Conditions for the difference set of a central Cantor set to be a Cantorval

Let $C(\lambda )\subset \lbrack 0,1]$ denote the central Cantor set generated by a sequence $\lambda = \left( \lambda_{n} \right) \in \left( 0,\frac{1}{2} \right) ^{\mathbb{N}}$. By the known trichotomy, the difference set $C(\lambda )-C(\lambda )$ of $C(\lambda )$ is one of three possible sets: a finite union of closed intervals, a Cantor set, and a Cantorval. Our main result describes effective conditions for $(\lambda_{n})$ which guarantee that $C(\lambda )-C(\lambda )$ is a Cantorval. We show that these conditions can be expressed in several equivalent forms. Under additional assumptions, the measure of the Cantorval $C(\lambda )-C(\lambda )$ is established. We give an application of the proved theorems for the achievement sets of some fast convergent series.Comment: 19 page

Cognition as shaking hands with the world. Introduction

One of the most common questions in today’s cognitive studies is the one regarding embodied cognition. The answer to this question draws our attention to many factors, including bodily actions, which also work to embody cognition. With this in mind, enactivism is included in discussions of embodiment

Problematyka niezawodności procesów logistycznych. Proces magazynowania

The article presents the role and meaning of the storage process in a company's functioning. The Author defines the notion of process' reliability and refers it to the storage process. The publication includes also detailed valuation of the storage process and an analysis of system's unreliability reasons

Zapiska o chrzcie Mieszka I na tle rozwoju najstarszych roczników polskich. U źródeł piastowskiej ideologii władzy

The note about the baptism of Mieszko I comes from the oldest Polish annals, created at the court of Bolesław Chrobry from around 984/985. It covered the years 965–990. The note was later added to it under the year 1013. The annals remained in Cracow during the Czech invasion of 1038 and was not taken to the Czech state. It was different from the one kept by Archbishop Gaudente in Gniezno

Historia mieszkańców Bydgoszczy do 1772 roku

The first residents of the newly chartered city, who were brought by the founders (1346), were of German origin. However in the second decade of the 15th century, they were a minority. It resulted from the fact that the German settlers stopped coming to the area after the first colonization campaign and migration of Polish people from the nearby villages and towns. In the late 16th century, population of Bydgoszcz and its suburbs was around 4,000-5,000. In the period between the end of the 14th century and the beginning of the 15th century many residents, including vogts and mayors, were involved in trade activities in the State of the Teutonic Knights. The commercial character of the city, related to its location on the Brda River and connection of the Vistula River with Gdańsk, dominated until the first partition of Poland. The most popular trade craft was beer making. Brewing developed outside the guild. The first craftsmen who established local guilds were butchers and potters. In the late 16th century, there were 12 guilds in the city. Religious buildings (monasteries, churches and hospitals) funded by the local townsmen confirm their affluence. Church fraternities flourished and religious celebrations were popular pastimes, although secular leisure activities were also common. The fall of the city was initiated by the war with Sweden in 1626-1629 and the earlier plaque (1623). Numerous calamities (fires) took place during the war of 1655-1660. Population size dropped most likely to 2,000. There were even fewer residents after the takeover of power by Prussia in 1772 (between 1,000 and 1,500)

Methods of enhancing the performance and security of web applications

. This article presents the methods of enhancing the performance and security of web applications. It also evaluates them and describes how to use them. The methods have been implemented in a test application. The article presents results of the research comparing state of the application before and after implementation of every listed method. Conclusions about impact of the methods of enhancing performance and security of web applications are based on the results of the research