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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Concerning $a_{f(n)} = \displaystyle\sum_{i=0}^m{c_i a_{g_i(n)}}$

Suppose $f(n)$ is $2n$. Suppose $g_i(n)$ is $n+i$. In other words, we have a recurrence given by $a_{2n} = c_0 a_n + c_1 a_{n+1} + \dots c_m a_{n+m}$ (1) What is known about this? (2) What type of ...
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6answers
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Intuition behind conjugation in group theory

I am learning group theory, and while learning automorphisms, I came across conjugation as an example in many textbooks. Though the definition itself, (and when considering the case of abelian ...
2
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2answers
378 views

Insidious exponential integral

I hope that someone's up for the challenge; I'm attempting to solve this via computer: \begin{equation} \int_{-\pi}^\pi{\displaystyle \frac{e^{i\cdot a\cdot t}(e^{i\cdot b\cdot t}-1)(e^{i\cdot c ...
0
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1answer
259 views

Complex integrals and the possibility of avoiding parametric equations

I've been playing around with this equation: $\displaystyle\int_{-\pi}^\pi{\displaystyle\frac{1-e^{3it}}{1-e^{it}}dt}$ Now it seems to me that we can (possibly) split the integral into four ...
11
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3answers
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Intuition for uniform continuity of a function on $\mathbb{R}$

I understand the formal definition of uniform continuity of a function, and how it is different from standard continuity. My question is: Is there an intuitive way to classify a function on ...
13
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3answers
333 views

What can be gleaned from looking at a domain-colored graph of a complex function?

Functions from $\mathbb{C} \rightarrow \mathbb{C}$ are hard to visualize because of their 4-dimensional nature. One nice way of looking at them is by what's called domain coloring. An example from the ...
2
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2answers
494 views

Is this integration approximation method known/used?

I'm approximating an integral with only exponentials. i.e., it is equal to $\displaystyle \int_{-\pi}^\pi{\frac{\displaystyle\sum_{j=a}^b{c_j e^{i\cdot d_j \cdot t}}}{\displaystyle\sum_{k=a}^b{r_k ...
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6answers
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How are eigenvectors/eigenvalues and differential equations connected?

In school and at university we never had eigenvalues nor differential equations, so these concepts were really giving me a hard time. Now I developed some intuition for both concepts. I learned that ...
12
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4answers
2k views

Importance of Cayley's theorem

I along with one of my friends were just discussing some basic things in group theory, when this question came up: What are some fundamental results in group theory? We happened to list out some: ...
5
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3answers
777 views

Convergence of the series $\sum \limits_{n=2}^{\infty} \frac{1}{n\log^s n}$

We all know that $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^s}$ converges for $s>1$ and diverges for $s \leq 1$ (Assume $s \in \mathbb{R}$). I was curious to see till what extent I can push the ...
2
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1answer
164 views

Using derivatives to evaluate an integral of exponentials

Reading a book on fractional calculus reminded me that I'd like to know more on the following method/idea. Given an integral: $\displaystyle \int_{-\pi}^\pi{\frac{\displaystyle\sum_{j=a}^b{e^{i\cdot ...
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6answers
4k views

Intuition explanation of taylor expansion?

Could you provide a geometric explanation of taylor expansion?
0
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1answer
71 views

Constructing finite versions of arbitrary series

I've been wondering if/when it's possible to "truncate" a series. Example 1 For example, the closed form for the series of naturals is: $\frac{1}{(x-1)^2}$ = $1 + 2z + 3x^2 + \cdots$ The ...
5
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1answer
553 views

Harmonic mean and logarithmic mean

The harmonic mean of a finite set of positive real numbers $\{x_1, x_2, \ldots, x_n\}$ is defined to be $$H(\{x_1, x_2, \ldots, x_n\}) = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + ...
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2answers
892 views

Intuitive explanation of the difference between waves in odd and even dimensions

Motivation: In odd dimensions, solutions to the wave equation: $u_{tt}(x,t)=\nabla u(x,t)$, $u_t(x,0)=0$, $u(x,0)=f(x)$, ($t\geq 0, x\in \mathbb{R}^n$) have the nice property that the value of ...
0
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1answer
130 views

Is there an easier formulation for the Hadamard product of certain pair of series?

I'm interested in finding a recursion or simple representation for a "Hadamard product" of two power series. The Hadamard Product The Hadamard product is defined on generating functions $f(x)$ ...
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0answers
293 views

How effective is this alternative to integration?

I have a function that is difficult to integrate. So I elect to work with power series representations. Suppose the power series representation for this function is the following: $f(x) = ...
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844 views

The determinant is the integral of algebra. The integral is the determinant of analysis

This is probably an obvious parallel that most people are aware of, but I only just noticed it the other day and it made me quite excited. The determinant in algebra has a lot in common with the ...
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3answers
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Intuitive way to understand covariance and contravariance in Tensor Algebra

I'm trying to understand basic tensor analysis. I understand the basic concept that the valency of the tensor determines how it is transformed, but I am having trouble visualizing the difference ...
3
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2answers
318 views

Intuition Behind Balanced Sets

Suppose $B \subset X$ where $X$ is a vector space. $B$ is called balanced if $\alpha B \subset B$ for every $\alpha \in \Phi$ with $|\alpha| \leq 1$. Note that $\Phi = \textbf{R}$ or $\Phi = ...
2
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1answer
110 views

Efficient Sampling

I'm trying to sample a lot of points efficiently. I'm wondering if the following method is possible. I sample points of a function (evaluate the function) mod $n$. I.e. I calculate f(element one), ...
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2answers
418 views

Why does an equiangular spiral become logarithmic (intuitively)?

One of the most famous 2D-curves are logarithmic spirals (or Spira mirabilis). They can be constructed by using a machinery that ensures a constant angle between the tangent and the radial lines all ...
21
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4answers
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You are standing at the origin of an “infinite forest” holding an “infinite bb-gun”

I use stories like these to develop intuition... or perhaps to destroy it. I have my own answers in mind, but I want to see if I have made any mistakes... You are standing at the origin of an ...
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4answers
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Meaning of convolution?

I am currently learning about the concept of convolution between two functions in my university course. The course notes are vague about what convolution is, so I was wondering if anyone could give ...
48
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5answers
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Connection between Fourier transform and Taylor series

Both Fourier transform and Taylor series are means to represent functions in a different form. My question: What is the connection between these two? Is there a way to get from one to the other (and ...
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3answers
496 views

Simple Solution to Diffusion equation

A very simple solution to the diffusion equation is $u(x,t)=x^2+2 t$ My question: How can this be a solution to the diffusion equation when nothing really diffuses, but just stays the same - see ...
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2answers
161 views

Can a function describe an area on a graph like an integral?

Am I correct to say: A difference between an integral and a function is: that an integral can describe an area on a graph, while a function can't? Or am I completely off course here?
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392 views

Schwarzian Derivative and One-Dimensional Dynamics - how are they connected?

During the summer, I did an REU where we focused primarily on one-dimensional dynamics and more specifically kneading theory. One thing that I was always confused about is why the Schwarzian ...
14
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8answers
716 views

Elevator pitch for a (sub)field of maths?

When I first saw the title of this question, I forgot for a moment I was on meta, and thought it was asking about quick, catchy, attractive, informative one-or-two-liner summaries of various fields of ...
20
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7answers
14k views

Laplace transformations for dummies

Is there a simple explanation of what the Laplace transformation do exactly and how they work? Reading my math book has left me in a foggy haze of proofs that I don't completely understand. I'm ...
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3answers
281 views

Intuition behind tensor expansions of linear maps

Given finite-dimensional vector spaces $V,W$, there is an isomorphism $\text{Hom}(V,W) \rightarrow V^* \otimes W$. In particular, any linear map $\phi : V \rightarrow W$ has a tensor expansion $\sum ...
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Orientability of $\mathbb{RP}^3$

I was wondering if there is a nice way to see that $\mathbb{RP}^{3}$ is orientable without using tools of algebraic topology, like homology. The only think I could think of was to argue that ...
2
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2answers
542 views

Understanding of convergence of intersections of sets

If you start with an infinite set, you can have a sequence of nested sets which converge to a single point. (ie Intersection of $\left(\large\frac{-1}{n}, \frac{1}{n}\right)$ as $n\to \infty$) ...
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4answers
2k views

Intuitive explanation of variance and moment in Probability

While I understand the intuition behind expectation, I don't really understand the meaning of variance and moment. What is a good way to think of those two terms?
3
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2answers
190 views

Relationship between torsion modules and topology

I was reviewing my class notes and found the following: "The name 'torsion' comes from topology and refers to spaces that are twisted, ex. Möbius band" In our notes we used the following definition ...
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Motivation behind standard deviation?

Let's take the numbers 0-10. Their mean is 5, and the individual deviations from 5 are -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 And so the average (magnitude of) ...
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Example: Function sequence uniformly converges, its derivatives don't.

Could anyone give an example of a sequence of differentiable (real) functions that uniformly converge to a differentiable function, but the derivatives of which don't converge to the derivative of the ...
2
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1answer
178 views

A “geometrical” representation for Ramsey's theorem

The [infinite] Ramsey theorem states that Let $n$ and $k$ be natural numbers. Every partition $\{X_1,\ldots ,X_k\}$ of $[\omega]^n$ into $k$ pieces has an infinite homogeneous set. Equivalently, ...
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The intuition behind generalized eigenvectors

An ordinary eigenvector can be viewed as a vector on which the operator acts by only stretching (without rotating) it. Is there a similar intuition behind generalized eigenvectors? EDIT: By ...
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1answer
361 views

Is this interpretation of Stieltjes integration correct?

If $f$ is a positive function, the intuitive interpretation of the Riemann integral $\int_a^b f(x) dx$ is the area under the curve $f$ between $a$ and $b$. Suppose $f$ and $g$ are smooth positive ...
18
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7answers
2k views

Why do we require a topological space to be closed under finite intersection?

In the definition of topological space, we require the intersection of a finite number of open sets to be open while we require the arbitrary union of open sets to be open. why is this? I'm assuming ...
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12answers
5k views

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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4answers
3k views

Finding all normal subgroups of a group

On my homework today, we had to find all the normal subgroups of $D_{n}$, the dihedral group of order 2n. I solved the problem by looking at how the conjugacy classes change based on whether n is even ...
49
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4answers
6k views

Why do we care about dual spaces?

When I first took linear algebra, we never learned about dual spaces. Today in lecture we discussed them and I understand what they are and everything, but I don't really understand why we want to ...
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6answers
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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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7answers
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Intuition for the definition of the Gamma function?

In these notes by Terence Tao is a proof of Stirling's formula. I really like most of it, but at a crucial step he uses the integral identity $$n! = \int_{0}^{\infty} t^n e^{-t} dt$$ coming from ...
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Could you explain why $\frac{d}{dx} e^x = e^x$ “intuitively”?

As the title implies, It is seems that $e^x$ is the only function whoes derivative is the same as itself. thanks.
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7answers
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Can this standard calculus result be explained “intuitively”

Recently I stumbled upon someone who said he wanted to understand why $\arctan x = \int\dfrac{dx}{1+x^2}$ At first I was confused. This is an easy result in any integral calculus course. But then he ...
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6answers
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Best intuitive metaphors for math concepts (of any level)

Frequently, we introduce a new concept with a formal definition, then immediately say "Intuitively, what this means is..." What are the absolute best metaphors you've seen (for concepts of any level)? ...
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Elementary proof of the Prime Number Theorem - Need?

Although I am very much new to "Analytic Number Theory", there are some non mathematical questions which puzzle me. First of all, why was G.H.Hardy so much keen to have an elementary proof of the ...