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

**10**

votes

**0**answers

182 views

### Derivation of Blattner's conjecture in the Beilinson-Bernstein picture

On the last page of Schmid's article "Discrete Series", he says
"In the Beilinson-Bernstein picture, discrete series modules are attached to closed $K$-orbits in $X$... the $K_{\mathbb R}$-structure ...

**6**

votes

**1**answer

377 views

### Is there a generalized Birkhoff ergodic theorem?

Is there a Birkhoff ergodic theorem for two measure preserving transformations $T$ and $S$ where $S\circ T= T \circ S$ so that $\frac{1}{n+1}\frac{1}{m+1}\sum_{i=0}^{n}\sum_{j=0}^{m}f \circ T^{i}\circ ...

**10**

votes

**1**answer

588 views

### Is there an analog of the Birch/Swinnerton-Dyer conjecture for abelian varieties in higher dimensions?

I am wondering if there is a multi-dimensional analog of the
Birch/Swinnerton-Dyer (BSD) conjecture.
The recent famous result inching toward resolution of that conjecture is:
Bhargava, Manjul, and ...

**5**

votes

**0**answers

89 views

### Reference request: Optimal $L^p$-decay for nonhomogenous heat equation in $\mathbb R^d$

Let $u$ be a classical solution for the nonhomogeneous heat equation in $\mathbb R_+ \times\mathbb R^d$:
$$
\begin{cases}
\partial_tu(t,x)-\Delta u(t,x) = f(t,x), \\
u(0,x)=u_0(x).
\end{cases}
$$
...

**4**

votes

**2**answers

179 views

### The unit tangent bundle of 2- or 4-manifolds as a principal $S^{1}$- or $S^{3}$-bundle

What type of obstructions have been studied so that the unit tangent bundle of a Riemannian 2-(4-)manifold have a structure of a principal $S^{1}$-($S^{3}$-)bundle?

**3**

votes

**0**answers

115 views

### Conics over number fields

I am looking for a reference for the following fact.
Let $k$ be a number field and let $S$ be a finite set of places of $k$ of even cardinality. Then there exists a unique conic $C$ over $k$ such ...

**2**

votes

**0**answers

102 views

### Questions about transformation or integral transformation

I have asked several mathematicians about the following questions,but all of them think they are good questions,but can not give a complete answer.Now I have to come here to ask mathematicians all ...

**4**

votes

**1**answer

157 views

### $C^0$ estimate for solutions of elliptic PDE with Neumann BC

I am interested in a reference for (or counterexample to) a particular
$C^0$ estimate for solutions of the Laplace equation with Neumann
boundary conditions. More precisely, let $(M,g)$ be a ...

**0**

votes

**1**answer

90 views

### Books on the analysis of hyperbolic partial differential equations

Most of the present books on pde analysis deal with the elliptic partial differential equations. Is there some book related to rigorous analysis with hyperbolic pdes, and especially hyperbolic systems ...

**2**

votes

**0**answers

51 views

### Name of a difference of continuants

I am getting ready to publish the manuscript
http://arxiv.org/pdf/1408.4631v2.pdf
and I am trying to do due diligence on a quantity I study before it gets published. (This is cross-posted from ...

**2**

votes

**2**answers

81 views

### Background for Kierstead terms

I was looking at some slides of John Longley's here, where he mentions "the Kierstead functional"
$$\lambda f.f(\lambda x.f(\lambda y.x)) \ ,$$
(where $f$ should be of type $2$, and $x,y$ of ground ...

**1**

vote

**0**answers

63 views

### Bounds re Asymptotic Formula for the Sum of Largest Prime Factors

I have a reference request related to the result :
$\sum_{n=2}^{x} P(n)$ ~ $\frac{\pi^2}{12}\frac{x^{2}}{log(x)}$ as $x \rightarrow \infty$
where $P(n)$ is the largest prime factor of the positive ...

**9**

votes

**2**answers

352 views

### on the center of a Lie group

I'm trying to set straight my various pieces of knowledge about the center of a compact Lie group, and I'm running in circles...
First some definitions:
• Let $G$ be compact, simple, and simply ...

**3**

votes

**0**answers

78 views

### birational classification of rationally connected 3-folds

What is the birational classification of (smooth projective) rationally connected 3-folds (over algebraically closed fields of characteristic $0$ or even $\mathbf{C}$, if $\mathrm{char}(k) = p > ...

**0**

votes

**1**answer

120 views

### Question on the partial differential equations in complex space

As is known most of the theory now developed for partial differential equations is in the real space, especially function space like Sobolev space, BMO space, $L^p$ space, etc. However is there some ...

**6**

votes

**1**answer

130 views

### Constructing sums of squares identities

Recall that a sum of squares formula for $[r,s,n]$ over a field $F$ is an identity of the form
$$ ( x_{1}^{2} + \cdots + x_{r}^{2})( y_{1}^{2} + \cdots + y_{s}^{2})
= ( z_{1}^{2} + \cdots + ...

**2**

votes

**2**answers

82 views

### Complexity of Untwisting Polygons

What is the complexity of the following task:
given a sequence $p_1, ..., p_n, p_1$ that defines a closed polyline in the euclidean plane,
what is the complexity of finding a reordering of the points, ...

**5**

votes

**0**answers

57 views

### Diophantine approximation in $\mathbb{G}_m^r$ with approximants restricted to a finiteley generated subgroup

Faltings, in the same paper (Diophantine approximation on abelian varieties, Ann. Math. 1991) in which he proved his famous ``big theorem,'' proved also that at any place $v$ of a number field $K$ and ...

**6**

votes

**1**answer

190 views

### On a weak tree property for inaccessible cardinals

Suppose that $\kappa$ is inaccessible and consider a tree of height $\kappa$ whose levels have size strictly below some cardinal $\gamma < \kappa$. Does this type of tree always have a ...

**1**

vote

**0**answers

45 views

### Asymptotic decay for the inhomogeneous Schrödinger equation

Let $H$ be an Schrödinger operator $(-\Delta+V(x))$ with a radially symmetric and smooth potential and $H$ is e.s.a. on $C^\infty_0(\mathbb{R}^n)$, furthermore let $\lambda$ be an element of the ...

**1**

vote

**1**answer

134 views

### L-function of twist

I'd like to ask the following easy question, since I can't find a reference.
Let $X/\mathbb{Q}$ a smooth projective variety. How does one express the $L$-function of the twist, $L(H^i(X)(r), s)$ in ...

**1**

vote

**1**answer

137 views

### Solution or Reference Request for a Closed Form of the Sum

I have been working for quite a while on finding a closed formula for the Legendre Symbol. Inspite of my best efforts I can't come anything better with a formula for the symbol ...

**1**

vote

**0**answers

213 views

### Learning roadmap in Algebra [closed]

I am a senior undergraduate student in mathematics, I have a sound knowledge in the following areas:
a) Commutative Algebra
b) Field Theory and Galois Theory
c) Homological Algebra
My question is ...

**4**

votes

**1**answer

123 views

### Inequality due to Siegel (assumptions) and upper bounds on number field discriminants

In Siegel's 1969 paper, Abschätzung von Einheiten, on page 73, he states the inequality
$$\log\sqrt d\le n-1+{n\over 2}\log\pi+r_2\log 2\qquad (*)$$
and compares with the bound due to Minkowski that
...

**0**

votes

**1**answer

49 views

### Reference for finite number of Weyl groups of reductive groups of rank $r$

I'm posting this question on behalf of someone without access to mathoverflow:
Can anybody give me a reliable reference (not a proof) to the following statement?
Up to isomorphism, there are only ...

**1**

vote

**0**answers

160 views

### Davenport's proof that almost all integers are the sum of 4 cubes

Where can I find a pdf that describes Davenport's proof that almost all integers are the sum of $4$ cubes?

**2**

votes

**2**answers

371 views

### Is there a von Koch-type theorem for the generalized Riemann hypothesis?

Helge von Koch proved in 1901 that the Riemann hypothesis is equivalent to the error term in the prime number theorem having the bound
$$
\mid\pi(x)-\textrm{li}(x)\mid=O(\sqrt{x} \log x).
$$
Q1: ...

**4**

votes

**0**answers

131 views

### Partial differential Equation over characteristic p

I want some references on partial differential equations over characteristic $p$. If we have a first order partial differential equation, how can we check whether there exists polynomials or rational ...

**-1**

votes

**0**answers

27 views

### Finding the source for minimizer of a functional for all $C^2$-curve $x(t_0)=x_0$ and $x(t_1)=x_1$

I am trying to find where this problem comes from and its corresponding proof for my students, but I cannot find the source anywhere. If anyone can find the source of this, or has any ideas where I ...

**9**

votes

**1**answer

228 views

### What is known about the distribution of eigenvectors of random matrices?

Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular:
How are individual eigenvectors ...

**0**

votes

**0**answers

62 views

### F-splitting and F-purity from commutative algebra viewpoint

First I define two terms:
Let $R$ be a commutative ring with identity,let char$R$ = $p$, let $F:R\rightarrow R$ be the Frobenius ring homomorphism. This makes $R$ into an $R$-module with respect to ...

**4**

votes

**0**answers

155 views

### Reference for Hodge decomposition

Let $U$ be a bounded open subset of $\mathbb{R}^d$ with Lipschitz boundary, and $g \in L^2(U,\mathbb{R}^d)$ be a solenoidal vector field (i.e. $\nabla \cdot g = 0$). Then $g$ can be written in the ...

**2**

votes

**2**answers

334 views

### Does this simple inequality have a name?

Let $x_{1},\ldots,x_{n}$ be nonnegative numbers such that $m \leq x_{i} \leq M$. Let
$$
S=\sum_{i=1}^{n}{x_{i}}
$$
and
$$
Q=\sum_{i=1}^{n}{x_{i}^{2}}.
$$
Then
$$
Q \leq S(M+m)-nMm.
$$
This has ...

**0**

votes

**1**answer

341 views

### For what nonnegative measures $\mu$ does $\mu*e^{-|\cdot|}\in L^{\infty}$?

I am trying to characterize all measures on $\mathbb{R}$ such that
$$
\sup_{x\in\mathbb{R}} \: (\mu*f)(x)<+\infty,
$$
where $f(x)$ is some specific integrable functions, such as $f(x)=e^{-|x|}$, ...

**5**

votes

**2**answers

225 views

### Using Discrete Morse Theory to represent hom classes

In "vanilla" Morse theory, you can construct cycles representing integral homology classes of a smooth manifold from a Morse function on the manifold (by looking at the flow into/out of the critical ...

**2**

votes

**0**answers

79 views

### Jackiw-Pi identity

In their paper http://journals.aps.org/prd/abstract/10.1103/PhysRevD.42.3500 (Classical and quantal nonrelativistic Chern-Simons theory) Jackiw and Pi introduced an unusual identity involving ...

**0**

votes

**1**answer

121 views

### Is the span of those vectors dense in $\ell_2$?

For all $x \in \mathbb{R}^n$ and $\alpha \in \mathbb{Z}_{\geq 0}^n$ let $x^\alpha=x_1^{\alpha_1} \cdots x_n^{\alpha_n}$. Let $$\ell^2=\{z=(z_\alpha)_{\alpha \in \mathbb{Z}_{\geq 0}^n}:\, z_{\alpha} ...

**1**

vote

**0**answers

79 views

### Reference request on operator matrices [closed]

I'm looking for a reference on linear, bounded, self-adjoint operators defined on the product space, $T:E\times F\to E\times F$ such that
$$Tx = \begin{pmatrix}A & B \\
C & D
...

**8**

votes

**2**answers

474 views

### Prove that the Dirichlet eta function is monotonic

Let us consider $\eta(p):= \sum\limits_{n=1}^\infty \frac{(-1)^{n+1}}{n^p}$ for $p>0$. Has anyone come along with an elementary proof that $\eta(x)$ is monotonically increasing on this set? By ...

**5**

votes

**1**answer

115 views

### Ham sandwich theorem for discrete measures - reference request

A discrete version of the ham sandwich theorem states as follows (see for instance "Common Hyperplane Medians for Random Vectors" - Hill):
For every $\mu_1,...,\mu_n$ discrete (i.e., purely atomic) ...

**3**

votes

**1**answer

280 views

### Category of modules over commutative monoid in symmetric monoidal category

Let $\left(\cal{C},\otimes ,I\right)$ be a symmetric monoidal category (not necessarily closed) and $A$ a commutative monoid in $\cal{C}$. In his DAG III (page 95), Lurie writes:
In many cases, ...

**0**

votes

**0**answers

36 views

### construction of four dimensional regular convex polytopes

Could anybody give me a reference book about the explicit construction of the 6 regular four dimensional convex polytopes. I cannot easily find Schläfli's original paper so I am looking for a modern ...

**2**

votes

**2**answers

235 views

### Distribution of a random walk on a directed line

Is there a closed formula for the distribution of $x_t$ in the following random process, describing a random walk on a directed line?
$x_0 = n$
$x_t$ is a uniformly random integer between 1 and ...

**1**

vote

**0**answers

106 views

### Searching for surprising equation connecting distant mathematical fields [closed]

I hope this is not too off topic on this site, if so i excuse myself.
Some time ago i read an article about important equations (as most lists feature prominent unequations) in math and the unsolved ...

**6**

votes

**6**answers

287 views

### Reference on representations of knot groups

Recently, I was studying the knot group and I want to learn some more material about it (e.g. its representations).
"Knots" by Burde and Zieschang discusses some material but it is not entirely ...

**7**

votes

**1**answer

220 views

### Higher coherent multiplicative structures on S-algebras

In their book, Elmendorf, Kriz, May and Mandell describe a useful category of spectra, called S-modules, where S is the sphere spectrum. Ring objects in this category can be identified with spectra ...

**5**

votes

**1**answer

218 views

### Harish-Chandra isomorphism for compact symmetric spaces

I would be interested to have an explicit description of the algebra of invariant differential operators on functions on a compact symmetric space $G/K$. A reference would be especially useful.
...

**6**

votes

**0**answers

318 views

### Any reason to believe that $NP \neq P$ is unprovable in ZFC

We know $NP \neq P$ from a lot of point of view like empirical reason,or theoretical reasons such as finite model theory or descriptive complexity.Although we find so many reasons to believe $NP \neq ...

**1**

vote

**0**answers

111 views

### Ext Quivers and their applications to Representation Theory

I am looking for references that provide an overview of the following two topics (it can be multiple references if necessary):
How to compute the Ext-quiver of a (locally finite or finite) ...

**1**

vote

**2**answers

159 views

### Looking for a reference for a paper by Mordell

On page 384 of the book "Number Theory:Volume 1:tools and Diophantine Equations" by Henri Cohen there is reference to the fact that: "It has been proved by Schinzel, Mordell nd successors that such an ...