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

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3
votes
0answers
66 views

Variational Principle for a System of Differential Equations

I am studying a differential operator of the form $$ L\left(\begin{array}{c} u \\ v \end{array}\right) = -\Delta \left(\begin{array}{c} u \\ v \end{array}\right) + V(x)\left(\begin{array}{c} u \\ v ...
6
votes
1answer
99 views

An analogue of cabling for configuration spaces

There is a well-known operation known as cabling for knots, and also for braid groups, where it is a homomorphism $$\beta_k \times \beta_\ell \longrightarrow \beta_{k\ell}$$ given by thickening up the ...
0
votes
0answers
112 views

$L^{p}(\mathbb R)\subset L^{1}(\mathbb R) \ast L^{p}(\mathbb R), (1< p< \infty)$?

Let $\mathbb T$ be a circle group. In 1939, Salem, has shown that, every member of $L^{1}(\mathbb T)$ can written as a product(convolution) some other two members of $L^{1}(\mathbb T),$ that is, ...
2
votes
2answers
200 views

Lifting to char 0, references and questions

Suppose that I have a surface $S$, smooth proper over an algebraically closed (perfect?)field $k$ that lifts algebraically to some $S_W$ defined over a field of char 0. I am interested in properties ...
10
votes
1answer
375 views

Harrington's unpublished note “The constructible reals can be anything”

Around 1974, Leo Harringto wrote an unpublished note entitled "The constructible reals can be anything", in which he proved that it is consistent that being $\Delta^1_n$ is the same as being ...
0
votes
0answers
64 views

A question related to the Grauert semi-continuity theorem

Let $f\colon X\to Y$ be a proper holomorphic map of complex analytic manifolds. Assume $f$ to be submersive for simplicity, but probably it is not important. Let $\mathcal{F}$ be a coherent sheaf on ...
1
vote
0answers
108 views

Does $G\times H$ have a dual when $G$ and $H$ have?

Let $G$ and $H$ be two groups with duals. Does $G\times H$ have a dual? A group $G$ has a dual iff the lattice of its subgroups is order-isomorphic to the dual of the subgroup lattice of some other ...
21
votes
2answers
999 views

Do's and don'ts of writing survey papers

I am not sure if this is the appropriate forum to ask as it is not directly related to a research level (math) problem, but I figured it was worth a try. I recently attended a conference and felt that ...
2
votes
0answers
117 views

Kloosterman-like sum with inverse to different moduli

In some recent work, the following strange-looking exponential sum arose: $$ \sum_k \sum_r \sum_s e\bigg( \frac{r \bar s^{(r)} \bar k^{(r^2+s^2)}}{r^2+s^2} \bigg). $$ Here $e(x) = e^{2\pi i x}$ as ...
1
vote
1answer
112 views

Ratner theorem and dense geodesic planes in hyperbolic manifolds

Suppose we have a closed hyperbolic $3$-manifold $M$. For any $x\in M$ and plane $\pi$ in $T_xM$ we consider $P$ the geodesic plane exp$(\pi)$ originating from $\pi$. For any $p\in \pi$ we consider ...
2
votes
0answers
89 views

Bloch Kato Exponential as formal lie group exponential

Let $K$ be a $p$-adic field and $V$ a $p$-adic representation. In their paper on tamagawa numbers of motives, Bloch and Kato define an exponential map as the connecting homomorphism $$DR(V) ...
14
votes
0answers
586 views

New(?) reciprocity law

Consider three functions $f, g$ and $h$ on a smooth curve $X$ over $\mathbb{C}$. I have found the following equality: $$\sum (res(f\frac{dg}{g})\frac{dh}{h}-res(f\frac{dh}{h})\frac{dg}{g})=0.$$ Here ...
2
votes
0answers
58 views

Asymptotic formula for restricted partition function

Let $p(n)$ be the partition function. Hardy and Ramanujan - and Uspensky, independently proved the asymptotic formula $$(1) \quad p(n) \sim \frac1{4\sqrt{3}} \frac{e^{c_0\sqrt{n}}}{n} \text{ as } n ...
1
vote
1answer
102 views

Global version of the Picard-Lindelöf theorem [closed]

Let $I\subseteq \mathbb{R}^{n}$ be an arbitrary (not necessarily closed) intervall and $f:I\times \mathbb{R}^{n}\to \mathbb{R}^{n}$ a continuous function such that in $I\times \mathbb{R}^{n}$ ...
3
votes
1answer
84 views

Do Heegaard Floer homology detect fibred knot in general oriented 3-manifold?

Do Heegaard Floer homology detect fibred knot in general oriented 3-manifolds other than $S^3$? If the answer is yes could you give a reference.
4
votes
1answer
572 views

solvable word problem without algorithm

Let $G$ be a finitely generated group. I wonder if there are examples where: 1) The word problem is known to be solvable in $G$ but there is no algorithm known. 2) The word problem is known to be ...
6
votes
3answers
268 views

Union of conjugates of a subgroup

Let $G$ be a finite group, $H \leq G$ a proper subgroup. It is well known that the union of the conjugates of $H$ does not cover $G$. I would like to know of more precise results (even in special ...
2
votes
1answer
166 views

A parametrization of subsets

Suppose $I\subseteq\{1,\dots,n\}$, and let $\{1,\dots,n\}\setminus I=\{j_1,\dots,j_m\}$ be an enumeration of the complement of $I$ with $j_r<j_{r+1}$ for each $r\in\{1,\dots,m-1\}$. To $I$ I can ...
0
votes
0answers
43 views

The set of (property) elements of a locally compact group is closed

For which properties $(P)$ is the following statement known to be true? In any locally compact group $G$, the elements of $G$ that satisfy $(P)$ form a closed subset of $G$. In other words, the ...
1
vote
0answers
31 views

Generalizing Concepts of Planar Euclidean Geometry to Symmetric TSP-Instances

To me it seems possible, to successfully look at symmetric TSP instances from a geometry-point of view. Examples are: the diagonals of the convex hull of a set of points in the euclidean plane; ...
2
votes
1answer
204 views

Proof of the Friedlander–Iwaniec theorem

Does anybody know where I could find the proof of the Friedlander–Iwaniec theorem. The link that I find when I search for it is http://www.pnas.org/content/94/4/1054.full.pdf+html, but this seems more ...
1
vote
0answers
93 views

Retracts of 2-categories

Let $\mathbf{C}$ be a strict $2$-category and let $M = \{(x_\alpha,y_\alpha)\}$ be a collection of object-pairs so that each hom-category $\mathbf{C}(x_\alpha,y_\alpha)$ has an initial element ...
0
votes
0answers
129 views

Reference: A nowhere vanishing section of a vector bundle is locally split

Well-known fact: If $(A, \mathfrak{m})$ is a local Noetherian ring, $E$ is a finitely generated free $A$-module, and $e\in E$ is an element not contained in $\mathfrak{m}E$, then $E/eA$ is also a ...
1
vote
0answers
60 views

Structure of the zero set of analytic maps (Lojasiewicz’s Structure Theorem for Varieties)

The Lojasiewicz’s Structure Theorem for Varieties states that for $\Phi:\mathbb{R}^n\rightarrow \mathbb{R}$ real analytic with $\Phi(0)=0$ the set $\Phi^{-1}(0)$ is locally a union of subvarieties of ...
2
votes
0answers
77 views

Is there a coordinate free proof of the Morrey--Kohn--Hormander identity?

The Morrey--Kohn--Hormander identity is the key to proving vanishing/existence results on bounded pseudoconvex domains in $\mathbb{C}^n$, or more generally, Stein domains. See, for instance, the ...
1
vote
3answers
271 views

A special case of the uniformization theorem

I am interested in a proof of the following fact : Suppose that $X$ is a Riemann surface homeomorphic to the Riemann sphere. Then $X$ is conformally equivalent to the Riemann sphere. Of course, this ...
2
votes
0answers
108 views

Cohomology of Bott-Samelson varieties?

How is the cohomology of Bott-Samelson varieties (desingularizations of Schubert Varieties ) usually calculated? Let's fix the Lie group to be $GL_n(\mathbb{C})$ or $SL_n(\mathbb{C})$ here. Is there ...
2
votes
0answers
81 views

Pure braid groups of the complement of a lattice in the complex plane: generators and relations

Where can I find a presentation (by `natural' generators and relations between them) of the pure braid groups $PB_n(S)$ (for $n>0$) of $S=\mathbb C\setminus (\mathbb Z\oplus i \mathbb Z)$? Thanks ...
1
vote
1answer
210 views

A generalization of the Grauert direct image theorem

EDIT: Let $f\colon X\to Y$ be proper holomorphic submersive map of complex analytic manifolds. Let $\mathcal{F}$ be the sheaf of holomorphic sections of a holomorphic vector bundle over $X$. Assume ...
9
votes
2answers
450 views

Bound on gcd of two integers

Well this is a problem I was fiddling with. I came up with it but it probably is not original. Suppose $a\in \mathbb{N}$ is not a perfect square. Then show that : ...
9
votes
2answers
555 views

Functions that Calculate their $L_p$ Norm

are there any examples of functions $f:x\in\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ and intervals $(a,b), 0\le a \lt b \le \infty$ , for which $$\Big(\int_a^b{|f(x)|^p dx}\Big)^\frac{1}{p} = f(p)$$ ...
7
votes
2answers
232 views

Arveson's extension for normal completely positive maps

My question deals with a version of Arveson's extension theorem (for the standard version, see, e.g., Paulsen's book Completely Bounded Maps and Operator Algebras). Let $\mathcal A$ be a von Neumann ...
2
votes
0answers
75 views

Primality Criterion for Specific Class of Numbers of the Form kb^n-1

Let $N=k\cdot b^n-1$ where $b$ is an even integer , $3\nmid b$ , $3\nmid N$ , $k \equiv 1,5 \pmod{6}$ , $k< b^n $ and $n>2$ . Let $S_i=P_b(S_{i-1})$ with $S_0=P_{k\cdot b/2}(P_{b/2}(4))$ , ...
16
votes
1answer
352 views

Covering a set with geometric progressions

Consider the set $S_n=\{1,2,\cdots ,n\}$. What is the minimum number of distinct geometric progressions that cover $S_n$? Let us call this number $a_n$. I was wondering about this number after doing a ...
6
votes
1answer
151 views

Abel's five terms relation from Yang-Baxter equation?

Can the famous Abel's five terms relation satisfied by the dilogarithm be derived from (a particular case of) the theory of Yang-Baxter equations? If yes, how? Thanks for any help.
2
votes
0answers
85 views

Primality Criterion for Specific Classes of Generalized Fermat Numbers

Let $F_n(b)= b^{2^n}+1 $ where $b$ is an even integer , $ 3\nmid b , 5\nmid b $ and $n\ge2$ Let $S_i=P_b(S_{i-1})$ with $S_0=P_{b/2}(P_{b/2}(8))$ where ...
9
votes
4answers
758 views

Ordinary Generating Function for Bell Numbers

In the OEIS entry for Bell numbers, there appears a generating function $$\sum_{k=0}^\infty B_k t^k = \sum_{r=0}^\infty \prod_{i=1}^r \frac{t}{1-it}$$ However, I could not locate any proof of ...
16
votes
2answers
534 views

Central Limit Theorem(s) for irrational rotation

Let $\alpha$ be irrational and $T: S^1 \rightarrow S^1$ be the rotation by $\alpha$. I'm interested in what type of Central Limit Theorem (if any) can hold for sums $Y_n = ...
5
votes
1answer
248 views

A geometric characterization of smooth points of a complex algebraic variety

Let $X^m\subset \mathbb{C}^n$ be an irreducible $m$-dimensional complex algebraic subvariety. Let $\mathbb{C}^n$ be equipped with the standard Hermitian metric. Fix an arbitrary point $p\in X$. Let ...
5
votes
1answer
215 views

Literature about metapolynomials

We say that a function $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is a metapolynomial if, for some positive integer $m$ and $n$, it can be represented in the form $$f(x_1,\cdots , x_k ...
1
vote
0answers
187 views

The integral of torsion

I found the following * exercise(exercise *9) in page 407 of the book of Do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced in the book ...
3
votes
0answers
127 views

Compact set covered by two opens

The following lemma about locally compact (but not necessarily Hausdorff) spaces or continuous lattices appears frequently but without citation. It is easy to prove but important in proofs. If a ...
0
votes
0answers
74 views

Transversal theory in ergodic theory

I was taking a glance at the original paper by Donald Ornstein, Bernoulli Shifts with the Same Entropy are Isomorphic, and I came across The Marriage Problem and a paper with the same name by P. ...
0
votes
1answer
116 views

Length-preserving Analogue of Riemann's Mapping Theorem

The Riemann mapping theorem (cf e.g. http://en.wikipedia.org/wiki/Riemann_mapping_theorem) essentially guarantees the existence of a biholomorphic mapping of a simply connected, open subset of the ...
3
votes
2answers
140 views

Hamilton-Ivey pinching in dimension 4

I've heard it said (e.g., in the accepted answer to this MO question) that a major obstacle to an effective theory of Ricci Flow in dimension 4 is the absence of the Hamilton-Ivey pinching phenomenon. ...
3
votes
1answer
162 views

Vanishing theorems for pluri-canonical bundle

I would like to know if Grauer-Riemenschneider vanishing theorem is still true in the setting of pluri-canonical bundle, i.e. the power of canonical bundle. Let me recall Grauer-Riemenschneider ...
4
votes
2answers
183 views

Inherent ambiguity of the context-sensitive language $L = {a^ib^ic^id^je^jf^j \bigcap a^ib^jc^id^je^if^j} $ or $a^nb^nc^nd^ne^nf^n$

What is the definition of ambiguity of context-sensitive grammar?This is relevant to the definition of inherent ambiguity of context-sensitive language.And any proof for the inherent ambiguity of ...
6
votes
0answers
92 views

Bundles over Function Spaces

Is there any reference on bundles over function spaces? In particular, I am interested in Banach-bundles over function spaces like $W^{k,r}(M)$, where $M$ is a Riemannian manifold. Separable ...
0
votes
0answers
15 views

Estimating Missing Shortest-Path-Tree Distances from Point Coordinates

Problem: given a finite set of points $\mathcal{P}:=\lbrace p_i\in\mathbb{R}^2 \mid i\in \lbrace 0,...,n\rbrace\wedge i\neq j\Leftrightarrow p_i\neq p_j\rbrace$ a function ...
4
votes
2answers
133 views

Visibility spaces and Gromov hyperbolicity

I would like to ask the community for a reference on the following subject: is there some thing as an equivalence between the definitions of Uniform Visibility manifolds and Gromov $\delta$-hyperbolic ...