Most active questions
215 questions from the last 7 days
0
votes
2
answers
140
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
2
votes
1
answer
111
views
Second order differential equation with non constant coefficient
Is it possible to solve the differential equation for $y(t)$ the following ODE?
$$
y^{\prime \prime}(t)+ \frac{f^{\prime}(t)}{2 f(t)} y^{\prime}(t) + k^{2} y(t) = 0
$$
It can also be rewritten as
$$
\...
- 45
2
votes
1
answer
109
views
Neumann problem for the Laplacian with Dirac delta functions
I have encountered a problem while dealing with the adjoint method in potential flow that is also described, in a similar fashion, in (eq. 39) of this paper. The problem is essentially this:
$$\begin{...
- 21
8
votes
0
answers
255
views
+300
Maps with small fibers between manifolds of equal dimension
The following question is an attempt to revise this one into what I intended.
Important revisions are shown in bold.
Are there any known examples of a compact Riemannian manifold $M$ with (possibly ...
- 501
8
votes
0
answers
206
views
Can $\mathbb{C}$ have a "doppelganger" in $L(\mathbb{R})$ with countable automorphism group?
Working in $\mathsf{ZFC}$ + large cardinals (a proper class of Woodins, to be precise), is there a field $F\in L(\mathbb{R})$ such that $V\models F\cong\mathbb{C}$ and $L(\mathbb{R})\models\vert\...
- 20.5k
3
votes
1
answer
163
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
3
votes
1
answer
158
views
How to maximize the variance of a subset of integers?
$\DeclareMathOperator{\Var}{Var}$Given the set of numbers $\Omega := \{1, \ldots, n\}, n \in \mathbb{Z}^+$, how can I choose a subset, $A$ of $\Omega$ , such that $\min(\Var(A), \Var(\Omega \setminus ...
- 41
4
votes
0
answers
190
views
Cell structure on the function space $\operatorname{Hom}(X,Y)$
By the Theorem of Milnor in his paper "On spaces having the homotopy type of a CW-complex", the function space $\operatorname{Hom}(X,Y)$ (with the compact-open topology) is homotopy ...
- 150
-4
votes
0
answers
212
views
Can a mathematics research paper have just propositions & corollaries? [closed]
I am writing a research paper, in which I am proving some properties of new convolution operation $\star$ for some transform, like linearity, associativity, commutativity,distributivity, shift ...
3
votes
1
answer
180
views
Can one say that there are equal numbers of sets satisfying formulas in Second Order Arithmetic?
Is there a way of saying in second order arithmetic that the number of sets $X$ such that $\phi$ equals the number of sets $X$ such that $\psi$, where $\phi$ and $\psi$ are formulas with $X$ free, and ...
- 2,555
3
votes
0
answers
174
views
What are non-archimedean norms on $\mathbb{R}$, whose restriction to $\mathbb{Q}$ is trivial?
I wonder if there is any classification result on non-archimedean norms on $\mathbb{R}$, with trivial restriction to $\mathbb{Q}$? Any references or examples would be welcomed!
Some examples of such ...
- 241
0
votes
1
answer
70
views
Is there a characterization of monoids that distribute over each other?
Let $(M, e_1, \times_1, e_2, \times_2)$ be an algebraic structure such that
$(M, e_1, \times_1)$ and $(M, e_2, \times_2)$ are monoids
$x \times_1 (y \times_2 z) = (x \times_1 y) \times_2 (x \times_1 ...
- 631
3
votes
1
answer
76
views
Tangential Sobolev spaces
Let $Ω⊂R^n$ be a smooth domain, define $U_s=\{x∈Ω | d(x,∂Ω)<s\}$; let $f∈W^{1,p}(Ω)∩W_{\mathrm{loc}} ^{2,p}(Ω)$; let $v$ be the unit normal to $Ω$; consider $v$ to be smooth with bounded ...
4
votes
1
answer
89
views
Invariant theory for unitary groups $\mathcal{U}(n)$
I'm trying to understand the invariant theory of the unitary groups $\mathcal{U}(n)$ on tensor powers of their standard representations $V^{\otimes p} \otimes (V^*)^{\otimes q}$. Let $\mathcal{U}(n)$ ...
- 1,124
4
votes
0
answers
159
views
Are the natural powers of two conservatively embedded in $\mathbb{C}$?
This is a followup to this question.
Consider $\mathbb{C}$ as a structure - in the sense of first-order logic - with the graphs of addition and multiplication. Let $\mathcal{X}$ be the substructure ...
- 20.5k
3
votes
1
answer
142
views
Forcing equivalence and equal generic extensions
Two forcing notions $\Bbb P$ and $\Bbb Q$ could be defined to be forcing equivalent if the associated complete Boolean algebras are isomorphic (so, the CBA's formed by considering the regular opens of ...
- 393
2
votes
0
answers
173
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
1
answer
89
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
3
votes
1
answer
88
views
How irregular can the set of points of non-differentiability for an L1 function's primitive F get, before the FTC fails?
A Fundamental Theorem of Calculus for Lebesgue Integration, J. J. Koliha begins with the passage
Lebesgue proved a number of remarkable results on the relation between integration and differentiation....
- 833
3
votes
0
answers
140
views
A closed formula for a sum involving hypergeometric functions
Can we find a closed formula for this sum:
$$\sum_{p,q\geq 0} (p+q+1)r^{p+q} \frac{{}_1F_1(1+p;2+p+q;r^2)}{{}_1F_1(1+p;2+p+q;1)}$$
where
$$_1F_1(a;c;z) = \sum_{n=0}^{\infty} \frac{(a)_n}{(c)_n n!} z^...
- 531
1
vote
0
answers
136
views
Measurability of a map involving probability measures
Let $X$ be a metrizable topological space and $\mathscr B_X$ the Borel $\sigma$-algebra on it. Let $\Delta X$ denote the set of probability measures on $(X,\mathscr B_X)$, and let $\mathscr B_{\Delta ...
- 405
4
votes
0
answers
150
views
Lemma in Roth's Theorem for Primes
I am reading Ben Green's paper Roth's Theorem in the Primes and I don't follow the proof of Lemma 6.1. I am not sure where the fact there are no more than $n^{3/4}$ elements $x\in A_0$ with $x\leq n^{...
- 271
0
votes
1
answer
71
views
Formula for $P(d) = \sum_{\sum_{i=1}^m k_i = d, k_i \in \mathbb{N}_+} \left( \prod_{i=1}^m k_i^{a_i} \right) $
Let $\mathbb{N}_+ = \{ 1, 2, \dots\} $. For a given sequence of elements $\{a_i \}_{1 \leq i \leq m} $in $ \mathbb{N}_+ $, we define
\begin{equation}
P(d) = \sum_{\sum_{i=1}^m k_i = d, k_i \in \...
- 89
0
votes
0
answers
134
views
A system of nonlinear Diophantine equations whose positive solutions are not coprime
Consider the following system of Diophantine equations:
$$v_1k_1=k_1^3-k_2^3+k_3^3 \\
v_2k_2=k_1^3+k_2^3-k_3^3 \\
v_3k_3=-k_1^3+k_2^3+k_3^3 \tag{1}$$
where $v_1,v_2,v_3$ and $k_1,k_2,k_3$ are integer-...
- 303
5
votes
0
answers
123
views
+50
Dimension of the intersection of the commuting variety with a particular subspace
Let $\mathcal C$ denote the commuting variety of pairs of matrices in $M_n(\mathbb{C})$, defined as:
$$
\mathcal C = \{ (A, B) \in M_n(\mathbb{C})^2 \mid [A, B] = 0 \}.
$$
It is well known that $\...
- 269
3
votes
1
answer
140
views
+50
Order of $\mathbb{E}[ \max_i |x_i + z_i| - \max_i |z_i|]$
Let $z_1, \dots, z_n$ be iid standard Normal, and let $x \in \mathbb{R}^n$. Put $\|u\|_\infty = \max_i |u_i|$.
Define
$$
F(x) = \mathbb{E}\Big[\|x + z\|_\infty - \|z\|_\infty\Big]
$$
If $\|x\|_\infty \...
- 380
2
votes
0
answers
124
views
Derived tensor products and regular sequences
Let $R \to A$ be a homomorphism of commutative rings, and let $x\in R$ be an element (or a sequence of elements in $R$, if you prefer) that is both $R$-regular and $A$-regular. Then we have
$$
A\...
- 125
3
votes
0
answers
127
views
Orbit space of the action of $\mathrm{GL}(V)$ on the Grassmannian of $V\wedge V$
$
\newcommand{\K}{\mathbb{K}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\N}{\mathbb{N}}
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\Grass}{Grass}
$Consider $\K\in\{\R,...
- 126
7
votes
0
answers
148
views
Cardinal characteristics and $\mathfrak{c} < \aleph_\omega$
Let $\mathsf{R}$ denote some finitely many relations about finitely many cardinal characteristics (e.g. $\mathfrak{a} \leq \mathfrak{s}$, $\mathfrak{a} < \mathfrak{d} = \mathfrak{r}$, $\mathfrak{b} ...
- 1,442
-1
votes
0
answers
115
views
Stability of flow map
$\DeclareMathOperator\Diff{Diff}$Setting:
Let $(M,g)$ be a compact and connected $C^{\infty}$-Riemannian manifold. Let $d_g$ denote the induced shorted path metric and equip $C^{\infty}(M)$ with the ...
- 5,405
0
votes
1
answer
119
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 ...
7
votes
0
answers
132
views
On the optimal strength of Goodstein's theorem
Goodstein's theorem is a famous example of an arithmetical statement that is unprovable in $\mathsf{PA}$ but provable in a stronger theory. It is well-known that Goodstein's theorem implies the ...
- 3,042
1
vote
1
answer
85
views
Question on gamma matrices
Let $(M,g)$ be a pseudo-Riemannian spin manifold and let us denote by $S$ the spinor bundle, i.e. the associated vector bundle with respect to the spin representation. Usually, the "gamma ...
- 1,171
0
votes
0
answers
116
views
How near are a groupoid and its 'preorderification'?
As remarks, a groupoid is a category with only (categorical) isomorphisms as its morphisms and a preorder is a category only having one morphism between each object. If we choose one isomorphism by ...
2
votes
0
answers
116
views
Uncertainty principle: minimize $\int_{-\infty}^\infty |t| |\widehat{f}(t)|^2 dt$ for $f$ of compact support
This is a question of uncertainty-principle type stemming from Eigenvalue of a convolution and a restriction?
Let $f:\mathbb{R}\to \mathbb{R}$ be even, absolutely continuous and supported in $[-\frac{...
- 20.2k
12
votes
0
answers
112
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
2
votes
0
answers
112
views
Action of torus on Laurent polynomials
Let $F$ be an algebraically closed 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 ...
- 376
3
votes
0
answers
134
views
Galois cohomology and Levi subgroups
Let $F$ a field and $G$ a smooth connected reductive group with a Levi subgroup $M$. Under what assumptions is $H^1(F, M) \to H^1(F, G)$ injective? In the case $F$ is nonarchimedean local I believe ...
- 605
7
votes
0
answers
117
views
Langlands correspondence of coverings of $\mathrm{SL}_2(\mathbb R)$ and modular forms with fractional weights
$\DeclareMathOperator\SL{SL}$Let $G \to \SL_2(\mathbb R)$ be a finite covering of degree $d \geq 2$. Then $G$ is a connected Lie group with semisimple Lie algebra $\mathfrak{g}=\mathfrak{sl}_2$ and ...
- 6,622
-2
votes
1
answer
61
views
On the correspondence between infinitesimal and integral description of connections
It is the title of an article by Petko Nikolov Triste Sissa 1981. I cannot access this pdf yet I remember that it was once avaliable on libgen and now I cannot find it. Please help.
- 19
-3
votes
0
answers
157
views
A presentation for the group $GL(n,\mathbb{Z}_p)$
Let $n\ge 2$. Let $p$ be a prime and $\mathbb{Z}_p$ denote the finite field with $p$ elements.
I want to know about the presentation for the group $GL(n,\mathbb{Z}_p)$ consisting of its generators and ...
- 103
2
votes
0
answers
99
views
+100
Inequalities for norm of centered Gaussian and uncentered Gaussian
Let $g$ denote a standard Gaussian vector in $\mathbb{R}^n$, and $\|\cdot\|$ a norm.
Let $x \in \mathbb{R}^n$ and define
$$
F(x) = \mathbb{E}[\|x + g\| - \|g\|].
$$
I am wondering if it is possible to ...
- 380
1
vote
0
answers
102
views
Curious congruences modulo $4$ involving primes
We define
$$S(n)=\sum_{a=2+(n\pmod 2)}^{n-2}
\sharp(\{j,1\leq j<n \pmod{a},(a,j)=1\})\ .$$
(Searching the OEIS yielded no results.)
For $n>2$ we have the following experimental observations (...
- 17.9k
2
votes
0
answers
134
views
Effective Bombieri-Lang conjecture
The Bombieri-Lang conjecture is the following well-known conjecture:
Let $X$ be a projective variety defined over a number field $K$. Suppose that $X$ is general type. Then $X(K)$, the set of $K$-...
- 26.9k
1
vote
1
answer
80
views
A contradiction involving HoTT function extensionality in higher inductive types
I'm having difficulty resolving what seems to be a contradiction between two lemmas in the HoTT Book and am seeking help to determine where I'm going wrong. Below, all types may be higher inductive ...
- 13
2
votes
1
answer
103
views
Number of disjoint set triplets in a union-closed family
Let $\mathcal{F}$ be a family of $n$ finite sets. In this case, the family can be regarded as a multiset, since it is allowed to contain multiple instances of the same set. Let $U(\mathcal{F})$ be the ...
- 1,000
3
votes
0
answers
102
views
Understanding the Lie derivative by multivector fields
For a vector field $X$ on a manifold there are two ways to define a Lie derivative: an algebraic one using Cartan's formula $\mathcal{L}_X \alpha = i_X d \alpha + d i_X \alpha$ and a dynamical one ...
- 161
2
votes
0
answers
93
views
Connectedness of equivariant Hilbert schemes of points of affine spaces (or as orbifolds)?
Let $G$ be an abelian finite group act on $\mathbb C^n$, when the equivariant Hilbert scheme $\mathrm{Hilb}^{R}(\mathbb C^n)^G=\mathrm{Hilb}^{R}([\mathbb C^n/G])$ is connected? Now $R$ is a ...
- 249
3
votes
0
answers
91
views
About BMO space on smooth open bounded domain
Let $\Omega$ be any open domain in $\Bbb R^d$.
Define the $\text{BMO}(\Omega)$ space as
$$ \text{BMO}(\Omega)= \big\{u\in L^1_{loc}(\Omega)\,\,:\,\, |u|_{\text{BMO}(\Omega)} <\infty \big\},
$$
...
- 1,101
0
votes
0
answers
103
views
On the form of algebraic numbers belonging to a specific field extension
Let $m>1$ be an integer and set $\theta=10^{-1/m}$. For a $\gamma\in \mathbb{Q}(\theta)$, there exists $a_0,\ldots,a_{m-1}\in \mathbb{Q}$ such that
$$
\gamma=a_0+a_1\theta+\cdots+a_{m-1}\theta^{m-1}...
- 515