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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$\frac{x}{10!} = \frac{1}{8!} + \frac{1}{9!}$

I have a pretty simple straightforward question. Q) Find the value of $x$ in the following: $$\frac{x}{10!} = \frac{1}{8!} + \frac{1}{9!}$$ Instinctively, I do the quickest thing I know how to ...
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59 views

Prove limit of modulus of quotient of two functions is infinity (S.A. pp 144 question 5.3.9)

We need to prove for all $M > 0$, there exists d such that $0 < |x−c| < d \implies |\frac{f(x) }{ g(x) }| ≥ M.$ Choose $d_1$ so that $0 < |x−c| < d_1 \implies \color{brown}{|f(x) − ...
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If $g'(c) \neq 0$, show $g(x) \neq g(c)$ for all $x \in V_d(c)$ (S.A. pp 144 q5.3.8)

Let $g : (a, b) \to R$ be differentiable at a point $c \in (a, b)$. $V_d(c) := \{ x \in R : |x - c| < d \}$. First Case $g'(c) > 0$: We cannot use the mean value theorem since we only know ...
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52 views

Length of a curve by integration: why won't flat segments do?

Maybe my question is a duplicate, but I guess I don't know the right terminology to find it elsewhere. I would be happy to delete it if someone can point out a duplicate. From elementary calculus, ...
1
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92 views

Cauchy's Generalized Mean Value Theorem. Required function. (S.A. pp 140 t5.3.5)

Cohen, Henle. Calculus pp 827, (http://www.vias.org/calculus/09_infinite_series_10_06.html) I revised the footnote in pp 14 http://www.math.uga.edu/~pete/2400calc2.pdf. This theorem can be ...
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87 views

Intuition. If f differentiable and $\lim_{x\to0} f'(x) = L$, then $f'(0) = L$. (S.A. pp 137 q5.2.8c,d) [duplicate]

True/False. (c) If $f$ is differentiable on an interval containing zero and if $\lim_{x\to0} f(x) = L$, then $f(0) = L$. (d) Drop the assumption that $f'(0)$ necessarily exists. Not a duplicate of ...
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180 views

If f' exists and f'(c) > 0 then f'(x) > 0 for all |x - c| < d for some d. (S.A. pp 137 question 5.2.8b)

If $f'$ exists on an open interval, and there is some point $c$ where $f'(c) > 0$, then there exists a d-neighborhood $\{x \in \mathbb{R} : |x - c| < d\} = V_d(c)$ around c in which $f'(x) > ...
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65 views

Function on $\mathbb{R}$ differentiable at a single point (S.A. pp 136 question 5.2.3)

By imitating the Dirichlet constructions in Section 4.1, construct a function on R that is differentiable at a single point. Tried and assayed. 1. How can you calculate this construction? Where ...
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65 views

Show that “$\Gamma \models S \Rightarrow \Gamma \vdash S$” entails “if $\Gamma \nvdash P \And \sim P$ then $\Gamma$ is satisfiable”

Show that "$\Gamma \models S \Rightarrow \Gamma \vdash S$" entails "if $\Gamma \nvdash P \And \sim P$ then $\Gamma$ is satisfiable" I'm primarily confused with the notation being used here. In ...
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58 views

Why there is this relation between $k$-vectors and $k$-forms?

I've been trying to understand the geometrical meaning of $k$-vectors and $k$-forms on some vector space $V$ of finite dimension $n$ over a field $\Bbb K$. Indeed, as I understood, a $k$-form $\omega ...
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47 views

Why always $\delta = 1/n$? Negation of Continuity and Uniform Continuity? (S.A. pp 117 T4.4.6)

These proofs about negation of continuity and uniform continuity proofs always invoke $d = 1/n$. Where did this emanate from? I know $\lim _{n\rightarrow \infty }\frac {1} {n}=0$. Why not something ...
2
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56 views

Action of a group on itself by conjugation is faithful $\iff$ trivial center

p. 5: A group action of G on X is called faithful (or effective) if different elements of G act on X in different ways: when $g_1 \neq g_2$ in G, there is an $x \in X$ such that $g_1 \cdot x \neq ...
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36 views

How small can we make a modulus and still perform linear algebra on these pairs?

We can work with numbers of the form $(a^n + a^m)$, where $a$, $n$, and $m$ are all naturals, and $-v \le m \le v$ and $-v \le n \le v$. There is one more possibility: $a^n$ could be replaced by $0$, ...
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129 views

A Nonabelian group of order of product of primes G has a trivial center - Fraleigh p. 153 15.18

Using Exercise 37, show: A nonabelian group G of order pq where p and q are primes has a trivial center. Reference: http://users.humboldt.edu/pgoetz/Homework%20Solutions/Math%20343/hw...
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Proof blueprint - If $G/Z(G)$ cyclic then $G$ Abelian - Fraleigh p. 153 15.37

(1.) Why didn't Fraleigh state the result in the direct form like in my title? Why state it with the negations and then prove the contrapositive? Isn't this extra unnecessary work? (2.) How do you ...
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188 views

What insight is supposed to be gained from this complex analysis exercise?

Let $C_0$ denote the circle centered around some point $z_0\in\mathbb{C}$ with radius $R$. We can parametrize this circle like this: $$\begin{array}{cc} z(\theta)=z_0+Re^{i\theta}, & \theta \in ...
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Apple game question

Player A and Player B play a game. On the middle of the table there is a pot full of $N$ apples of different weights. Player A starts first and chooses an apple and starts eating it. Losing no time ...
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55 views

Commutator subgroup of a simple group - Fraleigh p. 152 15.19(h)

True or False: (19h.) The commutator subgroup of a simple group G must be G itself. Answer: http://www.auburn.edu/~huanghu/math5310/alg-hw-ans-i think 3.pdf ...
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29 views

Need explanation for clustering coefficient formula

I need some explanation for clustering coefficient formula itsef firstly and why it can be used for detecting communities in a social network! Also I would like to know why it is not a good method for ...
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Center/Commutator Subgroup of Direct Product = Direct Product of these Subgroups - - Fraleigh p. 64 Theorem 6.14

(1.) What's the intuition? Full proof for Center Subgroups (2.) What's the proof blueprint? I know proof's using $A = B \iff A \subseteq B \wedge B \subseteq A$. But where did $(ga,hb)$ in ...
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32 views

Intuition for this explicit formula for the number of ways of putting N labeled balls in K unlabeled boxes?

In its article on "Stirling numbers of the second kind", Wikipedia gives this formula for $S(n, k)$ -- the number of ways of putting $n$ distinct balls into $k$ boxes (where the boxes aren't ...
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164 views

Intuitively when to use the wedge product?

When I first learned the dot product and the cross product in $\mathbb{R}^3$ I spent some time understanding when I would like to use them. After some time I understood that the dot product usefulness ...
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Intuition - $fr = r^{-1}f$ for Dihedral Groups - Carter p. 75

Name $r$ = clockwise 90 deg. rotation and $f$ = flip across the square's vertical axis = the brown $\color{brown}{f}$ in my picture underneath. Zev Chonoles's $f$ is different. Carter fleshes out why ...
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87 views

Intuition, Questions on Commutator Subgroup $\neq$ Set of All Commutators - Fraleigh p. 150 Theorem 15.20

This is too advanced for me. Not asking about proofs here. Theorem 15.20: The set of all commutators $= \{aba^{-1}b^{-1} : a,b \in G \} $ generates $ \color{red}{\text{ and hence $\neq$} }$ (but ...
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60 views

Intuition and Proof - $H$ is a maximal normal subgroup of $G \iff$ $G/H$ is simple. - Fraleigh p. 150 Theorem 15.18

I don't understand some steps in the proof by B.S.. Start with some definitions. http://en.wikipedia.org/wiki/Maximal_subgroup#Maximal_normal_subgroup: $H \unlhd G$ is a maximal normal subgroup ...
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80 views

Intuition - Theorem - A group homomorphism preserves normal subgroups - Fraleigh p. 149. Theorem 15.16

p. 128, 129. Theorem 13.12. Let $h$ be a homomorphism of groups $G \to G'$. III. If $S \le G$, then $h[S] \le \color{red}{G'}$. IV. If $S' \le G'$, then $h^{-1}[S'] \le G$. p. 149. ...
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Intuition - Quotient Group of Direct Products - Fraleigh ch. 15

Tried http://www.proofwiki.org/wiki/Quotient_Group_of_Direct_Products Proof on p. 3 and 4 . For the case $n = 2$. Define $h: A_1 \times A_2 \rightarrow \dfrac{A_{1}} {B_{1}} \times \dfrac ...
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60 views

Collapsing a Factor to the identity element - Fraleigh p. 14 Theorem 15.8

p. 146: We should acquire an intuitive feeling for this theorem in terms of $\color{red}{collapsing}$ one of the factors to the identity element. p. 147 15.8 Theorem: $\hat{H} = \{(h, e) ...
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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 ...
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57 views

Motivation for Conjugate transpose of a matrix

I'am currently going through a self study of Linear algebra . I'am finding it difficult to grasp the intuition behind the concept of Conjugate transpose of a matrix .Why take the complex conjugate of ...
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70 views

Intuition behind (statistical) completeness

I was wondering if any of the members of the MSE community would like to share his/her intuition about completeness in statistics. For the sake of "completeness", here's the definition, taken from ...
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33 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 ...
5
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181 views

Equivalences of continuity, sequential convergence iff limit (S.A. pp 106 t4.2.3, 110 t4.3.2)

1. This post became too long, ergo I moved this here. 2. I questioned anew here. How does $\color{red}{(I) \implies (III)}$? This contradicts $a \le b \not \implies \Leftarrow a < b$. 3. ...
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139 views

Prove nth root of k exists with supremum. (Abbott pp 27 1.4.6b) [closed]

(Ulrich Daepp. Reading, Writing, and Proving. edition 2. pp 133 Theorem 13.2) Modus Operandi. The basic idea is that the nth root = supremum of $A = \left\{ w\in \mathbb{R} ^{+}:w^{n} < ...
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57 views

How to motivate vectors as derivations?

In a manifold it's easy to motivate the definition of vectors as equivalence classes of curves. On the other hand the definition as derivations is harder to motivate. I know how to show that the space ...
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87 views

Intuition on formal neighborhood in a scheme

Let $X$ be a Noetherian scheme, $x \in X$ a closed point. Denote by $\hat X$ the completion of $X$ along $x$. Now assume that two coherent modules $F, G$ on $X$ coincide over $\hat X$, i.e. $i^*F = ...
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Intuition: If $a\leq b+\epsilon$ for all $\epsilon>0$ then $a\leq b$?

I am reading Tom Apostol's Analysis and come across this theorem. Should $a \leq b$ if $a\leq b+\epsilon$ for all $\epsilon >0$? I don't doubt the proof in the book but I don't understand the ...
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64 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 ...
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117 views

Surface area of a Hypersphere

Hypersphere in 4 dimensions, I am having problem with finding the surface area of it. please help. I know that surface area will have 3 dimensions in 4 dimensional space, I am having trouble to ...
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219 views

Direct proof. Square root function uniformly continuous on $[0, \infty)$ (S.A. pp 119 4.4.8)

(http://math.stanford.edu/~ksound/Math171S10/Hw8Sol_171.pdf) Prove for all $e > 0,$ there exists $d > 0$ : for all $x, y \ge 0$, $|x - y| < d \implies |\sqrt{x} - \sqrt{y}| < e$. (a) ...
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1answer
60 views

Nontrivial Homomorphism(s) from $\mathbb{Z_3}$ to $S_3$ - Fraleigh p. 134 13.37

Reference: http://users.humboldt.edu/pgoetz/Homework%20Solutions/Math%20343/hwsome number 1 to 17 that I forgotsolns.pdf There are exactly two nontrivial ...
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53 views

Characterize normal subgroups - Find all subgroups of $S_3$ conjugate to $\{id, (1,3) \}$ - Fraleigh p. 143 14.29

(27.) A subgroup H is conjugate to a subgroup K of a group G (viz. p. 141 $K \le G$ is a conjugate subgroup of $H$), if $i_g[H] = gHg^{-1} =K$ for some $g \in G$. Show that conjugacy is an ...
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1answer
213 views

In a finite cyclic group of order n, number of solutions to $x^m = e$ - Fraleigh p. 68 6.53,54

(53.) Show that in a finite cyclic group G of order n, written multiplicatively, the equation $x^m = e$ has exactly m solutions $x$ in G for each $m \in \mathbb{N}$ that divides n. (54.) With ...
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80 views

Why might one be inclined to think that polynomials of the form $\cos(n\arccos{x})$ would minimize error in Lagrange interpolation?

I was first introduced to Chebyshev polynomials (of the first kind) in the form $T_n(x)=\cos\left(n \operatorname{arccos}(x)\right)$. The usual recurrence relation was then derived from using trig ...
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32 views

Can we find an $n$ that minimizes this function?

If we suppose that we have positive integers $k$, $c$, and $v$, can we find the $n$ that minimizes: $$k^n \frac{\log{2^v}}{\log{v}}v^{\log_2{(k \cdot v \cdot c/n)}}$$
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Intuition — If $k \in \mathbb{Z}$ and $n \ge 2$, then the n$^{th}$ root of k is either an integer or irrational.

Origin — Elementary Number Theory — Jones — p25 — Exercise 2.4 (1) How do you prefigure the answer? Proofwiki start after prefiguring it. (2) What's the intuition? This answer ...
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50 views

Intuition for an open mapping

What is an intuitive picture of an open mapping? The definition of an open mapping (a function which maps open sets to open sets) is simple sounding, but it's really not as easy to picture as the ...
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22 views

How large does $m$ have to be to get unique values with high probability? [duplicate]

We can suppose we are given two naturals, $r$ and $n$. We can then pick $n$ unique naturals: $\{x_0, x_1, \dots, x_n\}$. The following function is important: $$\prod_{k=1}^n{(x_k)^{y_k}} (\mod m)$$ ...
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105 views

How to change variables in a surface integral without parametrizing

This is a doubt that I carry since my PDE classes. Some background (skippable): In the multivariable calculus course at my university we made all sorts of standard calculations involving surface ...
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77 views

How hard is finding values such that

We can work with powers of some naturals $(x_k)^{m_k}$. Here we have $n$ naturals, and $m_k$ is an integer in the range $-r$ to $r$. My question is, how small can $p$ be so that ...