Questions tagged [gm.general-mathematics]

Questions about mathematics which don't fall into the other arXiv categories. If you have a general question about mathematics but it is not research level, it's off-topic but it might be welcomed on Mathematics Stack Exchange.

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An ambitiouser binomial coefficients sum

I asked how to calculate $$\sum_{i = 0}^b(-1)^i\binom{b}{i}\binom{a+b-i-1}{a-i}$$ and got amazing answers. A bit later, however, I figured I needed something rather more complicated: I need to find ...
Valentino's user avatar
  • 369
-1 votes
3 answers
300 views

Binomial Coefficients sum [closed]

Any idea on whether or not $$\sum_{i = 0}^b(-1)^i\binom{b}{i}\binom{a+b-i-1}{a-i}$$ has a closed formula on $a$ and $b$ (and on what it is, in case it does)? It is supposed that $b \le a$.
Valentino's user avatar
  • 369
23 votes
14 answers
4k views

Math talk for all ages

I've been asked to give a talk to the winners of a recent math competition. The talk can be entirely congratulatory, or it can contain a bit of actual mathematics. I'd prefer the latter. I'd also ...
7 votes
1 answer
1k views

Do mathematicians use notebooks to keep their results these days? [closed]

Mathematicians work a lot and are usually inspired by many things. In their lifetimes they get to publish only portions of their results. There have been stories of how Gauss, Euler, Ramanujan, ...
27 votes
6 answers
5k views

Has any open/difficult problem in ordinary mathematics been solved only/mostly by appeal to set theory?

We know that many (if not all) mathematical notions can be reduced to the talk of sets and set-membership. But it nevertheless sounds like a grueling task (if at all possible) to actually get advanced ...
7 votes
1 answer
471 views

Abandoned notions in mathematics? [duplicate]

I'm looking for examples of abandoned or demised notions/concepts in mathematics, preferably (but not necessarily) after the age of foundations. To be clear: I'm not looking for abandoned ideas or ...
qk11's user avatar
  • 505
1 vote
0 answers
303 views

Online courses for mathematics [closed]

I'm sorry if I'm posting this in the wrong forum. My background is in biology and medicine. I am looking to re-learn undergraduate-level mathematics, in particular discrete mathematics, calculus, and ...
Ansel Lim's user avatar
43 votes
5 answers
15k views

How to deal with an advisor that offers you nearly no advising at all?

I am a young PhD student (24) at a Germany university and I am not sure whether this is the right place to ask this kind of question. If not feel free to move it elsewhere or delete it completely. ...
15 votes
9 answers
2k views

Tools from other disciplines useful to mathematics research?

Obviously, mathematics provides essential tools for physicists, biologists, economists, engineers and many others to use in their research. Equally obviously, physics, biology, economy and engineering ...
70 votes
22 answers
10k views

Small ideas that became big

I am looking for ideas that began as small and maybe naïve or weak in some obscure and not very known paper, school or book but at some point in history turned into big powerful tools in research ...
1 vote
1 answer
183 views

Examples of conjectures whose direct falsity implies different consequences than indirect falsity

Mathematics several times has statements of form $$\mathsf{Statement A}\implies\mathsf{Statement B}$$ where $\mathsf{Statement A}$ and $\mathsf{Statement B}$ are conjectures while the implication is ...
VS.'s user avatar
  • 1,816
38 votes
8 answers
5k views

Counterexamples against all odds

What are some examples of conjectures proved to be true generically (i.e. there is a dense $G_{\delta}$ of objects that affirm the conjecture) but are nevertheless false? Also, it would be cool to see ...
2 votes
0 answers
166 views

Dreaming of mathematics [duplicate]

Ramanujan said that his mathematical inspirations came in dreams (https://en.wikipedia.org/wiki/Namagiri_Thayar, https://www.foxnews.com/science/100-year-old-deathbed-dreams-of-mathematician-proved-...
user avatar
77 votes
15 answers
13k views

Each mathematician has only a few tricks

The question "Every mathematician has only a few tricks" originally had approximately the title of my question here, but originally admitted an interpretation asking for a small collection ...
163 votes
46 answers
31k views

Every mathematician has only a few tricks

In Gian-Carlo Rota's "Ten lessons I wish I had been taught" he has a section, "Every mathematician has only a few tricks", where he asserts that even mathematicians like Hilbert ...
18 votes
1 answer
2k views

Uses of Zorn's Lemma when the thing is actually unique

There is a revised version, which I might substitute for this one, but I would like to keep this as evidence of priority for the "special condition". Are there uses of the sledgehammer Zorn'...
Paul Taylor's user avatar
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0 votes
0 answers
56 views

"Anti-Leibniz order"

It seems that some people use the term "anti-Leibniz order" for what I'd call the "diagrammatic order" of composition: writing $f;g$ for the composition of $f$ and $g$ instead of $g\circ f$. (I have ...
Uli Fahrenberg's user avatar
26 votes
12 answers
2k views

Examples of improved notation that impacted research?

The intention of this question is to find practical examples of improved mathematical notation that enabled actual progress in someone's research work. I am aware that there is a related post ...
1 vote
0 answers
184 views

Studying the vast world of Number Theory [closed]

I'm a high school student, interested in mathematics, especially in number theory. While preparing for the IMO test, and thinking about generalizations or the root of many olympiad problems led me to ...
Junsukim's user avatar
  • 141
34 votes
5 answers
4k views

The origin(s) of the word "elliptic"

The word elliptic appears quite often in mathematics; I will list a few occurrences below. For some of these, it is clear to me how they are related; for instance, elliptic functions (named after ...
Jens Reinhold's user avatar
13 votes
2 answers
627 views

Number triangle

This question arose just out of curiosity. Note the triangle of 0-1's below, whose construction is as follows. Choose any number, say 53 as done here. The first line of the triangle is the binary ...
DSM's user avatar
  • 1,196
8 votes
1 answer
774 views

Recreational mathematical papers [closed]

Sometimes it is nice to get a less technical paper on mathematics to read and learn something different for a change. These papers often make us discover some new curiosity, to think about the process ...
IamWill's user avatar
  • 3,151
2 votes
0 answers
361 views

Advice for graphic tablet for math [closed]

With the current Coronavirus disease (COVID-19), many of us had to switch all our activity to full online mode. I am wondering whether some of you had the chance to use graphic tablets. I am looking ...
Raziel's user avatar
  • 3,153
-3 votes
1 answer
460 views

PCP theorem to check hard proofs [closed]

Is it technically possible to check formidable proofs like Mochizuki's using PCP theorem before mathematicians spend time in understanding the mechanics of the proof? If so why have mathematicians not ...
VS.'s user avatar
  • 1,816
9 votes
1 answer
908 views

How often do you put your research into trash?

A soft question. I am a PhD student, at early stages of my academic career; and have personally experienced the following many times. Sometimes you come up with a result, that you are not quite ...
0 votes
1 answer
87 views

Rule to determine rotationally invariant orders of the points of arbitrary 2d splines

I would like to find a rule to determine the order of the points of arbitrary 2d splines, which should be invariant with respect to rotation (as far as possible). To illustrate the problem, let us ...
Nos's user avatar
  • 103
3 votes
1 answer
389 views

Journey into a strange wilderness [closed]

W. S. Anglin wrote Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the ...
11 votes
3 answers
2k views

How can I simplify this sum any further?

Recently I was playing around with some numbers and I stumbled across the following formal power series: $$\sum_{k=0}^\infty\frac{x^{ak}}{(ak)!}\biggl(\sum_{l=0}^k\binom{ak}{al}\biggr)$$ I was able ...
Susp1cious's user avatar
81 votes
6 answers
11k views

Is data science mathematically interesting?

I have seen a plethora of job advertisements in the last few years on mathjobs.org for academic positions in data science. Now I understand why economic pressures would cause this to happen, but from ...
0 votes
1 answer
566 views

What exactly does it mean for a definition/example to be informal in Math? [closed]

Came across "(Informal)" while reading Analysis I by Tao . What exactly constitutes an example or a definition that is formal ? Definition 3.1.1. (Informal) We define a set A to be any unordered ...
SV98's user avatar
  • 3
1 vote
5 answers
638 views

Famous conjectures named after a mathematician that were resolved in their lifetimes [closed]

This is a question that I thought about recently, and I thought would be interesting to the MO community. What are some famous conjectures, more specifically those that attracted a lot of attention ...
2 votes
3 answers
961 views

Why does $\sqrt 5$ occur in manageable situations of these scenarios? [closed]

Banach-Mazur distance between $P_5$ and $P_3$ is $d(P_5,P_3)=1+\frac{\sqrt5}2$ where $P_n$ is regular polygon in $n$ sides https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7968198&tag=...
VS.'s user avatar
  • 1,816
24 votes
5 answers
6k views

Is the field of q-series 'dead'? [closed]

I had a discussion with my advisor about what am I interested as my future research direction and I said it is special functions and q-series. He laughed and said that the topic is essentially dead ...
5 votes
5 answers
269 views

Peculiarities in low dimensions or low order or etc

I have been pondering about certain conjectures and theorems viewed as either low vs high dimensions, or smaller vs larger primes, or anything of the sort "low vs high order". Let me mention a couple ...
2 votes
0 answers
74 views

How can I prove that the following function is increasing according to x1?

Suppose that $0 \le {X_1} < {X_2} < {X_3}$ . How is it possible to prove the following function is increasing based on ${X_1}$ in the range of $0 \le {X_1} < {X_2}$ ? $f({X_1},{X_2},{X_3})...
Hamed's user avatar
  • 21
0 votes
1 answer
293 views

Integrals I am curious about [closed]

Let $C_n(x) = \frac{n!}{\Gamma(x)\cdot \Gamma(n-x)}$ Is the following true: $\int_{0}^{n} [C_n(x)\cdot y^x \cdot (1-y)^{n-x}dx] = 1$?? just wondering In generality for continuous functions $f,g$ ...
user13953's user avatar
  • 113
4 votes
0 answers
589 views

What solutions to useful computational problems could be rewarded through cryptocurrency smart contracts?

What kinds of cryptocurrency smart contracts could be used to reward people for solving specific kinds of useful computational problems? Background In this question, I asked for proposals for useful ...
Joseph Van Name's user avatar
4 votes
0 answers
224 views

Looking for U.K. problem column (?) from 1980s

While digging through some dusty corners of my file cabinet, I found a photocopied sheet of eight (handwritten) problems from 1985 that I recall receiving from my secondary school mathematics teacher ...
Timothy Chow's user avatar
  • 78.1k
22 votes
1 answer
2k views

What is known about the common knowledge of mathematicians outside their field?

When giving a talk or writing a paper intended for non-specialist (i.e., mathematicians not specializing in the topic being discussed), the question inevitably occurs of what one can assume to be "...
19 votes
2 answers
731 views

Permanent archival of errata/corrigenda for published papers

(Note: This question might be off-topic for MO but the only other plausible alternative that comes to mind is Academia Stack Exchange, and there are some features of this question that are, to some ...
Timothy Chow's user avatar
  • 78.1k
5 votes
1 answer
308 views

Applications of De-Bruijn Sequences in "Pure Mathematics"

I know of a few applications of De-Bruijn Sequences and De Bruijn Graphs in combinatorics, applied mathematics, Engineering and computer science. But I have only found one application of De Bruijn ...
Serge the Toaster's user avatar
3 votes
1 answer
426 views

Improvements to one's own theorems

What are some notable (famous?) instances where the following has occurred. A particular author proves: Every P which satisfies Q has property Z. A few years later (roughly speaking) the same author ...
9 votes
0 answers
295 views

List of modern points of view simplifying or clarifying classical topics

There are many modern mathematical achievements which greatly clarify or (and) simplify classical important topics. I believe a list of such achievements, among other benefits, would be a big help for ...
0 votes
1 answer
675 views

"Mathematics is the science of the infinite" [closed]

The title is the first sentence of Hermann Weyl's 1930 essay, "Levels of Infinity." He focuses on "the distinction between actuality and potentiality, between Being and Possibility." He opines ...
Joseph O'Rourke's user avatar
5 votes
1 answer
332 views

A variant of Cauchy-type functional equation conjecture

Let $f:\mathbb{C}\to \mathbb{C}$ be a complex function such that $$|f(x-y)|=|f(x)-f(y)|,\qquad x,y\in\mathbb{C}.$$ Is it true that $$f(x+y)=f(x)+f(y),\qquad x,y\in\mathbb{C}?$$ The answer is ...
math110's user avatar
  • 4,230
14 votes
1 answer
1k views

Research semester in math

Admittedly, this is a soft question. My own experience in mathematical research has been of long periods of research, mostly characterized in long "blocks" and sporadic breakthrough. How does this ...
3 votes
2 answers
267 views

smallest square containing k non-overlapping equal rectangles at any orientation

This seems like something that should have a known answer, but I haven't found it after some time alternating between searching and generating multiple pages of algebra. I'm interested in $k=4$ and $...
rectangle packer's user avatar
-1 votes
2 answers
366 views

Is this expression always irrational? [closed]

Is it right that $$\sqrt[a]{2^{2^n}+1}$$ for every $$a>1,n \in \mathbb N $$ is always irrational?
Zoetrope's user avatar
3 votes
2 answers
166 views

Terminology: product on strict preorders corresponding to direct product of preorders?

I’ve had trouble finding a well-established term for the following very obvious and elementary construction on strict partial orders (i.e. transitive, irreflexive relations): Given two strict partial ...
Peter LeFanu Lumsdaine's user avatar
93 votes
20 answers
10k views

Short papers for undergraduate course on reading scholarly math

(I know this is perhaps only tangentially related to mathematics research, but I'm hoping it is worthy of consideration as a community wiki question.) Today, I was reminded of the existence of this ...

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