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 of a Submanifold

Could someone explain the intuition behind a submanifold. When, for example, is it appropriate to work with immersed submanifolds vs embedded submanifolds? Why is it important for a submanifold to be ...
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definition of separation axioms in topology

I am learning the Separation Axioms and came across the definition of regular space. In the definition they say, "Suppose the one point sets are closed in $X$" My question is: how can one point sets ...
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Are these statements correct? $A \subseteq f^{-1} \circ f(A)$ and $ f \circ f^{-1}(B) \subseteq B$

We wrote these two statements in class: $A \subseteq f^{-1} \circ f(A)$ $ f \circ f^{-1}(B) \subseteq B$ where $A$ and $B$ are sets and $f(A)= \lbrace f(x):x \in A \rbrace $ and $f^{-1}(B)= \lbrace ...
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170 views

Physical Meaning of Symplectic Vector Fields

The mathematics of symplectic (as well as Hamiltonian) vector fields is something that has been quite clear to me for some time, but recently I have been thinking much more about what certain ...
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124 views

Two equivalent definitions of convergent sequences?

I know that: Definiton 1. The sequence $(x_n)$ in the metric space $(X,d)$ is said to converge to the point $x_0\in X$ if $$\forall\epsilon>0, \exists n_0\in\mathbb{N} \text{ such that } \forall ...
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Why do class-sized models escape the completeness theorem?

Assume ZFC is consistent. Then its not true that every model $(M,\in')$ of ZFC has an element $x \in M$ such that $M$ believes that $x$ is a model of ZFC. Otherwise, by the semantic completeness ...
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132 views

Why does $e$ seem to be an intuitive number? [closed]

I often find two numbers roughly "in the same ballpark" if they are within a factor of about $e$ of each other. For example, if I know computers generally cost upward of $\$1000$, then $\$2700$ would ...
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53 views

Do there exist faithful functors $\mathrm{Grp} \rightarrow \mathrm{Set}$ that aren't naturally isomorphic to the underlying set functor?

I'm trying to get some intuition for the notion of natural isomorphism. To that end, my question is: do there exist faithful functors $\mathrm{Grp} \rightarrow \mathrm{Set}$ that aren't naturally ...
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2answers
261 views

Problem in understanding a proof there are five Platonic solids.

Thanks to several comments by Gerry Myerson, it is now clear that I wasn't clear, up to a state where I seriously confused myself. In a renewed attempt: Recently, I've been thinking about Platonic ...
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115 views

Problem with structure of a semisimple ring theorem

Structure of semisimple ring (Wedderburn-Artin) in Rings and Categories of Modules - Frank W. Anderson, Kent R. Fuller (auth.) Proof: Please explain that: "Now $_RR$ is direct sum off these ...
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How to really understand the tensor algebra?

If $V$ is a vector space over $F$, then we define $T^r_0(V)=V^{\otimes r}$, then we define the algebra of contravariant tensors to be $$T(V)=\bigoplus_{r=0}^\infty T^r_0(V)$$ together with the ...
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58 views

Why it makes sense to think of multivectors as “paralelograms”?

Let $V$ be a vector space over the field $\mathbb{K}$ and let $T(V)$ be it's tensor algebra. We usually define the exterior algebra $\Lambda (V)$ by the following process: we consider the bilateral ...
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4answers
943 views

What really is an indeterminate form?

We can apply l’Hôpital’s Rule to the indeterminate quotients $ \dfrac{0}{0} $ and $ \dfrac{\infty}{\infty} $, but why can’t we directly apply it to the indeterminate difference $ \infty - \infty $ or ...
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256 views

Adjunctions via Universal Arrows: Understanding a Proof.

I've having trouble understanding something in Turi's Category Theory Lecture Notes from The University of Edinburgh, which can be found here. It's the proof of Theorem 7.1, part (3). Here's the ...
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Intuitive reasoning why are quintics unsolvable

I know that quintics in general are unsolvable, whereas lower-degree equations are solvable and the formal explanation is very hard. I would like to have an intuitive reasoning of why it is so, ...
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42 views

Green's formula: $ \int_{a}^{b} [uL(v)-vL(u)]dx=p\left( u\frac{du}{dx}-v\frac{du}{dx}\right)\big|_{a}^{b}$

I'm reading a book on PDE's and they introduce ''Green's formula'' in a rather (to me) abrupt way. It is used to derive that for a steady state heat PDE $u(x)$ is given by: $u(x)=\int_{a}^b ...
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What's the geometrical intuition behind differential forms?

This question can look like a duplicate of this one, but it's kind of different. I'm trying to relate some geometrical meanings I've seem in some books to the definition of differential forms in ...
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114 views

Intuition behind $\vec{e_i} \times \vec{e_j}=\epsilon_{ijk} \vec{e_k}$ (Levi Civita)

Let $\vec{e_i}$ denote a unit vector. Then we can write: $\vec{e_i} \times \vec{e_j}=\epsilon_{ijk} \vec{e_k}$, where $\epsilon_{ijk}$ is the Levi Civita symbol. Can someone intuitively explain me ...
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68 views

Formal proof for $\lim_{x\to\infty}x\exp(-x) =0$

I intuitively understand that $$\lim_{x\rightarrow\infty} xe^{-x}=0$$ as the $e^{-\infty}$ approaches zero faster than $x$ approaches infinity. But this requires one to have a knowledge of the ...
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102 views

Counterexample for $A,B\triangleleft G,G/A\cong B \Rightarrow G/B\cong A$

Let $A,B\triangleleft G$. Give counterexample for the claims: a. $G/A\cong B \Rightarrow G/B\cong A$ b. $G/A\cong G/B\Leftrightarrow A\cong B$ I don't know from where to start. Can you ...
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Geometric intuition behind the Hasse principle

Let $f(X,Y) \in \mathbb{Q}[X,Y]$ be a quadratic polynomial. The Hasse-Minkowski theorem says that $f(X,Y) = 0$ has a solution $(x,y) \in \mathbb{Q}^2$ iff it has a solution in $\mathbb{R}^2$ and ...
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290 views

What does SVD Entropy Capture?

Looking at different definitions and types of Entropy, I run into the concept of SVD Entropy, which is defined as explained below. What is the intuition behind the SVD spectrum? What do different ...
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133 views

Linear (in)dependence of $\sin (x), \sin (x+1), \sin (x+2)$

I need to discuss the linear independence of the following given vectors: \begin{align} \sin(x), \sin(x+1), \sin(x+2)\end{align} there are many similar questions on math.SE but most of which I have ...
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Trigonometry confusion

I was doing a bit of trigonometry, as I have been for a couple of years and it suddenly dawned on me that I don't really understand the trigonometric functions, at all. You first learn the basic trig ...
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396 views

Class group and factorizations

There is a common characterization of the class group ${\rm Cl}(R)$ as a kind of measure of how badly factorization fails to be unique. The most obvious justification for this sentiment is that the ...
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154 views

Uniform convergence of $\sum_{n=0}^{\infty} \frac{(-1)^n 2i}{(2n+1)z^{2n+1}}$ on squarewith vertices $\pm6\pm6i$?

Can someone explain me how I can check whether the convergence of $\sum_{n=0}^{\infty} \frac{(-1)^n 2i}{(2n+1)z^{2n+1}}$ is uniform on the (boundary of) square $A$ with vertices $\pm6\pm6i$?
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268 views

How to recognize removable singularity and how to remove it

I don't understand the idea of a removable singularity yet. Can someone explain me how to recognize a removable singularity and how to remove it? Example: $g(z)=f(z)/z$. Is $z=0$ then a removable ...
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81 views

Understanding concept of an operation being well defined for an equivalence relation

Let $I$ be an ideal in a ring $R$. Define the relation (congruence modulo $I$) by $a \equiv b$ if $b - a \in I$ Denotes the equivalence class containing $a$ by $\bar{a}$. Define $$\bar{a} + ...
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Lebesgue Measure Definition

Given a subset $A \subset \mathbb{R}$ with the length of an open interval $\mu_L(I_k) = b_k -a_k : I \doteq [a_k,b_k]$ The lebesgue measure is defined as $$ \lambda^{\ast} (A) \doteq \inf \Big\{ ...
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Definition of trignometric functions

Why is the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse? How this definition came? Similar question for other five trigonometric function? Why they ...
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219 views

Convergence of alternating nested radicals

Last evening, after reading a couple of questions about nested radicals, I started to wonder about problems involving what I will term "alternating nested radicals;" below is an example, which I found ...
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101 views

Exponent Upon Exponent? [duplicate]

I can't understand this. Can you please make a clearer explanation?
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Intuitive understanding of Lemma of Jordan

Can somebody give me an intuitive understanding of the Lemma of Jordan, which is: $$\lim \limits_{R\rightarrow \infty} \int_{\gamma} \exp(i \omega z)\,\rm dz=0 $$ if: $$\lim \limits_{z\rightarrow ...
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0answers
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Finding the derivative of a piecewise function of a complex variable

I am working on an assignment that involves finding the derivative evaluated at zero of this piecewise function of a complex variable: Let $g:\mathbb{C}\rightarrow\mathbb{C}$ be defined by: ...
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262 views

When does infinite intersection preserve a closed property?

There are two statements well known in Math and Computer Science: Intersection of infinite number of regular languages is not regular. Intersection of infinite number of convex sets is convex. ...
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69 views

How to reason about congruences? If $x^2 \equiv a$ (mod $m$) and $y^3 \equiv a$ (mod $m$), then $\gcd(a,m) = 1$

Generally, I have no high level conception of what is going on in my number theory class. It feels like a loose collection of theorems and techniques that you can use on some problems, but I have ...
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145 views

Intuition for Geometric Transformations

I've been making a lot of effort over the past few hours to gain some intuition into the art of geometric transformation but to little avail. I would really like to be able to look at a transformation ...
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71 views

Solving limits of form $\infty \times \infty$

I need help with solving $f(x) \lim_{x \to \infty} e^x\times x^3$ I know that $\to \infty$ is the answer but I don't know how to get there. Using since $e^x \to \infty$ and $x^3 \to \infty$ I get ...
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56 views

On subspace verification

I am struggling with the following Problem: \begin{align}Y= \lbrace (x^4-y^4,0,0,0) \mid x,y \in \mathbb{R} \rbrace \subset \mathbb{R}^4 \end{align} Question, is the given Set a subspace of ...
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55 views

Significance of the DEGREE of Differential equation

Can anybody gives the idea why degree of a differential equation is important?. Every differential equation book writes the definition of "degree of a differential equation". But, why do we care about ...
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83 views

If $A$ is the adjacency matrix of a graph, why does the $(i,j)$ entry of $A^n$ give the number of $n$-step walks from $i$th vertex to $j$th vertex?

Let $A$ be the adjacency matrix of some directed graph with $m$ vertices labeled as $v_1, v_2, \ldots, v_m$. So here $A_{ij} = 1$ if there is an edge from $v_i$ to $v_j$, and $A_{ij} = 0$ otherwise. ...
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How to think about the change-of-coordinates matrix $P_{\mathcal{C}\leftarrow\mathcal{B}}$

I've taken a linear algebra course in the past, but I feel my understanding of coordinate change is very superficial. For example this exercise (4.7.1 from Lay's "Linear Algebra and its Applications" ...
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Geometric interpretation of the addition of linear equations in general form

I have a very simple question: suppose I have two 2D linear equations in general form $$ a_1x + b_1y + c_1 = 0$$ $$ a_2x + b_2y + c_2 = 0$$ I'd like to know what's the (intuitive) geometric ...
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Could we reasonably classify finite groups?

So I have been reading some work on $p$-groups, and I noticed a particularly disturbing sentence: "There is no hope of finding a finite set of invariants that will define every p-group up to ...
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288 views

Problem with definition of regular surface in classical differential geometry

I am reading Do Carmo's differential geometry book and the definition of a regular surface in the second chapter is given to be this: I have few doubts about this definition: 1) Why we need to find ...
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354 views

Reflexivity: How can something be related to itself?

Background: I'm a philosophy student. I'm comfortable with math, but don't have much of a background in it. One of the topics I'm writing about (I-relation in theories of identity) closely mirrors ...
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207 views

Geometric Intuition of Gaussian Curvature

Curvature of a curve at a point can be understood as how rapidly the curve tries to move away from the tangent of the curve at that point. And for curved surfaces we have defined the Gaussian ...
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164 views

Visualizing Markov and Chebyshev inequalities

I am helping a class on introductory probability covering Markov and Chebyshev's inequalities. I would like to give the students a nice visualization for why they are true or at least to show what ...
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316 views

What's the idea behind the covariant derivative?

I'm learning differential geometry from what I find on the Internet (to eventually find a grasp on General Relativity too). Right now I playing with a sphere. I have 3 functions ($x$, $y$, $z$) that ...
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285 views

Intuition behind definition of homotopic equivalence and distinction with homeomorphism

I am a physics student and have come across the definition of homotopic equivalence of two spaces as existence of two functions $f:X \to Y,g: Y \to X$ such that $g \circ f$ and $f \circ g$ are ...