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Symmetries on sets of strings

My question is a reference request about where to find work on symmetries on sets of strings. I'm not a mathematician, so the terminology I use below is probably very non-standard. My apologies. ...
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2answers
32 views

How to make an expression manifestly symmetric

Believe it or not, the following expression is symmetric under the exchange of the indices $j$ and $k$, i.e. $R_{kj}=R_{jk}$: $$ R_{jk}=j s_js_k-\sum_{n=1}^{\min(N-k,j)}(k-j+2n)s_{j-n}s_{k+n} $$ Where ...
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0answers
30 views

Is every finite group of isometries a subgroup of a finite reflection group?

Is every finite group of isometries in $d$-dimensional Euclidean space a subgroup of a finite group generated by reflections? By "reflection" I mean reflection in a hyperplane: the isometry fixing a ...
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36 views

Symbols to represent each distinct symmetry of polyhedra

Is there a pictorial or symbolic way to represent each distinct symmetry of a polyhedron?
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1answer
23 views

Reading the sphere diagrams in point groups on wikipedia

How do you read/make sense of the sphere diagrams shown here: http://en.wikipedia.org/wiki/List_of_spherical_symmetry_groups What do the yellow shaded areas represent? What are the red ...
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1answer
53 views

Visualizing Platonic Solid group symmetries

How do you visualize the rotation symmetries, to classify a icosahedron for example as Ih, H3, [5,3], (*532)
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12 views

For an app teaching about polyhedra, what are some core characteristics to include?

For fun: I'm building a 3d app that teaches about polyhedra. What should I include? The obvious didactic elements for each polyhedron would be: Fundamental polygon's Vertices 
Edges
 Faces
 (and ...
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0answers
26 views

Vanishing Fourier terms

Which Fourier coefficients vanish for a periodic function $ f(\theta) $ of period $ 2\pi $ satisfying $ f(\theta) = f(\pi − \theta) $? What about $ f(\theta) = - f(\pi − \theta) $ 􏰖Hint: ...
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178 views

Identities for Sieve of Eratosthenes collisions.

Edited to define the last two tables Three Questions: 1) Is all notation correct? 2) Is there a symbol for flatten? 3) How would we prove the identities: the sum of the divisors in the symmetric ...
5
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3answers
249 views

Is symmetry a valid option in inequalities?

Consider two questions: Q1. $$a+b+c+d+e=8$$ $$a^2+b^2+c^2+d^2+e^2=16$$ $$a,b,c,d,e\in\mathbb{I^+_0}$$ Find maximum value of 'e'? My answer: Since when e is maximum when all other variables are equal ...
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1answer
56 views

Programmatically recognizing symmetries of a polyhedron

I'm programming something, but I'm stuck at something which more math-oriented people probably can help me with. I am giving a polyhedron in the following form: for each vertex I get the cyclic order ...
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0answers
32 views

What is the symmetry of the Penrose tiling?

What is the symmetry of the Penrose tiling? Simply C5 or bigger? Any simple proof that the tiling is a complete cover of the plane?
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2answers
88 views

Realizing $\mathbb{Z}/n \mathbb{Z} \oplus \mathbb{Z}/n\mathbb{Z}$ as an isometry group

Every finite group arises as the isometry group of a subspace of an euclidean space (Albertsona, Boutin, Realizing Finite Groups in Euclidean Space). What are natural examples of spaces realizing the ...
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1answer
60 views

Prove or disprove: if a set of 2d-points have symmetry axises, then at least one of the axises is eigenvector

Let's assume we have a set of 2D-points. My claim is that if that group has at least one valid symmetry axis, then at least one of those axises is equivalent to an eigenvector of the covariance matrix ...
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0answers
15 views

symmetry ratio and symmetry index

Can someone explain how to calculate "symmetry ratio" and "symmetry index" when the following points in the Cartesian plane is given ? ...
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0answers
24 views

Association of financial phenomena/indications with the conservation laws of Black Scholes equation

For a while I've been doing research on methods of obtaining conservation laws via the symmetries of DEs. I'm presently doing research on identifying financial indicators/phenomena that can be ...
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0answers
22 views

Characterizing a function regarding symmetry

Let us suppose a function $f \colon \mathbb{N} \times \mathbb{N} \to \mathbb{R}$, such that $$\neg\left(\forall a,b \,|\, a \in N \land b \in N \implies f(a,b)=f(b,a)\right)$$ That is $$\left(\exists ...
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3answers
55 views

Definition of Equals

DISCLAIMER: This is a first time Math.SE post from a 30-something who is only now learning math. I have read the Rules and this may not satisfy the "Questions with too many answers" criteria or ...
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1answer
45 views

2x2 symmetric matrix is a subspace of vector space.

Can you kindly check my proof of the problem and correct if possible. The following $S=\{A\in M_{2,2} | AA^T=A^TA\}$ is a subspace of $V=M_{2,2}$ all real $2\times2$ matrices. My proof: S ...
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2answers
35 views

Orientation of rectangle on conic section

Consider a conic section. There are 2 rectangles such that all of the 8 vertices of the 2 rectangles lie on the conic section. Further assume that the 2 rectangles have different orientation (ie. a ...
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1answer
18 views

Mollifying while conserving symmetries

Suppose $B = B(z,v)$ is a function in $L^1_\rm{loc}(\mathbb{R}^d \times S^{d-1})$ depending only on the values of $|z|$ and $|z \cdot v|$ (you don't make any assumptions on how $B$ depends on $|z|$ ...
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Does “Spherical Symmetry” as defined in General Relativity imply a Foliation of Spheres?

In Carroll's "spacetime and geometry" he defines a spherical symmetrical spacetime as a spacetime $(M,g)$ for which there exists a Lie algebra homomorphism between the Lie algebra of a subset of the ...
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1answer
30 views

Proof of axis of symmetry [duplicate]

How do you prove -b/2a the Axis of symmetry equation using the Quadratic formula?
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51 views

Groups of isometries

I recently read the popular scientific book "Symmetry and the Monster" and it emphasizes groups as the sets of symmetries of geometrical objects. So I was wondering, do all groups appear as symmetry ...
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2answers
46 views

Why must isometry of regular polygon fix origin?

Here is the question: Suppose $\varphi\colon\Bbb R^2\to\Bbb R^2$ is an isometry and $\varphi(\pi_n)=\pi_n$, where $\pi_n$ is the regular $n$-gon with center at origin. Why must $\varphi$ fix the ...
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29 views

Reflection Symmetry for Non-Linear Differential Equations

We are given the equations: \begin{align} \dot{x}& =\mu \, x +y+y^3 \\ \dot{y}& =2x-2y+xy^2+\gamma \, x^2y \end{align} The question at hand is to determine whether there is some sort of ...
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0answers
17 views

Ordering binary matrices for reflection/rotation

I have a collection of $n\times n$ binary matrices and I would like to reduce it for symmetry ($D_4$ -- reflections and rotations). The naive method of testing each pair is very slow because the ...
4
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2answers
73 views

Intersection of two tetrahedra, point reflexion

We are given a regular tetrahedron $ABCD$ ($ABC$ is its` base and $D$ is its vertex) and we reflect it through the middle of its height (point reflexion) - and thus we obtain a congruent regular ...
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2answers
28 views

How to prove that $n$ is prime if an $n$-Venn diagram has $n$-fold rotational symmetry

I was reading this article on "The Search for Simple Symmetric Venn Diagrams" by Frank Ruskey, Carla D. Savage, and Stan Wagon and on the first page page they prove that $n$ is prime if an $n$-Venn ...
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1answer
25 views

Rotation-Reflection-Symmetries

I have the following exercise: Do a rotation of $2 \cdot 72 ^\circ$. Then do a reflection of the axis $d4$. Then do a reflection of the axis $d3$. Then do a rotation of $2 \cdot 72 ^\circ$. ...
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1answer
36 views

Rotation and Symmetries of Equilateral Triangle

Let $\Sigma=\{ 1,2,3 \}$ be the set of the vertices of an equilateral triangle. Let $f=\sigma$ be the rotation of level with center of rotation $O$ over an angle of $\frac{2 \pi}{3}$ radians or ...
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1answer
22 views

Symmetries of the set of points $S$

Consider that we have a set of points $S$ in the plane $\mathbb R^2$ or in the space $\mathbb R^3$ and we also consider the one to one mappings $f:S \to S$ which have the following property: They ...
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1answer
57 views

Connection Between Fundamental Solution and Symmetries of PDE

The typical derivation of the fundamental solution of Laplace's equation is to look for a radially symmetric solution because the Laplace equation has radial symmetry, and a similar heuristic can be ...
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1answer
30 views

Conventional unit mesh

I'm trying to find and outline a non-primitive conventional unit mesh, I'm not sure how to go about it. I'd also like to find any mirrors of planes and rotional symmetry axes. Would this look ...
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2answers
53 views

Map a half sliced unit disk to upper half plane

"half sliced unit disk" Can somebody tell me how to map this conformally to the upper half plane? I think the symmetry principle should be applied here but stuck on that for hours. Pardon my hasty ...
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0answers
50 views

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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2answers
25 views

Symmetry in Space

Is it possible for a non-co-planar set of points to be symmetric about a point but not symmetric about a plane? I am pretty sure this is true but I can't think of an example. Things that I think ...
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0answers
29 views

Symmetry groups of discrete functions

I'm looking for basic information about symmetry groups of discrete functions. It is difficult to search for such information, because searching for "symmetry group" gives results that refer almost ...
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1answer
111 views

Help with Autonne-Takagi factorization of a complex symmetric matrix.

Let $A=A_1i+A_2$ with $A$ non singular. Now let $$B =\begin{bmatrix} A_1 & A_2\\ A_2 & -A1 \end{bmatrix}$$ With $A_1$, $A_2$ and $B$ symmetric. Is it true that: 1) $B$ is non singular 2) ...
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2answers
38 views

Is $u^TAu \geq 0$ true for all symmetric matrices $A$?

we know from the definition of inner product and norm, that $u^Tu$ is always larger than zero, except the case where $u=0$ at which case it is zero. I came across a question that infers that $u^TAu ...
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0answers
14 views

Is the autocorrelation of a function the same if one term is flipped on the y axis?

I have some questions about autocorrelation. They are very related, so I thought that one single post was appropriate for the topic. The first question is already illustrated in the subject: if I ...
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1answer
50 views

Symmetry Definition and Equation [closed]

I need some help to understand Inversian Symmetry, Conformal Symmetry, and Scale Symmetry. Could you give me some guideline?
1
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1answer
30 views

Proving that the Moment Tensor is super-symmetric

The Carathéodory theorem in the image bellow is the one about convex hull, isn't it? Would you please explain why can the tensor F be rewritten as that sum? From that representation the author ...
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2answers
38 views

How to show properties of a given relation?

I am given that R is a relation on the given set X, and I have to show if the relation is (i) reflexive, (ii) symmetric, (iii) transitive, (iv) asymmetric, and (v) give an example of an element of ...
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2answers
27 views

Create a new function using symmetry w.r.t a point

Let's say I have a function $f$ which is defined on an interval $[0,1]$ . How can I create a function $g$ defined on $[0,2]$ where: $f(x)=\sqrt{x^{a} - x^b}$ ,$0<a<b$ ,$a$ and $b$ are constants ...
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2answers
44 views

Understanding the shape of $\phi''(x)=F(\phi(x))$

Hello I've got a question and no idea to get a solution. Maybe someone can give me an advice. The following problem is given: There is given a function $\phi \in C^2([a,b])$. Furthermore there is a ...
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0answers
12 views

Name for symmetric irreflexive binary relation

I have an irreflexive relation $\prec$ called unpreference: if $x\prec y$ then I say $x$ is unpreferred (or not preferred) to $y$. I wish to give a name to the symmetric part of the relationship, ...
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1answer
85 views

Lie algebra $\implies$ Lie group?

Lie's third theorem says that every finite-dimensional Lie algebra g over the real numbers is associated to a Lie group G. So say I have an $r-$ parameter group of symmetries whose tangents at the ...
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156 views

What to call this kind of symmetry in a sphere?

Geometrically, if the two hemispheres of a spherical distribution of some kind (let's say a spherical gas cloud) are similar such that the properties of the gas as seen by a person standing on a ...
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57 views

Symmetry of the pentagon and even permutation

I was doing part (iii). For the first part of that questions, $ D_{10} = \{e, \rho, \rho^2, \rho^2, \rho^4, \sigma\rho, \sigma\rho^2, \sigma\rho^3, \sigma\rho^4 \} $ where $\rho = (1 \ 2 \ 3 \ 4 \ ...