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-2
votes
1answer
117 views

Correction symbols used for mathematical texts [closed]

When proof reading and correcting a mathematical text, I sometimes see people use special notation symbols in the margin to indicate correction, deletion, replacement and so on. Is there any standard ...
8
votes
1answer
678 views

On the parity of $[x^n]$

I am trying to find a problem which appeared years ago in the American Mathematical Monthly. It went something like this: There was a Putnam Competition question which asked to show that there is a ...
22
votes
4answers
1k views

Expert, Intuitive, Organizing Analogies

In learning a new area it is very helpful to have high-level intuitive analogies that keep track of the various parts of an important argument or strategy. Experts have a store of such things, and ...
39
votes
25answers
7k views

Where can square roots come from when they are not distances?

In a recent survey "Supergeometry in Mathematics and Physics", Kapranov points out cases in which observable quantities of immediate interest are represented as bilinear combinations of more ...
7
votes
23answers
3k views

A search for theorems which appear to have very few, if any hypotheses [closed]

I'm interested in theorems which appear to have very few, if any hypotheses. Essentially a search for unexpected regularity or pattern in a relatively unstructured situation. By "few hypotheses" I ...
-1
votes
1answer
54 views

Idempotent solutions to the implict function theorem other than the identity?

I am interested in the following problem. Assume that an (anti)symmetric function $g:\mathbb{R}^2 \to \mathbb{R}$ satisfies the implicit function theorem. That is, $g(x,y) = \pm g(y,x)$ and $g(x,y)=0$ ...
2
votes
2answers
358 views

A logarithmic cotangent inequality

I must be a terrible googling searcher but I cannot find a reference to the following inequality: $$ \forall_{\phi\in(0;\frac \pi 4)}\ \ln(\cot(\phi)))\, <\, \cot(2\!\cdot\!\phi) $$ I have just ...
58
votes
15answers
6k views

Sophisticated treatments of topics in school mathematics

Sophisticated mathematical concepts typically shed light on sophisticated mathematics. But in a few cases they also apply to elementary mathematics in an interesting way. I find such examples ...
2
votes
1answer
128 views

Conic sections in high dimensions

Can every $n$-dimensional ellipsoid be obtained as a (spherical) conic section? This is false for generic quadrics but seems true for ellipsoid. Does anybody have any references?
40
votes
9answers
6k views

How does a mathematician choose on which problem to work?

Main question: How does a mathematician choose on which problem to work? An example approach to framing one's answer: What is a mathematical problem - big or small - that you solved or are ...
1
vote
1answer
95 views

Discrete summation of Gaussian functions. Decay time problem

I am facing the following problem. I have a function which is defined through a discrete sum of Gaussians $$F_M(t) = 2\sum\limits_{n=1}^{M}e^{-t^2 \sigma^2 n^2}\times \sum\limits_{k=n}^{M}p_k p_{k-n} +...
19
votes
4answers
1k views

What is the term for combining functions $f_1,f_2,\dots,f_n$ into a tuple $(f_1,\dots,f_n)$?

This is an embarrassingly simple question, but I was not able to find a definitive answer from literature search. Suppose one has some collection of functions $f_1: X \to Y_1, \dots, f_n: X \to Y_n$ ...
8
votes
1answer
272 views

Geometric meaning of unimodular matrix

Rotations are given by unitary matrices. What is the geometric meaning of unimodular matrices that are not unitary?
2
votes
0answers
63 views

How to find moment condition for generalized method of moments?

Consider a scalar system with $2K$ outputs and $K+2$ unknowns: $y_{k,1}=x_ka_1+n_{k,1} \quad y_{k,2}=x_ka_2+n_{k,1}$. The variables $n_{k,\ell}$ are zero mean noise variables. To estimate $a_1$ and $...
6
votes
1answer
748 views

Windows into new mathematical worlds [closed]

Yitang Zhang's Annals of Mathematics primes-gap result opened a new window, which Polymath's reduction from $70\times 10^6$ to $246$ attests. Perhaps Harald Helfgott's celebrated proof of the odd ...
6
votes
1answer
179 views

Solution to $(A+x^2)e^x=B$ with Lambert W function

Is it possible to obtain a analytical solution for $(A+x^2)e^x=B$, where we want to solve for $x$ with $A,B$ as constants?
3
votes
1answer
499 views

What is the rate of convergence? [closed]

How quickly does the series defined by $$x_0 = 0, \ x_{n+1} = \frac{x_n^2+1}{2}$$ converge to $1$?
15
votes
7answers
2k views

Is this a rational function?

Is $$\sum_{n=1}^{\infty} \frac{z^n}{2^n-1} \in \mathbb{C}(z)\ ?$$ In a slightly different vein, given a sequence of real numbers $\{a_n\}_{n=0}^\infty$, what are some necessary and sufficient ...
4
votes
2answers
191 views

Find the expansion of the exact solution (beyond Taylor)

In a paper by Kitagawa & Ueda Squeezed spin states they give an argument that the minimum variance in one-axis twisting Hamiltonian scales like $V_{min} \propto S^{-2/3}$. I will shortly describe ...
17
votes
1answer
2k views

Why do people use “formal calculation” to describe informal calculations?

Many times, I see the word formal being used to describe a calculation that is not rigorous. I would think that such calculations should rather be termed informal than formal. What is the explanation ...
1
vote
0answers
122 views

General procedure to find the determinant of an operator?

I want to learn to find the determinant of an operator. I am given an operator like $\Sigma _{\alpha\beta}=-k^2g_{\alpha\beta}+i\theta\epsilon_{\alpha\beta\gamma} k^\gamma$ $k^2=k^μk_μ$, $g^{αβ}$ ...
19
votes
3answers
1k views

Which way for reading the proofs?

I am a master student in mathematics. For me a large part of doing mathematics is thinking about, reading and verifying the proof of theorems that I find them in my field of study. I can do this ...
16
votes
2answers
1k views

What are the applications of operator algebras to other areas?

Question: What are the applications of operator algebras to other areas? More precisely, I would like to know the results in mathematical areas outside of operator algebras which were proved by ...
3
votes
1answer
965 views

Mathematics of Computer science and AI [closed]

Computer science and Artificial Intelligence have been fertile grounds for research for decades, not only for Engineers but particularly for Mathematicians. What kinds of Mathematics have emerged ...
2
votes
2answers
684 views

Popular books written by great mathematicians [closed]

I read: H. Poincare. Value of science F. Klein. Development of Mathematics in the 19th Century J.E. Littlewood. A Mathematicians Miscellany G.H. Hardy. A Mathematician’s Apology R. Courant, H....
6
votes
1answer
173 views

Compiling self-referential forms

Fix $1\leq d\in\mathbb{N}$ and set $D:=\{0,1,\ldots,d-1\}$. Consider the system of equations \begin{equation} x_i=c_i + \sum_{j\in D}\delta_{x_j,i} \end{equation} with $c_i\in D$ given and $x_i\in D$ ...
25
votes
5answers
2k views

Collaboration or acknowledgment?

This post is a sequel of: When should a supervisor be a co-author? This time the topic is about the interaction between two professional mathematicians (in particular junior-senior, but not ...
3
votes
1answer
127 views

Extension of an involutive automorphism

Suppose that $g$ is a complex semi-simple Lie algebra and $g'$ its reductive subalgebra. If $\tau$ is an involutive automorphism of $g'$, can $\tau$ be extended to an involutive automorphism of $g$ ...
2
votes
2answers
203 views

Websites for Math Shopping [closed]

I was wondering if anyone knows about good websites or stores to buy math related products. On etsy there are normal distribution plushes and famous mathematicians in coasters. However when I search ...
1
vote
2answers
164 views

iterative solution better than analytic solution? [closed]

My supervisor and I were discussing a specific optimisation problem this afternoon. To be simple: solve for $R$ in the equation $Rx=y$, where $x$, $y$ are made of samples in two difference ...
1
vote
1answer
222 views

Concept of synchronizability

This thread is about the concept of synchronizability. It's a concept I tried to formalize in its most general sense but without success. The goal of this thread is therefore to try to formalize it in ...
0
votes
1answer
178 views

How to solve $e^{f(x)} + a f(x) + bx = 0$ [closed]

How should determine solutions to equations of this form? $$e^{-f(x)} + b f(x) = ax$$ Here $f(x)>0$ is real valued. Also $a>0$, $b>0$.
8
votes
2answers
1k views

The impact of large cardinals in mathematics [closed]

What are the main applications of large cardinals in ordinary mathematics, and what is the philosophy behind using them. In particular: Question 1. What is the philosophy behind accepting large ...
4
votes
1answer
228 views

Constructing a function from preimages

This question was inspired by Can we build a continuous function from "fibers"/preimages defined over a topological base? Let $X,Y$ be sets and $L\subseteq \mathcal{P}(Y)$. Suppose $L$ has ...
-4
votes
1answer
220 views

geometry formula adjacent neighbour of a N dimensional cube [closed]

A question from a curious. How many adjacent N dimensional cube there is when they are regularly distributed on a "grid". By N dimensional cube I mean: a point is a OD cube a segment is a 1D cube ...
33
votes
6answers
3k views

Negative impact of wrong or non-rigorous proofs

The recent talks of Voevodsky (for example, http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/2014_IAS.pdf), which describe subtle errors in proofs by him as well as others, as well ...
9
votes
5answers
1k views

Proof or citation?

I'm writing an article. I suppose that I'll submit it to a more or less decent journal (in English). I have doubts about the following: I have a lemma (with quite a trivial proof). I don't want to ...
2
votes
0answers
339 views

Conjectures on fractions where each digit appears once in numerator and denominator

This is a highly redacted version of a question that was asked before. Please see Criteria of considering relevance of the question to the domain of research topics for details. Some numerical ...
7
votes
6answers
466 views

Do you have examples of such “transitive” elements?

(I've asked the same question at the MSE, so far with no answers, so I thought I'd try it here as well. If there's some clash with any site rules, please let me know and I'll abide.) Let $A$ be a set ...
8
votes
2answers
517 views

Are there any organized websites for seminar/conference videos?

These days, there are many conference centers and universities recording seminars and conference talks and make them available on the web. Some examples: http://www.fields.utoronto.ca/video-archive ...
67
votes
10answers
8k views

Have you solved problems in your sleep?

I have hit upon major (for me—relative to my trivial accomplishments) insights in my research in various sleep-deprived altered states of consciousness, e.g., long solo car-drives extending ...
2
votes
0answers
98 views

Jacobi triple product for multidimensional lattices

The Jacobi triple product identity gives as a special case a product formula for the theta function of a 1-dimensional lattice. Is there a more general product formula for the theta function of an ...
6
votes
1answer
507 views

Sources of Theorem drafts by the original author

When I look at first time to a theorem and I try to understand it or when I try to memorise a useful theorem I always have difficulties (I am not the only one. For example: I read a question: I always ...
-1
votes
1answer
565 views

What is an example of a non-axiomatic mathematical system? [closed]

In this wikipedia article on the foundations of mathematics, it says: In practice, most mathematicians ... do not work from axiomatic systems Is this correct? If so, what is an example of this?
35
votes
33answers
4k views

Structures that turn out to exhibit a symmetry even though their definition doesn't

Sometimes (often?) a structure depending on several parameters turns out to be symmetric w.r.t. interchanging two of the parameters, even though the definition gives a priori no clue of that symmetry. ...
2
votes
1answer
130 views

Talking about properties of “random” elements

I asked this in MSE but did not get a satisfying answer. I apologize in advance if this is not appropriate for MO. Suppose that we have some set X and we want to say that a "random" (or generic) ...
10
votes
5answers
710 views

Accessible proofs of contemporary results in mathematics

Are there strong results in contemporary mathematical research (last 20 years) which have a proof which every mathematician (holding a PhD) can completely understand within a few days? -- If yes, ...
3
votes
1answer
1k views

Equal signs with fancy marks

Some people use $\stackrel{\mathrm{def}}{=}$, $:=$ or $\stackrel{\Delta}{=}$ for definitions. In more informal contexts, I have also seen $\stackrel{?}{=}$, for "I wish to prove this equality, which ...
9
votes
4answers
749 views

Is there any monoid in which the product of two non-invertible elements could be invertible?

I think the title speaks for itself. Thus I just explain the story behind the question. Of course, you may want to skip the story. Story: Currently, I teach a course in linear algebra and matrices ...
30
votes
2answers
1k views

Applications of Lawvere's fixed point theorem

Lawvere's fixed point theorem states that in a cartesian closed category, if there is a morphism $A \to X^A$ which is point-surjective (meaning that $\hom(1,A) \to \hom(1,X^A)$ is surjective), then ...