Questions tagged [tag-removed]

This tag is to be used only when re-tagging highly(!) off-topic questions where none of the actual tags would make sense; all actual tags the questioner has used are removed and something is needed to have some tag, which is enforced by the software, so this tag is used. However note that this tag is also used in the process of moderators deleting tags. Thus a question having this tag does most of the time not mean that somebody found it off-topic.

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44 votes
6 answers
13k views

Best tablet computer for mathematics [closed]

I'm not sure if this is completely appropriate, but I thought I'd ask here. I'm in the market for a tablet computer. Unfortunately, my (mathematical) needs are very different from the needs of the ...
39 votes
6 answers
5k views

What is the simplest, most elementary proof that a particular number is transcendental?

I teach, among many other things, a class of wonderful and inquisitive 7th graders. We've recently been studying and discussing various number systems (N, Z, Q, R, C, algebraic numbers, and even ...
Justin Lanier's user avatar
36 votes
0 answers
1k views

Two-convexity ⇒ Lefschetz?

Assume that $\Omega$ is an open simply connected set in $\mathbb R^n$ (two-convexity) if 3 faces of a 3-simplex belong to $\Omega$ then whole simplex in $\Omega$. Is it true that any component of ...
Anton Petrunin's user avatar
35 votes
1 answer
2k views

Mr. G.P.K.'s questions [closed]

WARNING: An acquaintance of mine, Mr.Goosepond Prhklstr Kratchinabritchisitch, has requested permission to post his questions under my username. When I asked him why he didn't do it under his own ...
32 votes
2 answers
2k views

Gently falling functions

I wonder if it is possible to characterize the class of gently falling functions, which I would like to define as follows. Let $g(x)$ be a $C^2$ function defined on an interval $R \subseteq \mathbb{R}$...
Joseph O'Rourke's user avatar
26 votes
2 answers
2k views

Euler characteristic and universal cover

Let $M$ be a compact manifold, let $\tilde{M}$ be its universal cover, and suppose that the Euler characteristic $\chi(\tilde{M})=0$. My question is: does this imply that $\chi(M)=0$? This is clear if ...
CuriousUser's user avatar
  • 1,420
24 votes
9 answers
8k views

How to motivate and present epsilon-delta proofs to undergraduates?

This would seem to be a common question, but I am surprised not to see it already asked and answered on MO! I am teaching an undergraduate course, and I want to teach them to construct basic epsilon-...
24 votes
8 answers
20k views

Interesting Applications of the Classical Stokes Theorem?

When students learn multivariable calculus they're typically barraged with a collection of examples of the type "given surface X with boundary curve Y, evaluate the line integral of a vector field Y ...
24 votes
16 answers
4k views

functions satisfying "one-one iff onto"

Hello Everybody. I need some more examples for the following really interesting phenomenon: A function from the class ... is one-one iff it is onto. Some ...
22 votes
5 answers
2k views

Homological algebra and calculus (as in Newton)

This question reminded me of a possibly stupid idea that I had a while back. On page 2 of this paper, while discussing Euclid's axioms of plane geometry and spatial geometry, Manin makes an extremely ...
Kevin H. Lin's user avatar
  • 20.7k
22 votes
6 answers
3k views

Formal consequences of Riemann-Roch (multiple answers welcome)

This question aims to pin down what Riemann-Roch can tell us about a divisor on a curve, without any "geometric thinking". It can be annoying to wonder if there is some clever trick you're missing ...
Andrew Critch's user avatar
21 votes
1 answer
12k views

Non-diagonalizable complex symmetric matrix

This is a question in elementary linear algebra, though I hope it's not so trivial to be closed. Real symmetric matrices, complex hermitian matrices, unitary matrices, and complex matrices with ...
Qfwfq's user avatar
  • 22.7k
21 votes
1 answer
1k views

Red-blue alternating paths

Suppose we have two simple graphs on the same vertex set. We will call one of them red, the other blue. Suppose that for $i=1,..,k$ we have $deg (v_i)\ge i$ in both graphs, where $V_k=\{v_1,\ldots,v_k\...
domotorp's user avatar
  • 18.3k
20 votes
0 answers
2k views

Can the similarity between the Riesz representation theorem and the Yoneda embedding lemma be given a formal undergirding?

For example, by viewing Hilbert spaces as enriched categories in some fashion? (I suppose the same idea of considering the inner product of a Hilbert space as a generalized Hom-set has also been ...
Sridhar Ramesh's user avatar
19 votes
7 answers
2k views

Generalizations of "standard" calculus

We have the usual analogy between infinitesimal calculus (integrals and derivatives) and finite calculus (sums and forward differences), and also the generalization of infinitesimal calculus to ...
Zev Chonoles's user avatar
  • 6,712
18 votes
2 answers
3k views

The non-traveling mathematician problem

This is a career question. I have just begun a research postdoc position in Southern California. It has been hard, but I've enjoyed teaching my first graduate courses and working on research and ...
18 votes
1 answer
4k views

reference for "X compact <=> C_b(X) separable" (X metric space)

I know (and am able to prove via Stone-Čech compactification) that the following is correct: Theorem: A metric space is compact if and only if its space of bounded, continuous, real-valued ...
Wolfgang Loehr's user avatar
16 votes
13 answers
4k views

Do you find your students are less competent in basic algebra and arithmetic, and, if so, do you believe that this is due to overuse of calculators at an early level? [closed]

So first I gave my class the quiz problem: Compute $$\lim_{h\rightarrow 0} \frac{\frac{1}{3+h} - \frac{1}{3}}{h}.$$ Upon finding that they could not do that (no real surprize) I asked them to compute $...
16 votes
3 answers
3k views

How do i solve this : $\displaystyle \ f'=e^{{f}^{-1}}$?

Let $f$ be a function such that :$f:\mathbb{R}\to \mathbb{R}$ and $f^{-1}$ is a compositional inverse of $f$. I would'd like to know how do I solve this class of differential equation : $$\...
zeraoulia rafik's user avatar
16 votes
5 answers
4k views

Where to publish a paper on the Mafia game?

I wrote a research paper "A mathematical model of the Mafia game" (arXiv:1009.1031 [math.PR]). However, I do not know where to publish it. As an undergraduate studying majorly physics, I have little ...
Piotr Migdal's user avatar
  • 1,592
16 votes
2 answers
1k views

Independence of Leibniz rule and locality from other properties of the derivative?

The following is meant to be an axiomatization of differential calculus of a single variable. To avoid complications, let's say that $f$, $g$, $f'$, and $g'$ are smooth functions from $\mathbb{R}$ to $...
user avatar
16 votes
2 answers
1k views

There is mathematics behind the 1989 Tour de France !

The $1989$ Tour was won by Greg Lemond (USA, $1961$ - ), who beat Laurent Fignon (France, $1960$ - $2010$) by $8''$. Yes, eight seconds! The closest tour in history. Let me recall a few rules ...
Denis Serre's user avatar
  • 51.5k
16 votes
1 answer
670 views

Question about product topology

Suppose $S\subset\mathbb{R}$ is dense without interior point, and for every open interval $I,J\subset\mathbb{R}$, $I\cap S$ is homeomorphic to $J\cap S$. Is $S\times S$ homeomorphic to $S$? By Luzin ...
duodaa's user avatar
  • 163
16 votes
0 answers
640 views

Codimension Two Embeddings in Goodwillie-Weiss Manifold Calculus, and the Difficulty of Fundamental Groups

In manifold calculus, there are various analyticity estimates which run into trouble for codimension two embeddings. For instance, the functor $\operatorname{Emb}(M,N)$ is analytic in $M$ if $\dim M \...
Hiro Lee Tanaka's user avatar
15 votes
3 answers
771 views

Recognizing the 4-sphere and the Adjan--Rabin theorem

The problem of recognizing the standard $S^n$ is the following: Given some simplicial complex $M$ with rational vertices representing a closed manifold, can one decide (in finite time) if $M$ is ...
Malte's user avatar
  • 817
15 votes
3 answers
3k views

Model category structure on Set without axiom of choice

There is a model category structure on Set in which the cofibrations are the monomorphisms, the fibrations are maps which are either epimorphisms or have empty domain, and the weak equivalences are ...
Reid Barton's user avatar
  • 24.8k
15 votes
2 answers
664 views

Curve integral of exponent of superharmonic function.

Let $\phi$ be a real smooth superharmonic function on unit disc $D$ in $\mathbb C$; i.e. $\triangle \phi\le 0$. Then there is a curve $\gamma$ from the center of $D$ to its boundary such that $$\...
Anton Petrunin's user avatar
14 votes
11 answers
34k views

Why does undergraduate discrete math require calculus?

Often undergraduate discrete math classes in the US have a calculus prerequisite. Here is the description of the discrete math course from my undergrad: A general introduction to basic ...
14 votes
2 answers
1k views

Hopf Algebra for a physicist

Hello, for my bachelor's thesis I need to understand the Hopf Algebra of Feynman Diagrams. As I have only litte knowledge in Algebra by now I wanted to ask where I could start and what preknowledge I ...
understandhopf's user avatar
14 votes
1 answer
2k views

Anything going on for a mathematician stuck at New York?

First of all, apologies for the really non-standard question/announcement. I know this is not what MO was intended for, but in this situation it is the easiest way to reach (perhaps) the right person. ...
13 votes
4 answers
11k views

Fourier transform of Analytic Functions

Forgive me if this question does not meet the bar for this forum. But i would really appreciated some help. I'm trying to construct a function according to some conditions in the frequency domain of ...
jonalm's user avatar
  • 277
13 votes
2 answers
5k views

Definition of Function

What is authoritative canonical formal definition of function? For example, According to Wolfram MathWorld, $$isafun_1(f)\;\leftrightarrow\; \forall a\in f\;(\exists x\exists y \;\langle x,y\rangle =...
13 votes
2 answers
6k views

What is "Seetapun Enigma"?

A friend of mine just asked me this very question. While I had some training in combinatorics, I have never heard of the "Seetapun Enigma", which, supposedly, is related to the Ramsey's theorem. A ...
ssquidd's user avatar
  • 1,101
13 votes
1 answer
502 views

Permanent of a matrix of odd integers

It is clear that the permanent of an $n\times n$ matrix which entries are odd integers, is an even number, as it is the sum of $n!$ odd numbers. I am interested in finding the highest power of $2$ ...
Tal H's user avatar
  • 273
12 votes
4 answers
1k views

Coincidences amongst classifying spaces and Eilenberg Mac-Lane spaces

Given that $$\mathbb{R}P^{\infty} = B O(1) = K(\widehat{O(1)}, 1)$$ $$\mathbb{C} P^{\infty} = B U(1) = K( \widehat{U(1)}, 2)$$ is there any way to make sense of $$\mathbb{H}P^{\infty} = B Sp(1)$$ in a ...
Alexander Moll's user avatar
12 votes
2 answers
2k views

Can formally differentiating give a derivative of a discrete function?

When I teach calculus, I really try to stress the importance of knowing the domain of a function. One example that I sometimes like to use to show students the importance of inspecting the domain is ...
Steven Gubkin's user avatar
12 votes
1 answer
1k views

What are the applications of Dowker's theorem?

Let $R \subset X \times Y$ be any relation between sets $X$ and $Y$. CH Dowker constructed two simplicial complexes $K$ and $L$ associated to $R$: a simplex in $K$ consists of finitely many elements $...
Francis Snapper's user avatar
12 votes
2 answers
784 views

Matrices into path algebras

I was thinking about quivers recently, and the following idea came to me. Let ei,j denote the matrix unit in Mn for 1 ≤ i,j ≤ n. Let Γ denote the complete quiver on vertices {1, …,...
Sammy Black's user avatar
  • 1,746
11 votes
4 answers
2k views

Is the function $e^{x^2/2} \Phi(x)$ monotone increasing?

Hello, Here is an interesting problem. It looks elementary, but it has taken me some efforts without solving it. Let $$ h(x) = e^{x^2/2} \Phi(x),\qquad \text{with}\quad \Phi(x):=\int_{-\infty}^x \...
Anand's user avatar
  • 1,619
11 votes
2 answers
752 views

Interesting result on the Euler-Maschroni constant - what is the background?

Today I entered the following expression in maple: $$a_i = H_{10^i} - ln(10^i) - \gamma$$ Here $H_j$ equals $\sum_{k=1}^{j} 1/k$ and $\gamma$ is the Euler-Mascheroni constant. When I computed $a_n$ ...
tobias's user avatar
  • 377
11 votes
1 answer
881 views

What is happening to Martin Gardner's files?

Martin Gardner kept voluminous correspondence with amateur and professional mathematicians worldwide throughout his career. His files are a treasure trove of information about all areas of ...
Timothy Chow's user avatar
  • 78.1k
11 votes
2 answers
668 views

"Probabilistic ultrafilters?"

A naive question. Let $S$ be a set and let $[0,1]^S$ the set of functions from $S$ to the closed interval $[0,1]$. Suppose given some function $P \colon [0,1]^S \to [0,1]$ satisfying the following ...
JSE's user avatar
  • 19.1k
11 votes
1 answer
445 views

Determination of a symmetric convex region by parallel sections

This question is partly inspired by a problem in Stewart's Calculus: "Find the area of the region enclosed by $y=x^2$ and $x=y^2$." Suppose $f\colon [0,1]\to [0,1]$ is a convex increasing function ...
10 votes
2 answers
3k views

Continuous function from $[0,1]$ to $[0,1]$

Does there exist a continuous function $f:[0,1]\rightarrow [0,1]$ such that $f$ takes every value in $[0,1]$ an infinite number of times?
Cristos A. Ruiz's user avatar
10 votes
4 answers
1k views

Applications of full integral weight modular forms in elementary number theory

Except for Eisenstein series having the divisor functions as their Fourier coefficients, is there any other full integral weight modular form (of some level, preferably full) having arithmetic ...
Eren Mehmet Kiral's user avatar
10 votes
2 answers
2k views

Roadmap to Computer Algebra Systems Usage for Algebraic Geometry

I've decided it's time to start learning how to use a computer to do calculations... I've used Singular to some small extent so far, but I want to start relying on computer algebra systems more. ...
10 votes
2 answers
3k views

Convergence and non-convergence of left-point and mid-point Riemann sums

In standard calculus it is a well known fact that left-point and mid-point Riemann sums do become equal in the limit. When it comes to stochastic integration this is no longer the case. Taking the ...
vonjd's user avatar
  • 5,855
10 votes
4 answers
1k views

Reference for working with the implicit function theorem

I just had a student come to my office hours and ask me a ton of questions, the answer to all of which was "that's a slight variant to the implicit function theorem, which is proved by formal ...
David E Speyer's user avatar
10 votes
1 answer
677 views

A problem concerning $L^2([0,1]\times[0,1])$

Trying to solve a conjecture in differential geometry, I am leaded to the following problem (which may seem weird to a analyst). I wonder if anyone know some techniques that happen to solve it. Let $...
Xin Nie's user avatar
  • 1,764
10 votes
1 answer
730 views

An exponential polynomial with at least one bounded positivity component

In a forthcoming paper on nodal domains of Gaussian random functions, we (I and Misha Sodin) have a statement that is, roughly speaking, the following: if bounded nodal domains are possible at all, ...
fedja's user avatar
  • 59.5k

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