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1answer
53 views

Is this claim true$(\xi \circ k)(s)=(k \circ \xi )(s)=0$ $\implies$ $k(s)=\zeta(s)=0 $ is true if and only if RH is false?

It is well known that $\xi(s)=\xi(1-s)$ is a verified functional equation for all complex $s$, where $\xi(s) = s(s-1) \pi^{s/2} \Gamma\left(\frac{s}{2}\right) \zeta(s)$. let $k(s)=\xi(1-s)$ and ...
0
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1answer
14 views

Definition of symmetry

Let $X \subseteq R^2$. A symmetry of $X$ is isometry $f: R^2 \to R^2$ such that $f(X) = X$. For example, square has $8$ symmetries one of which is $R_{90}(a, b) = (-b, a)$. Is an element of $X$ ...
1
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0answers
11 views

Polygons with fixed number of axes of symmetry

Let $k$ be a positive integer. Suppose a polygon has exactly $k$ axes of symmetry. How many sides may the polygon have? A regular $n$-gon has $n$ axes of symmetry, so one answer is $k$. What are ...
2
votes
1answer
17 views

Maximum of a generalized Rayleigh quotient

Given two symmetric positive definite matrices $A,B\in\mathbb{R}^{n\times n}$ and $x\in\mathbb{R}^n$. How do I prove that the generalized Rayleigh quotient $R(A,B,x):=\dfrac{x\cdot A\cdot x}{x\cdot ...
0
votes
1answer
21 views

Progresive diagonalisation of a symmetric matrix

Given a symmetric real matrix $A$ there is an orthogonal transformation that brings $A$ to a block diagonal form, where $a \in \mathbb{R}$: $$ RAR^{T} = \left[ {\begin{array}{cc} a & 0 \\ 0 ...
0
votes
0answers
16 views

How to make a polar symmetry proof

I have an exercise that ask for a symmetry proof of the function: $$r=\sin(3\theta)$$ and $$r^2=2\cos(2\theta)$$ But i really dont know what they ask and what to do.
1
vote
1answer
27 views

Symmetric matrix if symmetric linear transformation

I want to proof the following theorem: With respect to any orthonormal basis, if the 2 $\times$ 2 matrix $\bigl(\begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \bigr)$ represents a ...
6
votes
2answers
109 views

Need help to visualise a set (reading Abelian groups)

I am reading Abstract Algebra. I cannot visualise the following example: Let $n$ be a positive integer, and consider the set $S_n$ of all permutations from the set $n = {1, 2, \ldots , n}$ to ...
2
votes
0answers
19 views

Symmetric Fibonacci images

I was playing with the Turtle module in Python and decided to try plotting the Fibonacci series with the following scheme where $f_n$ is the $n^{th}$ Fibonacci number: Rotate the turtle $k (f_n ...
-2
votes
1answer
117 views

determining reflective, symmetric, transitive, anti-symmetric properties and describing equivalence classes

The question: determine if p is reflective, symmetric, transitive and/or anti-symmetric, if p is an equivalence relation, describe the equivalence classes A = Z , and $apb$ if and only if $5 | (2 a + ...
3
votes
1answer
51 views

Fitting a grid inside a circle, is the solution always symmetric?

Let us construct a grid consisting of rectangles of height $h$ and width $w$. When we place a circle of radius $r$ over this grid, there is a certain amount of rectangles that completely lie within ...
1
vote
1answer
49 views

Hungarian Algorithm on Symmetric Matrix

I have a complete and weighted graph with an even number of vertices. I would like to separate all the vertices into pairs such that the sum of all the edge weights for each edge connecting the ...
0
votes
1answer
49 views

How can I compute angles & lengths of the following tiling shape

a while back I created a tile made of arrows: I did it using a vector graphics software, without really understanding the properties of this shape. Now, let's say I want to write a program to ...
2
votes
0answers
15 views

Visualisation of representations and their decomposition into irreps

A question in a Representation Theory midterm got me thinking, and made me realise I didn't really understand irreps. The question was on the subject of reps of $S_4$, and went: An obvious ...
-1
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2answers
40 views

Proving a matrix is always symmetric [duplicate]

$B$ is a square matrix of real numbers. Show that the matrix $BB^T$ is always symmetric.
1
vote
1answer
44 views

How is $r(\theta) = \sin \frac\theta2$ symmetric about the x-axis?

I understand how it is symmetric about the $y$-axis. because $r(-\theta) = \sin \left(-\frac\theta2\right)=-\sin \left(\frac\theta2\right)=-r(\theta)$ But how is it symmetric about $x$-axis?
2
votes
1answer
101 views

Why is the dihedral group closed under composition?!

I've been obsessing over this all day now. I understand associativity, presence of inverse elements and identity, but I don't get why a composition of a reflection with a rotation or other reflexions ...
1
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1answer
27 views

Symmetry of polar equations

In your opinion how to show symmetry in polar equations without graphing. i thought of these methods :- converting to cartesian then test . check the period of the function . please help me any ...
1
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0answers
24 views

Point symmetry group, identification.

Good day. Please, tell me information about the algorithm of identification point symmetry groups for two-dimensional data (timeseries)? Maybee some book ? I have a data file like this: ...
1
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0answers
34 views

How to find the center of of rotational symmetry in a Penrose tiling?

The center of Helsinki sports a gorgeous Penrose tiled pavement: Could anybody give me an algorithm that will permit me to find its 'center' - that is, the point around which the five-fold ...
0
votes
2answers
61 views

Prove symmetry of natural logarithm

Prove that $f(x)=\ln\sqrt{x^2+1}$ is symmetrical in $x=0$. $\ln\sqrt{(x-a)^2+1}=\ln\sqrt{(x+a)^2+1}$ $\sqrt{(x-a)^2+1}=\sqrt{(x+a)^2+1}$ $(x-a)^2+1=(x+a)^2+1$ $x^2-2ax+a^2+1=x^2+2ax+a^2+1$ ...
3
votes
1answer
53 views

Symmetries on sets of strings

My question is a reference request about symmetries on sets of strings. I'm not a mathematician, so the terminology I use below is probably very non-standard. My apologies. Terminology. Let $[n] = ...
1
vote
2answers
43 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 ...
3
votes
1answer
105 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 ...
5
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0answers
45 views

Symbols to represent each distinct symmetry of polyhedra

Is there a pictorial or symbolic way to represent each distinct symmetry of a polyhedron?
1
vote
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 ...
1
vote
1answer
59 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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0answers
13 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 ...
8
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0answers
272 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
votes
3answers
267 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 ...
3
votes
1answer
60 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
37 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?
4
votes
2answers
92 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 ...
0
votes
1answer
65 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
27 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 ? ...
2
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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 ...
1
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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 ...
3
votes
3answers
56 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 ...
2
votes
1answer
156 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 ...
1
vote
2answers
51 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 ...
0
votes
1answer
19 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 $|z|$ ...
2
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0answers
22 views

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 ...
0
votes
1answer
43 views

Proof of axis of symmetry [duplicate]

How do you prove -b/2a the Axis of symmetry equation using the Quadratic formula?
3
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1answer
70 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 ...
1
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2answers
48 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 ...
0
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0answers
36 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 ...
2
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0answers
26 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
votes
2answers
115 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
31 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 ...
1
vote
1answer
32 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 $2 \cdot 72 ^\circ$. ...