Podstawy matematyki bez aktualnej nieskończoności

Contemporary mathematics significantly uses notions which belong to ideal mathematics (in Hilbert’s sense) – which is expressed in language which essentially uses actual infinity. However, we do not have a meaningful notion of truth for such languages. We can only reduce the notion of truth to finitistic mathematics via axiomatic theories. Nevertheless, justification of truth of axioms themselves exceeds the capabilities of the theory based on these axioms. On the other hand, we can easily decide the truth or falsity of a statement in finite structures. The aim of this dissertation is to identify the fragment of mathematics, which is of the finitistic character. The fragment of mathematics which can be described without actual infinity. This is the part of mathematics which can be described in finite models and for which the truth of its statements can be verified within finite models.We call this fragment of mathematics with a term introduced by Knuth – the concrete mathematics. This part of mathematics is of computational character and it is closer to our empirical base, which makes it more difficult. We consider concrete foundations of mathematics, in particular the concrete model theory and semantics without actual infinity. We base on the notion of FM–representability, introduced by Mostowski, as an explication of expressibility without actual infinity. By the Mostowski’s FM–representability theorem, FM–representable notions are exactly those, which are recursive with the halting problem as an oracle. We show how to express basic concepts of model theory in the language without actual infinity. We investigate feasibility of the classical model– theoretic constructions in the concrete model theory. We present the Concrete Completeness Theorem and the Low Completeness Theorem; the Concrete Omitting Types Theorem; and Preservation Theorems. We identify the constructions which are not admissible in the concrete model theory by showing stages of these constructions which are not allowed in the concrete framework. We show which arguments from the axiomatic model theory fail in the concrete model theory. Moreover, we investigate how to approximate truth for finite models. In particular we study the properties of approximate FM–truth definitions which are expressible in modal logic. We introduce modal logic SL, axioms of which mimic the properties of a specific approximate FM–truth definition. We show that SL is the modal logic of any approximate FM–truth definition. This is done by proving a theorem analogous to Solovay’s completeness theorem for modal logic GL.Współczesna matematyka w znaczącej mierze posługuje się pojęciami, które należą do matematyki idealnej (w sensie Hilberta) -- wyrażona jest w języku istotnie wykorzystującym aktualną nieskończoność. Dla tego typu języków nie posiadamy sensownego kryterium prawdziwości. Jesteśmy w stanie jedynie redukować je do matematyki skończonościowej poprzez teorie aksjomatyczne. Niemniej uzasadnianie prawdziwości samych aksjomatów znajduje się poza zasięgiem teorii na nich opartej. Z drugiej strony w strukturach skończonych jesteśmy w stanie w prosty sposób rozstrzygać prawdziwość i fałszywość twierdzeń. Celem niniejszej rozprawy jest identyfikacja fragmentu matematyki, który ma skończonościowy charakter. Fragmentu matematyki, do którego opisu nie jest niezbędna aktualna nieskończoność, a wystarczy jedynie nieskończoność potencjalna. Jest to ta część matematyki, której pojęcia można wyrazić w modelach skończonych oraz prawdziwość twierdzeń której można w nich zweryfikować. Tę część matematyki, za Knuthem, nazywamy matematyką konkretną. Ma ona obliczeniowy, kombinatoryczny charakter i jest bliższa naszemu doświadczeniu niż matematyka idealna, a co za tym idzie jest trudniejsza. Rozważamy konkretne podstawy matematyki, w szczególności konkretną teorię modeli oraz semantykę bez aktualnej nieskończoności. Opieramy się na wprowadzonym przez Mostowskiego pojęciu FM--reprezentowalności, jako eksplikacji wyrażalności bez aktualnej nieskończoności oraz twierdzeniu o FM--reprezentowalności identyfikującym FM--reprezentowalne pojęcia z tymi, które są obliczalne z problemem stopu jako wyrocznią. Pokazujemy w jaki sposób można zinterpretować podstawowe pojęcia teorii modeli w języku bez aktualnej nieskończoności. Następnie badamy klasyczne konstrukcje teoriomodelowe pod kątem ich wykonalności w obszarze matematyki konkretnej. Prezentujemy twierdzenie o konkretnej pełności oraz twierdzenie o łatwej pełności, twierdzenie o omijaniu typów oraz twierdzenia o zachowaniu. Przedstawiamy konstrukcje, które są niewykonalne dla modeli konkretnych, identyfikując etapy konstrukcji teoriomodelowych, które nie są wykonalne w teorii modeli konkretnych. Identyfikujemy argumenty z aksjomatycznej teorii mnogości, które nie są dopuszczalne w konkretnej teorii modeli. Ponadto, badamy możliwość przybliżania prawdy arytmetycznej w modelach skończonych. W szczególności rozważamy te własności przybliżonych predykatów prawdy dla modeli skończonych, które wyrażalne są w logice modalnej. Wprowadzamy logikę modalną SL, której aksjomaty odzwierciedlają własności przybliżonych predykatów prawdy. Pokazujemy, że logika SL jest logiką przybliżonych predykatów prawdy -- dowodzimy twierdzenia analogicznego do twierdzenia o pełności dla logiki GL udowodnionego przez Solovaya

Polarization of tau leptons in semileptonic B decays

Analytic formulae for the one-loop order QCD corrections to the differential width of the semileptonic b decay are given with the tau polarization taken into account. Thence the polarization of tau is expressed by its energy and the invariant mass of the tau + antineutrino system. The non-perturbative corrections by Falk et al. are incorporated in the calculation.Comment: one footnote and one reference have been added; the paper is going to be published in Nucl. Phys.

Tractive Performance of Tyres in Forest Conditions – Impact Assessment of Ground and Tyres Parameters

This article deals with the assessment of traction properties of tyres on forest grounds. The research was carried out on skid trails located in pine stands. The tested grounds were different due to the cover of the soil and its mechanical properties. The study also deals with the evaluation of ways to improve traction by reducing the inflation pressure and using the tyre chain. The research was carried out using a specialized traction test stand for two tyres (9.5–24 and 400/55–22.5) different in width and tread pattern. The studies showed significant effect of ground conditions on traction. As a result of changes in the ground conditions, the values of drawbar force, rolling resistance and tractive efficiency were altered by 25%, 23% and 6%, respectively. The higher values of the drawbar force and tractive efficiency on all tested trails were obtained for 400/55–22.5 tyre. Both the use of tyre chains and the reduction of inflation pressure resulted in the increase in drawbar force and tractive efficiency. A better way to improve traction properties was the reduction of the tyre inflation pressure, which caused the increase in drawbar force and tractive efficiency. The use of tyre chains caused an increase in drawbar force over the entire slip range, while an increase in tractive efficiency has only been shown for the slip larger than 15%

Damage to Soil and Residual Trees Caused by Different Logging Systems Applied to Late Thinning

This paper concerns the evaluation of logging systems in terms of the damage to the forest ecosystem. Damages to trees and soil during late thinning conducted in foothills areas in Poland using tree-length and cut-to-length logging systems were assessed. In both stands, the test plots were located within the primary and secondary skid trails. In the study, areas occupied by skid trails were determined as well as the depth of ruts. In order to determine changes in the soil properties at selected measurement points, a soil penetration resistance and a maximum shearing stress were measured. For each logging system, the share of trees damaged during harvesting operations and location of injuries were determined. The studies have shown that a 70% larger area was required to form technological trail with CTL than with TL. After CTL, skid trails were scarred by shallow ruts, and the share of ruts with the depth between 0.16 and 0.25 was three times smaller than after TL. The average increase in penetration resistance of soil in the ruts after TL was 324% and 302% and after CTL 308% and 220%, respectively, for primary and secondary skid trail, in comparison to the values obtained in measurement points located 5 m from the trails. In TL, comparable changes of soil properties were caused by skidder wheels and by hauled wood. The research has shown a greater share of damaged trees after TL. In both logging systems, the most damage was found within the root collar and lower parts of the bole

Doming Modes and Dynamics of Model Heme Compounds

Synchrotron far-IR spectroscopy and density-functional calculations are used to characterize the low-frequency dynamics of model heme FeCO compounds. The “doming” vibrational mode in which the iron atom moves out of the porphyrin plane while the periphery of this ring moves in the opposite direction determines the reactivity of oxygen with this type of molecule in biological systems. Calculations of frequencies and absorption intensities and the measured pressure dependence of vibrational modes in the model compounds are used to identify the doming and related normal modes

Rozwój ICT w krajach Grupy Wyszehradzkiej - w poszukiwaniu przewag polskiego rynku teleinformatycznego

Recent years show an increase in the use of modern technologies, including infor­mation and communication technology ICT. This confirms the indicator analysis: The Networked Readiness Index (NRI), The Global Competitiveness Index (GCI), The Digital Economy and Society Index (DESI). Poland leads in digital public services and mobile broadband among the Visegrad Countries

FICS 2010

International audienceInformal proceedings of the 7th workshop on Fixed Points in Computer Science (FICS 2010), held in Brno, 21-22 August 201

Vergence eye movements in bipolar disorder

Aim. With respect to bipolar disorder (BD), previous research have demonstrated saccadic eye movements abnormalities, manifested mainly as an increase in reaction time (latency) in both prosaccadic and antisaccadic task. So far, there were no studies related to vergence eye movements in subjects with BD. Our primary aim was to evaluate vergence tracking performance in this clinical group.  Methods. 30 patients with BD in remission and 23 healthy controls were enrolled. Subjects underwent optometric examination where near point of convergence was measured by the use of Wolff Wand. Instrumented convergence measurements were performed using infrared eye tracker and dedicated vergence stimuli generator. Results. BD patients presented significantly higher average error between eyes’ convergence and convergence required to fixate the target and higher number of saccadic intrusions compared with healthy controls group. Principal component analysis performed on oculometric parameters revealed differences between BD patients and healthy controls. Significant correlations between the vergence disturbances and saccadic intrusions were found. Conclusions. BD patients showed the alterations of the vergence eye movements similar to the disturbances of eye movements in the fronto-parallel plane. While the abnormalities of vergence eye movements in some mental disorders have been reported, we have for the first time objectively measured this phenomenon in BD