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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Is there a physical interpretation of the alternating property?

A map from lists to list-elements is called "alternating" if any list with repeated elements is mapped to zero. This has statistical significance: regressions on collinear data are bad, dependent ...
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Intuition - For every % point that rates rise, a bond’s value will decline by its duration in years.

[Source:] Generally speaking, for every percentage point that rates rise, a bond’s value will decline by its duration (stated in years). So if rates climb by one percentage point, the value of a ...
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106 views

A math puzzle about slow clock

You have the misfortune to own an unreliable clock. This one loses exactly 20 minutes every hour. It is now showing 4:00am and you know that is was correct at midnight, when you set it. The clock ...
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45 views

Show that $\frac {\partial B} {\partial T} =$ $\frac{c}{(\exp\frac{hf}{kT}-1)^2}\frac{hf}{kT^2}$

Find an expression for $\frac {\partial B} {\partial T}$ applied to the Black-Body radiation law by Planck: $$B(f,T)=\frac{2hf^3}{c^2\left(\exp\frac{hf}{kT}-1\right)}$$ The correct answer is $\frac ...
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24 views

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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28 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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40 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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13 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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50 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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53 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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58 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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42 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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56 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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47 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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40 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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36 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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32 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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21 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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35 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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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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66 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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49 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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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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653 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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71 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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34 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
46 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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117 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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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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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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30 views

Taylor Polynomial - intuition

How do adding higher derivatives of the function on the same point gives a better approximation?
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101 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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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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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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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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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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29 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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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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64 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
69 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
84 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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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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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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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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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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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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166 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 ...