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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203 views

What happens after the cardinality $\mathfrak{c}$?

While having measure theory this year the following came in my mind: When we go from finite objects to infinite we "lose" a lot of properties. For example the summation isn't well defined ...
0
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4answers
156 views

A basic question on the derivative of a continuous function

Is the following condition necessary for the existence of derivative of a continuous function at point $x$: $$\lim_{h \to 0^+}\frac{f(x+h)-f(x)}{h} = \lim_{h \to 0^-}\frac{f(x)-f(x+h)}{h}$$
3
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4answers
423 views

A basic intuition on a probability problem

Two players take turns shooting at a target, with each shot by player $i$ hitting the target with probability $p_i$, $i=1,2$. Shooting ends when two consecutive shots hit the target. Let $\mu_i$ ...
4
votes
1answer
280 views

Intuition of law of iterated logarithm

Let $X_i$ be iid random variables with $EX_i = 0$ and $Var X_i=1$ and $S_n=X_1+\cdots+X_n$. Then the law of the iterated logarithm says almost everywhere we have ...
10
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3answers
803 views

Intuition of Addition Formula for Sine and Cosine

The proof of two angles for sine function is derived using $$\sin(A+B)=\sin A\cos B+\sin B\cos A$$ and $$\cos(A+B)=\cos A\cos B-\sin A\sin B$$ for cosine function. I know how to derive both of the ...
2
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3answers
431 views

What is the relation between vectors in physics and algebra?

Vector math is something I find very interesting. However, we have never been told the link between vectors in physics (usually represented as arrows, e.g. a force vector) and in algebra (e.g. ...
2
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3answers
146 views

Compact subsets of function spaces, geometry

The subset is called compact when every open cover contains a finite subcover. In Euclidean spaces, it is easy to visualize this by imagining some open ball that contains this set, thinking about the ...
10
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3answers
339 views

How to write $\pi$ as a set in ZF?

I know that from ZF we can construct some sets in a beautiful form obtaining the desired properties that we expect to have these sets. In ZF all is a set (including numbers, elements, functions, ...
0
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2answers
143 views

Does $P(A\cap B) + P(A\cap B^c) = P(A)$?

Based purely on intuition, it would seem that the following statement is true, when thinking of the events as sets: $$P(A\cap B) + P(A\cap B^c) = P(A)$$ However, I am not sure if this is true, and ...
8
votes
1answer
309 views

Why is the Derangement Probability so Close to $\frac{1}{e}$?

A derangement is a permutation of $\sigma$ of $\{1,2,3,\dots,n\}$ such that $\sigma(i) \neq i$ for every $i$. A common application of inclusion/exclusion in undergraduate combinatorics and ...
6
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1answer
196 views

What is the physical meaning of fractional calculus?

What is the physical meaning of the fractional integral and fractional derivative? And many researchers deal with the fractional boundary value problems, and what is the physical background? What ...
0
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1answer
112 views

Best way to write the characteristic polynomial

In linear algebra I've seen that there are two major different ways to write the characteristic polynomial of a mapping $f$: As $$ ...
4
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1answer
105 views

Geometric intuition behind the Uniform Boundedness Principle

Is there a way to visualize why the Uniform Boundedness Principle should be true? I understand the statement of the theorem but I'm having a hard time seeing a picture of it in my head.
1
vote
1answer
354 views

Prove that any circuit contains a cycle

This is a practice question (not HW) Prove that any circuit in a graph must contain a cycle AND that any circuit that is not a cycle contains at least two cycles. Note : This is for a first course ...
0
votes
2answers
135 views

What exactly is the complex plane, and how is it useful?

A lot of functions are defined on the complex plane, like the Gamma function: the Lambert W function, etc. But I have no idea about what the complex plane means and how it's useful, or just ...
1
vote
1answer
269 views

Geometric representation of product rule?

At time 1:06 of this video by minutephysics, there is a geometric representation of the product rule: However, I don't understand how the sums of the areas of those thin strips represent $d(u\cdot ...
6
votes
1answer
167 views

How to understand $\frac{d}{dt}\{(\exp(tX))_*(Y)\}|_{t=0}=[X,Y]$?

Let $G$ be a Lie group on which $X$ and $Y$ are two vector fields. Let $G\xrightarrow{\exp(tX)} G$ be the (Lie theory) exponential map corresponding to $X$. Then of fundamental importance is ...
2
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0answers
65 views

Intuition behind criterion for an irreducible Markov chain to be transient

I have been looking over my notes for Markov chains, and I have come across the following: Theorem: An irreducible Markov chain is transient iff for some state $i$ there exists a nonzero vector $y$ ...
13
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1answer
477 views

How to improve mathematical creativity?

To introduce myself: I'm an undergraduate mathematics student in Germany. Currently I'm studying in the second semester and until now I'm doing well, but I still got the feeling that my ability to ...
4
votes
1answer
150 views

Volume of a hypersphere

We know that the area of a circle (2-D) =$\pi r^{2}$ and the volume of a sphere (3-D)= $\dfrac{4}{3}\pi r^{3}$. Question:What is the "volume"(or whatever that is called) of a n-dimensional sphere? ...
13
votes
6answers
433 views

$\pi$ from the unit circle, $\sqrt 2$ from the unit square but what about $e$? [duplicate]

If one wants to introduce $\pi$ to a not mathematically savvy person, the unit circle would be a good choice. The unit square would be the way to go for $\sqrt 2$. But what about $e$? I've reviewed ...
11
votes
3answers
433 views

It is possible to define our intuitive notion for probability in subsets of $[0,1]$

I've always heard and read the sentence: If you pick a real number $x\in[0,1]$ at random, the probability to obtain a rational number is $0$. What is the meaning for that? Is this the "real" ...
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0answers
58 views

What to take from representation of $S_d$?

I am reading about group representations, and books I read all contain the representation theory for symmetric groups $S_d$. However none of them presents the material in a friendly way. After reading ...
4
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4answers
517 views

the role of logic in math and education

My question is somewhat related to this discussion: Is Mathematics one big tautology? I have a computer science background and I have always approached math from the logic point of view ...
0
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1answer
112 views

How information works?

I am really confused after reading wikipedia... What I don't get is how can something "bring" information, and in mathematics, how a mathematical object (like a set) can "have" information. For ...
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1answer
172 views

Logical explanation of Euler's formula

This question is a about (if not proving) at least guessing the Euler's formula. I don't want the proof using the infinite sums. We can guess by logic that for example that the equation ...
7
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2answers
460 views

Algebraic geometry in representation theory?

I heard that today algebraic geometry plays some significant role in representation theory, which is a little surprising because when I learnt representation theory it is basically algebra, topology, ...
2
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2answers
350 views

The thought process of derivatives explained (intermediate calculus) “derivatives with respect to what”

My intention here is to contribute, if there is a problem with my solution or explanation--if it is wrong--please add a comment and don't just down vote. My answer represents my understanding and I ...
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1answer
60 views

Intuitive bernoulli numbers

Can somebody explain me or give me a link with a intuitive point of view of Bernoulli numbers? I mean, somebody just saw a typical sequence of numbers that appears in some taylor expansions, and them ...
3
votes
1answer
202 views

The relationship between inner automorphisms, commutativity, normality, and conjugacy.

An inner automorphism of a group $G$ is defined to be a function $f: G \to G$ such that for $x\in G$ $f(x) = a^{-1}xa.$ I have three somewhat broad questions about this: Why is it related to ...
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1answer
104 views

Weak homotopy equivalence

I know a continuous map $f:X\to Y$ between topological spaces is a weak homotopic equivalence if it induces isomorphisms on the corresponding homotopy groups, but what kind of information do I get ...
5
votes
3answers
288 views

Verifying a proof that if $x,y,z \geq 0$ and $x+y+z = 1$, then $0 \le xy + yz + zx - 2xyz \le \frac{7}{27}$

I was working some recreational problems from a book (The Art and Craft Of Problem Solving, Zeitz) and came across one from the '84 IMO: Suppose that $x, y, z$ are non-negative reals, with $x + y ...
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1answer
111 views

An intutive explanation of natural density (asymptotic density)

I was wondering if someone can provide an intuitive explanation to natural density. I understand the concept very basically (pretty much the definition) but I can't seem to understand what natural ...
2
votes
1answer
119 views

Unclear on relationship between different dimensionalities of Fourier transform

This is probably a silly question, but it's one that's directly relevant to a project of mine and I figured this was the place to go. I have some objects that contain a 1d and a 2d array of double ...
11
votes
5answers
937 views

Why König's lemma isn't “obvious”?

I keep facing König's lemma "Every finitely branching infinite tree over $\mathbb{N}$ has infinite branch". Why it is not taken "obvious" but needs a careful proof? It seems somewhat obvious, but I ...
2
votes
1answer
847 views

Confusion about Banach Matchbox problem

While trying to solve Banach matchbox problem, I am getting a wrong answer. I dont understand what mistake I made. Please help me understand. The problem statement is presented below (Source:Here) ...
3
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1answer
79 views

Basic Question about linearity of expectation

I am going through some introductory notes on probability here http://www.stat.berkeley.edu/~aldous/134/gravner.pdf In Chapter 8, page 89, there is a problem where you get a bag containing 10 Black, ...
1
vote
1answer
308 views

Schonhage–Strassen algorithm

After brief intro to Fourier series, CFT, DFT and their basic properties I enjoyed implementing forward and backward FFT algorithm in complex numbers. I was happy to, at least, have an idea how is it ...
4
votes
2answers
747 views

Intuition behind the difference between derived sets and closed sets?

I missed the lecture from my Analysis class where my professor talked about derived sets. Furthermore, nothing about derived sets is in my textbook. Upon looking in many topology textbooks, few even ...
1
vote
1answer
194 views

Intuition behind symmetric and antisymmetric tensors

I've been studying multilinear algebra on Kostrikin's "Linear Algebra and Geometry" and he says the following. If $V$ is a linear space, $T^q_0(V)=V^{\otimes q}$ and if $f_\sigma :T^{q}_0(V)\to ...
0
votes
3answers
197 views

Help! Doubt About Uniqueness in Mathematics

Many times in mathematics, as for example when we find the solution of an ODE, we can not claim uniqueness just by construction, instead we have to use a theorem. The reasoning behind this is that ...
6
votes
2answers
171 views

Discreteness of eigenvalues for certain operators - can this approach be made rigorous?

I was idly thinking about why one might naïvely expect a discrete spectrum of eigenvalues for a linear operator $L$ when I dreamt up the following argument (which I expect isn't new instead - ...
2
votes
1answer
86 views

Wrong reasoning for uniqueness of solution of ODE?

Sometimes I have seen this argument to prove that a differential equation has an unique solution, but I think it's wrong. Suppose the differential equation: $$\mathscr{D}[y(t)]=f(x)$$ where ...
0
votes
0answers
67 views

expression that constrain the range of x to a positive interval

For any $x \in R$, I used the exponential $f(x)=e^x$ to constrain the value of $f$ to a positive interval. While serving this purpose, it happens that I cannot use the exponential for some other ...
5
votes
2answers
238 views

Intuition behind the Axiom of Choice

Why is it different to make one choice or many choices than to make infinite choices from a theoretical point of view in which indeed you are not going to do any? How could that be different from ...
2
votes
1answer
67 views

Is this intuition behind product manifolds correct?

I've been studying differential geometry on Spivak's books and recently I proved that the cartesian product of manifolds is another manifold. Right, however, what's the intuition behind this? I've ...
18
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1answer
525 views

What is duality?

I have seen some examples of duality. Sometimes applied to theorems, as for example Desargues theorem and Pappus theorem. Sometimes applied to spaces, for example the dual space of a vector space. ...
2
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1answer
475 views

proof for the general rule of conversion from base 10 to other bases

I just begin reading the book "what is mathematics" by Richard Courant. He states the general rule for passing from the base ten to any other base B is to perform successive divisions of the number z ...
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1answer
154 views

Turning an ellipse into a parabola

Today I was discussing circles, ellipses, hyperbolas, and parabolas in my precalculus class. We did the usual: completing the square, finding the center and radius (radii), etc. etc. But I like to ...
2
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1answer
89 views

Maths branch of logics or vice versa?

Is it logics a branch of maths or vice versa? From a the point of view of the definition of a logical system, logics is a 'calculus' which has axioms and rules as any branch of maths. However it ...