# All Questions

139,132
questions

0
votes

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55
views

### Can an upper bound for $r_{0}(n)$ be reached from a duality principle about the distinct primes $n$ "defines"?

Under Goldbach's conjecture, denote by $r_{0}(n)$ the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are prime and by $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n-r_{0}(n))$, so that $k_{0}(n)$ ...

0
votes

0
answers

233
views

### How to prove that the function $\lambda \mapsto \langle T_{\lambda}f; g\rangle$ is holomorphic?

Let $\langle;\rangle$ be the usual scalar product on $L^2(\Bbb R^2)$.
How to prove that the function $\lambda \mapsto \langle T_{\lambda}f; g\rangle$ is holomorphic on $\Bbb C^+_*=\{z\in\Bbb C:\text{...

7
votes

0
answers

124
views

+200

### Does determinacy imply unravellability for the Borel sets (over a weak base theory)?

As far as I know, the only way we currently know how to prove Borel determinacy in $\mathsf{ZFC}$ is to go through unravelability (a rather technical property whose definition can be found in Martin's ...

2
votes

0
answers

88
views

### Evaluate action of $f(\frac{d}{dx})$ using the Fourier/Laplace transform

Consider a function $f(x)$ that is numerically defined in $-1 \leq x \leq 1$ interval (assume $N$ samples). I am trying to compute the action of $f(d/dx)$ on a function $g(x)$ using the Fourier ...

2
votes

0
answers

32
views

### Extending $G$-closed sets to permutation bases of a permutation $RG$-module

I'm curious if there are any papers or results about the following scenario:
Let $R$ be a commutative ring (I'm interested in particular in the $R = \mathbb{Z}$ case, but fields are okay too), $G$ a ...

0
votes

0
answers

41
views

### Problem in understanding Theorem $6.2.9$ from Chari and Pressley

The theorem I am referring to here says that if we start with a Lie bialgebra $\mathfrak g$ determined by some skew-symmetric element $r \in \mathfrak g \otimes \mathfrak g$ satisfying classical Yang-...

1
vote

1
answer

36
views

### elaboration on the equation of directional derivative that lead to steepest gradient descent [closed]

I am reading the book
Ian Goodfellow and Yoshua Bengio and Aaron Courville, Deep Learning, MIT Press, 2016.
I am reaching to the point about directional derivative. Given the $u$ as the unit vector ...

0
votes

0
answers

15
views

### Is $\phi(t)=\|P(w+td)-w\|_X/t$ nonincreasing if $X$ is "only" a uniformly smooth and uniformly convex reflexive Banach space?

For a Hilbert space $X$ it is known that the function $\phi(t)=\frac{1}{t}\|P(w+td)-w\|_X$ with $t>0$ is nonincreasing. Here, $P:X\to C$ denotes the projection operator and $w \in C, d \in X$ are ...

1
vote

1
answer

71
views

### Are finite-dimensional real representations of semisimple real Lie algebras completely reducible?

Suppose $\mathfrak{g}$ is a real form of a semisimple Lie algebra $\mathfrak{g}_\mathbb{C} = \mathfrak{g} \otimes_\mathbb{R} \mathbb{C}$. Then we have the following:
There is an equivalence of ...

1
vote

0
answers

54
views

### irreducible subfactor inclusion and commutativity of induced projections

Let $N\subset M$ be an irreducible subfactor inclusion, i.e., $N'\cap M =\mathbb C1$, acting on a Hilbert space $H$.
Let $\Omega\in H$.
Does it follow that the projections onto $[N\Omega]$ and $[M'\...

5
votes

1
answer

613
views

### Can a smooth manifold be realised as the image of a smooth function?

Consider, $M$, a smooth $m$ dimensional submanifold of $\mathbf R^n$. Does there exist a smooth map $X: \mathbf{R}^m\to\mathbf R^n$ such that $M=X(\mathbf R^m)$?
$X$ may have points at which the ...

1
vote

0
answers

29
views

### Around similar inequalities than an inequality due to Nicolas, that involve products of consecutive Ramanujan primes

This is cross-posted (and this post is a version to ask just around the veracity of Conjecture 1) as the post with identifier 3594907 and same title), that I've edited on Mathematics Stack Exchange ...

1
vote

0
answers

148
views

### Projective scheme over the integers

Let $X$ be a projective scheme over $Spec(\mathbb{Z})$. Let $X_{p}$ be the reduction at $p$ of $X$. If for any prime $p$, $X_{p}$ is normal, can we deduce $X$ is normal? Or any counterexamples?

0
votes

1
answer

39
views

### Solution to non-autonomous delay differential equation?

If you define a special function called the Lambert W function, you can explicitly solve the classic delay differential equation $x'(t) = x(t - a)$ by supposing the solution is some $\exp(\lambda t)$ ...

1
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0
answers

55
views

### Diffeomorphism induced by small perturbation

Consider the surface $S_{\epsilon}$ defined as:
\begin{align}
%S&=\{\vec x \in \mathbf{R}^3: x=0\}, \\
S_{\epsilon}&=\{\vec x \in \mathbf{R}^3:\epsilon (x^2 + y^2 + z^2 - 1) + x=0\}.
\end{...

4
votes

1
answer

60
views

### Searching for cofinal subsets of directed sets subject to finite constraints

Let $(P,\leq)$ be a directed set with uncountable cofinality. For every element $p\in P$, we are given a finite set $c_p\subset P\smallsetminus \{p\}$ of "incompatible elements". We say that ...

4
votes

0
answers

58
views

### Counting the number of free bases of $F_n$ with elements of bounded length

Let $F$ be a free group of finite rank and fix a free generating set $X$ of $F$. Let $P_r$ denote the set of all free generating sets of $F$ whose elements have length bounded by $r$ (when considered ...

1
vote

0
answers

82
views

### Non vanishing of a cohomology class associated to a nef vector bundle

Lemma. Let $E$ be a rank $r$ nef vector bundle over a polarized smooth complex projective variety $(X,H)$ of dimension $n\leq r$. Then for any $t\in\mathbb{R}_{\geq0}$:
$$
\sum_{k=0}^nt^{n-k}\int_Xc_k(...

4
votes

0
answers

111
views

### Surface with $\Omega_X$ globally generated and singular Albanese image

This question is inspired by abx's comment to my previous question MO430933.
Let $X$ be a complex surface of general type, and denote by $$a \colon X \to \operatorname{Alb}(X)$$ the Albanese map of $X$...

-3
votes

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answers

30
views

### How to decide a threshold so that I can make small off-diagonal values of a diagonally dominant matrix to zero [closed]

I have a matrix C which is diagonally dominant. Most of the off diagonal values of C are of the order of 1e-15 or smaller. I need to replace these small values by zeros with some limiting threshold. ...

1
vote

0
answers

23
views

### Associating a matroid to a uniform hypergraph

For a fixed ground set $[n]=\{1,\ldots,n\}$, and for any matroid $M$ on $[n]$, specified as a collection of bases $B_M$, the corresponding matroid basis polytope $P_M$ is defined to be the convex hull ...

0
votes

0
answers

67
views

### find all $q$ such $p\mid\left(\left(\dfrac{p-1}{q}\right)!\right)^q+1？$

let$p$ be prime number, following is well known
$$p|\left(\left(\dfrac{p-1}{2}\right)!\right)^2+1$$
proof:link1
and this post have prove link2
$$p\nmid \left(\left(\dfrac{p-1}{3}\right)!\right)^3+1$$
...

4
votes

1
answer

71
views

### Extending a metric in a bi-Lipschitz way

Suppose we are in the following situation: $(X,d)$ is a metric space and $Y$ is a subspace of $X$. Furthermore we have a different metric $\delta$ defined on $Y$ such that $\delta$ is bi Lipschitz ...

1
vote

0
answers

22
views

### Non-vanishing of generalized minors on T-stable unipotent subgroups

Let $G$ be a complex simply connected algebraic group, $T$ a maximal torus of $G$ and $B$, $B^-$ Borel subgroups which are opposite with respect to $T$ and let $U$ (resp. $U^-$) be the unipotent ...

5
votes

1
answer

250
views

### Day and Lack's "Limits of small functors": Lemma 2.3

I've been trying to understand the (4 line!) proof of Lemma 2.3 of Limits of small functors, on small functors into copresheaf categories $\mathbf{Set}^\mathcal C$. To me it seems to be using that the ...

5
votes

0
answers

123
views

### Is it true that the $\mathbb{F}_p$-rank of a linear combination of matrices is usually not smaller than its $\mathbb{Q}$-rank?

Consider fixed $3 \times 3$ integer matrices $A_1,A_2,A_3$ and the $\sim H^3$ linear combination matrices $A(\mathbf{h})=h_1A_1+h_2A_2+h_3A_3$ where $h_1,h_2,h_3$ are integers with $\vert h_i\vert \le ...

2
votes

1
answer

68
views

### Pair of laminations that fill on a closed surface

Let $S$ be a hyperbolic surface of genus $g \geq 2$.
A discrete geodesic lamination on $S$ is a set of disjoint, simple, closed geodesics.
Let $L_{1}$ and $L_{2}$ be two discrete geodesic laminations ...

0
votes

0
answers

39
views

### A decision problem of an inverse problem in finite group theory

A finite group $G$ is called integral if there is a finite group $H$ such that $G\cong H'$.
In Araujo, Cameron, Casolo, Matucci's paper, integrals of groups, they tried to solve a problem as following:...

0
votes

0
answers

26
views

### Eigenvalues of orthogonal group element

Let $q$ be a quadratic form over a nonarchimedean local field $F$, and let $\operatorname{O}(q)$ be the corresponding orthogonal group. Let $g\in\operatorname{O}(q)$ be semisimple.
Can we know ...

0
votes

0
answers

30
views

### Intersection of certain subsets in a split connected reductive group $G$ related to affine open cover of $G/B$

Let $k$ be a field of characteristic zero and $G$ a split connected reductive group over $k$. Moreover, let $T$ be a split maximal torus of $G$ and $B\supset T$ a Borel subgroup. Additionally, we ...

2
votes

0
answers

44
views

### Explicit estimates on summability kernels

A "summability kernel" is a sequence of functions $k_n:[0,1)\to \mathbb C$ such that
$$ \int_0^1 k_n(t) \mathrm d t =1,$$
$$ \int_0^1 |k_n(t)| \mathrm d t =O(1),$$ with an implied constant ...

0
votes

0
answers

116
views

### Relation between $3$-term Plücker relations and more than $3$-term Plücker relations

$\DeclareMathOperator\Gr{Gr}$Let $\Gr(k,n)$ be the Grassmannian variety of $k$-planes in an $n$-dimensional vector space. The coordinate algebra $\mathbb{C}[\Gr(k,n)]$ is generated by Plücker ...

0
votes

0
answers

51
views

### Is the element in the connected component?

I posted this question at stack exchange, got two upvotes but no answer. If it doesn't belong here, please let me know.
In the algebraic group $G$ = PGL$_{8}$($\mathbb{C}$), there are two involutions $...

3
votes

1
answer

299
views

### Trigonometric Diophantine equation

Is there a general method to solve the equation $P(x_1,x_2,...,x_n)=0$ with $P$ is a polynomial in $n$ variables with integer coefficients and $x_k=\cos(q_k\pi)$ with $q_k$ is a rational number?
This ...

-3
votes

0
answers

45
views

### Is there a closed form of the net present value formula? [closed]

The net present value formula is as follows:
$$
npv(q, n, r)=\sum_{i=0}^{n}q\left(1-r\right)^i
$$
where $n$ is the number of periods, $q$ is the payment in each period, and $r$ is future discount ...

0
votes

0
answers

40
views

### How to prove this inequality for the norm $ \|\cdot\|_{1,\infty} $?

Let $ \{a_k\} $ is a positive sequence. For $ 0<p<\infty $, space $ L^{p,\infty} $ is defined by
$$
\left\{f:\|f\|_{p,\infty}=\inf\left\{C>0:a_f(\lambda)\leq C/\lambda^p\right\}\right\}
$$
...

0
votes

1
answer

49
views

### $\omega$-homogenous space which is not $\omega_1$-homogenous

Consider a metric space $(X,d)$ and let $\kappa$ be a cardinal. We say that $(X,d)$ is $\kappa$-homogenous, if every (surjective) isometry $h:X_1 \to X_2$ between subspaces of $(X,d)$ of size $< \...

4
votes

1
answer

75
views

### Is there any example of a Lindelöf space that has no Menger dense subspaces?

A space $X$ is said to be Menger if for each sequence $(\mathcal{U}_n)$ of open covers of $X$, there is a sequence $(\mathcal{V}_n)$ such that $\mathcal{V}_n$ is a finite subcollection of $\mathcal{U}...

1
vote

0
answers

151
views

+200

### Hodge theory of the AJS category (proof of Lusztig's conjectures in positive characteristic)

Recently, in this survey paper (https://arxiv.org/abs/1212.0791) Elias-Williamson describe a Hodge theoretic approach to the proof of Kazhdan-Lusztig conjectures; it is essentially equivalent to the ...

7
votes

1
answer

256
views

### Does the limit of $x_n$, defined by $x_{n+1}=1/(m+1-nx_n)$ exist?

Let $m$ be positive integer, and consider the recursion
$$x_{n+1}=\frac{1}{m+1-nx_n}.$$
Does the limit of $x_n$ exist?
I'm guessing the limit doesn't exists for any $m$.

4
votes

0
answers

139
views

+50

### $\Sigma_*$-product is not $\sigma$-countably compact

In Arhangel'skii's book "Topological function spaces" there is a part where the author uses that, if $\kappa>\omega$ is a cardinal number, then the space $$\Sigma_*(\kappa):=\left\{x\in \...

0
votes

0
answers

74
views

### Compact coadjoint orbits

The following statement is from the article Compact Coadjoint Orbits by John Rawnsley:
If $\mathcal{O}$ is a compact coadjoint orbit for the group $G$ then there is a closed normal subgroup $H$ of $G$...

6
votes

1
answer

141
views

### Square-root lattices: where do they appear?

As an experimental physicist working on crystallography I'm often dealing with the reconstruction of an object from intensity data that emerge from an imaging device. In mathematics the problem is ...

5
votes

1
answer

411
views

### Solve system of logical equations

I need a general method for solving systems of logical equations like:
$$
\begin{equation*}
\begin{cases}
c_{0} = a_{0} \land b_{0}\\\\
c_{1} = a_{0} \land b_{1} ⊕ a_{1} \land b_{0}\\\\
c_{2} ...

6
votes

1
answer

427
views

### Topology change induced by small perturbation

Consider the surface $S_{\epsilon}$ defined as:
\begin{align}
%S&=\{\vec x \in \mathbf{R}^3: x=0\}, \\
S_{\epsilon}&=\{\vec x \in \mathbf{R}^3:f_{\epsilon}(\vec x)\equiv\epsilon ((x^2 + y^2 - ...

2
votes

1
answer

367
views

### Arrows, furnished by Yoneda

What are some examples of 'important arrows' in a category that are significantly easier to define via fullness of the Yoneda embedding than in the base category?
The example that brought this to ...

4
votes

2
answers

371
views

### Length of a product of conjugates of an element in a free group

Let $G$ be a free group generated by a set $S$. For $g\in
G$, let $l(g)$ be the length of $g$ with respect to $S$.
Now for $a\in G$ and $g_1,\dotsc,g_n\in G$, let $$T=g_1^{-1}ag_1g_2^{-1}ag_2\dotsm ...

1
vote

0
answers

51
views

### A statement on completeness of complex exponentials

I'm currently reading a paper by Olevskii on almost integer translates: https://www.sciencedirect.com/science/article/pii/S0764444297878731
In this paper the author considers for a given sequence $\{ \...

1
vote

1
answer

72
views

### A 'natural' enumerable metric space with integral distances which is essentially the Euclidean space

It is easy to construct a metric space $E_d$ such that all points
of $E_d$ are at mutually integral distance and such that there is a map $\varphi$ from $E_d$ into the $d$-dimensional Euclidean space ...

1
vote

0
answers

40
views

### Uniform norm bounds for linear approximation of 1-Lipschitz functions

This problem seems like it should be quite easy/standard, but I've not found a solution written down anywhere.
Consider the set of 1-Lipschitz functions on the $[0,d]$ interval. Define the linear ...