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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Limit Point of a Set

Definition. A point $x$ is a limit point of a set A if every $\varepsilon$-neighborhood of $x$ intersects the set A in some point other than $x$. I understand the definition in that $x$ is our limit ...
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74 views

Can we get the formula for $\prod\limits_{k=0}^n{(1+2^k)^2}$ in terms of $n$?

Can we get the formula for $\prod\limits_{k=0}^n{(1+2^k)^2}$ in terms of $n$?
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73 views

Intuitive idea of axiom of choice

I'm currently reading a book on set theory and it gives the following formulation of the axiom of choice: Let $X$ be a non-empty set. Then there is a function $g: ...
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5answers
125 views

Why don't taylor series represent the entire function?

Say, I have a continuos function that is infinitely differentiate on the interval $I$. It can then be written as a taylor series. However, taylor series aren't always completely equal to the function ...
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94 views

“Poissonization” and intuition

In a french book, "Calcul des probabilités" from Foata and Fuchs, I found this theorem, which they call "Poissonization". "Let $(I_k)_{k \in \mathbb{N}}$ be a sequence of independent variables with ...
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55 views

Intuitive basis of Mobius inversion?

If we're given $f(n)= \sum_{d|n}g\left(\frac{n}{d}\right),n \in \mathbb{N},$ then Mobius inversion gives $$g(n)=\sum_{d|n}\mu \left( d\right) f \left( \frac{n}{d}\right).$$ Also, the generalised ...
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1answer
49 views

What does it mean for a function to “quickly” approach $0$?

We can talk about how "quickly" an infinite series approaches $0$ by talking about an asymptotic bound on its terms - a series that is $O(1/x)$ converges more slowly than one that is $O(1/x^2)$, etc. ...
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39 views

Degree of a divisor for algebraically closed fields vs not closed ones

Suppose we have an algebraically closed field $F$ and we consider the projective space $\mathbb{P}^1$ over $F$. If we consider some divisor $D = n_P P + n_Q Q +n_s S$, then we say the degree of $D$ ...
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132 views

University-level books focusing on intuition?

I help some students with difficulties in Mathematics and Physics (especially math, physics, and engineering majors). While in high school they usually don't study, or are not interested, etc., in ...
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51 views

Seeking intuitive explanation of Clifford Algebra

Is there a simple intuitive graphical explanation of Clifford Algebra for the layman? Since Clifford Algebra is a Geometric Algebra, surely the best way to present those concepts is with graphical ...
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35 views

Intuition analysis-deconstruction-reconstruction.

The following question is a refinement of this question, which caused a lot of people to give answers that were missing the point entirely, probably because the question was not clear. Being human, ...
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1answer
46 views

Which math discipline should i learn to become familiar with rewriting equations?

In my self study of calculus, I've found that there are examples in the books i read where the author rewrites an equation or expression either as part of a logical step in a proof, or to simplify it ...
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102 views

Axiom of Choice and Zorn's Lemma Equivalence: some intuition

$$ \text{Axiom of Choice $\Rightarrow$ Zorn's Lemma } $$ $$\text{Axiom of Choice $\Leftarrow$ Zorn's Lemma } $$ I feel mathmatically immature to go through these proofs now. My quesiton therefore is: ...
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102 views

Books (and supporting material) that are useful in deconstructing one's intuition?

I recently came across the following problem from Paul Zeitz's book The Art and Craft of Problem Solving. Given the image below, can you find a way to connect corresponding blocks (i.e. A to A, B to ...
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56 views

what does the sine function tell you about an input

Intuitively, I'm trying to understand the significance of the input in a sine function. I'm currently, trying to develop intuition behind sinusoids and what the input tells you about the output and ...
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3answers
98 views

What is an irrotational vector field intuitively?

I understand that, by definition, a vector field is irrotational if the rotation is zero, but what does this intuitively mean? I have an idea of what it could physically be, which I've concluded by ...
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1answer
33 views

Is it possible to always get the optimal formula regardless of the derivation method?

Today I've tried to solve a geometric problem (collision point between two circles in a specific situation). I found a working solution but I'm not sure if it was optimal (maybe my solution took more ...
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81 views

How to explain the significance of $\pi$ to a child? [closed]

In honor of $\pi$ Day, I thought I would pose this question. How would you explain the significance of $\pi$ to a child of, say, 9 years of age? While that's certainly an age that is old enough to ...
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28 views

Determine weaker hypotheses for Evaluation Fundamental Theorem of Calculus (Abbott p 202 T7.5.3)

(p 200 T7.5.1) If $f:[a,b] \to R$ is integrable and $F:[a,b] \to R$ satisfies $f(x)= f'(x) \; \forall x \in [a,b]$, then If $g$ is integrable on $[a,b]$, then $\int_a^b f = F(b) - F(a)$. ...
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109 views

What does Riemann-Stieltjes integral calculate when $\alpha(x) \neq x$?

When we get Riemann-Stieltjes integral becomes standard Riemann integral which calculates area under the curve. We have that $$ s(f,\alpha,P)=\sum_{k=1}^nm_k\Delta\alpha_k \ \text{ and }\ ...
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1answer
18 views

Population average age decreases with births AND deaths (kind of)?

When a baby is born, it's easy to see the average age of the population decreases. Intuitively, therefore, when a person dies, the average age of the population must increase to compensate. However, ...
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Visualize $f(b) - f(a)$ withOUT Mean Value Theorem (Stewart p 282 figure 4) [closed]

How can we visualize $\color{green}{f(b) - f(a)}$ withOUT the Mean Value Theorem or rewriting it as $\color{dodgerblue}{\dfrac{ f(b) - f(a) }{ b - a }} $ ? I'm trying to understand ...
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2answers
81 views

Modus operandi for proving Evaluation Fundamental Theorem of Calculus (Abbott p 200, Spivak p 272 T14.2)

1. How can we presage to use Mean Value Theorem to start the proof? 2. Mean Value Theorem engenders a point in an open interval. Shouldn't this be $x_i \in (t_{i - 1}, t_i) $? After ...
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Choices of epsilons in proof : $(b_n) \to b$ implies $\left\{\frac{1}{b_n}\right\} \to \frac{1}{b}$ (Abbott pp 47 T2.3.3.iv) [closed]

Original became long, ergo I ask anew. The trick is to look far enough out into the sequence $(b_n)$ so that the terms are closer to b than they are to 0. Consider the particular value $e_0 = |b|/2$. ...
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Is this a counterexample to “continuous function…can be drawn without lifting” ? (Abbott P111 exm4.3.6)

I'm au courant with http://math.stackexchange.com/a/288133 and http://math.stackexchange.com/a/422001. They're both Abbott P111 exm 4.3.6 which proves "a continuous function is sometimes described, ...
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18 views

Relation of common divisors leading to integer results

When dividing an integer $a$ by 3 and 7 both results in an integer answer, I intuitively feel that $a/A$ with $A=21$ would also be integer, which seems related to the fact that $3\times7=21$. ...
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66 views

What's the intuitive explanation that the volume of a solid is $\frac{1}{3} A_{base} h$?

I can see why the area of a triangle is $A = \frac{1}{2} bh$ because it's half of a rectangle with sides $b$ and $h$, but I fail to see the intuitive explanation for this general volume formula. (Yes, ...
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189 views

Why do statisticians like “$n-1$” instead of “$n$”?

Does anyone have an intuitive explanation (no formulas, just words! :D) about the "$n-1$" instead of "$n$" in the unbiased variance estimator $$S_n^2 = \dfrac{\sum\limits_{i = 1}^n ...
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58 views

Derivative of exponential functions

Can anyone present an intuitive reason for why the derivatives of exponential functions, lets say, as apposed to polynomials, grow more rapidly than the functions themselves? i.e. $$ y = e^{x^2}\\ ...
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When is a series expansion related to its derivative by a polynomial equation?

Is there some common theory behind the following two examples? Example 1. Let $p(t) = \sum_{n \geq 0} (-1)^k t^{2k}/(2k)!$, and $x = p(t), y = p'(t)$. Then $x^2 + y^2 = 1$ identically. Example 2. ...
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585 views

Why are we interested in irreducible representation but not faithful representation?

I am reading some materials of representation theory (of a group). The motivation of representation theory is to represent (by a homomorphism $h: G \to GL(V)$, from the group $G$ to a vector space ...
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1answer
50 views

How to presage Prove by Contrapositive, for Sequential Characterizations of Limit and Continuity? (Abbott pp 106 t4.2.3, 110 t4.3.2)

Dafinguzman answered consummately this question initially but it became too long. I want to question for different beliefs. 1. $(ii) \implies (i)$ in both Theorems 4.12 and 4.19 posit sequences ...
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1answer
57 views

Intuition and counterexamples for higher-order derivative test

In the higher-order test we keep differentiating a function till we find the n'th derivative (n being even) to be greater than or less than zero thereby identifying it as a minimum or maximum. My two ...
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Geometry of $k$-forms and $k$-vectors

In this question I was trying to see why $k$-forms are selected as the way to generalize vector calculus rather than $k$-vectors and a comment providing links to other questions made me end up with ...
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How would one arrive at the formulas for divergence and curl?

It has been some years since I've taken multivariable calculus now, but there's something I really never understood: how people would discover the expressions for divergence and curl. I mean, the ...
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Intuition for $\inf(AB) = \inf(A)\sup(B)$. Difference for sets and functions? (Abbott pp 199 q7.4.5)

1. What's the intuition for $\inf(AB) = \inf(A)\sup(B)$? Figure please? I know I must posit $A,B \subseteq R$ as bounded sets. If they're unbounded, $\sup$ doesn't exist. I believe $\inf(AB) = ...
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So we don't need to choose delta, epsilon, or $N \in \mathbb{N}$ in delta-epsilon or sequence convergence proofs?

(http://math.stackexchange.com/a/700667/85079) I would write the proof with all my bounds $\eta$ and then choose $\eta$ to make the conclusion match the arbitrary $\epsilon$. ...
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1answer
73 views

If $g \ge 0$ is continuous on $[a,b]$ and $g(x_0) > 0$ then $\int^{b}_a g > 0$ (Abbott pp 199 q7.4.4c)

True or False. If $g \ge 0$ is continuous on $[a,b]$ and $g(x_0) > 0$ for $\ge 1$ point $x_0 \in [a,b]$, then $\int^{b}_a g > 0.$ 1. Need determine if true or false. Ergo do we need ...
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93 views

If $\int^{b}_a f > 0$ then there is some interval and $\delta > 0$ on which $f(x) \ge \delta$ (Abbott pp 199 q7.4.4d)

True or False. If $\int^{b}_a f > 0$, then $\exists \; [c,d] \subseteq [a,b]$ and $\delta > 0$ such that $f(x) \ge \delta$ for all $x \in [c,d]$. 1. We need to determine if true or false. ...
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Differential forms turn infinitesimal stuff rigorous?

First of all, I know that infinitesimals are not well defined in standard analysis and they have rigorously nothing to do with differential forms. My doubt is on the intuition between one relationship ...
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Proof. sup{ f(x) } - inf{ f(x) } $\ge$ sup{ |f(x)| } - inf{ |f(x)| } (Abbott pp 198 q7.4.1)

Let f be a bounded function on a set A, and set $S = \sup\{f(x) : x ∈ A\}, I = \inf \{f(x) : x ∈ A\},$ $S' = \sup\{|f(x)| : x ∈ A\}, I' = \inf \{|f(x)| : x ∈ A\}.$ Show that $S - I ≥ S' - I'$. ...
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When does one proof of one direction of an If and Only If proof suffice?

Would someone please explain when this is admissible (please expound on $\color{darkred}{sometimes}$)? In advance of starting an Iff proof, how would one divine/previse if this convenience (of a ...
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Within If and Only If Proofs, why can the proof for one direction be easier than the other?

For $ P \iff Q$, my initial sentiment is that because P and Q are equivalent, the total of two proofs (one for each direction) should entail the equivalent level of "difficulty" or "exertion", as well ...
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Intuition. 3 Equivalences of Riemann integrability (Abbott pp 191 q7.2.4)

Not questioning about proofs. For this entire question, posit $f$ is a bounded function on $[a,b]$. ♪ f is integrable signifies $\inf \{ \, U(f, P) \, \} = \sup \{ \, L(f, P) \, \}$ where $P$ is any ...
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Intuition. Cauchy criteron for Riemann integrability (Spivak pp 239, S. Abbott pp 189 thm 7.2.8)

1. Why $\inf U(f,P') \le U(f, P)$ and $\sup L(f, P') \ge L(f,P) $? I tried to research but I can't find where Spivak defined it $P'$? 2. Why are there two partitions P', P''? Not the same? ...
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107 views

Easiest proof $\sup A + \sup B ≤ \sup(A + B).$ No epsilons, sequences. (S.A. pp 18 q1.3.9d)

(question 2. http://webcache.googleusercontent.com/search?q=cache:DohoRC3-bU8J:www.maths.usyd.edu.au/u/UG/IM/MATH2962/r/PDF/tut01s.pdf) Essay By definition of A + B and sup(A + B), for all a ∈ A and ...
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If sup A < sup B, there exists an element b ∈ B that's an upper bound for A. (S.A. pp 18 q1.3.8)

My Figure: By definition of $\sup B$, $\sup B$ is an upper bound for $B$. Set $e = \sup B − \sup A > 0$. By Lemma 1.3.7, there exists an element $b ∈ B$ satisfying $\begin{align} & \sup B − ...
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39 views

Intuition behind failure rate.

The failure rate of the exponential distribution is a constant, $\lambda$, as the exponential distribution is memoryless. So say we have that $\lambda = \frac{1}{10}$. What is that telling us? The ...
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94 views

Deep Understanding of Independence of Probabilities

I really want to have a deep understanding of the independent probabilities of two events. That means to me that I just do not want to use and know the definition. I want to fully understand the why. ...
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24 views

Can we find the $n$ that minimizes $\log{(c/n)}\log{(v)}(k^n v^2)^{\log{(c/n)\log{(v)}}}$?

Can we find the $n$ that minimizes: $$\log{(c/n)}\log{(v)}(k^n v^2)^{\log{(c/n)\log{(v)}}}$$ ...Here $c$, $n$, $k$ and $v$ are all naturals. I tried taking the derivative, and then setting it equal ...