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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What is the intuition of conjugacy classes?

How can I fully understand what are conjugacy classes are in groups? I know the definition, that $a$ and $b$ are conjugate if $gag^{-1}=b$ for some $g\in G$. But what is the intuition? Using a ...
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Intuition behind Matrix Multiplication

If I multiply two numbers, say $3$ and $5$, I know it means add $3$ to itself $5$ times or add $5$ to itself $3$ times. But If I multiply two matrices, what does it mean ? I mean I can't think it ...
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68 views

Explaining the meaning of equality

I've been tasked with explaining to a group of people what the notion of equality means in mathematics, I've come up with a working explanation, but would appreciate some input, suggestions etc. ...
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245 views

Problem in understanding a proof there are five Platonic solids.

Thanks to several comments by Gerry Myerson, it is now clear that I wasn't clear, up to a state where I seriously confused myself. In a renewed attempt: Recently, I've been thinking about Platonic ...
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How do I visualize this quotient space?

If $V = [0,1] \times [0,1] \subset \mathbb{R}^2$. We define the equivalence relation $\sim$ on $V$ as follows: every element $(x,y) \in V$ is equivalent with itself and besides that the three ...
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40 views

Verify $\frac {\partial B} {\partial T} =$ $\frac{c}{(e^\frac{hf}{kT}-1)^2}\frac{hf}{kT^2}e^\frac{hf}{kT}$

Find an expression for $\frac {\partial B} {\partial T}$ applied to the Black-Body radiation law by Planck: $$B(f,T)=\frac{2hf^3}{c^2\left(e^\frac{hf}{kT}-1\right)}$$ The correct answer (I believe) ...
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43 views

Show that $\frac {\partial B} {\partial T} =$ $\frac{c}{(\exp\frac{hf}{kT}-1)^2}\frac{hf}{kT^2}$

Find an expression for $\frac {\partial B} {\partial T}$ applied to the Black-Body radiation law by Planck: $$B(f,T)=\frac{2hf^3}{c^2\left(\exp\frac{hf}{kT}-1\right)}$$ The correct answer is $\frac ...
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1answer
294 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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61 views

Is there a physical interpretation of the alternating property?

A map from lists to list-elements is called "alternating" if any list with repeated elements is mapped to zero. This has statistical significance: regressions on collinear data are bad, dependent ...
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661 views

Intersection of Normal Subgroups is Normal in Subgroup but Not Group - Fraleigh p. 143 14.35

Show that if H is a subgroup of a group G, and N is a normal subgroup in G, then $H \cap N$ is normal in H. Show by an example that $H \cap N$ need not be normal in G. I can condone the proof hence ...
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Intuition - For every % point that rates rise, a bond’s value will decline by its duration in years.

[Source:] Generally speaking, for every percentage point that rates rise, a bond’s value will decline by its duration (stated in years). So if rates climb by one percentage point, the value of a ...
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1answer
69 views

A math puzzle about slow clock

You have the misfortune to own an unreliable clock. This one loses exactly 20 minutes every hour. It is now showing 4:00am and you know that is was correct at midnight, when you set it. The clock ...
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20 views

Mixed vs Equal Characteristic Local Rings

This is more of a vague, intuitive type of question, so perhaps there isn't anything too concrete anyone can offer. I am trying to get a sense of precisely why working with local rings of equal ...
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1answer
22 views

Solving for a Binary Matrix: A somewhat unusual method needs justification, and mabye interpretation.

Introduction: Define a "Bit Map" to be a matrix whose entries can only be $0$ or $1$. Then numbers above and beside each column and row indicates how many entries are "filled" with a one. For ...
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37 views

Diagonally dominant matrix — geometric interpretation

I like to have a visual interpretation of mathematical concepts. This is simple for many important kinds of matrices: orthogonal matrices are rotations, diagonal matrices scale along the natural ...
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1answer
40 views

Tangent space as derivations exercise

Thinking of the tangent space to a manifold as derivations is a concept which just kind of eludes me. I am comfortable thinking about tangent vectors as equivalence classes of curves and with the ...
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11 views

Examples of Relation Algebras

Would anyone please direct me to a host of examples of relation algebras. Is there an intuition for what these algebras are to model? That is, groups, for example, model a notion of symmetry; ...
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41 views

(Visual) Intuition: Division and complex fractions

When treating division as "groups of the numerator" (sorry, I don't know the technical term -- see image), why does a complex fraction in the denominator get added together to produce a 1 (number of ...
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13answers
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Intuitive explanation of Cauchy's Integral Formula in Complex Analysis

There is a theorem that states that if $f$ is analytic in a domain $D$, and the closed disc {$ z:|z-\alpha|\leq r$} contained in $D$, and $C$ denotes the disc's boundary followed in the positive ...
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1answer
44 views

Uses of stalks of sheaves and germs

I am trying to understand the motivation behind defining stalks of sheaves, but I suppose my complex geometry is a little weak. I know they are meant to represent germs of holomorphic functions at a ...
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How to intuitively understand prolongations

This question is concerned with the algebraic side of the theory of prolongations as explained in this paper by V. Guillemin and S. Sternberg. Let me first introduce my notation. We're working with a ...
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If sup A < sup B, there exists an element b ∈ B that's an upper bound for A. (S.A. pp 18 q1.3.8)

My Figure: By definition of $\sup B$, $\sup B$ is an upper bound for $B$. Set $e = \sup B − \sup A > 0$. By Lemma 1.3.7, there exists an element $b ∈ B$ satisfying $\begin{align} & \sup B − ...
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Losing a dimension when finding intersection between subspaces

Let $F=\mathbb Z_3, V=F^4$. Let $U=sp\{(1,0,0,0),(1,0,1,0),(0,1,1,1) \} \\W=sp\{(0,0,1,0),(-1,1,0,1),(1,1,1,1) \}$ Find $dim (U\cap W)$ we have $v\in U \text{ and } v\in W$ so $v=v$ ...
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Looking for an intuitive explanation why the row rank is equal to the column rank for a matrix

I am looking for an intuitive explanation as to why/how row rank of a matrix = column rank. I've read the proof at http://en.wikipedia.org/wiki/Rank_of_a_linear_transformation and I understand the ...
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Intuition behind generic point of a scheme?

I've been reading a little about algebraic geometry and how there seems to have existed this notion of "generic point" on a variety which wasn't carefully defined at first. But often times, ...
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1answer
43 views

How to geometrically interpret intertia of primes in field extensions?

I am trying to understand the intuition of thinking about number theoretic ideas in terms of geometric ones. For example, ramification is something that happens when a "covering" space of a Riemann ...
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Determine angle $x$ using only elementary geometry

Using only elementary geometry, determine angle x. You may not use trigonometry, such as sines and cosines, the law of sines, the law of cosines, etc.
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What is the Direction of a Zero (Null) Vector?

To be more precise, I am interested in knowing if the intuition that a Euclidean zero vector does not have a particular direction is actually correct, and if there is a rigorous formulation that would ...
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4answers
260 views

Why are we defining the norms on certain vector spaces the way they are?

What's the intuition behind defining $\|x\|_{\infty} = \max_{1 \le i \le n}\{|x_i|\}$ on the space of ordered $n$-tuples of complex numbers? I'm asking because I've been asked to find a norm on the ...
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“Poissonization” and intuition

In a french book, "Calcul des probabilités" from Foata and Fuchs, I found this theorem, which they call "Poissonization". "Let $(I_k)_{k \in \mathbb{N}}$ be a sequence of independent variables with ...
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Motivation for the name “vertical subspace” in the context of fiber bundles.

Let $p:E\to B$ be a smooth fiber bundle with fiber $F$. Consider the vector spaces $V_u=\{x\in T_uE: p_*(x)=0\}$. We call $V_u$ the vertical subspace of the tangent space $T_uE$. How can we see that ...
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Problem in deducing gradient in spherical coordinates.

I know the differential displacement in spherical coordinate as $$dr \cdot \widehat{r}+ r d\theta\cdot\widehat{\theta} + r\sin\theta d\phi\cdot \widehat{\phi}$$. But I can't figure out how the ...
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What is the intuitive meaning if multiplying by fractional 1?

first post ever on stack exchange in years of using it. Can anyone provide a historical or logical deduction of the reasoning behind multiplication by 1 via a fraction? For instance, in finance ...
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19 views

Custom equation guidance

i was having a tough time deciding on which SE site to post this rather unique question being that it would help me with a program I'm writing. I decided to come to you guys, not hoping that you would ...
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1answer
31 views

Need some help understanding the condition of the implicit function theorem

The condition for the implicit function theorem is that the (smooth) map $f: \mathbb R^n \to \mathbb R^m$ is locally a (smooth) map of $n-k$ variables if there are locally smooth maps $g_i , i \in ...
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Why do statisticians like “$n-1$” instead of “$n$”?

Does anyone have an intuitive explanation (no formulas, just words! :D) about the "$n-1$" instead of "$n$" in the unbiased variance estimator $$S_n^2 = \dfrac{\sum\limits_{i = 1}^n ...
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The definition of metric space,topological space

I have read some books in analysis,all of them define metric space,topological space or vector space directly,without any reason. Therefore, I want to know the background of the definition, the ...
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62 views

Why should we expect duality to give useful concepts in category theory?

Why should we expect the abstract notion of flipping arrows in a category to generate useful concepts from other useful ones? What exactly does flipping the direction of arrows mean and why is it ...
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What is the correct analogue of $\mathbb N$ in a ring of integers?

Question: Let $K$ be a number field. The proper intuitive motivation for the ring of integers $\mathcal O_K$ is that $\mathbb Z$ is to $\mathbb Q$ as $\mathcal O_K$ is to $K$. But what plays the role ...
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Proof of an odd function plus an even function

I was looking at the wiki page http://en.wikipedia.org/wiki/Even_and_odd_functions#The_sum_of_even_and_odd_functions and it says that to prove an even function plus an odd function, we first have to ...
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16answers
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Intuitive Understanding of the constant “$e$”

Potentially related-questions, shown before posting, didn't have anything like this, so I apologize in advance if this is a duplicate. I know there are many ways of calculating (or should I say ...
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3answers
110 views

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

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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The shortest path in a metric space with a given metric

My questions seem to be very basic and intuitively correct but I can't formally prove them. Before learning metric spaces, for $R^2$, we always define the distance between 2 points as $d_2 = ...
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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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What is the idea behind a projection operator? What does it do?

I know what a projection operator is, but I am unable to explain it in words without using mathematical symbols. Can anyone help me? I don't need examples or the definition - I want to know why and ...
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617 views

What is the difference between necessary condition & sufficient condition?

My book says : For having extreme point $a$ of function $f$, the necessary condition is that $f'(a) = 0$. However, it isn't a sufficient condition. Now, what is the difference between necessary ...
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60 views

Is there any intuitive way to think about the gamma function?

Is there a way to realize the gamma function intuitively? My first (and probably correct) guess is no, because, for example, $\Gamma(\frac 12)=\sqrt{\pi}$ doesn't make any intuitive sense at all. ...
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2answers
361 views

Questions on Proofs - Equivalent Conditions of Normal Subgroup - Fraleigh p. 141 Theorem 14.13

(1.) Why did Fraleigh shirk the proof for $(2) \implies (1)$? By dint of Arthur's comment, $(2) \iff \color{crimson}{gHg^{-1} \subseteq H} \quad \wedge \quad gHg^{-1} \supseteq H \implies ...
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Understanding induced representations

Let $G$ be a group and $H$ be a subgroup. Let $\phi:H\rightarrow GL(V)$ be a representation of $H$. There are three constructions in Wikipedia, but I am not really convinced by these. My question is: ...
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Why $\omega$ in $x = \cos(\omega t + \alpha)$ , $\omega$ isn't considered an arbitrary constant?

We know the SHM differential equation is of second-order $$\dfrac{d^2 x}{dt^2} = -{\omega}^2 x$$ . So, the solution of this equation must contain two arbitrary constants. And also we know that $x = ...