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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Can we perform this operation on block matrices?

We have a block matrix: $$ \left[\begin{array}{c|c|c} A & 0 & 0 \\ \hline 0 & B & 0 \\ \hline 0 & 0 & C \end{array}\right] $$ Here $A$, $B$ and $C$ are all permutation ...
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Why is compactness so important?

I've read many times that 'compactness' is such an extremely important and useful concept, though it's still not very apparent why. The only theorems I've seen concerning it are the Heine-Borel ...
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238 views

Intuition behind the definition of a measurable set

This week I saw the definition of a measurable set for an outer measure. Let $\mu^*$ be an outer measure on a set $X$. We call $A \subseteq X$ measurable if $$\mu^*(E) = \mu^*(A\cap E) + ...
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561 views

Intuition for Absorption and Distributive Laws in Elementary Logic

$P ∧ (Q ∨ R) \equiv (P ∧ Q) ∨ (P ∧ R) \tag{Distributive Law 1}$ $P ∨ (Q ∧ R) \equiv (P ∨ Q) ∧ (P ∨ R) \tag{Distributive Law 2}$ $P ∨ (P ∧ Q) \equiv P \tag{Absorption Law 1}$ $P ∧ (P ∨ Q) ...
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59 views

Clarification on Formal Definition of Functions

In my Abstract Algebra class, the professor defined a restriction as Given $ X\xrightarrow{f} Y $ and a non-void subset $S$ of $X$ define $ f \mid S\xrightarrow{S} Y $ by $(f \mid S )(s) = f(s), ...
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220 views

Matrices Intuition

I am currently studying matrix algebra. The axioms and theorems of this form of algebra are a bit different from the high school algebra I did. However one knows that one is dealing with real numbers ...
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345 views

Intuition - The Shortest Curve Between Two Points is a Line

The references below aver that the following is not crudely trivial: The shortest curve between two points is a (straight) line. An elementary school teacher construed it as follows ...
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107 views

Monty Hall vs. Card Example

In class, while illustrating the topic of conditional probability, my professor presented the following card example: You have 3 cards that have been randomly shuffled: card1, card2, and card3. ...
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166 views

Why does it make sense to pair existence and uniqueness?

Edit. September 8th, 2013. I've added another another section to the question, which should hopefully constrain its scope. Please see below the horizontal line!! Lets write the statement 'there is at ...
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1answer
57 views

Finding Similar Sequences

Can we find two sequences: $$\{a (b^0), a (b^1), a (b^2), a (b^3), \dots, a(b^n)\} \bmod p_1$$ $$\{c (d^0), c (d^1), c (d^2), c (d^3), \dots, c(d^n)\} \bmod p_2$$ that differ by only one number? ...
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165 views

Geometric Interpretation: Parallel forms are harmonic

Let $(M,g)$ be a Riemannian manifold. The canonical volume form $\mu=\sqrt{\det g_{ij}}\mathrm{d}x^1\wedge\dots\wedge\mathrm{d}x^m$ is parallel w.r.t. the induced Levi-Civita conection $\nabla$ ...
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925 views

Do all vectors have direction and magnitude?

I go by Vector. It's a mathematical term, represented by an arrow with both direction and magnitude. Vector! That's me, because I commit crimes with both direction and magnitude. Oh yeah! For ...
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507 views

Explain a surprisingly simple optimization result

The following optimization problem came to my attention as an idealization of the silly browser game Cookie Clicker, but is representative of a range of strategy games: You have an initial ...
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1answer
286 views

Foundation of category theory

In the first pages of "category theory for the working mathematician" Saunders claims that category can be introduced, without set theory, as objects and arrows without some "operations" satisfying ...
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149 views

Wilson's theorem intuition

Wilson's Theorem: $p$ is prime $\iff$ $(p-1)!\equiv -1\mod p$ I can use Wilson's theorem in questions, and I can follow the proof whereby factors of $(p-1)!$ are paired up with their (mod $p$) ...
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91 views

What's the intuition behind this definition of ordered pair in the $\lambda$-calculus?

On this page, we have the following definitions. pair = λabf.fab first = λp.p(λab.a) second = λp.p(λab.b) So I tried computing "first (pair a b)," and sure ...
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178 views

Are statements like “Every time I've done X, Y has happened” (vacuously) true if I've never done X?

I've recently been wondering about vacuous truths. I know a statement like "I've never been beaten in a race" is true if I've never been in a race, but what I'm wondering is if the following ...
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1answer
58 views

$|f(x)|<|g(x)|$ and $\int g(x)<\infty\Rightarrow\ \int f(x)<\infty$

Let f,g continious functions on $[0,\infty)$ s.t $\forall x\in[0,\infty), |f(x)|\le|g(x)|$. Prove or give counterexample thtat if $\int_ 0^\infty g(x)dx<\infty$ then $\int_0^\infty ...
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327 views

Confused about differentiation

I'm new to calculus and have been taught that $\displaystyle \frac{dy}{dx}$ is the rate of change of y with respect to x. Does $\displaystyle \frac{dy}{dx}$ show how much the variable y changes as x ...
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6answers
414 views

'If you neglect your homework, then you’ll fail': Can it be expressed as $P \vee \lnot Q $?

Source: p 46, How to Prove It, by Daniel Velleman Please beware that although the author writes the original apodosis as 'You’ll fail the course', I shorten it to 'You'll fail', for convenience. ...
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67 views

Expand $\left(1-\frac2x\right)^{\frac12}$ to find a value of $\sqrt{99}$ and $\sqrt{101}$

Given that $|x|\gt2$ find the first four terms in the series expansion of $\left(1-\frac2x\right)^{\frac12}$ in descending powers of $x$. By taking $x$ = 200 use the series to find a value of ...
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195 views

Definition of weak solutions from geometrical point of view

Why are weak solutions defined like: A function $u \in H^1(\Omega)$ is a weak solution of $$ Lu:=\operatorname{div}(A\nabla u)+b\cdot\nabla u+cu=f+\operatorname{div}F, \;\text{in } \Omega $$ if ...
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387 views

A basic question in the definition of limit point

For any subset of $R$ with the usual distance metric, any point inside it is a limit point. Only when the set is discrete there may be a point inside it which is not a limit point. Is this correct ? ...
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654 views

Visual intuition partial/directional derivative

I've had some trouble with the (visual) intuition behind the directional derivatives so I decided to take a step back and look up the visual intuition behind partial derivatives, which I think I do ...
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1answer
57 views

How many chess games are required to be played if 9 players win 2 games against each other player?

How many chess games are required to be played if 9 players win 2 games against each other player? Would the answer = 1152? I got this because each player plays 8 games in 1 iteration (he cannot ...
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1answer
153 views

Understanding the “Birthday Problem”

I found on this website http://www.cut-the-knot.org/do_you_know/coincidence.shtml proof that the probability of two people in a room having the same birthday equates to 50% when when there are 23 ...
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239 views

singleton null vector set linearly dependent, but other singletons are linearly independent set

Why the set $\{\theta_v\}$ where $\theta_v$ is the null vector of a vector space is a dependent set intuitively (what is the source of dependence) and the singleton vector set which are non-null are ...
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404 views

Function many to one

Recently I come across this definition of function. "A function F is a set of ordered pairs $( x , y)$, no two of which have the same first member. That is, if $(x, y) \in F$ and $(x, z) \in F$, then ...
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196 views

What is a fiber bundle? (for non-mathematicians)

How can I explain the concept of a fiber bundle to someone with no mathematical background?
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86 views

A basic intuitive question on basis

From Zorn's lemma, basis can be thought of as a maximal independent set as well as minimum cover (covering all the vectors). Is this observation correct ? Can this observation be related to the usual ...
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1answer
233 views

How is percolation defined and measured in social networks?

From the wikipedia article on percolation it appears that the theory is applicable to graphs in general, and this presentation describes the theory nicely. This review article on complex networks ...
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85 views

A basic doubt on linear dependence and basis vectors

I see that linear independence/dependence is defined for a finite set of vectors in books. But, basis vectors are always independent and they need not be finite. Is the definition consistent ?
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55 views

A basic question on unconstrained optimization

I am going through an introductory textbook on optimization where the following is said : "Optimization within a subspace or linear variety can often be reformulated as unconstrained optimization, ...
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5answers
209 views

$\sum_{k=1}^nH_k = (n+1)H_n-n$. Why?

This is motivated by my answer to this question. The Wikipedia entry on harmonic numbers gives the following identity: $$ \sum_{k=1}^nH_k=(n+1)H_n-n $$ Why is this? Note that I don't just ...
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Intuition behind multiplication

I recently read this post and the highest voted comment and it got me thinking. How does think about multiplication if it is decimals? For example, if we have $3.9876542 \times 2.3156479$ then how ...
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3answers
99 views

How to define an irrational to the power an irrational

How to define an irrational number to the power an irrational number ?. At max I can understand the definition of an irrational to the power rational, but how to define an irrational number to the ...
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2answers
477 views

Intuition/Understanding of Inverse and Determinants

This is not homework, but extends from a proof in my book. EDIT We're given an $m \times m$ nonsingular matrix $B$. According to the definition of an inverse, we can calculate each element of a ...
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1answer
93 views

Integral Polyhedra: Integer on each face

The general topic is unimodular matrices and integral polyhedra. I am really new to this field and I am studying for an exam in an advanced operations research course. In this case we are always ...
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148 views

Geometrical interpretation of a group action of $SU_2$ on $\mathbb S^3$

Background There're some nomenclatures from Michael Artin's Algebra to explain. 3-Sphere, or $\mathbb S^3$, is the locus of $x_0^2+x_1^2+x_2^2+x_3^2=1$, where $(x_0,x_1,x_2,x_3)\in\mathbb R^4$. ...
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856 views

On the origins of the (Weierstrass) Tangent half-angle substitution

The Weierstrass substitution is great for transforming complex trig integrals into simpler rational functions. Wikipedia suggests that it wasn't invented by Weierstrass, since Euler was already ...
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43 views

Can we prove that this formula correctly describes the $n$th least significant digit?

So if we take a look at the binary representation of numbers, we can see that the digits follow a pattern: $$0 = 0000$$ $$1 = 0001$$ $$2 = 0010$$ $$3 = 0011$$ $$4 = 0100$$ $$\dots$$ i.e. the least ...
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822 views

How to Garner Mathematical Intuition

Motivated by Why Is Intuition so Important to Mathematicians but Missing from Mathematics Education? $^{1}$ by Leona Burton, I would like to learn about specific ideas or strategies to attain ...
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1answer
55 views

Geometric $k$-blades

What are the purposes of $k$-blades? Why is it important to have a oriented area, or an oriented volume? I'm referring to $k$-blades in such a way that $$\hat{v_1} \wedge \cdots \wedge \hat{v_k} $$ ...
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892 views

Calculating limits using the $\epsilon$-$\delta$ definition.

Suppose you have a function $f(x)=( x^2-4)/(x-2)$. How then do we find the limit as $x\to2$ in accordance with the epsilon delta definition? I mean suppose we don't know how to calculate limit and we ...
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529 views

What is the epsilon-delta definition of limits, exactly?

I am a bit confused with infinitesimals, and want to know why they were discarded and the epsilon-delta definition is being used? What is the epsilon-delta definition of limit? What is the intuition ...
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1answer
58 views

Gauge fractions with exponents - No Calculator

How does one (without the use of Calculator) determine that $5/6$ is less than $(35/36)^6$? How is this done mentally?
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260 views

Why is a projection matrix symmetric?

I am looking for an intuitive reason for a projection matrix of an orthogonal projection to be symmetric. The algebraic proof is straightforward yet somewhat unsatisfactory. Take for example another ...
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2k views

What is the meaning of infinitesimal?

I have read that an infinitesimal is very small, it is unthinkably small but I am not quite comfortable with with its applications. My first question is that is an infinitesimal a stationary value? It ...
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2k views

What is the difference between a discrete function and a continuous function

Intuitively it seems that both concepts should be disjoint because if a function is discrete then it has some holes on it and if a function is continuous then it doesn't have holes. But now I'm not ...
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179 views

What does it mean for a coalgebra to be cogenerated by a subspace?

The usual definition of an algebra being generated by a subspace is as follows: Let $A$ be an algebra, $X \subset A$ a subspace, $\mathrm{Alg}(X)$ the free algebra generated by $X$. Then $A$ is ...