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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Serge Lang´s remarks on the superiority of algebra. What it actually means? [closed]

I read two comments of Lang that basically places algebra over other math subjects. One of this comments is on his calculus book preface (see Remark 1 below); I am not finding his other comment, but ...
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What makes elementary functions elementary?

Is there a mathematical reason (or possibly a historical one) that the "elementary" functions are what they are? As I'm learning calculus, I seem to focus most of my attention on trigonometric, ...
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399 views

What is the intuitive meaning of “conspiracy” in number theory?

Assuming very little number-theoretic background from my part, could you please explain me what is the intuitive meaning behind "conspiracy" in number theory? There is no formal entry on Wikipedia and ...
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315 views

what does following matrix says geometrically

Let $M\subset \mathbb C^2$ be a hypersurface defined by $F(z,w)=0$. Then for some point $p\in M$, I've $$\text{ rank of }\left( \begin{array}{ccc} 0 &\frac{\partial F}{\partial z} ...
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1answer
115 views

Lorentz reflection

What is a Lorentz reflection of $\mathbb R^3$? Is there a way to visualize it? Suppose I have a plane, P, what would (Lorentz) reflecting in it differ from (Euclid) reflecting in it? I know that the ...
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98 views

Geometric explanation of the product metric

Can someone describe to me the geometric intuition behind using a mapping $$ ((x_1,y_1),(x_2,y_2)) \mapsto \frac{d_1(x_1,y_1)}{1+d_1(x_1,y_1)} + \frac{1}{2} \frac{d_2(x_2,y_2)}{1+d_2(x_2,y_2)} $$ to ...
3
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1answer
332 views

Intuitive interpretation of these differential forms

Let $\pi: S^2-\{N\}\to \mathbb R^2$ be the stereographic projection map. Let $\sigma:\mathbb R^2\to S^2-\{N\}$ be its inverse. Let $p\in S^2-\{N\}$ and $x_1,x_2\in$ the tangent space of $S^2$ Would ...
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Explain for students: Why does 0 mod n equals 0 (zero)?

I told my students that the mod operator basically gives the remainder of division, so upon seeing: 0 mod 10 Some students (apparently) reasoned that, "10 goes ...
5
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2answers
240 views

The area problem!

We have to find area of the quadrilateral formed by joining the point of intersection of the four quarter circles that are drawn from each vertex in a unit square. $\hspace{4cm}$ The challenge is ...
6
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$\wedge,\cap$ and $\vee,\cup$ between Logic and Set Theory always interchangeable?

In "$\wedge,\cap,\times$ and $\vee,\cup,+$ are always interchangeable?" It has been shown that arithmetic shouldn't be included. So the new modified question is: The analogy of $\wedge,\cap$ and ...
4
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262 views

$\wedge,\cap,\times$ and $\vee,\cup,+$ are always interchangeable?

Update : Should have left the Arithmetic out of this question, the new modified question is posted here : $\wedge,\cap$ and $\vee,\cup$ between Logic and Set Theory always interchangeable? ...
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Line integration in complex analysis

In normal line integration, from what I understand, you are measuring the area underneath $f(x,y)$ along a curve in the $x\text{-}y$ plane from point $a$ to point $b$. But what is being measured with ...
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551 views

Organization of the Learning Process

Sorry for off topic. I'll delete this topic immediately when community decides it's useless, however if anyone finds it's interesting, share your opinion with us. I just want to know your opinion ...
15
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5answers
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What is the best way to develop Mathematical intuition?

I want to develop my pure mathematics knowledge and would like to know what is the best way to develop mathematical intuition? I am going through exercises that ask for proofs and I don't have the ...
6
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5answers
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What's an intuitive way of looking at quotient spaces?

I understand the concept of $\mathbb{Z}/n\mathbb{Z}$, but I am having a really hard time understanding how this concept of quotients applies to vector spaces. Suppose $V = \mathbb{F}[x]$ is a vector ...
3
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1answer
262 views

Avoiding algebraic integration by geometric arguments

Is there a geometric way of seeing why the integral $\int\limits_{-\infty}^\infty (x^2+y^2+z^2)^{-{3\over 2}}dz={2\over x^2+y^2}$? Otherwise what is a good way of evaluating it algebraically?
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475 views

Importance of 'smallness' in a category, and functor categories

I feel like, having spent a little time doing category theory now, this is probably a silly question, but I keep coming up to many things (definitions, examples etc.) where smallness is required. I ...
16
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1answer
972 views

Intuition behind homology with general coefficients

We just went over homology with general coefficients in topology and did some of the usual examples ($\mathbb{Z}_2$ for projective space and manifolds being the big examples) which led me to wonder ...
4
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0answers
87 views

Ways to think about one-relator groups

What are some intuitive ways to think about one-relator groups? I am aware of the Freiheitsatz, and Bass-Serre theory. What I'm interested in are ways people who work extensively with one-relator ...
18
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3answers
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Differential equations and Fourier and Laplace transforms

Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and ...
2
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3answers
204 views

Games with human edge [closed]

Which are some two- or one-player games, where humans far outperforms the best computer programs? And how does the relative edge scale with time allowed to think? (In time frame 1 sec to 8 hours per ...
6
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1answer
339 views

Ways to visualize the real numbers?

I was just wondering if there are any diagrams for visualizing subsets of the real numbers, or totally 'radically' different ways of looking at them as a real line? The model of the line relies on ...
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Significance of $\sigma$-finite measures

From Wikipedia: The class of $\sigma$-finite measures has some very convenient properties; $\sigma$-finiteness can be compared in this respect to separability of topological spaces. Some ...
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5answers
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Why “characteristic zero” and not “infinite characteristic”?

The characteristic of a ring (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. Otherwise, we ...
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Why are polynomials defined to be “formal”?

Despite the fact that $\forall n, n^3 + 2n \equiv 0 \pmod 3$, I understand that $n^3 + 2n$ (considered as a polynomial with coefficients in $\mathbb Z/3\mathbb Z$) is not equal to the zero polynomial. ...
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Motivation for solution to constructing a set of 1983 distinct integers such that no three are consecutive terms of an arithmetic progression

Problem: Is it possible to choose $1983$ distinct positive integers, all less than or equal to $100,000$, no three of which are consecutive terms of an arithmetic progression? (Source: IMO 1983 Q5) ...
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1answer
172 views

Visualizing the flat complex conic

Consider a conic $X^2 + Y^2 - Z^2 = 0$ in $\mathbb{C}P^2$. In an affine chart $Z \neq 0$ it is supposed to look like a circle (however it looks in $\mathbb{C}^2$), but the deceptiveness of imagining ...
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511 views

What's the connection between derivatives and boundaries?

The (second) fundamental theorem of calculus says that $$\int_a^b f'(x) dx = f(b) - f(a)$$ which can also be stated, if one knows enough about what's coming next, as: The integral of the ...
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Covering spaces - why are they useful?

As someone who trained as a physicist, I have known for ages that $\operatorname{SU}(2)$ is a double cover of $\operatorname{SO}(3)$, so, during an idle day at the office I decided to look up what ...
6
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2answers
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Why is this constant of integration taken as $\log A$ instead of just $C$?

Suppose we solve $$\frac{dy}{dx} = \frac{1 + y}{2 + x} .$$ Which can be written as the following and integrating both sides w.r.t. $y$ and $x$: $$\int\frac{1}{1 + y}dy = \int\frac{1}{2 +x}dx ,$$ we ...
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170 views

Why is $0$ excluded in the definition of the projective space for a vector space?

For a vector space $V$, $P(V)$ is defined to be $(V \setminus \{0 \}) / \sim$, where two non-zero vectors $v_1, v_2$ in $V$ are equivalent if they differ by a non-zero scalar $λ$, i.e., $v_1 = ...
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480 views

Question Regarding The Power Series For $e^x$

Currently I'm reading Higher Engineering Math by John Bird and under exponential function he talks about obtaining the value of $e$. He begins by saying The value of $e^x$ can be calculated to ...
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2answers
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Green's function

Why does the Green's function $G(r,r_0)$ of the Laplace's equation $\nabla^2 u=0$, the domain being the half plane, is equal $0$ on the boundary? How can I interpret the Laplace's equation physically? ...
4
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1answer
250 views

Visualizing homologous elements

For the fundamental group it's easy to visualize when two loops are homotopic. I was wondering if there are any ways to look at the equivalent problem for homology? I guess this might be tricky for ...
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Intuition for graphing Sine/Cosine

So I'm working my way through some basic trig (Khan Academy) - I'm trying to get a better intuition for what graphs of sine and cosine represent. I've seen the nice unit circle animations that do ...
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3answers
246 views

Choice of $\xi$ [duplicate]

Possible Duplicate: Rational Numbers Suppose $\{x \in \mathbb{Q}|x>0,x^2<2\}$ has a supremum. Call this supremum $c$. In order to show that this cannot be the case, we learned that we ...
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1answer
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Archimedes Method To Find The Area Under A Curve

I'am reading Tom Apostol's Calculus volume-1 text (page 3 and 4),where he talks about calculating the area under a curve which eventually leads to the concept of the definite integral.In the below ...
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356 views

How to think about derivatives in an abstract fashion?

Derivatives seem easy to understand abstractly as the rate of change of something, higher order derivatives are the rate of change of the rate of change of something, and so on. I, however, have ...
5
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1answer
424 views

Intuition about the size of $\aleph_k$ with $k>1$

Assuming CH for simplicity, I know of some more or less intuitive way to think about difference in sizes of $\aleph_0$ and $\aleph_1$. The most straightforward is the distinction of natural/rational ...
6
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1answer
420 views

Arnold's Trivium problem 52

Calculate the first term of the asymptotic expression as $k \to \infty$ of the integral $$ \int_{-\infty}^{+\infty}\frac{e^{ikx}}{\sqrt{1+x^{2n}}}dx $$ May I bother you to explain what the ...
6
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1answer
446 views

Intuition for Poincaré duality and Cap product

Can you provide me with any intuition behind the Cap product of a cohomology class and a homology class? What is its geometric meaning? Can you also give me an intuition why the Poincaré duality is ...
0
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1answer
143 views

How can a point have a slope? [duplicate]

Possible Duplicate: Approaching to zero, but not equal to zero, then why do the points get overlapped? You get the derivative of $f(x)$ by getting the limit as $h$ tends to $0$ of ...
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2answers
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what is the geometrical interpretation to positive definite matrix

What is the geometrical interpretation of positive definite matrix ? (not necessarily symmetric) if $A$ is positive definite, what does it do to a vector $x$ (i.e. $Ax$)?
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3answers
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Bilinearity: what does it mean?

What does bilinear really mean? Everytime I heard the word, I think it should be "linear in 2 ways?" For example, from the definition of inner product (taken from Appendix A of "Wavelets For ...
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Why the emphasis on Projective Space in Algebraic Geometry?

I have no doubt this is a basic question. However, I am working through Miranda's book on Riemann surfaces and algebraic curves, and it has yet to be addressed. Why does Miranda (and from what little ...
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1answer
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What is the intuition behind the proof of Abel-Ruffini theorem in abstract algebra?

Is there a way to explain this proof in Wikipedia without knowing the abstract algebra too much or deep function experience? In addition, I don't how the abstract algebra work even after I look at an ...
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2answers
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Propositional Logic Proof of $\vdash \lnot (p \supset q) \supset (p \land \lnot q)$

$\vdash \lnot (p \supset q) \supset (p \land \lnot q)$ I need to prove the above proposition via intuitionistic logic rules and/or natural logic rules. I guess it is not possible to prove with ...
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4answers
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Why is the derivative multiplication by frequency in Laplace transform?

Why is the time-domain derivative equivalent to multiplication by frequency ($s$) in the Laplace transform? Why is the time-domain integral equivalent to division by frequency ($\frac{1}{s}$) in the ...
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7answers
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Algebraic Intuition for Homological Algebra and Applications to More Elementary Algebra

I am taking a course next term in homological algebra (using Weibel's classic text) and am having a hard time seeing some of the big picture of the idea behind homological algebra. Now, this sort of ...
6
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1answer
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Geometric invariants of a scheme

Following my previous question about sheaf cohomology, I'd like to ask about its applications to algebraic geometry. I have now learned a little about homological algebra and I can see that for ...