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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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 ...
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423 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, ...
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340 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
58 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 ...
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193 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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102 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 ...
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283 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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108 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 ...
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1answer
115 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 ...
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895 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 ...
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811 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) ...
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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, ...
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291 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 ...
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718 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 ...
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183 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 ...
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196 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 ...
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169 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 - ...
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1answer
84 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 ...
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66 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 ...
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237 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 ...
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1answer
64 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 ...
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498 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. ...
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443 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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147 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 ...
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87 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 ...
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293 views

What are central automorphisms used for?

A central automorphism is an automorphism $\theta$ for which $x^{-1}\theta(x)\in Z(G)$ for each $x\in G$. It's not difficult to prove that the set of central automorphisms forms a subgroup of ...
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Geometrical Interpretetion of Half Derivative [duplicate]

How would you understand in a intuitive way the meaning of: $$D^{\frac{1}{2}}x^2=\frac{\Gamma(3)}{\Gamma(\frac{5}{2})}x^{\frac{3}{2}}=\frac{8}{3\sqrt{\pi}}x^{\frac{3}{2}}$$ or ...
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133 views

Differentiation confusion

I've been reading my textbook, and it tells me how to go about differentiating from first principles, it goes something like this: $\eqalign{ & \mathop {\lim }\limits_{h \to 0} {{f(x + h) - ...
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199 views

Intuition behind compact subspaces of a metric space

I've read up on compactness in a metric space and have found a few definitions (let $X$ be a metric space and $E \subset X$ in all the following): $E$ is compact in $X$ if for every open covering of ...
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236 views

Whats the connection between functions with curl 0 and holomorphic functions

When I first saw the Cauchy-Riemann differential equations they remind me on the conditions for the curl of a function to be zero. Here some notation I will use: $$\frac{\partial f}{\partial x} = ...
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3k views

Why Markov matrices always have 1 as an eigenvalue

Also called stochastic matrix. Let $A=[a_{ij}]$ - matrix over $\mathbb{R}$ $0\le a_{ij} \le 1 \forall i,j$ $\sum_{j}a_{ij}=1 \forall i$ i.e the sum along each column of $A$ is 1. I ...
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2k views

Why does Newton's method work?

I find many sites explaining how to use Newton's method, but none explaining why it works. Could someone give me the intuition behind it? Thanks.
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Intuition behind “Non-Archimedean” — two senses of “non-archimedean”.

There appear to be two senses of the qualifier "Archimedean" for fields. One is for ordered fields, and one is for "valued fields" (fields with an absolute value function defined). In the first case, ...
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300 views

Why is an irrational number's algebraic complexity the opposite of its Diophantine complexity?

Definition 1. Given $x \in \Bbb{R}$, the algebraic degree of $x$ is the degree of the minimal polynomial of $x$ over $\Bbb{Q}$. If $x$ is transcendental, we will define its algebraic degree to be ...
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53 views

Intuition Behind Krantz Theorem

The theorem I'm referring to is as follows: Let $z_0$ be a root of a nonzero holomorphic function $f$ , and let $n$ be the least positive integer such that, the $n$-th derivative of $f$ evaluated ...
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594 views

Intuition behind Maschke's theorem

I'm an undergraduate learning about group representations and Young tableaux, and have came across Maschke's theorem stating; If $G$ is a finite group and $F$ is a field who's characteristic does ...
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114 views

Is the canonical projection of $G$ onto $G/\ker f$ a lift of $f$?

I'm not sure I understand what a lift is and why it is called that. If I understand the definition correctly, then if I have a group homomorphism $f:G\to H$, then the canonical projection $\pi:G\to ...
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1k views

How Bernoulli differential equation arise naturally?

A Bernoulli differential equation is a non-linear differential equation of the form $$ \frac{dy}{dx} + P(x)y = Q(x)y^n. $$ I understand this is special; Because its exact solution is known though ...
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212 views

Intuition on group actions

I'm trying to get more intuition on this definition: Let $(G,\circ,e)$ be group. A group action is a mapping $G×X→X:(g,x) ↦g.x \,$ such that: \begin{align*} ∀x∈X &: e.x=x \tag{1}\\ ∀g,h∈G,∀x∈X ...
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56 views

conditional convergence

This is an practice question from "Advanced Calculus, Folland" Chapter 6.3, Q.2 (not HW) I am not sure how to go about this question :: suppose $\sum { { a }_{ n } } $ is conditionally convergent. ...
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407 views

dim$(V)$ = $n$, dim$(W)$ = $m$ $\implies$ dim($L(V,W)$) = $nm$

I am reading Hoffman & Kunze's chapter on linear transformations, with a view towards understanding dual spaces. (I primarily want to read Calculus on Manifolds; in the first chapter of that book, ...
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75 views

Examples of convergence of series

These questions are practice questions from the text "Advanced Calculus, Folland" chapter 6.2 (not HW) I am working on some exercises on convergence of series and I feel that I understand it well but ...
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305 views

Can someone please explain Morera's Theorem and the Schwarz Reflection Principle?

I have been reading Complex Analysis, Third Edition by Joseph Bak and Donald J. Newman and am stuck with chapter 7 which deals with the mentioned theorem. I have read through the chapter at least ...
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Intuition behind Fourier coefficients

Actually I'm trying to dive into Fourier series and have some trouble understanding the idea behind the Fourier coefficients. Let's have a Fourier series $$f(x) = a_0 + ...
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284 views

Rigour vs intuition

Researcher David Tall has written in chapter one of Advanced Mathematical Thinking that ...
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568 views

Intuition for $\omega^\omega$

I'm trying to understand the ordinal number $\omega^\omega$ and I'm having a hard time. I think I understand what $\omega^2$ is. It's what I would get if I took countably many copies of $\omega$ and ...
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114 views

Graphically, what is positive semidefinite-ness?

Suppose that we are trying to minimize a function $f$ on $\mathbb{R}^n$ and we apply Newton's method, updating: \begin{align} \mathbf{x}_{n+1} = \mathbf{x}_n - [\nabla^2 f(\mathbf{x}_n)]^{-1} \nabla ...
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319 views

Intuitive Explanation of Morphism Theorem

Is there an intuitive explanation for the morphism theorem from introductory abstract algebra? First Morphism Theorem: Let $K$ be the kernel of the group morphism $f: G \to H$. Then $G/K$ is ...
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239 views

Geometric intuition for Jordan normal forms (invariant subspaces, shearing, scaling, etc.)

I'm trying to visualize what a linear operator does to a vector space if that operator can be put into Jordan normal form. For concrete motivation, let's take $V = \mathbb{R}^3$, with some linear ...
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63 views

Expected Value of Students on a Bus

There's a question in my probability book that says there are $148$ students on $4$ buses containing $40, 33, 25, 50$ students, respectively. If we let $X$ denote the number of students that were on ...