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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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How can this open map characterization be explained or interpreted?

I hope you're having a good day. I'm an undergrad mathematics student, I took a general topology course a few months ago, and I'm now reviewing topological spaces to prepare for functional analysis. I ...
MHDFrags's user avatar
0 votes
1 answer
70 views

Commutative algebra from Hungerford’s algebra [closed]

I just finished a course in abstract algebra (group theory, ring and module theory, field and Galois theory) from Hungerford’s algebra GTM. I want to study algebraic geometry, and commutative algebra ...
user264745's user avatar
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4 votes
1 answer
138 views

(When) are recursive "definitions" definitions?

This is a "soft" question, but I'm greatly interested in canvassing opinions on it. I don't know whether there is anything like a consensus on the answer. Under what conditions (if any) are ...
ac2357's user avatar
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0 votes
0 answers
51 views

Shall proofs be externalised?

I might have a reached a point in a proof I’m writing in which I can take two different directions: I can complete my proof by relying on someone else’s theorem; I can complete my proof without ...
EoDmnFOr3q's user avatar
  • 1,226
-2 votes
0 answers
27 views

Real analysis (series) practice questions.

I have an analysis 1 exam coming up this week, and I am still not particularly comfortable proving series convergence / divergence as well as continuity using the epsilon delta definition. I’m looking ...
klonedrekt's user avatar
0 votes
0 answers
28 views

Connection between twisted graded modules and twisted sheaves

I came across the definitions of graded twisted modules while reading about the syzygy theorem. In the meantime I also attended an algebraic geometry course where twisted sheaves occured. For both ...
Flynn Fehre's user avatar
1 vote
0 answers
56 views

SDEs and irreversibility in statistical mechanics

In statistical mechanics, the equations governing particle behavior are typically deterministic ordinary differential equations (ODEs). However, in real-life particle systems, these systems are never ...
Zhang Yuhan's user avatar
2 votes
0 answers
68 views

Is it a bit circular to say that every thing is equal to exactly one thing?

This is a theorem of first-order logic with equality: $(\forall x)(\exists! y)x=y$, where the $\exists!$ means the unique existential quantifier. However, this theorem seems a bit circular, or at ...
user107952's user avatar
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1 vote
1 answer
71 views

Putnam 2008 B2 -- Soft Question

The 2008 Putnam exam has the following question for B2: Let $F_0(x) = \ln x.$ For $n \geq 0$ and $x>0,$ let $F_{n+1} (x) = \int _0 ^xF_n(t) \, dt$. Evaluate $$\lim _{n \rightarrow \infty} \frac{n! ...
Eli Yablon's user avatar
3 votes
1 answer
77 views

Do exists an inverse Cayley–Dickson construction for deducing lower-dimensional number systems?

Is there a known inverse or reverse Cayley–Dickson construction that enables deduction of numbers in the reverse order, from higher-dimensional to lower-dimensional sets? For example, starting from ...
wepajakeg's user avatar
0 votes
1 answer
28 views

Redundancy in the definition of uniform spaces

Let me paraphrase Wikipedia's definition of uniform spaces: Definition A. A set $X$ endowed with a nonempty collection $\Phi$ of subsets of $X \times X$ is a uniform space if the following conditions ...
Dannyu NDos's user avatar
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3 votes
1 answer
70 views

Topics for undergraduate student seminar on algebraic geometry

I am a masters student, and chatting with undergraduates and a couple professors we agreed on (trying to organize) a student-level seminar on algebraic geometry. Our main motivation is that, while we ...
qmv.01's user avatar
  • 108
6 votes
3 answers
804 views

Math heavy programming challenge book.

I know how to code and some math. I love Project Euler because it combines both math and programming. Please recommend some math heavy programming challenge books as I can't seem to find any on Google....
Harshit Bujar Baruah's user avatar
1 vote
2 answers
41 views

Fermat's last Theorem and elliptic curve cryptography

AFAIR, elliptic curve cryptography became popular soon after Fermat's last Theorem had been proven. Is it just a coincidence, or some important cryptographic properties of elliptic curves follow from ...
Roman Maltsev's user avatar
5 votes
0 answers
164 views

Conjectures involving $\Lambda(n)$

As the title suggests, I am looking for conjectures involving the Von Mangoldt function, $\Lambda(n)$. I understand this is not a rigorous mathematical question, however if reference requests for ...
Mako's user avatar
  • 702
2 votes
0 answers
53 views

Computational framing of topological counterexamples [duplicate]

Bit of a soft question here, but bear with me: Topology is infamous as a source of weird counterexamples. Pretty much anyone who has been through a traditional introductory topology course can recall ...
user3716267's user avatar
  • 1,378
0 votes
0 answers
51 views

Challenging yet accessible problems for middle school students

I'm currently working for a Math summer camp and am tasked with providing 7th grade students with math problems that are challenging yet accessible problems that could be solved by most students ...
Michael's user avatar
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0 votes
1 answer
37 views

Should I finish a proof assuming a (possibly not independent) additional axiom?

I am writing a proof and I do not know how to proceed. In the paper I’m writing, I have assumed four Axioms that I suspect imply one very particular thing. However, the proof contains a step that I ...
EoDmnFOr3q's user avatar
  • 1,226
9 votes
4 answers
960 views

Proof by Contradiction: "Bad Form" or "Finest Weapon"? Reconciling Perspectives [duplicate]

G.H. Hardy famously described proof by contradiction as "one of a mathematician's finest weapons." However, I've encountered claims that some schools of thought consider proof by ...
Nagaraju Chukkala's user avatar
1 vote
1 answer
55 views

Inductive Definition vs Inductive Proof vs Recursive Definition

I read some answers here that tried to explain the difference between "Inductive Definition" vs "Recursive Definition". But I couldn't really understand the difference between the ...
Junsu Kim's user avatar
  • 140
1 vote
1 answer
57 views

Short notation for $(u_1u_2\cdots u_n)' =u_1'u_2\cdots u_n + u_1u_2'\cdots u_n+\cdots+u_1u_2\cdots u_n'$

Looking for a Short hand notation for $(u_1u_2\cdots u_n)' =u_1'u_2\cdots u_n + u_1u_2'\cdots u_n+\cdots+u_1u_2\cdots u_n'$ Or some type of a rolling signifier notation that might already exists in ...
jimjim's user avatar
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0 votes
0 answers
29 views

Are there graphs with multiple edge sets?

I've tried looking this up here and on google, but I think I may not be using the right words. I'm not talking about a mutligraph (multiple edges between the same nodes) or a hypergraph (multiple ...
Sofia's user avatar
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1 vote
0 answers
43 views

Degree with which a polynomial changes with some small change

Soft question: I was curious as to how one could measure the degree with which a polynomial is perturbed. More formally, let $P(x) \in \mathbb{C}$ be a polynomial and $\epsilon$ be a very small number,...
MokutekiJ's user avatar
  • 166
1 vote
0 answers
25 views

Reverse question of superposition of waves

This is a soft question. Consider two Gaussian like pulse waves with different heights, means and variances, and there is an overlap between them. By the principle of superposition, they form a new ...
guoran guan's user avatar
1 vote
0 answers
63 views

What makes a mathematical theorem research paper worthy? Is it purely subjective or are there objective criteria? [closed]

There are infinitely many theorems of mathematics, but only a tiny portion of them are interesting enough to be publishable in a research journal. Is the criterion for which theorems are interesting ...
user107952's user avatar
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1 vote
1 answer
94 views

Are there terminologies for "one-to-one" but not "onto" functions, and "onto" but not "one-to-one" functions?

One-to-one (injective) functions are not necessarily not onto (not surjective). Similarly, onto functions are not necessarily not one-to-one. So, a function can be one-to-one and onto (bijective). $f(...
Hussain-Alqatari's user avatar
1 vote
0 answers
41 views

(Im)possibility of closed-form expression of Clausen functions

When I started learning Riemann zeta function, I was fascinated that $\zeta(2n)$ can be expressed with finite integers and $\pi$ while $\pi$ has no obvious relation with the sum-$\zeta(2n)$ but no &...
Quý Nhân Đặng Hoàng's user avatar
2 votes
1 answer
68 views

How to write an inequality? [duplicate]

I have a very simple doubt regarding the writing of inequalities. Consider any arbitrary finite set $X$ and three functions $f,g,h:X\to\mathbb{R}$. Fix any element $x\in X$ and suppose the following ...
EoDmnFOr3q's user avatar
  • 1,226
3 votes
0 answers
82 views

How to restudy functional analysis

I am re-studying functional analysis to prepare for my next year's PDE courses, I've taken it this year but I feel like I am not very good at it. May I ask what is the study routine/strategy for ...
HIH's user avatar
  • 451
2 votes
0 answers
83 views

Comparing mathematical objects by the "rigidity" of their definitions

A loose interest of mine recently has been ordering mathematical objects by how "combinatorial" their study is, in broad terms. I consider the study of a mathematical object more “...
safsom's user avatar
  • 497
1 vote
4 answers
273 views

How to use "given" in expository writing.

In "How to write mathematics", Halmos says the following. Two digressions about “given”. (1) Do not use it when it means nothing. Example: “For any given $p$ there is a $q$.” (2) Remember ...
Henrique Fonseca's user avatar
0 votes
1 answer
32 views

What is the definition of and quantifier order in consistency for one-step methods?

Consider the usual Cauchy IVP $y = f(x, y)$ with $y(x_0) = y_0$, satisfying assumptions of Picard's theorem in a rectangle $D$. A one-step method is essentially $$y_{n+1} = y_{n} + h \Phi(x_n, y_n; h),...
Linear Christmas's user avatar
0 votes
1 answer
68 views

How to express the originals system of equation in terms of its Groebner bases?

As a network engineer I need to explain some mathematical stuff to my fellow coleagues. Particularly, I need to explain the fact the the Groebner Basis will create an equivalent system. One particular ...
Tuong Nguyen Minh's user avatar
1 vote
1 answer
40 views

References and suggestions for calculus-1 B.Tech [closed]

I have just cleared the entrance examination and am planning on doing Mathematics and computing (B.Tech). The semester has not yet started but I am planning to pre-initiate the studies to get a lead ...
Yash Shrivastava's user avatar
1 vote
0 answers
117 views

Comments on the book Topology by James Dugundji

This post asks for comments on the book "Topology" by James Dugundji. Before I was given a copy of this book, I never read it in detail. When I did, I think this is a very good book. However,...
Ho Man-Ho's user avatar
  • 637
4 votes
1 answer
130 views

I'm turning 18 soon, and I don't know how to do anything regarding college applications at all. [closed]

I'm sorry for how long, whiny, and pointless this will feel. I have not been able to find anyone aside from therapists to discuss any of this with, and I need advice from people who actually know what ...
Nikki's user avatar
  • 67
1 vote
0 answers
35 views

How to accurately average a function with a nonlinear response?

I am a physics PhD student working in optics and I have a bit of a weird problem that I am trying to sort out and I'm hoping you math folks can help me with. Without boring you with the experimental ...
UltrashortGiraffe's user avatar
1 vote
0 answers
38 views

Relation between Arnold tongues and Virasoro algebra

My knowledge in many mathematics topics is far from completeness, so please consider my question as so-called soft question. I know that Viroso group is a central extension of group of circle ...
Artem Alexandrov's user avatar
0 votes
1 answer
118 views

Where exactly are we using the fact that $f$ is an odd function in the proof of the Borsuk-Ulam Theorem? [closed]

Consider the proposition that $f:S^n \to S^n$ is an odd map implies $f$ has an odd degree, which is essentially the proposition used in Borsuk-Ulam theorem's proof as given in Hatcher. Now one point ...
Kishalay Sarkar's user avatar
0 votes
0 answers
49 views

conventional notations for subsets/super-sets and proper subsets/proper super-sets

Background $A\neq B\quad \tag{1}$ Notations: $\subset,\subseteq, \subseteqq,\nsubseteq,\nsubseteqq,\subsetneq,\subsetneqq,\varsubsetneq,\varsubsetneqq \tag{2}$ $\supset,\supseteq, \supseteqq,\...
Seth's user avatar
  • 3,683
3 votes
3 answers
214 views

Book Recommendations for Learning Python for Mathematics.

Lately, I've been finding that I often need to compute various things and graph some pretty complicated functions. I've realized that learning to program, especially in Python, could be really helpful ...
Mathematics enjoyer's user avatar
0 votes
1 answer
88 views

Is there any Complex Analysis book that is very "analysis"? [closed]

I have learned Complex Analysis for once, and I'm trying to review it. Then I found out that most Complex Analysis books are not very "analysis". That is, for typical analysis books(...
M_k's user avatar
  • 1,921
0 votes
1 answer
99 views

Are mathematical journals generally readable for graduated students? [closed]

I'm an undergraduate-level self-taught mathematics student. I'm estimating I might be around the second year of the degree on mathematics in terms of general knowledge. While I love the subject, i'm a ...
Simón Flavio Ibañez's user avatar
1 vote
1 answer
106 views

Why do limits work? [closed]

Started basics of calculus as a high school student , and Limits already confuse me . Can somebody explain it to me why limits actually work . If the value of x tends to infinity ,which is undefined ...
preesha's user avatar
  • 21
0 votes
0 answers
47 views

Book recommendation of complex analysis in several variables

I want to learn complex analysis. I would like a book that teaches about complex analysis in several variables without assuming that the reader knows complex analysis in one variable. The only book I ...
noob's user avatar
  • 43
1 vote
0 answers
98 views

Why does learning theory study generalization bounds?

Disclaimer: I know that mathematics needs no external motivation to be developed, and that such view is (in the long term) helpful even for applications. Nonetheless, I believe it is crucial for ...
Alek Fröhlich's user avatar
1 vote
1 answer
207 views

Do we have complete understanding of $\mathbb N$?

We have some understanding of natural numbers. We have PA theory and we believe that $\mathbb N$ is one of the PA models. But PA can't prove some statements about $\mathbb N$ even though they are true ...
user341's user avatar
  • 157
2 votes
0 answers
57 views

How much notation should there be in a formal proof? Is there a general guideline?

I am writing a conference paper in formal language theory with an involved proof and I've found myself struggling with notation. In particular, I don't know when to favor notation and when to favor ...
JonasPK's user avatar
  • 31
3 votes
0 answers
139 views

Is there a website that has all the special functions? [closed]

There are a lot of special functions, and I wonder if there is a website that collects all of them, similar to how the Encyclopedia of Triangle Centers collects information on triangle centers. ...
pie's user avatar
  • 6,620
4 votes
0 answers
82 views

Are there other usual ways of justifying the passage of differentiation into integral?

I am recently reading Evan's Partial Differential Equations. In the book, the author sometimes passage differentiation into integrals (i.e. $\frac{\text{d}(\int_{\Omega} f(\mathbf{x},y)\text{d}\mathbf{...
Asigan's user avatar
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