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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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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24 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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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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23 views

Convergence of a monotone sequence of real numbers [on hold]

I am trying to get a better intuition about this theorem. I will use an example: is there a monotone sequence with a max/min that converges to the max/min? by definition there need to be infinite ...
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25 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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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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42 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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56 views

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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40 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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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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14 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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81 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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21 views

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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106 views

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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19 views

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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145 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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51 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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80 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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42 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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818 views

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

$\require{cancel}$ 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 ...
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52 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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86 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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202 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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81 views

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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28 views

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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75 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
46 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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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 ...
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363 views

Geometric intuition behind pullback?

I am having hard time with forming a geometric intuition of pullback and pushforward. The definition the book gives is like this: There are two open sets, $A$ and $B$. There is a dual transformation ...
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151 views

Ellipse inscribed on a quadrilateral

The problem is: Given that an ellipse is inscribed on a convex quadrilateral and each one of it's diagonals pass through one foci of the ellipse show that the product of the opposite sides ...
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30 views

Why Delta Complex structure?

I am a bit confused about delta complex structure and CW complex structrue. I was wondering is n-dimensional disks homeomorphic two n dimensional triangles? I mean if that is the case then I can make ...
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1answer
102 views

Visualization of Lens Spaces

I am trying to visualize lens spaces geometrically. While I am aware of the fact that most manifolds which cannot be embedded in $\mathbb{R}^3$ are hard to visualize because of the obvious ...
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What does the Dedekind Rule `say'?

In Relation Algebra, the modal law or dedekind rule $$R;S \,\cap\, T \;\subseteq\; (R \cap T;S^\circ);S$$ appears often and I wonder what is the motivation behind it. Moreoever, what does it "say". I ...
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Intuition behind variation of parameters method for solving differential equations

I have used the variation of parameters method (and have been taught it, although not hugely in depth) and I was wondering if I've understood the intuition behind it. In particular I've been thinking ...
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41 views

What is the different between a product of gaussian random variables and their PDFs

Multiplying two univariate Gaussian PDFs $$ X \sim \mathcal{N}(\mu_X,\sigma_X) \\ Y \sim \mathcal{N}(\mu_Y,\sigma_Y) \\ Z = X*Y $$ results in closed form equations for $\mu_Z$ and $\sigma_Z^2$: $$ ...
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Puzzle: Players A,B,C,D are in a line

Players A,B,C,D stands in a line. Players A, D do not move. round 1: player B moves one distance closer to the midpoint of A,C round 2: player c moves one distance closer to the midpoint of B,D ...
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1answer
50 views

What is the idea behind “representability” in a first order theory?

I've been reading through Enderton's logic, this notion is introduced and is given special attention as it's said that they are crucial in the proof of incompletness theorems. I grasp the formal ...
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115 views

Why is $P \to Q \equiv \neg P \vee Q$?

By truth table, we know that $P \to Q$ is equivalent to $\neg P \vee Q$. But I'm trying to understand why this work? How can connective "or" be implication. I tried some examples but I still can't ...
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36 views

definition of multiplication intuition (Set theory)

We define how to multiply two integers using set theory as follows [(a,b)][(c,d)] = [(ac + bd,ad + bc)] what is the intuition behind defining multiplication like this ?
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Why do graphs sometimes have unevenly spaced intervals in axis e.g. logarithmic

Why do some graphs not have evenly spaced out units on the x or y-axis? For example they have log on one or both of the axis? Here's an example from this page I guess the page does give a bit of an ...
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Finding angles plane geometry

$\Delta ABC$ is obtuse on $B$ with $\angle ABC = 90 + \frac{\angle BAC}2$ and we have a point $D \in AC$ (in the segment, I mean D is in between A and C) such that $\angle BDA = \angle ABD + ...
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105 views

Problematic lack of intuition for number theory

I Have a problem. I mostly do mathematics because I find it fascinating and enjoy doing it. Now whenever I skim through a book a number theory I always find myself thinking 'I wish I would understand ...
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49 views

What is the variance of self-information (or surprisal)?

The self-information of an outcome $x_i$, or surprisal, is defined as: $$ I(x_i)=-\log P(x_i), $$ where $P$ means probability. This way, the Shannon entropy can be seen as the "average" or "expected" ...
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45 views

Intuitive difference between a continuous map and a homeomorphism

I know the formal definitions of both continuous map and a homeomorphism between two spaces. If two spaces are homeomorphic intuitively they can be thought of spaces which can converted to each other ...
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61 views

Is it correct to say that if $\lim\limits_{x \to a}f(x) = 0$ it is an Infinitesimal?

I think I'm misuderstanding something here, because to my understanding the definition of infinitesimal given in my textbook does not convey the same thing as in other sources. I've read the ...
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The support and the non-vanishing set of a function on a scheme

I have some confusion regarding the two concepts: Let $(X,\mathscr{O}_X)$ be a scheme, let $f\in \Gamma(\mathscr{O}_X,X)$ and define the support of $f$ to be $$\operatorname{Supp}(f) : = \{p\in X: ...