# All Questions

**3**

votes

**3**answers

655 views

### Connection between properties of Dynamical and Ergodic Systems

Hi All
While studying Topological and Ergodic Dynamics, I've got quite preplexed by the different Properties a system might have (minimality, regionally recurring, transitivity, mixing, ergodic, ...

**1**

vote

**1**answer

330 views

### Variation formula of a metric [closed]

In Terry Tao's notes on the Poincare Conjecture, he makes a jump I can't understand.
From differentiating the identity $g^{\alpha \beta}g_{\beta \gamma} = \delta^\alpha_\gamma$ we obtain the ...

**5**

votes

**3**answers

498 views

### Any rigorous way to claim that sums with repeat summands are few?

Let $B \subset \mathbb{Z}^+$. Define $r_{B,h}(n)$ to be the number of ways of writing $n$ as the sum of $h$ elements of $B$ and $R_{B,h}(n)$ the number of ways to write $n$ as the sum of $h$ DISTINCT ...

**0**

votes

**1**answer

87 views

### Is it possible to represent non-linear ranking type constraints as equivalent linear constraints?

I have formulated a linear program with binary indicator variables $z_i(a)$ which is equal to $1$ if the $i^{th}$ document is of rank $a$ and $0$ otherwise.
The other variables in the linear ...

**12**

votes

**2**answers

584 views

### Are there any natural recursively but not primitive-recursively axiomatized theories?

In principle, we could have a recursively axiomatized theory for which the property numbers-an-axiom (even relative to some routine Gödel numbering scheme) is recursive but not primitive recursive. ...

**3**

votes

**6**answers

1k views

### Source needed (at final-year undergrad level) for the double cover of SO(3) by SU(2)

This is a bit of an ill-defined question, and I feel I should have been able to resolve it by combining Google with a few library trips, but I'm having difficulty narrowing down the search results to ...

**22**

votes

**4**answers

2k views

### In what sense is the étale topology equivalent to the Euclidean topology?

I have heard it said more than once—on Wikipedia, for example—that the étale topology on the category of, say, smooth varieties over $\mathbb{C}$, is equivalent to the Euclidean topology. I have not ...

**5**

votes

**5**answers

2k views

### Definition of infinite permutations

I've been trying to find a definition of an infinite permutation on-line without much success. Does there exist a canonical definition or are there various ways one might go about defining this?
The ...

**8**

votes

**5**answers

3k views

### Methods for solving Pell's equation?

It is known that the minimum solution of Pell's equation $x^2-dy^2=\pm1$ can be found from the continued fraction expansion of $\sqrt d$. Are there other methods for finding the minimum (or any other) ...

**6**

votes

**6**answers

895 views

### The relationship between low dimensional topology and dynamics

I am just curious how dynamics get connected with low dimensional topology. Or it is just that we have now powerful computing machines therefore it is natural to use them on topological problems. What ...

**7**

votes

**6**answers

3k views

### Picard group, Fundamental group, and deformation

One of the most elementary theorems about Picard group is probably $\mathrm{Pic} (X \times \mathbb{A}^n) \cong \mathrm{Pic} X$ and $\mathrm{Pic} (X \times \mathbb{P}^n) \cong \mathrm{Pic} X \times ...

**12**

votes

**1**answer

1k views

### The Rabinowitz Trick

The recent question about problems which are solved by generalizations got me thinking about the Rabinowitz trick, which is used to prove a statement of Hilbert's Nullstellensatz, specifically, the ...

**11**

votes

**5**answers

1k views

### Intended interpretations of set theories

In his Set Theory. An Introduction to Indepencence Proofs, Kunen develops $ZFC$ from a platonistic point of view because he believes that this is pedagogically easier. When he talks about the intended ...

**7**

votes

**1**answer

338 views

### Reference request for Plancherel measure

I need a good reference for the basic definitions of the dual of locally compact group (not necessarily abelian), its natural topology, $\sigma$-algebra, and the Plancherel measure on it (when they ...

**3**

votes

**4**answers

871 views

### Values of Dirichlet L-funcions at natural numbers

I want to know about reference of formulas for
$$
L(s,D)=\sum_{n=1}^\infty \left(\frac{D}{n}\right)\,n^{-s}
$$
for $s$ a positive integer number and $D$ a fundamental discriminant. For $s=1$ we have ...

**14**

votes

**3**answers

3k views

### sets with positive Lebesgue measure boundary

Consider a compact subset $K$ of $R^n$ which is the closure of its interior. Does its boundary $\partial K$ have zero Lebesgue measure ?
I guess it's wrong, because the topological assumption is ...

**16**

votes

**1**answer

1k views

### Voevodsky's counterexample to the existence of a motivic t-structure

I have been trying to unravel some of the known relationships between various ideas on mixed motives. I find the litterature quite hard to follow -"from experts, for experts".
Voevodsky in ...

**18**

votes

**2**answers

2k views

### What is the relationship between integrable systems and toric degenerations?

Given an integrable system on a Kahler manifold X, is there a way to associate a toric degeneration of X whose Milnor fibers are related to the fibers of the integrable system?
An integrable system ...

**7**

votes

**4**answers

1k views

### Geometric interpretation of Universal enveloping algebras

Given a complex Lie algebra $\mathfrak g$, we can form its universal enveloping algebra and interpret it as a noncommutative space.
Is this perspective useful? What does this space "look like"?
How ...

**1**

vote

**1**answer

393 views

### Form of even perfect numbers [closed]

From the list of even perfect numbers
http://en.wikipedia.org/wiki/List_of_perfect_numbers
it can be observed that all of them have either 6 or 8 as a last digit.
Is this true for all even perfect ...

**4**

votes

**1**answer

93 views

### Transfinitely iterated limit computability

Call a real $x$ limit computable iff there is a Turing machine $T$ such that, for any $i\in\omega$, there is $t(i)\in\omega$ such that the $i$th entry on the tape is not changed after time $t(i)$ and ...

**8**

votes

**3**answers

976 views

### Kunen's use of Countable Transitive Models

Hi,
I have a doubt concerning Kunen's exposition of forcing in his classical book (arguably $the$ book on forcing). When dealing with Countable Transitive Models to set up the forcing machinery, ...

**5**

votes

**3**answers

768 views

### Explicit isomorphism between distributions and universal enveloping algebra

The universal enveloping algebra of a Lie algebra $\mathfrak{g}$ is isomorphic to the algebra of distributions on the Lie group $G$ with support at the identity. The proof I have of this fact uses the ...

**7**

votes

**1**answer

2k views

### Where to start reading into p-adic non-abelian Hodge theory?

I'm curious about Faltings' "A p-adic Simpson correspondence ". Do you know more detailed, introductory, expositions, surveys, texts of seminars on that?
Edit: Annette Werner's survey "Vector ...

**7**

votes

**2**answers

696 views

### An elementary question about the cut locus

Let $M$ be a Riemannian manifold, $x$ and $y$ are two points in $M$.
Assume that $x$ is not in the cut locus of $y$. Does there exist a neighborhood $U$ of $x$ and a neighborhood $V$ of $y$ such ...

**4**

votes

**1**answer

960 views

### Number field analogue of the Goldbach Conjecture

Is there a generalization of Goldbachs conjecture for prime ideals in number fields?

**14**

votes

**2**answers

462 views

### Outer automorphisms of finite extensions

Let $H$ be a finite index subgroup of a finitely generated group $G$. Assume that $Out(H)$ is finite. Can $Out(G)$ be infinite?

**15**

votes

**1**answer

757 views

### About an exercise in Serre's “Trees”

Following section 1.4, which is entitled "Constructions Using Amalgams" is the innocent-enough looking exercise:
Show that the group defined by the presentation (presumably on the generators $x_1, ...

**10**

votes

**2**answers

803 views

### How Does a Borel Subgroup Know Which Weights Are Dominant

Let $G$ be a simple group (say $SL_n$) and let $B$ be a Borel subgroup (say upper triangular matrices). Then all irreducible representations of $G$ are induced from one-dimensional representations of ...

**1**

vote

**1**answer

231 views

### Turning a measurable function to a bijection

Let $f:(0,1)\rightarrow (0,1)$ be a borel measurable function such that for every $y$ in $(0,1)$ , $f^{-1}(y)$ is a borel set and $\mu(f^{-1}(y))=0$ and also $\mu (f((0,1)))=1$ where $\mu$ is the ...

**5**

votes

**1**answer

818 views

### Inverting a covariance matrix numerically stable

Given an $n\times n$ covariance matrix $C$ where $n$ around $250$, I need to calculate $x\cdot C^{-1}\cdot x^t$ for many vectors $x \in \mathbb{R}^n$ (the problem comes from approximating noise by an ...

**5**

votes

**1**answer

504 views

### Infinite direct products and derived subgroups

Suppose $G_1, G_2, \dots, G_n, \dots$ are groups (I use countable sequences, though the question is also applicable for uncountable collections of groups). Suppose G is the unrestricted external ...

**16**

votes

**1**answer

880 views

### constants in Gamma factors in functional equation for zeta functions.

Usually the Riemann zeta function $\zeta(s)$ gets multiplied by a "gamma factor" to give a function $\xi(s)$ satisfying a functional equation $\xi(s)=\xi(1-s)$. If I changed this gamma factor by a ...

**0**

votes

**0**answers

4 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 ...

**0**

votes

**0**answers

26 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 ...

**0**

votes

**0**answers

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 ...

**-1**

votes

**0**answers

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 ...

**6**

votes

**0**answers

74 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 ...

**2**

votes

**2**answers

100 views

### Is the boundary of an open, regular, bounded, path-connected, and simply connected set a Jordan curve

Trying to find weakest condition on an open bounded set to apply Carathéodory's theorem.
My bounded open sets can be assumed to be pretty well-behaved, but I wonder if the above conditions are ...

**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 ...

**0**

votes

**1**answer

126 views

### Model over DVR for smooth projective curves

Let $C$ be a smooth, projective, geometrically irreducible curve of genus at least $2$ over a complete discrete valued field $F$ of characteristic zero (not necessarily algebraically closed). Let $R$ ...

**1**

vote

**1**answer

90 views

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

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

**2**

votes

**0**answers

49 views

### TTF triples are recollements

The notion of recollement
$$
\mathcal{A}'
\stackrel{\overset{i^*}{\longleftarrow}}{\stackrel{\overset{i_*}{\longrightarrow}}{\underset{i^!}{\longleftarrow}}}
...

**-1**

votes

**0**answers

46 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**

votes

**0**answers

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 ...

**1**

vote

**0**answers

46 views

### Beauville's Integrable System with singular spectral curves

Let us consider Beauville's Integrable System. So, we live on $\mathbb{P}^1$. There is the moduli space of matrices $M_r(d)/\mathrm{PGL}(r)$ with polynomial entries of degree less than or equal $d$. ...

**0**

votes

**0**answers

67 views

### Normal subgroupoid? [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 $H$ ...

**1**

vote

**0**answers

64 views

### All relations among degree n monomials in n variables

In the course of my work, I have run into the problem of finding exactly all relations among degree $n$ monomials in $k[x_1,\dotsc,x_n]$, with specific interest in the case $n=3$ (e.g. $x_1^2 x_2 ...

**1**

vote

**2**answers

123 views

### Subsets of $\mathbb{N}$ whose lower density respects complements

The lower density of $A\subseteq\mathbb{N}$ is defined to be $\lambda(A)=\lim\text{inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}$. We set $${\cal C} = \{A\subseteq \mathbb{N}: ...

**0**

votes

**0**answers

175 views

### Getting back into advanced mathematics [on hold]

I hope that this is the correct forum to post my question on. I'm a 37 year old IT professional working in the Banking Sector who previously studied for a PhD in Computattional Fluid dynamics back in ...