Difference between theory of computational complexity and theory of complexity.

I read that the theory of complexity is the study of complex systems. Although I associate the theory of computational complexity with the classification of problems.

Theory of gr.group – Measured in cosets in a group?

This should be possible for any group, assuming I have data on the correct extent of the measures.

Leave $ mathcal {B} $ be the Boolean algebra generated by cosets of finite index subgroups $ G $. Then there is a finite additive (in fact $ G $-invariant) measure $ mu $ in $ G $ so that if $ C $ It is a coset of $ H $Y $[G:H]= n < infty $, so $ mu (C) = 1 / n $. Specifically, if $ A in mathcal {B} $ then there is a finite index subgroup $ H $ of $ G $, such that $ A $ is a union of cosets of $ H $. Yes $ n $ is the index, and $ m $ is the number of cosets in the union then it is established $ mu (A) = m / n $. You can directly verify that this is a well-defined measure as desired.

Now one must be able to extend $ mu $ arbitrarily to a finely additive measure in $ mathcal {P} (G) $ (which will not be $ G $-invariant necessarily). I think this follows from Section 457 of Fremlin's "Theory of Measure".

Observation 1: the initial measure $ mu $ it is in fact the only $ G $-invariant finite probability measure additive in $ mathcal {B} $, and can be constructed from Haar's measure in the profinite completion of $ G $. In particular, if $ mathcal {N} $ Denotes the collection of normal subgroups of finite index of $ G $, then the profinite ending is $ hat {G} = varprojlim _ { mathcal {N}} G / N $. We can write elements of $ hat {G} $ as $ (C_N) _ {N in mathcal {N}} $, where $ C_N $ It is a coset of $ N $. Given a set $ A $ in $ mathcal {B} $define $ X_A $ to be the set of $ (C_N) _ {N in mathcal {N}} in hat {G} $ such that $ C_N cap A neq emptyset $ for all $ N en mathcal {N} $. So $ X_A $ is closed and you can check that $ mu (A) $ Make the measure of $ X_A $.

Observation 2: Perhaps it is also worth mentioning that if $ G $ is susceptible (for example, abelian) then there is a $ G $-invariant finite probability measure additive in $ mathcal {P} (G) $, which must satisfy the desired conditions directly by finite additivity.

Database Theory – Limitations of the ** reservoir sampling algorithm ** in a DBMS

I'm trying to fully understand the idea of Reservoir sampling algorithm. The article in Wikipedia gives the following example:

If we have a sample size of $ s = 10 $, then the probability that the $ 9 ^ {th} $ The article that will be added to the sample is $ 1 $ as its still there is space. When the $ 11 ^ {th} $ The article arrives, the probability of adding and replacing an old article (random position) is $ frac {10} {11} $. Here the probability that an item in a specific position is replaced is $ frac {1} {10} $. Again when the $ 12 ^ {th} $ the article arrives, the probability of replacing an existing article is $ frac {10} {12} $. From this it is clear that with each additional article that comes, after the $ 10 ^ {th} $ article, the probability of it being added to the sample decreases, while the probability that the previous sample remains the same increases. From this I can assume that when the $ 1101 ^ {th} the article arrives, the probability that it will be added to the sample will be $ frac {10} {1101} = 0.0090826521 % $.

In another scenario if our sample size $ s = 100,000 $ and we are processing the $ 25,898,750 ^ {th} $ article, the probability that the new article will be added to the sample and replace an existing article at random will be $ frac {100,000} {25,898,750} = 0.0038611902 % $

First of all, I think that my understanding of the subject is correct but, in addition, this method is a viable solution when we have a very large sample size, say $ s = 10,000,000,000 $? What restrictions are imposed then?

ring theory – Rank of a module $ R $ $ M $ over $ R & # 39; $, where $ R $ is a projective algebra of $ R & # 39; $.

I'm stuck trying to solve Exercise 7.9.2 in Jacobson Basic Algebra II.[AlltheringswillbecommutativeJacobsonsaysamodule[AllringswillbecommutativeJacobsonsaysamodule[TodoslosanillosseránconmutativosJacobsondiceunmodulo[AllringswillbecommutativeJacobsonsaysamodule$ M $ finished $ R $ it has rank $ n $ yes for any ideal ideal $ P $, $ M_P $ It's free $ R_P $-range module $ n $].

State the following. Leave $ M $ be a projecting finger (finely generated) $ R $-module that has a range. Tell $ R $ is a $ R & # 39; $-algebra that is practical and projective as a module, and has a range (again as $ R & # 39; $-module). Then try $ M $ has a range over $ R & # 39; $, and that is equal to

$$ rk_ {R & # 39;} (M) = (rk_RM) (rk_ {R & # 39; R). $$

So far I can not even prove that $ M $ it is projective as $ R & # 39; $-module, for which I would greatly appreciate any help.

ct.category theory – Enough sets of colimits in small categories

Leave $ C $ Be a small category, and consider the kind of diagrams. $ G: D a C $, with $ D $ A small category, which has colimits in. $ C $. This is an appropriate class even when $ C $ It is very small, eg. when $ D $ has a terminal object $ t $, any functor $ G: D a C $ has a colimit $ G (t) $, and there is an appropriate class of small categories with a terminal object.

However, these colimits feel somewhat "trivial"; in some cases, at least, we can find a small set of diagrams that "carry all the non-trivial information" on the colimit diagrams in $ C $. For example, yes $ C $ it is a poset, then it is enough to consider injective functors $ G $ (and we can also take $ D $ to be discrete too), and these form an essentially small set. For a non-posetal $ C $ we can not restrict ourselves to injective functors, since co-products are not idempotent, but there may be some other restriction that works. Note that according to Freyd's theorem, a small non-posetal category has a limit on the cardinality of the coproducts it can admit; but this does not answer the question itself, since a particular colimit can exist even if the coproducts that would be necessary to build it from co-products and co-factors do not.

Here are two ways to ask the precise question:

  1. Given a small category $ C $, there is a small set $ L $ of diagrams $ G: D a C $ With colimits such that for any diagram. $ G & # 39 ;: D & # 39; a C $ with a colimit, there is a $ (D, G) in L $ and a final functor $ F: D to D & # 39; $ such that $ G = G & # 39; circ F $?

  2. Given a small category $ C $, there is a small set $ L $ of diagrams $ G: D a C $ with colimits such that if a functor $ H: C a E $ Keep the colimits of all the diagrams in. $ L $, then it keeps all the colimits that exist in $ C $?

Any solution to question 1 is also a solution to question 2, but I'm not sure if the opposite is true. The mention of Freyd's previous theorem suggests that a solution might require a classical logic; I would find it more surprising if such a set existed for a small complete non-posetal category, although I do not immediately see an argument that I can not.

Of course, you can also ask similar questions for rich categories, internal categories, $ infty $-categories, and so on. Bonus points go to a response that applies more generally in such contexts.

Graph theory – Definition of a cut

This is going to be obvious, but the correction of an exercise made me doubt about the definition of the capacity of a cut in a flow:

Exercise

I thought that the capacity of a cut was the sum over the incoming and outgoing edges of the capacity, where we multiply the capacity of the incoming edges by $ -1 $ and outgoing by $ 1 $. So this would produce $ 1 cdot 2-1 cdot1 + 1 cdot 2 $ for this cut = 3.

I must be doing something wrong …

nt.number theory – Extending prime numbers digit by digit while maintaining primality

I looked at a table of prime numbers and observed the following:

If we choose $ 7 $ Can we concatenate a digit to the left to form a new prime number? If concatenate $ 1 $ to get $ 17 $. We can do the same with $ 17 $? If concatenate $ 6 $ to get $ 617 $. And with $ 617 $? If concatenate $ 2 $ to get $ 2617 $. Then we can form $ 62617 $. And I could not continue since the table gives cousins ​​with the last entry. $ 104729 $.

Now a little terminology. Call a prime number $ a_1 … a_k $ a survivor of order $ m $ if they exist $ m $ digits $ b_1, …, b_m $ (all different from zero) so that the numbers $ b_1a_1 … a_k $ Y $ b_2b_1a_1..a_k $ and and $ b_mb_ {m-1} … b_1a_1 … a_k $ They are all prime numbers.

Call a prime number $ a_1 … a_k $ a survivor of order $ + infty $ Yes $ a_1 … a_k $ is a survivor of order $ m $ for each $ m en mathbb N $.

I would like to know:

There is survivor of order $ + infty $?

(This question, with exactly the same title and content, was formulated in the MSE about an hour ago, and I think I should apologize for having asked the same question here and there in such a short time, but, as I thought someone will come up very quickly with an argument about what question will be decided, and that did not happen, I decided to ask it here as well, so that this question also gets attention here, yes, it has a recreational flavor, but I hope you like it.)

Numerical theory nt. Euler characteristics in the range one of the cases.

Suppose $ E $ is an elliptic curve over a numerical field with a good ordinary reduction in primes over a fixed odd cousin $ p $. We are interested in Iwasawa's theory of cyclotomics. $ mathbb {Z} _p $ extension under the additional assumption that $ E $ It has a point of infinite order and also that $ L (E, s) $ has a simple zero in $ s = 1 $.

Is there a definition of a modified Euler characteristic for the Selmer group about the cyclotomic extension that can be related to the derivative? $ L & # 39; (E, 1) $ In the framework of the BSD conjecture? Can someone give me some precise references?

nt.number theory – Explicit bivariate quadratic polynomials in which Coppersmith is better than the standard solver?

http://www.numbertheory.org/pdfs/general_quadratic_solution.pdf provides a general method for solving the divariate bivariate quadratic equation while Coppersmith introduced a method to solve bivariate polynomials that work demonstrably and have been shown to break $ RSA $ system if half of the low significant bits of either $ P $ or $ Q $ They are known.

The equation that comes out is $$ (2 ^ ku + v) (2 ^ ku & # 39; + v & # 39;) = PQ $$ where if we assume $ v $ It is known. So $ vv & # 39; equiv PQ bmod 2 ^ k $ gives $ v & # 39; $.

So we have a quadratic diophantine equation. $$ 2 ^ kuu & # 39; + (uv & # 39; + u & # 39; v) = frac {PQ-vv & # 39;} {2 ^ k}. $$

Why do I need the Coppersmith method to solve this? Can not a regular Diophantine solver work here, and therefore there are explicit polynomials in which Coppersmith is better than the standard solver in a bivariate quadratic case?

Theory of elementary sets – Ordnance sum inequality – Test verification

Motto Given three ordinals $ alpha $, $ beta $Y $ gamma $, so
$$
alpha < beta to gamma + alpha < gamma + beta.
$$

Test
Dice $ alpha < beta $, so $ alpha subsetneq beta $, $ alpha $ is an adequate initial segment of $ beta $A) Yes
begin {meets *}
{0 } times gamma cup {1 } times alpha subsetneq
{0 } times gamma cup {1 } times beta,
end {meets *}

and the l.h.s. is an initial segment of r.h.s., so the claim is as follows:
$$
gamma + alpha
= operatorname {ord} left ( {0 } times gamma cup {1 } times alpha right) <
operatorname {ord} left ( {0 } times gamma cup {1 } times beta right)
= gamma + beta
$$