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 geometric meaning of representability?

Representable functors play a large role in algebraic geometry when developed through the 'functor of points' approach. One finds schemes represent Zariski sheaves and this gives access to the great ...
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24 views

Multiplication principle and permutation

Definition of Multiplication principle in Principles and Techniques in combinatorics by Chuan-Chong, Khee-Meng is given as: Let $$\prod_{i=1}^rA_i=A_1\times\dots\times A_r=\{(a_1,\dots,a_r) | ...
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What does $3+2i$ apples mean? Can the simple counting analogy with apples be extended to complex numbers? [closed]

Please read, or at least skim the question. Past attempts at answering the question have ignored vital constraints provided below. Natural Numbers Imagine I have $n$, a positive natural number, of ...
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2answers
122 views

Intuition for orthogonality in $\{0, 1\}^n$

In the beginning of [Kanerva 1988] a boolean algebra over $$ \{0, 1\}^n $$ with bitwise OR and AND is introduced. Example for bitwise OR: $$101 + 001 = 101$$ Example for bitwise AND: $$101 * 001 = ...
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50 views

Intuitively, why is compounding percentages not expressed as adding percentages?

I pursue only intuition; please do not answer with formal proofs. I already know the theoretical reason: because each percentage expresses a different base. $1.$ But why not intuitively? My ...
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74 views

What was the genesis of Hua's identity?

Many resources I have read prove Hua's identity more-or-less mechanically. I have seen there is more than one raison d'être for Hua's identity: e.g. its connection to the fundamental theorem of ...
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78 views

Is a series of successive derivatives known/useful?

So, while trying to find something else, it looks like I've found, for many $f(x)$: $$f(x) + f'(x) + f''(x) + f^{(3)}(x) + \dots + f^{(n)}(x)$$ Assuming that there is an easy way to find this sum ...
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58 views

How to understand intuition behind compactness? [duplicate]

I have taken a course in general topology this semester.while solving problems,i find it difficult to go by the definition which says that a space is compact if every open cover of it has a finite ...
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1answer
61 views

Intuitive proof that $U(n)$ isn't isomorphic to $SU(n) \times S^1$

One way to prove this is by comparing their centers. However, I do not feel that this proof gives me much insight into the structures of the groups. (It would make me very happy if I were to be ...
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1answer
38 views

Can anyone help to explain Benfords Law [duplicate]

Okay so recently I have heard of what is known as Benford's law, and this is the first time I absolutely cannot think of my own inutuion about how this is true. Just trying to think about it makes my ...
32
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4answers
883 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 ...
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44 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 ...
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2answers
62 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 ...
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3answers
94 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
79 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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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 ...
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20 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. ...
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80 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 ...
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92 views

How to compute double summations where the two summands are not independent?

Edit: From the vote counts I see that people want this question closed as it seems unclear what I was asking, so I have tried to word it a bit better to avoid closure. I hope this helps, please ...
4
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1answer
39 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
143 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 ...
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1answer
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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 ...
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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 ...
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45 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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70 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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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 ...
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114 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 ...
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181 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 ...
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1answer
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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 ...
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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 ...
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1answer
54 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 ...
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108 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 ...
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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 ...
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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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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
121 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 ...
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135 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 ...
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1answer
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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 ...
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1answer
45 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 ...
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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 ...
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66 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 ...
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477 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 ...
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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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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 ...
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2answers
131 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 ...
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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 ...