All Questions
150,047
questions
0
votes
0
answers
2
views
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]):
$$K(k) = \int_{0}^{1} \frac {dx} {\sqrt{(1-x^{2})(1-k^{2}x^{2})}}...
- 4,700
4
votes
1
answer
228
views
Con(PA) via non-well-foundedness?
Lumsdaine made the following interesting
comment:
if Con(PA) fails in a non-standard model, it means it contains a
“proof of non-standard length” of a contradiction from PA. With a
little work, one ...
- 33.1k
0
votes
0
answers
4
views
What is the Goldie dimension of the ring of stable stems?
Let $p$ be a prime, and let $\pi_\ast^{(p)}$ be the ring of stable homotopy groups of spheres localized at the prime $p$. This is a nonnegatively-graded-commutative ring with $\mathbb Z_{(p)}$ in ...
- 58.3k
0
votes
0
answers
16
views
Circle action on manifolds with boundary
Are there some natural examples of smooth manifolds WITH BOUNDARY having a smooth action of the circle?
- 10.6k
0
votes
0
answers
55
views
Is this set, defined in terms of an irreducible representation, closed under inverses?
$\DeclareMathOperator\GL{GL}$Let $ H $ be an irreducible finite subgroup of $ \GL(n,\mathbb{C}) $. Define $ N^r(H) $ inductively by
$$
N^{r+1}(H)=\{ g \in \GL(n,\mathbb{C}): g H g^{-1} \subset N^r(H) \...
- 59.1k
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 $\binom{n}{0},\binom{n}{1},\binom{n}{n-1},\binom{n}{n}$.
...
- 2,133
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 ...
- 33.1k
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 ...
- 21.5k
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 ...
CommunityBot
- 1
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 ...
- 60.9k
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{...
- 172k
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? ...
- 24.8k
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 $X$...
- 4,165
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 ...
- 59.1k
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{...
- 141
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 + ...
- 617
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 ...
- 149k
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 ...
- 723
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 ...
- 11.2k
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 ...
CommunityBot
- 1
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/...
- 4,588
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 ...
- 26.3k
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 $(...
- 4,259
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$ ...
- 66k
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\...
- 77
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 ...
- 371
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 $\eta(s)$ ...
- 906
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 ...
- 553
-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 ...
- 19.7k
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
$...
- 4,259
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 $x^S=\Pi_{i\in S}x_i$ , $\tilde{f}(S)\in\mathbb{R}$. (...
- 49
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, \...
- 101
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 ...
- 2,662
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
$$a(2n+1)=a(n), a(2n)=a(n)+a(n-2^{q(n)})+a(...
- 2,886
0
votes
0
answers
77
views
Units of the group algebra of a free group
Let $K$ be a field of characteristic zero and $F_n$ be a free group of rank $n$. What is known about the group of units $K[F_n]^\times$?
In the case of $n=1$, there are only trivial units: $K[F_1]^\...
- 327
8
votes
1
answer
365
views
Are “most” bounded derivatives not Riemann integrable?
Given $a,b\in\mathbb R$ with $a<b$. Let
$$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 ...
- 3,097
-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\...
- 59.1k
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 ...
- 5,473
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
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, ...
- 133
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
$\mu_n = \frac 1n \sum_{i=1}^n \delta_{X_i}$, $n \geq 1$, ...
CommunityBot
- 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 ...
- 14.8k
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 ...
- 12.4k
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 ...
- 8,739
-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 ...
CommunityBot
- 1
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 $\...
- 2,118
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 $\...
- 6,605