# All Questions

**1**

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2 views

### Reference Request for Hilbert Schemes

I'm a physicist working on Fractional Quantum Hall effect. The mathematical subjects of study are symmetric, translational invariant, homogeneous polynomials on $\mathbb{C}$. Very early in my study I ...

**2**

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**0**answers

23 views

### Lie Algebra of Aut(GL(n,R))

What is the Lie Algebra of $Aut(Gl(n,F))$ when $F$ is either $\mathbb{R}$ or $\mathbb{C}$?
Is it enough to consider the injection via Hochschild:
$Aut(GL(n)) \to Aut(\mathfrak{gl}(n))$?
In ...

**0**

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**0**answers

18 views

### How to find the number of possible sub rectangles touching the edge of a larger one? [on hold]

http://i.imgur.com/iUDIeMG.png
If there exists a rectangular matrix of order M by N then how to find the number of ways to pick a sub-rectangle matrix of any size which is a multiple of 1 square ...

**3**

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**0**answers

46 views

### Graph spectra and topology

This is a somewhat vague question, but I'm wondering if there has been any research into connections between the spectrum of a graph and some notion of the "topology" of that graph.
To give an ...

**3**

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**2**answers

40 views

### Basic question about polytope duals

The following must be well known. Is there a beginning or midlevel
text where the answer is discussed? Thanks.
Along with a polytope one has the notion of its dual which is officially
defined via ...

**-4**

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**0**answers

16 views

### first order linear systems differential equations [on hold]

This is an easy question on one dimension but when moving into a system of equations I can’t find the exact solution using matrixes and vectors.
given $\dot {\vec x} =A{\vec x}+\vec b$ (where A is a ...

**17**

votes

**2**answers

268 views

### Recent observation of gravitational waves [on hold]

It was exciting to hear that LIGO detected the merging of two black
holes one billion light-years away. One of the black holes had 36
times the mass of the sun, and the other 29. After the merging the
...

**6**

votes

**1**answer

68 views

### Are these two quotients of $\omega^\omega$ isomorphic?

Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$. For $f,g\in\omega^\omega$ we say $f\simeq_{\text{fin}} g$ if there is $n\in \omega$ such that $f(k) = g(k)$ for all $k\geq n$.
...

**-4**

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**0**answers

30 views

### lebegue integral

we suppose that $f:R^+\rightarrow R^+$ is a positive lebegue integrable function and for every $\epsilon >0$, $\int_0^\epsilon f(s)ds >0$ .if we have $\sqrt[n]{\int_0^a f(s)ds} <1$ ...

**1**

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23 views

### For a division ring $D$, does $[D:C_D(a)]_{right}$ vary when $D$ is enlarged?

In a commutative field $K$, the Zariski dimension of an algebraic subset of $K^n$ over $K$ does not vary if one enlarges $K$ if I understood well. In particular, for two Zariski-closed vector spaces ...

**3**

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39 views

### Quantization of $S^2$ as $C^*$-algebra?

The general context for the question - is belief that quantization of compact symplectic manifolds can be endowed with the structure of $C^*$-algebra (see MO230695).
The particular question is about ...

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**0**answers

73 views

### Express $ \int_0^1 \frac{dz}{\sqrt{x(x^2 - 1)(x - \lambda)}} $ as hypergeometric function

How do we express the following as hypergeometric function? Let $\lambda > 1$:
$$ \int_0^1 \frac{dz}{\sqrt{x(x^2 - 1)(x - \lambda)}} $$
Is this still of the ${}_2F_1$ type? How to find the ...

**-1**

votes

**0**answers

51 views

### Maschke Theorem descomposition [on hold]

When I was doing an exercise to illustrate to myself the decomposition of the famous Maschke´s Theorem, I realized I didn't understand how was the decomposition stated in the theorem.
This is the ...

**1**

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**0**answers

58 views

### Intuition behind the Duistermaat-Guillemin version of Weyl's law

The theorem in question (see this paper), after a modification by Ivrii (see this paper) states the following:
Let $(M, g)$ be a compact Riemannian manifold of dimension $n \geq 2$. Assume that the ...

**-3**

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**0**answers

66 views

### Ramsey theory and logic [on hold]

i need literature which contains formal proof of finite Ramsey theorem in PA, possibly, available on- line.

**0**

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**0**answers

12 views

### combination of field and particle methods for fluid dynamics

in numerical fluid dynamics there are field methods like finite-volume, finite-element, etc. and particle methods like Smoothed-Particle-Hydrodynamics – SPH and others. Both approaches have advantages ...

**2**

votes

**0**answers

37 views

### Resolution of indeterminacies for a map to Grassmannian of planes

Let $X\subset \mathbb P_k^N$ be a $n$-dimensional smooth projective variety ($n\geq 2$) and $\phi_l:X^l\dashrightarrow Gr(l,N+1)$ ($l\leq n+1$) be the natural rational map which associates to a ...

**1**

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**0**answers

9 views

### Least Width of Planar Unimodal Curves with Unit Diameter

I am currently trying to find a way to define some notion of "roundness" for subtours in graphs and that definition should only be based on the comparing (sums of) edge length and on the order in ...

**1**

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**0**answers

74 views

### Verdier duality on excellent schemes

Let $f:X\rightarrow Y$ be a regular morphism between $k$-schemes which are noetherian and excellent with a funcion of dimension.
In the book by Illusie-Laszlo-Orgogozo, there is a theorem (4.4.1 in ...

**12**

votes

**1**answer

345 views

### What is a Futaki invariant, what is the intuition behind it, and why is it important?

As the question title suggests, what is a Futaki invariant, what is the intuition behind it, and why is it important?

**0**

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**0**answers

39 views

### On Lie theory with special functions. [on hold]

I research in Lie theory with special functions.
But I saw a lot of research on the use of lie theory in hyper-geometric and hermit and other ..
Is there a new kind of functions, not considered ...

**5**

votes

**1**answer

114 views

### Weight filtration on certain Galois representations

Let $G$ be the absolute Galois group of a number field $K$. Let $\ell$ be a prime number. There are representations $\mathbb{Z}_\ell(n)$ of $G$ on the group of $\ell$-adic integers given by the ...

**-5**

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**0**answers

13 views

### When comparing different bars on a bar chart, can you use percentage difference/change? [on hold]

I wanted to know if you can use percentage difference for discontinuous data

**1**

vote

**2**answers

161 views

### Zariski-closed subgroups of ${\mathbf G}_{\mathbf a}^n$

Let's work over an algebraically closed field $K$. A $1$-dimensional Zariski-closed connected subgroup of ${\mathbf G}_{\mathbf a}^n$ is isomorphic to ${\mathbf G}_{\mathbf a}^1$. If $K$ has ...

**6**

votes

**8**answers

589 views

### What are fun elementary subjects in probability?

I have to read several lectures on probability or applications of probability for high school students (of high level). There is no necessary part I must lecture, that is, my aim is just ...

**7**

votes

**1**answer

92 views

### Why do we want $p$-permutation modules in splendid equivalences?

First Rickard (in Splendid Equivalences: Derived Categories and Permutation Modules ) and then Rouquier (Block theory via stable and Rickard equivalences, Appendix A.1) define splendid equivalences ...

**1**

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**0**answers

39 views

### How do the heat kernel coefficients depend on the curvature tensor

this is a crosspost, the same question was asked first here: http://math.stackexchange.com/questions/1640092/polynomial-in-the-components-of-the-curvature-tensor
Since I have not received any answers ...

**5**

votes

**2**answers

135 views

### Rational homology sphere that is not Seifert manifold

I wonder if there is an example of rational homology sphere that is not a Seifert manifold. If there is, how can one construct such a rational homology sphere from a surgery of a knot in $S^3$?

**2**

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**0**answers

48 views

### The weak-star closure of closed left ideals corresponding to pure states

I asked this question at math.stackexchange and received no comment.
Let $A$ be a C*-algebra and $\phi$ be a positive linear functional on $A$. Let $\tilde{\phi}$ be its unique $w^*$-continuous ...

**-1**

votes

**0**answers

12 views

### Problem Formulation for SGD as Applied in Non-stationary Optimziation

This post is about how to formulate a time-varying (loss) problem. Basically I am looking for a target-tracking (time-varying) model which is amenable (error can be controlled) for stochastic ...

**1**

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**0**answers

35 views

### Suggestion for books in Pertubation theory with an emphasis on the theory

As the title suggest I am looking for another good coverage of the theory of Pertubation theory.
Currently I am working through Murodock's book: Pertubations: Theory and Methods.
But I am rest assure ...

**1**

vote

**1**answer

71 views

### When the reduced $C^*$-algebra of $\Gamma$ admits character then $\Gamma$ is amenable [on hold]

Suppose that $C^*_r(\Gamma)$ admits some character (homomorphism into $\mathbb{C}$)-here $\Gamma$ is discrete group and $C^*_r(\Gamma)$ is the closure of the image of the group ring $\mathbb{C}\Gamma$ ...

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85 views

### Are there any ways we can determine whether the $\Xi_x$-classes of natural numbers upto $\frac{1}{2}p^2_x -2$ exvert all non-trivial $\Xi_x$-classes? [on hold]

This question follows from the information provided below.
Are there any ways we can determine whether the $\Xi_x$-classes of natural numbers up to $\frac{1}{2}p^2_x -2$ exvert all non-trivial ...

**6**

votes

**1**answer

193 views

### Is there a consistent arithmetically definable extension of PA that proves its own consistency?

I asked this on stackexchange with no answer.
The negation would be the obvious generalization of Gödel's second incompleteness from r.e. extensions of PA to any arithmetically definable extension of ...

**10**

votes

**1**answer

321 views

### “Set theory” founded on lists rather than sets

On a computer, sets are often represented rather "indirectly / implicitly", e.g. in terms of some properties that they or their members satisfy. But some sets can be represented more "directly / ...

**-3**

votes

**0**answers

21 views

### minimizing sum of distances [on hold]

I have three points A(-3.5, 0), B(2,0), C(0.3).
Looking for D(0,d) such that AD + BD + CD is minimal. Fermat does not work here due to D lying on the y-axis.
I thought I could just minimize the sum ...

**1**

vote

**1**answer

117 views

### Poincaré lemma for distributions

Let us consider a current on $\mathbb R^n$, that is a differential form whose coefficients are distributions. For simplicity, let us check the case of a $1$-form
$$
u=\sum_{1\le j\le n} u_j dx_j,\quad ...

**21**

votes

**2**answers

511 views

### Elementary congruences and L-functions

In a recent article, Emmanuel Lecouturier proves a generalization of the following surprising result: for a Mersenne prime $N = 2^p - 1 \ge 31$, the element
$$ S = \prod_{k=1}^{\frac{N-1}2} k^k $$
...

**2**

votes

**1**answer

98 views

### About the characteristic polynomial of Frobenius of the Jacobian of a genus 2 hyperelliptic curve

I was looking for some information related to the values of the characteristic polynomial $\chi(t)$ of the Frobenius of a Jacobian of a hyperelliptic curve $C$ of genus 2 over $\mathbb{F}_q$ and in ...

**0**

votes

**0**answers

18 views

### Find optimal value for a regularization parameter in generalized eigenvalue problem

Consider the generalized eigenvalue problem :
$ \Sigma_{XY} \Sigma_{YX} {W} = \lambda \Sigma_{XX} {W} $
where $\Sigma_{XX} $ and $\Sigma_{XY}$ are sample covariance matrices are of the matrices ...

**0**

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**0**answers

24 views

### Gradient Descent with L2 Norm Regularization [on hold]

So I've worked out Stochastic Gradient Descent to be the following formula approximately for Logistic Regression to be:
$
w_{t+1} = w_t - \eta((\sigma({w_t}^Tx_i) - y_t)x_t)
$
$p(\mathbf{y} = 1 | ...

**11**

votes

**1**answer

205 views

### Mixed Hodge structure on sheaf cohomology of a variation of Hodge structures

I'm new here. I hope to do it right!
I am interested in studying mixed Hodge structures over complex algebraic surfaces and their generalizations.
Let us take a smooth complex variety $X$ and a ...

**1**

vote

**0**answers

21 views

### Divergence of a second order tensor [on hold]

I think that I have found 2 seemingly conflicting sources relating to the divergence of a second order tensor. I am not sure which is correct.
Suppose you would like to compute the components of a ...

**2**

votes

**2**answers

115 views

### Recover Poisson bracket on $C[G]$ using the Lie cobracket $\delta: g \to \Lambda^2 g$

By a theorem of Drinfeld, there is a one to one correspondence between Lie bialgebras and Poisson Lie groups. Therefore given a Lie cobracket $\delta: g \to \Lambda^2 g$, there is a Poisson bracket on ...

**2**

votes

**0**answers

64 views

### Classifications of Lie bialgebras

What is the current status of the classifications of Lie bialgebras? In particular, has the following problem been solved? Let $gl_n$ be the general linear Lie algebra. Classify all Lie cobrackets ...

**1**

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**0**answers

109 views

### if 0→A→A⊕B→B→0 is an exact sequence of finitely generated modules over a commutative Noetherian ring, then the exact sequence does split [duplicate]

Here, Martin Brandenburg says it is not true that "Every short exact sequence of the form $0 \to A \to A \oplus B \to B \to 0$ splits." Then Mohan says in comments that "As a positive result,
If ...

**1**

vote

**0**answers

69 views

### to what extent is a reductive group hyperbolic?

The group $SL(2,F)$ where $F$ is a local nonArchemidian field is hyperbolic. Various generalizations of the notion of hyperbolicity have been studied in the literature (I've seen terms like ...

**3**

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**0**answers

103 views

### For which fields are the 1-dimensional algebraic groups known?

Given an algebraically closed field, or even a perfect one, a connected 1-dimensional algebraic group $G$ over the field $K$ is isomorphic to either $\mathbf G_a$ or $\mathbf G_m$.
For which fields ...

**0**

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**0**answers

33 views

### Equation of a curved line from a graph [on hold]

I am trying to calculate an equation to represent the graph attached to this question. It's an extract from a take-off performance graph used in aviation.
The second graph shows how it is used. The ...

**-2**

votes

**0**answers

32 views

### On Incidence structure of finite Projective plane [on hold]

Consider a finite projective plane $\mathcal{P}$ over a finite field $F_q$, $q$ a prime power. Is it possible to define a map $f:\mathcal{P}\times \mathcal{P}\rightarrow \mathcal{P}$ such that
(i) ...