**13**

votes

**3**answers

528 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

**1**answer

332 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 )
...

**0**

votes

**1**answer

237 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

**0**answers

744 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

**2**answers

665 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

**1**answer

482 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

**1**answer

416 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

**2**answers

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

**0**answers

237 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

**0**answers

198 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

**1**answer

430 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

**16**answers

3k 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

**1**answer

189 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

**9**answers

5k 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

**1**answer

329 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

**1**answer

553 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

**1**answer

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

**0**answers

581 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

**1**answer

452 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

**1**answer

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

**0**answers

219 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

**1**answer

425 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 ...

**2**

votes

**2**answers

732 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

**1**answer

508 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

**1**answer

363 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

**2**answers

693 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

**1**answer

164 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

**3**answers

587 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

**1**answer

393 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 ...

**16**

votes

**8**answers

8k 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

**3**answers

273 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

**2**answers

386 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

**2**answers

443 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

**1**answer

257 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

**2**answers

270 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

**1**answer

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

**1**answer

326 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

**1**answer

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

**1**answer

457 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

**1**answer

595 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 ...

**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 ...

**2**

votes

**1**answer

139 views

### Spline fit with bounded derivations

How can I do a Spline Fit with bounds on some derivations?
Problem
Given:
Set of data points $t_k, x_k$
Set of nodes $n_i$
order $D$ of the spline (in my case $D=5$)
lower and upper bounds ...

**1**

vote

**1**answer

752 views

### A question about density character of Banach spaces. [closed]

Let $\langle M_i:i<\theta\rangle$ be an increasing chain of Banach spaces, where each $M_i$ has density character $\mu$ (i.e.,the mininum cardinality of a dense subset of $M_i$ is $\mu$). Let ...

**0**

votes

**3**answers

710 views

### How to isolate $f(x)$ in $f(x+a)=f(x)+a\times g(x)$?

$a \in \mathbb{R}$
$f:\mathbb{R} \rightarrow \mathbb{R}$
$g:\mathbb{R} \rightarrow \mathbb{R}$
For generic functions $f$ and $g$, how isolate $f(x)$ in the equation below?
$f(x+a)=f(x)+a\times ...

**0**

votes

**1**answer

1k views

### How to determine whether a multivariate function is bounded or not

Suppose there is a function $f:\mathbb{R}_+^n\mapsto \mathbb{R}$. Are there any systematic ways to determine whether the range of $f$ is bounded or not?
For example, there is a function ...

**2**

votes

**2**answers

617 views

### Power function inequality

Let $x$ and $p$ be real numbers with $x \ge 1$ and $p \ge 2$ . Show that $(x - 1)(x + 1)^{p - 1} \ge x^p - 1$ .
I recently discovered this result. I am sure it is known, but it is new to me. It is ...

**0**

votes

**0**answers

220 views

### Properties of morphisms induced by divisors on curves

There are a few properties from Hartshorne IV on curves that I am trying to verify. Let $D$ be an effective divisor on a curve (integral scheme of dimension 1, proper over $k$, regular) $X$, $\dim ...

**2**

votes

**1**answer

568 views

### A singular value inequality

Let $s_1,s_2: \mathbb{R}^{2\times 2} \mapsto \mathbb{R}_+$,
$s_{1}\left(\cdot\right)\ge s_{2}\left(\cdot\right)\ge 0$, be the
singular values of a $2\times2$ matrix. Is it true that
...

**1**

vote

**1**answer

350 views

### two polynomials

If $p,q:\mathbb R^3\rightarrow\mathbb R$ are two polynomials, such that $\{p=0\}\cap\{q=0\}$ is two-dimensional, does it follow that $p$ and $q$ have a common factor? (I believe it does.) How to prove ...

**1**

vote

**0**answers

139 views

### Morphisms, inducing isomorphisms on global functions

Let $f:X\to Y$ be a surjective morphism of complex varieties with $Y$ affine. Assume that every fiber of this morphism has the property that all the global functions are constant.
What else do I need ...