# All Questions

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84 views

### Coefficients of Hilbert polynomials

Recall that we can define the Hilbert series of a graded commutative algebra
$$\displaystyle S = \bigoplus_{n \geq 0} S_n$$
over a field $K$ by
$$\displaystyle \mathcal{H}_S(t) = \sum_{n=0}^\infty ...

**2**

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50 views

### Reference for exercises with solutions for affine Lie algebras

I am doing self-study of affine Lie algebras, and while I have all recommended reference material, I find it hard to prepare for the exam.
It was quite easy to study finite-dimensional simple Lie ...

**-1**

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**0**answers

26 views

### Ky Fan norms and nuclear norm [on hold]

Ky Fan norms and the nuclear norm seem to be very relevant to my research so I would like to be familiar with what is already known.
Can anybody recommend a reference discussing any aspects of these ...

**10**

votes

**1**answer

263 views

+150

### Open problems in Berkovich geometry

I would like to know if there is a state of the art recent reference on non-archimedean analytic spaces mentioning/listing open problems, conjectures, unresolved questions in the theory (*). I have ...

**3**

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**0**answers

29 views

### Probability of a giant component existing in a $G(n,p)$ random graph with $p=\omega(\frac 1n)$

Consider a random $G(n,p)$ graph where $p=\omega(\frac 1n)$, and let $x$ denote the probability that the graph has a connected component of size linear in $n$.
It is well known that $x$ tends to $1$ ...

**2**

votes

**1**answer

74 views

### Is a matrix element of a norm continuous representation always a trigonometric polynomial?

I asked a similar question for the case of compact groups not long ago in math.stackexchange. Now I understand that the answer was "yes", and I want to modify that question. This is also related to my ...

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votes

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46 views

### Extending a model to a given compactification of its generic fiber

Let $R$ be a discrete valuation ring and $K$ its field of fraction. Let $X$ be a proper $K$-variety, $U$ a dense open and consider an $R$-model $\mathcal{U}$ of $U$.
Can we embed $\mathcal{U}$ in a ...

**6**

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**0**answers

112 views

### Exactness of pure functors

I can't prove this lemma in "Notes on motivic cohomology, Beilinson, Macpherson, Schechtman":
Lemma. A pure functor is exact.
Definitions: A mixed category $\mathcal{M}$ is a ...

**5**

votes

**2**answers

191 views

### Geodesics and Riemannian submersions

Let $X,Y$ be Riemannian manifolds, and $f\colon X\to Y$ be a Riemannian submersion.
Let $\gamma$ be a geodesic on $X$ starting at a point $x\in X$ and which is orthogonal to the fiber $f^{-1}(f(x))$.
...

**6**

votes

**1**answer

117 views

### Trigonometric polynomials on non-compact and non-abelian groups

I asked this initially in math.stackexchange, but it disappeared almost immediately, so I hope it will be proper to aks this here.
Hewitt and Ross define trigonometric polynomial on a locally compact ...

**1**

vote

**0**answers

88 views

### general formula for volume of a simplex? [migrated]

I am looking for a general formula to calculate the volume of a euclidean simplex in any number of dimensions. On Wikipedia I found that a formula similar to Heron's formula can be applied to ...

**2**

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**0**answers

46 views

### Divisibility of the degree of an extension by the degree its residual field

Let $A$ be and integrally closed domain whose quotient field is $K$, $L$ be a finite Galois extension of $K$, and $B$ be the integral closure of $A$ in $L$. Let $M_A$ be a maximal ideal of $A$, and ...

**7**

votes

**2**answers

167 views

### Recent trends in effective analysis

The references listed at http://en.wikipedia.org/wiki/Computable_analysis have all been published 30-15 years ago. Are the approaches which these references expose still up-to-date and relevant to the ...

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40 views

### About the selection of reals $u_0,u_1$ such that $u_{n}$ is a positive integer

Let $r\geq 4$ and $n≥1$ be two positive integers. Let us consider the sequence $(u_{n})$ defined by:
$$u_{n}=r^{n^2}\sum_{m=1}^{n}\frac{u_1-ru_0+2(m-1)}{r^{m^2}}+u_0$$ where $u_0,u_1$ are real ...

**0**

votes

**2**answers

233 views

### On independent sets of graph

Given $G$ a regular graph on $n$ vertices, denote $\alpha(G)>1$ to be independence number.
Denote $\Gamma(G)$ to be collection of possible subset of independent vertices in $G$ of cardinality ...

**3**

votes

**1**answer

156 views

### References for general Hasse-Weil zeta function

Most research on the Hasse-Weil zeta function focuses on some particular type of algebraic variety, and general surveys usually deal mostly with the better understood elliptic curve case.
I am ...

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76 views

### Tensor product of complexes [on hold]

Let $A$ be a ring and let the modules that are involved be left and right A-modules (not necessarily bimodules over A).
I'll denote as $\mathcal{E}^n_R(M, N)$ the category of n-fold extensions of M ...

**-2**

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**0**answers

31 views

### calculating E(Xt^2,Xt-h^2) with Xt normal(0,sigma^2) [on hold]

the problèm is
let {X_t} all normal N(0, sigma^2)
défine rho_X(h)=cov(X_t,X_t-h)/var(X) and Y_t=(X_t)^2
proove that rho_Y(h)=[rho_X(h)]^2
i know that i have to use expected value E(E(Y/X)) but i don't ...

**2**

votes

**0**answers

87 views

### Comparing the size of two sums

Let $q$ be a prime power and $\mathbb{F}_q$ be the field of $q$ elements.
Let $\mathbb{F}_q[t]$ be the polynomial ring over $\mathbb{F}_q$.
I am working on a research project, where I bounded a ...

**1**

vote

**1**answer

65 views

### 1-wasserstein distance v.s. total variation distance

Suppose that $\mu_1$ and $\mu_2$ are two distributions defined on $\mathbb{R}^n$ and $\gamma$ is a symmetric distribution (around $0$) on $\mathbb{R}^n$ with compact support. Let $\gamma_x$ denote the ...

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votes

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45 views

### Complementary polynomials

Denote $S=\{0,1\}^n$.
$\mathsf{MLP}_{d,n}=\{p\in\Bbb R[x_1,\dots,x_n]:p\mathsf{\mbox{ is mutilinear with total degree}}(p)=d\}$.
Is there an $n\geq d^2+1$ such that there exists distinct polynomials ...

**2**

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**0**answers

56 views

### Analogous to a PDE but where independent variable is a function

Consider, as an example for my question, a density function $u(\boldsymbol{x},t)$ on a vector field $\boldsymbol{x}$ at some time $t$. The flow velocity vector of the density is given by ...

**2**

votes

**0**answers

154 views

### Is $\mathcal{K}(H)$ injective $\mathcal{B}(H)$-module?

Does anyone know if the right Banach $\mathcal{B}(H)$-module $\mathcal{K}(H)$ is injective?
The same question for $\ell_\infty$-module $c_0$. Both these modules are not dual, so standard arguments ...

**10**

votes

**1**answer

154 views

### Choosing a metric in which homeomorphism is Holder continuous

Let $X$ be a compact metrizable space, and let $f:X \to X$ be a homeomorphism. Is it always possible to choose a compatible metric on $X$ in which $f$ is Holder continuous? I've tried some simple ...

**5**

votes

**2**answers

270 views

### Identity for Power Series and Binomial Coefficients

This question concerns a combinatorial identity obeyed by power series coefficients. Throughout we let $[x^{M}]\{\phi(x)\}$ denote the coefficient of $x^{M}$ in a power series $\phi(x)$.
Let $k$ be ...

**-3**

votes

**0**answers

37 views

### Matrix Algebra reduction [on hold]

I am trying to reduce the following:
$x$ and $y$ column vectors
$y^t$ is the transposed column vector
$(I - \frac{1}{(1+ y^t x)} * x y^t) (I + x y^t) = I$
I am stuck at $x y^t * y^t X = x y^t (x ...

**9**

votes

**1**answer

263 views

+50

### On the boundary of the twindragon

Let $\mathcal T$ be the famous twindragon, i.e.,
$$
\mathcal T=\left\{\sum_{n=0}^\infty a_n\left(\frac{1+i}2\right)^n : a_n\in\{0,1\}\right\}.
$$
Then, as is well known, $\mathcal T$ has a ...

**9**

votes

**2**answers

175 views

### Vanishing of certain periodic series: A question related to $L(1 , \chi) \neq 0$.

Fix $q$ to be a positive integer. Let $$f : \mathbb{N} \to \{-1 ,0, 1\}$$ be a $q$-periodic arithmetic function such that $$\sum_{n = 1}^q f(n) = 0.$$ If $f$ is not identically zero, is it true that ...

**2**

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**0**answers

65 views

### Asymptotic expansion for the Bell numbers

The Bell numbers $B(n)$ (that is, the numbers that count the set partitions of a set, and have exponential generating function $\exp(e^x -1)$ ) admit the asymptotic expansion
$$\frac{\log B(n)}{n} = ...

**1**

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34 views

### Measurability of eigenelements $s \mapsto (\varphi_k(s), \lambda_k(s))$ of Laplace-Beltrami on $M_s$

For each $s \in [a,b]$, let $M_s$ be a compact Riemannian manifold with no boundary.
Under what conditions on $s \mapsto M_s$ do the eigenvalues and eigenfunctions $(\varphi_k(s), \lambda_k(s))_{k ...

**-3**

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**0**answers

40 views

### Maximum - Minum area [on hold]

The problem is listed by this link because i couln't post images here (something about reputation).
http://math.stackexchange.com/questions/1117894/maximum-minumum-area

**0**

votes

**1**answer

94 views

### Schematic image of a relative Cartier divisor of a fiberwise dense open

Let $S$ be a scheme and $A$ an abelian $S$-scheme, i.e., $A \rightarrow S$ is a proper smooth $S$-group scheme whose fibers are $g$-dimensional abelian varieties. Suppose that one has a fiberwise ...

**0**

votes

**1**answer

55 views

### Help with notation for the state of a dynamical system defined by a PDE

Before my question let me briefly describe a simplified version of the dynamical system I'm working with. Suppose that I have a density function $m(\boldsymbol{x},t)$, that describes the abundance of ...

**5**

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**0**answers

94 views

### Groupoid cardinality of DM stack and point counting on coarse moduli spaces

Let $X$ be a finite type DM stack over a finite field $k$ with a coarse moduli space $X_c$. (We only assume $X_c$ is an algebraic space and $X$ might have infinite inertia stack.)
Under which ...

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**0**answers

73 views

### Moduli space of points - Gorenstein ideal

I've been working on algebraic covers, $\varphi\colon X\rightarrow Y$, ($\varphi_*\mathcal{O}_X$ is a locally free $\mathcal{O}_Y$-algebra of rank d).
I'm more interested in the algebraic point of ...

**2**

votes

**1**answer

80 views

### Statistical distance between discrete and continuous distributions

Are there any statistical distance functions that are capable of comparing a continuous and a discrete distribution? From reading this list
http://en.wikipedia.org/wiki/Statistical_distance
the only ...

**2**

votes

**0**answers

79 views

### Reference for cdh topology

Let $f:X\rightarrow Y$ be a proper surjective morphism over some base scheme $S$ of finite type, suppose $f$ restricts to an isomorphism over some open $U$ of $X$, we also suppose both $X$ and $Y$ are ...

**2**

votes

**1**answer

136 views

### Rankin-Selberg convolution and product of degrees

As I'm kinda obsessed with the Selberg class and because of the general converse conjecture, I'm still trying to get a rough idea of what automorphic representations and their L-functions as well as ...

**2**

votes

**1**answer

156 views

### Difference between Frobenii on Tate modules of special and generic fibre

Let $E$ be elliptic curve over $\mathbb Q$ and $p$ a prime of good reduction for $E$. Fix $\ell \neq p$.
If $E_p$ is ordinary then we have Frobenius $F_p$ on $E_p$. Assume $F_p$ lifts to ...

**1**

vote

**0**answers

96 views

### What is the technical difference between a deformation and a perturbation?

What is the technical difference between a deformation and a perturbation? Do they exist in somewhat different categories?

**1**

vote

**1**answer

122 views

### Sequence of smooth maps converging to the identity

Let $\{g_{n}:B_{1}(0) \rightarrow \mathbb{R}^{2}\}_{n\in \mathbb{N}}$ be a sequence of smooth maps where $B_{1}(0)=\{x\in \mathbb{R}^{2} \mid |x|<1\}$ is the unit ball in $\mathbb{R}^{2}$. Assume ...

**-6**

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**0**answers

138 views

### New algorithm discovered to find prime numbers [on hold]

I think that I have discovered a new algorithm to find prime numbers. It uses all prime numbers less than a particular number to find prime numbers within a range. I am not sure whether it generates ...

**0**

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**0**answers

189 views

### Can mathematics get from other sciences what it got from physics? [on hold]

Throughout history, physics has been an unparalleled source of '' inspiration'' for discovering/inventing mathematical ideas, which is due to its ability to describe the physical world. But can this ...

**3**

votes

**3**answers

236 views

### Survey papers of results (relevant to mathematics) produced during research on the relationship between mathematics and music theory

It is well-known that there is a relationship between music and mathematics, and there are many references that explore this topic (for example, Benson's book).
However, I would like to ask if there ...

**2**

votes

**1**answer

293 views

+50

### Bounds for constructing $n!$ with additions, subtractions, and multiplications starting from $1$

Found this on Complexity Zoo warning expired certificate
check NP Over The Complex Numbers.
[BCS+97] show the following striking result. For a positive integer $n$, let $t(n)$ denote the minimum ...

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vote

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54 views

### Relations in a space generated by indicator functions

Simple Question
I ran into the following seemingly simple question.
For an arbitrary set $M$ consider the real vector space generated by indicator functions $\chi_A$ of all subsets $A\subset M$. ...

**1**

vote

**0**answers

71 views

### Why are they called 'pernicious' numbers?

The definition of a pernicious number:
In number theory, a pernicious number is a positive integer where the Hamming weight (or digit sum) of its binary representation is prime.
The meaning of ...

**1**

vote

**1**answer

47 views

### Discrete Maximum Entropy Distribution with given mean

For a given mean $\mu$, what is the entropy maximizing probability distribution on the nonnegative integers?
Different sources indicated either the geometric or the Poisson distribution for this. As ...

**1**

vote

**2**answers

499 views

### Are hyperreal numbers isomorphic to formal power series?

I wonder whether hyperreal numbers isomorphic with formal Laurent series?
It seems that any hyperreal number can be represented in the form of Laurent series over $\omega$. For instance,
...

**7**

votes

**1**answer

145 views

### A question on the twistor space of a manifold

Let $M$ be either (a) self-dual conformal 4-manifold, or (b) hypercomplex $4n$-manifold.
In either case one can construct the twistor space $Z$ (in the case (b) $Z=\mathbb{C}\mathbb{P}^1\times M$ as a ...