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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-2
votes
0answers
47 views

Can someone help me with a Towers of Hanoi problem? [on hold]

I have 9 disks which is hard to keep track of, and I want to know... out of the 2^9 -1 moves... how many positions are there when all three posts are occupied, and is there a short formula describing ...
-7
votes
1answer
130 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
4answers
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 ...
-3
votes
1answer
94 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 ...
47
votes
3answers
4k 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
1answer
180 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
1answer
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
0answers
144 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 ...
5
votes
3answers
746 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
2answers
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
1answer
476 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
2answers
238 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
1answer
263 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
1answer
198 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
1answer
225 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
1answer
353 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
4answers
429 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 ...
5
votes
1answer
451 views

Showing a filter with a certain property on the power set of $\mathbb{Z}$ is a one point filter

Let $\mathcal{P}_0(X)$ the Power set of $X$ without the empty set and let $\dot{x}:=\{A\subseteq X: x \in A\}$ the one point filter generated by $x$. Furtermore let $$ \mathcal{A} := \{ f \in ...
1
vote
1answer
136 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
6answers
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
1answer
227 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
1answer
98 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
3answers
1k 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
0answers
123 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
1answer
176 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
1answer
264 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
1answer
446 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
2answers
710 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
2answers
336 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
0answers
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
1answer
293 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
1answer
382 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
1answer
532 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
1answer
107 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
16answers
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
1answer
182 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
1answer
244 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
1answer
454 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
9answers
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
1answer
138 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
1answer
167 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
0answers
182 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
2answers
282 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
2answers
343 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
1answer
187 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
0answers
748 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
1answer
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
1answer
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
2answers
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 ...
-1
votes
1answer
203 views

Group or manifold ? [closed]

I have a question in seeing this $$U(n)=\frac{U(n)}{U(n-1)} * \frac{U(n-1)}{U(n-2)}*\cdots *\frac{U(2)}{U(1)}*U(1)$$ So, group U(n) is written as product of quotient spaces. Is quotient space, for ...