Related papers: Independence and Induction in Reverse Mathematics
We consider strong combinatorial principles for sigma-directed families of countable sets in the ordering by inclusion modulo finite, e.g. P-ideals of countable sets. We try for principles as strong as possible while remaining compatible…
Effectively inseparable pairs and their properties play an important role in the meta-mathematics of arithmetic and incompleteness. Different notions are introduced and shown in the literature to be equivalent to effective inseparability.…
If the universe follows a specific design, then a central question is which cost function is optimized by the observed forces. This is the problem of inverse optimal control, or inverse reinforcement learning, in which a reward function is…
For a dust-like self-similar set (generated by IFSs with the strong separation condition), Elekes, Keleti and M\'{a}th\'{e} found an invariant, called `algebraic dependence number', by considering its generating IFSs and isometry invariant…
There exist a number of results proving that for certain classes of interacting particle systems in population genetics, mutual invadability of types implies coexistence. In this paper we prove a sort of converse statement for a class of…
In finite problems comprising objects, situations, and an object- and situation-contingent payoff function, we study the comparative statics of the set of undominated objects, meaning those for which there exists no mixture over objects…
Let "ex" be the cardinality of the smallest independent family of subsets of omega (independent means that all nontrivial Boolean combinations are infinite) which cannot be extended to a homogeneous independent family. "Homogeneous" means…
A family of random matrices is said to converge strongly to a limiting family of operators if the operator norm of every noncommutative polynomial of the matrices converges to that of the limiting operators. Recent developments surrounding…
We establish several results concerning the expected general phenomenon that, given a multiplicative function $f:\mathbb{N}\to\mathbb{C}$, the values of $f(n)$ and $f(n+a)$ are "generally" independent unless $f$ is of a "special" form.…
We show three basic properties on the image Milnor number $\mu_I(f)$ of a germ $f\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ with isolated instability. First, we show the conservation of the image Milnor number, from which one…
Maximal ancestral graphs (MAGs) are used to encode conditional independence relations in DAG models with hidden variables. Different MAGs may represent the same set of conditional independences and are called Markov equivalent. This paper…
We present a unified theory for formal mathematical systems including recursive systems closely related to formal grammars, including the predicate calculus as well as a formal induction principle. We introduce recursive systems generating…
This paper analyzes independence concepts for sets of probability measures associated with directed acyclic graphs. The paper shows that epistemic independence and the standard Markov condition violate desirable separation properties. The…
In [CMRM24], it was proved that it is relatively consistent that \emph{bounding number} $\mathfrak{b}$ is smaller than the uniformity of $\mathcal{MA}$, where $\mathcal{MA}$ denotes the ideal of the meager-additive sets of $2^{\omega}$. To…
A matroid is a notion of independence in combinatorial optimization which is closely related to computational efficiency. In particular, it is well known that the maximum of a constrained modular function can be found greedily if and only…
Statistical independence is a notion ubiquitous in various fields such as in statistics, probability, number theory and physics. We establish the stability of independence for any pair of random variables by their corresponding Brockwell…
The smooth development of large parts of mathematics hinges on the idea that some sets are `small' or `negligible' and can therefore be ignored for a given purpose. The perhaps most famous smallness notion, namely `measure zero', originated…
We construct an additional independent integral of motion for a class of three dimensional minimally superintegrable systems with constant magnetic field. This class was introduced in [J. Phys. A: Math. Theor. 50 (2017), 245202, 24 pages]…
In this thesis, the main objects of study are probability measures on the isomorphism classes of countable, connected rooted graphs. An important class of such measures is formed by unimodular measures, which satisfy a certain equation,…
Let $G$ be a graph and $\mathcal{F}$ a family of graphs. Define $\alpha_{\mathcal{F}}(G)$ as the maximum order of any induced subgraph of $G$ that belongs to the family $\mathcal{F}$. For the family $\mathcal{F}$ of graphs with…