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10
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0answers
309 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
210 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 ...
8
votes
0answers
258 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 ...
7
votes
0answers
114 views

Infer distance from a point to a line, from the distance from a point to a plane [Stewart P793 12.4.44]

I'm able to prove $44$, but how would one deduce $43$ from it without further industry, forthwith? $43$ seems like a reduced, 2D version of $44$? I'm not enquiring about individual proofs. $44.$ ...
6
votes
0answers
95 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 ...
5
votes
0answers
117 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 $N_G(H)$, ...
5
votes
0answers
194 views

Colored Picture for Equivalence Classes, Relations, Partitions, ..

Origin — A Book of Abstract Algebra — Charles Pinter — p120. I'm trying to sketch a colored picture for the ideas from equivalence classes, equivalence relations, partitions, etc... underneath. ...
5
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0answers
78 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
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0answers
63 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
votes
0answers
57 views

How to visualise Bollobas' 1965 theorem?

Theorem $[n]=\{1,\ldots,n\}$. Let $\lbrace (R_i, S_i), i \in I \rbrace, R_i, S_i \subset [n]$ be such that $R_i \cap S_i = \emptyset, R_i \cap S_j \ne \emptyset (i \ne j)$. Then $$\sum_{i \in I} ...
4
votes
0answers
48 views

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

Let $H \le$ group G and $a, b \in G.$ Prove or give a counterexample. If $aH= bH,$ (30.) then $Ha= Hb.$ (33.) then $a^2 H = b^2 H.$ I understand p. 3: Let $G = S_3$ and $H = \{(1), (1,3)\}$. ...
4
votes
0answers
47 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
186 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
265 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
votes
0answers
59 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
78 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
110 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
37 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
138 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
161 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
204 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
147 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
132 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
104 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
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0answers
39 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
52 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. ...
2
votes
0answers
177 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
88 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
130 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
35 views

Explaining Spin(3)

I’m going to discuss the action of Spin(3) on Euclidean vectors. This thing has several alternative names: “versors”/“rotation quaternions”, “quaternionic adjoint representation”, “quaternion action ...
1
vote
0answers
75 views

Geometrical application of generation function for permutation

It is quite well known that the generation function for permutations is represented as $$(1+x)(1+x+x^2)\dots(1+x+x^2+x^3...+x^{n−1})$$ (See, e.g., question The generating function for permutations ...
1
vote
0answers
20 views

Different ways of visualizing a certain classes of single variable complex functions

Single variable complex functions of a real variable are ubiquitous in engineering contexts such as control engineering and signal processing, and visualizing them is of utmost importance in designing ...
1
vote
0answers
98 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) = ...
1
vote
0answers
39 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
31 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
67 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
100 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
229 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
102 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
61 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
8 views

Are there tools for presentation and vizualization of deduction?

I read that Kalish and Montague introduced a natural deduction method (http://en.wikipedia.org/wiki/Donald_Kalish), which can be easily implemented in software. Any other tools who can help a logician ...
0
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0answers
9 views

Draw an ellipse corresponding to a bivariate normal distribution

Let $$\mu=\left(\begin{array}{c}\mu_1 \\ \mu_2\end{array}\right)$$ and $$\Sigma=\left(\begin{array}{cc}\Sigma_{1,1} & \Sigma_{1,2} \\ \Sigma_{2,1} & \Sigma_{2,2}\end{array}\right)$$ be ...
0
votes
0answers
20 views

Affine Transformation and Continuous Deformation

How do these two concepts relate? Thus far I have a (what I think is a) good intuitive idea of a continuous deformation- the visual basically looks like the boundary being stretched so that it never ...
0
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0answers
23 views

Nyquist limit explanation

Kindly explain Nyquist in easy words. The actual question is as follows. We can attempt to display sampled data by simply plotting the points and letting the human visual system merge the points into ...
0
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0answers
84 views

Intuition/Picture - Matrix Multiplication - Product of [Row or Column Vector] and Matrix [Lay P95]

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 picture, and no proofs please. $1.$ Intuitively, in ...
0
votes
0answers
37 views

What kind of graph was github's impact graph?

Github used to have a graph called an impact graph. It feels almost like a Sankey diagram and almost like a stacked area chart. What is the name of this kind of graph? I couldn't find a better ...
0
votes
0answers
42 views

Computer-aided study of elementary geometry

As a beginning student of elementary (euclidean plane) geometry, so far, I have gotten the impression that there are two major approaches to geometries: naive vs axiomatic. Being a humanities student ...
0
votes
0answers
13 views

Normal Gradient z component is too small?

I am trying to draw arrows on a graph to show the normal gradient of a function. By gradient, I mean the arrows should follow the surface of the function, not being perpendicular to it. For example ...
0
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0answers
40 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
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0answers
39 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 ...