This tag is used if a reference is needed in a paper or textbook on a specific result.

**4**

votes

**2**answers

192 views

### Well-complemented copies of $\ell_p^n$

This must be surely known but I couldn't locate this problem in the literature. It popped out in a priori unrelated approximation problem but if true, would help me greatly.
Let $p\in (1,\infty)$.
...

**5**

votes

**0**answers

171 views

### Version of Stone Weierstrass for functions not vanishing at infinity

I am trying to see what is known about uniform density of function spaces in $C(\mathbb{R}^n)$ or $C_b(\mathbb{R}^n)$ (bounded continuous functions on $\mathbb{R}^n$). By uniform density, I mean ...

**2**

votes

**0**answers

28 views

### Continuity of expected hitting value of diffusion

Let $W$ be a $d$-dimensional Brownian motion and $X$ the strong solution to
$$\mathrm{d} X = \mu(X)\mathrm{d} t + \sigma(X)\mathrm{d} W,$$
starting from some $x$, where $\mu$ and $\sigma$ are ...

**5**

votes

**0**answers

105 views

### Minimal possible cardinality of a $(a_1, …, a_k)$-distributable multiset

Suppose we have a multiset $M$ of positive rational numbers. Sum of $M$ equals $1$. We'll call this multiset $n$-distributable for some $n\in \mathbb{N}$, if there exists a partition $M_1 \sqcup ... ...

**3**

votes

**1**answer

202 views

### Example of a triangulable topological manifold which does not admit a PL structure

I know there are some examples of manifolds which don't admit a PL structure (combinatorial triangulation), and that it has been recently proven that in dimension $n\geq5$ there are manifold which are ...

**2**

votes

**2**answers

337 views

### Reference for higher categorical analogue of algebraic cycle? [closed]

Are there higher categorical analogues of algebraic cycles?
What are some references?
This question arise in an attempt to generalize algebraic cycles towards higher dimensional algebra. Has there ...

**2**

votes

**1**answer

272 views

### Intuitive functional analysis book

I want to know functional analysis book like Terence tao's real analysis and measure theory book, full of intuition. I am aware of linear algebra, real analysis, measure theory, Probability theory.

**0**

votes

**1**answer

74 views

### Forbidden Tripartite Graphs

I was looking at extremal graph theory. I have understood the proofs of upper bounds for the Zarankiewicz problem which basically states: What can you say about the edges of a graph with $n$ vertices ...

**1**

vote

**2**answers

107 views

### Is there a name for a partial order in which there is a countable chain which “dominates” the whole space?

Is there a name for a partial order $\preceq$ on a set $X$ with the following property: "there exists a countable set $S \subset X$ such that for all $x \in X$ there exists $y \in S$ with $x \preceq ...

**1**

vote

**0**answers

14 views

### Properties of Optimal k-Tour Spanners

Let the edge set of a Optimal k-Tour Spanner of a graph $G$ be equal to the edges of $G$ that lie on at least one optimal tour through exactly $3<k<n$ distinct vertices of $G$.
I would like ...

**2**

votes

**0**answers

94 views

### What sorts of weights for perverse sheaves were or can be computed?

I am studying certain weights for (triangulated categories of relative) motives. Those are interesting; yet one can hardly say that they are very much explicit or effectively computable. So, I would ...

**2**

votes

**0**answers

117 views

### Existence of universal extension between two modules?

I need a reference for the following fact:
Let $R$ be (for simplicity) an algebra over the field $k$, let $A, B$ be $R$-modules. Let $E = Ext^1_R(B, A)$. There is a natural isomorphism between ...

**5**

votes

**1**answer

468 views

### reference for “Topological algebra of Grothendieck”

I would like to have some references for Grothendieck's theory of "Topological algebra": a synthesis of homotopical and homological algebra,
with special emphasis on topoi.

**6**

votes

**1**answer

230 views

### Conway's subprime Fibonacci sequences

I want to be certain I have the latest information on
Conway's subprime Fibonacci sequences,
arXiv-posted a year ago; I am referencing the status in
a review.
To wit, starting with $(0,1)$:1
$$
0, 1, ...

**3**

votes

**3**answers

213 views

### Convergence of expansion for fractional iteration

I was reading about tetration here. The site mentions that the convergence of the expansion for fractional iteration is unproven. However, I was interested in reading more literature about convergence ...

**1**

vote

**0**answers

100 views

### Classification of Automorphism set of a Regular graph

Let $A$ be the adjacency matrix of an $r$-regular graph $G$ with $n$ vertices (Not complete or cycle graph) . Also, let $Aut(G)$ be the set of all its automorphisms (i.e. set of permutation ...

**1**

vote

**0**answers

62 views

### Point Spectrum of a Second Order System of Differential Equations

Consider the following operator acting on $H^1(\mathbb{R})$
$$
\mathcal{L} \left(\begin{array}{c} \phi \\ \psi \end{array}\right) = -\left(\begin{array}{c} \phi \\ \psi \end{array}\right)'' + V(x) ...

**17**

votes

**0**answers

491 views

### Reference request: Parallel processor theorem of William Thurston

Sometime in the 1980's or 1990's, Bill Thurston proved a theorem regarding the existence of a universal parallel processing machine, using a certain class for such machines having finite deterministic ...

**0**

votes

**1**answer

21 views

### Supremum of centered jointly generalized chi-square random variables

Let $\zeta_n$ be a sequence of centered jointly generalized chi-square random variables, i.e. $\zeta_n = \sum_{k=1}^{m_n} a_{k,n}(\xi_{k,n}^2 - E[\xi_{k,n}^2])$, and $\xi_{k,n}$ are centered jointly ...

**0**

votes

**0**answers

62 views

### Suitable reference on smooth manifolds for qualifying exam study?

Is there a single suitable reference to study for the smooth manifold (geometry) half of a typical Topology/Geometry PhD preliminary exam at an average AMS group I school? (Note: I know details about ...

**42**

votes

**1**answer

2k views

### What were the main ideas and gaps in Yoichi Miyaoka's attempted proof (1988) of Fermat's Last Theorem?

Out of sheer curiosity I have been reading Stewert and Tall's "Algebraic Number Theory and Fermat's Last Theorem" (2001). As it contains various bits of history, I found out to my own shame that I was ...

**2**

votes

**0**answers

46 views

### Points are removable for weakly differentiable functions

If $\Omega \subseteq \mathbb{R}^N$ is an open set and $N \ge 2$, then any point $a \in \Omega$ is removable for weakly differentiable maps: for each function $u \in W^{1, 1} (\Omega \setminus \{a\})$, ...

**0**

votes

**0**answers

96 views

### reference for groupoid cohomology

In nLab (groupoid cohomology) says:
"Under the homotopy hypothesis theorem, plain (non-internal) groupoid cohomology is the same as the cohomology of homotopy 1-types."
Are there references for ...

**1**

vote

**0**answers

155 views

### One parameter family of elliptic equations

Consider the following 2nd order nonlinear elliptic equation on $\mathbb{R}^n$: $$-\Delta \varphi + \sum_i a_i(x, \varepsilon)\partial_i \varphi + \varphi = N(\varphi),$$
where $N$ is a smooth ...

**-2**

votes

**0**answers

62 views

### Are there papers or books that explain why Bernhard Riemann believed that his hypothesis is true?

This question accross this question from SE such that no answer ,I think every
mathematician should know Wy B.Riemann believed that his hypothesis is true .I'm affraid
to ask this question in M.O ...

**5**

votes

**1**answer

315 views

### soft: Reference/ Suggested Read: Homological Algebraic techniques in PDEs

I was reading this article on wikipiedia and was interested by the apparent link between Homological Algebra and PDEs. What is an accessible reference which showcases the link between these topics? ...

**1**

vote

**1**answer

189 views

### reference on Dirichlet theorem on primes in arithmetic progression

I appreciate if you could help me to find a reference (and a proof).
Combining Dirichlet theorem on primes in arithmetic progression with Chebotarev densitiy theorem, we know that given two positive ...

**4**

votes

**2**answers

125 views

### Reference for (co)limit-preserving functor $X\mapsto R^X$

Fix a commutative ring $R$. There's a contravariant functor from finite sets to finite $R$-algebras sending $X$ to $R^X$. Viewed as a covariant functor $\text{set}^{op}\to R\text{-alg}$, this functor ...

**4**

votes

**1**answer

131 views

### Are automorphism groups of polarized varieties of finite type

It is "well-known" that the stack of polarized varieties is an algebraic stack with quasi-compact and separated diagonal.
In particular, if $(X,L)$ and $(Y,M)$ are polarized schemes over a scheme ...

**0**

votes

**1**answer

283 views

### Particular case of Beal's Conjecture

Is it known that there exist no coprime positive integers $A$, $B$ and $C$ such that $A^3+B^4=C^3$? This is a particular case of Beal's Conjecture.

**2**

votes

**2**answers

223 views

### Hausdorff Dimensions of Limit set of subgroups of SL(2,Z)

In a recent paper by Bourgain, Sarnak, Gamburd [1] talks about subgroups of $SL(2,\mathbb{Z})$.
Let $\Lambda$ be a finitely generated non-elementary subgroup of $SL(2,\mathbb{Z})$ with Hausdorff ...

**1**

vote

**0**answers

64 views

### Examples for differential operators first order

Currently I am dealing with the problems which involve the general first order differential operator, i.e., for some open domain $\Omega\subset\mathbb{R}^n$ with certain regular bondary and a function ...

**4**

votes

**2**answers

290 views

### What does it mean that the Hessian is proportional to the metric?

Let $(M,g)$ be a smooth manifold equipped with a metric tensor $g$, and $f\in C^\infty(M)$ a regular function (i.e., with nowhere vanishing differential).
Denote by $\mathrm{Hess}_g(f):=\nabla df$ ...

**5**

votes

**0**answers

73 views

### Fock Space Proof of $(g(x)\phi^4)_2$ Mass Gap?

Is there a proof that does not depend on Euclidean methods? Is this a proof? :
$V(g)$ can be written as $P+R$ where $P$ is non-negative and $R$ is $N$-bounded (and hence $(H_0+\lambda P)$-bounded). ...

**6**

votes

**4**answers

427 views

### SO$(4)$ (& SO$(n)$) characterization?

I believe it is the case that
any finite subgroup of SO$(3)$
(the $3 \times 3$ orthogonal matrices of determinant $1$)
is either a cyclic group $C_n$,
or a dihedral group $D_n$, or one of the groups ...

**5**

votes

**1**answer

84 views

### Matroids similar to the cycle matroid

Let $G=(V,E)$ be a graph (loops and multiple edges are permitted). Three following systems of dependent sets in $E$ define matroids:
1) Set $A\subset E$ is dependent if $A$ contains cycle. This is a ...

**0**

votes

**1**answer

97 views

### quasi-projective and separated as topological properties

Let $X$ be a non-reduced noetherian scheme over $\mathbb{Z}$ or $\mathbb{C}$. Assume that $X^{red}$ is quasi-projective and separated, does the same hold for $X$ ?
(By the way, projective implies a ...

**2**

votes

**1**answer

118 views

### Counting elements with certain word length in abelian groups

Given a (finite) abelian group $G = \langle S \mid R \rangle$, has the problem of counting the number of elements which can be expressed as a word (in $S$) of length $\leq k$ been studied? If so, ...

**1**

vote

**0**answers

55 views

### concentration inequalities for quadratic forms of correlated random vectors

Let $\mathbf{n}$ is a Gaussian random vector with mean $\mathbf{0}$ and co-variance matrix $\mathbf{H}$. Let $\mathbf{r} = Sign(\mathbf{n})$, where $Sign(n_i) = 1$ if $n_i>0$ and $Sign(n_i) = -1$ ...

**1**

vote

**0**answers

44 views

### Reference request - Compact embedding of intermediate space

Given two Banach spaces $X_0$ and $X_1$ with norms $\|\cdot\|_0$ and $\|\cdot\|_1$, respectively, such that $X_0\subset X_1$ and $X_0\hookrightarrow X_1$, i.e., $X_0$ is continuous embedded in $X_1$.
...

**0**

votes

**1**answer

237 views

### Supplementary notes to Mumford's The Red Book of Varieties and Schemes

I am a graduate student with good mathematical maturity (I took advanced courses like category theory, commutative algebra...). I want to study algebraic geometry from Mumford's red book. I find it ...

**0**

votes

**0**answers

59 views

### The spring Markov chain on $\mathbb{N}$

I'm trying to understand and learn more about "almost surely bounded" Markov chains on countable state spaces. I'm looking for references where I can learn how to work with more complicated examples ...

**6**

votes

**0**answers

147 views

### Bundles over Grassmanian with given top Chern class

So, I have been working on Chern classes for my master thesis and apparently (My proofs could be wrong and a few things are still vague) I was able to give a construction method and exhibit, via ...

**4**

votes

**0**answers

105 views

### Reference for the Banach Manifold structure of $C^k(M,N)$

I'm completely new to the subject of banach manifolds and I'm looking for a reference of the following:
Let $M$,$N$ be smooth (=$C^\infty$) finite-dimensional compact manifolds. Consider the set ...

**0**

votes

**0**answers

88 views

### Universal property of complete linear systems

Let $X$ be a projective scheme over a field $k$ and $S$ a $k$-scheme. Fix a closed immersion $i:X \to \mathbb{P}^n$ for some $n$ and denote by $\mathcal{O}_X(1):=i^*\mathcal{O}_{\mathbb{P}^n}(1)$. Let ...

**6**

votes

**1**answer

148 views

### Zeta-Determinant for shifted Laplacians on the circle

Consider on the circle $S^1$ the operator
$$L := - \frac{\partial^2}{\partial \theta^2} + c$$
for some constant $c \in \mathbb{R}$.
What is its $\zeta$-regularized determinant?
This should be ...

**1**

vote

**1**answer

48 views

### Distribution of Poles of solutions to the first Painleve equation

In the introduction of this paper the distribution of the poles of solutions to the first Painlevé equation is discussed.
In particular it is said that the poles form a deformed lattice that ...

**-1**

votes

**1**answer

222 views

### Class forcings and elementary embeddings

In the Hamkins-Kirmayer-Perlmutter paper "Generalizations Of The Kunen Inconsistency", they prove the following theorem:
"Theorem 7: In any set forcing extension $V[G]$, there is no nontrivial ...

**1**

vote

**1**answer

76 views

### Essential surfaces in the Exterior of Montesinos knots

Hatcher and Oertel computed the boundary slopes of essential surfaces of Montesinos knots in this paper. But they do not consider surfaces that do not intersect the boundary of the exterior. An ...

**3**

votes

**0**answers

118 views

### What does “control of a deformation problem” mean?

Is the expression "control of a deformation problem' ever defined? There are of course many examples relating a dg-Lie or L-infty algebra to a deformation problem, and the phrase is evocative. Is it ...