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 ...

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6
votes
2answers
616 views

Conservative differential equations “in the wild”

Dear MO world, I'm teaching an undergraduate course on "fun with chaos". As part of a test (on bifurcations in differential equations), I asked students to sketch phase portraits for a family of (2d) ...
2
votes
2answers
409 views

closed form solution to $x\ln (\frac{2}{x})=k(a-x) \ln (\frac{2}{a-x})$

$x\ln (\frac{2}{x})=k(a-x) \ln (\frac{2}{a-x})$ where $a$ and $k$ are positive constants. $a$ is usually small, say, $0< a<0.1$ and $x\in (0,a)$. There are ways to calculate numerical ...
0
votes
1answer
143 views

invariance under dilations

we have that the function (for suitable f) $ F(x)= \sum_{-\infty}^{\infty}f(x+n) $ is INVARIANT under any integer traslation $ y=x+n$ for integer 'n' however my question is can we find a lattice ...
5
votes
2answers
961 views

Construction of Serre Spectral Sequence

I'm trying to follow Hopkins' construction of the Serre Spectral Sequence, but some "obvious" things are not that obvious to me. He starts with considering a double complex $C_{\bullet,\bullet}$ with ...
0
votes
0answers
444 views

“closed form” finite sum

If a finite sum has a definite integral representation, for which it can be proved the underlying indefinite integral is not an elementary function, then does this imply the original finite sum can ...
5
votes
0answers
179 views

Does every stack with a connection admit an atlas with a connection?

Dear all, Let $S$ be a scheme in characteristic $0$, and let $\mathscr{X}/S$ mean a crystal in Artin stacks over $S$ in the sense of this handout, page 4, Definition 0.5, where we replace the scheme ...
11
votes
2answers
4k 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 ...
12
votes
3answers
488 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 ...
1
vote
1answer
299 views

Simple and general relation between continuant polynomials

Continued fraction $[a_0,a_1,...,a_n]$ may be expressed as quotient of two polynomials of $(a_0,a_1,...,a_n)$, named continuants (see http://en.wikipedia.org/wiki/Continuant_%28mathematics%29 ) ...
3
votes
1answer
243 views

Are piecewise linear curves dense among Hölder curves?

Consider for some $0 < \alpha \le 1$ the space functions $x:[0,1] \to \mathbb{R}^n$ such that $x(0) = 0$ and $\sup_{s,t} \frac{\|f(t)-f(s)\|}{|t-s|^{\alpha}}$ is finite. There are at least two ...
0
votes
1answer
230 views

derivative of a special function in integral form

What is the derivative of $Q_m\left(\frac{\alpha}{x^a},\frac{\beta}{x^b}\right)$ with respect to $x$, i.e, $$\frac{\partial}{\partial x}Q_m\left(\frac{\alpha}{x^a},\frac{\beta}{x^b}\right), \quad ...
6
votes
0answers
652 views

Homotopy groups of a Bouquet of n-spheres

Let $$X=\vee_{\alpha\in A} S_{\alpha}^n$$ be a bouquet of $n$-spheres. Q: How does one compute the homotopy groups $\pi_k(X)$?
11
votes
2answers
655 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 $a_n$ ...
11
votes
1answer
472 views

Is exponent of discrete-analytic function also discrete-analytic?

Lets define a discrete analytic function such a function that is equal to its Newton series: $$f(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k f\left (0\right)$$ Is function $g(x)=e^{f(x)}$ also ...
2
votes
1answer
354 views

An iterated tensor product integral

In "Differential equations driven by rough paths" (Terry Lyons, et al) section 1.4.2 it's claimed that the symmetric part of the tensor: $\int_{0 \le u_1 \le \cdots \le u_j \le t} \mathrm{d}X_{u_1} ...
9
votes
1answer
1k views

Naive definition of surface area doesn't work?

A first stab at a definition of surface area might go like this: Let S be a surface. Select finitely many points from S and make a bunch of triangles having these points as vertexes. Add up the ...
9
votes
2answers
2k views

How to find Colin Day's PhD Thesis

A week or so ago, I was saddened to read Jim Stasheff's post on the AlgTop mailing list, announcing the passing of Colin Day, after a long bout with cancer. I was thinking of reading Colin Day's PhD ...
2
votes
0answers
234 views

Monge Ampere and Calculus

[ I posted the question on Math StackExchange but didn't get any reply nor comment, so I'm trying here ] I am learning about mass transportation theory and the Monge-Ampere equation, to transport a ...
1
vote
0answers
195 views

Fractional Radon - Nikodym derivative

Given $f$ a function measurable in $[0,\infty]$ defined as $$\frac{d\mu}{d\nu}$$ such that, for every measurable set $A$ we have: $$\mu(A)=\int_A{}fd\nu$$ the function $f$ is called $Radon - ...
0
votes
1answer
402 views

Sequence of smooth functions converging to sgn(x)

I'm looking for a sequence of smooth functions $f_i(x)$ converging to Sign$(x)$, each of which additionally have the following property: \begin{equation} f_i(x_1+x_2) = g_i(x_1, f_i(x_2)) ...
22
votes
16answers
2k 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 ...
1
vote
1answer
182 views

About one series. Are there some related special functions?

Hello, I have the following series: $$ \sum_{n=2}^\infty \frac{t^n}{\Gamma(a n)} = ?,\qquad t\ge 0, $$ where the parameter $a\in (0,1]$, $\Gamma$ is the Gamma function. When $a=1$, the above sum ...
18
votes
9answers
4k 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 ...
7
votes
1answer
325 views

Labeling a Square Array

Suppose that the $n^2$ cells of an $n\times n$ array are labeled with the integers $1, \dots, n^2$. Under the traditional left-to-right and top-to-bottom labeling, the labels of horizontally adjacent ...
1
vote
1answer
490 views

A question on gauge functions

In the second paragraph on Page 71 of the book Matrix Analysis by Bhatia, 1997, it says ``as a consequence of (III.12) we have Theorem III 4.4''. How can one get the inequality in Theorem III 4.4 from ...
0
votes
1answer
212 views

Maximum value of a function with iterated logarithms

Consider this function $f(x)=-x\log\log(bx)+x\log\log\log(bx)+ax$, where $a$ and $b$ are positive constants, and the logarithms are 2-based. Is it possible to find the maximum value (or even with ...
2
votes
0answers
537 views

Tamagawa number for functional fields

Let $G$ be a split semi-simple simply connected group over a global field $F$ and let $\omega$ be a top-degree differential form on $G$ without zeroes (defined over $F$). It is well known that ...
-1
votes
1answer
438 views

Is this manifold orientable? [closed]

Let $C$ be the set of points $(a,b,c,d) \in \mathbb{C}^4$ which satisfy 1) $ \left|a\right|^2+\left|c\right|^2=\left|b\right|^2+\left|d\right|^2 =1 $. 2) $ a\bar{b}+c\bar{d}=0 $ There is a ...
1
vote
1answer
143 views

integral of a rational function (1+a_i s)^-1/prod((1+a_j s)^k)

Is there any closed-form expression for the following integral: $\int_0^\infty \frac{1}{(1+a_i s) \prod_{j=1}^n (1+a_j s)^k} ds $ where the ai are >0 and k is a positive integer. And, if k is not an ...
1
vote
0answers
214 views

Homotopy-Fibre Sequence of Classifying Spaces

Let $G$ be a topological group and $H$ be a normal subgroup of $G$ (I think $H$ is required to be admissible in the sense that the quotient map $G\to G/H$ is a principal $H$-bundle, am I right?). Then ...
2
votes
1answer
416 views

A real algebraic curve in $\mathbb{R}^3$ is the intersection of zero sets of polynomials in $\mathbb{R}[x,y,z]$. Can we choose the polynomials in a way, that, seen in $\mathbb{C}[x,y,z]$, the intersection of their zero sets is a complex alg. curve?

By "algebraic curve" here (both in $\mathbb{R}^3$ and in $\mathbb{C}^3$), I do not only mean that the dimension of the set as an algebraic variety is 1, but also that its dimension is 1 in a ...
1
vote
2answers
640 views

expectation of log(1+x) if x is a gamma random variable

I would like to know if there is a closed form expression for the expectation of log(1+x) when x is a gamma random variable. Thank you.
0
votes
1answer
451 views

Maximum sum of 3 consecutive numbers in a permutation [closed]

Given that $X = \{0, 1, 2, ..., 7, 8, 9\}$, and $P$ is a permutation on $X$. Let $M(P)$ be the maximum sum of 3 consecutive elements. For example, if $P = (0, 2, 4, 1, 5, 7, 9, 3, 8, 6)$, then $M(P)$ ...
0
votes
1answer
324 views

Minimum distance between two data sets

Suppose we have two sets of data, $X$ and $Y$, each of which contains $10$ positive numbers. Now let us order the data sets $X=\left\{ x_{1},\cdots,x_{10}\right\}$, $x_{1}\ge\cdots\ge x_{10}>0$ and ...
6
votes
2answers
583 views

Computational complexity of integration in two dimensions

I have an algorithm for solving a certain problem that requires that I compute two-dimensional integrals as a subroutine, and I'd like to make some kind of statement about its running time. Suppose ...
3
votes
1answer
160 views

bounding the probability that a polynomial is near 0

Given a polynomial $p(x_1,\ldots, x_k)$ in $k$ variables with maximum degree $n$, and $x_1,\ldots, x_k \in [0,1]$. Suppose $\max_{x \in [0,1]^k} p(x) = 1$, can we get an upper bound on the probability ...
8
votes
3answers
576 views

Variable-centric logical foundation of calculus

Since calculus originated long before our modern function concept, much of our language of calculus still focuses on variables and their interrelationships rather than explicitly on functions. For ...
3
votes
1answer
387 views

Definition of Pontrjagin Classes

I am studying characteristic classes recently and find some quesions about Pontrjagin classes. Firstly the definition of Pontrjagin classes is not that natural. When we talk about Pontrjagin classes ...
15
votes
8answers
6k 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 ...
4
votes
3answers
263 views

Operation on Isospectral graphs

Suppose $G$ and $H$ are two isospectral connected graphs. Can we say anything about isospectrality of graphs that obtain by binary operation between $G$ and $H$? For example,in special case, is ...
2
votes
2answers
380 views

branch points of modular parametrization of an elliptic curve

Let $E$ be an elliptic curve over a number field K. Then there is a morphism $\phi:X_0(n) \to E$. Consider composition $f:X_0(n)\to \mathbf{P}^1_K$, where we compose with degree 2 cover $E\to ...
1
vote
2answers
434 views

Uncountability of the “Peculiar” sets:

I call a set PECULIAR, if its elements are uncountable, pairwise disjoint subsets of R (the real number system). As for example, the set {(0,1),(3,5),[8,9]\Q},where Q denotes the set of rationals, is ...
1
vote
1answer
242 views

spectrum and degree sequence

We have the spectrum and the degree sequence of one graph. Can we uniquely determine the graph with these given information?
1
vote
2answers
266 views

Abelian subgroups of ball quotient

Let $X$ be a compact complex surface of general type which a ball quotient. Is it true that $\pi_{1}(X)$ can not contain ${\mathbb{Z}}^{2}$ as a subgroup? What kind of infinite abelian groups can ...
0
votes
1answer
1k views

Fourier transform of a differential operator

I have a differential operator defined by its Fourier transform: $\left(\alpha k_x^2 + \beta k_y^2 + \gamma k_x k_y \right)^\delta \hspace{20pt} \alpha,\beta,\gamma,\delta \in \mathbb{R}$ I don't ...
0
votes
1answer
318 views

A form of Lefschetz duality

Let W be a manifold with boundary such that \partial W is a union of two compact manifold A,B attached along their boundary. Does poincare duality hold for (W,A) and (W,B)?
22
votes
1answer
2k views

Why do we use $\epsilon$ and $\delta$?

My understanding (from a talk by Rob Bradley) is that Cauchy is responsible for the now-standard $\epsilon{-}\delta$ formulation of calculus, introduced in his 1821 Cours d’analyse. Although perhaps ...
5
votes
1answer
429 views

Localizability of differential operators a la Grothendieck

Hello, Maybe this question is trivial, so sorry Let $A$ be a (comm. with 1) $k$-algebra, where $k$ is a ring (comm. with 1). Then we can define the module of differential operators $D^{\leq n} ...
10
votes
1answer
590 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 ...
9
votes
4answers
937 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 ...