Wyniki wyszukiwania

1

Some Remarkable Identities Involving Numbers

100%

Ziobro R.

Formalized Mathematics

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2014

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tom 22

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nr 3

205-208

EN

The article focuses on simple identities found for binomials, their divisibility, and basic inequalities. A general formula allowing factorization of the sum of like powers is introduced and used to prove elementary theorems for natural numbers. Formulas for short multiplication are sometimes referred in English or French as remarkable identities. The same formulas could be found in works concerning polynomial factorization, where there exists no single term for various identities. Their usability is not questionable, and they have been successfully utilized since for ages. For example, in his books published in 1731 (p. 385), Edward Hatton [3] wrote: “Note, that the differences of any two like powers of two quantities, will always be divided by the difference of the quantities without any remainer...”. Despite of its conceptual simplicity, the problem of factorization of sums/differences of two like powers could still be analyzed [7], giving new and possibly interesting results [6].

2

An equational basis in four variables for the three-element tournament

100%

Grätzer G., Kisielewicz A., Wolk B.

Colloquium Mathematicum

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1992

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tom 63

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nr 1

41-44

3

Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik

Hyperidentities in transitive graph algebras

75%

Poomsa-ard T., Wetweerapong J., Samartkoon C.

Discussiones Mathematicae - General Algebra and Applications

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2005

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tom 25

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nr 1

23-37

EN

Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type (2,0). We say that a graph G satisfies an identity s ≈ t if the corresponding graph algebra A(G) satisfies s ≈ t. A graph G = (V,E) is called a transitive graph if the corresponding graph algebra A(G) satisfies the equation x(yz) ≈ (xz)(yz). An identity s ≈ t of terms s and t of any type t is called a hyperidentity of an algebra A̲ if whenever the operation symbols occurring in s and t are replaced by any term operations of A of the appropriate arity, the resulting identities hold in A̲ . In this paper we characterize transitive graph algebras, identities and hyperidentities in transitive graph algebras.

4

Identity, Equality, Nameability and Completeness

63%

Manzano M., Moreno M. C.

Bulletin of the Section of Logic

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2017

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tom 46

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nr 3/4

EN

This article is an extended promenade strolling along the winding roads of identity, equality, nameability and completeness, looking for places where they converge. We have distinguished between identity and equality; the first is a binary relation between objects while the second is a symbolic relation between terms. Owing to the central role the notion of identity plays in logic, you can be interested either in how to define it using other logical concepts or in the opposite scheme. In the first case, one investigates what kind of logic is required. In the second case, one is interested in the definition of the other logical concepts (connectives and quantifiers) in terms of the identity relation, using also abstraction. The present paper investigates whether identity can be introduced by definition arriving to the conclusion that only in full higher-order logic a reliable definition of identity is possible. However, the definition needs the standard semantics and we know that with this semantics completeness is lost. We have also studied the relationship of equality with comprehension and extensionality and pointed out the relevant role played by these two axioms in Henkin’s completeness method. We finish our paper with a section devoted to general semantics, where the role played by the nameable hierarchy of types is the key in Henkin’s completeness method.

5

Identity, equality, nameability and completeness. Part II

63%

Manzano M., Moreno M. C.

Bulletin of the Section of Logic

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2018

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tom 47

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nr 3

EN

This article is a continuation of our promenade along the winding roads of identity, equality, nameability and completeness. We continue looking for a place where all these concepts converge. We assume that identity is a binary relation between objects while equality is a symbolic relation between terms. Identity plays a central role in logic and we have looked at it from two different points of view. In one case, identity is a notion which has to be defined and, in the other case, identity is a notion used to define other logical concepts. In our previous paper, [16], we investigated whether identity can be introduced by definition arriving to the conclusion that only in full higher-order logic with standard semantics a reliable definition of identity is possible. In the present study we have moved to modal logic and realized that here we can distinguish in the formal language between two different equality symbols, the first one shall be interpreted as extensional genuine identity and only applies for objects, the second one applies for non rigid terms and has the characteristic of synonymy. We have also analyzed the hybrid modal logic where we can introduce rigid terms by definition and can express that two worlds are identical by using the nominals and the @ operator. We finish our paper in the kingdom of identity where the only primitives are lambda and equality. Here we show how other logical concepts can be defined in terms of the identity relation. We have found at the end of our walk a possible point of convergence in the logic Equational Hybrid Propositional Type Theory (EHPTT), [14] and [15].

6

Edukacja geograficzna z elementami edukacji antydyskryminacyjnej w szkole waldorfskiej

51%

Zatorski M.

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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2016

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tom 10

179-189

EN

During the primary school geographical education students are not aware that the acquired knowledge and skills are those necessary for the proper perception and observation of social and natural phenomena. In waldorf pedagogy geography is regarded as one of the two most important school subjects that mark the path of learning by the students of the reality surrounding them. Realized in this school elements of the antidiscrimination education it would make geography – a school subject reconciliation between nations, because, as used to say Professor Stanislaw Pawłowski “One is the Earth and the person on it” (Pawłowski, 1939: 1). Implementation of antidiscrimination education components to implemented geographic content in public schools, will strengthen practical and educational role of geography. It would also be a response to a systemic solution to the lack of anti-discrimination education in Polish schools.

PL

W trakcie podstawowej szkolnej edukacji geograficznej uczniowie nie mają świadomości, że nabyte wiadomości i umiejętności są niezbędne do właściwej percepcji i obserwacji zjawisk społecznych oraz przyrodniczych. W pedagogice waldorfskiej geografia traktowana jest jako jeden z dwóch najważniejszych przedmiotów szkolnych, które wyznaczają drogę poznawania przez uczniów otaczającej ich rzeczywistości. Realizowane w tej szkole elementy edukacji antydyskryminacyjnej czynią z geografii naukę pojednania pomiędzy narodami, w myśl słów prof. Stanisława Pawłowskiego: „Jedna jest Ziemia i jeden człowiek na niej” (Pawłowski, 1939: 1). Wdrażanie elementów edukacji antydyskryminacyjnej do realizowanych treści geograficznych w szkołach publicznych, pozwoliłoby na wzmocnienie praktycznej i wychowawczej roli geografii. Byłaby to również odpowiedź na systemowe rozwiązanie braku edukacji antydyskryminacyjnej w polskich szkołach.

Ograniczanie wyników

2 Bulletin of the Section of Logic

1 Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

1 Colloquium Mathematicum

1 Discussiones Mathematicae - General Algebra and Applications

1 Formalized Mathematics

2 Manzano M.

2 Moreno M. C.

1 Grätzer G.

1 Kisielewicz A.

1 Poomsa-ard T.

1 Samartkoon C.

1 Wetweerapong J.

1 Wolk B.

1 Zatorski M.

1 Ziobro R.

1 2018

2 2017

1 2014