Recently Active Questions
159,036 questions
11
votes
0
answers
820
views
Infinite-dimensional affine space in algebraic geometry and algebraic topology
In topology, there are of course many different infinite-dimensional topological vector spaces over $\mathbb R$ or $\mathbb C$. Luckily, in algebraic topology, one rarely needs to worry too much about ...
- 111
28
votes
4
answers
3k
views
Slick proof of Stirling's Formula?
In Upper Limit on the Central Binomial Coefficient, Noam Elkies and David Speyer have given a nice proof that the central binomial coefficient $\binom{2n}{n} \sim \frac{4^n}{\sqrt{\pi n}}$. This can ...
- 6,367
0
votes
0
answers
8
views
Longest TSP in the unitary disc
I have the unitary disc $D=\{(x,y) \in R^2: x^2 + y^2 \leq 1\} $, and an integer $n \geq 2$. I want to select $n$ points in $D$ to maximise the length shortest path that connects them all. In other ...
1
vote
0
answers
101
views
Specific regularity in bipartite graphs
Let $G(A,B)$ be a bipartite graph with $|A| = |B| = n$, where $n$ is sufficiently large. The edge density of $G$ is $d = \frac{e(A,B)}{n^2}$, where $e(A,B)$ denotes the number of edges between sets $A$...
- 359
5
votes
1
answer
240
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 ...
- 12.9k
6
votes
1
answer
353
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/...
- 12.9k
6
votes
1
answer
65
views
What is the consistency strength of $L(\mathbb{R})\models$ "$\omega_1$ has tree property"?
Motivated by this question. For us "$\kappa$ has tree property" means any tree that has underlying set $\kappa$, height $\kappa$ and levels $<\kappa$ has a branch.
The strength of $\...
- 949
7
votes
2
answers
422
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 $\mathbb{F}$-...
- 796
-1
votes
0
answers
54
views
Quantitative formula for number of invertible square matrices over a finite field? [closed]
Let $M_n(F)$ denote the set of $n\times n$ matrices for a value $n\in \mathbb{N}$ with components in a field $F$ with finite cardinality . Let $$I=\{A\in M_n(F):~~ \det(A) \neq 0 \}.$$ What is the ...
- 5,429
5
votes
3
answers
909
views
What are the criteria for an elementary function to be infinitely integrable in elementary functions?
What features of elementary functions define a class of functions whose consecutive indefinite integration also gives an elementary function?
Is there a way to check whether a given elementary ...
- 21
-6
votes
0
answers
49
views
Request for Peer Review and Guidance to Refine Mathematical Manuscript [closed]
Dear Colleagues and Enthusiasts,
I am seeking peer review and constructive feedback on my manuscript titled "From Chaos to Order: A Study of Balance Chaos Mathematics and Absolute Mathematical ...
4
votes
1
answer
129
views
Why do most eigenspaces of a Lie algebra automorphism have finitely many orbits?
I'm interested in understanding the following lemma, which Vogan states (Lemma 4.8) in his paper The Local Langlands Conjecture (omitting the "well-known" proof).
Suppose $G$ is a complex ...
- 12.9k
12
votes
2
answers
2k
views
Fold-and-cut problem in three dimensions
The fold-and-cut theory states that "Any shape with straight sides can be cut from a single (idealized) sheet of paper by folding it flat and making a single straight complete cut. Such shapes include ...
- 151k
0
votes
0
answers
16
views
Mean for product of random submatrix and its regularized inverse
I consider the following random circulant matrix
$$
H_w = F^* \mathrm{diag}(w_1, \dots w_p)F, \, w_i\overset{iid}{\sim}\Gamma(1,1),
$$
where $F$ is the matrix for discrete Fourier transform, $F^*$ ...
- 537
3
votes
0
answers
77
views
Unramified Galois cohomology
Let $k$ be a local field with absolute Galois group $\Gamma_k$, inertia subgroup $I_k \subset \Gamma_k$, and residue field $\mathbb{F}$. Let $M$ be a finite Galois module over $k$.
The unramified ...
- 21.4k
18
votes
3
answers
2k
views
When does a CW-complex of dimension 2 embed in $\Bbb R^4$?
Let $X$ be a finite CW-complex of dimension two having just one 0-cell
(+ finitely many 1-cells + finitely many 2-cells).
Is it true that X can be embedded in $\Bbb R^4$?
If true, is it due to ...
- 13.6k
0
votes
0
answers
55
views
Hom functor and Cohen-Macaulay modules
Let $A$ be a local Gorenstein $\mathbb{C}$-algebra (not necessarily regular). Let $M,N$ be maximal Cohen-Macaulay $A$-modules. Is Hom(M,N) a maximal Cohen-Macaulay A-module?
Note that I had asked this ...
- 1,040
4
votes
1
answer
38
views
Stability of stochastic differential equations
Consider an SDE of the form
$$dX^\mu_t = a(t, X^\mu_t) dt + \sigma(t, X^\mu_t) dB_t$$
with initial condition $X^\mu_0 \sim \mu$, where $\mu$ is some measure on $\mathbb{R}^d$.
I am searching for ...
- 1,914
11
votes
1
answer
461
views
Closed formula for the factorial over reals
How to prove that there is no formula for $n!$ that does only use the binary operations $+,-,*,/$ on real numbers, powers of real numbers, and fixed real numbers?
Similar question have been asked ...
- 13.1k
1
vote
0
answers
122
views
Publishing alone may be counterproductive?
I am a (hopefully) soon-to-be PhD graduate, and by the time I graduate I should have 4 papers, 3 of them with only me as the only author and another one in collaboration with other people. I recently ...
- 739
2
votes
0
answers
23
views
Avoiding class/unit group computation when computing $p$-Selmer groups
Let $K$ be a number field, $S$ be a finite set of places of $K$, and $K(S,p)$ be the $p$-Selmer group of the $S$-integers of $K$, that is the set of nonzero elements of $K$ modulo $p$-th powers whose ...
2
votes
0
answers
53
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 $a\in[0,1)$:
$$(a^2-1)Q''(...
- 890
1
vote
0
answers
43
views
Closed form of $\frac{1}{{\pi}^n}\sum_{k\in \mathbb{Z}^n}\prod_{j=1}^n\frac{1}{1+(m_{j}^{T}k)^2}$
Given vectors $m_{j}\in\mathbb{Z}^{n},M=(m_{1} \ldots m_{n}),\det(M)\not =0$. Is it possible to find a closed form of:
$$S=\frac{1}{{\pi}^n}\sum_{k\in \mathbb{Z}^n}\prod_{j=1}^n\frac{1}{1+(m_{j}^{T}k)^...
- 241
6
votes
1
answer
131
views
Obstruction theory for specializing perfect complexes?
I'm considering a problem around the moduli of perfect complexes.
Let's consider a $X$ smooth proper over Spec($R$), $R$ is a DVR, $char=0$, equipped with a perfect complex $\mathcal F$ on $X_{K(R)}$.
...
- 16.2k
11
votes
3
answers
1k
views
Quantitative and elementary proofs of the Prime Number Theorem
I would like to know two things: one, whether the best quantative bounds in the Prime Number Theorem are still basically those given by the Vinogradov-Korobov zero-free region? and two, whether there ...
- 1,356
5
votes
3
answers
366
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 $[0,1]$. ...
- 149k
6
votes
2
answers
130
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 ...
- 60.6k
2
votes
1
answer
86
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 $Y$, ...
- 22.5k
2
votes
0
answers
30
views
Is rank of the length spectrum of a closed negatively surface/manifold infinite?
Suppose that $(S,\mathfrak{g})$ is a closed negatively curved Riemannian surface =(or more generally a manifold). Negative curvature guarantees that the non-trivial conjugacy classes $\text{conj}(\...
- 109
1
vote
1
answer
55
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 $B$?
- 12.9k
1
vote
0
answers
63
views
Could this closed-form expression for the integral of the Riemann $\xi$-function along the critical line provide new insights?
The Riemann $\xi$ and $\Xi$-functions are respectively defined as:
\begin{align}
\xi(s) &= \frac{s\,(s-1)}{2}\, \pi^{-s/2} \,\Gamma\left(\frac{s}{2}\,\right) \zeta(s) \qquad s \in \mathbb{C} \\
\...
- 21
0
votes
0
answers
68
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
-1
votes
0
answers
42
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
$$\bigl(O(a)\bigr)_n ~{}={}~ \...
- 349
1
vote
0
answers
62
views
Why do we require that all successors model this formula?
I'm reading Fitting's Intuitionistic Logic, Model Theory and Forcing. This occurs in Chapter 7.15.
The aim is to prove that a certain intuitionistic model is an intuitionistic model of ZF. I ...
- 149
4
votes
0
answers
58
views
An algebraic group $G$ over $L^+$ such that $G_{\mathbb{R}}$ is compact for almost all embeddings
Does there exist a reductive group $G$ of type $E_7$ over a given totally real field $L^+$ such that for every embedding $\tau:L^+\to \mathbb{R}$ except one , $G_{\tau,\mathbb{R}}(\mathbb{R})$ is ...
- 12.9k
2
votes
0
answers
122
views
Action of torus on Laurent polynomials
Let $F$ be an algebraically closed field and suppose that the torus $(F^*)^n$ acts on the Laurent polynomial ring $L$ in $n$ variables $X_1, \dots, X_n$ defined by $X_i \dashrightarrow a_iX_i$ for ...
- 376
3
votes
1
answer
136
views
Estimating a solution to Euler-type ODE #2
This is a similar question to this but with a different ODE.
Let $f$ be a continuous function in $L^2([1,\infty)$ satisfying $\sup_{r\geq 1} r|f(r)| <\infty$. Let $\ell$ be a positive integer, $R&...
- 969
0
votes
0
answers
143
views
A system of nonlinear Diophantine equations whose positive solutions are not coprime
Consider the following system of Diophantine equations:
$$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}$$
where $v_1,v_2,v_3$ and $k_1,k_2,k_3$ are integer-...
- 303
2
votes
0
answers
193
views
Is a triangulated category admitting a tilting object algebraic or even equivalent to the derived category of some ring?
Let $\mathcal{T}$ be a triangulated category having all infinite coproducts(such triangulated category is sometimes said to be cocomplete or satisfying the TR5 axiom). We call an object $G$ tilting if
...
- 35.2k
0
votes
1
answer
104
views
Curious about methods for finding Goldbach pairs for large even numbers
I am exploring the question of efficiently identifying two prime numbers that sum to a given large even number, particularly for even numbers exceeding 100 digits. While brute force and precomputed ...
- 1
0
votes
1
answer
130
views
Can we construct an isomorphism between $\mathrm{BS}(1,n)$ and $\mathbb{Z}[1/n]\rtimes\mathbb{Z}$ such that it preserve the order?
It is given in Regular left-orders on groups that the solvable Baumslag-Solitar group $\mathrm{BS}(1,n)=\langle a, b\mid aba^{-1}=b^n\rangle $ is isomorphic to $\mathbb{Z}[1/n]\rtimes \mathbb{Z}$ for ...
- 1,829
5
votes
1
answer
147
views
Do the order statistics give a good approximation of uniform random variables?
Let $\{X_i\}_{i \geq 1}$ be a sequence of iid uniform random variables on $[0, 1]$. Define, for each $n$, the order statistic $O_n$ of $X_n$ by
$$O_n := \frac{1}{n}\#\{1 \leq k \leq n \, \, | \, X_k \...
- 410
6
votes
1
answer
343
views
If $t \to \lVert f(\cdot,t) \rVert_{L^2_x}^2$ is absolutely continuous, can we interchange the spatial integral and time derivative? (from MSE)
I originally posted this question on MSE. But it seems more nontrivial than expected, so I guess MO is a more appropriate place to ask.
I repeat the question for the sake of completeness:
Let $f(x,t) ...
CommunityBot
- 1
1
vote
0
answers
21
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
53
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
3
votes
1
answer
148
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 \...
- 7,559
1
vote
0
answers
69
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
7
votes
1
answer
237
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 &...
- 7,559
0
votes
0
answers
65
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
35
votes
8
answers
3k
views
Examples of integer sequences coincidences
For the time being, the OEIS website contains almost $300000$ sequences. Each of these sequences is the mark of a specific mathematical concept. Sometimes two (or more) distinct concepts have the ...
