located at position $x_{I}$. (For simplicity, we can consider one spatial dimension, or just ignore an index labeling the different ...