# All Questions

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7 views

### How to prepare a radical change of research field after the PhD

I am in the middle of my PhD in functional analysis. My undergraduate studies were focused on pure theory and so it was logical to continue in this direction. However, recently I got into contact with ...

**-1**

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20 views

### Freedom of speech in scientific discussions - An invitation to more tolerance in Scientific debates [migrated]

I hope this post enjoys some tolerance, and don't get closed or put on hold immediately.
I believe that freedom of speech in scientific discussions is one of the key values which enriches the debates ...

**2**

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**1**answer

38 views

### Lower bound for $ \sum_{i=1}^n x_i f(x_i)$ when $\sum_{i=1}^{n}x_i = K$

Considering,
the set of all n dim. vectors $\{x_i\}_{i=1,...,n} $ such that $x_i \geq 0 $ and $\sum_{i=1}^{n}x_i = K$
Any continuous and strictly increasing function $f^+(x)$ : $ \mathbb R^+ \to ...

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35 views

### picard group of a cyclic cover

I want to find any result about the picard group of a ramified double cover of a projective plane.
Isn't there any general result about this case unlike ruled surface?
Can you recommend any good ...

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33 views

### Log smooth models for abelian varieties

Let $K$ be a field which is complete for a discrete valuation. Assume that the residue field has characteristic $p > 0$. Let $A$ be an abelian variety over $K$ having the property that (for some ...

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**1**answer

46 views

### Bijection between dominant rational maps and morphisms of function fields?

Let $X$ and $Y$ be two integral schemes of finite type over a field $k$. Consider the function fields $K(X)$ and $K(Y)$.
Do we have a bijection between:
(a) Dominant rational maps $X \rightarrow Y$
...

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38 views

### lie derivative on complex manifold

Let $M$ be a complex manifold with complex structure $J$.
Then for any smooth vector field $X$ on $M$ can be splited into its holomorphic and anti-holomorphic components $X=Z+\overline{Z}$. For any ...

**3**

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34 views

### Rectifying the definition of a closed category

The definition of a closed category I'm using is here.
Suppose $V$ is a closed category and that for each object $b\in V$, $[b,-]$ has a left adjoint $- \otimes b$. The result is nearly a monoidal ...

**1**

vote

**1**answer

27 views

### Integrals involving the Tricomi hypergeometric function

I am looking for a reference for the two following equalities involving the Tricomi function $U$ and the Meijer function $G$. I have found these formulas on the website http://functions.wolfram.com/, ...

**1**

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32 views

### How small can the Mumford-Tate group of hypersurface be?

Is there some way of giving a lower bound on the dimension of the Mumford-Tate group of a hypersurface? Let's say it's of general type, say, of degree $10$ inside $ \mathbb{P}^3$.
I would expect the ...

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21 views

### line complex in projective space [on hold]

In a paper of "R.H.Dye", which you can find here: http://link.springer.com/article/10.1007%2FBF02413785#page-1, I face with a mathematical object "line complex in projective space $PG(2n-1,q)$, I need ...

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33 views

### Rank of a particular matrix

Denote $P_{2n}$ to be collection of homogeneous total degree $2$ real polynomials in exactly $2n$ variables such that coefficient of every monomial is either $1$ or $-1$.
Split variable set into ...

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38 views

### Orientation form on the blow up of a Kaehler manifold

Let $(X,\omega)$ be a complex Kaehler manifold of (complex) dimension $d$, and let $Y\subset X$ a complex submanifold of dimension $k$. Evidently $[\omega]^d\in H^{2d}(X,{\mathbb{R}})$ is always ...

**4**

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**1**answer

60 views

### Counting equivalence relations with marked classes

The number of equivalence relations on a set of $n$ elements is the Bell number $B_n$.
If we wish to count the number of equivalence classes on a set of $n$ elements where one of the classes is ...

**2**

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**1**answer

110 views

### Existence of Hecke operators with distinct eigenvalues?

Consider the space of modular forms $M_k(N)$. Any modular form $f \in M_k(N)$ is determined by a finite number of Fourier coefficients (e.g., Sturm's bound), thus there is a finite set of Hecke ...

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59 views

### Visibility in a prime orchard

This suggests a variant on Polya's orchard problem.
That problem asks1
for which radius $\epsilon$ of trees at each lattice point within a distance $R$ of the origin block all lines of sight to the ...

**6**

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**2**answers

215 views

### Stability of Solar System [on hold]

My question is simple:
Is the Solar System stable?
You can see this Wikipedia page.
In May 2015 i was in the conference of Cedric Villani at Sharif university in Iran with this title: "Of ...

**3**

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108 views

### Goldbach's problem in algebraic number fields [duplicate]

Are there any results on the representation of numbers in algebraic number fields as the sum of primes in the ring of integers in that field? There are some results for Waring's problem in other ...

**6**

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153 views

### Easiest way to see that $\zeta_{\mathbb{Z}[i]}(s) = \zeta(s) L(s, \chi)$?

As the question suggests, what is the easiest way to see that$$\zeta_{\mathbb{Z}[i]}(s) = \zeta(s)L(s, \chi)?$$Here, $\chi$ is the homomorphism $(\mathbb{Z}/4\mathbb{Z})^\times \to \mathbb{C}^\times$ ...

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**1**answer

77 views

### How to prove the existence of the polytope in $\mathbb{R}^d$ with a given number of faces, minimizing the isoperimetric ratio?

This is the isoperimetric type question. We know that in $\mathbb{R}^d$, balls are the sets that minimize the isoperimetric ratio $\frac{S^{d}}{V^{d-1}}$, where $S$ is the surface area and $V$ is the ...

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45 views

### Non-Forking and Related Concepts

Is the importance of developing forking machinery in the way we set it up, or is it in the fact that it allows us to come up with a notion of independence via the properties of non-forking? I'm ...

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144 views

### Set as a (strict) infinite-category?

First, let me say that I have no idea if such a post has its place here. However, I believe that the ideas I'm going to present are important. The goal of this thread is three fold:
1) trying to ...

**2**

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**1**answer

87 views

### Upper bound for number of prime numbers in a range

Theorem 3.2 in http://arxiv.org/pdf/1405.2593.pdf shows that for any $x$ there are $\gg x\exp(-\sqrt{\log x})$ integers $x_0 \in [x; 2x]$ such that $\pi(x_0 + \log x) - \pi(x_o) \gg \log\log x$.
Is ...

**2**

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**1**answer

276 views

### $\pi_8(S^5)=\pi_8(SO(6))=\mathbb{Z}/24$?

I am currently thinking about a physics model related to framed bordism $\Omega_3^{fr}=Z/24=\pi^s_3$, and the first stable example is $\pi_8(S^5)$, so I was curious about the generator, and happened ...

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55 views

### Sheaf cohomology argument

Consider in a domain $X$ in $\mathbb{C}^n$ the following problem: Does there exist a holomorphic function $f$ on $X$ such that $f(p_j) = a_j$ with $\{p_j~:~j\in \mathbb{N}\}$ given and distinct ...

**4**

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**1**answer

60 views

### Analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$, does $L_c(s, \chi)$ necessarily equal $1$?

Consider an analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$. Let $g \in \mathbb{F}_q[T]$, $g \neq 0$, let $\chi: (\mathbb{F}_q[T]/(g))^\times \to \mathbb{C}^\times$ be a homomorphism, let $c ...

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**1**answer

56 views

### Symplectic form on the third symmetric power of a plane

Let $V$ a vector space of dimension $2$ over a field $k$ of characteristic different from $2$ and $3$. Let $S^{3}V$ the third symmetric power of $V$.
How to construct a symplectic form on $S^{3}V$ ...

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**1**answer

97 views

### Probability a random matrix contains a short integer vector in its kernel

Consider a random $m$ by $n$ matrix $M$ with $m \leq n$, chosen uniformly over all those whose elements are in $\{0,1\}$ (or $\{-1,1\}$ if it is any easier). Is there any mathematical theory that ...

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40 views

### Convergence of the Lyndon - Hochschild - Serre spectral sequence [on hold]

I have some trouble understanding the notion of convergence of a spectral sequence conceptually (in general). More specifically, I'm trying to understand the convergence of the Lyndon - Hochschild - ...

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**1**answer

108 views

### Lightning strike fractal formula [on hold]

I need to generate random gold ore channels for a game, I was thinking they would look kinda like lightning strikes. Anyone know any good fractals (recursive functions) that looks like it? Or ...

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43 views

### Does it require Reedy fibrancy when we want the totalization to be weakly equivalent to the homotopy limit?

This question arises when I am reading the last two Chapter of Hirschhorn's "Model categories and their localizations"
In Part (2) of Theorem 19.8.4 of that book it says
If ...

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65 views

### How can I (iteratively) solve these equations? [on hold]

I am by no means a mathematician at all (programmer) so I need some pointers on how to solve the following equations - if someone could point me to a method that would work, that would be very ...

**-2**

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**1**answer

28 views

### generate analytically bivariate correlated data [on hold]

How does one generate correlated binomial data when one is given marginal probabilities of each and also the correlation coefficient.
The following code in SAS for example works best when we want ...

**-3**

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37 views

### How can we find the presentation of a group? [on hold]

Is it possible that to find the presentation of a group $G$ such that it is extension of $\mathbb{Z}_2\times \mathbb{Z}_2$ by $\mathbb{Z}_2$?

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23 views

### Isotropic subspaces in a symplectic vectorspace over $GF(q)$

Let $V$ be a symplectic vectorspace of dimension $2n$, and $r\mid n$. Is this statement true?"There is an isotropic spread of $r$ dimensional subspaces in $V$". By an isotropis subspace I mean a ...

**2**

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72 views

### Is a specific sequentially closed subset of $M([0,1])$ closed?

Let $M([0,1])$ be the set of finite signed measures on $[0,1]$
(with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all ...

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42 views

### Does null geodesic flow live on a natural compact bundle?

Let $(M,g)$ be a compact pseudo-Riemannian manifold (closed or with boundary).
A geodesic $\gamma:(a,b)\to M$ is called null if $g_{ij}\dot\gamma^i\dot\gamma^j=0$.
The geodesic flow can be seen as a ...

**4**

votes

**1**answer

125 views

### Prove that the Log-Euclidean distance is negative-definite

Let $\Bbb{S}_{++}^n$ be the $\frac{n(n+1)}{2}$-dimensional Riemannian manifold of the symmetric positive definite (SPD) $n\times n$ real matrices.
The Log-Euclidean distance between two points of ...

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59 views

### Integration by parts? [on hold]

Let $f:{\mathbb R}\rightarrow {\mathbb R}_+$ be a density function with finite expectation. This is,
$$\int_{\mathbb R}x f(x)dx<\infty.$$
Suppose that we want to integrate $I(a)=\int_a^{\infty} x ...

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35 views

### Euler equation formula [on hold]

when I am using Euler equation for Fourier transform integrals of type $
\int_{-\infty}^{\infty} dx f(x) exp[ikx] $ I am getting following integrals:
$\int_{-\infty}^{\infty} dx f(x) cos(kx)$ (for ...

**2**

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82 views

### Twisting by a multiplicative Character in Katz, Perversity and Exponential sums

Let $C(x_1,\ldots,x_n)$ be a nonsigular cubic form with integral coefficients.
In his Proof that $C$ fulfills the Hasse-Principle, if $n\geq 9$, Hooley used the following estimate that was provided ...

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75 views

### $F[[T]] \times F[[1/T]]$ fundamental domain, show compactness

Let $p$ be a prime number. What is the easiest way to see that $(\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T)))/\mathbb{F}_p[T, 1/T]$ is compact? Here $\mathbb{F}_p[T, 1/T]$ is embedded in ...

**3**

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**2**answers

147 views

### Jacobson-Morozov theorem

Jacobson-Morozov theorem for a semisimple algebraic group $G$ (presumably I am working over algebraically closed field) states that: given a unipotent u, there exists a homomorphism $\phi$ from $SL_2$ ...

**0**

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**1**answer

50 views

### Comparison of Lp norm of matrix and its entry wise norm. [on hold]

I need to know the relation between operator norm of a matrix i.e. $ \Vert A\Vert_p$ for case of p=1 and 2 and its entry wise Frobenius norm $ \Vert A\Vert_F$.

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122 views

### What is the symmetry of SU(3) - when seen as a group / algebra? [on hold]

Simply asked: is it more correct to state that the symmetry of the SU(3) manifold is $Z_3$ or $S_3$? Or neither of the two?
SU(3) has a kind of threefold symmetry; but which one exactly? When ...

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55 views

### Prove or disprove a claim about covering a polytope by convex polytopes in a certain way

Here is the claim:
Given a polytope $K$ in a unit ball in $\mathbb{R}^d$, there exists a universal constant $C(d)>0$ depending only on $d$ and a countable collection of convex polytopes ...

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65 views

### How do i show that the fixed points of this dynamics $ x_{n+1}=x_{n}^2-x_{n-1}^2 $ are stable? [on hold]

Is there somone who can show me how do i show that the fixe point of this
dynamics $$ x_{n+1}=x_{n}^2-x_{n-1}^2 $$ are stable ?
$x_{0}+x_{1}>0 $,$x_{0}=0,x_{1}=\frac{1}{2}$
*My attempt only I ...

**-1**

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**0**answers

48 views

### Normal Sub-groupoid [on hold]

Is there a definition of normal groupoid?
For normal sub-quasi-group I found two:
The first one: a sub-quasi-group $H$ is called normal if there exists a normal congruence $\theta$ such that $H$ ...

**3**

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41 views

### Expected size of determinant of $AA^T$ for non-square random Toeplitz $A$

If $A$ is chosen uniformly at random over all possible $m$ by $n$ Toeplitz (0,1)-matrices, what is the expected size of the value of the determinant of $AA^T$? We can assume $m \leq n$ and all ...

**2**

votes

**2**answers

91 views

### Identities involving sums of Catalan numbers

The $n$-th Catalan number is defined as $C_n:=\frac{1}{n+1}\binom{2n}{n}=\frac{1}{n}\binom{2n}{n+1}$.
I have found the following two identities involving Catalan numbers, and my question is if ...