Trending questions
159,025 questions
5
votes
2
answers
45
views
On the continuity a function given by evaluating compact subsets of smooth functions
Let $M$ be a compact connected smooth manifold. Write $C^{\infty}(M)$ for the Frechet space of the smooth real-valued functions on $M$ equipped with the usual $C^{\infty}$-topology.
Given a compact ...
- 557
2
votes
0
answers
13
views
On the solution sets to the ODE $(a^2-1)Q''(a) + \left(3a-\frac{d-1}{a} \right)Q'(a) + \frac{3}{4}Q'(a)=0$
Looking for self-similar, radial solutions to a certain nonlinear wave equation in $\mathbb{R}^{d+1}$ leads to studying the following linear ODE in the self-similar variable $a\in[0,1)$:
$$(a^2-1)Q''(...
- 890
3
votes
1
answer
77
views
Few doubts about 'A new elementary proof of the Prime Number Theorem" by Richter
I'm working on Richter's "A new elementary proof of the Prime Number Theorem" paper.
I have some doubt about the proof of proposition 3.1
Here's the reference to the paper: https://arxiv.org/...
- 95
0
votes
0
answers
27
views
Generic reducedness of geometric generic fibre
Let $f:X\to Y$ be a surjective morphism between two projective schemes over a field of characteristic $p>0$. Also assume that $X$ is smooth,$Y$ smooth & irreducible and $f_*\mathcal{O}_X=\...
- 6,038
4
votes
1
answer
109
views
$\ell$-adic analogue of Kedlaya-Mochizuki
There is a well-known analogy between holonomic $\mathcal{D}$-modules on complex algebraic varieties and $\ell$-adic perverse sheaves on varieties over finite fields. Many theorems in one setting have ...
- 761
0
votes
1
answer
17
views
relationships between two stochastic matrices
Consider two $n \times n$ stochastic matrices $A$ and $B$. If for any two probability vectors $x$, $y$ in $R^n$, we have $xA=yA$ implies $xB=yB$, what can we say about the relationship of $A$ and $B$?
- 1
0
votes
1
answer
79
views
Curious about methods for finding Goldbach pairs for large even numbers
I am exploring the question of efficiently identifying two prime numbers that sum to a given large even number, particularly for even numbers exceeding 100 digits. While brute force and precomputed ...
- 1
0
votes
0
answers
13
views
Action of point stabilizers in finite doubly transitive groups
Suppose that $(H,X)$ is a finite faithful doubly transitive permutation group (where $H$ acts on the set $X$). Moreover, suppose that $H$ also acts doubly transitively (and faithfully) on a set $Y$, ...
- 4,547
0
votes
1
answer
107
views
Can we construct an isomorphism between $\mathrm{BS}(1,n)$ and $\mathbb{Z}[1/n]\rtimes\mathbb{Z}$ such that it preserve the order?
It is given in Regular left-orders on groups that the solvable Baumslag-Solitar group $\mathrm{BS}(1,n)=\langle a, b\mid aba^{-1}=b^n\rangle $ is isomorphic to $\mathbb{Z}[1/n]\rtimes \mathbb{Z}$ for ...
1
vote
0
answers
13
views
References for Hilbert Space Structure and Density of Smooth Functions in Weighted Sobolev Spaces on $ \mathbb{R} $
I am looking for references and materials that discuss the following aspects of weighted Sobolev spaces $ W^{k,2}_\rho(\mathbb{R}) $ defined on the entire real line $ \mathbb{R} $:
Hilbert Space ...
- 131
1
vote
0
answers
37
views
Generalization of Connes metric on state space
Let we have a spectral triples $(A,H,D)$
The Connes distance on the space of states of $A$ is the following:
$$d(\phi,\psi)=sup_{ |[D,a]|\leq 1} |\phi(a)-\psi(a)|\quad (*)$$
Is this metric ...
- 366
5
votes
1
answer
74
views
Do the order statistics give a good approximation of uniform random variables?
Let $\{X_i\}_{i \geq 1}$ be a sequence of iid uniform random variables on $[0, 1]$. Define, for each $n$, the order statistic $O_n$ of $X_n$ by
$$O_n := \frac{1}{n}\#\{1 \leq k \leq n \, \, | \, X_k \...
- 6,313
0
votes
0
answers
57
views
Copy and repeat or copy and sum integer coefficients
Let
$$
\ell(n) = \left\lfloor\log_2 n\right\rfloor.
$$
Let $T(n,k)$ be an integer coefficients with row length $f(n)$ (number of zeros in the binary expansion of $n$ plus $2$ for $n>0$ with $f(0)=1$...
- 4,938
5
votes
2
answers
253
views
Asymptotics for minimum of a sequence of random variables
This is a question that I'm sure has been investigated before, but I have found no good search terms for.
Let $X_i$ be a sequence of random variables, independent and uniformly distributed in $[0,1]$. ...
- 28.2k
-1
votes
0
answers
27
views
Tightest decreasing majorant
I had asked this question here but received no answer.
Let $O$ be an operator that maps sequences to sequences such that the elements of the sequence $O(a)$ are given by
$$\bigl(O(a)\bigr)_n ~{}={}~ \...
- 349
1
vote
0
answers
46
views
Computing with the Picard group of non-integral curves
Let $C$ be a curve over $\mathbb{Q}_p$ and let $\mathcal{C}$ be a regular model of $C$ over $\mathbb{Z}_p$, with $\mathcal{J}$ the Neron model of the Jacobian of $C$. Raynaud's theorem asserts that $\...
- 111
-2
votes
0
answers
81
views
Can this prime pyramid reveal deeper insights into prime distribution? Has someone seen this pattern before?
Definition
The Burz Prime-Number Pyramid is a triangular arrangement of consecutive integers, structured such that the length of each row corresponds to a prime number, except the first row which ...
- 1
3
votes
0
answers
47
views
Semisimple elements and fixed points
The following statement seems to be well-known:
Let $X$ be a variety on which an affine algebraic group $H$ acts with
finitely many orbits and let $s \in H$ be semisimple. Then $H_s = \{h
\in H \mid ...
- 53
0
votes
0
answers
47
views
Translation Invariants of Polynomials
The function $f(k)$ is a (numerical) polynomial in $\mathbb{Q}[k]$, and the set
$ S_f = \{ f_d : d \in \mathbb{N} \} $
is a set associated with $ f $, where $f_d(k)=f(k+d)$.
I am interested in finding ...
- 89
30
votes
1
answer
3k
views
Closed formula for the factorial over naturals
How to prove that there is no formula for $n!$ that does only use the binary operations $+,-,*,/$ on natural numbers, powers of natural numbers, and fixed natural numbers?
The same question over the ...
- 19k
-2
votes
0
answers
27
views
Convergence of $ \vec{x_{n+1}} = A^{-1} f(\vec{x_{n}},\vec{y_{n}}), \:\:\: \vec{y_{n+1}} = A^{-1} g(\vec{x_{n}},\vec{y_{n}}) $
I have two systems
$$ A \vec{x} = f(\vec{x}, \vec{y}), \:\:\: A \vec{y} = g(\vec{x}, \vec{y}) $$
Both have the same constant, square, invertible matrix $A$. I implemented an iterative algorithm with ...
- 97
7
votes
1
answer
300
views
Kobayashi-Nomizu "Foundations of differential geometry" on page 117 wrong?
$\DeclareMathOperator\GL{GL}$Let $M$ be a smooth manifold, $G$ a Lie group and $P(M,G)$ a principal $G$-bundle and $\rho: G \to \GL(V)$ of $G$ a representation with $V$ finite-dimensional $\mathbb{F}$-...
- 331
-1
votes
0
answers
42
views
How to prove the following theorem by distribution function and series
Let $g$ be nonnegative and measurable function in $\Omega$ and $\mu_{g}$ be its distribution function, i.e.,
$$
\mu_{g}(t)=\left|\left\{ x\in \Omega:g(x)>t\right\}\right|,\; t>0.
$$
Let $\eta>...
- 1
2
votes
0
answers
34
views
An algebraic group $G$ over $L^+$ such that $G_{\mathbb{R}}$ is compact for almost all embedings
Does there exists a reductive group $G$ of type $E_7$ over a given totally real field $L^+$ such that for every embedding $\tau:L^+\to \mathbb{R}$ except one , $G_{\tau,\mathbb{R}}(\mathbb{R})$ is ...
- 95
1
vote
0
answers
83
views
Specific regularity in bipartite graphs
Let $G(A,B)$ be a bipartite graph with $|A| = |B| = n$, where $n$ is sufficiently large. The average degree of $G$ is $d = \frac{e(A,B)}{n}$, where $e(A,B)$ denotes the number of edges between sets $A$...
- 359
5
votes
1
answer
79
views
Measure dependance of groupoid von Neumann algebra
Let $(G,\mu)$ be a measured groupoid and denote by $\nu,\nu^{-1}$ the measures on all of $G$ induced by $\mu$ and Haar system $\{\lambda^x\}$.
I have a question regarding the dependance of the ...
- 311
-1
votes
0
answers
18
views
What is the expected value of the set when N elements are chosen from the same probability distribution?
Suppose we have a parameter that follows some probability distribution $f(x)$. When simulating an $N$-body with that parameter as an attribute, how should values of the parameter be chosen?
Let each ...
- 1
6
votes
2
answers
391
views
closed form for an alternating cosecant sum
Is there any closed form for the following finite sum
$$\sum_{j=1}^{n-1}\frac{(-1)^j}{\sin (\frac{j\pi}{n})}$$
where $n$ is an even number?
Any comment or reference is welcome.
- 701
12
votes
0
answers
108
views
When could a diligent calculus student compute all Picard iterates algebraically?
As is well known, in the typical proof of the Picard–Lindelöf theorem, one shows the existence of a solution of the initial value problem $y'(t) = f(t,y(t))$, $y(t_0) = y_0$ by considering the Picard ...
- 13k
-1
votes
0
answers
41
views
Homomorphism from field of hyperreals to field of reals?
I am curious if it is possible to construct a homomorphism from a field of hyperreal numbers to the field of real numbers? (Similarly, a homomorphism from the surreals to the reals?)
Assuming that ...
- 55
16
votes
3
answers
1k
views
Is there a natural topology for sets of topological spaces?
The Gromov–Hausdorff metric makes a set of compact metric spaces into a metric space itself. I am wondering what some natural generalizations there are for arbitrary topological spaces. Namely, is ...
- 719
4
votes
1
answer
179
views
Projective automorphisms of a plane cubic curves
Let $E$ be a smooth cubic curve in $\mathbb{P}^2$ over an algebraically closed field $k$.
What is the group of the projective transformations preserving $E$ ?
In characteristic $0$ the answer is known ...
- 486
1
vote
0
answers
57
views
Non metrizable uniform spaces
Bourbaki's book on general topology states that a uniform space is metrizable if it is Hausdorff and the filter of entourages of the uniformity has a countable basis. However, he doesn't provide an ...
- 111
13
votes
4
answers
2k
views
The ten most fundamental topics in geometric group theory
What are the ten most fundamental topics in geometric group theory?
This is a pedagogical question prompted by the fact that I am teaching geometric group theory to undergraduates. They are expected ...
0
votes
2
answers
137
views
Is $1+\left (\frac{m}{m-1}\right )^{\frac{\log(p-1)}{\log(2)}}\cdot(p-1)^{\frac{\log(m)}{\log(2)}+1}<\frac{m}{m-1}p^{\frac{\log(m)}{\log(2)}+1}$?
For $p>2,m>2$, is $$1+\left (\frac{m}{m-1}\right )^{\frac{\log(p-1)}{\log(2)}}\cdot(p-1)^{\frac{\log(m)}{\log(2)}+1}<\frac{m}{m-1}p^{\frac{\log(m)}{\log(2)}+1}$$
?
Motivation:
I am trying to ...
- 3,074
4
votes
1
answer
710
views
Can the Pythagorean theorem be proved using imaginary numbers?
Can the Pythagorean theorem be proved using imaginary numbers? The proof must avoid circular reasoning, of course.
I asked essentially the same question at MSE, but did not receive a definitive answer,...
- 3,567
60
votes
72
answers
9k
views
When is 2 qualitatively different from 3?
I'd like to get a list of instances in mathematics where a problem with two parameters (or some parameter set to $2$) is qualitatively different from the instance of that problem with the value set to ...
5
votes
0
answers
57
views
Underlying noncommutative topologies of noncommutative complex varieties
Let $X$ be a (separated) complex algebraic variety. Then we can view its analytification $\newcommand\topo{\text{top}}X^{\topo}$ as a locally compact Hausdorff space. I wonder whether the same ...
- 2,856
7
votes
1
answer
216
views
Size doubling amoeba
Note: Here time is assumed to be discrete and indexed by the naturals $\mathbb N$.
A curious species of amoeba undergoes the following evolution rule - at each time step, with probability $0 < p &...
- 6,313
2
votes
0
answers
162
views
Is a triangulated category admitting a tilting object algebraic or even equivalent to the derived category of some ring?
Let $\mathcal{T}$ be a triangulated category having all infinite coproducts(such triangulated category is sometimes said to be cocomplete or satisfying the TR5 axiom). We call an object $G$ tilting if
...
- 21
0
votes
0
answers
45
views
Fractal dimension using wavelets [closed]
I'm trying to estimate the fractal dimension of a function.
I created the log(energies) Vs log(scales) plot and I'm computing the Fractal Dimension (D) from the slope using the relation
$$
\alpha = -...
2
votes
0
answers
97
views
Action of torus on Laurent polynomials
Let $F$ be a field and suppose that the torus $(F^*)^n$ acts on the Laurent polynomial ring $L$ in $n$ variables $X_1, \dots, X_n$ defined by $X_i \dashrightarrow a_iX_i$ for suitable scalars $a_i$.
...
- 376
0
votes
0
answers
100
views
Algebraic relations for $\Gamma$ function
Let $N$ and $n$ be positive integers with $\mathrm{GCD}(n,N)\ne1$. I want to prove the following claim:
$\Gamma\left(\frac nN\right)$, $\pi$ and the $\Gamma\left(\frac uN\right)$ ($u\in[1,N-1]$, $\...
- 3,998
0
votes
0
answers
72
views
Can a generalized root formula exist for polynomials with finite degrees? [closed]
Let $p\in\mathbb{Z}[x]$, set $d = \textbf{Deg}(p)$, and write $$p(x) = \sum_{i=0}^{d}{a_ix^i}$$ for some sequence $\{a_0,a_1,a_2,...., a_{d-1}\}$. Is there a mapping $\mathcal{F}$ so that $$\mathcal{F}...
9
votes
1
answer
328
views
An elementary proof of the equivalence of the Bol and Moufang identities
By a well-known result of Bol (1937) and Bruck (1946), for any loop the following two identities are equivalent:
B: $x(y(xz))=((xy)x)z$
M: $(xy)(zx)=(x(yz))x$.
A proof of the equivalence (B)$\...
- 42k
2
votes
0
answers
84
views
Is there a natural topology for subsets of a fixed topological space?
This question is an extension/clarification of the question: Is there a natural topology for sets of topological spaces?
The Hausdorff distance assigns a distance to any two subspaces $X, Y$ of a ...
- 719
-4
votes
1
answer
163
views
What are all the complex structures on $\mathbb{R}^2$ which live inside $\mathrm{SL}_2(\mathbb{Z})$? [closed]
By "complex structure" I am referring to 2x2 matrices which square to $-\mathrm{Id}_2$. I need to know those with integer entries and determinant equal to 1.
Thank you
- 3
2
votes
0
answers
34
views
Maximum number of connected components of surfaces in three dimensions, what is known?
Part of Hilbert's 16th problem is:
It seems to me that a thorough investigation of the relative positions of the upper bound for separate branches is of great interest, and similarly the ...
- 21
1
vote
0
answers
94
views
Examples of nontrivial morphism between simple bundles but not isomorphism
We know stable bundles have a good property:
If $f: E\longrightarrow E'$ is a nontrivial morphism where $E,E'$ have the same rank and degree, then $f$ must be an isomorphism.
I'm wondering does this ...
- 111
3
votes
3
answers
401
views
Huygens' principle or finite speed of propagation?
I am reading the paper Large global solutions for energy supercritical nonlinear wave equations on $\mathbb{R}^{3+1}$ by Krieger and Schlag and am confused by one of their steps.
For context, $v(t,r)$ ...
- 890