## Model theory and logic of notions of programming methodology

Linear Programming corresponds to first order theory of reals with addition and order. What do notions such as semidefinite programming, second order cone programming and convex programming correspond to?

## database theory – Find candidate keys – What are the steps

I have these following functional dependencies I figured out:

``````DM -> RA
RDT -> AM
``````

I got with a software to calculate what the candidate keys were to this:

``````{R, D, T}
{A, D, T}
{M, D, T}
``````

But I don’t know HOW i should do this manually to figure out this. Not to use the actual software. What the steps are to solving this. First I thought I should do something like this to figure out the candidate keys:

``````DM+ = DMRA
RDT+ = RDTAM
``````

But from what I understand is that only the RDT+ is giving all the attributes for it to be a candidate key? I am so confused by this. How should I think to pick it out from these functional depedencies?

## probability theory – Show that : \$operatorname{Var}left(sum_{i=1}^{n} I_{A_{i}}right) leq c sum_{i=1}^{n} Pleft(A_{i}right) \$

Let $$A_1,dots,A_n$$ be a sequence of pairwise independent probability events, show that there exists $$c$$ such that :

$$operatorname{Var}left(sum_{i=1}^{n} I_{A_{i}}right) leq c sum_{i=1}^{n} Pleft(A_{i}right)$$

computing the LHS, we get :

$$operatorname{Var}left(sum_{i=1}^{n} I_{A_{i}}right) =sum_{i=1}^{n} P(A_i)+2left(sum_{ineq j}^{1dots n} P(A_i)P(A_j)right) – left(sum_{i=1}^{n} P(A_i)right)^2$$

$$=sum_{i=1}^{n} P(A_i) – sum_{i=1}^{n} P^2(A_i) =sum_{i=1}^{n} P(A_i)(1-P(A_i))$$

My question is : can $$c$$ be $$max_{i} (1-P(A_i))$$ ?

## group theory – Conjugacy classes of a set of subgroups of \$S_5\$?

Let $$G$$ be the symmetric group $$S_5$$ of permutations of five symbols. Consider the set $$S$$ of subgroups of $$G$$ that are isomorphic to the non-cyclic group of order $$4$$. Let us call two subgroups $$H$$ and $$K$$ belonging to $$S$$
as equivalent if they are conjugate (that is, there exists $$g in G$$ such that $$gHg^{-1}=K$$). How many equivalence classes are there in $$S$$?

What I know: I know that a non-cyclic group of order $$4$$ in $$S_5$$ is $$K={e, (1 2), (3 4), (1 2)(3 4)}$$. Now all the subgroups of order $$4$$ which are not generated by $$4$$-cycles are isomorphic to $$K$$. How to think further?

## gr.group theory – When is the Natural Map of Tate Cohomolgy is an Isomorphism?

First of all I want to say that I am not at all an expert in Group cohomology . Recently I attended a seminar where the speaker mentioned about something called Tate Cohomology Groups which in someway relate Group homology and Group cohomology in one sequence.

My Question is the following:

Is there any result towards the characterization of the finite Groups $$G$$ and $$G$$-Modules $$A$$ such that the natural map $$N:H_0(G,A) rightarrow H^{0}(G,A)$$ appearing in Tate Cohomology is an Isomorphism of Groups?

(My reference is https://en.wikipedia.org/wiki/Tate_cohomology_group)

It seems interesting to me because when $$N$$ is an isomorphism it represents some sort of notion of Duality between group homology and group cohomology at the zeroth level.

I asked this question to the speaker but did not get satisfactory answer . So I am asking here for an answer / Partial answer to my question.

I apologise in advance if my question sounds Stupid.

## nt.number theory – On numbers that are sums of reciprocals of integers

Suppose $$r$$ is a positive real number. When is there a set $$I$$ (finite or infinite) of positive integers such that $$sumlimits_{iin I} i^{-1} =r$$? In other words, which positive real numbers can be realized as sums of reciprocals of integers, without multiplicity? I think there are uncountably-many sets $$I$$ for which the sum does converge, and of course divergence of the harmonic series means we can add elements to $$I$$ to increase the sum without bound, so I don’t see any obvious ways to rule out any significant ranges of $$r$$. On the other hand I’d be rather surprised if the answer was “always.”

## probability theory – Is the sample space for a stochastic process defined as the set of possible sample paths that may be realised by it?

For example, were I to conduct an experiment where I constructed a stochastic process (defined at 2 time-steps) by tossing a coin twice. Would the sample space be defined as the set of possible outcomes for each toss
$$Omega = { H, T }$$
or the set of sample paths that may be taken by the process
$$Omega = { HH, HT, TH, TT }$$

## computability theory – Computable functions with limited domains

In the developments I’ve seen of primitive recursive and computable functions, the functions always have codomain $$mathbb{N}$$, but are allowed to have domain $$mathbb{N}^{m}$$ for any natural number $$m$$. This seems odd to me—treating the domains and codomains as fundamentally different.

One solution would be to allow functions $$fcolon mathbb{N}^mto mathbb{N}^n$$ for any natural numbers $$m,n$$. Of course, such a function is really just an $$n$$-tuple of functions $$(f_1,f_2,ldots, f_n)$$ where $$f_i$$ is just the $$i$$th coordinate of $$f$$, and computability for $$f$$ would amount to computability for each $$f_i$$.

However, I’m more interested in the opposite direction: limiting the domain to always be $$mathbb{N}$$. This seems to match, more naturally, what an idealized machine is doing by taking in a single natural number and spitting out a single natural number (or not halting). Of course, a fortiori, one could develop the recursive functions as usual, look at the subclass of functions whose domains are $$mathbb{N}$$, and called these the limited domain computable functions, and then show that from these we can reconstruct the non-limited functions in a simple way.

My question is if there is a more natural approach. Just as the recursive functions are built up from some starting functions, using very limited and natural operators, I wonder if there are ways to build up the “limited domain computable functions” similarly, in a non-“ad hoc” way. (For instance, it would be nice if we could do it without the need to develop a universal Turing machine first.)

In other words: Does the extra generality in the domain necessarily simplify the development of computable functions?

## set theory – I’m looking for a conjecture on a possible solution in an infinite set

Recently, I came across a conjecture by two mathematicians about something along the lines of “if an infinite set contains a solution modulus an ever increasing power, then a finite solution should exist in the set even though the set appears to only contain ever increasing numbers”. I’m winging it here. It’s something along these lines. Any help would be greatly appreciated.

I checked my browser history since it was less than a month ago, but not a single hit came up.

Thanks,

Bassam

## gr.group theory – Question about generalizing Cauchy identity

One of the Cauchy identities says that
$$prod_{i,j}(1+x_iy_j) = sum_lambda s_lambda (x_1, cdots,x_m) s_{lambda’} (y_1, cdots,y_n)$$

Where $$lambda$$ is a Young diagram, $$lambda’$$ is the transpose diagram, and $$s$$ is the Schur polynomial.

I would like to elevate $$x_i$$ and $$y_j$$ to $$d$$ dimensional vectors $$vec x_i$$ and $$vec y_j$$. Is there any generalized Cauchy identity, or any other nice closed form, for $$prod_{i,j}(1+vec x_icdot vec y_j),$$ with $$d>1$$?