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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Why this space is homeomorphic to the plane?

I'm trying to see why this picture below is homeomorphic to the $\mathbb R^2$. It's really hard, please I need an intuitive idea of this. This seems very weird for me, I need help. Thanks a lot
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1answer
142 views

Questions Based On A Couple Of Weird Limits

When evaluating certain limits,i get an answer with which i'am not fully convinced despite following steps which i claim is correct.Answers given by analytic method and L'Hopital's rule differs! I ...
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503 views

Intuitive interpretation of the adjacency matrix as a linear operator.

Naturally we can describe graphs via tables of "yes there is an edge" or "no there is not" between each pair of vertices, so the definition of an adjacency matrix is easily understood. Thinking of ...
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Intuitive proof of multivariable changing of variables formula (jacobian) without using mapping and/or measure theory?

iWhat is a intuitive proof of multivariable changing of variables formula (jacobian) without using mapping and/or measure theory? I was thinking that textbooks make the proofs over complicate. If ...
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This quotient space is homeomorphic to the Möbius strip?

Let $G:\mathbb R \times [-1,1]\to \mathbb R \times [-1,1]$ be a map defined by $G(x,y)=(x+1,-y)$ This space $Q=\mathbb R\times [-1,1]/\sim$, where $(x_1,y_1)\sim (x_2,y_2)$ if and only if there is ...
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Understanding induced representations

Let $G$ be a group and $H$ be a subgroup. Let $\phi:H\rightarrow GL(V)$ be a representation of $H$. There are three constructions in Wikipedia, but I am not really convinced by these. My question is: ...
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217 views

The Method Of Separation Of Variables

Can anybody explain me why the method of separation of variables for linear homogeneous PDE works ? thanks
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2k views

Centralizer, Normalizer and Stabilizer - intuition

What is the motivation/intuition behind these concepts? What notion/property of a group do they capture? Or what is the scenario of application. Thanks.
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682 views

Intuition behind isomorphism of algebraic varieties

Let $X \subset \mathbb A^n$, $W \subset \Bbb A^m$ be two algebraic sets. A function $\phi:X \rightarrow W$ is a morphism if there exist $m$ polynomial functions $f_1,\ldots,f_m \in K[X]$ such that for ...
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2answers
423 views

Lie group and SO3 visualisation

Maybe I'm asking a very vague question but I'd like to know if there are some visualisation tools available already that explain lie algebra exponential map or logarithm? I'd like to be able to ...
8
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2answers
289 views

What was Klein working on when he “replaces his Riemann surface by a metallic surface”?

I am reading The Value of Science by Poincare, and the following paragraph from Chapter I seems rather interesting: Look at Professor Klein: he is studying one of the most abstract questions of ...
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154 views

On the differential equation $y''+y=0$

Consider the differential equation $$\frac{d^{2}y}{dx^{2}}+y=0$$ with initial conditions $y(0)=0$ and $y'(0)=1$. The solution is well known - $y=\sin(x)$. I know how to derive this solution, since the ...
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804 views

Labeled/unlabeled balls in unlabeled boxes

I was hoping I could receive some clarification into the the four cases: Placing labeled balls in unlabeled boxes with repetition. Placing labeled balls in unlabeled boxes without repetition. ...
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3k views

What's the Clifford algebra?

I'm reading a book on Clifford algebra for physicists. I don't quite understand it conceptually even if I can do most algebraic manipulations. Can some-one teach me what the Clifford algebra really ...
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1answer
60 views

Using induction to prove P(k) with P(k-1)

Use induction to show $$1+\frac14+\frac19+...+\frac1{n^2} < 2-\frac1{n}$$ Assume $P(k-1)$: $$1+\frac14+\frac19+...+\frac1{(k-1)^2} < 2-\frac1{k-1}$$ Show $P(k)$: I tried to show that ...
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230 views

Finding a shortest path between a set of points

Suppose: $x_1,x_2,x_3,x_4$ is a shortest path from $x_1$ to $x_4$ $x_2,x_5,x_6,x_7,x_9$ is a shortest path from $x_2$ to $x_9$ $x_{10},x_5,x_8,x_3,x_9$ is a shortest path from $x_{10}$ to $x_9$ ...
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7k views

How many ways are there to choose 10 objects from 6 distinct types when…

(a) the objects are ordered and repetition is not allowed? (b) the objects are ordered and repetition is allowed? (c) the objects are unordered and repetition is not allowed? (d) the objects are ...
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1answer
116 views

Expressing a relationship in a graph using quantified logic

Express the following using quantified formulae for a simple undirected graph $G = (V,E)$. The predicate P({u,v}) is true when $\{u,e\}\in E$ and false otherwise. The diameter of $G$ is at most 2. ...
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160 views

Finding recurrence and an algorithm to represent it

You find yourself in a country with integer coin denominations $c_1 < c_2 < ... < c_r$, where $c_1 = 1$. Unfortunately, the greedy algorithm is not guaranteed to find the optimal way to ...
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302 views
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313 views

If $f(n) = \Theta (g(n))$, why does $g(n) = \Omega (f(n))$?

Why is this the case? I understand that if $f(n) = \Theta (g(n))$ then $c_1g(n)<f(n)<c_2g(n)$, but why does this show that $g(n)$ is bounded below by $f(n)$? I would think that it would be ...
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How does intuition fail for higher dimensions?

From this answer: Now, Algebraic Geometry is one of the oldest, deepest, broadest and most active subjects in Mathematics with connections to almost all other branches in either a very direct ...
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1answer
81 views

If A $\propto$ B and A $\propto$ C while keeping each other constant, then why is A $\propto$ BC? [duplicate]

Possible Duplicate: Proportional to 2 Separate Variables vs. Proportional to Product of 2 Variables I guess some people may find this obvious, but I really don't. My question is: If ...
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4answers
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Can someone intuitively explain what the convolution integral is?

I'm having a hard time understanding how the convolution integral works (for Laplace transforms of two functions multiplied together) and was hoping someone could clear the topic up or link to sources ...
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659 views

Intuitive explanation of Residue theorem in Complex Analysis

The residue theorem that states that if a) $U$ is a simply connected and open subset of the complex plane, b) $a_1,\dots,a_n$ are finitely many points of $U$, c) and $f$ is a function which is ...
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647 views

What is the intuitive meaning of the adjugate matrix?

The definition of the adjugate matrix is easy to understand, but I have never seen it used for anything. What is the intuitive meaning of this matrix? Are there examples of applications which may ...
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134 views

What's the most elegant way of rotating a 3-dimensional co-ordinate system?

For two dimensional rotation of $x$ and $y$ axes anticlockwise by $\varphi$, the equation that transforms $P(x,y) \rightarrow P(x',y')$, $x'=x \cos(\varphi)+y \sin(\varphi)$ and $y'=y \cos(\varphi)- ...
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Does a four-variable analog of the Hall-Witt identity exist?

Lately I have been thinking about commutator formulas, sparked by rereading the following paragraph in Isaacs (p.125): An amazing commutator formula is the Hall-Witt identity: ...
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256 views

What's the intuition behind non-integer exponents/powers

Consider some $a \in \mathbb{R}$ and $x \in \mathbb{R}\backslash \mathbb{N}$. Is there some intuition to be had for the number $a^x$? For example the intuition of $a^2$ is obvious; it's $a*a$ which ...
6
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3answers
670 views

What's the intuition of the transpose of a matrix? [duplicate]

I know the transpose is to swap the columns and rows of a matrix. And $A^T$$A$ is a symmetric matrix which elements are the inner product of each column of $A$. But I didn't understand the intuition ...
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603 views

Understanding the three isomorphism theorems

I have learnt the following three isomorphisms for a while but without true understanding: A group homomorphism $\phi:G\to G'$ can be decomposed into ...
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216 views

How can it happen to find infinite bases in $\mathbb R^n$ if $\mathbb R^n$ does not admit more than $n$ linearly independent vectors?

How can it happen to find infinite bases in $\mathbb R^n$ if $\mathbb R^n$ does not admit more than $n$ linearly independent vectors? Also considered that each basis of $\mathbb R^n$ has the same ...
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0answers
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A gratifying re-encounter with a piece of math that was out of my mind

A series of real numbers is said to be conditionally convergent if it is convergent but not absolutely convergent. By rearranging the terms of a conditionally convergent series we can make the ...
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2answers
430 views

Explain convertion algorithm from bytes to Kb, Mb, Gb.

I was trying to convert file size from bytes to human understandable value and found one interesting solution. I will provide it on php with explanation. ...
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294 views

How to 'analyze' problems in analysis; Computing $\int_0^{2\pi}\frac{1}{(a+b\cos(\theta))^2}d\theta$

If $a, b \in \mathbb{R}$ with $a > b > 0$, compute this ungodly thing; $$\int_0^{2\pi}\frac{1}{(a+b\cos(\theta))^2}d\theta$$ I'm really not a fan of complex analysis... I can't visualize ...
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Motivating (iso)morphism of varieties

I am reading course notes on algebraic geometry, where a morphism of varieties is defined as follows ($k$ is an algebraically closed field): Let $X$ be a quasi-affine or quasi-projective ...
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2answers
94 views

What is the CONCEPT when we speak of maximum entropy?

What is an intuitive interpretation of the concept of maximum entropy? I want to understand this concept better but what I'm finding is too "advanced" right now. Can anyone simplify it ... imagine I'm ...
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Intuition on the Orbit-Stabilizer Theorem

The Orbit-Stabilizer says that, given a group $G$ which acts on a set $X$, then there exists a bijection between the orbit of an element $x\in X$ and the set of left cosets of the stabilizer group of ...
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Tricks to remember Fatou's lemma

For a sequence of non-negative measurable functions $f_n$, Fatou's lemma is a statement about the inequality $\int \liminf_{n\rightarrow \infty} f_n \mathrm{d}\mu \leq \liminf_{n\rightarrow ...
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1answer
299 views

Harmonic function.

The function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ given by $f(x) = \|x\|^{2-n}$, where $\|~\|$ denotes the Euclidean norm, is harmonic. This is just a simple computation. My question is: why ...
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Intuition on group homomorphisms

So I'm studying for finals now, and came across the idea of homomorphisms again. This is not a new idea for me at all, having seen them in groups, rings, fields ect. However, on reevaluating them I ...
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3answers
290 views

Why the principle of counting does not match with our common sense

Principle of counting says that "the number of odd integers, which is the same as the number of even integers, is also the same as the number of integers overall." This does not match with my ...
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4answers
534 views

What is the significance of multiplication (as distinct from addition) in algebra & ring theory?

In higher math, operators are defined over a set of objects; and these operators are usually denoted as addition and multiplication with a distribution rule. Assuming multiplication is not repeated ...
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375 views

Math Courses involving clever integration techniques

I am a third year undergraduate mathematics student. I learned some basic techniques for simplifying sums in high school algebra, but I have encountered some of the more interesting techniques in my ...
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520 views

Variance of binomial distribution

Why for $X\sim B(n,p)$ is $Var(X)=np(1-p)$? $Var(X)=\sum x_i^2 p_i -(\sum x_i p_i)^2=\sum_{r=0}^n r^2 \binom{n}{r}p^r(1-p)^{n-r}+( \sum_{r=0}^n r \binom{n}{r}p^r(1-p)^{n-r} )^2$ In my ...
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232 views

Cauchy's Theorem for Groups

Specifically: If $p$ is a prime divisor of the order of a finite group $G$, then there exists an element of order $p$ in $G$ So I'm looking for a little intuition behind this idea. I understand how ...
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1answer
674 views

Intuition why the volume and surface area of the unit sphere eventually decrease

The volume formula for a unit sphere, $$\frac{\pi^{n/2}}{\Gamma{(1 + n/2)}},$$ and the surface area formula, $$\frac{2\pi^{n/2}}{\Gamma{(n/2)}},$$ both attain maximum values for finite $n$. We can ...
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28answers
26k views

Too old to start math [closed]

I'm sorry if this question goes against the meta for posting questions - I attached all the "beware, this is a soft-question" tags I could. This is a question I've been asking myself now for some ...
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6answers
3k views

Combinatorial proofs: having a difficult time understanding how to write them out

Can someone explain how combinatorial proofs work? I've included an example questions that's been giving me a hard time. Any insight on the topic would be great. $$\sum_{k=1}^{n}k{n \choose k} = ...
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1answer
549 views

Multiple choice questions on relations and some of their properties

I'm confused about these 3 selected problems. I have the solutions for each, if necessary, but I'm much more interested in understanding the material. If anyone can offer a clear, concise, and ...