Articles of reference request

Linear combination of natural numbers with positive coefficients

I’m searching for a reference for the following result, so as to avoid writing a full proof in a paper. Alternatively, if a one-liner exists, I’d be glad to know it! Theorem: Let $a, b$ be two positive integers. Then there is a finite set $N$ of positive integers smaller than $\mbox{lcm}(a, b)$ such that: […]

Can I construct a complete (as a Boolean algebra) $\aleph_0$ saturated elementary extension of a given Boolean algbera?

This is a follow-up to my previous question: Let $B$ be an arbitrary Boolean algebra. Can one construct a $\aleph_0$-saturated $B^* \succ B$ that is complete, i.e., all joins and meets exist in $B^*$? For my previous question, which involved uncountable cardinal numbers instead of $\aleph_0$, the answer is no. What happens we only need […]

Is there a log-space algorithm for divisibility?

Is there an algorithm to test divisibility in space $O(\log n)$, or even in space $O(\log(n)^k)$ for some $k$? Given a pair of integers $(a, b)$, the algorithm should return TRUE if $b$ is divisible by $a$, and FALSE otherwise. I understand that there is no proof that divisibility is not in $L$ since that […]

Reference request, self study of cardinals and cardinal arithmetic without AC

I’m looking for references (books/lecture notes) for : Cardinality without choice, Scott’s trick; Cardinal arithmetic without choice. Any suggestions? Thanks in advance.

Has it been proved that odd perfect numbers cannot be triangular?

(Note: This question has been cross-posted to MO.) Euclid and Euler proved that every even perfect number is of the form $m = \frac{{M_p}\left(M_p + 1\right)}{2}$ where $M_p = 2^p – 1$ is a prime number, called a Mersenne prime. Thus, an even perfect number is triangular. On the other hand, Euler showed that an […]

Euler characteristic of a quotient space

I have a question relating to an answer on The cited answer says: Let $X$ be a topological space for which [the Euler characteristic] $\chi(X)$ is defined and behaves in the expected way for unions, Cartesian products, and quotients by a finite free action. … [Then] $$\chi(X^{(2)}) = \frac{\chi(X \times X) – \chi(\operatorname{Diag}(X))}{2} + […]

Group action on a manifold with finitely many orbits

I’m looking for a result along the lines of the following: Let $G$ be a group acting on a set $X$. If the action partitions $X$ into finitely many $G$-orbits, then $\dim G \geq \dim X$. For this to even make sense, it seems like $G$ and $X$ should have a vector space/manifold structure, but […]

Reference request for ordered groups

I’ve been reading Pete Clark’s notes on commutative algebra, and I especially liked section 17 on valuation rings, and ordered groups in particular. I’m looking for more introductory material regarding the ordering of groups, monoids and vectorspaces. Could anyone suggest something? I’m interested in printed works as well as online notes and courses. I read […]

From a deterministic discrete process to a Markov chain: conditions?

When will a probabilistic process obtained by an “abstraction” from a deterministic discrete process satisfy the Markov property? Example #1) Suppose we have some recurrence, e.g., $a_t=a^2_{t-1}$, $t>0$. It’s a deterministic process. However, if we make an “abstraction” by just considering the one particular digit of each $a_t$, we have a probabilistic process. We wonder […]

Recommended Reading on Regression Analysis?

For a university project, I am implementing an automated regression analysis tool. However, I have very little background in statistics. So what books / articles / material would you suggest I could use to brush up on this topic, based on your experiences? Thanks