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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 ...
  • 99
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 ...
  • 41
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 ...
  • 11
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 ...
  • 51
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'\...
  • 207
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 ...
  • 343
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 vote
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{...
  • 343
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 ...
  • 250
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
0 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
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 $$p\nmid \left(\left(\dfrac{p-1}{3}\right)!\right)^3+1$$ ...
  • 3,644
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 ...
  • 23
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 ...
  • 551
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 ...
  • 911
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 ...
  • 2,675
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 ...
  • 5,885
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 ...
  • 95
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 $m$.
  • 3,644
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 \...
  • 888
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 $G$...
  • 53
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 ...
  • 61
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} ...
  • 73
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 - ...
  • 343
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 ...
  • 125
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 $\{ \...
  • 157
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 ...

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