## 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:

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:
$$gamma + alpha = operatorname {ord} left ( {0 } times gamma cup {1 } times alpha right) < operatorname {ord} left ( {0 } times gamma cup {1 } times beta right) = gamma + beta$$