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10
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0answers
215 views

Visualizing a Calabi Yau

I would like to understand how I can visualize the quintic threefold $$ z_1^5 + z_2^5 + z_3^5 + z_4^5 +z_5^5 - 5\psi z_1z_2z_3z_4z_5 = 0$$ For a similar problem, Hanson proposes the following: ...
9
votes
0answers
201 views

Visualizing the Partition numbers (suggestions for visualization techniques)

So Ken Ono says that the partition numbers behave like fractals, in which case I'd like to try to find an appropriately illuminating way of visualizing them. But I'm sort of stuck at the moment, so ...
7
votes
0answers
189 views

Kakeya Needle problem video

I'm intruiged by the Kakeya Needle problem, described here on Wikipedia. Wikipedia has a nice animation of a needle turning through a hypo-cycloid: What I'm searching for is a visualisation of the ...
5
votes
0answers
40 views

What group are the group of symmetries of these figures isomorphic to - Fraleigh p. 85 Theorem 8.23, 24, 26

In this section we discussed the group of symmetries of an equilateral triangle and of a square. In Exercises 23 through 26, give a group that we have discussed in the text that is isomorphic to the ...
5
votes
0answers
59 views

An Auto-Generated Cartography of Mathematical Theories: Has it been done already?

While looking for a way to visualize the logical structure of mathematical theories a graph-like depiction came to my mind, where propositions are represented by vertices. An edge goes from ...
4
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0answers
56 views

Visualize normal subgroup, normalizer, cosets.

A few important aspects of the relationship $H \lhd N_G(H) \le G$ are highlighted in Figure 7.31. First, the size of $N_G(H)$ is some multiple of |H|, and the size of G is some multiple of $N_G(H)$, ...
4
votes
0answers
30 views

Magical Counterexamples to Nonidentities - Power of Cosets and Right Coset - Fraleigh p. 103 10.30, 33

Let H be a subgroup of a group G and let $a, b \in G.$ Prove the statement or give a counterexample. If $aH= bH,$ (30.) then $Ha= Hb.$ (33.) then $a^2 H = b^2 H.$ I understand p. 3: Let $G = S_3$ ...
4
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0answers
30 views

Visual Solution - Find All (Cyclic) Subgroups of $D_4$ generated by 1, 2, … elements - Fraleigh p. 84 8.19

Verify that the subgroup diagram for $D_4$ shown in Fig. 8.13 is correct by finding all (cyclic) subgroups generated by one element, then all subgroups generated by two elements, etc. Here, $p_i$ mean ...
4
votes
0answers
70 views

Visual Proofs of Series Summations

I'd like to put together a compilation of visually geometric proofs of series summations. I have three famous 2D examples to clarify what I mean below, but other "visually geometric" proofs of an ...
4
votes
0answers
152 views

Visualization of immersed submanifold

I am trying to visualize the difference between immersed submanifold and embedded submanifold. At first, I thought that, for example, if I can embed manifold $M$ in $\mathbb{R}^4$ and if my friend can ...
4
votes
0answers
247 views

Visualizing the domain of the square root

I would like to show someone the domain of the complex square root function (the 2-sheeted riemann surface). Is there a good interactive visualization software for this? I would like some sort of ...
3
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0answers
27 views

Intution, Figure. Negation of Continuity and Uniform Continuity (S.A. pp 117 T4.4.6)

Every time I need negation, I have to write out all the logical symbols to negate manually. I know how to determine these negations myself. But I want to compehend intuition or figure like ...
3
votes
0answers
47 views

Intuition on Axiom of Completeness

♪ (J. Stewart. Calculus 6th ed. pp 682) Axiom of Completeness = AoC = A nonempty set of real numbers that has an upper bound has a least upper bound. AoC is an expression of the fact that there ...
3
votes
0answers
39 views

Visualize cosets of kernel of homomorphism, normal subgroup

Question 1. 'Since we know that the codomain is a group, this cannot happen.' I don't understand. Can someone elaborate? I know all homomorphisms are functions but not vice versa. Functions are ...
3
votes
0answers
27 views

Visualize every quotient map follows a pattern, subgroup and its left cosets

page 167. Because of the Fundamental Homomorphism Theorem, Nathan Carter calls non-embedding homomorphisms quotient maps. This is one of the key facts about homomorphisms: they come in ...
3
votes
0answers
28 views

Intuition or Picture for Results on Preimages, Images of Functions, their unions, intersections [Chartrand 3E P235 9.12]

What's the intuition behind the following results? In view of their number, a more helpful question might be how to intuit/naturalise them. Here's a more thorough collectanea. I'm also interested ...
3
votes
0answers
123 views

Visualizations of ordinal numbers

I find this picture of the ordinal numbers up to $\omega^\omega$ rather hard to grasp: I wonder if the following might be a more compelling way to visualize ordinal numbers up to $\omega^\omega$: ...
3
votes
0answers
110 views

Power-set in Hypercube: historical background of indexing each term like Hasse Diagram?

My instructor wants references about the indexation over the hypercube, related question here, he does not believe that I was the first who used it -- [update] thanks to a comment, the name is Hasse ...
3
votes
0answers
142 views

Imagining four or higher dimensions and the difference to imagining three dimensions

I’m very interested in how people envision four or higher dimensions. And I’m especially interested in how geometers and topologists who actually work in four dimensions do. Now I know of the video ...
3
votes
0answers
133 views

Visualizing and manipulating 4-dimensional data with 3D technology

It is possible to visualize 3 dimensional data (like a scatter plot) by projecting it on a 2 dimensional screen in a way that allows to interact with it in an intuitive way. Is it possible to ...
3
votes
0answers
121 views

Illustrations of a line and a curve intersecting for complex field

Are there nice illustrations on the Net of say $y=a·x+b$ and $y=x^2$ intersecting where x and y are complex? I'm thinking of the amplitude of y being depicted as height above the complex plane with ...
2
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0answers
24 views

Intuition/Picture - Matrix Multiplication - Product of [Row or Column Vector] and Matrix [Strang P59]

From P59 of Intro to Lin Alg, 4th Ed by Strang This question is not a duplicate of the original, in which user Shuchang proved the question. Presently I'm asking about further intuition or a ...
2
votes
0answers
45 views

Intuition of Picture - Collapse, Factor Group, Homomorphism, Normal Subgroup - Fraleigh p. 144 Figure 15.1

Let $N \unlhd G$. In the factor group $G/N$, the subgroup $N$ acts as identity element. Regard N as being collapsed to a single element, to the identity element. This collapsing of N together ...
2
votes
0answers
28 views

When do Entries Remain after and despite Matrix Multiplication? [Strang P92 2.5.41]

Suppose $E_1, E_2, E_3$ are 4 by 4 identity matrices, except $E_1$ has $a, b, c$ in column 1 and $E_2$ has $d, e$ in column $2$ and $E_3$ has $f$ in column 3 (below the $1$ s). Multiply $L = ...
2
votes
0answers
29 views

How to Intuit/See Matrix Factorisation [GStrang P250 Ex 5.1A]

I beg leave for your forgiveness over the colours. Please enlighten me if there's a more efficient way. How is the determinant of the checkerboard sign pattern matrix, $ \begin{bmatrix} a(1, ...
2
votes
0answers
155 views

Looking for proof-without-words of Bezout's identity

I'm looking for a "proof-without-words" of Bezout's identity (for integers). Does anyone know of one?
2
votes
0answers
80 views

Help me to vizualise this falling ball on spinning Earth

The earth rotates. The ball falls in an latitude, not equator, let say in Germany. I am trying to understand how to express the ball in terms of the angular velocity on the planet. The constant ...
2
votes
0answers
125 views

Turning real roots into curves (for visualisation)

One can obviously map a set of real numbers $x_1, x_2, \ldots x_N$ to a curve in 2-D via $y=(x-x_1)(x-x_2)\ldots(x-x_N)$. Thinking about data visualisation, one can portray a set of $N$ observations ...
1
vote
0answers
13 views

Problem using Stokes's Theorem - Boundary Curve, Unit Normal Vector [Stewart P1097 16.8.5]

$\Large{1.}$ How does one determine the boundary curve, denoted as C, to be the plane $z = -1$? I’m flummoxed because $S$ here is given as bottomless. I'm not enquiring about formal or rigorous ...
1
vote
0answers
25 views

Visualizing a complex curve

This may be as much a question about computers as about math. Let $C=\{f(r,s,t)=0\}$ be a curve in $\mathbb{CP}^2.$ By forgetting about the points at infinity we can view $C$ as a surface in ...
1
vote
0answers
22 views

Motivation behind Definition of Projection [Poole P27]

In the long paragraph above equation $(2)$, http://mathinsight.org/dot_product avers: This leads to the definition that the dot product $\mathbf{a⋅b}$, divided by $∥\mathbf{b}∥$ (= magntitude of ...
1
vote
0answers
51 views

Complex analysis visualization (Cauchy Theorem, Residue Theorem)?

I usually think of complex functions on the complex plane like vector fields. So basically what I have problems with is visualizing firstly Holomorphic functions. I have also read and successfully ...
1
vote
0answers
80 views

Applet to find least-crossings drawing for an input graph

Is there a convenient online applet that allows me to draw a graph, after which it outputs a plane drawing of an isomorphic graph that has (approximately) the least number of crossings among all ...
1
vote
0answers
43 views

Visually apealing holologous transformation of a given contour

There is this problem which roughly says: You want to put a framed picture onto the wall with a cord to the picture frame. The cord is a single one, and both ends are attached to the frame. ...
1
vote
0answers
168 views

Help to understand the concept of diminishing returns

Suppose I have this function: $q=f(k,l)=600k^2l^2-k^3l^3$ Then, $f_l=1200k^2l-3k^3l^2$ $f_k=1200kl^2-3k^2l^3$ $f_{ll}=1200k^2-6k^3l$ $f_{kk}=1200l^2-6kl^3$ $f_{kl}=f_{lk}=2400kl-9k^2l^2$ Now, ...
1
vote
0answers
99 views

Which space this space drawn in this picture is homeomorphic?

Based in this question Why this space is homeomorphic to the plane? I would like to know which space this space is homeomorphic, any help or an intuitive idea are welcome. [Context of Image: ...
1
vote
0answers
57 views

Visualising the derivative/slope $f'(x_0)$ of $f:\mathbb{R} \rightarrow \mathbb{R}$ as a line segment

A function $f:I\subseteq \mathbb{R} \rightarrow \mathbb{R}$ that is differentiable at $x_0 \in I$ obeys the following equality for all $h\in (I-x_0)$ (i.e. for all $h\in \mathbb{R}$ such that $x_0+h ...
0
votes
0answers
5 views

Literature on 3 dimensional image segmentation

Currently I am working on a paper on 3d image segmentation and finding my way into the topic. However I can not find good mathematical literature on the topic. I am looking for a books or scripts that ...
0
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0answers
31 views

Examples of spectral sequences

Where can I find geometric examples of spectral sequence calculation? For instance, calculation of the spectral sequences of a filtration on torus induces by an enough nice Morse function is quite a ...
0
votes
0answers
46 views

Intuition for $\inf(AB) = \inf(A)\sup(B)$. Difference for sets and functions? (Abbott pp 199 q7.4.5)

1. What's the intuition for $\inf(AB) = \inf(A)\sup(B)$? Figure please? I know I must posit $A,B \subseteq R$ as bounded sets. If they're unbounded, $\sup$ doesn't exist. I believe $\inf(AB) = ...
0
votes
0answers
15 views

Can an arbitrary network of nodes be effectively visualized as a circular “treemap”?

We all know that a treemap is effective for visualizing hierarchical tree data (i.e. where there are only 1 to many relationships like in a computer file system): But how difficult algorithmically ...
0
votes
0answers
25 views

logic - how to model or diagram conditional probabilities of multiple related scenarios.

I am interested in modeling questions and specific outcomes so that i can evaluate conditional probabilities and mathematical expectation. I am looking for a way to diagram or otherwise describe the ...
0
votes
0answers
25 views

Graphing complex plane curves

This may be just as much a question about computers as a question about math. Suppose we have a complex curve $C\subset\mathbb{CP}^2,$ given by some $f(r,s)=0.$ Picking an affine chart, we can view ...
0
votes
0answers
48 views

Parametric equations and specifications of a triskelion (triple spiral)

I haven't been able to find the parametric equations and specifications to form a triskelion, a triple spiral (this is made of three interlocked couples of spirals). Using the parametric equation of ...
0
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0answers
42 views

Must the “n” in mod(n) always be prime?

I'm experimenting with mod(n) and have the following questions even after reading the Wiki page and numerous articles about the subject. Must mod(n) always be prime for cryptographic purposes? Is ...
0
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0answers
23 views

How to visualize the result of Low-rank decomposition?

In a numerical low rank decomposition, whether it is non-negative matrix factorization(NMF), or binary matrix factorization(BMF), or non-negative sparse PCA, we have two low-rank matrices to ...
0
votes
0answers
69 views

Visualize trajectories of high dimensional system in a two dimensional space

I have high dimensional data from a dynamical system and I would like to visualize its trajectory. I came across a paper where they used classical multidimensional scaling to plot the evolution of a ...
0
votes
0answers
27 views

How to visulize surface link in four dimension?

I am now facing a problem with "surface link" in four dimension. I have heard that three 2-torus can be linked in four dimension. And I have created a movie by cutting four dimensional space with ...
0
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0answers
65 views

Xi function plotter ??

i would like to know what software i need to plot the RIemann Xi function $ \xi(s)= \frac{1}{2}s(s-1)\Gamma (s/2)\zeta (s) $ on a given interval , for example from $ s=0 , s=1000 $ or from $ s=0 , ...
0
votes
0answers
180 views

The geometry of functions $\mathbb{R}^2\rightarrow \mathbb{R}$ that satisfy the norm axioms

What are constraints on the "looks" of a function $f:\mathbb{R}^2\rightarrow \mathbb{R}$, if $f$ satisfies $$f(x_1+x_2)\leq f(x_1) + f(x_2), \ \quad x_1,x_2\in \mathbb{R}^2 \quad \quad (1)$$i.e. the ...