Questions tagged [big-list]
Questions asking for a "big list" of examples, illustrations, etc. Ask only when the topic is compelling, and please do not use this as the only tag for a question.
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names for $f^{-1}(0)$
What are commonly-used names for $f^{-1}(0)$, where $f: X \to Y$ for $Y$ some algebraic structure with $0$?
I am particularly thinking of the case where $Y = \mathbb{R}$, where I was expecting that ...
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Integrals with massive amount of cancellation (positive and negative portions of integrand cancel out almost perfectly)
In Gamelin's Complex Analysis, there are exercises/examples (pg. 201-202) of the form
$$\int_{-\infty}^\infty \frac{P(x)}{Q(x)} \cos(ax) dx$$
The first "trick" is to use complex analysis: ...
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What are examples of Halmos's claim that a single small concrete special case can capture every instance of a concept of great generality?
Paul Halmos states:
It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case.
What are examples of ...
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Category-Theory notions and examples in Algebraic Topology
I never really dived in to category theory, you see. I'm familiar with core concepts at a basic level, but I really lack context and depth. Sadly, it is unlikely that I'll have the time to relearn it ...
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List of closed form special cases and transformations of Wolfram language’s inverse beta regularized $\text I^{-1}_x(a,b)$.
The Wolfram Language’s Inverse Beta Regularized $\text I^{-1}_z(a,b)$ is a quantile function. This applicable yet obscure function appears in Excel as BETA.INV and a special case of it as the Inverse ...
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Recommendation for Examples (calculations) to many topics in maths.
do you know good websites/books/PDFs in which many examples to topics are discussed? What I mean by that is, I am looking for sources where to each topic there are plenty of detailed discussed ...
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Examples of random variables with discrete distribution
I'm studying probability theory with the concepts of measure theory. Now I'm learning about discrete distribution, however I'm having a hard time trying to find examples of random variable (measurable ...
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Topical example of a universal arrow to and from a functor
If $ S\colon \mathsf D\to \mathsf C $ is a functor between categories and $ c\in \mathsf C $, Mac Lane in his Categories for the Working Mathematician defines a universal arrow from $ c $ to $ S $ as ...
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Unique prime numbers
This is a question out of curiosity. For example:
$2$ is the ONLY even prime number.
$5$ is the ONLY prime number whose last (or only) digit is $5$.
$73$ is the ONLY prime number which satisfies ...
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Counterintuitive instances of intransitivity
Recently, I read a fascinating article by E. Klarreich on intransitive dice. Among other things, the authors give an example of a triplet of dice $A$, $B$, and $C$ such that the probabilities of $A$ ...
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What are all known formulas for the determinant of a matrix?
Suppose that $A = (a_{i, j})_{i, j \in [n]}$ is any square matrix. This question is about collecting the known formulas for $\det A$ (which don't depend on $A$ having a special form or dimension) in ...
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Examples of classes of structures which are "surprisingly" axiomatizable.
This is a bit of a soft question, but I am interested in a list of classes of structures (in the sense of model theory) which are "surprisingly" first-order axiomatizable classes. Meaning, ...
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Interesting ways to write 2023
The year 2023 is near and today I found this nice way to write that number:
$\displaystyle\color{blue}{\pi}\left(\frac{(\pi !)!-\lceil\pi\rceil\pi !}{\pi^{\sqrt{\pi}}-\pi !}\right)+\lfloor\pi\rfloor=...
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Finger's Constant
A fingers'constant is defined as to have as first decimal :
$$1.2345...$$
I ask for the most emblematic example .
For example :
$$2-\prod_{k=1}^{\infty}\left(1-\frac{1}{\left(k+1\right)e^{k}}\right)$$
...
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Simple results that are clarified by abstract algebra
There are many deep results whose proofs rely heavily on the abstraction of modern algebra (e.g., the unsolvability of the quintic). But what are some simple results (i.e., results that can be proven ...
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What properties characterize squarefree integers?
It is known that, if a positive integer $m$ is squarefree, then the following properties hold:
If $m \mid n^2$ holds, then $m \mid n$ is true (where $n$ is a positive integer).
The equation $m = \...
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Results/Objects with Multiple Generalisations
The Fundamental Theorem of Calculus has at least two generalisations: Stokes's Theorem, and the Radon-Nikodym Theorem. As far as I know, there is no meta-generalisation, from which both can be derived,...
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(Fast) Algorithms for decomposing polyominoes into rectangles
Here are some polyominoes:
The general definition of a polyomino allows for holes (I think). This question restricts itself to those polyominoes that are hole-free.
It would be nice if the algorithms ...
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Usage examples of Jordan canonical form
I would like to know what the Jordan canonical form is useful for.
This answer says that “It simplifies many abstract proofs to assume a matrix in the proof is in Jordan canonical form”.
Can you give ...
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What different ways are there to construct the Lebesgue measure?
I have recently learned of different approaches (which I've included below) to constructing the Lebesgue measure, and I'm somewhat startled by how much each approach can illuminate the theory as a ...
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Theorems about finite sets the proof of which require the notion of infinite set
I believe that there should exist theorems about finite sets which are not provable without the notion of infinite sets.
I am curious if I am right.
What are the examples of such theorems if they ?
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Any "reasonable" and "natural" false graph theory conjectures for which the Petersen graph is not a counterexample?
I heard that the Petersen graph is a good test of a graph theory conjecture. As in, if you want to know whether a graph theory conjecture is false, the Petersen graph is likely to be a counterexample. ...
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Applications of non-symmetric bilinear forms?
There are numerous applications of symmetric bilinear forms: They show up in the Hessian matrix in optimisation, for instance. What I'm curious about is where asymmetric bilinear forms occur.
I know ...
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How to scale down a list of integers
I have a list, that can be of various lengths (less than 100 or more than 1.800.000) and I need to
"shrink down" the list to 800 values.
I have tried taking a value for every step, where a ...
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What are some slick proofs of basic algebraic facts? [closed]
Couple of years ago, I saw a proof in Horn's "A Second Course in Linear Algebra" of the fact that if a matrix (over a field) has a right inverse then it is automatically invertible, ...
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Examples of inappropriate credit in mathematics [duplicate]
What are some known cases of inappropriate credit in mathematics?
Here, "inappropriate credit" means a common attribution that is not totally fair because someone who also discovered the ...
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Novelty topics for the dissemination of research mathematics on paper
In this post I was inspired* in the article [1] (in Spanish), and some books that I know, books as the Spanish edition of Birth of a Theorem: A Mathematical Adventure due to Cédric Villani or The ...
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Chronological list of Algebraic Systems
In "A Book of Abstract Algebra", Second Edition by Charles C. Pinter, he states
"Other exotic algebras arose in a variety of contexts, often in connection with scientific problems. ...
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What should you do if you have a research idea outside your area of expertise?
Background
Sometimes, I have ideas for research in mathematical subjects about which I don't know much. Let me describe an example to make it more concrete.
In his 2009 article "The ...
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What are some extensively studied mathematical objects NOT yet proven to exist?
I am currently making a presentation for some undergraduate students entering a mathematics B.S. program. I'd like to add a slide about the fun of math, and part of that (to myself, at least) is that ...
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Famous or semi-famous cases of math professors making errors
I don't know if this question is best suited to this stack exchange. If it isn't, feel free to migrate it or close it. This question was inspired by a mistake I saw in a math class. I corrected the ...
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Mathematical applications of Egyptian fractions
Background
In chapter 5 of the book "Number Theory in Science and Communication" by Manfred E. Schroeder, the author goes into continued, Egyptian and Farey fractions. On p. 65, he writes:
&...
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Are $\cosh(1), \sinh(1), $ and $e^{1/r}-1 $ the only known constants with regular Engel expansions?
Background
If we define the Engel expansion of a number $x$ as the unique increasing sequence $x_{E}:= \{a_{1}, a_{2}, a_{3}, \dots \} $ such that $$x = \frac{1}{a_{1}} + \frac{1}{a_{1}a_{2}} + \frac{...
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Reference request: which theorems are "interesting" to mathematicians?
Disclaimer: this question is more about philosophy of mathematics than technical mathematics.
Mathematicians always need to choose what to focus their work on. Many pure mathematicians like to say ...
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Graph Transformation
Knowing dealing with graph transformations come handy MANY times. I searched on google to get a comprehensive graph transformation list but couldn't find one. Some good while back I learned them all ...
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Examples of well-known unsolved problems, in subjects don't related to number theory, that have several equivalent formulations
I know statements for some unsolved problems related in some way to number theory, being stated (I refer for each one of these unsolved problems) several equivalent formulations, for example: these ...
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Venn diagram for basic types of operators
This Venn diagram is an attempt to visually classify densely-defined linear operators between Banach spaces (self-adjoint operators are an exception, defined between Hilbert spaces). Operators are ...
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Examples of cases where it is easier to define the negation of a property than the property itself
Are there cases in mathematics where it is easier to define the negation of a property $P$ than property $P$ itself? I would like several answers giving examples of such cases. Basically, I am looking ...
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What purely real analytic techniques are there to evaluate $\int_{-\pi/2}^{\pi/2}\frac{1}{1+\sin^4(x)}\,\mathrm{d}x$?
$\newcommand{\d}{\,\mathrm{d}}$Last night, I evaluated the following integral: $$\begin{align}I:&=\int_{-\pi/2}^{\pi/2}\frac{1}{1+\sin^4(x)}\d x\\&=\int_{-1}^1\frac{1}{(1+x^4)\sqrt{1-x^2}}\d x\...
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List of Naturally Isomorphic Real Vector Space Pairs
Let's consider the category of finite dimensional real vector spaces (VS) / inner product spaces (IPS).
Which of the following pairs of isomorphic vector spaces (given appropriate dim constraints), ...
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Is there a list of logics?
There are a lots of logics. Some of them are:
Propositional logic
Predicate logic
Second order logic
$n$ order logic
Fuzzy logic
Modal logic
Multivalued logic
etc
So I`d like to know whether there ...
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Is there any relationship between a Dirichlet series and the same series with the sequence "shifted" by one term?
Suppose
$$ F(s) := \sum_{n=1}^{\infty} \frac{a_{n}}{n^{s}} $$
is a Dirichlet series for the sequence $a_{1}, a_{2}, \ldots\in\mathbb{C}$. Then let
$$ G(s) := \sum_{n=1}^{\infty} \frac{a_{n}}{(n+1)^{s}}...
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Build a "rich" first-order logic within a given category
I would like to know a mathematical framework with an internal logic where isomorphic objects can be considered equal.
For example, consider the rationals $\mathbb{Q}$. With this set we can construct ...
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A Collection of Bogus Proofs
Hello M.S.E. people,
This question is just for fun, don't take it seriously :). We have all encountered Bogus Proofs, which seem logical and reasonable, but they prove some claims which are completely ...
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Strong ("second principle") mathematical induction: non-trivial examples
The fundamental theorem of arithmetic (FTA) can be proved using the following:
if a statement is true for $n=1$, and its truth for $n=1,2,\ldots,k$ implies its truth for $n=k+1$, then it is true for ...
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Obscure Results and Areas of Mathematics
My degree is in economics although I did relevant mathematics in college and used algebraic topology in my thesis. Over the Pandemic I've been reading a lot of mathematics that I didn't get a chance ...
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Mathematics hiding in plain sight
What are some basic math facts (say, secondary or early undergraduate level) that somehow went unnoticed by you for a long time, and when you realized they made you wonder how you could have missed ...
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Good articles/books to learn about tensors / tensor products?
Currently a mathematics graduate student at a state university. I'm looking for articles or books (preferably free PDF links) that nicely explain tensors and tensor products at a graduate student ...
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Examples of nontrivial proofs by cases in which only one case is ever realized
For teaching reasons, I'm looking for examples of proofs that use a nontrivial case breakdown in which only one case is ever realized, and yet it is very hard to prove which case is the "real&...
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Mathematicians endorsing platonism -- examples?
Take platonism to be the view that there are abstract mathematical objects which exist independently of us as mathematicians and our language, thought, and practices.
Looking at the Stanford ...
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