Mathematical intuition is the instinctive impression regarding mathematical ideas which originate naturally without regard to formal mathematical proofs. It may or may not stem from a cognitive rational process.

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Intuition behind normal subgroups

I've studied quite a bit of group theory recently, but I'm still not able to grok why normal subgroups are so important, to the extent that theorems like $(G/H)/(K/H)\approx G/K$ don't hold unless $K$ ...
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Intuition or figure for Reverse Triangle Inequality $||\mathbf{a}| − |\mathbf{b}|| ≤ |\mathbf{a} − \mathbf{b}|$ (Abbott p 11 q1.2.5)

I acquiesce to Wikipedia's picture for Triangle Inequality. But without referring to Triangle Inequality at all, is there intuition or figure please for Reverse Triangle Inequality for all ...
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48 views

Intuition - Normal Subgroup Test - Fraleigh p. 141 Theorem 14.13

(1.) Not querying proofs or formality. I do this in my other question. Normal Subgroup Test says H is normal in G $\iff gH{g}^{-1}\subseteq H$ for all $g \in G$. What's the intuition of this ...
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40 views

Directional derivatives in any direction do not imply continuity?

I found an example where a function from R^2toR has directional derivatives at a point p at any direction however the function isn't continuous at p. I found this very weird because I thought that ...
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Equivalence Relation definitions of Coset - looks like 1-step Subgroup Test? [Fraleigh p. 97 theorem 10.1]

p. 4 We are especially interested in the case where the set is a group, and the equivalence relation has something to do with a given subgroup. That is, we want to partition a group G into subsets, ...
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How do we arrive at the definite integral to find area approximated by a sum of rectangles?

The area enclosed by a one variable function from a to b can be approximated by $n$ rectangles$$A \approx \sum_{i=1}^{n} f(x_i)(x_i-x_{i-1})$$ and if we let $n \rightarrow \infty$ we get $$A = ...
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60 views

Kolmogorov's $0-1$ law and constant RV

Kolmogorov's $0-1$ Law: For any terminal event $A$ we have that either $\mathbb{P}(A)=1$ or $\mathbb{P}(A)=0$. Alternatively any $F_{\infty}$ measurable random variable (so basically a terminal ...
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121 views

Prove That the Second Moment is Minimized with a Circle Packing

Graham and Sloane studied the problem of minimzing the second moment of disks on the plane, i.e. minimize $$ U = \frac{1}{d^2} \sum_{i=1}^{n} || \mathbf{p}_i - \bar{\mathbf{p}} ||^2 $$ s.t. ...
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85 views

Any group of prime order is cyclic - Proof blueprint [Fraleigh p. 100 Cory 10.11] [closed]

Not querying the proof or formality. I include only part of the proof. The order of the group is a prime number. Call it p. Hence by means of the definition of prime number, $p > 1$. Since the ...
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41 views

Intersection of Groups is a Group? Is a Union of Groups? - Fraleigh p. 66 Exercise 6.32h

This is a true or false question, hence are the answers supposed to follow quickly? Because the empty set has no identity element, hence $\emptyset$ is not a group. Hence I'm inquiring for ...
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Meaning of Normal Vector in Surface Integration

Is there a good interpretation of what the normal vector (and its magnitude) $$\mathbf{N}=\frac{\partial \mathbf{X}}{\partial s}\times\frac{\partial\mathbf{X}}{\partial t}$$ to the parametric surface ...
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123 views

Picture - Equivalence Relation & Classes, Partitions, Quotient Set, & other related ideas

To get intuition for them and to remember them, I'd be grateful for a picture that combines and embodies the key definitions regarding Equivalence Relations & Classes, Quotient Sets, and ...
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88 views

Show that $f(x,y)= \|x-y\|_2^2$ is differentiable

Problem: Show that $f: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ with $f(x,y)=\|x-y\|_2^2$ is differentiable and compute its differential at every point in the domain of $f$Note: $\| \cdot ...
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49 views

Physical significance of knot vector in B-spline.

A B-spline blending curve formulation is: $P(u)=\sum_{k=0}^np_k B_{k,d}(u)$ Given $n+1$ control points, B-spline blending functions are polynomials of degree $d-1$, $(1<d<=n+1)$. ...
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96 views

What is the relationship between the second isomorphism theorem and the third one in group theory?

The second isomorphism theorem [wiki] in group theory is as follows: Let $G$ be a group. $H \triangleleft G, K \le G$. Then: $HK \le G$, $(H \cap K) \triangleleft K$, and $K/(H \cap K) ...
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99 views

How does this amount change if we add to it?

We can iterate over the naturals, with zero included. Here the focus is on the numbers from $0$ to $2^s - 1$, inclusive, in binary. So we have the numbers as: $$0000, 0001, 0010, \dots 1111 \text{ ...
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86 views

Cauchy-Schwarz Inequality - Proof using Projections [Lay P379 Thm 6.7.16]

t If $u=0$, then the inequality becomes $ 0 \le 0 $, which is true. See Practice Problem 6.7.1 on P382. If $u\neq 0$, let $W$ be the subspace spanned by $u$. $1.$ How would one determine to ...
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integrate a difficult function

I can't solve it. please help! I tried everything. Integration by parts - doesn't work. but maybe I didnt do it right. I tried to substitute , but I'm stuck. $$\int \frac{x}{\cos x}\sin(\tan ...
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216 views

What is defined by rate of change at a single point?

Rate of change measures how fast a process is going when it moves from one point to another. It measures the change of, say, $Y$ when $X$ moves from $X$ to $X + \Delta X$. But my problem arises when ...
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How to understand “Union of balls centered at rational numbers is way less than $\mathbb{R}$

A few month ago I had to prove $\lambda(\mathbb{Q}) = 0$ (where $\lambda$ is the one-dimensional Lebesgue measure). The idea: Let $\varepsilon \gt 0, r_n := \frac{\varepsilon}{2^n}$ and $\mathbb{Q} = ...
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24 views

Understanding a complex set description

I am trying to understand the following math expression, describing the possible sets of a 15 Puzzle: The expressions for tile and ...
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133 views

First and Second Fundamental Form Intuition

I was just wondering what various quantities relating to the first and second fundamental forms of a regular surface mean intuitively. First of all, another explanation as to what the first and second ...
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Is $(-1)^{1/8} + (-1)^{7/8}$ ever a value whose real component is $0$?

Is $$(-1)^{1/8} + (-1)^{7/8}$$ ever a value whose real component is $0$? Is this ever true in modular arithmetic, hypercomplexes, and/or both?
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explanation for a combinatorial identity involving the binomial coefficient

I am looking for an intuitive explanation for the identity: $$\binom{n}{h}\binom{n-h}{k} = \binom{n}{k}\binom{n-k}{h}$$ Thanks!
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131 views

An intutive way to think about odd and even numbers. [closed]

What is an intuitive way to think about odd and even numbers? And about divisibility also...
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47 views

What is a zero morphism in an abelian category

I am trying to familiarize myself with some basic category theory and I am getting confused with what a $0$-morphism is. If we are in category of say $k$-vector spaces then I am guessing ...
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78 views

Intuition - Countable iff Surjection iff Injection [Velleman P310 Thm 7.1.5]

Define $I_n = \{1, 2, ..., n \} $. Let $A$ be a nonempty set. TFAE : (i) $A$ is finite (ie: a bijection $h:A\rightarrow I_{N}$ exists) or A is countably infinite (ie: a bijection ...
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50 views

Intuitive significance of harmonicity

I'm nearing the end of the semester of an introductory-level complex variables class. (Very introductory -- it's the version of the class that's only required for engineering and physics majors, as it ...
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66 views

Proof strategy for $(=>)$: If $g \circ f = id_A$, then f onto $\iff$ g 1-1. [Chartrand 3Ed P239 9.72]

For nonempty sets A and B and functions f : A → B and g : B → A, suppose that $g \circ f =$ the identity function on A. $(♦)$ (d) $(=>)$ Assume that $f$ is onto. This means there exist ...
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Intuition behind Descartes' Rule of Signs

I have read several places that Descartes' Rule of Signs was familiar to both Descartes and Newton, and that both considered it too "obvious" to merit a proof. I know how to prove it, but I would like ...
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Descartes' Rule of Signs

I have read several places that Descartes' Rule of Signs was familiar to both Descartes and Newton, and that both considered it too "obvious" to merit a proof. I know how to prove it, but I would like ...
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Intuition - Divisibility results - If c divides some integers, then c divides any linear combination of them.

Not querying about proofs here. I don't want to memorize, thence are there intuitions or illustrations for them? As a student, how else can I remember these results? Origin - Elementary Number ...
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Motivation behind steps in proof of Hoeffding Inequality

The lemma that is proved for proving Hoeffding's inequality is: If $a\leq X\leq b$ and $E[X]=0$, $E[e^{tX}] \leq e^{\frac{t^2(b-a)^2}{8}}$ Here's a link to the proof: ...
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140 views

Why if $a = qb + r$, then $\operatorname{gcd}(a,b) = \operatorname{gcd}(b, r)$ intuitively?

Origin - Elementary Number Theory, Jones, p$5$, Lemma $1.5$ Are there any illustrations? I tried wikipedia's article and the first picture to the right, but I think this delineates Euclid's ...
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Intuition about the where the beta distribution has its maximum

I've stumped myself trying to develop an intuition about why the beta distribution should have its maximum where it does. I can differentiate easily enough, and I can manage a simple argument based on ...
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239 views

Polar coordinates for $xz$-plane: $z = r\sin\theta$ ? [Stewart P1091 16.7.25]

$1.$ The unit disk is projected onto the xz-plane, so shouldn’t $x = 1\cos \theta$ and $\color{red}{z = 1 \sin \theta} $? User Semsem below kindly identified the problem: The normal to the ...
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Intuition behind a convergent subsequence of $\sin(n)$

$\let\eps\varepsilon$ I was looking through a Real Analysis exam paper one day and was stuck on a question; fortunately there is a solution provided which I will sketch below, but I have no intuition ...
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Existence of a regular uncountable $\aleph_{\alpha}$ without $\mathsf{AC}$

Set theory (Jech) $\text{p.}\;27:$ It is an open problem whether one can prove without the axiom of choice that there exists a regular uncountable $\aleph_{\alpha}\;($the informed guess is that ...
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67 views

what problem do quadratics solve?

Areas of Math were developed to deal with problems which couldn't be tackled with preexisting methods. So, for example, 2D geometry was used to solve measuring problems with land, basic algebra for ...
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1answer
40 views

What values or form of values can we get for these multiplications modulo a prime?

If we have four complex values, all of the form $a + b i$, for integers $a$ and $b$, we can label them $c$, $d$, $e$ and $f$. Now if we want to find $g$ and $h$ such that $$g \equiv ce \equiv df ...
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111 views

What does the Yoneda lemma say for the identity functor and finite sets?

So I try to plug in the simplest arguments into the Yoneda lemma and see how to interpret it. I'll try it for the identity functor and the category of finite sets, in particular, I use an three ...
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61 views

Mentally visualizing functions of complex numbers

I've recently been learning about functions of complex numbers (to complex numbers), and I can't quite fit them into my head. When I think about real functions, I tend to mentally visualize them as ...
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Maximizing the Magnitude of the Resultant Vector

Given a set of $n$ two-dimensional unit vectors: $\left\{ \mathbf{v}_1, \dots, \mathbf{v}_n \right\}$, I want to find the coefficients $\left\{ \alpha_1, \dots, \alpha_n \right\}$, $0 \leq \alpha_i ...
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27 views

Seeing complex roots on the graph of a polynomial

When I sketch the graph for a general second degree polynomial $y = ax^2 + bx + c$ it is easy to "see" its roots by looking at the points where $y=0$. This is true also for any $n$-degree polynomial. ...
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36 views

Borsuk–Ulam theorem for $n=2$

How one can intuitively prove the following statement: At any moment there is always a pair of antipodal points on the Earth's surface with equal temperatures. What about a rigorous proof?
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Am I wrong in thinking that $e^{i \pi} = -1$ is hardly remarkable?

I believe my trouble is that the identity, $e^{i \pi} = -1$, comes down to the definition of the exponentiation of $i$, which seems rather obscure to me. This is my current understanding of ...
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25 views

How can we prove that we can get any $f$ modulo a prime $p$, that satisfies these equations?

We can suppose that we have four naturals not equal to zero: $a, b, c, d$. Further, we're working modulo a prime $p$. Now if we find $a, b, c, d$ that satisfy: $$f \equiv a \cdot c \equiv b \cdot d ...
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Is there an intuition for why power sets come in powers of $2?$

My title is a bit sloppy. First let me say that I perfectly understand the proof that $|P(S)|=2^{|S|}$, I am not asking for an easy to understand proof. My question is more whether there is an ...
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About the hyperplane conjecture.

I have recently heard about the hyperplane conjecture and I would like to understand better the problematic behind this conjecture. The hyperplane conjecture: There exists a universal constant ...
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Limit Point of a Set

Definition. A point $x$ is a limit point of a set A if every $\varepsilon$-neighborhood of $x$ intersects the set A in some point other than $x$. I understand the definition in that $x$ is our limit ...