**-3**

votes

**0**answers

254 views

### 27 years old. Working in silicon industry. Considering PhD in Math [on hold]

All,
I am 27 years old. I am working in the silicon industry. Did my Masters in Electrical Engineering.
My personal life has impeded me and got in the way of my decision making in academics. ...

**0**

votes

**0**answers

107 views

### proof for unsolved problem [on hold]

I have proofs for non-existence of magic square of squares and perfect cuboid, I have submitted them for AMS and they rejected soon without any consideration. Here I am not professional to decide ...

**4**

votes

**3**answers

673 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 ...

**16**

votes

**1**answer

459 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

192 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

231 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. ...

**2**

votes

**2**answers

377 views

### Defining a topology in the Power Set

I have the follwing question:
Given a topological space $T$ is possible in general to give a topology to $2^T$ (the power set of $T$) such that this topology in $2^T$ is related to $T$.
If the ...

**0**

votes

**1**answer

191 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 ...

**5**

votes

**1**answer

207 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 ...

**3**

votes

**3**answers

297 views

### Embedding Theorem for topological spaces, and in general

There are many examples throughout mathematics of abstracting the formal properties of a "familiar" structure, but then having a theorem stating that all models of the abstract axioms embed into one ...

**7**

votes

**3**answers

855 views

### What is the definition of continuity of set-valued functions?

According to the wiki of Kakutani's fixed-point theorem, A set-valued mapping $\varphi$ from a topological space $X$ into a powerset $\wp(Y)$ called upper semi-continuous if for every open set $W ...

**5**

votes

**1**answer

279 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 ...

**3**

votes

**4**answers

372 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 ...

**6**

votes

**1**answer

239 views

### On the cardinality of perfect spaces with the countable chain condition

QUESTION: Does every regular perfect space with the countable chain condition have cardinality bounded above by the continuum? Is this at least true for perfectly normal ccc spaces?
Recall that a ...

**8**

votes

**0**answers

239 views

### Can a composition with itself of a universal self-map be non-universal?

I have formulated (and published) the notion of a universal map (and of a universal morphism), and the problems below, in the early 1960-ies.
DEFINITION A continuous map $u: ...

**5**

votes

**1**answer

444 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

**1**answer

113 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 ...

**2**

votes

**1**answer

225 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

92 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

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

**0**answers

118 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

166 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

246 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

377 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

656 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 $\mathcal{T}$ ...

**2**

votes

**1**answer

241 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

259 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

388 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 ...

**0**

votes

**1**answer

77 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, ...

**3**

votes

**1**answer

176 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

239 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 ...

**1**

vote

**2**answers

303 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 ...

**0**

votes

**1**answer

129 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

**0**answers

133 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 ...

**4**

votes

**1**answer

412 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} ...

**2**

votes

**2**answers

258 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 ...

**2**

votes

**1**answer

164 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

**1**answer

64 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 ...

**2**

votes

**0**answers

491 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}
...

**-1**

votes

**1**answer

196 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 ...

**2**

votes

**1**answer

287 views

### How do minimal polynomials relate?

How does the idea of a "minimal polynomial" for a matrix (i.e. for a matrix $A$, the polynomial, $\mu (x)$, of least degree, such that $\mu (A) =0$) relate the the "minimal polynomial" for some ...

**4**

votes

**2**answers

315 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

191 views

### How many flavors should a notational system offer for rank-1 tensors?

The notation for tensors is like the plumbing in a very old Vermont farmhouse. It may once have been intentionally designed, but after that it just evolved. As an example, it seems that depending on ...

**2**

votes

**1**answer

251 views

### Homotopy groups of K3

Let X be a K3 surface and $Y=X/\mathbb{Z}_2$, an Enrique surface.
Long exact sequence of homotopy groups corresponding to fiberaion $\pi:X\to Y$, says that $\pi_2(X)=\pi_2(Y)$, while we know $H_2(X)$ ...

**1**

vote

**0**answers

330 views

### Properties of a rational function of multiple variables

Suppose you are given a multivariable rational function f(x0,x1,x2,x3,..,xn), so the only four operation are +,-,*,/.
Assume that all constants and exponents are integers within certain range.
I ...

**12**

votes

**1**answer

305 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 $2$ ...

**11**

votes

**2**answers

753 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 ...

**6**

votes

**1**answer

705 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 ...

**0**

votes

**1**answer

149 views

### Boundary Problem with an Area Constraint

Consider a boundary given by vertices (0,a), (0,0) and (1,0) (an 'L' shaped boundary).
The problem is to find the equation that passes between the endpoints (0,a) (1,0) of minimum length that ...