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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Conic sections and common functions

Is there a intuitive proof/reason of why plots of some common functions like y=x^2 are shaped like cross sections of a seemingly unrelated 3D object like a cone? ...
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30 views

Intuition behind combinatorics problems?

I have a hard time understanding when a problem is a combination or permutation. Especially when using the multinomial theorem. For example: three boxes numbered 1,2 and 3 for k = 1 2 and 3, box k ...
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Conceptual question about independence and stopping times

Let $\{X_i\}_{i\in \mathbb{N}}$ be a sequence of i.i.d. random variables with common distribution function $\mu$. Consider a property $A$, such that $\mu(A)>0$. Define $T$ to be stopping time ...
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Is it possible to derive the sum of the tangent or cotangent from this?

So we can suppose that we can find the finite sum $$\sum_{x=a}^b{ \tan{(x)} + \cot{(x)} } \tag{1}$$ for essentially all integer values of $x$. I'm wondering, can we derive either: $$\sum_{x=a}^b{ ...
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What does it really mean when we say that the probability of something is zero? [duplicate]

Conventionally, people will say a probability of zero is equivalent as saying that the event is impossible. But when we look at the probability from a mathematics perspective, probability is defined ...
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Intuitively, how do you explain the concept of Flux?

Lately in my physics and mathematics classes, I've come across the concept of Flux. And although I've been able to define them mathematically and figure out how to use them. I'm still not entirely ...
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69 views

Main goal of algebraic topology and progress towards the main goal… [closed]

In algebraic topology we use algebraic structure such as group to classify some topological spaces. To do that mathematician invented homotopy group,homology,co-homology etc.One can apply these tools ...
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counting occurence of subgraphs by counting their occurence in larger subgraphs

I have a mental block in fully understanding the following notion. Let $G$ be a graph of order $n$ and $H$ a fixed small graph of order $k \le n$. Suppose that there are $d$ copies of $H$ as an ...
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What is the d in the formula of a plane in $ R^3$

In algebra the formula for a line is $y=ax+b$ the $b$ moves the position of the line up and down the y axis. The formula for a plane is given to me as $ax+by+cz+d=0$ the $d$ must move the position of ...
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Why are matrices written as such?

Another thread has talked about the purpose of a matrix. Dr. Math roughly summarized it as: A matrix is just a compact notation, which allows you to specify several linear equations at once ...
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246 views

What do rationals represent?

While learning about the construction of number systems, I realized that I had many misunderstandings of crucial concepts which I was learning intuitively. I recently learned about the construction of ...
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Not clear on what we mean with numbers with infinite digits

I am confused on a rather simplistic question. 1/3 = 0.333333333333 to infinity. So it has infinite digits. How is it possible to multiply such a number with another one and get a finite number? 6/3 = ...
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41 views

Understanding completness of a metric space

I have a question about completeness of a metric space. The lecture notes that I am reading on my own says the following. PROPOSITION  1. Let X be a complete metric space. Let Y be a closed subset ...
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269 views

How do you go about formalizing a concept?

I am reading Godel Escher Bach. I love it. In the first few chapters, the author shows what a formal system is and gives examples that eventually lead to a typographical formal system of strings that ...
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Intuition behind the definition of Measurable Sets

I started studying "Measure Theory and Integration" and went through the first section which talks about Lebesgue Outer Measure of a set. All was well until I started with the second section which ...
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38 views

On estimating monthly credit card payment amounts (some pragmatic constraints inside)

Right off the bat, I do hope this question doesn't attract a bunch of derisive comments about my personal affairs. I give the lengthy personal anecdote because I don't have the mathematical training ...
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2answers
52 views

What is the line integral in higher dimensions?

Given a function $\ f:\mathbb{R}^2\to\mathbb{R}$ and some curve $\ \gamma:[a,b]\to\mathbb{R}^2$ it is my understanding that the integral of $\ f$ over $\ \gamma$ is the area of the region "between" $\ ...
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33 views

Which functions lies in $H^{loc}_{s}\setminus H_{s}$?

We put $H^{s}=$The Sobolev spaces, and $H^{loc}_{s}=$The localized Sobolev spaces. We note that, $H_{s}\subset H^{loc}_{s};$ also this. Bit roughly speaking, I am interested in knowing that how big ...
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65 views

Differential Geometry Intuition Question

Apologies if I get the notation wrong. Still learning this stuff. Suppose I have a 2 dimensional Riemannian manifold $\mathcal{M}$ that is covered by a single chart: $\phi: \mathcal{M} \rightarrow ...
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Can we characterize the space of functions which is real analytic but not real entire?

A complex valued function $F,$ defined on an open set $E$ in the plane $\mathbb R^{2}$, is said to be real-analytic in $E$ if to every point $(s_{0}, t_{0})$ in there corresponds an expansion with ...
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163 views

Why should the generalization of a 'sequence' be called a 'net'?

The title says it all, really. Reading through Reed & Simon's book on functional analysis, I have now reached the chapter on topological spaces, and the notion of a net is introduced there to ...
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82 views

Analogy of ideals with Normal subgroups in groups.

I've started with Ideals in ring theory but still not comfortable with the analogy it has with normal subgroups in group theory.Like we can visualize normal subgroups as Is there some good intutive ...
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23 views

About a Property of maximal solutions of separable ODE's $y'=g(x)h(y)$ for locally Lipschitz $h : U\to\mathbb R$, $U$ open

Theorem: Let $\varphi : (a,b) \to \mathbb R$ be a maximal solution of the IVP $$ y'(x) = g(x) \cdot h(y(x)), \quad y(x_0) = y_0 \quad (1) $$ with continuous functions $g : I \to \mathbb R$ and $h : U ...
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Positive Linear Transformations: What good for?

Positivity is a concept appearing quite frequently in the study of algebras and its related spectral theory. Positive elements naturally give rise to an ordering and therefore allows to construct ...
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59 views

Why must closest approach occur when relative velocity is perpendicular to motion?

The first part i) I can solve correctly, but I need some advice and intuition on how to solve the second part ii). Here is the mark-scheme for the question: But for part ii) I do not understand ...
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48 views

What is a geometric shape?

I thought that the concept of "gemoetric shape" is clear enough - squares, ellipses, triangles, you know. But then I found several papers, such as this one, which define "shape" as "an ...
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Why does $n$ have to be a perfect square for me to construct an equilateral triangle out of equal smaller ones?

If I have $n$ unit squares and want to build a bigger square out of the ones I already have, it is obvious that $n$ itself has to be a perfect square. But after doing some elementary math it turned ...
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66 views

Intuition behind the definition of linear transformation

I have studied that given vector spaces $V_1$ and $V_2$, a function $T:V_1 \rightarrow V_2$ is called a linear transformation of $V_1$ into $V_2$, if following two properties are true for all $u, v ...
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35 views

At the instant of release of an object from rest. Is the only force that can act its weight? [closed]

This is the third question from a mechanics exam past paper: I can do parts i) and ii) but for iii) in finding the angular acceleration, i used $C=I\alpha$, where $C$ is the applied couple or ...
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40 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 ...
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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 ...
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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 ...
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54 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 ...
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25 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 ...
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65 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 ...
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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 ...
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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 ...
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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 ...
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36 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. $ ...
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108 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.
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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 ...
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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 ...
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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 ...
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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] ...
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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 ...
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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 ...
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37 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 ...
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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 $-1 + \sqrt{2a-1}$ = ...
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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 $$ ...
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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} ...