# All Questions

150,590
questions

0
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### Comparing the Stacks Project Homotopy limit with limits in the $\infty$-category

In the Stacks project Tag 08TC, there is a definition of a homotopy limit in a derived category, and I expect it to compare with a limit in the $\infty$-categorical enhancement. I guess this is also ...

0
votes

0
answers

22
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### Is $\sum\limits_{k=1}^nk^i=S_3(n)\times\frac{P_{i-3}(n)}{N_i}$ for odd $i>1,\sum\limits_{k=1}^nk^i=S_2(n)\times\frac{P_{i-2}'(n)}{N_i}$ for even $i$?

I asked this question here
When I was in high school, I was fascinated by $\displaystyle\sum\limits_{k=1}^n k= \frac{n(n+1)}{2}$ so I tried to find the general solution for $\displaystyle\sum\...

-4
votes

0
answers

21
views

### Examiner si vous savez ça sinon laisser [closed]

Bonjour, SMIC ou Solution Maths Institute Clay
19+19=38
38+38=76
380+380=760
20×38=760
2×38=76
38=2×20
10X=11
10111=23
23=10123
1)10.868673767=(2.713+2.718)×2
2)10.868673767=2.713+2.718+2.718281828+2....

1
vote

0
answers

64
views

### Show me that I have not simplified the proof of the Adian-Rabin theorem

I am not a mathematics researcher but I am concerned that this question, posed with slightly different wording on math.stackexchange, may be too esoteric for that forum since it concerns the details ...

0
votes

0
answers

20
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### Estimate for the operator $A A_D^{-1}$

Let $O\subset\mathbb{R}^d$
be a bounded domain of the class $C^{1,1}$
(or $C^2$
for simplicity). Let the operator $A_D$
be formally given by the differential expression $A=-\mathrm{div}g(x)\nabla$
...

-1
votes

1
answer

63
views

### Why do we define independence for zero-probability events?

I am learning about probability and the definition of pairwise independence is given as $P(AB) = P(A)P(B)$. My textbook motivates this definition as one to capture the intuition where the knowledge of ...

3
votes

1
answer

111
views

### Psychological test for Euclidean geometry

There is the so-called FCI test.
It contains a list of questions such that anyone who can speak will have an opinion.
Based on the answers one can determine if the person knows elementary mechanics.
I ...

0
votes

0
answers

46
views

### Commutative/ symmetric second covariant derivative

Consider a smooth manifold $M$ together with an affine connection (or covariant derivative) $\nabla$ on the tangent bundle $TM$.
Is it possible to have an affine connection, possibly with non-zero ...

-1
votes

0
answers

49
views

### Possibility of upper bounding $\|f\|_{L^2}$ by the weighted $L^2$ norm of its Laplace transform

Assume that $f$ is a smooth and bounded function defined on $[0,\infty)$ such that its mean value is zero (i.e., $\int_{\mathbb{R}_+} xf(x) \mathrm{d}x = 0$). We denote by $$\hat{f}(\xi) = \int_{\...

0
votes

0
answers

24
views

### Connection of eigenspace of finite Hilbert matrix and its continuous operator counterpart

I am trying to understand the connection between the eigenspace of the continuous operator
$$
H(x,y) = \frac{1}{x+y}
$$
which is nothing but the square of the Laplace operator, and its discrete ...

-5
votes

0
answers

38
views

### short exact sequence of groups [closed]

We have a short exact sequence as
$$0 \rightarrow \mathbb{Z}_2\rightarrow G \rightarrow \mathbb{Z}_2\rightarrow 0,$$
can we conclude that the group $G$ is isomorphic to $\mathbb{Z}_2 + \mathbb{Z}_2$ ...

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111
views

### Does there exist an $L$-function for any subset of $\mathbb{N}$?

Consider the following prime sum:
\begin{aligned}
\sum _{p}{\frac {\cos(x\log p)}{p^{1/2}}}
\end{aligned}
whose spikes appear at the Riemann $\zeta$ zeros as shown here.
Taking these detected spikes (...

1
vote

1
answer

88
views

### Two spectral sequences arising from a simplicial spectrum

Let $X_\bullet$ be a simplicial spectrum, and let $X = |X_\bullet|$ be the geometric realization.
Let's assume each $X_n$ is connective.
From this situation, we can form two filtrations on $X$: the ...

0
votes

0
answers

21
views

### Lattice not contained in any connected subgroup is not contained in any positive dimensional subgroup

Let $ G $ be a simple Lie group and let $ \Gamma $ be a lattice in $ G $. If $ \Gamma $ is not contained in any connected subgroup of $ G $ does that imply that $ \Gamma $ is not contained in any ...

1
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0
answers

63
views

### How does the behaviour of a hyperderived functor of many variables change if you use $\prod$-totalisation instead of $\oplus$-totalisation?

$\newcommand{\tot}{\operatorname{Tot}}\newcommand{\A}{\mathscr{A}}\newcommand{\L}{\mathbb{L}}\newcommand{\R}{\mathbb{R}}$Say $T$ is is a functor $\A_1\times\A_2\times\cdots\times\A_n\to\A$ of Abelian ...

3
votes

0
answers

132
views

### What exactly is a Tannakian subcategory?

I've searched all the standard references (Deligne--Milne, Saavedra-Rivano) and cannot find a definition of Tannakian subcategory. What I find is many authors who discuss the Tannakian subcategory ...

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votes

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answers

160
views

### Is the integer factorization into prime numbers normally distributed?

Let $P_1(n) := 1$ if $n=1$ and $\max_{q|n, \text{ }q\text{ prime}} q$ otherwise, denote the largest prime divisor of $n$.
Let us define some rooted trees $T_{n,m}$ for $1 \le m \le n$ by:
$T_{n,m}$ ...

2
votes

1
answer

87
views

### Does $L^1$ boundedness and convergence in probability imply convergence in probability of the Cesaro sums?

Let $X_n$ be a sequence of random variables with uniformly bounded $L^1$ norm. Suppose $X_n$ converges in probability to $X \in L^1$.
Is it true that the Cesaro sums $Y_n := \frac{1}{n} \sum_{i = 1}^n ...

3
votes

0
answers

46
views

### $R$-recursion for the A249833 (similar to A235129)

Let $a(n)$ be A249833 i.e. an integer sequence such that its exponential generating function $A(x)$ satisfies
$$
A(x) = 1 + \int A(x) + (A(x))^2\log A(x)\,dx
$$
The sequence begins with
$$
1, 1, 2, 7, ...

-4
votes

1
answer

54
views

### I need help in solving the integral [closed]

It looks like this: sqrt(1+24cos²(3π*x/4))*dx. I am interested in the approach to solving this task and similar ones. Or you can suggest a formula that should be used to solve it.

0
votes

0
answers

44
views

### Is monotonicity redundant in this definition of Tarskian logics?

Given a logic over a language $L$, which has a consequence relation $\vdash$. This logic is Tarskian if for every $\Gamma \cup \Delta \cup {\alpha} \subseteq L$:
If $\alpha \in \Gamma$, then $\Gamma \...

2
votes

0
answers

77
views

### Trace class operators

There is a notion of trace class operator in a Hilbert space.
Is there a notion of trace class operator in arbitrary Banach space? locally convex space?
A reference will be helpful.

-3
votes

0
answers

57
views

### Permutations history [closed]

I am writing an article about permutations groups and I would like to introduce something regarding permutation's and transposition's history and their applications. I searched a lot but I am not 100% ...

4
votes

0
answers

100
views

### About a formula in Lawrence-Venkatesh's proof on Mordell conjecture

In Lawrence-Venkatesh, the lemma 2.10 states that
For number fields $L/K$, and a representation $\rho:G_L\to GL_n(\mathbb{Q}_p)$ that is crystalline at all primes above $p$ and pure of weight $w$, ...

-1
votes

0
answers

59
views

### Reversing heat transfer with respect to time

Fact: One can easily compute heat dispersion in a plane using the heat equation.
Question: Has any research been done on computing the process in the reverse time direction?
That is, given a heat map $...

1
vote

0
answers

116
views

### Infinity stacks

I was going through some notions of stacks and higher stacks on nLab. $\infty$-stacks are usually $(\infty,1)$-sheaves which take values in $\infty$-groupoids. Now to recall, $(\infty,1)$-sheaf is a ...

2
votes

0
answers

100
views

### Computing the Dieudonné module of $\mu_p$ from Fontaine's Witt Covector

In Groupes $p$-divisibles sur les corps locaux, Fontaine introduced a uniform construction of Dieudonné modules through the definition of the Witt covector. Consider a perfect field $k$ of ...

2
votes

0
answers

39
views

### From exact triangles in the stable category of maximal Cohen--Macaulay modules to short exact sequences

Let $R$ be a local Gorenstein ring. Let $\underline{\text{CM}}(R)$ be the stable category of maximal Cohen--Macaulay modules, it is known to carry a triangulated structure. My question is: If $M\to N\...

-1
votes

0
answers

62
views

### Weak derivative of $|u|^s$ $(s>1)$ [migrated]

I often come across instances in texts where people calculate the weak derivative of $|u|^s$ for $s>1$ as $s|u|^{s-1} \operatorname{sign}(u) \partial_x u$ for some $u\in W^{1,s}(\Omega)$.
However, ...

-2
votes

0
answers

39
views

### Difference between linear discriminant analysis and Fisher discriminant analysis

I've always thought that Fisher discriminant analysis (FDA) was simply a multi-class generalization of linear discriminant analysis (LDA). But a recent paper I was reading states that the methods are ...

0
votes

0
answers

113
views

### Solve NP-hard type problems with linear programming

I would like to know if there is any way to solve an NP-hard type problem, for example, the TSP, sum of subsets or knapsack problem, by using linear programming and not by brute force.
I ask this ...

4
votes

0
answers

153
views

### Is the Taylor map continuous?

(Skip to the bolded theorem below for my question, if you'd like)
Some context on asymptotic expansions and the Taylor map
In the setting of irregular singularities of meromorphic connections on the ...

-4
votes

0
answers

47
views

### How to get a general formula for the roots of x^(x-1)-n=0 [closed]

Let n be a real number, and let n≥1.
How can we get a general solution to x^(x-1)=n

5
votes

2
answers

198
views

### How much Dependent Choice is provable in $Z_2$? And what about Projective Determinacy?

So, second order arithmetic, $Z_2$, is capable of proving quite a few things. One thing which would be of use is dependent choice for $\mathbb{R}$.
Basically, dependent choice on $\mathbb{R}$ says ...

4
votes

0
answers

99
views

### Has anyone studied factoring as a CO-product?

In factorization, like integer factorization, you start with an integer and end up with a kind-of list of pairs of other elements, namely the factors.
I want to explore the "Co-ness" of this....

0
votes

0
answers

172
views

### On fifth powers forming a Sidon set

We call a set of natural numbers $\mathcal S$ to be a Sidon Set if $a+b=c+d$ for $a,b,c,d\in \mathcal S$ implies $\{a,b\}=\{c,d\}$. In other words, all pairwise sums are distinct.
Erdős conjectured ...

-3
votes

0
answers

109
views

### Extended dominated convergence theorem in Kallenbergs book [closed]

In the book of Olav Kallenberg Foundations of Modern Probability there is stated a extenden version of the dominated convergence theorem. Its proof and the statement goes exactly:
Theorem 1.23. (...

17
votes

1
answer

989
views

### Is this an instance of the snake lemma?

I recently had need of the following fact (in the category of abelian groups, but I'm pretty sure it holds for all abelian categories): given a commutative diagram of the form
(quiver link), thus $k \...

2
votes

0
answers

77
views

### A property of the Riemannian metric on the symmetric space $X_\textsf{d}=\textsf{SL}(d,\Bbb R)/\textsf{SO}(d)$

Consider the symmetric space $X_\textsf{d}=\textsf{SL}(d,\Bbb R)/\textsf{SO}(d)$. For any $M\in X_d$, by transporting the Killing form on $T_I(X_d)$ (the space of symmetric matrices with trace zero) ...

3
votes

0
answers

72
views

### Smallest dominating set

Given a graph $G$, we say $S$ is a dominating set if $S\cup \{N(x):x\in S\}=V(G)$. Let $d(n,k)$ be the smallest integer $s$ so that every $n$-vertex graph $G$ with minimum degree $k$ has some ...

0
votes

0
answers

34
views

### Smoothness of solutions to wave equation in a bounded domain

Consider the wave equation
\begin{equation}
\partial_t^2 u - \sum \partial^2_{x_i} u =0
\end{equation}
in a bounded domain $M$ with $C^\infty$ boundary, and the boundary conditons
\begin{equation}
u(...

0
votes

1
answer

89
views

### The sequence has a stationary accumulation point

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be a smooth (continuously differentiable), convex function with a non-empty set of minimizers and $\{x^k\}$ be a sequence such that
(a) $\{x^k\}$ has an ...

1
vote

0
answers

67
views

### On the square of the Ornstein-Uhlenbeck process

The generator of the Ornstein-Uhlenbeck process is $\Delta-x.\nabla$. I'm looking to understand the Markov process whose generator is $(\Delta-x.\nabla)^2$. What is such a process called? Is there any ...

0
votes

0
answers

51
views

### Ask for intuitive explanation of relation between entropy and Kolmogorov complexity

The following is Theorem 8.1.2 from introduction to Kolmogorov Complexity and it's Application:
Let $P$ be a computable probability mass function and $H(P) = \infty$. There
exists a sequence of ...

-1
votes

0
answers

88
views

### Maximizing a function involving minimums with variable parameters

I am working on a problem where I have a set of $m$ data points, each represented as a triplet $(a_i, b_i, c_i)$ for $ i = 1, 2, \dots, m $. I am interested in a target function defined as:
$f(t) = 2a ...

0
votes

0
answers

56
views

### Is the solution of the nonlinear elliptic PDE unique for a given energy?

It is well-known that the following semilinear problem
$$ -\Delta u= f(x, u)\text{ in } \Omega,~ u=0 \text{ on } \partial\Omega$$
on a smooth bounded domain $\Omega$ admits infinitely many $H_0^1(\...

-3
votes

0
answers

149
views

### A Conjecture on R(3,k) [closed]

R(3,3)=6，each point has 2 red edges
R(3,5)=14，each point has 4 red edges
R(3,9)=36，each point has 8 red edges
I guess R(3,2^n+1)=2^n+3^n+1.The Critical circulant Ramsey graphs of R(3,2^n+1), where ...

0
votes

0
answers

98
views

### What is the logic behind the Extended Euclidean Algorithm procedure? [closed]

Thank you beforehand for reading my question.
In the terms that I'd want to understand the Extended version of the Euclidean Algorithm, I understand the Euclidean Algorithm as follows:
You find the ...

-1
votes

0
answers

40
views

### An example of a geodesic in the Wasserstein space $(\mathcal P_2 (\mathbb R^d), W_2)$ which does not have a constant speed

Let $(\mathcal P_2 (\mathbb R^d), W_2)$ be the Wasserstein space of Borel probability measures on $\mathbb R^d$ with finite second moment. We fix $\mu, \nu \in \mathcal P_2 (\mathbb R^d)$ and let $\...

-1
votes

1
answer

89
views

### Overview resources for (rigorous) critical phenomena

I recently came across this overview which discusses some results in the theory of critical phenomena. It is already quite old and I would like to know if there are other (more recent) overviews in ...