Equational Reformulations of Intuitionistic Propositional Calculus and Classical First-order Predicate Calculus
Equivalent Expressions of Direct Sum Decomposition of Groups1
In this article, the equivalent expressions of the direct sum decomposition of groups are mainly discussed. In the first section, we formalize the fact that the internal direct sum decomposition can be defined as normal subgroups and some of their properties. In the second section, we formalize an equivalent form of internal direct sum of commutative groups. In the last section, we formalize that the external direct sum leads an internal direct sum. We referred to [19], [18] [8] and [14] in the...
Equivalent fuzzy sets
Necessary and sufficient conditions under which two fuzzy sets (in the most general, poset valued setting) with the same domain have equal families of cut sets are given. The corresponding equivalence relation on the related fuzzy power set is investigated. Relationship of poset valued fuzzy sets and fuzzy sets for which the co-domain is Dedekind-MacNeille completion of that posets is deduced.
Erblich euklidische Körper.
Errata to the paper "Remarks of the elementary theories of formal and convergent power series" Fundamenta Mathematicae 105 (1980), pp. 229-239
Erratum to “Hyperfinite and standard unifications for physical theories”.
Erweiterte Fassung eines von Clebsch aufgestellten Uebertragungsprincips und deren Anwendung
Essai sur la généralisation naturelle de la logique usuelle (premier mémoire)
Essai sur les principes fondamentaux de la géométrie et de la mécanique [Book]
Estensioni contraddittorie della teoria Ampia
Dopo una breve presentazione della teoria , una teoria non riduzionista ed autoreferenziale dei fondamenti della matematica proposta da Clavelli, De Giorgi, Forti e Tortorelli nel 1987, si mostra l'inconsistenza di estensioni della teoria ottenute aggiungendo forti assiomi su relazioni e operazioni (ad es. assiomi che danno la composizione di operazioni, la congiunzione di relazioni, ecc.) e/o assiomi che forniscono qualche relazione "combinatoria".
Étude et implémentation d'un système de déduction pour logique algorithmique
Euler’s Partition Theorem
In this article we prove the Euler’s Partition Theorem which states that the number of integer partitions with odd parts equals the number of partitions with distinct parts. The formalization follows H.S. Wilf’s lecture notes [28] (see also [1]). Euler’s Partition Theorem is listed as item #45 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/ [27].
E-unification by means of tree tuple synchronized grammars.
Evaluating many valued modus ponens
This paper deals with many valued case of modus ponens. Cases with implicative and with clausal rules are studied. Many valued modus ponens via discrete connectives is studied with implicative rules as well as with clausal rules. Some properties of discrete modus ponens operator are given.
Event-Based Proof of the Mutual Exclusion Property of Peterson’s Algorithm
Proving properties of distributed algorithms is still a highly challenging problem and various approaches that have been proposed to tackle it [1] can be roughly divided into state-based and event-based proofs. Informally speaking, state-based approaches define the behavior of a distributed algorithm as a set of sequences of memory states during its executions, while event-based approaches treat the behaviors by means of events which are produced by the executions of an algorithm. Of course, combined...
Events of Borel Sets, Construction of Borel Sets and Random Variables for Stochastic Finance
We consider special events of Borel sets with the aim to prove, that the set of the irrational numbers is an event of the Borel sets. The set of the natural numbers, the set of the integer numbers and the set of the rational numbers are countable, so we can use the literature [10] (pp. 78-81) as a basis for the similar construction of the proof. Next we prove, that different sets can construct the Borel sets [16] (pp. 9-10). Literature [16] (pp. 9-10) and [11] (pp. 11-12) gives an overview, that...
Exploring a Syntactic Notion of Modal Many-Valued Logics.
Exponential Objects
In the first part of this article we formalize the concepts of terminal and initial object, categorical product [4] and natural transformation within a free-object category [1]. In particular, we show that this definition of natural transformation is equivalent to the standard definition [13]. Then we introduce the exponential object using its universal property and we show the isomorphism between the exponential object of categories and the functor category [12].
Extended Real-Valued Double Sequence and Its Convergence
In this article we introduce the convergence of extended realvalued double sequences [16], [17]. It is similar to our previous articles [15], [10]. In addition, we also prove Fatou’s lemma and the monotone convergence theorem for double sequences.
Extending probability spaces and adapted distribution