Quantum information as the information of infinite collections or series

The quantum information introduced by quantum mechanics is equivalent to a certain generalization of classical information: from finite to infinite series or collections. The quantity of information is the quantity of choices measured in the units of elementary choice. The “qubit”, can be interpreted as that generalization of “bit”, which is a choice among a continuum of alternatives. The axiom of choice is necessary for quantum information. The coherent state is transformed into a well-ordered series of results in time after measurement. The quantity of quantum information is the transfinite ordinal number corresponding to the infinity series in question. The transfinite ordinal numbers can be defined as ambiguously corresponding “transfinite natural numbers” generalizing the natural numbers of Peano arithmetic to “Hilbert arithmetic” allowing for the unification of the foundations of mathematics and quantum mechanics.

Trudności w opanowaniu odmiany liczebników w nauczaniu języka polskiego jako obcego na przykładzie studentów ukraińsko i rosyjskojęzycznych.

The aim of this paper is to describe the most common errors made by Ukrainian- and Russian-speaking people while learning infl ection of numerals for person and their causes. The applied comparative method permits a closer look at formal and syntactic relations as well as collocations of numerals and specifi c lexeme classes (this regards primarily agreement between cardinal numbers and masculine personal verbs, compound structures with the numeral jeden (one) and numerals showing adjectival infl ection). This study concentrates on a comparison of cardinal and ordinal numbers in Ukrainian, Russian and Polish, which allowed an analysis of the causes of errors made while learning the infl ection of numerals by Ukrainian- and Russian-speaking students.

Set Theory

The chapter is a concise, practical presentation of the basics of set theory. The topics include set equivalence, countability, partially ordered, linearly ordered, and well-ordered sets, the axiom of choice, and Zorn’s lemma, as well as cardinal numbers and cardinal arithmetic. The first two sections are essential for a proper understanding of the rest of the book. In particular, a thorough understanding of countability and Zorn’s lemma are indispensable. Parts of the section on cardinal numbers may be included, but only an intuitive understanding of cardinal numbers is sufficient to follow such topics as the discussion on the existence of a vector space of arbitrary (infinite) dimension, and the existence of inseparable Hilbert spaces. Cardinal arithmetic can be omitted since its applications later in the book are limited. Ordinal numbers have been carefully avoided.

Arithmatics in Serat Centhini

Serat Centhini (Suluk Tambanglaras or Suluk Tambangraras-Amongraga) is one of the greatest literary works in New Javanese literature. Its construction began in 1742 Java (1814 AD) during Sunan Pakubuwana IV. The main reference is the Jatiswara book which was made in 1711 Java or 1783 AD during Sunan Pakubuwana III. This article tracks the arithmetic information contained in Serat Centhini, including cardinal numbers, ordinal numbers, Javanese calendar, and candrasengkala. The research was carried out by using the literature study method of Serat Centhini which is translated into Indonesian and transliterated in Latin script. The results showed that Serat Centhini recorded quite a lot of mathematical information which was used in the daily life of Javanese people. The results of this study can be used as teaching materials for ethnomathematics.

A Comparative Morphological Approach to Class Maintaining Derivational Affixes in English and Kurdish

This paper is a comparative morphological study of some class maintaining derivational affixes that do not alter the grammatical categories lexemes in Standard English and Central Kurdish from the standpoints of Generative Morphology. For the comparative analysis of the two languages, some of the derivational affixes that form new meanings from the existing lexemes and retain the grammatical categories of the newly derived lexemes have been classified. The main aim of the study is to identify the points of similarity and difference of class maintaining derivational affixes in both languages. The findings indicate that in the addition of nominal affixes, English and Kurdish are similar in that ‘concrete nouns’ could remain concrete nouns, as well as could convert into abstract nouns by adding certain affixes. In English, a prefix can also be added to a concrete noun to derive a new concrete noun, whereas in Kurdish, only a prefix can be added to an abstract noun to form a concrete noun. In the addition of adjectival affixes, both languages are similar in that adjectives can derive new adjectives by attaching some prefixes and some suffixes to the existing lexemes. In English, the cardinal numbers remain cardinals when the suffixes –teen and –ty are attached to them, whereas in Kurdish the only rare case can be seen when the suffix –a is attached to the two cardinal numerals hawt/ haft ‘seven’ and hašt ‘eight’. The suffixes –th in English and -(h)am and -(h)amin in Kurdish can be attached to the cardinal numbers to form the ordinal numbers.

PRIME NUMBER LAW. DEPENDENCE OF PRIME NUMBERS ON THEIR ORDINAL NUMBERS AND GOLDBACH – EULER BINARY PROBLEM USING COMPUTER

The article considers the methods of defining and finding the distribution of composite numbers CN, prime numbers PN, twins of prime numbers Tw and twins of composite numbers TwCN that do not have divisors 2 and 3 in the set of natural numbers - ℕ based on a set of numbers like Θ = {6∙κ ± 1, κ ∈ ℕ}, which is a semigroup in relation to multiplication. There has been proposed a method of obtaining primes by using their ordinal numbers in the set of primes and vice versa, as well as a new algorithm for searching and distributing primes based on a closedness of the elements of the set Θ. It has been shown that a composite number can be presented in the form of products (6x ± 1) (6y ± 1), where x, y ℕ - are positive integer solutions of one of the 4 Diophantine equations: . It has been proved that if there is a parameter λ of prime twins, then none of Diophantine equations P (x, y, λ) = 0 has positive integer solutions. There has been found the new distribution law of prime numbers π(x) in the segment [1 ÷ N]. Any even number is comparable to one of the numbers i.e. . According to the above remainders m, even numbers are divided into 3 types, each type having its own way of representing sums of 2 elements of the set Θ. For any even number in a segment [1 ÷ ν], where ν = (ζ−m) / 6, , there is a parameter of an even number; it is proved that there is always a pair of numbers that are elements of the united sets of parameters of prime twins and parameters of transition numbers , i.e. numbers of the form with the same λ, if the form is a prime number, then the form is a composite number, and vice versa.

Ordinals by Abstraction

The neo-Fregean programme in the philosophy of mathematics seeks to provide foundations for fundamental mathematical theories in abstraction principles. Ian Rumfitt (2018) proposes to introduce ordinal numbers by means of an abstraction principle, (ORD), which says, roughly, that ‘the ordinal number attaching to one well-ordered series is identical with that attaching to another if, and only if, the two series are isomorphic’. Rumfitt’s proposal poses a sharp and serious challenge to those seeking to advance the neo-Fregean programme, for Rumfitt proposes to save (ORD) from threatening paradox by avoiding dependence on an impredicative comprehension principle. However, such a principle is usually taken to be required by the neo-Fregean account of the cardinal numbers. Thus if neo-Fregean foundations for elementary arithmetic are to be saved, we must explain how we can avoid paradox for (ORD) in another way. In this chapter, the prospects for doing so are explored.

Joint Optimization of Well Locations, Types, Drilling Order, and Controls Given a Set of Potential Drilling Paths

Summary Much work has been performed on the optimal well placement/control problem, including some investigations on optimizing well types (injector or producer) and/or drilling order. However, to the best of our knowledge, there are only a handful of papers dealing with the following problem that is sometimes given to reservoir-engineering groups: given a potential set of reasonable drilling paths and a drilling budget that is sufficient to drill only a few wells, find the optimal well paths, determine whether a well should be an injector or a producer, and determine the drilling order that maximizes the net present value (NPV) of production over the life of the reservoir. In this work, the optimal choices of drilling paths, types, and drilling order are found using the genetic algorithm (GA) with mixed encodings. A binary encoding for the optimization variables pertaining to well-location indices and well types is proposed to effectively handle a large amount of categorical variables, while the drilling sequence is parameterized with ordinal numbers. These two sets of variables are optimized both simultaneously and sequentially. Finally, control optimization using a stochastic simplex approximate gradient (StoSAG) is performed to further improve the NPV of life-cycle production. The proposed workflow is tested on two examples: a 3D channelized reservoir where the potential well paths are either vertical or horizontal, and the Brugge model where only vertical wells are drilled. Both numerical examples indicate that GA combined with StoSAG is a viable solution to the problem considered.