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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861 views

How to get intuition in topology concerning the definitions?

Most topology texts go on directly to give definition of topology, then they give some examples and that's it, like they directly tell you right Let $X$ be a set and let $τ$ be a family of subsets ...
3
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1answer
39 views

Difference between “real functions” and “real-valued functions”

According to my textbook: A function which has either $\mathbb R$ or one of its subsets as its range is called a real valued function. Further, if its domain is also either $\mathbb R$ or a ...
0
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2answers
59 views

Proper solution of the limit of $\sin(x)/\tan(x)$ as $x \to 0$

In one of my math book, I have a problem where I need to compute $\lim_{x\to0}{\frac{\sin(x)}{\tan(x)}}$ I came up with a solution that I am not able to write formally. The reasoning is the following ...
2
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3answers
66 views

Basic question on the infinitely many solutions of a linear system Ax=b,

I just want to verify the geometry of solutions to $Ax=b$, for the case when we have infinitely many solutions: If say for a $3\times 3$ matrix, after Gaussian Elimination, I have two pivot variables ...
3
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1answer
72 views

Violating Cauchy's Integral Theorem

With regards to utilizing Cauchy's Integral Theorem for integration over closed contours: https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem In particular the result that $\int_\gamma ...
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0answers
41 views

Understanding manifolds (asking just for confirmation)

In lecture we used the following definition of manifolds: A subset $ M \subset \mathbb{R}^n $ is called a k-dimensional manifold of the class $C^\alpha$, if $ \forall a \in M $ there is an open ...
0
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0answers
18 views

Taking the square of an image in Fourier domain, why not square of real part?

In my quest to understand Math during the Christmas holidays I'm working on Fourier transforms today. I understand that a single point in Fourier space corresponds to line in normal 2D image space. ...
4
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0answers
77 views

Mathematics Wallpapers

I know that this sounds very silly. But I don't know where else to ask. Is there a good free site for mathematics wallpapers , pictures etc ? Most of the time it is very difficult to find exact ...
4
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1answer
35 views

Number of solutions to $|ax - bx| = a \;\text{or}\; b$?

While watching basketball tonight, I noticed that for 3, 4, and 6, $(6 \times 3) - (4 \times 3) = 18 - 12 = 6$. I thought this was a cool relationship and it led me to the following question: For ...
4
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1answer
134 views

Tricky proof that the weighted average is a better estimate than the un-weighted average:

The following is a word for word copy of a tough question and the solution to it. I have marked $\color{red}{\mathrm{red}}$ the parts of the solution for which I do not understand and the parts marked ...
15
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1answer
293 views

Understanding proof by algebraic geometry, Fermat's last theorem for polynomials when $n = 3$.

This is a followup to my question here. See here. The question is as follows. How do we see that there do not exist nonconstant, relatively prime, polynomials $a(t)$, $b(t)$, and $c(t) \in ...
7
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2answers
123 views

Intuition for Kuratowski-Mrowka characterization of compactness

Fact. A space $X$ is compact iff for every space $Y$, the projection $X\times Y\rightarrow Y$ is a closed map. The finite subcover definition of compactness seems reasonably intuitive: finite covers ...
0
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0answers
32 views

What is the Lagrange remainder in a Taylor series expansion

I know what a Taylor series expansion is and I know how to find the Lagrange remainder but what does it mean intuitively? I need an explanation of what the Lagrange remainder represents in terms of ...
1
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1answer
35 views

Understanding Abel-Ruffini

I'm wondering of anyone can point me towards a proof of why we can't have a quintic formula, using concepts from basic group theory. In particular, I understand that there is some connection with ...
2
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2answers
74 views

Motivating $y'=y \implies y=Ce^x$

Is there some intuitive reason why one should think that a function which is its own derivative should be of the form $Ca^x$ for some number $a$? Of course I can prove that the unique solution set to ...
1
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3answers
55 views

How to see that $\text{gcd}(a,b) = \text{gcd}(a-b,b)$?

I'm trying to understand why $\text{gcd}(a,b) = \text{gcd}(a-b,b)$. What is clear to me is that the $\text{gcd}$ divides $a,b$ and also $a-b$ (let's assume $a\ge b$). But then it seems to me we ...
2
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2answers
30 views

Is there a more intuitive way of coming up with matrices for problems like this? E.g. Find two matrices B and C with AB = AC, and B does not equal C

Let A = $\begin{bmatrix}1 & 0\\1 & 0\end{bmatrix}$ Find two matrices $B$ and $C$ with $AB = AC$, and $B$ does not equal $C$. I always have trouble with problems like this. Here, I ...
5
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2answers
64 views

Is there an intuitive way of viewing the Law of Total Expectation $\mathbb{E}\big[\mathbb{E}[X|Y]\big]=\mathbb{E}[X]?$

Law of total expectation If $\mathbb{E}\big[|X|\big]$ finite then for any $Y,\;\mathbb{E}\big[\mathbb{E}[X\mid Y]\big]=\mathbb{E}[X]$ I remember reading this for the first time and ...
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3answers
31 views

Summation of a constant using sigma notation

Apologies if this is a silly question, but is it possible to prove that $$\sum_{n=1}^{N}c=N\cdot c$$ or does this simply follow from the definition of sigma notation? I am fairly sure it's the ...
4
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0answers
97 views

Learning Galois theory geometrically?

Recently I started poking at algebraic geometry and commutative algebra. My background is basic category theory and basic algebraic topology. I don't know a lot of other mathematics. I noticed Galois ...
11
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149 views

How does the internal language of a topos come to be?

There are several books and articles on topos theory which mention the internal language, but I can't manage to see the big picture from any of them. I would like a soft explanation of how the ...
0
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1answer
25 views

Bijections and Visual Representations

Throughout math we learn about the cardinality of sets. We learn that the existence of a bijection between two sets imply that the cardinality of these two sets are equal. For each set there is ...
29
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5answers
929 views

What is an intuitive definition for “conjugate” in Group Theory?

In Abstract Algebra, I learned about "conjugation" in the context of a group $H$ being a 'normal' subgroup of $G$ if the element $xhx^{-1}\in H$ for any $x\in G$. But this is not the first time I've ...
3
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1answer
51 views

What is the Newton's general theory of diameters?

I was reading a book on Mathematics, which contained this topic. I was not able to grasp the concept. There was not much info on internet also. It was as follows: Let an $n$th order curve be ...
0
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0answers
93 views

More intuitive/easier explanation for Knuth's division algorithm needed

Knuth's division algorithm, here refers to the D algorithm mentioned in TAOCP written by Donald Knuth in Volume 2 (Semi-numerical algorithms) Section 4.3.1 I have been trying to understand the ...
0
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0answers
88 views

Left invariant Vector Field on $S^2$

How intuitively look like all left invariant vector fields on this manifold: the 2 dimensional unit sphere $S^2$ with the smooth structure inherited from $\mathbb R^3$? Why all left invariant vector ...
0
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1answer
39 views

What does Poincaré mean for intuition of pure number?

To what does Poincaré refer in this article http://www-history.mcs.st-andrews.ac.uk/Extras/Poincare_Intuition.html speaking about the intuition of pure number? My answer is that he may refer to a ...
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2answers
74 views

Intuitive or visual understanding of the real projective plane

If we take the definition of a real projective space $\mathbb{R}\mathrm{P}^n$ as the space $S^n$ modulo the antipodal map ($x\sim -x$), it is possible to see that $\mathbb{R}\mathrm{P}^1$ is ...
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4answers
73 views

Factorial in power series; intuitive/combinatorial interpretation?

It is well known that the terms of the power series of exponential and trigonometric functions often involve the factorial function, essentially as a consequence of iterating the power rule. My ...
3
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2answers
120 views

Intuition behind universal arrow construction of adjoint functors

I try to understand adjoint functors in category theory. I like the idea of thinking of a left adjoint as 'best approximation from above' and a right adjoint as 'best approximation from below.' More ...
3
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1answer
72 views

Higher homotopy groups meaning

I am developing intuition for higher homotopy groups but it's very hard for me to visualize what $\pi_2$ represents (and $\pi_n$ for that matter). I know that $\pi_2(S^2) \cong \mathbb{Z}$ and can ...
2
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1answer
43 views

Deterministicaly stops winning, yet if conditioned on time there is positive probability to win in the future.

In the answer to question http://mathoverflow.net/questions/12462/limsup-and-liminf-for-a-sequence-of-sets "has2" gives a concrete example for the use of $\limsup$ and $\liminf$ in form of a "card ...
0
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3answers
88 views

How are irrational numbers, fixed points on the number line?

Please, while answering/reading this question, only keep in mind my point of view only. The question is, that how come an irrational number on a number line is a fixed point. To make things more ...
2
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1answer
59 views

Alternative Geometries

In our world, the distance between two points (in 2d) is defined as $\sqrt{(\Delta x)^2 + (\Delta y)^2}$. Suppose that in an alternative geometry, it was defined as $\sqrt[p]{|\Delta x|^p + |\Delta ...
3
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7answers
466 views

What does it mean for a set to be countably infinite?

Why distinguish between countable and uncountable? What advantages does this property have? I haven't studied much set theory but I am writing about the set of algebraic vs transcendental numbers and ...
6
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1answer
98 views

Why is abelianness such a precious property?

My abstract algebra teacher said the other day that constructions like ideals and cosets and normal subgroups are "trying to capture a little bit of abelianness." He has used phrases like "magic ...
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2answers
89 views

Is an example of a PDF $\lambda e^{-\lambda t}$ or $\lambda e^{-\lambda t}\mathrm{d}t$?

The reason I ask is because according to this source: the $\fbox{$\color{blue}{\mathrm{PDF}}$}$ for the sum of two Exponential Density Functions is $$\rho(x_1,x_2)\mathrm{d}x_1 ...
4
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2answers
126 views

Which general physical transformation to the number space does exponentiation represent?

Addition and multiplication may be defined in two ways, one specific and one general: Addition specific: addition is repeated incrementation. This is specific and sub-optimal as while $2 + 4$ is ...
28
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7answers
3k views

Why does the fundamental theorem of calculus work?

I've known for some time that one of the fundamental theorems of calculus states: $$ \int_{a}^{b}\ f'(x){\mathrm{d} x} = f(b)-f(a) $$ Despite using this formula, I've yet to see a proof or even a ...
1
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2answers
65 views

What is the difference between a Poisson and an Exponential distribution?

For a Poisson distribution: $$\mathsf{P}(X=x)=\frac{e^{-\mu}\times \mu^x}{x!}$$ where $\mu$ is the mean number of occurrences. For an Exponential distribution: $$f(x;\lambda) = \begin{cases} ...
0
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1answer
23 views

The Banach–Tarski Paradox [duplicate]

Can a ball be decomposed into a finite number of point sets and reassembled into two balls identical to the original? What is the true nature of this paradox ? I don't really understand this ?
0
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0answers
14 views

Need explanation on normalising data between 0 and 1

I have two columns of data which are of very different size ranges. So I decided to normalise them to be within the range 0 and 1. Could someone explain what doing this actually means for my ...
1
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2answers
76 views

How to see symbol manipulation from an intuitive perspective in math?

I have recently started to develop my mathematical intuition. In the past I saw math as a mere game of symbol manipulation, whosoever was able to see patterns and cram formulas and apply them upon ...
2
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1answer
51 views

Geometric interpretation of monotone operators on a Hilbert space

Recall that a monotone operator is defined by the relationship as follows: $$\langle y - x, F(y) - F(x)\rangle \geq 0, \quad \forall x,y \in X$$ ($X$ is a Hilbert space) What is a good geometric ...
2
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1answer
85 views

Proof/intuition that any number can be expressed in binary form and every number will have a unique representation?

I was just thinking lately that how do we know that literally every number can be expressed in binary? And that too, with a unique representation? Clarification: With numbers, I mean whole numbers. ...
2
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1answer
42 views

Vector Integration - Intuition

I understand that an integral of a scalar valued function can be visualized as "signed area under the curve". But what about integration of a vector valued function by its parameter? Is there a ...
2
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1answer
52 views

What is the origin of the formula: $\rho_x (x)=\left|\frac{{d}x}{{d}\alpha}\right|^{-1}\rho_\alpha(\alpha)$ that relates random variables?

I'm trying to understand the origin of a certain formula used in the solution to the following question: This question relates to the position probability density for a classical particle ...
9
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3answers
315 views

Developing Mathematic Intuition

I'm an engineering student, currently working my way through the fundamental mathematics courses. I've done reasonably well so far—mostly A's and a couple of B's in Algebra, Statistics, ...
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1answer
56 views

Are all instances of torsion special cases of the same concept?

The concept of 'torsion' pervades mathematics. As far as I know the origin of the word is in algebraic topology where it was used to describe chains $\gamma$ which are not boundaries but such that ...
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1answer
24 views

Question about: How many partitions of $12$ have parts of size at most $5$?

If the parts are of size at most $5$, why are considering numbers greater than $5$? For example, why can $z_5$ take on $10, 15, \dots$?