# 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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### 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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### 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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### 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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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 $50/50$...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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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:... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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, ... 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 ... 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 ... 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$\hat B=90$... 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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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 ...