Wyniki wyszukiwania - Biblioteka Nauki

1

Trigonometric Series and Set Theory

100%

Kechris A. S.

Wiadomości Matematyczne

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2012

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tom T. 48, Nr 2

109--118

EN

The historical survey concerning the interactions between the theory of trigonometric series and descriptive set theory. We concentrate here on the area related to problems of uniqueness for trigonometric series. Detailed historical and bibliographical references can be found in the books and survey papers listed at the end.

2

Herbrand-satisfiability of a Quantified Set-theoretic Fragment

100%

Cantone D. , Longo C. , Nicolosi-Asmundo M.

Fundamenta Informaticae

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2017

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tom Vol. 150, nr 1

49--71

EN

In the last decades, several fragments of set theory have been studied in the context of Computable Set Theory. In general, the semantics of set-theoretic languages differs from the canonical first-order semantics in that the interpretation domain of set-theoretic terms is fixed to a given universe of sets. Because of this, theoretical results and various machinery developed in the context of first-order logic could be not easily applicable in the set-theoretic realm. Recently, the decidability of quantified fragments of set theory which allow one to explicitly handle ordered pairs has been studied, in view of applications in the field of knowledge representation. Among other results, a NEXPTIME decision procedure for satisfiability of formulae in one of these fragments, ∀0π , has been devised. In this paper we exploit the main features of such a decision procedure to reduce the satisfiability problem for the fragment ∀0π to the problem of Herbrand satisfiability for a first-order language extending it. In addition, it turns out that such a reduction maps formulae of the Disjunctive Datalog subset of ∀0π into Disjunctive Datalog formulae.

3

Wykorzystanie teorii zbiorów do badania konfiguracji organizacyjnych

80%

Kwiotkowska A.

Zeszyty Naukowe. Organizacja i Zarządzanie / Politechnika Śląska

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2010

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tom z. 53

163-175

PL

Badania konfiguracji organizacyjnych obejmują rozpatrywanie wielowymiarowych układów koncepcyjnie odrębnych cech charakterystycznych, które zazwyczaj występują razem. W artykule do badania konfiguracji organizacyjnych proponuje się wykorzystanie teorii zbiorów traktujących konfiguracje jak różne typy przypadków. Celem artykułu jest określenie istoty konfiguracji organizacyjnych oraz wskazanie praktycznych aspektów wykorzystania teorii zbiorów do badania konfiguracji organizacyjnych.

EN

The research of organizational configurations includes the investigation of multidimensional systems conceptually distinct characteristics, which usually occur together. In the article, to the research of organizational configurations was proposed to use set-theory as treating different types of configurations of the cases. The aim of this paper is to define the essence of organizational configurations and the indication of the practical aspects of using set-theory to the research the organizational configuration.

4

Embedding Cohen algebras using pcf theory

80%

Shelah S.

Fundamenta Mathematicae

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2000

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tom 166

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nr 1-2

83-86

EN

Using a theorem from pcf theory, we show that for any singular cardinal ν, the product of the Cohen forcing notions on κ, κ < ν, adds a generic for the Cohen forcing notion on $ν^+$.

5

Georg Cantor i idea jedności nauki

80%

Dadaczyński J.

Zagadnienia Filozoficzne w Nauce

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2009

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nr 44

84-99

PL

G. Cantor presented - in an unpublished paper (1884) - a vision of the unity of science. He argued all sciences can be reduced directly to the set theory. A source of this idea was for Cantor the unity of mathematics (on the basis of set theory). Cantor represented thesis about the unity of science irrespective of the representatives of positivism (E. Mach).

6

Set-theoretic mereology

80%

Hamkins J. D. , Kikuchi M.

Logic and Logical Philosophy

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2016

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tom 25

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nr 3 Mereology and Beyond (II)

285-308

EN

We consider a set-theoretic version of mereology based on the inclusion relation ⊆ and analyze how well it might serve as a foundation of mathematics. After establishing the non-definability of ∈ from ⊆, we identify the natural axioms for ⊆-based mereology, which constitute a finitely axiomatizable, complete, decidable theory. Ultimately, for these reasons, we conclude that this form of set-theoretic mereology cannot by itself serve as a foundation of mathematics. Meanwhile, augmented forms of set-theoretic mereology, such as that obtained by adding the singleton operator, are foundationally robust.

7

Strong covering without squares

80%

Shelah S.

Fundamenta Mathematicae

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2000

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tom 166

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nr 1-2

87-107

EN

Let W be an inner model of ZFC. Let κ be a cardinal in V. We say that κ-covering holds between V and W iff for all X ∈ V with X ⊆ ON and V ⊨ |X| < κ, there exists Y ∈ W such that X ⊆ Y ⊆ ON and V ⊨ |Y| < κ. Strong κ-covering holds between V and W iff for every structure M ∈ V for some countable first-order language whose underlying set is some ordinal λ, and every X ∈ V with X ⊆ λ and V ⊨ |X| < κ, there is Y ∈ W such that X ⊆ Y ≺ M and V ⊨ |Y| < κ. We prove that if κ is V-regular, $κ^+_V = κ^+_W$, and we have both κ-covering and $κ^+$-covering between W and V, then strong κ-covering holds. Next we show that we can drop the assumption of $κ^+$-covering at the expense of assuming some more absoluteness of cardinals and cofinalities between W and V, and that we can drop the assumption that $κ^+_W = κ^+_V$ and weaken the $κ^+$-covering assumption at the expense of assuming some structural facts about W (the existence of certain square sequences).

8

Yurii Rogozhin’s Contributions to the Field of Small Universal Turing Machines

80%

Woods D. , Neary T.

Fundamenta Informaticae

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2015

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tom Vol. 138, nr 1/2

251--258

EN

In the field of small universal Turing machines, Yurii Rogozhin holds a special prize: he was first to close off an infinite number of open questions by drawing a closed curve that separates the infinite set of Turing machines that are universal from a finite set of small machines for which we don’t yet know. Rogozhin did this by finding the smallest known universal Turing machines at the time, both in terms of number of states and number of symbols. This brief note summarises this and a few of Yurii’s other contributions to the field, including his work with Manfred Kudlek on small circular Post machines.

9

Polish set theorists and large cardinals

80%

Stillwell J.

Wiadomości Matematyczne

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2012

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tom T. 48, Nr 2

247--256

10

Remarks about the Replacement of School Induction Definitions by Normal Definitions

80%

Bryll G. , Rygał G.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2003

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tom Vol. 9

123--129

EN

Pupils and teachers often ask themselves a question: can induction definitions be replaced in an equivalent way by normal definitions? In this paper we present a method of replacement of induction definitions by normal definitions illustrating the given theorems by a few examples. From the viewpoint of the set theory operations and relations can be treated as certain sets. We discuss a method of replacement of an induction definition of the given set by a normal definition of this set. An induction definition of a set A has in general the following form (compare with [2]): D1. A set A is the least one from among the sets X satisfying the conditions: W1 (X) : a1,...,an ∈ X (the starting conditions), W 2 (X) : x1,...,xn ∈ X ⤇f (x1,...,xn) ∈ X (the induction conduction).

11

An Analysis of Probabilistic Approximations for Rule Induction from Incomplete Data Sets

70%

Clark P. G. , Grzymala-Busse J. W. , Hippe Z. S.

Fundamenta Informaticae

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2014

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tom Vol. 132, nr 3

365--379

EN

The main objective of our research was to test whether the probabilistic approximations should be used in rule induction from incomplete data. For our research we designed experiments using six standard data sets. Four of the data sets were incomplete to begin with and two of the data sets had missing attribute values that were randomly inserted. In the six data sets, we used two interpretations of missing attribute values: lost values and “do not care” conditions. In addition we used three definitions of approximations: singleton, subset and concept. Among 36 combinations of a data set, type of missing attribute values and type of approximation, for five combinations the error rate (the result of ten-fold cross validation) was smaller than for ordinary (lower and upper) approximations; for other four combinations, the error rate was larger than for ordinary approximations. For the remaining 27 combinations, the difference between these error rates was not statistically significant.

12

Filling assessment model of strategic cost management system (SCMS)

70%

Petrov O. , Minina Y.

Zeszyty Naukowe Wyższej Szkoły Ekonomii i Informatyki w Krakowie

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2018

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nr 14

260-276

EN

The article presents a filling assessment model of SCMS, developed for the reflection of actual content of enterprise strategic cost management system, which can be used in real-time mode. It is proposed to use the mathematical apparatus of set theory and tools for working with sets in hyperspace for practical construction and use of such model. Ref. 7.

PL

Artykuł przedstawia model oceny wypełnienia SCMS, opracowany w celu odzwierciedlenia rzeczywistej zawartości systemu zarządzania kosztami strategicznymi przedsiębiorstwa, który może być wykorzystywany w trybie czasu rzeczywistego. Proponuje się wykorzystanie aparatu matematycznego teorii mnogości i narzędzi do pracy z zestawami w hiperprzestrzeni do praktycznej konstrukcji i wykorzystania takiego modelu.

13

Research for fuzzy Duoma debt model and its fuzzy solution

70%

Bing-Yuan C.

Systems Science

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2003

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tom Vol. 29, nr 3

81-93

EN

Debt model is generalized first in this paper; and then an extensively used fuzzy debt model is built by a concept of inverse image defined by fuzzy functions. Third, a proof is demonstrated of the existence of fuzzy solutions as well as dependence upon initial values and parameters of solutions. Fourth, a series of solving methods is advanced with problems discussed on this system's stability. And finally, Duoma debt model with a triangle fuzzy function is solved.

14

Grounding languages in cognitive agents and robots

70%

Katarzyniak R. P. , Pieczynska-Kuchtiak A.

Systems Science

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2003

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tom Vol. 29, nr 2

113-127

EN

The symbol grounding problem is discussed for the case of simple language of formulas with modal operators and the cognitive agent. The language formulas are built from modal operators and logical connectives of conjunction, disjunction and exclusive disjunction. The cognitive agent carries out perceptions of an external world and stores their content in dedicated temporal database. The empirical experience contained in this database defines the scope of possible meaning that can ever be assigned to belief formulas. Two close but different approaches to implementing the idea of grounding in cognitive structures of the agent are presented.

15

Aksjomat wyboru w pracach Wacława Sierpińskiego

70%

Lewandowska K.

Zagadnienia Filozoficzne w Nauce

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2013

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nr 52

9-33

PL

This paper presents Wacław Sierpiński – the first advocate of the axiom of choice. We focus on the philosophical and mathematical topics related to the axiom of choice which were considered by Sierpiński. We analyze some of his papers to show how his results effected the debate over Zermelo’s axiom. Sierpiński’s impact on this discussion is of particular importance since he was the first who tried to explore consequences of the axiom of choice thoroughly and asserted its undoubted significance to mathematics as a whole.

16

Cantor’s paradise from the perspective of non‐revisionist Wittgensteinianism

70%

Gomułka J.

ARGUMENT: Biannual Philosophical Journal

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2020

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tom 10

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nr 2

351-372

EN

Cantor’s paradise from the perspective of non‐revisionist Wittgensteinianism: Ludwig Wittgenstein is known for his criticism of transfinite set theory. He forwards the claim that we tend to conceptualise infinity as an object due to the systematic confusion of extension with in‐ tension. There can be no mathematical symbol that directly refers to infinity: a rule is the only form by which the latter can appear in our symbolic operations. In consequence, Wittgenstein rejects such ideas as infinite cardinals, the Cantorian understanding of non‐denumerability, and the view of real numbers as a continuous sequence of points on a number line. Moreover, as he understands mathematics to be an anthropological phenomenon, he rejects set theory due to its lack of application. As I argue here, it is possible to defend Georg Cantor’s theory by taking a standpoint I call quietistic conventionalism. The standpoint broadly resembles Wittgenstein’s formalist middle period and allows us to view transfinite set theory as a result of a series of definitions established by arbitrary decisions that have no ontological consequences. I point to the fact that we are inclined to accept such definitions because of certain psycho‐ logical mechanisms such as the hypothetical Basic Metaphor of Infinity proposed by George Lakoff and Rafael E. Núñez. Regarding Wittgenstein’s criterion of applicability, I argue that it presupposes a static view of science. Therefore, we should not rely on it because we are unable to foresee what will turn out to be useful in the future.

17

Graphical Partitions and Graphical Relations

70%

Shaheen T. , Stell J. G.

Fundamenta Informaticae

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2019

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tom Vol. 165, nr 1

75--98

EN

We generalize the well-known correspondence between partitions and equivalence relations on a set to the case of graphs and hypergraphs. This is motivated by the role that partitions and equivalence relations play in Rough Set Theory and the results provide some of the foundations needed to develop a theory of rough graphs. We use one notion of a partition of a hypergraph, which we call a graphical partition, and we show how these structures correspond to relations on a hypergraph having additional properties. In the case of a hypergraph with only nodes and no edges these properties are exactly the usual reflexivity, symmetry and transitivity properties required for equivalence relations on a set. We present definitions for upper and lower approximations of a subgraph with respect to a graphical partition. These generalize the well-known approximations in Rough Set Theory. We establish fundamental properties of our generalized approximations and provide examples of these constructions on some graphs.

18

Specialized Predictor for Reaction Systems with Context Properties

70%

Barbuti R. , Gori R. , Levi F. , Milazzo P.

Fundamenta Informaticae

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2016

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tom Vol. 147, nr 2/3

173--191

EN

Reaction systems are a qualitative formalism for modeling systems of biochemical reactions characterized by the non-permanency of the elements: molecules disappear if not produced by any enabled reaction. Reaction systems execute in an environment that provides new molecules at each step. Brijder, Ehrenfeucht and Rozemberg introduced the idea of predictors. A predictor of a molecule s, for a given n, is the set of molecules to be observed in the environment to determine whether s is produced or not at step n by the system. We introduced the notion of formula based predictor, that is a propositional logic formula that precisely characterizes environments that lead to the production of s after n steps. In this paper we revise the notion of formula based predictor by defining a specialized version that assumes the environment to provide molecules according to what expressed by a temporal logic formula. As an application, we use specialized formula based predictors to give theoretical grounds to previously obtained results on a model of gene regulation.

19

DNA Tiles, Wang Tiles and Combinators

70%

Bellia M. , Occhiuto M. E.

Fundamenta Informaticae

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2014

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tom Vol. 133, nr 2/3

105--121

EN

We investigate the relation between Combinatory Logic and Wang Tiles with the aim of studying Combinators as a programming language for Self-Assembly and DNA computing. We introduce a subset of Combinatory Logic, SKI#, which is Turing Complete, includes simply Typed Combinatory Logic and contains only combinators whose computations require finitely many different redexes. Then, we define a language of Tiles, SKI-Tile, for the representation and the computation of the terms of SKI# in Self-Assembly. Moreover, we introduce a program development methodology that given any computable function, expressed in SKI#, provides a finite set of Tiles that self-assemble to return the computations of the function applications. Finally, the methodology is applied to the derivation of a SKI-Tile program that self-assemble to compute the factorial function.

20

On a problem of Steve Kalikow

70%

Shelah S.

Fundamenta Mathematicae

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2000

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tom 166

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nr 1-2

137-151

EN

The Kalikow problem for a pair (λ,κ) of cardinal numbers,λ > κ (in particular κ = 2) is whether we can map the family of ω-sequences from λ to the family of ω-sequences from κ in a very continuous manner. Namely, we demand that for η,ν ∈ ω we have: η, ν are almost equal if and only if their images are. We show consistency of the negative answer, e.g., for $ℵ_ω$ but we prove it for smaller cardinals. We indicate a close connection with the free subset property and its variants.