**11**

votes

**5**answers

1k views

### What is $\sum (x+\mathbb{Z})^{-2}$?

This is a simple question, but its been bugging me. Define the function $\gamma$ on $\mathbb{R}\backslash \mathbb{Z}$ by
$$\gamma(x):=\sum_{i\in \mathbb{Z}}\frac{1}{(x+i)^2}$$
The sum converges ...

**4**

votes

**0**answers

796 views

### integer solutions of $ (n!+1)=m^2$

Consider $4!=24$, if you add one you get $25=5^2$. The same occurs with $5! = 120 = 11^2 - 1$, and $7! = 5040 = 71^2 - 1$. Are there other solutions of the equation $n!+1 = m^2$?
I verified that no ...

**1**

vote

**0**answers

610 views

### Area Under Generalized Parabolas and Hyperbolas without Calculus

This is shorter and more specific version of certain questions about a rather simple quadrature method. The answers I got were great but not what I asked. The terms in the title for $y=x^p$ look ...

**-7**

votes

**1**answer

165 views

### The P VS NP problem as relates to factoring [closed]

Good Day all,
If someone could prove there is no simple way to factor an integer, and show searching is absolutely required, would that also prove P is not equal to NP?
Thanks in advance for your ...

**-4**

votes

**4**answers

3k views

### Where can I buy Napier's Bones/Rods? [closed]

I'm not sure this is the best place to ask, but it is math related. Where can I purchase Napier's Bones (rods)? I've been searching around the web and I haven't really been able to find anywhere that ...

**-4**

votes

**1**answer

108 views

### covering all the points on a real number line [closed]

So my question is really simple (and may be a bit naive):
The claim is "I can cover all the points in a region from their x, y projections by locating a grid in which they lie in and recursively ...

**48**

votes

**3**answers

5k views

### What is the status of the Gauss Circle Problem?

For $r > 0$, let $L(r) = \# \{ (x,y) \in \mathbb{Z}^2 \ | \ x^2 + y^2 \leq r^2\}$ be the number of lattice points lying on or inside the standard circle of radius $r$. It is easy to see that $L(r) ...

**2**

votes

**1**answer

190 views

### Classification of indecomposable inclusions $(H \subset G)$ with $G$ decomposable

Definition: A group $G$ is indecomposable if: $G = G_1 \times G_2 \Rightarrow \exists i \ G_i = 1$.
We can generalize the notion of indecomposable from groups to inclusion of groups as ...

**1**

vote

**1**answer

1k views

### The Jones-Sato-Wada-Wiens polynomial for prime numbers and differential calculus?

After works of Davis, Matijasevic, Putman and Robinson between 1960 and 1970, we know that every recursively enumerable set of numbers can be represented by a polynomial.
In particular, it's the case ...

**5**

votes

**0**answers

151 views

### Other applications of the 'increment' approach

I would like to hear about other instances of the so-called 'increments' approach, first used by Roth to prove that a subset of $\mathbb{N}$ of positive upper density contained infinitely many ...

**6**

votes

**3**answers

783 views

### When are maps between topological spaces homotopic?

I wanted to ask if there is any known mehod to quantify 'how many' homotopy classes of maps there are between two given topological spaces $X$, $Y$ (CW-complexes, say).
So far I had the following ...

**-8**

votes

**2**answers

1k views

### Filters and intersection of two binary relations

Let $\mathfrak{F}$ is the complete lattice of filters (including the improper filter) on some set, ordered
inverse to set-theoretic inclusion.
I will denote $\left\langle f \right\rangle \mathcal{X} ...

**16**

votes

**1**answer

484 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 ...

**2**

votes

**2**answers

263 views

### Elementary question about Isotopy (in the definition of a Teichmuller space)

Disclaimer - I don't have much experience in topology/complex geometry, so I apologize if what I'm asking is too elementary for this site.
Let $S$ be some orientable surface obtained by removing ...

**4**

votes

**1**answer

274 views

### Topological characterisation for a (closed irreducible) hyperbolic 3-manifold

Is there a topological characterisation of what a (closed irreducible) hyperbolic 3-manifold is? I don't know any Riemannian geometry and still want to understand what an exceptional Dehn surgery is. ...

**0**

votes

**1**answer

200 views

### Fibration in the 3 torus.

The Hopf fibration $S^1\rightarrow S^3\rightarrow S^2$ gives a decomposition of $S^3$ into 2-tori and to circles, so that the tori are foliated by circles of slope 1. If you take the region between ...

**6**

votes

**1**answer

236 views

### Is the homeomorphism class of a connected open set of C determined by its fundamental group?

Let $U,U'\subseteq\mathbf{C}$ be two connected open sets such that $\pi_1(U)\simeq\pi_1(U')$.
Q: Does this imply that $U$ is homeomorphic to $U'$?
In the case where the $\pi_1$'s are trivial then ...

**5**

votes

**1**answer

377 views

### Does there exist a space X whose suspension is homotopy equivalent to [0,1] rel ends but where X is not contractible?

As pointed out by David White in
when mapping cone is contractible
there exists an acyclic CW-complex $X$ which is not contractible but whose suspension is contractible. Namely, let $a$ and $b$ be ...

**2**

votes

**4**answers

441 views

### Picturing a Certain Torus and Klein Bottle

The other day I was explaining orientability to someone and we were walking through some of the statements about orientability on the Wikipedia page on the topic. While I was able to satisfy his ...

**1**

vote

**1**answer

145 views

### Finitely Generated Commutative Z-algebra.

Let $R$ be a commutative, finitely generated $\mathbb{Z}$-algebra, then the nil radical is equal to the Jacobson radical.
I am not able to make much traction on this, nor can I find this result in ...

**16**

votes

**6**answers

5k views

### Angle Maximizing the Distance of a Projectile

It is well-known that to maximize the horizontal distance traveled by a projectile fired from the ground at a given speed, one should fire it at a $45^\circ$ angle. What's less-known, though not too ...

**2**

votes

**1**answer

228 views

### Finding a good ordering of $\mathbb{Q}$

Oftentimes in density arguments we let $\{x_n\}$ be a dense sequence and this is sufficient to imply the desired result.
From a research question I am working on I have simplified the ...

**1**

vote

**1**answer

101 views

### Approximating rational generating functions

Suppose we have a initial segment $x_1,\ldots,x_N$ (for reasonably large $N$) of a sequence of natural numbers $(x_i)$. We have reason to believe the generating function $\sum_{i=0}^\infty x_iX^i$ is ...

**4**

votes

**3**answers

2k views

### Does the derivative of log have a Dirac delta term?

Dirac writes down the following formula on page 61 of his "Principles of quantum mechanics":
$\frac{d}{dx}\log x = \frac{1}{x} -i\pi\delta(x)$, see http://adsabs.harvard.edu/abs/1947pqm..book.....D ...

**3**

votes

**0**answers

125 views

### P-adic Weierstrass Lemma for several variables

The p-adic Weiestrass lemma asserts that a power series $f(z)$ with coefficients in the ring of integers of a local field can be factored as $π^n·u(z)·p(z)$ where u(z) is a unit in the ring of power ...

**0**

votes

**1**answer

182 views

### Are period domains ever contractible

Which simply-connected period domains are contractible?
Examples. Siegel upper-half space? Poincare upper-half plane? Universal cover of a Shimura variety?
Are these contractible?

**1**

vote

**1**answer

269 views

### Name convention for the composition of the preimage of a function and the function itself

Hi, given a function $f:X \rightarrow Y$, not necessarily invertible, is there a conventional name for the function $$g_f := f^{-1} \circ f:X \rightarrow \mathcal{P}(X),$$ where $\mathcal{P}(X)$ ...

**4**

votes

**1**answer

469 views

### Identity involving Fresnel integrals

In the paper E. Mehlum, Appell and the apple (nonlinear splines in space), Technical
Report No. 1676 (1981), Central institute for industrial research, Oslo (reproduced in the book Mathematical ...

**9**

votes

**2**answers

733 views

### Stone–Čech Compactification of $\mathbb{Z}$ with Fürstenberg Topology

The Stone–Čech Compactification of $\mathbb{N}$ as a discrete space has been extensively studied and can be represented using ultrafilters.
Consider $X=(\mathbb{Z},\mathcal{T})$, where $\mathcal{T}$ ...

**1**

vote

**2**answers

359 views

### Limit with theorem of dominated convergence

Let $f\in L^{2,s}(\mathbb{R}^3)=\bigg\lbrace u\bigg|\int_{\mathbb{R}^3}dx\,|u(x)|^2(1+|x|^2)^s<\infty\bigg\rbrace$ ($s>\frac{1}{2}$)
I have to calculate this limit
$$\lim_{|x-y|\to ...

**1**

vote

**0**answers

203 views

### Average weighted value of a linear functional over increasing bounded subsets of Z^n

Say you're working within the finite-dimensional free Z-module $\mathbb{Z}^n$, and you want to impose a "norm" on this module. By a "norm" I mean a function $\|·\|: \mathbb{Z}^n \to \mathbb{R}$ which ...

**2**

votes

**1**answer

308 views

### Continuous functions on path-connected subsets

Let $X$ be a topological space, and $PX$ the space of all paths on $X$. Then let $G\subset X$ be a path-connected subset and $p\in G$ a point. Let $\sigma:G\rightarrow PX$ be a continuous function ...

**3**

votes

**1**answer

428 views

### Possible to find a set of log-concave functions with log-concave sums?

While the set of log-convex functions is closed under addition, the set of log-concave functions is not. Yet if $f$ is log-concave, $\ln(k f) = \ln(k)+\ln(f)$, with $k \in \mathbb{R}^+$ constant, is ...

**-2**

votes

**1**answer

587 views

### sections of tensor product bundle ( tensor product of two vector bundles ) [closed]

Suppose we have a smooth manifold M and E--->M is a vector bundle. A connection on E is a linear map from the set of all smooth section on E into the set of smooth sections of the tensor product of E ...

**-1**

votes

**1**answer

121 views

### How to define an “anisotropic vector” for a given object?

Dear experts,
I am looking for a way to define an "alignment vector" (or anisotropy or orientation vector?) for a given geometrical object. I am not sure how to put this into correct technical terms, ...

**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 ...

**3**

votes

**1**answer

184 views

### Spectral synthesis for central functions on locally compact groups

There is a large literature on harmonic analysis on locally compact group, that
I am just beginning to discover. However I have not seen so far anything that emphasizes the central functions on $G$. A ...

**2**

votes

**1**answer

246 views

### symmetry of generationg function of PDE

We know that for finding the solutions of PDE equations, one of methods is "reduction of PDE", . For nonlinear equation
$v_t=(v^{-4/3}v_x)_x+\lambda v$ how can we compute the generators of Lie ...

**4**

votes

**1**answer

474 views

### Is there a closed form expression/series expansion for $\int_{\epsilon-i\infty}^{\epsilon+i\infty} e^{az+b^2z^2}\Gamma(z)\Gamma(1-z)dz$ ?

I've been trying to find a closed form expression/series expansion for the following integral without success:
$$F(a,b)=\int_{\epsilon-i\infty}^{\epsilon+i\infty} ...

**27**

votes

**9**answers

3k views

### Covering maps in real life that can be demonstrated to students

Edit: I've really enjoyed everyone's examples (especially the pictures!), but I was mostly looking for a general theorem. For instance, a similar statement to mine is, Can the mapping cylinder of ...

**-1**

votes

**1**answer

143 views

### Limit of a function in a weighted Sobolev space

I have a function $f(x)$ in the space $H^{2,-s}(\mathbb{R}^3)$; have this limit sense
$$\lim_{|x-y|\to 0} f(x)$$
? ($y$ is a fixed point)
If i have $f$ in $H^2$ I can say that
$$\lim_{|x-y|\to 0} ...

**2**

votes

**1**answer

172 views

### Lipshitz Constant of the convex extension of a submodular function

The title says it all :)
Given a submodular function (take the rank function of a matroid, for a concrete example) $f:\{0,1\}^n\rightarrow \mathbb{R}$, we can extend it to a convex function ...

**2**

votes

**0**answers

195 views

### Extension divergence-free, curl-converging vector field

Hi.
Consider a smooth open Set $\Omega\subset\mathbb{R}^3$ and a bounded sequence of vector fields $(u_n)_n \in L^2(\Omega)$ having $0$ divergence. I know how to extend this sequence to the whole ...

**2**

votes

**2**answers

288 views

### Homotopy Equivalences and Induced Correspondences between Fibre Bundles

Suppose that $f:X\rightarrow Y$ is a homotopy equivalence of manifolds. Given a manifold $F$, the pullback construction for $f$ yields a correspondence between isomorphism classes of fibre bundles ...

**4**

votes

**2**answers

354 views

### How to specify a finite group up to inner automorphism?

I want some finite set of data to which I can canoically associate a "group up to inner automorphism", and which can be constructed canoically from a "group up to inner automorphism". I have a few ...

**2**

votes

**1**answer

207 views

### A uniformity with a countable base is a pseudometric uniformity.

I need a proof for this proposition:
If a uniformity $\mathfrak U$ on $X$ has a
countable fundamental system of
entourages, then it can be defined by
a pseudometric on $X$.
which is the ...

**2**

votes

**0**answers

860 views

### Derivative of the regularized upper incomplete gamma function

Hello everyone!
I have a question about the derivative of the regularized upper incomplete gamma function. Considering the gamma function and the incomplete gamma function
\begin{eqnarray}
...

**2**

votes

**1**answer

71 views

### Computing a point of refraction

Oddball question: say I want to travel from $(a, b)$ where $b > 0$ to $(c, d)$ where $d < 0$ using the shortest path, where I can travel at velocity $v_1$ in the upper half-plane and at velocity ...

**1**

vote

**1**answer

421 views

### Is this min not less than a min

Let $\mathbf{D}$ be the unit disk, is
$$\inf_{\begin{array}{c}
v_{1},v_{2},v_{3},v_{4}\in\mathbf{D},\\
v_{0}\in\mbox{convexhull}\left(v_{1},v_{2},v_{3},v_{4}\right)
\end{array}}\max_{0\le ...

**6**

votes

**2**answers

2k views

### Generalizing square wheels rolling on inverted catenaries

It is not uncommon to see in a science museum a bicycle with
square wheels that rides smoothly over a washboard-like
surface made from inverted catenary curves (e.g., at the Münich museum).
The ...