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1
Content available O pewnych modyfikacjach teorii skierowanych liczb rozmytych
100%
Piasecki K.
Optimum. Studia Ekonomiczne
|
2017
|
tom 3(87)
PL
Skierowane liczby rozmyte zostały zdefiniowane w doskonały i intuicyjny sposób przez Witolda Kosińskiego. Z tej przyczyny skierowane liczby rozmyte coraz częściej określa się mianem liczb Kosińskiego. W pierwszej części tej pracy zaproponowano w pełni sformalizowaną definicję liczby Kosińskiego. Definicję tę następnie uogólniono do przypadku skierowanej liczby rozmytej z nieciągłą funkcją przynależności. Istotną wadą arytmetyki zaproponowanej przez Kosińskiego był brak zamknięcia przestrzeni skierowanych liczb rozmytych ze względu na podstawowe działania arytmetyczne, takie jak: dodawanie, odejmowanie, mnożenie i dzielenie. Głównym celem prezentowanej pracy jest taka modyfikacja działań arytmetycznych, aby przestrzeń liczb Kosińskiego była zamknięta z racji zmodyfikowanych działań arytmetycznych.
EN
Ordered fuzzy numbers have been defined in an excellent, intuitive way by Witold Kosiński. For this reason, they are increasingly referred to as Kosiński’s numbers. A fully formalized definition of a Kosiński’s number is proposed in the first part of this work. This definition is generalized so as to fit an ordered fuzzy number with an upper semi-continuous membership function. A significant drawback of Kosiński’s arithmetic is that the space of ordered fuzzy numbers is not closed under addition, subtraction, multiplication, or division. The main aim of this paper is to modify the arithmetic in such a way that the space of ordered fuzzy numbers is closed under the modified arithmetic operations.
2
Content available remote A Function Elimination Method for Checking Satisfiability of Arithmetical Logics
88%
Castiglioni V. , Lanotte R. , Tini S.
Fundamenta Informaticae
|
2016
|
tom Vol. 143, nr 1/2
51--71
EN
We study function elimination for Arithmetical Logics. We propose a method allowing substitution of functions occurring in a given formula with functions with less arity. We prove the correctness of the method and we use it to show the decidability of the satisfiability problem for two classes of formulas allowing linear and polynomial terms.
3
Content available remote Fregovo pojetí aplikace aritmetiky
88%
Sousedík P. , Svoboda D.
Filosofický časopis (Philosophical Journal)
|
2015
|
tom 63
|
nr 3, Special No: Sedmkrát z logiky a metodologie vědy
101-124
EN
The authors believe that the problem of applicability can be approached in two ways. One approach derives from the fact that the empirical world has been the source of many mathematical concepts, and claims that arithmetic captures reality in the same way as common empirical disciplines. Its miraculous applicability can then be explained by the greater universality of the concepts used. Such an approach is designated a poste¬riori. The other approach to the problem of applicability, designated a priori, assumes that arithmetic is not grounded empirically, in fact it is already there before all expe¬rience. Upon analysis, both approaches authors’ view, these merits and shortcomings were already noticed by Frege. Though his conception is to be classified as an a priori approach, he – unlike his predecessors – also learned much from proponents of a posteriori conceptions.
4
Content available remote Finite Arithmetics
75%
Krynicki M. , Mostowski M. , Zdanowski K.
Fundamenta Informaticae
|
2007
|
tom Vol. 81, nr 1-3
183-202
EN
The paper presents the current state of knowledge in the field of logical investigations of finite arithmetics. This is an attempt to summarize the ideas and results in this area. Some new results are presented - these are mainly generalizations of the earlier results related to properties of sl-theories and some nontrivial cases of FM-representability theorem.
5
Content available remote Influence of uncertainties and wrapping effect reduction in linear dynamical systems by affine arithmetic
75%
Stręk T.
Vibrations in Physical Systems
|
2004
|
tom Vol. 21
361--364
6
Upraszczanie afinicznych relacji i zbiorów w celu podniesienia wydajności generowanego kodu równoległego
75%
Pałkowski M.
Metody Informatyki Stosowanej
|
2008
|
tom nr 3 (Tom 16)
213--222
EN
A representation of relations and sets determines the amount of instructions and the form of loops in generated parallel code with using the Omega Calculator [1,2]. An unsimplified set of iterations may cause unnecessary instructions in parallel code and longer time of code execution. Algorithms of the simplification of relations and sets by means of the operation are presented. Under experiments, code of generated loops along with execution time is examined. The algorithms can be used also to reduce amount of one-conjunct relations and sets.
7
Content available remote Some aspects of teaching the concepts: arithmetic and weighted means at various levels of mathematical education
75%
Bugajska-Jaszczołt B. , Czajkowska M.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
|
2007
|
tom Vol. 12
163-170
EN
We refer to selected problems of formulating the definitions of arithmetic and weighted means in secondary school and at a higher level. The first problem discussed here are the results of theoretical research. We compare the manner of formulating the definitions, in particular, the level of formalization of mathematical language. We present the differences between various definitions, found in the analyzed textbooks, in such aspects as: generality, degree and kind of complication of logical structure, and intuitions provoked by them. We underline qualitatively different objects defined by them - arithmetic mean of numbers and arithmetic mean of scalable feature, emphasizing the consequences.
8
Content available remote Sekwencja Farey’a, czyli jak geolog znalazł miejsce w historii matematyki
75%
Biernacki W.
Wiadomości Naftowe i Gazownicze
|
2006
|
tom R. 9, Nr 9 (101)
14-18
9
Content available Matematyka nie opisuje świata, lecz wychodzi mu naprzeciw
71%
Mioduszewski J.
Zagadnienia Filozoficzne w Nauce
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2015
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nr 58
7-43
EN
In everyday experience mathematics rarely appears to us as a whole, and certainly never as a system in the sense of David Hilbert’s considerations from early 20th Century. Mathematical disciplines seem to be independent and autonomous. We do not see that specific deduction goes beyond particular convention applicable in given discipline. In the late 19th Century this view was shared by Felix Klein and Richard Dedekind. The latter’s work “What are numbers and what should they be?” (Was Was sind und was sollen die Zahlen?) was the inspiration for writing this article. This essay is an attempt to see mathematics not as a building, but as a living organism seeking its explanation.
10
Content available remote Uwagi o arytmetyce Grassmanna
63%
Hanusek J.
Diametros
|
2015
|
nr Diametros 45
107-121
EN
Hermann Grassmann’s 1861 work [2] was probably the first attempt at an axiomatic approach to arithmetic (of integers with a distinguished subset of positive ones). The historical significance of this work is enormous, even though the set of axioms has proven to be incomplete. Basing on the interpretation of Grassmann’s theory provided by Hao Wang in [4], I present its detailed discussion, define the class of models of Grassmann’s arithmetic and discuss a certain axiom system for integers, modeled on Grassmann’s theory. At the end I propose to modify the set of axioms of Grassmann’s arithmetic, which consists in adding an elementary sentence and removing a non-elementary one. I prove that after this modification the only model of the theory up to isomorphism is the standard model.
PL
Praca Hermanna Grassmanna z roku 1861 była pierwszą próbą aksjomatycznego ujęcia arytmetyki (liczb całkowitych z wyróżnionym podzbiorem liczb dodatnich). Znaczenie historyczne tej pracy jest ogromne, choć sama aksjomatyka okazała się niepełna. Opierając się na interpretacji teorii Grassmanna dokonanej przez HaoWanga [1957], przedstawiam szczegółowe jej omówienie i definiuję klasę modeli tej teorii. Na koniec podaję propozycję modyfikacji aksjomatyki arytmetyki Grassmanna, która polega na dodaniu pewnego zdania elementarnego i usunięciu zdania nieelementarnego. Przedstawiam dowód że po takiej modyfikacji teorii jej jedynym modelem z dokładnością˛ do izomorfizmu jest model standardowy.
11
Content available remote Gerbert a aritmetika : mezi filosofií čísla a počtářským uměním
63%
Otisk M.
Filosofický časopis (Philosophical Journal)
|
2019
|
tom 67
|
nr 4
503-520
EN
The attitude towards arithmetic in the Middle Ages was closely connected to antiquity’s opinion on the doctrine of the number, in whose framework practical arithmetic (calculating) and theoretical arithmetic (the theory or philosophy of numbers) were differentiated. The latter was traditionally assigned the greater importance in the area of philosophy, but the first had also been, from antiquity, perceived as a means that would lead to an understanding of the nature of numbers and which would cultivate the abstract and philosophical thinking of humans. This study discusses the example of the important, late 10th century thinker Gerbert of Aurillac (Pope Sylvester II) to show how it is possible to use arithmetic for philosophical research. First, the philosophy of numbers is presented, in which Gerbert (in connection with Boethius) reveals an arithmetically based reality, as numbers are also the thoughts of God. Practical arithmetic, for which Gerbert became especially famous (Western Arabic numerals and the abacus), then sharpened the human mind and facilitated the grasping of the nature of numbers.
CS
Středověký postoj k aritmetice úzce navazuje na antické nahlížení na nauku o čísle, v jejímž rámci byla rozlišována praktická (počtářství) a teoretická aritmetika (teorie či filosofie čísla). Druhé uvedené je v rámci filosofie tradičně připisována větší důležitost, ale také první je již od antiky vnímána jako prostředek vedoucí k pochopení povahy čísel, jež rozvíjí lidské abstraktní a filosofické myšlení. Tato studie ukazuje na příkladu významného myslitele konce 10. století, Gerberta z Remeše (papeže Silvestra II.), jak lze aritmetiku užít k filosofickému zkoumání. Nejprve je představena filosofie čísla, která u Gerberta (v návaznosti na Boëthia) odhaluje aritmetické založení reality, neboť čísla jsou zároveň myšlenkami Božími. Praktická aritmetika, v níž se Gerbert zejména proslavil (západoarabské číslice a abakus), pak bystří lidskou mysl a umožňuje pochopení povahy čísel.
DE
Der mittelalterliche Standpunkt zur Arithmetik ist eng mit dem antiken Blick auf die Zahlenlehre verbunden, in deren Rahmen die praktische Arithmetik (Rechenkunst) von der theoretischen Arithmetik (Theorie und Philosophie der Zahl) unterschieden wurde. Der zweitgenannten wird in der Philosophie traditionsgemäß größere Wichtigkeit beigemessen, aber auch die praktische Arithmetik wurde bereits in der Antike als Mittel zum Verständnis der Natur der Zahlen angesehen, was wiederum die Entwicklung des abstrakten und philosophischen Denkens fördert. In der vorliegenden Studie wird anhand des Beispiels eines bedeutenden Denkers vom Ende des 10. Jahrhunderts, Gerbert von Reims (Papst Silvester II.), gezeigt, wie die Arithmetik für die philosophische Erörterung verwendet werden kann. Dabei wird zunächst die Philosophie der Zahl vorgestellt, die bei Gerbert (in Anknüpfung an Boethius) das arithmetische Fundament der Wirklichkeit aufdeckt, da Zahlen gleichzeitig Gedanken Gottes sind. Die praktische Arithmetik wiederum, in der Gerbert insbesondere hervortrat (westarabische Ziffern und der Abakus), schärft den menschlichen Verstand und ermöglicht das Verständnis der Natur der Zahlen.
12
Content available Implementacja wyrażeń arytmetycznych w rekonfigurowalnych sterownikach logicznych
63%
Milik A. , Pułka A.
Pomiary Automatyka Kontrola
|
2011
|
tom R. 57, nr 8
842-844
PL
W artykule przedstawiono metodę odwzorowania operacji arytmetycznych przeznaczoną dla rekonfigurowalnych sterowników logicznych. Istotą opracowanej metody jest wykorzystanie własności układów sprzętowych oraz architektury FPGA. W procesie implementacji brane są pod uwagę czas realizacji obliczeń oraz ograniczone zasoby logiczne. W oparciu o metodę szacowania czasu propagacji zrealizowano metodę łańcuchowego łączenia operacji kombinacyjnych pozwalającą na wykonanie wielu operacji w cyklu obliczeniowym.
EN
The paper presents a package for arithmetic operation synthesis dedicated for reconfigurable logic controllers. Different representations (graphical or textual) commonly used are handled. The synthesis process starts from transforming algorithm representation into a data flow graph. The constant reduction and the tree height reduction optimization method are applied to the flow graph (Fig. 2). The developed method combines the ALAP and list allocation strategies with original elements. The main constraint is put to the number of available logic resources that can be allocated. The procedure attempts to allocate resources assuring it proper utilization in a calculation process. Together with resource allocation the operation scheduling is performed. During operation assignment the propagation time based concept of operation scheduling is used. The proposed method allows using sequential and combinatorial units. Operations are chained inside one state until total combinatorial propagation time does not exceed the assumed cycle time. This allows reducing the required number of calculation cycles by introducing combinatorial chains of operations (Figs. 3 and 4). Finally, an example of PID controller implementation is considered and compared with previous manual implementations (Fig. 5). Introducing the automatic implementation method allows reducing radically the calculation time (2.18 times) with little increase in hardware resources (+18%) (see Tab. 1).
13
On Writing and Handwriting
51%
Kučera M.
Journal of Pedagogy
|
2010
|
tom 1
|
nr 2
11-28
EN
Writing is often considered secondary to the spoken language, as it is only coded sound-by-sound. But other scholars have demonstrated that writing is similar to ‘arithmetic’: a cognitive structuring, a shift to the meta-level (‘for the eye’). Handwriting (referred to here as the cursive writing in the sense of joined up handwriting, of ‘écriture liée’) differs from writing (in the first analysis): it has its own grammar composed of paradigmatic gestemes and tracemes and its own syntagmatic rules that connect them. In emotional terms, handwriting is designed to provide a special pleasure by its own drive (instinct, ‘Trieb’). But there is also cognitive aspect to it: the rapidity and fluidity of a cursive writing could be (in professional writing, for instance) more important (at the climax of the creative process) than it being legible for all eternity. The project of the new handwriting reform for Czech schools, abolishing the liaison between letters, is shown to be a modern and technically simplified form of calligraphy.
14
Content available Synteza diagramu LD z operacjami arytmetycznymi przeznaczona dla układów FPGA
51%
Milik A.
Pomiary Automatyka Kontrola
|
2012
|
tom R. 58, nr 7
617-619
PL
W artykule przedstawiono automatyczną metodę syntezy układu sterowania danego w postaci diagramu stykowego LD lub listy instrukcji IL do sprzętowego układu sterowania implementowanego w układzie FPGA. Zaproponowana metoda pozwala uzyskać sprzętowy układ sterowania zachowujący sekwencyjne własności przetwarzania wynikające z zapisu LD i IL. Przedstawiony algorytm syntezy pozwala na dokonanie syntezy operacji logicznych i arytmetycznych. Istotnymi celami opracowanego algorytmu jest masowe przetwarzanie, redukcja cykli obliczeniowych oraz odwzorowanie w ograniczonej liczbie zasobów operacji arytmetycznych.
EN
The paper presents the synthesis algorithm of a ladder diagram (LD) or instruction list (IL) into a reconfigurable logic controller implemented in FPGA [5, 8, 9]. The algorithm incorporates synthesis of Boolean and fixed point arithmetic operations. It utilizes the intermediate form of the data flow graph (DFG) [4, 6]. PLCs introduce variable dependencies caused by serial processing of LD (Fig. 1). It has been proved that appropriate distribution of feedback signals allows implementing LD logic dependencies during a single calculation cycle (Fig. 2). The LD diagram is compiled into DFG that records variable dependencies. The presented optimization allows reducing the controller complexity and its response time in comparison to solutions presented in [2, 3] (Fig. 3). Arithmetic operations introduce larger implementation complexity and require more time to calculate than logic operation. The DFG generated from LD or IL is used for scheduling and mapping (Fig. 4). The scheduling and mapping procedure assumes the limited number of arithmetic resources while logic operations are allocated without constraints. The scheduling procedure takes into account operation execution timing (Fig. 4C). The obtained circuit after scheduling with arithmetic operations may require more than one cycle to complete all operations in comparison to the model limited only to logic operations. The presented synthesis procedure enables obtainment of fully functional hardware implementation of the controller given by LD or IL with massively parallel processing and a very short response time (1 to several clock cycles).
15
Content available remote Combinatorial Etude
51%
Stoenchev M. , Todorov V.
Annals of Computer Science and Information Systems
|
2021
|
tom Vol. 25
231--234
EN
The purpose of this article is to consider a special class of combinatorial problems, the so called Prouhet-Tarry Escot problem, solution of which is realized by constructing finite sequences of ±1. For example, for fixed p∈N, is well known the existence of np∈N with the property: any set of np consecutive integers can be divided into 2 sets, with equal sums of its p[th] powers. The considered property remains valid also for sets of finite arithmetic progressions of complex numbers.
16
Content available Does Science Progress Towards Ever Higher Solvability Through Feedbacks Between Insights and Routines?
44%
Marciszewski W.
Studia Semiotyczne
|
2018
|
tom 32
|
nr 2
153-185
EN
The affirmative answer to the title question is justified in two ways: logical and empirical. (1) The logical justification is due to Gödel’s discovery (1931) that in any axiomatic formalized theory, having at least the expressive power of PA (Peano Arithmetic), at any stage of development there must appear unsolvable problems. However, some of them become solvable in a further development of the theory in question, owing to subsequent investigations. These lead to new concepts, expressed with additional axioms or rules. Owing to the so-amplified axiomatic basis, new routine procedures like algorithms, can be reached. Those, in turn, help to gain new insights which lead to still more powerful axioms, and consequently again to ampler algorithmic resources. Thus scientific progress proceeds to an ever higher scope of solvability. (2) The existence of such feedback cycles – in a formal way rendered with Turing’s systems of logic based on ordinal (1939) – gets empirically supported by the history of mathematics and other exact sciences. An instructive instance of such a process is found in the history of the number zero. Without that insight due to some ancient Hindu mathematicians there could not arise such an axiomatic theory as PA. It defines the algorithms of arithmetical operations, which in turn help intuitions; those, in turn, give rise to algorithmic routines, not available in any of the previous phases of the process in question. While the logical substantiation of the point of this essay is a well-established result of logico-semantic inquiries, its empirical claim, based on historical evidences, remains open for discussion. Hence the author’s intention to address philosophers and historians of science, and to encourage their critical responses.
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