# All Questions

**11**

votes

**1**answer

2k views

### Kolmogorov Complexity and Proof Techniques

I'm interested in examples of theorems that employ the proof techniques that are utilized in the proof of the undecidability of Kolmogorov Complexity.
Definition:(Sipser) Let x be a binary string. ...

**3**

votes

**1**answer

820 views

### Dense sets in the space of continuous functions

Let $X$ be a compact metric space, and
let $C(X)$ be the Banach space of continuous real-valued function on $X$, with
the maximum norm.
Suppose $S\subset C(X)$ is a set of functions with the ...

**3**

votes

**2**answers

548 views

### Reference on an equivariant resolution of singularities

Let $X$ be an algebraic variety over $\mathbb{C}$ (or a normal complex space).
I found the word "equivariant resolution" in several papers on singularity theory or deformation theory. I think that it ...

**10**

votes

**2**answers

2k views

### Stokes' theorem etc., for non-Hausdorff manifolds

This question is prompted by another one.
I want to motivate the definition of a scheme for people who know about manifolds(smooth, or complex analytic). So I define a manifold in the following way.
...

**2**

votes

**1**answer

458 views

### Multinomial transformation for matrices

Suppose we have a vector of probabilities $\mathbf{p}=(p_1,...,p_n)$, where $p_i>0$ for $i=1,...n$ and $\sum p_i=1$. Define new vector $\mathbf{r}=(r_1,...,r_{n-1})$ in a following way:
...

**2**

votes

**0**answers

145 views

### Topological invariance for formally étale morphisms

If $f:X_0\rightarrow X$ is a closed immersion of locally noetherian schemes such that the topological spaces of $X_0$ and $X$ are identical (or, more generally, if $f$ is a universal homeomorphism), ...

**5**

votes

**0**answers

393 views

### morphism which is open but not universally open

In someone's note, I have seen such an example, but I can't show that it is not universally open. Here is the example:
Let $k$ be a field and $A = k[T]_{(T)}$, the discrete valuation ring obtained ...

**0**

votes

**0**answers

24 views

### The non-singular controls always in neighbourhood of singular controls?

Consider the case of a right invariant affine distribution: $D_{U} = \{ aU + \lambda bU | a,b \in \mathfrak{su}(n), \lambda \in \mathbb{R} \}$ on $SU(4)$.
Consider the equations:
...

**0**

votes

**0**answers

28 views

### How many triangulations of a polytope contain a given simplex [on hold]

Let $P$ be a full-dimensional convex polytope in $\mathbb R^n$, and let $v_1, \dots, v_m$ be its vertices.
A triangulation of $P$ is a set $\mathcal T$ of simplices, which (i) cover $P$, (ii) are ...

**0**

votes

**1**answer

1k views

### What is the orthonormal basis for the Bergman space on the disk?

[EDIT by YC: the original question's title asked about a basis for the Hardy space on the disk. It is clear from the actual question that what was meant was the Bergman space.]
In arXiv:0310.5297, ...

**4**

votes

**2**answers

807 views

### Why is the mapping class group of hyperbolic manifolds finite?

Hi! I'm trying to understand why a hyperbolic n-manifold has finite mapping class group if $n \geq 3 $. In books I'm reading it's said it's a consequence of Mostow's rigidity theorem:
"If M and N are ...

**2**

votes

**1**answer

466 views

### What are in units of an affinoid algebra?

Suppose that $K$ is a complete local field and $A$ is an affinoid $K$-algebra. Is there a known way to produce an explicit description of the units of $A$?
Here is what I already know: write ...

**3**

votes

**0**answers

79 views

### Application and relevance of Sobolev gradients

The Sobolev gradient concept has been developed in the 1970s, with a first publication in 1985, and an introduction can be found at: Ranka
I would like to learn how strong the impact of Sobolev ...

**5**

votes

**0**answers

129 views

### Non trivial rank 2 holomorphic vector bundles in complex dimensions greater than or equal 2

Does every compact complex manifold of complex dimension greater than or equal two
possess a nontrivial rank 2 holomorphic vector bundle?

**9**

votes

**3**answers

1k views

### Why are ring actions much harder to find than group actions?

I admit freely that the following question is a bit of a fishing expedition inspired by this lovely "definition" of a module as found on Wikipedia:
A module is a ring action on an abelian group.
...

**-2**

votes

**1**answer

334 views

### How many of Ramanujan's discoveries have had a practical application? [closed]

I was reading about the Indian mathematician Srinivasa Ramanujan who, before dying at the age of 32, independently compiled nearly 3900 results (this is from Wikipedia). So based on this he seems to ...

**5**

votes

**3**answers

280 views

### Can groups of twice-odd order have quaternionic representations?

Let $G$ be a finite group and $\phi\colon G\to \mathrm{GL}_d(\mathbb C)$ be an irreducible representation, with character $\chi$. Recall that
$\phi$ is complex type if $\chi$ is not real-valued,
...

**14**

votes

**2**answers

1k views

### An $L^0$ Khintchine inequality

Suppose that $\epsilon_1,\epsilon_2,\ldots$ are IID random variables with the Bernoulli distribution $\mathbb{P}(\epsilon_n=\pm1)=1/2$, and $a_1,a_2,\ldots$ is a real sequence with $\sum_na_n^2=1$. ...

**4**

votes

**0**answers

124 views

### Is there an example of a holomorphic vector bundle whose Atiyah class vanishes and does not admit a flat connection?

Let $E\to X$ be a holomorphic vector bundle over a Kähler manifold. The vanishing of the Atiyah class, $At(E)=0$, is equivalent to the existence of a holomorphic connection on $E$.
Moreover, it is ...

**7**

votes

**4**answers

940 views

### Prime numbers $p$ not of the form $ab + bc + ac$ $(0 < a < b < c )$ (and related questions)

If we ask which natural numbers n are not expressible as $n = ab + bc + ca$ ($0 < a < b < c$) then this is a well known open problem. Numbers not expressible in such form are called ...

**53**

votes

**2**answers

4k views

### Is every sigma-algebra the Borel algebra of a topology?

This question arises from the excellent question posed on math.SE
by Salvo Tringali, namely, Correspondence
between Borel algebras and topology.
Since the question was not answered there after some ...

**10**

votes

**3**answers

1k views

### $\omega$-topos theory?

I've been reading through Lurie's book on higher topos theory, where he develops the theory of $(\infty,1)$-toposes, which leads me to the following question: Is there any sort of higher topos theory ...

**216**

votes

**68**answers

102k views

### Proofs without words

Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results?
(One could ask if this is of interest to mathematicians, and I ...

**9**

votes

**4**answers

1k views

### Induction of tensor product vs. tensor product of inductions

This is a pure curiosity question and may turn out completely devoid of substance.
Let $G$ be a finite group and $H$ a subgroup, and let $V$ and $W$ be two representations of $H$ (representations are ...

**2**

votes

**1**answer

622 views

### Self intersection of blown up points and the lines which they lie on

I'm currently trying to understand the process of blowing-up, and a few things strike me as a little difficult to get an intuitive understanding of what's happening.
The current problem is on self ...

**-3**

votes

**0**answers

47 views

### The connecting morphism of a $\delta$-functor is natural [closed]

I am trying to prove the following:
An Introduction to Homological Algebra - C. A. Weibel (1994)
Exercise 2.1.1) Let $\mathcal{S}$ be the Category of short exact sequences
$$0\to A\to B\to C\to ...

**49**

votes

**51**answers

17k views

### Colloquial catchy statements encoding serious mathematics

As the title says, please share colloquial statements that encode (in a non-rigorous way, of course) some nontrivial mathematical fact (or heuristic). Instead of giving examples here I added them as ...

**8**

votes

**1**answer

716 views

### Rational singularities for fibered surfaces

This question consists of two parts. I will try to be as short and clear as possible.
Let $S$ be a Dedekind scheme of characteristic zero. The main examples are $\mathbf{P}^1_k$, with $k$ a field of ...

**5**

votes

**1**answer

430 views

### Generators of associated graded algebra

Suppose that $A = \bigcup_{n=0}^{\infty} A_n$ is a filtered algebra over a field $k$. The associated graded algebra is $\mathrm{gr} A = \bigoplus_{n=0}^{\infty} A_n/A_{n-1}$, where we define $A_{-1} ...

**32**

votes

**7**answers

4k views

### Why the Killing form?

I'm teaching a short summer course on algebraic groups and it's time to talk about the Killing form on the Lie algebra. The students are all undergrads of varying levels of inexperience, and I try to ...

**4**

votes

**3**answers

1k views

### Online estimation of covariance matrix

I am trying to dynamically estimate the (low-dimensional) covariance matrix ${\mathbb E}[{\bf x}_t{\bf x}_t^\top]$ of a stream of data points ${\bf x}_t\in{\mathbb R}^N$ online, without any memory. ...

**37**

votes

**3**answers

2k views

### What are the higher homotopy groups of Spec Z ?

The homotopy groups of the étale topos of a scheme were defined by Artin and Mazur. Are these known for Spec Z? Certainly π1 is trivial because Spec Z has no unramified étale ...

**13**

votes

**4**answers

2k views

### Other Homology Theories still Count Holes?

This may be a naive question, but since first learning homology I considered it as a tool which counts appropriate holes in your space (on top of orientation and torsion phenomena). Then I was ...

**8**

votes

**1**answer

2k views

### Ext groups and Serre duality

Hi,
I have a question related to Serre Duality:
if I have a smooth projective variety $X$ with dualizing sheaf $\omega$ and two coherent sheaves $F$ and $G$ on $X$, then how can I get a canonical map
...

**15**

votes

**3**answers

763 views

### Periodicity theorems in (generalized) cohomology theories

It is well-known that topological K-theory is blessed with the Bott periodicity theorem, which specifies an isomorphism between $K^2(X)$ and $K^0(X)$ (where $K^n$ is defined from $K^0$ by taking ...

**7**

votes

**3**answers

955 views

### A simple example where elliptic boundary regularity fails due to a kink in the domain

I'm seeking a simple example of where elliptic (preferably linear) boundary regularity fails due to a simple kink in the domain.
So far my gueses were to look at $-\Delta u = f$ on $[0,2\pi] \times ...

**4**

votes

**1**answer

94 views

### Closest point to a dual lattice point (in particular for root lattices!)

Given a lattice $\Lambda\subset \mathbb{R}^n$ and a point $p\in\mathbb{R}^n$ outside the lattice, then I known it is a hard question to determine the set $S\subset \Lambda$ of all lattice points with ...

**35**

votes

**4**answers

3k views

### Can the symmetric groups on sets of different cardinalities be isomorphic?

For any set X, let SX be the symmetric group on
X, the group of permutations of X.
My question is: Can there be two nonempty sets X and Y with
different cardinalities, but for which SX is
isomorphic ...

**1**

vote

**0**answers

26 views

### Fully residually free groups and completion

Let $G$ be a fully residually free group with a finitely generated profinite completion. Is $G$ necessarily finitely generated?

**6**

votes

**2**answers

392 views

### Information from Moment Polytopes

Let $T$ be a compact real torus, and $X$ a Hamiltonian $T$-manifold (which you may take to be a smooth complex projective variety) with moment map $\mu:X\rightarrow\frak{t}^*$. If ...

**3**

votes

**0**answers

97 views

### The distribution of the elements of an eigenvector of random matrices

Suppose a random matrix $A$ with its elements following Gaussian distribution with non-zero mean. We know that the eigenvalues of $A$ have two patches: one is at the real axis that is far away from ...

**4**

votes

**1**answer

158 views

### Canonical models of Teichmüller curves

It is well-known that Shimura varieties can be defined over number fields and that moreover they possess canonical model over number fields. On the other hand, Teichmüller curves can also be defined ...

**3**

votes

**1**answer

88 views

### Effects of many degree-2 variable nodes in the Tanner graph during the decoding of LDPC codes

Suppose that we have a LDPC code $C$ with a $(n -k )\times n $ parity check matrix $H$, and there exist approximately $ \sqrt n$ numbers of degree-2 columns. It means that there are approximately ...

**111**

votes

**37**answers

98k views

### Too old for advanced mathematics? [closed]

Kind of an odd question, perhaps, so I apologize in advance if it is inappropriate for this forum. I've never taken a mathematics course since high school, and didn't complete college. However, ...

**5**

votes

**2**answers

165 views

### What is this construction using iterated face maps of semisimplicial sets?

Let $X$ be a semisimplicial set (face maps but no degeneracy maps). Fix a positive integer $k$. Let $Y_n$ be $X_{(n+1)k}$ and then define $\partial^Y_i:Y_n\to Y_{n-1}$ by
$$\partial^Y_i = ...

**7**

votes

**1**answer

478 views

### Are there effective small intervals in which primes are dense?

As mentioned in Terry Tao's comment to this question, it is constructively known
that there are primes between sufficiently large cubes. $\:$ According to wikipedia,
"there exists a constant $\: ...

**1**

vote

**0**answers

46 views

### Has every Lusin vector space a stronger Polish vector space topology?

Let $X$ be a topological vector space or even a locally convex space such that its (vector space) topology is Lusin, i.e. there is some stronger Polish topology. Does there also exist a stronger ...

**2**

votes

**3**answers

389 views

### The number of submodules of $\mathbb{Z}_q^n$

Observe $\mathbb{Z}_q^n = \mathbb{Z}_q \times \cdots \times\mathbb{Z}_q$ as a module over $\mathbb{Z}_q\equiv\mathbb{Z}/q\mathbb{Z}$, for general $q$.
I am interested in the following questions:
How ...

**4**

votes

**2**answers

370 views

### What does it mean for a differential equation “to be integrable”? [duplicate]

What does it mean for a differential equation "to be integrable"?
Are "integrable" and "solvable" synonyms?
The first thing that comes to my mind is to say: it's integrable if we can find the ...

**11**

votes

**1**answer

223 views

### Practical advantages of univalent foundations

I'm interested in the machine translation of mathematics from informal to formal (a la Ganesalingam in The Language of Mathematics). As a first step, I am designing a computer language for expressing ...