This paper contains detailed description of two consistency proofs, which state that in the system called Peano arithmetic no contradiction can be obtained. The proofs were first published in 1936 and 1938 by the German mathematician Gerhard Gentzen. For the purpose of this paper, the proofs were read and studied from the original articles called "Die Widerspruchsfreiheit der reinen Zahlentheorie" and "Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie". The first mentioned proof is interesting from the historical point of view. Gentzen used a natural deduction sequent calculus and ordinal numbers in an unusual form he invented. The second proof is similar to the consistency proof, which is commonly known as a consistency proof for Peano arithmetic nowadays.

The aim of this thesis is Russell's analysis of Peano arithmetic. This analysis was presented by Russell and A. N. Whitehead in the book Prin- cipia Mathematica and then in Russell's book Introduction to Mathema- tical Philosophy which provided this approach in a more accessible form. The thesis focuses on Russell's critique of original Peano axioms and his effort to use only logical definitions instead of axioms. Another goal of the thesis is Russell's theory of classes and substitution of classes by propositional functions. Furthermore, the type theory for propositional functions is introduced and explained. All is converted into present-day logical notation. Moreover, the non-standard models of Russell's Peano arithmetic are studied. Finally, there are two particular arithmetic exam- ples illustrating the purposes of the thesis. Key words Bertrand Russell, Peano Arithmetic, classes, propositional funkction

of doctoral thesis Study of Arithmetical Structures and Theories with Regard to Representative and Descriptive Analysis Petr Glivický We are motivated by a problem of understanding relations between local and global properties of an operation o in a structure of the form B, o , with regard to an application for the study of models B, · of Peano arithmetic, where B is a model of Presburger arithmetic. We are particularly interested in a dependency problem, which we formulate as the problem of describing the dependency closure iclO (E) = {d ∈ Bn ; (∀o, o ∈ O)(o E = o E ⇒ o(d) = o (d))}, where B is a structure, O a set of n-ary operations on B, and E ⊆ Bn. We show, that this problem can be reduced to a definability question in certain expansion of B. In particular, if B is a saturated model of Presburger arithmetic, and O is the set of all (saturated) Peano products on B, we prove that, for a ∈ B, iclO ({a}×B) is the smallest possible, i.e. it contains just those pairs (d0, d1) ∈ B2 for which at least one of di equals p(a), for some polynomial p ∈ Q[x]. We show that the presented problematics is closely connected to the descriptive analysis of linear theories. That are theories, models of which are - up to a change of the language - certain discretely ordered modules over specific discretely ordered...

The thesis focuses on the subject Optimality of Maastricht fiscal criteria in the light of the economic theory. The first chapter presents the functions of public finances and the discrepancies in their understanding. It is aiming at some types of public deficits and their consequences. It introduces the topics concerning possiblities of public debt solving, Maastricht criteria and Stability and Growth Pact observance. The second chapter brings the opinions about unappropriate current fiscal criteria and offers many alternative possibilities to calculate them. The third chapter presents the European Commission prediction of public finances sustainability in Europe compared to the predictions in stability and convergence programmes of European countries. The second part of the last chapter focuses on the empirical analysis of different scenarios of public finances and other indicators development. The scenarios include fiscal arithmetic used for the Maastricht fiscal convergence criteria determination.

Magistriniame darbe ,,Formulių redukcija multiplikatyvioje aritmetikoje” nagrinėjamas sekvencinis multiplikatyvios aritmetikos variantas su lygybe. Šis skaičiavimas yra bazė, kuriant skaičiavimus,naudojamus automatizuojant įrodymus įvairiuose aritmetikos fragmentuose. Darbo tikslas- susipažinti su įrodymo teorija bei jos taikymu sekvenciniame multiplikatyviosios aritmetikos variante.Darbas padalintas į 3 skyrius : pargindinės sąvokos, pagalbinės lemos ir formulių redukcija. Pradžioje pateikiamas trumpas įvadas į Peano aritmetiką.Apibrėžiamas sekvencinis skaičiavimas K, turintis neloginių simbolių signatūrą {0,',P, *,=}.Savarankišką darbo dalį sudaro antrasis bei trečiasis skyriai.Bet kuriai bekvantorinei skaičiavimo K formulei A(x) randama tam tikros formos jai ekvivalenti normalioji disjunkcinė forma.Taip pat nagrinėjama sutvarkytųjų formulių redukcija. / In this postgraduate work “Reduction of formulas in the multiplicative arithmetic” the sequential variant with equality of multiplicative arithmetic is being analyzed. This calculus is a base when creating calculations which are used in different fragments of arithmetic. The aim of this work is to get acquainted with a proving theory and its application in sequential variant of multiplicative arithmetic. The work is divided into 3 sections: main conceptions, auxiliary lemmas and formula’s reduction. The short introduction into Pean’s arithmetic is given in the beginning. The sequential calculus K, which has non-logical symbol’s signature {0,`,P,.,=} is being described. Sections 2 and 3 are self-sufficient parts of this work. For any formula A(x) of calculation K the equivalent normal disjunctive form is found. Also the reduction of ordered formulas is analyzed.

Visais laikais aritmetika buvo svarbus mokomasis dalykas mokykloje: mokiniai susipažįsta su natūraliaisiais ir trupmeniniais skaičiais, išmoksta aritmetikos veiksmų, sudarančių sudėtingą pakopinę struktūrą. Kiekvienas paprastas veiksmas yra sudėtingesnių veiksmų sudėtinis elementas, todėl svarbu, kad kiekvieną paprastesnį veiksmą mokinys išmoktų atlikti greitai ir be klaidų, „negalvodamas“. Daugumai mokinių aritmetikos veiksmų atlikimo sunkumai yra pagrindinė priežastis, trukdanti mokytis ne tik aritmetikos, bet ir kitų matematikos temų. Todėl svarbu ieškoti būdų, metodų ir priemonių, kaip padėti mokiniams įveikti šiuos kliuvinius. Manoma, kad mokymo(si) turinį perkeliant į elektroninę erdvę, pavyks rasti būdų, leisiančių veiksmingiau organizuoti mokymo(si) procesą. Vis dažniau kalbama apie virtualiąsias mokymo(si) aplinka, kur mokymo(si) turinys pateikiamas nepriklausomais medžiagos gabalėliais – virtualiaisiais mokymo(si) objektais. Šiuo disertaciniu tyrimu siekta nustatyti, kokios yra virtualiųjų mokymo(si) objektų taikymo galimybės ir veiksmingumas, ugdant mokinių gebėjimus atlikti aritmetikos veiksmus IV–V klasėse. / In all times arithmetic has been an important subject at school: pupils familiarize themselves with natural and fractional numbers, learn arithmetic operations, acquire computation abilities. Actions of teaching and learning mathematics make up a complicated tiered structure. Every simple operation is a constituent of more complicated operations; therefore, the pupil has to learn to perform these operations quickly and without mistakes, „without thinking“. For the majority of pupils difficulties in performing arithmetic operations is the main reason hindering not only to learn arithmetic but also other topics of mathematics. Therefore, it is important to search for the ways, methods and means facilitating pupils to cope with these hindrances. It is supposed that as the content of teaching and learning moves to the electronic space, ways enabling to control and manage the teaching and learning process will be successfully found. More often virtual teaching and learning environments are mentioned, where the content of teaching and learning is presented in independent pieces of teaching material – virtual teaching and learning objects This dissertation research was designed to determine what is a virtual teaching and learning objects the application possibilities and efficiency, developing pupils' abilities to perform arithmetic operations in forms 4 and 5.

Visais laikais aritmetika buvo svarbus mokomasis dalykas mokykloje: mokiniai susipažįsta su natūraliaisiais ir trupmeniniais skaičiais, išmoksta aritmetikos veiksmų, sudarančių sudėtingą pakopinę struktūrą. Kiekvienas paprastas veiksmas yra sudėtingesnių veiksmų sudėtinis elementas, todėl svarbu, kad kiekvieną paprastesnį veiksmą mokinys išmoktų atlikti greitai ir be klaidų, „negalvodamas“. Daugumai mokinių aritmetikos veiksmų atlikimo sunkumai yra pagrindinė priežastis, trukdanti mokytis ne tik aritmetikos, bet ir kitų matematikos temų. Todėl svarbu ieškoti būdų, metodų ir priemonių, kaip padėti mokiniams įveikti šiuos kliuvinius. Manoma, kad mokymo(si) turinį perkeliant į elektroninę erdvę, pavyks rasti būdų, leisiančių veiksmingiau organizuoti mokymo(si) procesą. Vis dažniau kalbama apie virtualiąsias mokymo(si) aplinka, kur mokymo(si) turinys pateikiamas nepriklausomais medžiagos gabalėliais – virtualiaisiais mokymo(si) objektais. Šiuo disertaciniu tyrimu siekta nustatyti, kokios yra virtualiųjų mokymo(si) objektų taikymo galimybės ir veiksmingumas, ugdant mokinių gebėjimus atlikti aritmetikos veiksmus IV–V klasėse. / In all times arithmetic has been an important subject at school: pupils familiarize themselves with natural and fractional numbers, learn arithmetic operations, acquire computation abilities. Actions of teaching and learning mathematics make up a complicated tiered structure. Every simple operation is a constituent of more complicated operations; therefore, the pupil has to learn to perform these operations quickly and without mistakes, „without thinking“. For the majority of pupils difficulties in performing arithmetic operations is the main reason hindering not only to learn arithmetic but also other topics of mathematics. Therefore, it is important to search for the ways, methods and means facilitating pupils to cope with these hindrances. It is supposed that as the content of teaching and learning moves to the electronic space, ways enabling to control and manage the teaching and learning process will be successfully found. More often virtual teaching and learning environments are mentioned, where the content of teaching and learning is presented in independent pieces of teaching material – virtual teaching and learning objects This dissertation research was designed to determine what is a virtual teaching and learning objects the application possibilities and efficiency, developing pupils' abilities to perform arithmetic operations in forms 4 and 5.

Title: Complexity Theory in Feasible Mathematics Author: Ján Pich Department: Department of Algebra Supervisor: Prof. RNDr. Jan Krajíček, DrSc., MAE Abstract: We study the provability of statements and conjectures from Complex- ity Theory in Bounded Arithmetic. First, modulo a hardness assumption, we show that theories weaker in terms of provably total functions than Buss's theory S1 2 cannot prove nk -size circuit lower bounds for SAT formalized as a Σb 2-formula LB(SAT, nk ). In particular, the true universal first-order theory in the language containing names for all uniform NC1 algorithms denoted TNC1 does not prove LB(SAT, n4kc ) where k ≥ 1, c ≥ 2 unless each function f ∈ SIZE(nk ) can be approximated by formulas Fn of subexponential size 2O(n1/c) with subexponential advantage: Px∈{0,1}n [Fn(x) = f(x)] ≥ 1/2 + 1/2O(n1/c) . Unconditionally, V 0 does not prove quasipolynomial nlog n -size circuit lower bounds for SAT. Considering upper bounds, we prove the PCP theorem in Cook's theory PV1. This includes a formalization of the (n, d, λ)-graphs in PV1. A consequence of the result is that Extended Frege proof system admits p-size proofs of tautologies encoding the PCP theorem. Keywords: Circuit Lower Bounds, Bounded Arithmetic, The PCP theorem

The thesis is devoted to ancient Indian mathematics; it describes the mathe- matical knowledge, computational techniques and methods for solving various ari- thmetic, algebraic and geometric problems that the Indians knew and used. The thesis follows the development of Indian mathematics from the oldest knowledge contained in ancient Vedic texts to the knowledge originated from the classic me- dieval arithmetic and algebraic works. This is the first comprehensive text written in Czech which contains the translation of original problems and analysis of their solutions in the current mathematical formulation and symbolism. The sources are mainly English translations of ancient Sanskrit texts and their commentaries.

The main content of the thesis is the analysis of elective mathematics lessons including examples of the tasks. This work could be useful mainly to teachers of primary elective mathematics. It is divided into two parts. The first one deals with the main theory. The second part is based on the analysis of particular lessons and tasks. The work also contains a set of tasks suitable for primary pupils.