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This article is the first in a series formalizing some results in my joint work with Prof. Joanna Golinska-Pilarek ([12] and [13]) concerning a logic proposed by Prof. Andrzej Grzegorczyk ([14]).

We present some mathematical folklore about representing formulas in “Polish notation”, that is, with operators of fixed arity prepended to their arguments. This notation, which was published by Jan Łukasiewicz in [15], eliminates the need for parentheses and is generally well suited for rigorous reasoning about syntactic properties of formulas.

References Ausubel, D. (1963). The psychology of meaning verbal learning. 1st ed. New York: Grune and Stratton. Dobesova, Z. (2013). USING THE PHYSICS OF NOTATION TO ANALYSE MODELBUILDER DIAGRAMS. In Proceedings ofSGEM 2013, 1 3th International Multidisciplinary Scientific GeoConference (pp. 595-602). Sofia: STEF92 Technology. https:!/ Drotár, P. (2008). Vyt1Zivam informaCnich technologii ve ryuce. Praha: Obcanske sdruZeni SPHV. Eysenck, M. W., & Keane, M. T. (2008). Kognitivni psychologie. Praha: Academia. Hartlová, H


We translate the articles covering group theory already available in the Mizar Mathematical Library from multiplicative into additive notation. We adapt the works of Wojciech A. Trybulec [41, 42, 43] and Artur Korniłowicz [25].

In particular, these authors have defined the notions of group, abelian group, power of an element of a group, order of a group and order of an element, subgroup, coset of a subgroup, index of a subgroup, conjugation, normal subgroup, topological group, dense subset and basis of a topological group. Lagrange’s theorem and some other theorems concerning these notions [9, 24, 22] are presented.

Note that “The term ℤ-module is simply another name for an additive abelian group” [27]. We take an approach different than that used by Futa et al. [21] to use in a future article the results obtained by Artur Korniłowicz [25]. Indeed, Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters” in Isabelle/HOL [23, 10]. Our goal is to define the convergence of a sequence and the convergence of a series in an abelian topological group [11] using the notion of filters.


The article is an attempt to reconstruct the fundamental elements of Kazimierz Serocki’s musical language on the basis of his own statements concerning his music. Those statements come first and foremost from his lectures prepared for the Meisterkurs für Komposition at the Musik-Akademie in Basel (1976), whose manuscripts are now held in the Polish Composers’ Archive of the University of Warsaw Library. The lecture texts (Notations- und Realisationsprobleme, Klangfarben als Kompositionsmaterial, Chance der offenen Form) present a whole set of problems which Serocki considered as the most important for his method of composition. The central place among these problems is occupied by the idea of “composing with sound colour” (“mit Klangfarben komponieren”). Sound colour plays a decisive role in the creative process, as it constitutes self-sufficient material for composition. Sound colour has a form-shaping role in the musical work, since it can build sequences of sound structures in various configurations, which perform various functions in the piece. The idea of composing with sound colour is presented by the composer in the context of an adequate way of notating sound phenomena and the possibility of performing music from such notation. This idea was also related in the lectures to the principles of constructing polyvalent open forms (mehrdeutige Form) out of small- and large-scale components. Pitch organisation, on the other hand, remains of secondary interest in the composer’s commentaries. Serocki’s self-reflection provides us with original and innovative answers to the most important problems that contemporary composers have had to face in their work. It also provides significant and hitherto frequently little-known insights into the components of the unique style of the author of Pianophonie, and these insights can be effectively utilised in the course of future research on Serocki’s work.

very important in most racket sports (Abian-Vicen et al., 2013). Notational analysis is one of the methods that can be applied within the observational methodology to determine the characteristics of a badminton game ( Hughes et al., 2007 ) and research exists that uses it to investigate these characteristics in this sport ( Abdullahi and Coetzee, 2017 ). The objective of this study was to analyze and compare technical and timing variables in professional players between the individual and double modalities in outdoor badminton. Methods Participants Male individual


The aim of the article is to show the relations in the innovation process planning model. The relations argued here guarantee the stable and reliable way to achieve the result in the form of an increased competitiveness by a professionally directed development of the company. The manager needs to specify the effect while initiating the realisation of the process, has to be achieved this by the system of indirect goals. The original model proposed here shows the standard of dependence between the plans of the fragments of the innovation process which make up for achieving its final goal. The relation in the present article was shown by using the standard Business Process Model and Notation. This enabled the specification of interrelations between the decision levels at which subsequent fragments of the innovation process are planned. This gives the possibility of a better coordination of the process, reducing the time needed for the achievement of its effect. The model has been compiled on the basis of the practises followed in Polish companies. It is not, however, the reflection of these practises, but rather an idealised standard of proceedings which aims at improving the effectiveness of the management of innovations on the operational level. The model shown could be the basis of the creation of systems supporting the decision making, supporting the knowledge management or those supporting the communication in the innovation processes.


One of the key purposes of Business Process Model and Notation (BPMN) is to support graphical representation of the process model. However, such models have a lack of support for the graphical representation of resources, whose processes are used during simulation or execution of process instance. The paper analyzes different methods and their extensions for resource modeling. Further, this article presents a selected set of resource properties that are relevant for resource modeling. The paper proposes an approach that explains how to use the selected set of resource properties for extension of process modeling using BPMN and simulation tools. They are based on BPMN, where business process instances use resources in a concurrency manner.

Studies, 34, 223-232. 19. Carling, C., Williams, M. A., & Reilly, T. (1995). Handbook of Soccer Match Analysis. A systematic approach to improving performance . New York: Routledge 20. Darst, P. W., Zakrajsek, D. B., & Mancini, V. H. (1989). Analyzing physical education and sport instruction . Champaign, IL: Human Kinetics. 21. Hughes, M. D., & Franks, I. M. (2004). Notational Analysis of Sport. Second Edition. New York: Routledge.


Famous Uaxactun Mural paintings, which were found in Structure B-XIII, have been wellknown to Maya scholars for decades. They are considered as proof of Maya-Teotihuacan connection, and the importance of Uaxactun. On the other hand, the different scenes with musicians, nobles and warriors are beautiful source of Maya iconography. Below these paintings, also a short inscription in form of calendric notation was found. This inscription was not analyzed due to imperfect drawing, which was made by Antonio Tejeda in 1930’s. Fortunately, it was possible to create a new rendering of this inscription, thanks to preserved photographs of Carnegie Institute.

–514. Hughes, M., & Franks, M. (2004). Notational analysis of sport . London, UK: Routledge. Jonsson, G. K., Anguera, M. T., Blanco-Villaseñor, Á., Losada, J. L., Hernández-Mendo, A., Ardá, T., … Castellano, J. (2006). Hidden patterns of play interaction in soccer using SOF-CODER. Behavior Research Methods , 38 (3), 372–381. Kalamaras, D. (2014). Social Networks Visualizer (SocNetV): Social network analysis and visualization software. Social Networks Visualizer . Online Multimedia, Homepage: . Lago-Ballesteros, J., & Lago-Peñas, C. (2010