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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Mixed vs Equal Characteristic Local Rings

This is more of a vague, intuitive type of question, so perhaps there isn't anything too concrete anyone can offer. I am trying to get a sense of precisely why working with local rings of equal ...
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1answer
19 views

Solving for a Binary Matrix: A somewhat unusual method needs justification, and mabye interpretation.

Introduction: Define a "Bit Map" to be a matrix whose entries can only be $0$ or $1$. Then numbers above and beside each column and row indicates how many entries are "filled" with a one. For ...
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2answers
29 views

Diagonally dominant matrix — geometric interpretation

I like to have a visual interpretation of mathematical concepts. This is simple for many important kinds of matrices: orthogonal matrices are rotations, diagonal matrices scale along the natural ...
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11 views

Examples of Relation Algebras

Would anyone please direct me to a host of examples of relation algebras. Is there an intuition for what these algebras are to model? That is, groups, for example, model a notion of symmetry; ...
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1answer
40 views

Tangent space as derivations exercise

Thinking of the tangent space to a manifold as derivations is a concept which just kind of eludes me. I am comfortable thinking about tangent vectors as equivalence classes of curves and with the ...
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1answer
40 views

(Visual) Intuition: Division and complex fractions

When treating division as "groups of the numerator" (sorry, I don't know the technical term -- see image), why does a complex fraction in the denominator get added together to produce a 1 (number of ...
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1answer
43 views

Uses of stalks of sheaves and germs

I am trying to understand the motivation behind defining stalks of sheaves, but I suppose my complex geometry is a little weak. I know they are meant to represent germs of holomorphic functions at a ...
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1answer
40 views

Losing a dimension when finding intersection between subspaces

Let $F=\mathbb Z_3, V=F^4$. Let $U=sp\{(1,0,0,0),(1,0,1,0),(0,1,1,1) \} \\W=sp\{(0,0,1,0),(-1,1,0,1),(1,1,1,1) \}$ Find $dim (U\cap W)$ we have $v\in U \text{ and } v\in W$ so $v=v$ ...
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49 views

Intuition behind generic point of a scheme?

I've been reading a little about algebraic geometry and how there seems to have existed this notion of "generic point" on a variety which wasn't carefully defined at first. But often times, ...
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1answer
41 views

How to geometrically interpret intertia of primes in field extensions?

I am trying to understand the intuition of thinking about number theoretic ideas in terms of geometric ones. For example, ramification is something that happens when a "covering" space of a Riemann ...
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32 views

Motivation for the name “vertical subspace” in the context of fiber bundles.

Let $p:E\to B$ be a smooth fiber bundle with fiber $F$. Consider the vector spaces $V_u=\{x\in T_uE: p_*(x)=0\}$. We call $V_u$ the vertical subspace of the tangent space $T_uE$. How can we see that ...
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1answer
27 views

Problem in deducing gradient in spherical coordinates.

I know the differential displacement in spherical coordinate as $$dr \cdot \widehat{r}+ r d\theta\cdot\widehat{\theta} + r\sin\theta d\phi\cdot \widehat{\phi}$$. But I can't figure out how the ...
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30 views

What is the intuitive meaning if multiplying by fractional 1?

first post ever on stack exchange in years of using it. Can anyone provide a historical or logical deduction of the reasoning behind multiplication by 1 via a fraction? For instance, in finance ...
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19 views

Custom equation guidance

i was having a tough time deciding on which SE site to post this rather unique question being that it would help me with a program I'm writing. I decided to come to you guys, not hoping that you would ...
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1answer
31 views

Need some help understanding the condition of the implicit function theorem

The condition for the implicit function theorem is that the (smooth) map $f: \mathbb R^n \to \mathbb R^m$ is locally a (smooth) map of $n-k$ variables if there are locally smooth maps $g_i , i \in ...
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2answers
85 views

What is the correct analogue of $\mathbb N$ in a ring of integers?

Question: Let $K$ be a number field. The proper intuitive motivation for the ring of integers $\mathcal O_K$ is that $\mathbb Z$ is to $\mathbb Q$ as $\mathcal O_K$ is to $K$. But what plays the role ...
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1answer
62 views

Why should we expect duality to give useful concepts in category theory?

Why should we expect the abstract notion of flipping arrows in a category to generate useful concepts from other useful ones? What exactly does flipping the direction of arrows mean and why is it ...
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2answers
47 views

The shortest path in a metric space with a given metric

My questions seem to be very basic and intuitively correct but I can't formally prove them. Before learning metric spaces, for $R^2$, we always define the distance between 2 points as $d_2 = ...
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146 views

How to intuitively understand prolongations

This question is concerned with the algebraic side of the theory of prolongations as explained in this paper by V. Guillemin and S. Sternberg. Let me first introduce my notation. We're working with a ...
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5answers
617 views

What is the difference between necessary condition & sufficient condition?

My book says : For having extreme point $a$ of function $f$, the necessary condition is that $f'(a) = 0$. However, it isn't a sufficient condition. Now, what is the difference between necessary ...
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1answer
60 views

Is there any intuitive way to think about the gamma function?

Is there a way to realize the gamma function intuitively? My first (and probably correct) guess is no, because, for example, $\Gamma(\frac 12)=\sqrt{\pi}$ doesn't make any intuitive sense at all. ...
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3answers
26 views

Why $\omega$ in $x = \cos(\omega t + \alpha)$ , $\omega$ isn't considered an arbitrary constant?

We know the SHM differential equation is of second-order $$\dfrac{d^2 x}{dt^2} = -{\omega}^2 x$$ . So, the solution of this equation must contain two arbitrary constants. And also we know that $x = ...
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Intuition behind filter on a set

In "Counter-examples in topology" of Steen and Seebach, they define a filter on a set $X$ is a collection F of subsets of $X$ with the following properties: Every subset of $X$ which contains a set ...
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1answer
45 views

Horse race - once vs. twice

You select your horse for a race. The horse has a parameter $q \in [-0.3,+0.3]$. Its probability of winning depends on the weather, which can be either Rainy or Sunny: In Rainy weather, the ...
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What do people mean by “finite”?

Many arguments about the foundations or philosophy of mathematics centre on the question of whether or not there exist objects or entities (such as certain sets) which are not "finite". (For ...
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1answer
103 views

Couldn't there be a better definition of imaginary numbers? [closed]

It is a common known fact that i is defined as a number that satisfies the quality of $i^2 = -1$ or in other words, $i = \sqrt-1$. The intuition of this is often said to be that multiplying by $i$ ...
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What is the idea behind a projection operator? What does it do?

I know what a projection operator is, but I am unable to explain it in words without using mathematical symbols. Can anyone help me? I don't need examples or the definition - I want to know why and ...
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28 views

Need some help understanding this exercise about injective plane curve

Let $\gamma (t) = (x(t), y(t))$ be a smooth regular plane curve $\gamma: I \to \mathbb R^2$ where $I$ is some open interval. Now consider the following exercise: Let $\varphi (u,v) = (x(u), v + ...
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1answer
16 views

If points of mobius transformation are given, then how to determine the mapping?

A Mobius transformation is a map $$f(x)=\frac{rx+s}{tx+u}$$ where $ru-st \neq 0$. Suppose we have $f(a)=c, f(b)=d, f(c)=a, f(d)=b$, where $a,b,c,d \in \mathbb{R}$. Then from here, the answer given by ...
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1answer
28 views

Taylor Polynomial - intuition

How do adding higher derivatives of the function on the same point gives a better approximation?
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4answers
98 views

In every set of $14$ integers there are two that their difference is divisible by $13$

Prove that in every set of $14$ integers there are two that their difference is divisible by $13$ The proof goes like this, there are $13$ remainders by dividing by $13$, there are $14$ numbers ...
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2answers
102 views

Understanding why $a+b\sqrt {2}\neq \sqrt {3} $

I want to intuitively understand why $a+b\sqrt {2}\neq \sqrt {3} $ for $a, b \in \mathbb Q $ I really have no intuition regarding this matter, and have to deal with similar concepts regularly while ...
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1answer
42 views

Can an element of a power set $2^A$ be a subset of $2^A$?

This question is continued from a previous thread I started, but it had more than one question so I had to move the other question here. For this example consider an injective map $f: A \to 2^A$ then ...
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1answer
58 views

If $A$ is a non-empty set and $2^A$ is the power set of $A$. Is $2^A \subseteq A$?

I'm aware that if there exists an injective map $f: A \to 2^A$ then for each element $a\in A$ $\exists$ $f(a)\subseteq A$. But does this also mean $f(a)\subseteq 2^A$? I ask this because when ...
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4answers
69 views

Why Mendelson axiom schemas are true?

I'm taking course in logic. The book is available here I don't understand why is Mendelson axiom schemas are the way they are. For example implication creation schema $φ ⇒ (ψ ⇒ φ)$ My thoughts ...
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27 views

mean value theorem applications

it is said that the mean value theorem prove that the graph between $[a,b]$ has a point where it is equal to the average change of the graph, (sorry for not being accurate) is there more information ...
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167 views

What do groups and rings “look like”?

Taking undergraduate physics courses, I had to deal with Euclidean vectors often. In classes like Calc III, the concept was also there. I'm not sure if this is why, but I've always had a more ...
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1answer
57 views

Why Does The Taylor Remainder Formula Work?

I've been studying calculus on my own and have come across Taylor series. It is very intuitive until I came across the remainder part of the formula where things got fuzzy. I understand why the ...
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1answer
56 views

Why are power sets called power sets?

Why are power sets called power sets? What is so powerful about them?
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1answer
72 views

Trouble understanding what a measure-zero set is.

To begin with some context, I haven't had any exposure to measure theory yet. I solved the following problem. A set $A\subset \mathbb R$ such that $\forall \epsilon >0$, there exists countably ...
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2answers
33 views

What is the best way to explain setting a restriction on $\delta$ in $\epsilon$-$\delta$ proofs?

I'm trying to prepare a somewhat informal lesson striving to provide an intuitive understanding of why for some limit proofs, we have to set an upper bound on $\delta$. For example, here's part of ...
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1answer
29 views

Geometric proof for $|| u ||^2 + || v ||^2 = \frac{1}{2}||u-v||^2 + 2||\frac{u+v}{2}||^2$

Is there an geometric proof for the following identity? $|| u ||^2 + || v ||^2 = \frac{1}{2}||u-v||^2 + 2||\frac{u+v}{2}||^2$. The norm here is normal Euclidean norm, and $u,v$ are vectors.
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52 views

Physical analogies of a math concepts [closed]

In a post Terence Tao explained a very nice way to think about convolution and noted that "one should try to use physical intuition to model mathematical concepts whenever one can". I found this very ...
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3answers
50 views

Why is $\mathbb T\cup\mathbb A = \mathbb Q \cup \mathbb I =\mathbb R$?

Where $\mathbb T $ is the set of transcendental numbers, and $\mathbb I $ is the set of irrational numbers and $\mathbb A $ is the set of algebraic numbers. The sets $\mathbb Q$ and $\mathbb R$ have ...
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96 views

Ordinals - motivation and rigor at the same time

Can someone provide a description of ordinals within ZFC in a rigorous way that exhibits motivation? Every description or explanation I see in the literature or on the Internet is either too formal ...
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3answers
160 views

Understanding the use of the Cartesian Product in the proof of $|\mathbb R\times \mathbb R|=|\mathbb R|$

Where the Cartesian Product of two sets $\mathbb A$ and $\mathbb B$ is such that $\mathbb A\times \mathbb B=\{{ (a,b)|a \in \mathbb{A}, b \in \mathbb{B}\}}$ In trying to understand the proof that ...
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1answer
46 views

matrices with determinant equals to one

we already know what does it mean the determiant of a matrix is null, it's not invertible ! but what about matrices with determinant equals to $1$ ?! I know that the determinant of matrix is the ...
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1answer
58 views

What is the practical meaning of derivatives? [closed]

I mean practically integration means sum of all components, and the integral can be visualized as the area below a curve. Is there a similar intuition or geometric meaning of the derivative?
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119 views

Intuitively understanding $\sum_{i=1}^ni={n+1\choose2}$

It's straightforward to show that $$\sum_{i=1}^ni=\frac{n(n+1)}{2}={n+1\choose2}$$ but intuitively, this is hard to grasp. Should I understand this to be coincidence? Why does the sum of the first ...
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2answers
79 views

Proofs of theorems, where picture is sufficient

A while ago I have had the pleasure to come across those lectures of Topology & Geometry by Dr Tadashi Tokieda (I do recommend watching at least the first lecture, both parts). My question is ...