# All Questions

**0**

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2 views

### Computing Gauss Legendre Curvature for Large N

I've been scanning across the web, and haven't found a good method to compute the Gauss Legendre abscisas and weights $\{ x_j, w^j \} _j$ for large N. My question is how to do it, and why should it ...

**0**

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8 views

### One question on complex analysis about analytic functions

Let $\Omega$ be a bounded planar domain and let $O_1, O_2$ be two open subsets of $\Omega$ such that their closures are disjoint. Let $\mathcal{A}$ be the class of analytic functions on $\Omega$, is ...

**-2**

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20 views

### a Theory of Iterated Functions

I'm an amateur who's looking for a co-writer to publish articles relating to the question below and others, derived from a general result applicable to topics as diverse as prime or random numbers, as ...

**0**

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**0**answers

8 views

### Gromov Geometric Structures and Killing fields

Let's fix some notations: $M$ will denote a real smooth, $m$-dimensional, manifold, $F^k(M)$ is the k-th order frame bundle on $M$ and $Gl^k(m)$ is the space of $k$-jets of diffeomorphisms of $\mathbb ...

**0**

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31 views

### A question on (odd) perfect numbers

I have asked this question in MSE a few weeks back, but did not receive any responses. I have cross-posted it to MO, hoping that it is appropriate for this site.
Let $\sigma(x)$ be the (classical) ...

**1**

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35 views

### How to use Integrals to calculate the expected value of two-dimensional Gaussian distribution [on hold]

Given that I have the following joint density function (two-dimensional Gaussian):
$f(u,v)= \frac{1}{1\pi\sigma_1\sigma_2\sqrt{1-\rho^2}}e^{-\frac{1}{2}Q(u,v)}$
where
...

**3**

votes

**1**answer

37 views

### Average height of rational points on a curve

I am seeking a formalism to define the average height of
the rational points on a curve. This is straightforward
if the number of points is finite, but (to me) not straightforward
when the rational ...

**1**

vote

**0**answers

14 views

### how understand periodicity in a combination of power, gamma and zeta functions?

Riemann's functional equation may be written:
$$
\frac{\zeta(s)}{\zeta(1-s)} = 2^s \pi^{s-1} \sin(\frac{\pi s}2) \Gamma(1-s) \tag{1}
$$
and so by symmetry:
$$
\frac{\zeta(1-s)}{\zeta(s)} = 2^{1-s} ...

**1**

vote

**1**answer

46 views

### Density of polynomials with a prescribed number field extension

For any polynomial $f(x) \in \mathbb{Z}[x]$, let $K_f$ denote the minimum splitting field over $\mathbb{Q}$ which contains all of the roots of $f$. Let $n \geq 2$ be a fixed integer, and let $K$ be a ...

**1**

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9 views

### When is a convex program continuous in its constraint vectors?

Consider $$F(z)=\min ae^{-x}+b e^{-y} s.t. x\ge 0, y\ge 0\text{ and } x+y=z$$
I checked if this function is continuous, but it is not at $z=0$. $F(z)=2\sqrt{ab}e^{-z/2}$ when $z\ne 0$, and ...

**3**

votes

**2**answers

106 views

### Is anything known about the eigenspectrum of the regular representation of the permutation group?

I am looking for information like upper bounds on how many times any eigenvalue can occur or something like how many eigenvalues can be there in some given range. Is anything like this known?
The ...

**-3**

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**0**answers

34 views

### Summation of N Choose k [on hold]

I was working on a math problem that required me to figure out the general summation of N choose 0 + N choose 1 + ... + N choose k.
I know that if k = N, the answer is 2^N. But is there an answer for ...

**-3**

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**0**answers

36 views

### Integration over Lie groups [on hold]

Is it possible to build a notion of integration $ \int $ of ( Lie forms ??? ) forms over Lie groups in the same way that we define a notion of integration of differential forms over manifolds ?
Thanks ...

**1**

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**0**answers

35 views

### Two questions on hyperspace of a metric space

Let $(X,T)$ be a compact metrizable space. For every metric $d$ on $X$ which is $T-$ compatible, the Hausdorff metric on $2^{X}$ gives a topology on $2^{X}$.
(Up to homeomorphism) is this topology ...

**-5**

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30 views

### Definite Integrals [on hold]

Alright so I'm taking calc One and at the end of my semester prepping for the final. I understand basic derivatives and integrals really well, but when i run into this problem here on the final review ...

**6**

votes

**1**answer

116 views

### Volume of the unitary group

I saw a very remarkable asymptotic formula (or a conjecture?) for the volume of of the unitary group $ U(n)$ which is the following:
$$\log[\mathrm{Volume}(U(n))] \sim_{n\rightarrow \infty} ...

**1**

vote

**1**answer

31 views

### Expected number of leaf nodes in some theoretical graph models

If a leaf node of a graph refers to a node having the degree of 1, how can one compute the expected number of leaf nodes of:
(A) a random graph (e.g., Erdos-Renyi graph),
(B) a small-world graph ...

**1**

vote

**1**answer

58 views

### Verifying that a differential is surjective

I've been reading "Weakly commensurable arithmetic groups and locally symmetric spaces" (Prasad and Rapinchuk, 2009). I'm having some trouble showing the following fact:
Let $K_v$ be a local field, ...

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**0**answers

33 views

### How can nontrivial elementary embeddings of the universe to some inner model be surjective? [migrated]

Consider $\kappa$ to the least measurable cardinal, or equivalently $\kappa$ is the critical point for an elementary embedding $j:V \rightarrow M$ from the universe $V$ to an inner model $M$ (critical ...

**4**

votes

**1**answer

162 views

### A question on Hawaiian earring

I have asked this question in MSE but have not got any satisfactory answer, so I am asking it here. Any idea on how to approach this problem will be highly appreciated.
Consider the Hawaiian earring. ...

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**0**answers

23 views

### complexities aggregation [on hold]

I will ask a question, but I hope that it is not a stupid one,
I have an algorithm, and I want to calculate its complexity.
in fact it has two parts, the complexity of the first part is
O(p)
...

**3**

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**0**answers

39 views

### Recognizing Schwartz regular distributions

Are there characterizations of Schwartz regular distributions other than being locally integrable (which does not lend itself to easy manipulations)?
To be more detailed: if I want to show that some ...

**-1**

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**0**answers

36 views

### Specific examples or applications of homotopy coherent diagrams [on hold]

nlab gives the basic idea about homotopy coherent diagram 1. Please show me a specific example or application on how it is done, thanks !

**1**

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26 views

### Solving complicated equation involving integral of error functions

I've been trying to solve the following equation for $\sigma$
$$\sigma^2 = \int_0^1 \left\{ \frac{1}{2} \operatorname{erfc} \left[ \frac{x + B}{\sqrt{2 (Rp - 1)} \, \sigma} \right] + \frac{1}{2} ...

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24 views

### Centralizer of a non-regular Lie algebra element

It is well understood that the centralizer of a regular element $A$ of a Lie algebra of complex (square, diagonalizable) matrices consists of polynomials $p(A)$ in that element of degree less than $n$ ...

**2**

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**0**answers

13 views

### Comparing right and left quasi-representable bimodules

Let $\mathcal V$ be your favourite (closed, symmetric) monoidal model category. To fix ideas, set $\mathcal V = \mathrm{Ch}(k)$, the category of chain complexes over a fixed commutative ring. Given a ...

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18 views

### Nonlinear Schrödinger blow-up for non radial solutions

I am studying a paper of Frank Merle and Pierre Raphaël,
http://math.unice.fr/~praphael/Publications/blow-up-norme-critique.pdf.
The equations are
$$
i\partial_tu+\Delta u=-|u|^{p-1}u
$$
on ...

**-1**

votes

**0**answers

96 views

### Which classes of geometric results or problems can not be achieved by algebraic methods [on hold]

Algebra has been using when necessary as a tool to solve geometric and topological problems. I have seen algebraic results through geometric methods in the literature. Which classes of geometric and ...

**-2**

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47 views

### Can someone help me with a Towers of Hanoi problem? [on hold]

I have 9 disks which is hard to keep track of, and I want to know... out of the 2^9 -1 moves... how many positions are there when all three posts are occupied, and is there a short formula describing ...

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15 views

### question about lambda calculus [migrated]

I'm triyng to understanding lambda calculus but I have some difficulty espacially when websites or books I search starts to make things a bit more complicated.
what I've understood by now is:
given
...

**-1**

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**0**answers

47 views

### Lemma: (Path -Cantor Lifting)? [on hold]

Definition: Let $p:E\rightarrow B$ be a map. If
$f:X\rightarrow B$is a map, a lifting of is a map
$\widetilde{f}:X\rightarrow E$ such that $p\circ \widetilde{f}=f$
¿TRUE or FALSE?
"Let $C$= Cantor ...

**7**

votes

**2**answers

126 views

### approximate two different real numbers to order $\frac{1}{z^{3/2}}$

I took this result from Minkowski's book on Geometry of numbers:
Two arbitrary real quantitites $a$ and $b$ may be made to approach as near as we wish in value the two fractions $\frac{x}{z}$ and ...

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88 views

### Which base change preserves integrality of schemes

Let $f:X \to Y$ be a flat morphism of projective noetherian integral schemes. Is there any known condition on a morphism $Z \to Y$ under which the resulting fiber product $X \times_Y Z$ is still ...

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**0**answers

44 views

### algebraic closedness in in residue field [on hold]

If $A\subseteq B$ are affine doamins over an algebraically closed field of $k$ of characteristic zero, such that $Q(A)$ is algebraically closed in $Q(B)$, how can one show that $Q(A)$ is also ...

**1**

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152 views

### Mathematics in three minutes! [on hold]

Okay, this is not a research level math question and might be closed in less than three minutes. But, I'd like to give it a shot! Today, a friend of mine e-mailed me asking how she can define ...

**0**

votes

**1**answer

100 views

### irreducibility of general fiber

I would like to get a reference of the following fact.
Let $A\subseteq B$ be affine domains over an algebraically closed field of characteristic zero. If $Q(A)$ is algebraically closed in $Q(B)$, ...

**-2**

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**0**answers

50 views

### Fractal Generators and Symmetry [on hold]

Using the Mandelbrot set as a starting place, $z_{n+1} = z_n^2+c$, may be written and computed as:
$x_{n+1} = x_n^2 -y_n^2+Re(c)$
$y_{n+1} = 2.0*x_n*y_n+Im(c)$
Let f(x,y) and g(x,y) be ...

**0**

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47 views

### On the centroid of a triangle [migrated]

There's three different ways to see a triangle in the Euclidean plane:
as three non-collinear points, say $A$, $B$, $C$;
as the line segments connecting the three points, that we can parametrize as a ...

**2**

votes

**2**answers

42 views

### Animating a unitary transform [on hold]

For the purpose of showing how (quantum) unitary operators behave as a computer animation, I would like to create a function $A_U(t)$ of some (complex) unitary matrix $U$, such that $A_U(t)$ is ...

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38 views

### About equivalence of two fractional Sobolev/Hilbert spaces

Let $(\varphi_k, \lambda_k)$ be the eigenelements of the Neumann Laplacian. It's possible to define a space
$$H(\Omega) = \{ u \in L^2(\Omega) \mid \sum_{k \geq 1}\lambda_k^{\frac ...

**2**

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46 views

### behaviour of first eigenfunction near the boundary

Consider $\Omega$ a smooth bounded domain in $R^N$ and suppose $ \phi_1(x)>0$ is the first eigenfunction of $ -\Delta$ in $H_0^1(\Omega)$ normalized however one chooses.
My interest is in how $ ...

**4**

votes

**1**answer

113 views

### How to teach generalizing the induction hypothesis?

I just finished teaching a class on using proof assistants (in this case, Agda) to write provably correct programs. Reflecting on how it went, the biggest difficulty I noticed the students having was ...

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37 views

### Restriction of locally free sheaves and semi-stability on curves

Let $C$ be a stable curve and $\mathcal{F}$ be a locally free sheaf on $C$ such that the restriction of $\mathcal{F}$ to any of the irreducible component $C_i$ of $C$, $\mathcal{F}|_{C_i}$ is Gieseker ...

**4**

votes

**1**answer

75 views

### Multivariable function analysis

Sorry if the terms I'm going to use is not professional enough:) This is about the complexity analysis of an algorithm.
Let $\alpha$ be the largest zero root of the polynomial ...

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100 views

### Analogues of the Monster for central charges different from 24

One way to define the Monster group is to consider a conformal field theory (CFT) corresponding to central charge $c=24$ and look at the automorphism group of its vertex operator algebra. For one of ...

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26 views

### the associated action on the transition functions

Let $X$ be a curve with an involution $\sigma$ generically unramified, given a $G-$bundle $E$ of rank $r$, than we ca take its pull-back, I want to describe the action of $\sigma$ on $G$. Fix a ...

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44 views

### Probabilistic proof for expander existence [on hold]

I am new to probabilistic proofs and trying to understand them better. Apparently, a common probabilistic proof focuses on the existence of expanders (eg. vertex expanders).
I've been using the search ...

**3**

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51 views

### Rellich's theorem from compact resolvent

On a compact Riemannian manifold, we know that the Laplacian $\Delta$ has compact resolvent. In proving this, one typical way is to use Rellich's theorem about the compact embedding of $H^1(M)$ into ...

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20 views

### rank of a Lie group over a non-archimedean local field of positive characteristic

In the case of a Lie algebra over a non-archimedean local field of positive characteristic (I have been led to believe that) it is not necessarily true that all Cartan subalgebras have the same ...

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40 views

### Titchmarsh S function [on hold]

SO it is known that Titchmarsh S function $$ S(T)= \pi^{-1} arg\quad \zeta\bigg(\frac{1}{2}+iT\bigg)$$ under the assumption of riemann-hypothesis gives $$ S(T)=O(\frac{\log T}{\log \log T})$$ can ...