The main goal of these notes is to provide an elementary proof of Tychonoff’s theorem. Tychonoff’s Theorem. The product of arbitrarily many compact spaces is compact. It is easier to prove that the product of finitely many compact spaces is compact than it is to prove the general case. As May says in his notes [6] “The case of finite products is not difficult, but the general case is.” Schaum’s...

This paper presents several extensions of order-sorted logic based on the notion of property classification in formal ontology. The refined order-sorted language contains types (as rigid sorts), anti-rigid sorts, and unary predicates in order to distinctly express the following properties: substantial sorts, nonsubstantial sorts, and non-sortal properties. For many separated knowledge bases con...

We focus in this paper on some meta-level ontological distinctions among unary predicates, like those between concepts and assertional properties. Three are the main contributions of this work, mostly based on a revisitation of philosophical (and linguistic) literature in the perspective of knowledge representation. The first is a formal notion of ontological commitment, based on a modal logic ...

Let $G$ be a real Lie group, $\Lambda<G$ lattice and $H<G$ connected semisimple subgroup without compact factors with finite center. We define the notion of $H$-expanding measures $\mu$ on $H$ and, applying recent work Eskin-Lindenstrauss, prove that $\mu$-stationary probability $G/\Lambda$ are homogeneous. Transferring construction by Benoist-Quint drawing ideas Eskin-Mirzakhani-Mohammadi, we ...

We discuss some examples that illustrate the countability of the positive rational numbers and related sets. Techniques include radix representations, Gödel numbering, the fundamental theorem of arithmetic, continued fractions, Egyptian fractions, and the sequence of ratios of successive hyperbinary representation numbers. Ginsberg [7] gave an injective mapping of the rational numbers Q into th...

Recently, Rattan and the first author (Ann. Comb. 25 (2021) 697–728) proved a conjectured inequality of Berkovich Uncu 23 (2019) 263–284) concerning partitions with an impermissible part. In this article, we generalize upon considering t parts. We compare these whose certain parts appear frequency which is perfect $$t^{th}$$ power. addition, that study here have smallest part greater than or eq...

The subject of this discussion are the almost forgotten victims chemical weapons on Eastern Front World War I. Source data related to military operations, in particular German gas attacks region Bzura and Rawka rivers, basis for presenting problems estimation number poisonous chlorine period between May July 1915. Using example selected relating wave attack 31, 1915, I show specificity complexi...

The increasing significance of theorem proving-based formalization in mathematics and computer science highlights the necessity for formalizing foundational mathematical theories. In this work, we employ Coq interactive prover to methodically formalize language, semantics, syntax propositional logic, a fundamental aspect reasoning proof construction. We construct four Hilbert-style axiom system...