Questions tagged [soft-question]

For questions whose answers can't be objectively evaluated as correct or incorrect, but which are still relevant to this site. Please be specific about what you are after.

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36 views

The union (limit) of a set of sets and "potential" vs "realized" infinity

Let $\mathbb{N}$ be the set of natural numbers. Let $A_i$ be the set $\{0, 1, 2, ..., i\}$ of natural numbers up to $i \in \mathbb{N}$. Let $\{A_i | i \in \mathbb{N}\}$ be set of all $A_i$. (The set $\...
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0answers
40 views

Analogy between logical formula and random variable

A logical formula $\phi$ induces a function from the space of valuations to the set $\{0,1\}$. Similarly, a binary random variable is a function from a sample space $\Omega$ to $\{0,1\}$. If we think ...
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0answers
18 views

Is there a way to recover, without using integration, the basis of a rotating frame of reference , knowing the angular velocity vector?

Suppose one wants to " control" ( using, say, Geogebra) the rotation of a moving frame of reference $R' = O', \vec {i'}, \vec {j'}, \vec {k'}$ using an angular velocity vector $\vec {\omega}=...
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3answers
84 views

Recommendation for probabilities books

I am a university student, majoring in Mathematics, I have previously studied "a text book of convergence" by W. L. Ferrare, which is the best book I have studied , because it has many ...
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1answer
178 views

What is the intuition for why all of math can be developed using set theory?

I learnt that formal language of pure set theory allows one to formalize all mathematical notions and arguments. The language just has one non logical symbol(!) "the belongs to relation" ...
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1answer
55 views

The simplex category as a "free" category

Consider the simplex category $\Delta$. Its objects are the linearly ordered sets of the form $[n]=\{0<1<\dots<n\}$. Its morphisms are nondecreasing functions. There are two special classes ...
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2answers
116 views

How to toss a coin in your head. [duplicate]

This is quite a soft question but hopefully, this community has some nice answers. What is a good way to "toss a coin" in your head? That is an "algorithm" to generate heads $50/50$...
3
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1answer
43 views

Intuition degeneracy maps

I'm trying to get an intuitive understanding for the notion of a simplicial set. Roughly, a simplicial set consists of a set $S_n$ of $n$-simplices for each non-negative $n$, and families of face maps ...
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1answer
44 views

Relationship between $\aleph$ and the symbol $|A|$ or $\operatorname{card}(A)$

Aleph ($\aleph$) is the symbol used in mathematics to indicate the cardinality of the numerable. We know that an infinite set has cardinality $\aleph_0$ if there exists a bijection that relates it ...
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1answer
93 views

What is the difference between the integral$\int\limits_{}^{}$ and segments $\sum_{} ^{} $

I want to know, what is the difference between the integral and segments I know that، Integration came when scientists asked how can calcule the area of unusual shapes، and one of they scientists came ...
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0answers
26 views

Describe evolution of this surface over time using a differential equation [closed]

Think of a surface $S$ bounded by a unit cube, as a balloon with two openings for air at $(0,1,1)$ and $(1,1,0)$ and constant air flow from both openings. As time passes, the balloon will fill the ...
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0answers
32 views

Structure/Order of explaining distribution-theory

please tell me if this is too off-topic, then I will delete the post. In my master thesis I want to use (and explain) distributions in order to solve some PDE, but since covid still is present it is ...
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1answer
48 views

Is there more reliable notion of equivalence of two sets than bijection? [duplicate]

I know that common sense is to define set $A$ to be equivalent to a set $B$ if and only if there exists a bijection between these two. Using that definition I can easily state that $\mathbb{N} \equiv \...
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2answers
100 views

$\dim(W + U) = \dim(W) + \dim(U) - \dim(W \cap U)$ have a correlation with $|A\cup B|=|A|+|B|-|A\cap B|$ with the sets?

In mathematics, the Grassmann formula is a relation concerning the dimension of the vector subspaces of a vector space or of the projective subspace of a projective space. We know that the enunciation ...
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0answers
31 views

Notation for equivalence relations between pairs of numbers that have the same difference or the same quotient

I am looking for a sensible notation for the equivalence relations on pairs of natural numbers that can be used to introduce integers and rationals. Namely, I am using the following two equivalence ...
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1answer
31 views

How can I calculate how many possibilties there are for a password

If there is a password that is 1000 characters long consisting of uppercase and lowercase letters and numbers what would be the possibility of someone figuring it out/how long would it take for ...
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2answers
300 views

Why does the trick to remember the trigonometric table work?

When I was first introduced to trigonometric ratios, I learned the following trick to remember the trigonometric table of standard angles ($0^{\circ}$, $30^{\circ}$, $45^{\circ}$, $60^{\circ}$, $90^{\...
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2answers
97 views

Real numbers vs. the real number line

I don't know how to formulate this question precisely, so let me explain where I am coming from, noting that I know little about nonEuclidean geometry. I was thinking about how to explain how ...
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0answers
65 views

Should I read / learn math textbooks linearly or just what I need? [closed]

I found a book: How to Think Like a Mathematician by Kevin Houston. And I have been skiming it; I found a passage that says: Don’t read it like a novel *Do not read mathematics like a novel. You do ...
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0answers
40 views

Who first proposed the idea of "resolution of the identity." Was it by von Neumann? [closed]

Who first proposed the idea of "resolution of the identity"? Was it by von Neumann? In Japanese it translates as "resolution of the unit." I couldn't figure out why it was ...
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0answers
17 views

How to improve the precision of a meromorphic approximation iteratively?

Background : The other day I managed to find a numerical approach to estimate the position of poles and zeroes of a Meromorphic function. The issue is that sometimes it does not converge perfectly it ...
2
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1answer
26 views

Standard notation for writing probability - fraction vs percentage

I have a question about the usage of fractions and percentages in questions related to probability. I am wondering if a fraction is the same as a percentage. To better explain my question, I will ...
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0answers
32 views

Intuition behind the definition of the torsion of a curve.

I understand the intuition behind defining the curvature $k$ of a curve $\alpha :I\rightarrow \mathbb{R}^3$ as $$\frac{\big(|\alpha '||\alpha ''|-|\alpha ' \cdot \alpha ''|\big)^{1/2}}{|\alpha '|^3}$$ ...
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5answers
2k views

If instantaneous rates of change aren't that rigorous, how correct is the usage of instantaneous rates of change (like velocity) by physicists?

According to this answer, instantaneous rates of change are more intuitive than they are rigorous. I tend to agree with that answer because, in the Wikipedia article on differential calculus, they ...
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0answers
36 views

You can’t apply induction to reason about uncountable sets, can you?

My question is a soft one. Induction as a proof technique I admire, but to my knowledge it is strongly coupled with the “least element principle” and enumerability (labeling something uniquely with ...
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1answer
79 views

Let $f$ be the function... vs. Let $f(x)$ be the function...

When defining, or referring to, functions, I've seen both of the styles in the title. I was wondering which is considered to be more correct, or better style. I've always found it strange to refer to $...
2
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1answer
179 views

Why does 3blue1brown use the "around a point" to describe a derivative?

In this article (which includes a link to the video version of the article as well), Grant Sanderson aka 3blue1brown describes a derivative. He says at the end of the passage headed "The Paradox&...
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0answers
54 views

Numbers of the form $15k+2^n$ include an abundance of primes

Quite by accident, I noticed that numbers of the form $15k+2^n$ where $k$ is an odd number seem to feature an unusually high frequency of primes among them. Of course, such numbers have no factors of $...
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1answer
42 views

Formula to calculate percentage ( % ) discounted price - subtract vs multiply?

From a mathematician's perspective regarding the two formulas below to calculate discounted price... e.g. Apple is normally sold for $\$100$, but today you can get them for $20\%$ cheaper! FORMULA #1:...
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0answers
42 views

Is there a connection between the analytical properties of complex analytic functions and the complex numbers being complete algebraically?

I will first have to apologize that this will be a very fuzzy question. At this time I have no better way to formulate it. I am willing to reformulate it when I can better pinpoint what it is that ...
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0answers
51 views

What is the Borel Hierarchy?

I'm taking a measure theory class, and our professor mentioned in passing that the Borel sets "stratify" into a hierarchy. But I'm getting lost in the $\Pi$s, $\Sigma$s, and $\Delta$s on the ...
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0answers
82 views

Visually intriguing unsolved problems which are easy to explain

I have come across a list of visual proofs which are wrong (Visually deceptive "proofs" which are mathematically wrong) visual proofs which are not wrong (Proofs without words) visually ...
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1answer
71 views

Why are commutative diagrams called commutative diagrams?

Why are commutative diagrams called that way? I don't see what's commutative about them. Here is the commutative diagram for the Fundamental Theorem of Homomorphisms, in case a particular diagram is ...
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0answers
48 views

is there a formal way to say let "x be the i-th greatest element in a subset of [n]"

I was wondering if there is a formal way to say that "x is the i-th greatest element in a subset of $[n]$" (or maybe is this formal?). For example if i have $M=\{2,3,5,7,11,20\}$, how can I ...
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0answers
63 views

Does $2^1 \cdot 3^2 \cdot \ldots \cdot p_m^m$ have any interesting properties?

In my research in logic there is an example I like: Skolem arithmetic. I recently added to my quest the following constant: $2^1 \cdot 3^2 \cdot \ldots \cdot p_m^m$ for a fixed $m \in \mathbb{N}$ and ...
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1answer
28 views

About a simple system of equations slightly changed

Well maybe it's a weird question but let me propose it : Let the system : $$ax+by-c=0,ux+vy-d=0$$ This pair of equations is easy to solve but now what happends if : $$ax+by-c\simeq0,ux+vy-d\simeq0$$ I ...
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1answer
71 views

Using a conjecture to solve a conjecture

Is it mathematically correct to use a conjecture to prove another conjecture ? And if the second one is proved, does that mean that we'll only focus on proving the first ? There are many cases that ...
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1answer
141 views

Applied Math Major Roadmap [closed]

A little background for my question: I have been studying Math with a problem/solving strategy until very recent (mostly taking a bunch of interesting problems from contests like IMO and traying to ...
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2answers
56 views

How ro solve these kind of questions by alligation? [closed]

Question: "The ratio of expenditure and savings is 3:2. If the income increases by 15% and savings increases by 6 % then by how much per cent should his expenditure increase?" Doubt: Now ...
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0answers
28 views

Which methods exist for systematic pole&hole identification of unknown meromorphic functions?

The other day I looked back at old study material for complex analysis (one variable) and I remembered the residue theorem - one of the central parts of the course. I thought it could be interesting ...
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0answers
39 views

Prerequisites for V. Arnold's "Ordinary Differential Equations"

I'm looking on getting started with Arnold's famous textbook, however before diving in wanted to specify which prerequisites I should have before reading the book, so as to not get unnecessarily ...
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1answer
108 views

Books to read after AOPS Calculus?

I've been going through the AOPS Calculus textbook, and I genuinely really enjoy reading this textbook. I enjoy the format, where if I had to break it down: 1 - Introduce a problem, or a goal, ...
2
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1answer
69 views

Books/articles about étale cohomology of Grothendieck toposes

Toposes were invented to define étale cohomology (and with that, prove the Weil conjecture). All this is written down in SGA4 and SGA$4\frac{1}{2}$. However, it can be a bit overwhelming, to try to ...
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1answer
59 views

Formalize intuition about comaximal monoids in a commutative ring

One way to understand a ring $A$ is to think of it as the ring of nice global functions on a space $X$ into integral domains or fields. Following this geometric analogy one can think of certain ideals ...
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1answer
34 views

Is there a specific notation to denote which angle is the orthogonal angle of a right triangle?

If an exercise says "the right triangle $ABC$" is there an order of the letters to signify which angle is the orthogonal angle? For example $ABC$ is a right triangle means $\hat B=90$ ...
2
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0answers
35 views

Number of regular $n$-topes in $\mathbb{R}^n$ for $n\in\mathbb{R}$?

Ever since I've come across it, I have been puzzled by the sequence $(a_n)_{n\in\mathbb{N}_0}=(1,1,\infty,5,6,3,3,3,\cdots)$, describing the number of regular $n$-topes in $n$-dimensions, where $a(k)=...
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1answer
80 views

What questions Topology answers?

I was reading the great Thurston's article on Mathematical Education and one thing he points out is: People appreciate and catch on to a mathematical theory much better after they have first grappled ...
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0answers
98 views

Are special functions an active field? A mathematician's dilemma.

Studying analysis, I stumbled upon two branches which I loved: functional analysis and special functions. I began to study q-series and a bit of elliptic functions. However, soon after I talked to ...
3
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0answers
99 views

Easier Way to Display of Commutation Lemma for Limit Operator

By definition, the limit operator $\mathbb{L}$ in sequences with real terms $\{a_n\}$ commutes. For example, $$\color{red}{\lim_n} \color{blue}{\ln n} = \color{blue}{\ln} \color{red}{\lim_n n} \tag 1$...
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0answers
15 views

Reference for stochastic dominance

Is there a good book reference for stochastic dominance properties of random processes on $\mathbb{N}$? For example characterizations of the following property for a stochastic process X(n) on $\...

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