# All Questions

**3**

votes

**1**answer

254 views

### UFD and fundamental group

Let $C$ be the curve $x^2+y^2-1$, defined over $\mathbb R$. It is easy to see that $\mathbb R[C]$ is not a UFD, as witnessed by the identity $(1-x)(1+x)=y^2$. On the other hand, the real locus ...

**2**

votes

**0**answers

50 views

### Deformations of associative algebras and Hochschild cohomology

I am studying the deformation theory of associative algebras (and Poisson algebras) and came across a question for which I cannot find an answer:
Let $(A,\mu)$ be a commutative associative algebra ...

**1**

vote

**1**answer

142 views

### Moduli space of flat connections over a torus

Let us fix a principal bundle $G\hookrightarrow P\to T^{2}$, where $T^{2}$ is a torus. Is the moduli space of flat connections on $P$ known? At least, it is known for some particular gauge groups, ...

**-2**

votes

**0**answers

41 views

### Automorphisms of B_n [migrated]

I'm working on a proof for graph isomorphism involving a Poset diagram. If all automorphisms for B_n are inner automorphisms, then my proof will be complete. Is this the case? Or are some of the ...

**4**

votes

**1**answer

230 views

### Etale cohomology approach on $\tau(n)$

Ramanujan's $\tau$ conjecture states that $$\tau(n)=O_\epsilon(n^{\frac{11}2+\epsilon}),$$ which is a consequence of Deligne's proof of Weil conjectures. Answers in ...

**2**

votes

**0**answers

36 views

### Symplectic form on a toric manifold

I have a standard question about symplectic forms on toric manifolds:
Let $P$ be an $n$-dimensional Delzant polytope and let $X_P$ be the corresponding symplectic toric manifold with symplectic form ...

**7**

votes

**3**answers

305 views

### Freiling's Axiom of Symmetry Concretized

Freiling's Axiom of Symmetry says that for any function $f:[0,1]\to \mathcal{P}([0,1])$ such that for every $x\in [0,1]$ we have $|f(x)|=\aleph_0$, then there exist $y,z\in [0,1]$ such that $z\notin ...

**5**

votes

**2**answers

146 views

### Is an $\mathfrak{sl}_2$-triple determined up to Lie algebra automorphism by the adjoint representation?

Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra, and let $\phi_1:\mathfrak{sl}_2(\mathbb{C})\rightarrow\mathfrak{g}$ and ...

**0**

votes

**0**answers

37 views

### Two surfaces with zero gaussian curvature [on hold]

According to Hartman's article every surface $f(s,t)$ with zero gaussian curvature locally admits parametrization
$f = a_1(u) v + b_1(u),$
$s = a_2(u) v + b_2(u),$
$t = a_3(u) v + b_3(u).$
Now let's ...

**1**

vote

**0**answers

30 views

### On compactness in Sion's minimax theorem

Sions minimax theorem (wiki, paper) can be stated as follows:
Let $X$ be a compact convex subset of a linear topological space and $Y$ a convex
subset of a linear topological space. Let $f$ be a ...

**0**

votes

**1**answer

47 views

### Principal bundle associated to a Courant algebroid

A Courant algebroid is defined as a real vector bundle equipped with a product and a symmetric bilinear on its space of sections satisfying a particular set of conditions. What would be the definition ...

**-1**

votes

**0**answers

46 views

### Recent articles or textbooks on stochastic analysis/calculus [on hold]

I was directed from here. Where can I look for recent articles or journal publications (or what are they called) on stochastic analysis/calculus?
While I'm here, any recommendations for stochastic ...

**-2**

votes

**0**answers

44 views

### finite cyclic group have subgroup of prime index [on hold]

I'am trying to prove that if $G$ is finite cyclic group there is subroup of $G$ of prime index ,is it true?

**5**

votes

**0**answers

81 views

### For which sidelengths are there polyominos composed of three squares that tile the plane?

Given three naturals $a<b<c$. We consider polyominos, connected or not, which are composed of three squares of sides $a,b,c$.
How can one characterize all triples $a,b,c$ for which such a ...

**3**

votes

**0**answers

175 views

### Flatness over a perfectoid ring

I want to prove the following: Let $R$ be a perfectoid ring and $\varpi$ a pseudo uniformizer in $R$ which admits all $p$-th power roots, then a module over $R^\circ$ is flat if and only if it has no ...

**2**

votes

**0**answers

62 views

### References for the bicategory of ring-bimodule pairs

One of the standard examples of a bicategory is the bicategory of rings (with bimodules as 1-morphisms), which is sometimes denoted $\operatorname{Bim}$ and in other sources $\operatorname{Ring}$ (or ...

**-3**

votes

**0**answers

32 views

### Averages of bounded function [migrated]

For a continuous $f\colon\mathbb{C}\longrightarrow[0,1]$, what could be said about $\mathcal{F}=\{f_r\colon r>0\}$ where $$\forall\,r>0,\,\forall\,z\!\in\!\mathbb{C},\quad ...

**-4**

votes

**0**answers

42 views

### Differentiability of a function [on hold]

Can someone help me out to give a bound of the first derivative w.r.t. $x$ of the function $\min_{x,y\in\Omega}\{1,\frac{d(x,\partial\Omega)d(y,\partial\Omega)}{|x-y|^2}\}$?. ...

**1**

vote

**1**answer

62 views

### Source for equations involving congruences of Fibonacci and Lucas numbers

In a paper of Cohn (see here), he uses some formulae involving congruences of Lucas- and Fibonacci-numbers (equations 11,12,13 in the preliminaries section). Does anyone know a source for these (and ...

**4**

votes

**1**answer

188 views

### Differential forms along the fiber

Let $E \to M$ be a smooth fiber bundle. Instead of differential forms defined on the whole tangent bundle $TE$ one could also consider forms on the vertical tangent bundle $VE$, i.e. forms defined on ...

**9**

votes

**1**answer

123 views

### How can we interpret the eigenvalues and eigenvectors of Euclidean Distance Matrices?

I asked this question in Math Stack Exchange earlier here: http://math.stackexchange.com/questions/1199380/what-is-the-intuition-behind-how-can-we-interpret-the-eigenvalues-and-eigenvec and since I ...

**0**

votes

**0**answers

37 views

### Are normal deformations of an embedding open in the $C^{\infty}$-space of embeddings of a compact smooth manifold`

Let $M$ be a compact smooth manifold (closed, for simplicity), $n\in\mathbb{N}$ and equip the space of embeddings $Emb(M,\mathbb{R^n})$ wih the Whitney-$C^{\infty}$-topology. (The weak and the strong ...

**1**

vote

**0**answers

49 views

### is sufficient cohesion equivalent to the connectedness of subobject classifier?

I'm following Lawvere article Axiomatic Cohesion.
He states (Proposition VI.4) that sufficient cohesion is equivalent to the connectedness of subject classifier, but I can't follow the proof. I can't ...

**1**

vote

**0**answers

56 views

### Extension of scalars and projective limits

Consider a morphism of commutative rings $h\colon R\rightarrow S$. This gives rise to a functor $h^*\colon{\sf Mod}(R)\rightarrow{\sf Mod}(S)$, called scalar extension by means of $h$. This functor ...

**1**

vote

**0**answers

64 views

### Why is the oriented $G$-homotopy type of a $G$-complex uniquely determined by the periodicity generator?

Say we have a periodicity generator $e \in H^k(BG)$. I can show that we then have a $(k-1)$-dimensional $G$-complex $X$ with free $G$-action. It's also not that difficult to see that it has trivial ...

**6**

votes

**5**answers

260 views

### The maximal eigenvalue of a symmetric Toeplitz matrix

Let $0\le x\le 1$ be a real number. Denote by $A_n(x)=(a_{ij})$ the $n$ by $n$ matrix such that $a_{ij}=x^{|i-j|}$ and let $\lambda_n(x)$ be the maximal eigenvalue of $A_n(x)$.
Is there any ...

**1**

vote

**0**answers

121 views

### Connectedness of fibers for flat, proper morphism

Let $f:X \to Y$ be a flat proper morphism of noetherian schemes of finite type over a field. Assume that $Y$ is an integral scheme and the generic fiber of $f$ is irreducible of dimension $1$. Is it ...

**-1**

votes

**0**answers

26 views

### Game Theory question about a financial pyramid scheme. Begging for help [migrated]

Salut, fellow game theorists. I have to solve 6 Game Theory problems and fell almost hopeless. Would appreciate any guidance with this one.
A company Zest is actively promoting its cervices. ...

**-2**

votes

**0**answers

24 views

### Geometry of generalized Lipschitz functions [on hold]

Clearly, that geometry of Lipschitz function is a double cone whose vertex can be translated along the graph, so that the graph always remains entirely outside the cone (image from wiki).
I've found ...

**3**

votes

**0**answers

52 views

### Decidable properties of the Cayley complex of a presentation

Let $X= X(P)$ be the Cayley complex of a finite group presentation $P=<S | R>$. Are there geometric properties of $X$ that are known to be decidable by an algorithm that takes $P$ as input? For ...

**0**

votes

**0**answers

48 views

### Bounded-open topology vs norm on $L\left(X,Y\right)$

In general topology there is two ways of introducing a topology on the space of (continuous) maps between, say, metric spaces: set-open topology and uniform topology (it is a uniformity of uniform ...

**1**

vote

**0**answers

14 views

### Estimating overshoot in spline interpolation

Say I have a spline space $\mathcal S$ of dimension $n$ with a set of unisolvent points $(\xi_i)_{i=1}^n$, i.e., points at which I can unambiguously interpolate within the spline space. So, given ...

**0**

votes

**0**answers

16 views

### Bessel PDE solution discontinuity [on hold]

For the Bessel PDE defined as
$ x^{2}y^{ \prime\prime} +xy^{\prime}+(x^{2}-m^{2})y=0 $,
Is there any reason to conclude that its solution y will be discontinuous at $x = 0$ ?
Thanks in advance.

**1**

vote

**1**answer

120 views

### Does there exist an integer that is both solitary and almost perfect?

This question is an offshoot from the following MSE post. I hope that it is appropriate for this site.
Let $\sigma(x)$ be the sum of the divisors of $x$.
An integer $a$ is said to be solitary if ...

**0**

votes

**0**answers

70 views

### What is wrong with the argument that zero permanent is polynomial?

This Lecture summarizes some well known facts about $\#P$ completeness of permanent.
Given a CNF formula $\phi$ on $n$ variables, they construct
matrix $A$ such that:
$$perm(A)=4^{3m} \#SAT(\phi)$$
...

**0**

votes

**1**answer

124 views

### Hermitian form, fundamental $2$-form of Kahler structure on $\mathbb{C}^n$

I've come across the following (it is an excerpt of Stolzenberg's lecture notes 19):
Wirtinger's Inequality.
Let $L$ be a complex linear space and let $M$ be a real
even-dimensional ...

**0**

votes

**0**answers

39 views

### Differences between primitive central idempotents and primitive orthogonal idempotents [on hold]

If we have a complete set of primitive orthogonal idempotents of an algebra $A$, then we can obtain simple modules, indecomposable projective modules, indecomposable injective modules of $A$.
If we ...

**3**

votes

**0**answers

75 views

### why can't taylor series capture memory effects? [on hold]

I am trying to understand when to use Volterra series.I found this on wikipedia 'The Volterra series is a model for non-linear behavior similar to the Taylor series. It differs from the Taylor series ...

**2**

votes

**2**answers

66 views

### Continuous image relation on topological spaces

Let $\kappa$ be a cardinal, and let $\text{Top}(\kappa)$ be the set of topological spaces $(X,\tau)$ such that $X\subseteq \kappa$. We pre-order $\text{Top}(\kappa)$ by
for $X, Y \in ...

**4**

votes

**1**answer

185 views

### Do algebraic stacks satisfy fpqc descent?

It is known, thanks to Gabber, that algebraic spaces are sheaves in the fpqc topology:
Stacks project 03W8
Is the analogous statement for algebraic (Artin) stacks true? If not, is it true under ...

**7**

votes

**1**answer

186 views

### Homotopy of orthogonal groups in the unstable range

We fix an integer $n$ and consider the stabilization map $O(n)\to O$.
Using rational methods one can easily check that the map
$\pi_{4i-1}(O(n))\to \pi_{4i-1}(O)\cong\mathbb{Z}$ vanishes for ...

**-4**

votes

**0**answers

26 views

### How to Solve this Matrix Question [on hold]

So I've been given this question which is one two by two matrix, and I believe multipled by another two by two matrix which = the matrix
[ 1 0
0 1]
The ...

**4**

votes

**1**answer

206 views

### Is there a nonabelian finite simple group with Grothendieck ring of multiplicity one?

Let $G$ be a finite group. It admits finitely many irreducible complex representations $H_1, \dots, H_r$ which generate, for $\oplus$ and $\otimes$, the Grothendieck ring $\mathcal{G}(G)$ of $G$ (also ...

**2**

votes

**0**answers

102 views

### Relationship between coherent toposes/coherent logic and coherent sheaves

I've heard it claimed that the adjective "coherent" in logic/topos theory (i.e. coherent logic, coherent toposes, coherent categories) was adopted to fit in with the terminology of coherent sheaves in ...

**3**

votes

**1**answer

120 views

### Represent matrix immanants using Schur functions

For each irreducible character $\chi^\lambda$ of the symmetric group $S_n$, the immanant of an $n\times n$ square matrix $A$ is defined as
\begin{equation*}
d_\lambda(A) := \sum_{\sigma \in S_n} ...

**2**

votes

**1**answer

127 views

### Reading Ratner's paper “Ragunathan's conjectures for SL(2,R)”

Hello everyone (interested),
I am trying to read Marina Ratner's paper "Ragunathan's conjectures for $SL_{2}(R)$" (Israel Journal of Mathematics 80 (1992), 1-31). There is a claim right at the end of ...

**4**

votes

**0**answers

66 views

### Finitely generated projective modules over the algebra of sections of the Clifford bundle

Consider a (pseudo-)Riemannian manifold $(M,g)$ and the corresponding Clifford bundle $Cl_g(T^*M)$. Let $R$ be the algebra of sections of $CL_g(T^*M)$, with point-wise multiplication. What are the ...

**4**

votes

**1**answer

98 views

### Sufficient conditions for establishing a total order on a family of probability distributions?

Let $\mathcal{X}$ be some set of independent random variables. Define the ordering on $\mathcal{X}$ by $X_i \prec X_j$ if and only if $\mathcal{P}\left\{X_i \le X_j\right\} \ge \frac{1}{2}$. Are there ...

**-3**

votes

**0**answers

17 views

### series of numbers [migrated]

I asked this question some time ago but was closed, but, I think it is interesting and I will ask again. So, if we have the formula
$$
\frac{(n+1)n}{2}=1+2+\dotsb+n
$$
now, if the difference ...

**1**

vote

**0**answers

77 views

### Showing a wedge product is nonzero

Let $V$ be a complex vector space of dimension $n$, equipped with a Hermitian inner product whose Kahler form we denote by $\omega$. Let's set $P = \bigwedge^{2p} V^*$ and $Q = \bigwedge^{2q} V^*$ for ...