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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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
83 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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1k 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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1answer
744 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
287 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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1answer
102 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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414 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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87 views

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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5answers
145 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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1answer
305 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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2answers
500 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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4answers
8k 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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4k 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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169 views

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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14answers
1k views

Dot Product Intuition

I'm searching to develop the intuition (rather than memorization) in relating the two forms of a dot product (by an angle theta between the vectors and by the components of the vector ). For ...
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1answer
361 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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1answer
59 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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1answer
995 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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1answer
172 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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4answers
3k 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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1answer
348 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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1answer
61 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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1answer
634 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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84 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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1answer
462 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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2answers
443 views

Rigour vs intuition

Researcher David Tall has written in chapter one of Advanced Mathematical Thinking that ...
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4answers
790 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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1answer
166 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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432 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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0answers
339 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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1answer
94 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 ...
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1answer
479 views

What is the intuition behind the Lagrange multiplier?

I know that the minimum or maximum point is achieved when the gradient in the constraint function is parallel to the gradient on the $f$ function. But why the Lambda is called the Lagrange ...
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5answers
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The Meaning of the Fundamental Theorem of Calculus

I am currently taking an advanced Calculus class in college, and we are studying generalizations of the FTC. We just started on the version for Line Integrals, and one can see the explicit symmetry ...
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2answers
90 views

Non-isomorphic structures with equal cardinality

Let $\mathfrak{A}=(\mathbb{N},S,0)$ be a structure where $S$ is the sucessor function. Let $\mathfrak{B} =(\mathbb{N}\times \{0\} \cup \mathbb{Z} \times\{1\} ,S, 0)$ with $0 = (0,0)$ and $$ S(k,i) ...
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5answers
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Why does the power rule work?

If $$f(x)=x^u$$ then the derivative function will always be $$f'(x)=u*x^{u-1}$$ I've been trying to figure out why that makes sense and I can't quite get there. I know it can be proven with limits, ...
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12answers
10k views

Looking for an intuitive explanation why the row rank is equal to the column rank for a matrix

I am looking for an intuitive explanation as to why/how row rank of a matrix = column rank. I've read the proof at http://en.wikipedia.org/wiki/Rank_of_a_linear_transformation and I understand the ...
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1answer
720 views

How to understand convex duality intuitively

Is there an intuitive way to understand the convex duality? If the primal problem is minimization, the dual is maximization over another set of variables - but I would love to have a geometric ...
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2answers
172 views

Convergence of Improper Integrals

I am working on some exercises for Improper Integrals (not homework). The question is 1.c in Folland Advanced Calculus : $$\int_0^\infty x^2 e^{-x^2 } \, dx$$ It asks whether the above Improper ...
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4answers
2k views

Algebra: Best mental images

I'm curious how people think of Algebras (in the universal sense, i.e., monoids, groups, rings, etc.). Cayley diagrams of groups with few generators are useful for thinking about group actions on ...
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3answers
815 views

Intuition behind the Frattini subgroup

I am trying to get a better feel for what the Frattini subgroup really is, intuitively. Let $G$ be a group and denote its Frattini subgroup by $\Phi(G)$. I know that $\Phi(G)$ is the intersection of ...
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3answers
309 views

Are matrices best understood as linear maps?

Any linear map between finite-dimensional vector spaces may be represented by a matrix, and conversely. Matrix-matrix multiplication corresponds to map composition, and matrix-vector multiplication ...
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1answer
92 views

Transformations and coordinate Systems

I am working on some practice exercises (not homework) on transformations and need some intuition and help. One of the questions is: $(u,v)=f(x,y)$ where $ \quad u= { e }^{ x }\cos(y), \quad v = { e ...
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1answer
270 views

Article about primes.(Revised)

I'm trying to write a article about primes, and I'm curious whether I can really involve other topics (like complex numbers) and relate them to observe peculiar properties of primes. Or can I try ...
6
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1answer
250 views

Intuition behind $\nabla \times \mathbf{F}$

Is there a simple explanation why this form for the curl of a vector field $\mathbf{F}$, $$\nabla \times \mathbf{F}=\begin{vmatrix} \hat{x} & \hat{y} &\hat{z} \\ \frac{\partial}{\partial ...
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2answers
993 views

Intuition for the Frobenius method

I'm teaching a differential equations class now and I am hoping to give a reason for the Frobenius series method beyond simply "we guess these solutions". Now, for the Euler equation $$t^n x^{(n)}(t) ...
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4answers
826 views

How would you explain why $e^{i\pi}+1=0$ to a middle school student?

Hi I was asked by a friends child who is in middle school why $e^{i\pi}+1=0$. Now I couldn't think of a way to explain it so he would understand. Albert Einstein once said “If you can't explain it ...
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5answers
3k views

Intuition for the Importance of Modular Forms

I am learning about modular forms for the first time this term and am just starting to wrap my head around what might be the big picture of things. I was wondering if the following interpretation of ...
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3answers
856 views

Intuition on fundamental theorem of arithmetic

I'm sorry ahead if time if this is overly trivial for this site. Currently in school, much of what I enjoy is number theory - based. Currently, I lean pretty heavily on the FTA for a good deal of my ...
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1answer
415 views

Axioms vs. Universal Constructions/Properties

What (exactly) is the difference between defining a mathematical object by it's axioms and by a universal construction ? Please take my 3 opinions into consideration, as they also contain more ...