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 are the Cauchy-Riemann equations in polar form 'obivous'?

In my book on complex analysis I'm asked to prove the Cauchy-Riemann equations in polar form, which I did. However, at the end of the question the author asks why these relations are 'almost obvious'. ...
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6answers
160 views

Intuition: “If P then Q” = 'Not P or Q' [on hold]

I already understand, and so ask NOT about, the Conditional Law: $P \Rightarrow Q \; \equiv \;\lnot P \vee Q$. But what's the intuition? Because I ask only for intuition, please do NOT prove formally ...
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13 views

Linear functionals defines projection operators?

The way I always understood linear functionals on a vector space $V$ is to consider then as measuring objects which give projections when they are given vectors. Now I wanted to make this a little bit ...
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5answers
74 views

Intuition: Why is the biconditional true if both statements are false?

I already know that a false statement implies anything. Because I ask only for intuition, please do NOT prove this or use truth tables (which I already understand). Source: p 333, A Concise ...
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How to think/see point-set topology abstractly?

I've started learning point-set topology this semester. I've learned basic material about: topology on a set topological space open sets closed sets clopen sets closure neighborhoods interior point ...
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When is it sufficient to use logic as proof for an intuitive answer

Say I have the following limit $$\lim_{x\rightarrow\infty}\frac{3^x}{e^{x-1}}$$ In this case it's simple enough to write it as $\lim_{x\rightarrow\infty}{3^x}{e^{1-x}}$ and then show it approaches ...
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How to intuit 'unless'? [duplicate]

Foreword: The following (seeking only intuition) does NOT duplicate this (which explains with formal proofs.) I already know, and so ask NOT about, the proof of: $A$ unless $B$ = $A$ if not $B$ = ...
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31 views

Intuitive probability theory!?

Recently I saw something where someone had a paper with several lines on it and a needle. The length of the needle was the same as the distance between the lines. They then proceeded to say that when ...
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1answer
48 views

Why is a cartesian morphism called cartesian?

I am reading about fibred categories. After reading the definition of "vertical" morphism, I can imagine why they are named like that. What about "cartesian" morphisms? What is cartesian about them? I ...
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9k views

What did Alan Turing mean when he said he didn't fully understand dy/dx?

Alan Turing's notebook has recently been sold at an auction house in London. In it he says this: Written out: The Leibniz notation $\frac{\mathrm{d}y}{\mathrm{d}x}$ I find extremely difficult ...
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1answer
15 views

Barendregt's Substitution Lemma (lambda calculus)

I am struggling to put words on an idea used in Barendregt's Substitution Lemma's proof. (available here) The lemma states that: If x≠y and x not free in L and M, L are $\lambda$-terms: then ...
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1answer
32 views

Interpretation of alternative group structures on a given group

Let $(G,e,\circ) $ be a group with $e$ the identity element and $a \in G$ and $\circ$ the group operation. Then we can form a new group $(G_a,a,\circ_a)$ with the same underlying set as $G$ and $x ...
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14 views

Antiderivative of a position function.

Let's say you had some sort of position function, i.e. a moving ball, $f(x)=-2x^{2}+3x-7$. What is the intuitive (relates to the situation) definition of the antiderivative of this function? It ...
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2answers
43 views

Index in complex analysis is an integer : Intuition?

In complex analysis course, we prove that given a closed path $\gamma$ and $a\notin \gamma^*$ the following number: $$ \frac{1}{2i\pi}\int_{\gamma}\frac{dz}{z-a} $$ is an integer. The integral can ...
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43 views

How to show $k$-algebras are isomorphic in practice

I am working through some problems which require me to show when some $k$-algebra ($k$ a field) maps are isomorphisms. Unfortunately, I've got myself a bit confused with definitions and the like, and ...
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4answers
219 views

Why are we defining the norms on certain vector spaces the way they are?

What's the intuition behind defining $\|x\|_{\infty} = \max_{1 \le i \le n}\{|x_i|\}$ on the space of ordered $n$-tuples of complex numbers? I'm asking because I've been asked to find a norm on the ...
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1answer
65 views

What the curvature $2$-form really represents?

Let $(E,\pi,B)$ be a principal bundle with structure group $G$. The connection $1$-form can be thought of as a projection on the vertical part. It allows us to characterize the horizontal subspaces as ...
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connection among big-M, Lagrangian, Pentalty Method, and Augmented Lagrangian

In the context of solving linear programs, the big-M method refers to adding additional variables to the problem such that there is, as far as I understand it, a trivial basic feasible solution. In ...
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2answers
87 views

How to intuit 'only if'?

I already know, and so ask NOT about, the proof of:   $A$ only if $B$   =   $A \Longrightarrow B$. Because I ask only for intuition, please do NOT prove this or use truth tables. My problem: I try ...
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0answers
31 views

Motivation For Tensor Product of R-Modules

I have recently learned about tensor products of modules,specifically the material in Dummit and Foote chapter 10 section 4. My understanding is that the construction of tensor spaces is important ...
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1answer
61 views

pointwise convergence of a filter on $\mathbb{R}^\mathbb{R}$

In my topology lecture we have defined pointwise convergence for filters on function spaces, say $\mathbb{R}^\mathbb{R}$. A filter $\varphi$ on $\mathbb{R}^\mathbb{R}$ converges pointwise to ...
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3answers
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What are some illustrative examples that demonstrate how $\succ$ can differ in behavior from $>$ and/or $\geq$?

I really, really want to understand the generalization of metric spaces known as continuity spaces. Unfortunately, I always get tripped up right at the beginning. The problem is that I have little or ...
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1answer
34 views

intuition about cubic splines vs quadratic splines (degree 3 vs degree 2).

my intuition about quadratic(degree 2) splines is that by the help of its three variables (in each sub-interval) you can make a piecewise differentiable function on the whole interval. in the process ...
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1answer
105 views

What is an example of a proof that explicitly relies on the law of excluded middle?

I was talking with a friend about logic and I realized she might be an intuitionist. I was looking online for a proof that explicitly uses the law of excluded middle to see if she would have an issue ...
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1answer
51 views

Intuition behind the connection one-form

When we define a connection on one principal bundle $\pi: P\to M$ with structure group $G$ we define it as an association of one subspace $H_pP\subset T_pP$ for each $p\in P$ such that $T_pP = H_pP ...
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Where did the idea of hermite interpolation came from?

I am given the Hermite interpolation formula directly in my text book without ANY explanations about how it was first made (obviously it was somehow constructed for the first time with some sort of ...
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42 views

How to prove that $2^\sqrt2$ is irrational [duplicate]

I know the result is intuitive, but is there a way to mathematically prove it. I tried taking logarithm or tried proving by contradiction, but I don't know how to proceed. Thanks.
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1answer
28 views

What does the power of an ideal *mean*?

I am stumped trying to understand Silverman's definition of $\operatorname{ord}_P(f)$, the (normalized) valuation on $\bar K[C]_P$ (which denotes the localization of a curve $C$'s coordinate ring at ...
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15 views

$h$ invariant of forms and its relation to Birch rank

Let $k$ be a field. Given a form $F_i \in k[x_1,..., x_n]$ of degree $d$, the $h$-invariant $h_k(F_i)$ is defined to be the least positive integer $h$ such that $F_i$ can be written as $$ A_1B_1 + ...
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4answers
85 views

How to visualize permutations?

I'm getting a warning that this is a subjective question, and it very well probably is. But nevertheless, it is still a valid question that helps in the studying of mathematics from my point of view. ...
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Explanation of Gauss sums' preference for the North-East?

Define the Gauss sums algebraically as follows: $$\mathcal{G}_q = \sum_{a=0}^{q-1} e^{2 \pi i a^2 / q}$$ Then the result ends up being $\sqrt{q}, 0, i \sqrt{q}, (1+i)\sqrt{q}$ depending on $q \equiv ...
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Intuition about $v\otimes w$

In Physics and Differential Geometry usually tensors of type $(k,l)$ on a vector space $V$ over $\mathbb{F}$ are defined as multilinear functions $$f : \underbrace{V\times\cdots\times V}_{k \ ...
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Intuition behind spectral radius of a graph

Suppose that I have a graph G, along with its respective adjacency matrix A. The definition of how one can compute the spectral radius of this graph is not hard to grasp, but I was wondering about the ...
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2answers
153 views

$\int_{0}^{\infty}e^{-st}h(t)dt=0 \Rightarrow h(t)=0.$

Suppose $h(t)$ is continuous function and $\int_{0}^{\infty}e^{-st}h(t)dt=0 ~\forall~ s>s_{0}$, then prove that $h(t)=0$. I know "if a function is continuous, non-negative or non-positive, and its ...
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52 views

Intuition behind non-continuous derivatives

Although the intuition behind continuity and derivatives seem pretty obvious to me, I cannot figure out what it means to have a differentiable function with a non-continuous derivative. I'm trying to ...
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84 views

'If…then…' and '…if…' and '…only if…' and 'If… only then…' statements?

Suppose you have two statements A and B and "If A then B". I am trying to think of what this implies and alternative ways of writing this. I think "If A then B" = A$\rightarrow$B = "A is ...
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1answer
43 views

What is an intuitive explanation for Birkhoff's ergodic theorem?

If I'm not familiar with measure theory, what is a good way to understand the idea behind the definitions involved, the interpretation of the theorem, and the proofs thereof? Particularly, it's not ...
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14answers
903 views

How to explain to a 14-year-old that $\sqrt{(-3)^2}$ isn't $-3$?

I had this problem yesterday. I tried to explain to the kid this: $$\sqrt{(-3)^2} = 3,$$ and he immediately said: "My teacher told us that we can cancel the square with the square root, so it's ...
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3answers
54 views

How is the following a CW complex

My professor today draw on the board a sphere and attached to half a circle of the sphere half of the boundary of a disk so the shape looked like you glue a curvy half disk to a sphere. He then said ...
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1answer
92 views

Notions of consistency / heterogeneity in sets of vector values

The problem Let us consider a row vector u of size $n\in\mathbb{N}$, containing only binary values (0,1): $$u=(u_1 \cdots u_n), n\in\mathbb{N}$$ $$\forall i \in \{1\ldots n\}, u_i \in\{0,1\}$$ I ...
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What is the intuition behind / How can we interpret the eigenvalues and eigenvectors of Euclidean Distance Matrices ?

Given a set of points $x_1,x_2,...,x_m$ in the euclidean space $\mathbb{R}^n$, we can form a $m$ x $m$ Euclidean Distance Matrix $D$ where $D_{ij}={||x_i-x_j||}^2$. We know a little bit about these ...
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3answers
207 views

What is the intuition behind the definition of the differential of a function?

What is the intuition behind the definition of a differential of a function in differential geometry? i.e. $$df(p)(v_{p}) =v_{p} (f)(p) $$ where $v_{p} \in T_{p} M$ is a vector in the tangent space to ...
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2answers
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Why General Leibniz rule and Newton's Binomial are so similar?

The binomial expansion: $$(x+y)^{n} = \sum_{k=0}^{n} \binom{n}{k} x^k y^{n-k}$$ The General Leibniz rule (used as a generalization of the product rule for derivatives): $$(fg)^{(n)} = ...
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3answers
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Cosine and Sine Angle Addition Intuition [duplicate]

I am lacking in understanding in the cosine and sine angle addition formulas. I have seen several questions similar to this but I have not seen an answer that explains how this conclusion can be ...
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Definition of Mapping

Some formal characterization is seems very abstract for me. For instance: Let $X$ be a finite set of alternatives. We denote by $\chi$ (respectively, by $B$) the collection of all non-empty ...
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1answer
77 views

Help with intuition on Cardinal Arithmetic Problems

It happens a lot to me that when I find an intuitive model (picture) of a mathematical entity, the proofs left as exercises in books are very easy to solve. For example when dealing with filters and ...
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2answers
49 views

how to understand the definition of continuity in analysis?

Please have a look at the picture above. This is about the continuity in analysis. I don't really understand how to utilize this definition? It says that is statement is equivalent to f is ...
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Visualization of the fact that the integers defining lens spaces must be coprime

This is related to this question I asked: Visualization of Lens Spaces and is also related to this question by @Earthliŋ: Why are the integers appearing in lens spaces coprime? I understand the ...
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1answer
27 views

Is there an interpretation for writing a polynomial in $x$ as a polynomial in $(x-b)$?

Let $Q(x)$ be a polynomial in $x$ of order $n$. The Taylor polynomial of $Q(x)$ of order $n$ developed around $x=b$ (denoted by $P_{n,b}(x)$ ) corresponds to $Q(x)$ written in $(x-b)$. This can be ...
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Given a drawing of a parabola is there any geometric construction one can make to find its focus?

This question was inspired by another one I asked myself these days Given a drawing of an ellipse is there any geometric construction we can do to find it's foci? I think this is harder, I can't ...