# All Questions

150,047
questions

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votes

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### Elliptic integral as quantity associated with Riemann surface?

There are many elliptic integrals, so to show my point let me
just pick one of them (complete elliptic integral of the first
kind [1]):
$$K(k) = \int_{0}^{1} \frac {dx} {\sqrt{(1-x^{2})(1-k^{2}x^{2})}}...

4
votes

1
answer

228
views

### Con(PA) via non-well-foundedness?

Lumsdaine made the following interesting
comment:
if Con(PA) fails in a non-standard model, it means it contains a
“proof of non-standard length” of a contradiction from PA. With a
little work, one ...

0
votes

0
answers

4
views

### What is the Goldie dimension of the ring of stable stems?

Let $p$ be a prime, and let $\pi_\ast^{(p)}$ be the ring of stable homotopy groups of spheres localized at the prime $p$. This is a nonnegatively-graded-commutative ring with $\mathbb Z_{(p)}$ in ...

0
votes

0
answers

16
views

### Circle action on manifolds with boundary

Are there some natural examples of smooth manifolds WITH BOUNDARY having a smooth action of the circle?

0
votes

0
answers

55
views

### Is this set, defined in terms of an irreducible representation, closed under inverses?

$\DeclareMathOperator\GL{GL}$Let $ H $ be an irreducible finite subgroup of $ \GL(n,\mathbb{C}) $. Define $ N^r(H) $ inductively by
$$
N^{r+1}(H)=\{ g \in \GL(n,\mathbb{C}): g H g^{-1} \subset N^r(H) \...

1
vote

0
answers

20
views

### Does the interior of Pascal's triangle contain three consecutive integers?

This question defeated Math SE, so I am posting it here.
Consider the interior of Pascal's triangle: the triangle without numbers of the form $\binom{n}{0},\binom{n}{1},\binom{n}{n-1},\binom{n}{n}$.
...

1
vote

1
answer

344
views

### Composition of bibundles

I am reading Orbifolds as stacks?
Given Lie groupoids $\mathcal{G}$ and $\mathcal{H}$ there is a notion of what is called a bibundle from $\mathcal{G}$ to $\mathcal{H}$ which is supposed to be a ...

76
votes

49
answers

42k
views

### Examples of interesting false proofs

According to Wikipedia False proof
For example the reason validity fails may be a division by zero that is hidden by algebraic notation. There is a striking quality of the mathematical fallacy: as ...

4
votes

1
answer

595
views

### Difference between Chebyshev first and second degree iterative methods

Consider linear equation $Au = f$.
We want to solve it with iterative method (assuming $A$ is good).
First order iterative method is:
$$
u^{k+1} = u^k - \alpha_{k+1}(Au^k - f),
$$
The second degree ...

106
votes

0
answers

16k
views

### A proof of $\dim(R[T])=\dim(R)+1$ without prime ideals?

Please read this first before answering. This question is only concerned with a proof of the dimension formula using the Coquand-Lombardi characterization below. If you post something that doesn't ...

3
votes

2
answers

98
views

### Abel–Plana formula with fractional offset

The Abel–Plana formula compares the sum $\sum_{n=0}^\infty f(n)$ to the integral $\int_0^\infty f(x)\,dx$,
\begin{equation}
\sum_{n=0}^{\infty}f\left(n\right)-\int_{0}^{\infty}f\left(x\right)dx=\frac{...

0
votes

0
answers

46
views

### Collision among products of binomial coefficients

Let $k, \ell \geq 2$ be positive integers. Do there exist infinitely many positive integers $N$ such that the equation
$$\displaystyle \binom{m}{k} \binom{n}{\ell} = N$$
have more than four solutions? ...

3
votes

1
answer

43
views

### Question regarding the definition of linearization of line bundles

I'm reading Dolgachev's book 'Lectures on invariant theory'. In Chapter 7, the linearization of a group action is discussed. Let $G$ be a linear algebraic group acting on a quasi-projective variety $X$...

1
vote

1
answer

181
views

### Hahn-Banach theorem and ultrafilter lemma

I'm unable to understand a remark in "Two application of the method of construction by ultrapowers to analysis" by Luxemburg, which uses the ultrafilter lemma to prove the Hahn-Banach ...

0
votes

0
answers

39
views

### Expansion of metric near boundary of 3 dimensional Poincaré-Einstein/hyperbolic manifolds

In Mazzeo-Alexakis, there is a brief discussion that if $(M^3,g) \sim \mathbb{H}^3/\Gamma$ (for $\Gamma$ convex cocompact), then the metric can be expanded near the topological boundary as
$$g = \frac{...

0
votes

0
answers

24
views

### Does any warped product metric admit a function with hessian proportional to the metric?

It is known that the existence of a function with hessian proportional to the metric implies that the metric is a warped product metric. Is the reciprocal true as well? I.e, if $(B \times N, g = g_B + ...

11
votes

1
answer

459
views

### Has there been any progress on Conway's and Soifer's shortest paper?

In 2005 Conway and Soifer published the famous shortest ever paper, asking whether an equilateral triangle of sidelength $n+\varepsilon$ can be covered by $n^2+1$ unit equilateral triangles and ...

11
votes

1
answer

441
views

### A strengthening of base 2 Fermat pseudoprime

If $n$ is a prime then for all $k$ with $1 \le k \le [n/2]$,
$k$ divides ${n-1 \choose 2k-1}$ because of the identity
${n-1 \choose 2k-1} \frac{n}{k}=2{n \choose 2k}$. My question is whether
an ...

1
vote

1
answer

120
views

### Is $\sup_{f\in \mathcal{F}}\left|\int _Xfg \, d\mu\right|<\infty$ true for all $g\in L^\infty _\mathbb{C}(\mu )$?

Suppose that $(X,\mathcal{A},\mu )$ is a finite measure space. Let $\mathcal{F}\subseteq L^1_\mathbb{C}(\mu )$. If $\sup_{f\in \mathcal{F}}\left|\int _Xf\varphi \, d\mu\right|<\infty$ for all ...

9
votes

1
answer

594
views

### Sequence of real-rooted polynomials

I've been interested in proving a log-concavity result via proving that certain family of polynomials is real-rooted. By performing a sequence of transformations, I can reduce that problem to proving ...

0
votes

1
answer

80
views

### To find the convex planar region minimizing diameter when area and perimeter are given

The basic question is to find that planar convex region for which diameter is a minimum when area and perimeter are specified.
A partial answer is given here: http://nandacumar.blogspot.com/2012/11/...

0
votes

0
answers

223
views

### Stokes equation and Helmholtz decomposition

I apologize in advance for the long question, but it involves some work I been thinking about and would like help with. The Navier-Stokes equation in $\mathbb{R}^3$ subjected to no gravitational ...

3
votes

1
answer

114
views

### Frobenius and regular scheme

Let $X$ be a noetherian regular scheme over $\mathbb{F}_{p}$. Then by Kunz's theorem, the absolute Frobenius $F: X\rightarrow X$ is flat and integral. Can it be written as a projective limit of finite ...

0
votes

1
answer

58
views

### $c_0(2^{\kappa})$ does not embed in $\ell_\infty(\kappa)$?

How to prove that $(c_0(2^\kappa),\|\cdot\|_\infty)$ does not embed into $(\ell_\infty(\kappa),\|\cdot\|_\infty)$? Recall that $(c_0(2^\kappa),\|\cdot\|_\infty)$ is the Banach space of all families $(...

2
votes

3
answers

124
views

### Groups acting non-properly cocompactly on hyperbolic spaces

A group $G$ is hyperbolic if it admits a geometric (the action is proper and co-bounded) action on a geodesic hyperbolic metric space. Also, the definition can be given as follows, a group $G$ ...

0
votes

0
answers

176
views

+50

### Determine the rate of convergence

Let $(\mu_n)_{n\in \mathbb{N}}$ be a sequence of random variables in $[0,1]$. Let $\Theta$ be a subset of $\mathbb{R}^K$.
Definitions
D1: Define the set
$$
\Theta^*\equiv \Big\{\theta \in \Theta: \Pr\...

5
votes

1
answer

150
views

### Is this a typo in Macdonald's paper "The Poincaré Series of a Coxeter Group"?

I have a question about the proof of lemma 2.14 in Macdonald's paper The Poincaré Series of a Coxeter Group [1], where he used induction on $l(w)$ to prove that if $|E|=|R(w)|$, then $E=R(w)$. The ...

1
vote

0
answers

42
views

### Question on recursive formulas for $\eta(2 n+1)$ and $\beta(2 n)$ where $n\in\mathbb{N}$

This question is a refinement of my related MSE question which was asked over 2 years ago and no answers have yet been posted.
Consider the following formulas for the Dirichlet eta function $\eta(s)$ ...

2
votes

0
answers

68
views

### Extension of isotopies

In what follows $M$ will be a manifold (without boundary, for simplicity) and $C\subseteq M$ will denote a compact subset. In the paper Deformations of spaces of imbeddings Edwards and Kirby prove the ...

-1
votes

0
answers

17
views

### Is there a lower bound on finding an exact answer to a binary linear system of equations?

To find an exact answer to a binary linear system of equations Ax=b, I think the best idea is to use Gaussian elimination algorithm, since the involved mathematical operations are safe and don't ...

2
votes

1
answer

195
views

### Compactification of a product of manifolds

Let $M$ be a smooth manifold. We make the assumption that $M$ can be viewed as the interior of a compact manifold with boundary $\overline{M}$. In practice, for an explicit manifold, any ...

3
votes

3
answers

166
views

### Logarithm of a bounded operator

Let $\mathbb H$ be a Hilbert space and let $A\in \mathcal B(\mathbb H)$ such that the spectrum of $A$ does not meet a closed half-line issued from 0 in the complex plane. Then I guess that
$
A=\exp L
$...

2
votes

0
answers

68
views

+50

### Truncating the high degree part of a positive boolean function doesn't change the distance to positive functions too much

Given $n\in\mathbb{Z}^{+}$, suppose $f:\{-1,1\}^n\to[0,1], $ then $f$ has a Fourier expansion: $f(x)=\sum_{S\subseteq[n]} \tilde{f}(S)x^S,$ where $x^S=\Pi_{i\in S}x_i$ , $\tilde{f}(S)\in\mathbb{R}$. (...

0
votes

0
answers

16
views

### Thinning of (mixed) binomial point process

Let $N= \sum_{i=1}^M \delta_{X_i}$ be a mixed Binomial process over $(\mathbb X, \mathcal X)$. I.e., $M$ is a $\mathbb Z_+$ valued random variable with probability mass function $q_M(m)$, $m=0, 1, \...

3
votes

1
answer

57
views

### Convex optimization without Slater's condition

In nearly all convex optimization methods that I read about, it is assumed that the problem satisfies Slater's condition, that is, there is a point that strictly satisfies all constraints (the ...

3
votes

0
answers

152
views

### Closed form for $a(2^m(2^n-2^p-1))$

Let $q(n)$ be A007814, i.e., the number of trailing zeros in the binary representation of $n$. Here
$$q(2n+1)=0, q(2n)=q(n)+1$$
Let $a(n)$ be A329369. Here
$$a(2n+1)=a(n), a(2n)=a(n)+a(n-2^{q(n)})+a(...

0
votes

0
answers

77
views

### Units of the group algebra of a free group

Let $K$ be a field of characteristic zero and $F_n$ be a free group of rank $n$. What is known about the group of units $K[F_n]^\times$?
In the case of $n=1$, there are only trivial units: $K[F_1]^\...

8
votes

1
answer

365
views

### Are “most” bounded derivatives not Riemann integrable?

Given $a,b\in\mathbb R$ with $a<b$. Let
$$X=\{f\in C([a,b]): f \text{ is differentiable on } [a,b] \text{ with }f' \text{ bounded }\},$$
and
$$A=\{f\in X: f' \text{ is Riemann integrable}\}. $$
It ...

-2
votes

0
answers

36
views

### derivative with inner and outer summation

Hope this question is not that much stupid, but it really confuses me for many days...
Let $f$ be a function of $E_u$:
\begin{align}
f=&\sum_{u=1}^{K_u}\left(2\alpha_u\sqrt{w_u(1+\beta_u)E_u\...

16
votes

1
answer

364
views

### Types of generating functions (ordinary, exponential, ???) closed under substitution

A nice feature of ordinary and exponential generating functions is that they are closed under substitution: if $F(z)$ and $G(z)$ both have integer coefficients, then $F(G(z))$ also has integer ...

0
votes

0
answers

83
views

### An integral and a Jensen-type formula

For $f$ be analytic on the disc $\overline{D}(0,R)$ centred at $0$ with radius $R>0$ and such that $f(0)\neq 0$, then the following formula is well-known
\begin{align}
\frac{1}{2\pi}\int_{-\pi}^{\...

1
vote

1
answer

136
views

### How to conclude the quasi-projective case of the derived McKay correspondence from the projective case?

I am currently trying to understand the paper "Mukai implies McKay" from Bridgeland, King and Reid (cf. here). Let me sum up the setting we find ourselves in:
Let $M$ be a smooth quasi-...

3
votes

1
answer

165
views

### Is there a category theoretic definition of a cryptographic commitment scheme?

I'm trying to come up with a composeable framework for cryptographic commitment schemes, where inclusion proofs can be combined in different ways. I'm thinking this can be done with category theory, ...

2
votes

1
answer

637
views

### Convergence of empirical measures in Wasserstein distance

Let $X_1, X_2, \ldots$ be iid random variables with common distribution $\gamma$, the standard Gaussian distribution on $\mathbb {R}$, and let
$\mu_n = \frac 1n \sum_{i=1}^n \delta_{X_i}$, $n \geq 1$, ...

16
votes

3
answers

3k
views

### Who invented projective space $\mathbb{P}^n$?

Who invented projective space $\mathbb{P}^n$ as an extension of the usual affine space $\mathbb{A}^n$?
Who was the first person to consider projective closures of
plane affine algebraic curves (curves ...

3
votes

1
answer

104
views

### The $E$-(co)homology of $\mathrm{BGL}(R)^+$ and the algebraic $K$-theory of $R$

$\DeclareMathOperator\BGL{BGL}$In the paper, 'Two-primary Algebraic $K$-theory of rings of integers in number fields', Rognes and Weibel compute the $2$-torsion part in the algebraic $K$-theory of the ...

2
votes

1
answer

84
views

### Trace morphism for projective morphism on differentials forms

Let $k$ be a field, $X$ and $Y$ two connected $k$-varieties, and $f:X\rightarrow Y$ a dominant projective morphism of relative dimension $d$.
I would like to know under which condition there is a ...

-3
votes

0
answers

161
views

### Is there any odd natural number $n$ satisfying $n=e^2−f^2=g^2−h^2=k^2−l^2$ and $e^2f^2=g^2h^2+k^2l^2$ where $e,f,g,h,k,l~(>1)$ are natural numbers? [closed]

Is there any odd natural number $n$ satisfying $n=e^2−f^2=g^2−h^2=k^2−l^2$ and $e^2f^2=g^2h^2+k^2l^2$ where $e,f, g, h, k, l$ are natural numbers greater than 1? The problem is related to some famous ...

1
vote

0
answers

48
views

### Amenability of $\textrm{w}_0(L^1(G))$

Let $G$ be an infinite compact group and $A=L^1(G)$. It is known that $c_0(A)$ is amenable [Runde2020, p.80] while $\ell^{\infty}(A)$ is not [Daws2009] .
Let $\textrm{w}_0(A)$ denote the subspace of $\...

4
votes

1
answer

136
views

+100

### Gonality of specific Riemann surfaces $y^k=\tfrac{z^k-1}{z^k+1}$

The gonality of a compact Riemann surface $\Sigma$ is defined to be the lowest degree $d$ of a non-constant holomorphic map $f\colon \Sigma\to\mathbb CP^1.$ This means the gonality is 1 only for $\...