# All Questions

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### 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. ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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: ...
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### 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), ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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. ...
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### 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 ...
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### 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, ...
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### 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 $\sum_na_n^2=1$. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $\omega$-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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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