# All Questions

**0**

votes

**0**answers

8 views

### When a smooth algebra is regular?

Let $A \subseteq B$ be noetherian integral domains, $A$ regular (=every localization at maximal ideal is a regular local ring) and $B$ is a smooth $A$-algebra (for the definition of a smooth algebra, ...

**8**

votes

**1**answer

98 views

### Origin of the term “Diophantine equation”

It seems that the term "Diophantine equation" has been around at least since the second half of the 19th century, since the historian Hermann Hankel writes (polemically) in the chapter on Diophantus ...

**12**

votes

**1**answer

104 views

### When is there a submersion from a sphere into a sphere?

(First posted on math.SE, with no answers.)
That is:
For which positive integers $n, k \ge 1$ does there exist a submersion $S^{n+k} \to S^k$?
The discussion at this math.SE question has ...

**-2**

votes

**0**answers

15 views

### positiv Martingale using Itô

I would to like to prove that the process:
$$e^{\int_{0}^{T}\theta _{s}dW_{s}-\frac{1}{2}\int_{0}^{T}\theta _{s}^2ds}$$
is a martingale which is positiv and has a mean=1
$$\theta is continuous ...

**-4**

votes

**0**answers

33 views

### Matrix 5th rooth - how to find it? [on hold]

I kindly ask you to help me with an example of calculating 5th rooth of 3 over 3 matrix.
Thank you

**3**

votes

**0**answers

78 views

### What's the name of this branched covering?

I've come across a double cover of $\mathbb P_1(\mathbb C)$, ramified at $[1:1]$ in homogeneous coordinates, given as the quotient by the natural $\mathbb Z/2\mathbb Z$-action generated by ...

**4**

votes

**1**answer

32 views

### Numerical equality testing

I am working on developing an online homework system.
One thing I would like to have is something which compares a student's answer (like $2\sin(x)\cos(x)$) with the intended answer (maybe ...

**5**

votes

**0**answers

42 views

### Nontrivial finite group with trivial cohomology in prescribed degree

For any non-trivial finite group $G$ there exists some $j > 0$ such that $H^{aj}(G) \neq 0$ for all $a = 1,2,3,\dots$, see e.g. this question: Non-vanishing of group cohomology in sufficiently high ...

**-2**

votes

**0**answers

28 views

### How to prove that $h^*(A)$ exists and $h^*(A)=\{x\in ON : x \leq A\}$?

Definition : $h^*(A)$ is the least aleph such that $\not\leq^* A$.
$Z \not\leq^* X$ means theie is no map from $X$ onto $Z$.
How to prove that $h^*(A)$ exists and $h^*(A)=\{x\in ON : x \leq A\}$? ...

**2**

votes

**0**answers

44 views

### Naive G-spectrum representing geometric equivariant cobordism

Let $G$ be a finite group. By the transversality results of Wasserman $G$-equivariant bordism (say real) should be a naive homology theory, and as such it should be represented by a naive G-spectrum.
...

**0**

votes

**0**answers

34 views

### Sum of the series with Stirling numbers [on hold]

Yesterday I worked on one problem in discrethe math and in the process of decision I came across this series. Try to do it with generating functions, but there is no success for me.
So, what do you ...

**0**

votes

**0**answers

21 views

### Function series involving a suite of imprimitive Dirichlet characters and a zero of $L(\chi,s)$

I have difficulties to understand the behavior of following suite of functions near zero :
$$F_P(x)= \sum\limits_{n=P}^{\infty} \chi_P(n) f(nx)$$
With $f(x) = x^{-s_0} e^{-x}$ where $s_0$ is a non ...

**2**

votes

**1**answer

29 views

### Maximizing entropy under constraints

This question is about an extension of the variational principle in thermodynamical formalism when one adds linear constraints to the measures.
Consider the one-sided shift ...

**2**

votes

**0**answers

39 views

### On the number of irreducible components of an exceptional divisor

Let $X$ be a complex, affine variety and $Z\subseteq X$ a closed subset of $X$. Let $E$ be the exceptional divisor of the blow-up $\pi:\tilde X\to X$ of $X$ with center $Z$.
My question is: Under ...

**5**

votes

**0**answers

32 views

### Simultaneous approximation of different functions in $L^2(\mu)$ and Hölder space

Let $\mu$ be a probability measure on the circle $S^1=\mathbb{R}/\mathbb{Z}$ which is singular with respect to the Lebesgue measure $\lambda$. Consider the functions spaces $L^2(\mu)$ on the one hand, ...

**2**

votes

**0**answers

77 views

### Differences associated with differences of primes: are they all 1,2,3?

Let $d_k$ be the $k^{th}$ difference sequence of the primes; that is, $$d_k = \sum_{i=0}^{k} (-1)^i {k \choose i} P_{k+1-i},$$ where $P_i$ denotes the $i$-th prime number. Let $(s_n)$ be the ...

**0**

votes

**0**answers

55 views

### Problem with operator and Fourier transform

I am currently dealing with a problem in functional analysis where I want to show the following.
Let $\phi_{k+1}(x):=\sqrt{2} \sum_{n \in \mathbb{Z}} h(n) \phi_k(2x-n)$ and $\phi_0= \chi_{[0,1]},$
if ...

**-3**

votes

**0**answers

11 views

### Calculate coordinate transformation matrix [on hold]

Calculate coordinate transformation matrix A in the base [1; x; x2].
In the space R2 [x] of real polynomials highest rate? D is given a scalar product.
Sebi-adjoint transformation of A: R2 [x]! R2 ...

**2**

votes

**2**answers

122 views

### Convergence of a sum with the ranks of homotopy groups

Let $F$ be a (nontrivial) topological space that satisfies the following conditions: 1) $\pi_n(F)$ has a trivial action of $\pi_1(F)$ for $n>0$ and 2) its homology groups are finitely generated. ...

**0**

votes

**0**answers

15 views

### Definite integral involving Legendre Polynomial [on hold]

Does anyone know the answer to the following definite integral:
$\displaystyle \int_{0}^{\pi}P_{\ell}(\cos\theta)\sin^{k}\theta\, d\theta $,
for $k\geq 1$, where $P_{\ell}(x)$ is the $\ell$-th ...

**4**

votes

**1**answer

84 views

### Radial limit does not exist almost everywhere

Problem 4 in Chapter 4 of Stein's book "Real Analysis" says
$\sum_{n\geqslant 0}z^{2^n}$
doesn't have radial limit as $z$ approaches the unit circle from inside almost everywhere. It's fairly easy ...

**0**

votes

**0**answers

43 views

### cobordism and smoth-manifold [on hold]

Let M, N , N' and M' be smooth n-manifolds with nonempty boundaries , and suppose h:∂M→∂N , g:∂M'→∂N' are diffeomorphisms . Let M∪_h N be the adjunction space formed by identifying each xϵ∂M with ...

**4**

votes

**0**answers

25 views

### Multi-podal points

Two points $x,y \in \mathbb{R}^n$ are called antipodal if $x = -y$.
Stated differently, $x,y$ are antipodal if:
They have the same absolute value in each of their $n$ coordinates;
Each of their ...

**0**

votes

**1**answer

72 views

### Can a rigid CY threefold have infinitely many automorphisms

Let $X$ be a rigid Calabi-Yau threefold. Does $X$ have only finitely many automorphisms?
N.B. A smooth projective threefold $X$ over $\mathbb C$ is a rigid Calabi-Yau variety if $h^i(X,\mathcal O_X) ...

**0**

votes

**0**answers

26 views

### Is Wiener amalgam spaces $W^{2,1}(\mathbb R)\subset C_0(\mathbb R)$?

I have been learning Wiener amalgam spaces.
In Wiener amalgam spaces $W(X, L^2)$, I am taking $X=\mathcal{F}L^{1}=\{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}\},$ and $m(x)=1.$
Take $f(x)= ...

**0**

votes

**0**answers

87 views

### Finite trigonometric polynomial

I noticed by numerical and some explicit calculations for a few examples that for real-valued finitely supported functions $\phi \in L^2(\mathbb{R})$ we have that
$T(x):= \sum_{n \in \mathbb{Z}} ...

**1**

vote

**1**answer

92 views

### Cohomology of Homogeneous Complex Manifolds

Let $M$ be compact $G$-homogeneous manifold, equipped with the equivariant complex structure, when $G$ is a semi-simple algebraic group. The obvious example is every flag manifold. In that case, all ...

**2**

votes

**0**answers

31 views

### Is there any topological information encoded by the zero locus of a complex Hessian?

On a Kahler manifold one can always find a function $K$ so that (up to some constant) the Kahler form $\omega$ can be written as $\omega = \partial \bar{\partial}K$ (sometimes, under conditions that I ...

**1**

vote

**0**answers

45 views

### Fiber Bundle with a perfect bilinear map

Let $\pi:X\rightarrow Y$ a double cover of Riemann surfaces, and consider a $SL_n-$bundle $E$ with a perfect $\mathcal O_Y-$bilinear pairing $$\psi:\pi_*E\times\pi_*E\rightarrow\pi_*\mathcal O_X$$
...

**1**

vote

**0**answers

37 views

### Finitely generated ordered monoids and noetherian subsets

(This question was asked a long time ago on MSE but got no answer so far...)
Let $E$ be an additively written cancellable commutative monoid with no non-trivial units. We furnish $E$ with the order ...

**11**

votes

**1**answer

166 views

### How many traces are there on Temperley-Lieb, Fuss-Catalan, Iwahori-Hecke, Birman-Wenzl-Murakami-Kauffman, … algebras?

There is a theorem (I believe by Ocneanu) that the Markov trace on the tower of Temperley-Lieb algebras is (essentially) unique.
What about just traces on separate algebras? That is, take one of ...

**4**

votes

**0**answers

59 views

### Analytic equivalence relations whose classes are sometimes Borel

There are analytic equivalence relations for which the statement "All classes are Borel" is independent of $ZFC$. In all the examples I know about, the classes are non Borel in $L$ or $L[z]$ for some ...

**0**

votes

**1**answer

56 views

### If $q^k n^2$ is an odd perfect number with Euler prime $q$, are the following statements known to hold in general?

Let $\sigma$ be the classical sum-of-divisors function.
A number is said to be perfect if $\sigma(N)=2N$.
If $q^k n^2$ is an odd perfect number with Euler prime $q$ (i.e., $q \equiv k \equiv 1 \pmod ...

**2**

votes

**0**answers

9 views

### Discrete “difference” equations that involve changes in both shift and scale

A standard use of the Z-transform ($F(z) = \sum_n (f[n] \cdot z^{-n} )$) is to understand the effect of a difference equation on a signal. For instance:
$y[n] = x[n] + y[n-1]$
$Y(z) = X(z) + Y(z) ...

**3**

votes

**0**answers

41 views

### Meager subgroups of compact groups

Suppose we have an infinite compact (Hausdorff) group $G$, and a subgroup $H\leq G$ which is meagre.
Can $H$ always be covered by a countable family of nowhere dense sets $H_n$ such that $H_n^2$ is ...

**-2**

votes

**1**answer

51 views

### stable splitting into a wedge sum [on hold]

Suppose $X$ is a CW-complex such that there is a stable splitting of $X$ into wedge sum
$$
\Sigma^t X\cong \bigvee _{k=1}^\infty Y_k.
$$
(1). Does this imply
$$
X\to \Sigma^tX\to \bigvee ...

**0**

votes

**0**answers

49 views

### Hyperbolic manifold of dim 3 with finite volume.

The geometrization Theorem for 3-manifolds classifies all oriented compact (without boundary ) 3-manifolds. Is there a theorem which classifies all hyperbolic oriented 3-manifold (without boundary ) ...

**5**

votes

**1**answer

98 views

### Do non-normal states exist in the Solovay model?

Let H be an infinite dimensional Hilbert space. Then there exist non-normal states on B(H) in ZFC (i.e. states that are not represented by a density operator).
Is this also true in the Solovay model ...

**0**

votes

**0**answers

34 views

### Idempotent ideal in ring of continuous functions

Is there any equivalence conditions under which an ideal $I$ in ring of continuous functions be be an idempotent ideal?

**-5**

votes

**0**answers

19 views

### How to find the odds for events (two or more) using its probability [on hold]

i am currently using the below formula to calculate odds of event 1 win
event 1 winning odds = (p1)/(1-p1)/
(p2)(1-p2)
p1 = probability of winning of event1
p2 = ...

**0**

votes

**0**answers

93 views

### A diophantine equation

A few days ago I saw a question about the diophantine equation $p^2+p+1=q^\alpha$ in
What is prime power of this equation of p?
and later in
A Diophantine equation with prime powers
I want the ...

**3**

votes

**0**answers

46 views

### Reference for a lemma on the asymptotic upper density of special sets with large gaps and intervals

Update. Based on Anthony Quas' comment below, the proof can be made sensibly shorter and the lemma can be slightly generalized by weakening the old assumption (iii).
In a joint paper that I am ...

**6**

votes

**0**answers

60 views

### Equation of the curve corresponding to a polarization of an abelian surface

Let $\mathbb{C}^2/\Lambda$ be a polarized abelian surface. I think it is well-known how to write down the equation of the divisor corresponding to the polarization, in terms of theta functions etc. ...

**4**

votes

**1**answer

115 views

### understanding of rough path

A rough path is defined as an ordered pair
$ (X, \mathbb X)$, where $X$ is a path mapping from $[0,T]$ to some Banach space $V$
and $\mathbb X:[0,T]^2 \mapsto V^2$ is another mapping for additional ...

**4**

votes

**0**answers

40 views

### Concavity of mixed volumes and mixed discriminants

For $n\times n$ symmetric matrices $A_1, \ldots, A_n$, the mixed discriminant $D(A_1, \ldots, A_n)$ can be defined as $1/n!$ times the coefficient of $t_1\ldots t_n$ in the homogeneous polynomial ...

**5**

votes

**1**answer

68 views

### Pseudodifferential operators on spaces with boundary

Consider the upper half space $\mathbb{R}^n_{+} = \{x = (x_1,..,x_n) \in \mathbb{R}^n : x_n \geq 0\}$. Consider the Laplacian on this space with either the Dirichlet boundary condition or the Neumann ...

**8**

votes

**2**answers

264 views

### On the number of consecutive divisors of an integer

Define for $n \in \mathbb{N}$ the function $$\tau_1(n):=\sum_{\substack{d|n, \\ d+1|n}}1,$$ i.e. the number of consecutive divisors of an integer. The average of $\tau_1(n)$ is $1$ since $$\sum_{n\leq ...

**2**

votes

**1**answer

107 views

### Can the epsilon induction condition be presented with successor case and limit case, similar to transfinite induction?

The epsilon induction looks like this:
$\forall x \Big(\forall y (y \in x \rightarrow P(y)) \rightarrow P(x)\Big) \rightarrow \forall x P(x)$
Here, the quantifiers "run over" any sets and not only ...

**-3**

votes

**0**answers

44 views

### Computer Science/Maths Hamming Distance [on hold]

My professor told us to try and remember the equation used for an upcoming exam, however I'm struggling to fit the equation into the question:
http://i.stack.imgur.com/RoPYG.png
(need a high ...

**5**

votes

**0**answers

78 views

### Homotopy pullback preserving functor

In the paper http://arxiv.org/pdf/math/0101162.pdf, the authors claim during the proof of Prop. 4.2 that a functor $F:A \to B$ which preserves fibrations and weak equivalences preserves homotopy ...