Group theory gr: how do reflection groups relate to general point groups?

I always tried to understand how finite reflection groups from $ Bbb R ^ d $ (of some fixed dimension $ d $) relate to the point groups of the same space $ smash { Bbb R ^ d} $ (finite subgroup of the orthogonal group $ smash { mathrm O ( Bbb R ^ d)} $)

Initially, I had the impression that each group of points is a subgroup of a finite reflection group. This turned out to be incorrect, which is obvious in retrospect. Many reflection groups have symmetries in the placement of their mirrors that can be used to enlarge the group.

So let's take these expanded groups instead. From my geometric understanding, by that I mean the symmetry groups of the uniform polytopes. Then i will call you uniform dot groups. Most (or all?) Of uniform polytopes can be generated from a reflection group, and then have all the symmetries of this group, but could have more.

Question: Is each group of points a subgroup of a uniform group of points?

With regard to the answer to that question, I am open to any statement that sheds light on the location of reflection groups (or groups easily derived from them) within the family of general point groups.

nt.number theory – In $ sum _ { substack {p leq x \ p, p + 2 text {twin primes}}} frac (( log p) ^ m} {p} $, in the assumption of the first Hardy-Littlewood conjecture

I was wondering, inspired by a result of (1) (Proposition 17) what the asymptotic behavior of the sequence should be, assuming the first Hardy-Littlewood conjecture,

$$ sum _ { substack { text {primes} p leq x \ text {such that} p + 2 text {is prime}}} frac { log ^ m p} {p} $$

how $ x to infty $, where $ m geq 1 $ denotes a fixed integer. So here $ p $ denotes the smallest of twin cousins ​​(sequence A001359 of OEIS) and we assume that there is a First Hardy-Littlewood conjecture.

A reference to the first Hardy-Littlewood conjecture is this section of Wikipedia.

I don't know if this exercise is in the literature for any fixed integer $ m $. I would like to know the deduction for some integer $ m geq 1 $.

Question. Deduce by some integer $ m geq 1 $ And under the assumption that Hardy-Littlewood's first guess is true, what should be the asymptotic behavior of $$ sum _ { substack {p leq x \ p, p + 2 text {twin cousins}}} frac { log ^ m p} {p} $$
how $ x to infty $. If you are in the literature, feel free to refer the reference and try to find and read the result of the literature. Thank you.

References

(1) Christian Axler, About a family of functions defined on sums of cousins, Journal of Integer Sequences, Volume 22 (2019), Number 1, Article 19.5.7.

number theory: solve two variables for each n related to the Collatz conjecture

For this code, for each x I would like to solve all ranges of values ​​for c1 and c2 in a bounded range, that is, c1 and c2 in the range of real numbers + -100 for c1 and c2 for each x, which combined give "Length (stepsForEachN) == nRangeToCheck – 1". Here is the code so far, I'm not sure how to solve the two variables c1 and c2 for each x:

(*stepsForEachN output is A006577={1,7,2,5,8,16,3,19} if c1=c2=1*)
c1 = 1; 
c2 = 1;
nRangeToCheck = 10;
stepsForEachNwithIndex = {};
stepsForEachN = {};
stepsForEachNIndex = {};
maxStepsToCheck = 10000;

c1ValuesForEachN = {};

For(x = 2, x <= nRangeToCheck, x++,

 n = x;

 For(i = 1, i <= maxStepsToCheck, i++,
  If(EvenQ(n), n = Floor((n/2)*c1),
   If(OddQ(n), n = Floor((3*n + 1)*c2))
   );

  If(n < 1.9,
   AppendTo(stepsForEachN, i);
   AppendTo(stepsForEachNIndex, x);
   AppendTo(stepsForEachNwithIndex, {x, i});
   i = maxStepsToCheck + 1
   )
  )
 )
Length(stepsForEachN) == nRangeToCheck - 1

graph theory: is this a characterization of a tournament score?

For all $ n in mathbb N $, $ Delta_n = (0,2,3, …, n) $,$ D_n = X + X ^ 2 + …. X ^ $ Y $ I_n = mathbb N cap (0, n) $

For all
$ f in mathbb N ^ {I_n} $ strictly growing, we say that $ P = Sigma_ {i in I_n} a_i.X ^ {s_i} en mathbb Z (X) $ is 1-equivalent to $ P = Sigma_ {i in I_n} a_i.X ^ {$} Y 2 equivalents to $ P = Sigma_ {i in I_n} to _ { sigma (i)}. X ^ s_i + D_ {s_n} $ For any $ sigma $ permutation of $ I_ {s_n} $

$ Z subset mathbb Z (X) $ is the kind of $ 0 $ according to him $ 3-equivalence $ Witch is defined as the equivalence ratio generated by these two superior equivalence relationships.

We say that $ P = Sigma_ {i in I_n} a_i.X ^ {i} en mathbb Z (X) $,have braquetting property yes for every positive integer $ k leq n $
we have

$ Sigma_ {i in I_k} a_i geq 0 $ Y $ Sigma_ {i in I_n} a_i = 0 $

We say he has the property score If any $ Q $ that is to say $ 3 equivalent $ to $ P $ have the braquetting property

One can show that the elements of $ Z $ have the property score, and the question is the reciprocal:

Is any $ P in mathbb Z (X) $ which has the property of punctuation, in Z?

Note that $ (a_0, a_1 + 1, a_2 + 2, …, a_n + n) $ it is the list of exits of a complete digraph (tournament) if and only if $ a_0 + a_1X + a_2X ^ 2 + …. + a_nX ^ n $ have the property score

nt.number theory – Fourier transform of $ I_Y $, $ Y = { text {numbers with many prime factors} } $

Leave $ Y $ be the set of integers $ N <n leq 2 N $ with more than $ D log log N $ prime factors We can consider, say, $ D = ( log log N) ^ {1- epsilon} $.

We have fairly accurate approximations for the size of $ Y $ (I know chapter II.6 of the Tenenbaum book and the references therein). I wonder what work is available in the Fourier transformation $ widehat {1_Y} $ of the characteristic function of $ Y $.


I would wait $ widehat {1_Y} $ have spikes in the main arches (i.e. arcs around rational $ a / q $ with small denominator). This is because $ Y $ obviously it is "biased towards divisibility," so it should be overrepresented in the congruence class $ 0 $ mod $ d $ For any $ d $, in relation to other congruence classes mod $ d $.
A calculation at the end of the envelope suggests that the value at the peak around $ a / q $ must be approximately proportional to $ c ^ { omega (q)} / q $. But what is known?

set theory – Infinite generalizations of HOD

Let's say a set is ordinal$ _ { kappa lambda} $-definable if it is definable by a formula in the infinite language $ mathcal {L} _ { kappa lambda} $ with parameters of $ mathsf {On} $. Leave $ HOD _ { kappa lambda} $ be the class of hereditary ordinal$ _ { kappa lambda} $-definable sets. Definitely $ HOD _ omega omega} = HOD $; for a simple argument given in the quotation below, $ HOD _ { infty infty} = V $.

What can we say about $ HOD _ { kappa lambda} $ for other values ​​of $ kappa $ Y $ lambda $? Some questions that obviously arise: do we have $ mathsf {Con} (HOD _ { kappa ^ + omega} neq HOD _ { kappa omega}) $ for all $ kappa $? We have $ mathsf {Con} (HOD _ { kappa ^ + lambda} neq HOD _ { kappa lambda}) $ for all $ kappa, lambda $? We have $ mathsf {Con} (HOD _ { kappa lambda ^ +} neq HOD _ { kappa lambda}) $ for all $ kappa, lambda $?

(This question is inspired by the discussion in the Gödel Constructible Universe in Infinite Logic (A possible approach to the problem of HOD)).

test verification – Basic exercise on distribution theory

$ textbf {Exercise.} $ Allow $ u in mathfrak {D} & # 39; ( Omega) $ Y $ varphi in mathfrak {D} $ such that $ varphi | _ { text {supp} u} equiv 0 $. It is true that $ varphi u equiv 0 $?

It seems obviously true that $ varphi u equiv 0 $, but I would like to prove my intent because this exercise seems extremely simple to be an exercise. If I'm wrong, I'd like a clue to build a counterexample because I can't see how $ varphi u neq 0 $ in these hypotheses

What I thought is quite simple:

Case $ 1 $$ x in text {supp} u $:

$ ( varphi u) (x) = varphi (x) u (x) = 0 u (x) = 0 $.

Case $ 2 $$ x notin text {supp} u $:

$ ( varphi u) (x) = varphi (x) u (x) = varphi (x) 0 = 0 $,

Thus $ varphi u equiv 0 $. $ square $

complexity theory: is this equivalent to some famous problem of complete NP?

Consider a rectangular $ n times n $ array consisting of values $ x_ {ij} in {- 1, 1 } $. We want to minimize the following expression:

$$ sum_ {i, j} a_ {ij} x_ {ij} – sum_ {neighbors} x_ {ij} x_ {kl} $$

Where $ a_ {ij} $ the values ​​(positive or negative) are input parameters and adding to the neighbors means that we only include vertices that are directly adjacent to each other, that is, $ k = i ± 1; j = l $ or $ k = i; j = l pm 1 $.

My question is: is this problem equivalent to any NP-complete problem "famous"? I could find a series of similar minimization questions for MAXCUT type tasks or for formulations of Ising models in physics, but it is never in particular.

Edit to clarify: by minimization, I meant finding an optimal set $ x_ {ij} $ (output) given a certain set of $ a_ {ij} $ (entry).

field theory: how to combine symmetric tensile expressions into a single expression in xAct

Let's say I have the following expression in xAct
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where covariant derivatives are commutative, $ epsilon $ Y $ h $ they are symmetric tensioners

As we can see, this expression is equivalent to

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If we simulate it with respect to the indices $ i, j, k, l $

Is there any way to express the first expression as the second with some kind of SymmetryGroup?

Theory of complexity: algorithm to find a simple route with a maximum weight less than a constant in DAG

Given a weighted directed acyclic graph $ G = (V, E, W) $, where the weights are at the vertices. I am looking for a simple route of maximum total weight, but this total weight should not exceed a given constant $ K $.

Maybe my question is elementary, but I can't find any solution. In fact, it is well known that finding a simple path with maximum weight in $ G $ It is polynomial, but by adding the fact that this total weight should not exceed a given constant $ K $Will the problem remain polynomial? because we need to maintain in each node the set of route lengths that can reach the next vertices.