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.

learn more… | top users | synonyms (1)

0
votes
2answers
46 views

Why does resolving forces in one direction give a completely different answer to resolving the opposite way?

I can solve parts i), ii) and am able to show that $R=0$ for part iii). In this question $g$ is the acceleration of free fall taken to be $9.8$ Using Newtons 2nd law [$F=ma$] for the last part I ...
2
votes
3answers
77 views

Plausibility vs Probability

http://whatho.in/2013/plausibility-versus-probability/ refers to pp 155-156 of 533 of Thinking, Fast and Slow by Daniel Kahneman. I'll use one of Kahneman's other questions from p 156: A ...
1
vote
2answers
59 views

Interpretation of partial derivatives of vertical coordinate with respect to $x$ and time

My question is from my lecturers notes, this is what he wrote and I don't know what he is on about : What is a physical meaning of partial derivatives of $y(x,t)$? $y_x(x,t)$ is the rate of change of ...
0
votes
1answer
59 views

$1$ is not congruent because of Fermat's Last Theorem?

I would like someone to explain something I did not understand. I was reading a page called "nuking the mosquito" where they give very complex proofs for very simple results. The proof I want to talk ...
1
vote
1answer
31 views

Elements in the same coset and the Cayley Diagram

A question from Visual group theory, by Nathan Carter. In a Cayley diagram, if $aH$ is a coset of a subgroup $H$ of a group $G$ and $b$ belongs to $aH$, why is it that every node that can be reached ...
3
votes
1answer
66 views

Geometric idea behind equations of the form $|x-a|\pm|x-b|=c$

So let's say I want to solve $$|x-a|\pm|x-b|=c$$ Using the classic multiple cases approach, one can show that the solutions are given by $$x=\frac{a+b\pm c}2 $$ But how can one make sense of this ...
0
votes
3answers
85 views

Why does solving $\int \frac{v}{9.8-0.0025v^2}\mathrm{d}v=\int1{d}x$ for $v^2$ in terms of $x$ produce 2 completely different answers?

In this question $g=9.8$ (acceleration of free fall). You are also given that when $x=0$ $v=0$. My answer is $v^2=400g(1-e^\frac{x}{200})$. I obtained it by integrating both sides so that ...
3
votes
2answers
70 views

Intuition for the compactness of real projective space $\mathbb{R}\mathbb{P}^n$.

I want to have an intuition for why the $n$-dimensional real projective space defined as $$\mathbb{R}\mathbb{P}^n:=\mbox{set of 1-dimensional subspaces of }\mathbb{R}^{n+1}$$ is compact. I don't see ...
3
votes
1answer
38 views

What does local space of a given Banach space says intuitively?

We put, $\mathcal{D}(\mathbb R)=$ The space of $C^{\infty}-$ functions on $\mathbb R$ with compact support Example: For instance bump function is in $\mathcal{D}(\mathbb R)$ Let $E$ is a Banach ...
1
vote
1answer
80 views

The meaning of the connection between power spectral density and auto correlation

I know that if we have a signal $x(t)$, then its Fourier transform would be the signal in the frequency space, which I understand to be how much of each frequency exists in the x(t) signal. $ ...
2
votes
3answers
109 views

Show that $q^4+2pq^2 +p^2 = 2pq -(pq)^2 -1$ becomes $p^3+q^3+3pq-1=0$.

I know that these two are exactly the same equation but I can't seem to prove it. You are also given that $p+q=1$. This is a follow up from a similar question.
4
votes
1answer
66 views

By plugging $p=1-q$, into the $3$ equations show that $x=y=z$

By plugging $p=1-q$, into the 3 equations: $$\begin{cases} z=py+qx \\ x=pz+qy \\ y=px+qz \end{cases}$$ show that $\boxed{x=y=z}$ This is from the final part of question 7 in this STEP paper, and is ...
2
votes
1answer
42 views

Computing of the Gamma Function

I have stumbled upon Gamma functions when dealing with Gamma distributions on my studies with basic statistics. However, I have not understood how its computation expands factorials to real and ...
2
votes
3answers
50 views

How to determine value from willingness to pay?

I use the British pounds symbol instead of dollars because $ conflicts with Mathjax. Source: p 296, The Legal Analyst, Ward Farnsworth "... one time in a thousand we do lose the film; if you’re ...
0
votes
1answer
59 views

Series problem:

Parts i) & ii) I can solve. For part iii) I get $z=py+qx$ [For $n=0$] $x=pz+qy$ [For $n=1$] $y=px+qz$ [For $n=2$] leading to $(1-pq)x=(q^2+p)z$ [1] $(1-pq)z=(q^2+p)y$ [2] ...
0
votes
2answers
74 views

Sequential Algebraic Problem:

The first part i) I can do. For part ii) this is how far I can get: If n is odd then $y=px+qy$ If n is even then $x=py+qx$ After some rearranging i end up with $y(1-q)=px$ & $x(1-q)=py$ and ...
1
vote
4answers
99 views

Explaining multiplication of fractions

The best way I've been able to describe multiplication is as $$ a\times b = \sum^a_{i=1} b$$ But my definition does not account for things such as $2.99792458\times8.987551787$ and ...
0
votes
1answer
51 views

Everyday life examples of hyperbolic rotations

I'm trying to find some intuition in Lorentz transformations. I understand that they are basically rotations by imaginary angle of vector of the form $\{ict,x\}$ (for $1+1$ space-time dimensions), and ...
1
vote
1answer
19 views

Understanding Inequalities.

If $(\displaystyle\frac{2(a-1)}{a+1}\lt\ln a\lt\frac{2(a-1)}{2+\ln a})$ equals $(\displaystyle\frac{2(a-1)}{a+1}\lt\ln a \lt -1 + \sqrt{2a-1})$. Does that mean that $-1 + \sqrt{2a-1}$ = ...
1
vote
1answer
24 views

Show that $ \frac{2+\ln a }{2}\lt\frac{a-1}{\ln a} \lt \frac{1+a}{2}$ becomes $ \frac{2(a-1)}{a+1}\lt\ln a \lt -1 + \sqrt{2a-1}$

Show that $\displaystyle \frac{2+\ln a }{2}\lt\frac{a-1}{\ln a} \lt \frac{1+a}{2}$ becomes $\displaystyle \frac{2(a-1)}{a+1}\lt\ln a \lt -1 + \sqrt{2a-1}$ The closest I can get is $$ ...
0
votes
1answer
43 views

Trigonometric series problem: finding a second valid solution.

Given that I can do part of this question so here goes: Substituting $\theta=\frac{1\pi}{11}$ into LHS of given expression gives $$\cos\frac{1\pi}{11} + \cos\frac{2\pi}{11} + \cos\frac{3\pi}{11} ...
0
votes
2answers
43 views

Problem involving summing exponential series:

I can show the first part (i) (a), but the second part (b) i think it should be $S=\infty$ since the denominator is zero with that value of $\theta$. However, this is not the answer, any ideas? ...
0
votes
1answer
55 views

Using De-moivres to solve the following problem:

Part (i) I can solve and understand that the solutions are $Z=e^\frac{2ki\pi}{5}$ for $k = 0,1,2,3,4$ Its the part (ii) I cannot understand. Could someone kindly give me a ...
0
votes
1answer
56 views

Using De Moivre's theorem with relation to the argument of a complex number

Given that $Z^4 = 64(\cos\pi+ i\sin\pi)= 64(-1+0i) = -64$ I understand that the argument [$arg(Z^4)$] is $\pi$, now if instead given the form $Z^4 =64(-1+0i)$ and I desired to find the argument ...
3
votes
0answers
76 views

Distinction between nowhere monotone and nowhere differentiable

It is known that all functions that are continuous and nowhere differentiable are also nowhere monotone but that there is a function that is everywhere differentiable but nowhere monotone. I have ...
2
votes
1answer
66 views

Intuition concerning Riemann Sums

I have just started learning integrals, and I want to know the following: In the definition of a riemann integral, it states that the interval that the integral is to be evaluated, is partitioned ...
3
votes
2answers
90 views

The semidirect product as a deformation of the direct product

The way I think of the semidirect product is as a "deformation" of the direct product. Is there a way of making this intuition precise? Perhaps using some certain (co-) homology theory of groups?
0
votes
1answer
21 views

What nontrivial operations exhibit $\text{op}(f(x)) + \text{op}(f(x+1)) = \text{op}(f(x) + f(x+1))$?

What nontrivial operations exhibit $\text{op}(f(x)) + \text{op}(f(x+1)) = \text{op}(f(x) + f(x+1))$? For example, I know that summation, integration, and their inverses all exhibit this property. To ...
1
vote
6answers
184 views

Why is a raised to the power of Zero is 1? [duplicate]

Why is $a^0=1$ $\forall a \in Z, a\neq0$. I understand $2^4=2\cdot2\cdot2\cdot2$ How can I express $a^0$. I am serious about the practical proof of this
1
vote
2answers
43 views

Show elementarily that $\lim_{R\to\infty}\int_{\Gamma_1} \frac{e^{iz}}{z} = 0$

Context: I am trying to show that $\int_0^\infty x^{-1}\sin x dx = \frac{\pi}{2}$ using complex analysis, by first integrating $\oint_{\Gamma} z^{-1}e^{iz}$, where $\Gamma$ is a closed contour ...
1
vote
0answers
50 views

Why is it called a primitive root?

I am looking for a paper or reference that explains why primitive roots are called primitive roots. I know what they are but was wondering if there was a reason?
1
vote
4answers
180 views

What if we removed all the irrational numbers from the real number line? [closed]

(1) Imagine drawing the real number line and tippexing out all the irrational numbers. What would the resulting shape look like- would there even be a line? (2) And what about if we ...
10
votes
1answer
140 views

Geometric Intuition for Dihedral Group Automorphisms

I noticed the other day that the automorphism group of the dihedral group $D_{2n}$ (of order $2n$) is $\operatorname{Aff}(\mathbb Z/n\mathbb Z)$, the group of affine transformations of the $\mathbb ...
0
votes
1answer
41 views

Gambling interpretation of conditional probability

In Billingsley, when defining conditional probability the following property has been given a gambling interpretation : $$ \int_G P[A||\mathscr{G}]dP = P(A \cap G), G \in \mathscr{G} $$ where at ...
2
votes
2answers
39 views

Single variable justification for the multivariate chain rule.

I $\def\d{\mathrm d}\def\p{\partial}$am going to ask everyone to switch their paradigms to that of the real line. I am looking for a "lowbrow" explanation of the following phenomena. I am talking ...
19
votes
5answers
2k views

Why isn't the Cantor Set contradictory?

So you start with a 1-dimensional stick, remove the middle third of it, leaving 2 pieces. From each of these 2 pieces, remove the middle third. Etc. Whatever is left at the end of infinitely many ...
1
vote
3answers
234 views

Expressing the probability density function of $Ax$ in terms of the pdf of $x$

I understand that, for example, you might have a density function which measures the probability of observing an outcome in a certain interval measured in feet, but someone wishes to use meters ...
4
votes
5answers
129 views

Getting an intuitive feel for induced representations

I'm reading about induced representations for research. Particularly, I'm trying to get a firm grasp on the finite group case before venturing on to the locally compact case. I've been looking at ...
0
votes
0answers
73 views

Differentiable curves that are not smooth

We call a curve admitting a parameterization $t\to z(t)$, $t\in[0,1]$ differentiable if the vector function $z$ is differentiable. We call the curve smooth if it is differentiable and its derivative ...
4
votes
2answers
77 views

Intuitive interpretation of $\frac{\partial S(a,t)}{\partial t} = -\frac{\partial S(a,t)}{\partial a}$

I'm trying to visualize what the following equation is saying: $$\frac{\partial S(a,t)}{\partial t} = -\frac{\partial S(a,t)}{\partial a}$$ where $S$ is a probability-density, but I think you can ...
1
vote
1answer
28 views

Understanding arguments to functions in $\mathbb R^n$

Example of two theorems I have problems with: Mean value theorem: $U\subseteq\mathbb R^n$ open, $f:U\to\mathbb R^m$continuously differentiable, $x\in U$, $\xi\in\mathbb R^n$ such that $x+t\xi\in U$ ...
5
votes
2answers
105 views

$2^{1/4} \times 4^{1/8} \times 8^{1/16} \times 16^{1/32} \times \ldots\to2$

$2^{1/4} \times 4^{1/8} \times 8^{1/16} \times 16^{1/32} \times \ldots\to2$ How can I explain this to a school student who doesn't know what a limit is?
0
votes
2answers
111 views

Why do we denote $S^1$ for the the unit circle and $S^2$ for unit sphere?

Maybe a quite easy question. Why is $S^1$ the unit circle and $S^2$ is the unit sphere? Also why is $S^1\times S^1$ a torus? It does not seem that they have anything in common, do they?
3
votes
1answer
78 views

Intuition behind prism operators to prove homotopy invariance of homology

I'm trying to understand the proof of homotopy invariance of induced maps on homology. However, I do not really understand the intuition behind this proof and especially what the prism operators (as ...
1
vote
3answers
92 views

Can a statement in FOL be equivalent to two separate independent statements?

This may seem like a dumb question, and it certainly seems dumb to me asking it, but I'm running into a contradiction. I'm looking at the problem of finding a statement $\phi$ such that $\psi$ and ...
0
votes
1answer
29 views

Meaning of “Identify a set with another set” in group theory

There is a exercise problem that asks "Identify a set with another set ". I don't understand what I should do. Do I need to establish a bijection between them? Thanks EDIT-I: Actual question: G is a ...
1
vote
1answer
91 views

Arrows-only implication & disjunction in $\mathbf{Set}.$

Just before the truth-arrows in a topos subsection of Goldblatt's "Topoi: A Categorial Analysis of logic," descriptions of the truth functions $\Rightarrow$ and $\smallsmile$ are given in ...
5
votes
6answers
202 views

The physical meaning of ${n \choose k} = {n \choose n-k}$.

They say that $${n \choose k}={n \choose n-k}.$$ Can someone explain its physical meaning? Among many problems that use this proof, here is an example: The english alphabet has $26$ letters ...
2
votes
7answers
730 views

What are the most important functions every mathematician should know? [closed]

I am an undergrad in math and was wondering, what are for you the most important functions every mathematician should know? At the moment I think ...
0
votes
1answer
60 views

Factoring $x^4 -8a^2x^2 -48a^4 -8bx^3 - 32a^2 bx +16b^2x^2 +64a^2b^2$

The subject line pretty much says it all. In my geometry class today, the following equation came up: $$x^4 -8a^2x^2 -48a^4 -8bx^3 - 32a^2 bx +16b^2x^2 +64a^2b^2 = 0$$ Specifically, it was in the ...