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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Examples of self adjoint compact operators on Hilbert spaces

As the titles says, I'm looking for examples of self adjoint compact operators on Hilbert spaces. So far I know of the diagonal operator on $\ell^2$, $$ (Tx)_i = \alpha_ix_i $$ for some sequence $...
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1answer
44 views

How to find a prime $p$ from only knowing $x^2\bmod p$ for suitably chosen (small and few) value of $x$?

This question is related to the CodeChef problem Guess The Prime of the now closed July 2019 competition. It seems to have caused some stir-up recently during and even shortly after the contest. ...
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13answers
486 views

Where does $\pi^2$ appear spontaneously within Physical Phenomenon and Mathematics Equations?

The term $\pi$ is found to appear in many equations and natural phenomenon; however my question is related to $\pi^2$. While trying to figure out the reason for some $\pi^2$ terms appearing in ...
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35 views

List of possible ways to product two groups together.

Given three groups $A,B,C$ and two homomorphisms $f:A \rightarrow B$ and $g: B \rightarrow C$ such that $f$ is one-to-one, $g$ is onto and $Im(f) = ker(g)$. Then $$A \longrightarrow_{f} B \...
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1answer
74 views

Most elegant computation [closed]

A lot of enphasis in pure mathematics in placed on beautuful and instructional proofs, and for good reason. However, in some circles the quest for elegant formalism can come at the cost of neglecting ...
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46 views

Problems with physics-inspired solutions?

I'm looking for cool, competition-style problems that borrow ideas from physics in their solution. For example, one I found in the book Putnam and Beyond goes like this: Orthogonal to each face of ...
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1answer
27 views

real life usages of (x+abs(x))/2

I know for all $x<=0$ that $y=0$ and that for all $x>=0$ that $y=x$. I have been wondering if there are any real-life uses of this equation
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47 views

Has the rigour of real analysis shown some unexpected truths about real life that would have otherwise not be discovered?

It is a common question, "what is the use of real analysis", and the answer is usually "it adds rigour to our mathematical tools and machinery to make sure that they work without just saying they do". ...
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1answer
58 views

What interesting mathematical identities (or theorems) have been proven using quaternions?

I've heard about the four square theorem and how it was proven using quaternions, which I found to be extremely fascinating, I was wondering whether there are any other interesting theorems which seem ...
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1answer
69 views

Discrepancies in mathematical definitions. [closed]

What are some examples where there is a discrepancy in the mathematical definition of a term? For example : $\bullet$ Isosceles triangle: "exactly two sides are equal" or does it say "minimum two ...
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10answers
481 views

List of integrals or series for Gieseking's constant $\rm{Cl}_2\big(\tfrac{\pi}3\big)$?

Catalan's constant $K$ can be defined as, $$K = \text{Cl}_2\big(\tfrac{\pi}2\big) = \Im\, \rm{Li}_2\big(e^{\pi i/2}\big)= \sum_{n=0}^\infty\left(\frac1{(4n+1)^2}-\frac1{(4n+3)^2}\right)=0.91596\dots$$ ...
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2answers
215 views

List of not-especially famous problems in undergraduate level mathematics

I know lists of problems like these have been compiled before, but most tend to collect either extremely difficult problems ( like Collatz conjecture in a question about number theory ) or ...
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1answer
186 views

Closed forms of Nielsen polylogarithms $\int_0^1\frac{(\ln t)^{n-1}(\ln(1-z\,t))^p}{t}dt$?

(This summarizes my posts on Nielsen polylogs.) I. Question 1: How to complete the table below? Consider the special cases $z=-1$ and $z=\frac12$. Given the Nielsen generalized polylogarithm, $$S_{n,...
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1answer
179 views

Errata for Bott Tu Differential Forms

My book is Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott of which An Introduction to Manifolds by Loring W. Tu is a prequel. I am making this as suggested here An old "...
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9answers
216 views

Theorems about the number $5$? [closed]

I knew three famous theorems related number five in mathematics. Golden ratio. $\varphi=\frac{\sqrt{\boxed{5}}+1}{2}$. Consequence of euler characteristics. There are only $\boxed{5}$ platonic ...
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51 views

Need Reference calculus problems book

I research problems book in calculus/analysis(series/sequences and integrals) the subject is only technique/good problems.I give you some sample that i found.(sorry for the language): Exercices in ...
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0answers
822 views

Mathematics for the afterlife

Paul Erdős said this about the $3n + 1$ conjecture: Mathematics may not be ready for such problems. Similarly, there are parts of mathematics that I am not yet ready for. Some things, however, I ...
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3answers
62 views

Different ways of solving this simple modulus inequality

Consider the set $S$ of all $z\in\mathbb C$ for which \begin{equation}\tag{1}\label{1} \left|\frac{2+z}{2-z}\right| \le 1. \end{equation} One can easily find that $S=\{z\in\mathbb C \, :\, \Re(z) \...
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1answer
337 views

Applications of the 5/8 Theorem

The 5/8 theorem for compact groups says the following: Theorem (5/8 Theorem for Compact Groups) Let $G$ be a compact Hausdorff topological group with Haar measure $\mu$. If $G$ is not abelian then ...
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4answers
294 views

Big list: Free textbooks and resources

We have had multiple book-recommendation, but I have not found a question where the many free (yet legal) books are available under one umbrella. Therefore I propose a thread where all the best, free ...
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1answer
28 views

Optimal algorithm to find maximum value of dot product of two lists

I am looking for an optimal (fastest) way to find a maximum value of dot product of two lists. A list can be rearranged freely in order to maximize the result. My idea for this was to quicksort two ...
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3answers
488 views

Examples of subgroups where it's nontrivial to show closure under multiplication?

Usually when a subgroup is declared, it is trivial (or at least straightforward to a sophomore) to prove that it is a subgroup under multiplication. For example: Homomorphic image and preimage of a ...
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4answers
111 views

Examples of contravariant functors

I understand the definition and usefulness of the notion of functor. But I am worrying about the usefulness of the notion of a contravariant functor. Wikipedia writes: There are many constructions ...
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1answer
31 views

Is there a list of all the possible operations for a certain object? (Wording might be wrong)

I have been thinking about this for some time and I think this is the question I wanted to ask here, since it's more straightforward than asking "When is a mathematical operation/symbol manipulation ...
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2answers
189 views

Real Examples of Misleading Statistics

I need to give a presentation to a group of students on Tuesday about why one needs to be careful when examining statistics or mathematical results in the media or online. In his book How Not To Be ...
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1answer
68 views

Most important non-elementary functions in Math

I am looking for non-elementary function that is Well behaved over $\mathbb{R}$ and $\mathbb{C}$ like piecewise smooth, not like Weierstrass function Ubiquitous and well studied Simple to compute ...
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13answers
4k views

Are there any other methods to apply to solving simultaneous equations?

We are asked to solve for $x$ and $y$ in the following pair of simultaneous equations: $$\begin{align}3x+2y&=36 \tag1\\ 5x+4y&=64\tag2\end{align}$$ I can multiply $(1)$ by $2$, yielding $...
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2answers
85 views

Combinatorics problems that can be solved via infinite descent

I'm looking for high school problems that can be solved with the method of infinite descent. Usually, those problems are from number theory, but I would be very happy if someone could provide a ...
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0answers
192 views

Figures and Numbers: Relating properties of geometric shapes and their Fourier series

Consider two types of parametrized curves $\gamma:[0,2\pi]\rightarrow \mathbb{R}^2$ open curves $\gamma_\sim(t) = (t,a(t) + b(t))$ closed curves $\gamma_\bigcirc(t) = (a(t),b(t)) = a(t) + ib(t)$ ...
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0answers
52 views

Interesting applications of density to prove difficult theorems

If we wanted to prove certain statements for every element of a set $S$, a possible approach is to prove the statement for a certain dense subset $S'\subset S$ (with respect to a certain metric), then ...
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3answers
696 views

Equivalence of different ways of geometrical multiplication

There are at least five ways to multiply two natural numbers $a$ and $b$ given as integer points $A$ and $B$ on the number line by geometrical means. Two of them include counting, the others are ...
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Errata in Convex Analysis and Minimization Algorithms by Hiriart-Urruty and Lemaréchal.

Hiriart-Urruty and Lemaréchal have written Convex Analysis and Minimization Algorithms I and II. Is there an errata list for these books?
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437 views

Examples of the Pigeonhole Principle

As most of you might know, the Pigeonhole Principle basically states that If $n$ items are put into $m$ containers, with $n>m$, then at least one container must contain more than one item It ...
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0answers
35 views

List of Integration Formulas [duplicate]

While going through some of my notes, I came across a formula for evaluating an integral of the form: $$\int_0^\infty {\frac{dx}{x^n+1}} = \frac{\pi}{n\sin(\frac{\pi}{n})}$$ I was wondering if any ...
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0answers
96 views

Algebra & Artificial Intelligence (AI)

Artificial intelligence, especially deep learning & neural networks for image processing and classfication, are related to statistics and physics e.g. as decribed in below papers. Statistics and ...
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1answer
427 views

Category Theory & Artificial Intelligence (AI)

Category theory turns out to be useful in more and more areas. (see e.g. MSE - Category Theory & Biology) Question. Does anyeone know of some connection of category theory to (convolutional) ...
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1answer
85 views

Make a list of algebraic, topological and dynamical properties of the circle.

For example, there is only one cyclic group of order $m$ in the circle, the circle is compact and connected, all its proper compact and connected subspaces are arcs, is a minimal dynamical system ...
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2answers
73 views

Multiple proofs of $\sum_{d|n}{\phi(d)}=n$ [duplicate]

I am looking for multiple proofs of that statement: here $\phi(n)$ denotes the Euler’s totient $$\sum_{d|n}{\phi(d)}=n$$ Here’s one: By unique factorisation theorem: $n=\prod_{k=1}^{m}{p_k^{\...
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0answers
85 views

Collection of less well-known, non-trivial, elegant story proofs (ie, “double counting proofs”) of combinatorial identities

By story proof I mean proving a combinatorial identity by counting the number of elements of some carefully chosen set in two different ways to obtain the different expressions in the identity. The ...
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1answer
64 views

Pathological, continuous functions

Today in my introduction to measure theory course, the professor said that often when we think of continuity, what we're actually thinking about is smooth functions. We've studied the Cantor set and ...
12
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2answers
677 views

Important Olympiad-inequalities [duplicate]

As an olympiad-participant, I've had to solve numerous inequalities; some easy ones and some very difficult ones. Inequalities might appear in every Olympiad discipline (Number theory, Algebra, ...
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2answers
166 views

The “I wish I had read that first” textbook list

Here, I am trying to collect as many "I wish I had read that textbook first" textbook names as possible, for as many different topics in mathematics as possible. The "I wish I had read that textbook ...
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2answers
239 views

“Milk” the integral $\int_0^\infty\left(\frac{x^2}{x^4+2ax^2+1}\right)^r\frac{x^2+1}{x^2(x^s+1)}\mathrm dx$

I found the following integral in chapter $13$ of Irresistible Integrals, and I would like to see which conclusions you can reach from it. My goal in asking this question is to see which methods I can ...
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2answers
92 views

What natural ways of partitioning a group $G$ are there?

What natural, or at least useful, ways are there to partition a finite group $G$? The two examples that come to mind are: Partitioning $G$ into all the left (or right) cosets of a subgroup $H$ of $G$....
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1answer
79 views

Html5 Math Applets. Interactive free online

I was an aficionado at collecting links from websites with math java applets that allowed interaction to learn mathematical concepts visually and interactively. My favorite was http://www.ies-math.com/...
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1answer
323 views

Killing a butterfly with a bazooka [duplicate]

Let $n\ge3$. Prove that $\sqrt[n]2\notin\Bbb Q$. Let us suppose that $\sqrt[n]2=p/q$, that is $2q^n=p^n$, so $q^n+q^n=p^n$, against FLT. Do you know similar examples, in which simple problems are ...
12
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1answer
372 views

Collection: Results on stopping times for Brownian motion (with drift)

The aim of this question is to collect results on stopping times of Brownian motion (possibly with drift), with a focus on distributional properties: distributions of stopping times (Laplace ...
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0answers
25 views

Sequence $\rightarrow$ Generating function simple practice problems

I am trying to teach myself more about generating functions and how they "generate" sequences. Specifically I am trying to learn more about how to go from sequence to function and I would like some ...
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1answer
64 views

Forward-backward induction

I've seen the famous proof presented by Cauchy for the AM-GM inequality but what other neat proofs use forward-backward induction? Is it fundamentally inextricable from ordinary induction (are there ...
13
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9answers
2k views

Theorems in the form of “if and only if” such that the proof of one direction is extremely EASY to prove and the other one is extremely HARD [closed]

I believe this is a common phenomenon in mathematics, but surprisingly, no such list has been created on this site. I don't know if it's of value, just out of curiosity, I want to see more examples. ...