The Second Euler-Lagrange Equation of Variational Calculus on Time Scales

The fundamental problem of the calculus of variations on time scales concerns the minimization of a delta-integral over all trajectories satisfying given boundary conditions. In this paper we prove the second Euler-Lagrange necessary optimality condition for optimal trajectories of variational problems on time scales. As an example of application of the main result, we give an alternative and simpler proof to the Noether theorem on time scales recently obtained in [J. Math. Anal. Appl. 342 (2008), no. 2, 1220-1226].Comment: This work was partially presented at the Workshop in Control, Nonsmooth Analysis and Optimization, celebrating Francis Clarke's and Richard Vinter's 60th birthday, Porto, May 4-8, 2009. Submitted 26-May-2009; Revised 12-Jan-2010; Accepted 29-March-2010 in revised form; for publication in the European Journal of Contro

Metropolisation Processes In Contemporary Space Of Poland

Published in: Origin Spatial Development of Contemporary Poland in Łódź University Geographical Research, edited by T.MarszałThe last 20 years of scientific research on metropolises and metropolitan areas in Poland can be divided into three basic stages. The first one, observed at the turn of the 20 th century, focused on the implementation of the basic terminology and methodology from European and American scientific literature. That was also the time of the first attempts of identify Polish metropolises and delimit their functional areas. This period strongly influenced further achievements in this field, first of all, two basic approaches towards understanding the metropolis were applied then. The first one, according to the international terminology, refers to a big city which due to well-developed metropolitan functions is considered (or aspires) to be an important centre on the global scale (global metropolis). In Poland only the capital city, Warsaw, to some extent seems to meet such criteria. In such circumstances, this term more and more often was used to refer also to other regional and supra-regional centres. As a result, scientists who certainly acknowledged this nuance, began to equate “metropolitan areas” with “urban agglomerations”. The popularity of this approach could be explained in the context of prestige and political objectives. However, despite the fact that for many years numerous attempts to institutionalize such metropolitan areas have been made, still there is no agreement on the identification of metropolises and the delimitation of their functional areas, not mentioning other issues related to metropolitan governance. The second stage of research in the described field fell on the first decade of the 21 st century. The main focus was on identification of metropolitan functions in the biggest Polish cities, usually four or five of them. At that time a dual approach towards the understanding the idea of metropolis was clearly visible. Nowadays we experience the third stage of metropolitan research, in which the internal structure of metropolitan areas as well as the relations within them attract scientific attention. Simultaneously, numerous projects focusing on smaller territorial units (e.g. small towns located in metropolitan hinterland) are being carried out. It seems that in future this stage could be followed by focusing the interest on the institutionalisation of metropolitan areas. This ought to result in integration of transport policies, land management and performing other municipal tasks, and, in a broader perspective, in establishing complex metropolitan management. The example of considerable advancement in this field may already be found in Poznań, where local authorities, together with the scientific circ les, successfully stimulate integration of the metropolitan area. To conclude, metropolises and metropolitan areas are nowadays among the most important scientific research subjects in Poland. Not only does this refer to socio-economic geography, but also to economy, urban sociology and management. A considerable share of the scientific projects in this field has application values – their results are implemented in various planning documents prepared at each level of territorial organisation. This evidently proves the importance of research activities regarding metropolisation processes as well as the necessity of their continuation in future

Domesticating Mathematics: Taxonomic Diversity in Archaeozoological Assemblages

Understanding taxonomic richness is indispensable in studying the choices made in the exploitation of the local fauna such as those of the broad-spectrum revolution in the Near East. Such investigations, however, sometimes disregard the stochastic nature of human decision-making, while others present complex mathematical models. The basic nature of samples is rarely discussed. The fundamental thesis of our study is that assemblages of different sizes can be compared only exercising extreme caution. When species richness is studied as a function of assemblage size, the remains of birds, large and “micro-” mammals show strongly differing statistics. Diversity increases along with the number of identifiable bones in the case of birds, while the same trend is most protracted in the case of large mammals. This means that taxonomic diversity in animal bone assemblages is a function of not only assemblage size but also of taxonomic composition. The structure of “input” data is thus qualitatively dependent on assemblage size. The underlying factors influencing the archaeological representation of the fauna, including sampling, anatomical and taphonomic traits of the three types of vertebrates are discussed by animal groups. Taxonomically different animal remains should not be included within the same faunal list as their culture historical interpretations (animal keeping, hunting, random natural deposition) also differ. These phenomena must all be considered when a reliable interpretation of animal remains is attempted in various archaeological periods. A régészeti állatcsont-anyagok fajgazdagságának megértése nélkülözhetetlen olyan kérdések kutatásában, mint pl. a közel-keleti őskori közösségek által fogyasztott állatfajok rendszertani spektrumának szélessége, választékának kihasználása. Az ilyen vizsgálatok során sokan figyelmen kívül hagyják az emberi döntések sztochasztikus természetét, míg mások bonyolult matematikai modellekkel igyekeznek leírni azt. A minták alapvető természetére kevesen fordítanak figyelmet. Vizsgálataink alaptétele, hogy a különböző nagyságú minták csak rendkívüli körültekintéssel hasonlíthatók össze. Amikor a meghatározott fajok számát a mintanagyságok függvényében vizsgáljuk, a madarak, a nagytestű emlősállatok és a „mikrofauna” emlős fajai más-más értékeket mutatnak. A fajgazdagság nem lineárisan követi a csontok darabszámának alakulását: ahhoz viszonyítva a madarak esetében növekszik a legintenzívebben, leginkább pedig a nagytestű emlősök csoportjában marad el a mintanagyság gyarapodásától. Eszerint az egyes leletegyüttesek állattani változatossága nemcsak a mintanagyságtól, hanem a taxonómiai összetételtől is függ. Ez azt mutatja, hogy a régészeti értelmezésben már a „bejövő” alapadatok összetétele is minőségileg függ a mintanagyságtól. A tanulmányban körvonalazzuk a háttérben meghúzódó okokat, a három vizsgált gerinces állatcsoport reprezentáltságát meghatározó mintavételi, tafonómiai és anatómiai különbségeket. E maradványok értelmezése nem mosható egybe egyetlen „faunalistában”, hiszen kultúrtörténeti jelentésük (állattartás/vadászat, madarászat és véletlenszerű, természetes lerakódás) is változó. Mindez nem mellőzhető az állatok különböző régészeti korszakokban játszott szerepének hitelességre törekvő értékelésekor

Social linkages in Lodz Metropolitan Area

Social Linkages in Lodz Metropolitan Area. In this paper, the results of the research on territorial cohesion of Lodz Metropolitan Area, perceived in the context of social linkages, are presented. The author took under consideration commuting to places of work and education, as well as customer flows to the chosen service facilities (the example of cultural institutions located in Lodz). The analysis included in the article is based on varied data, characterized in its first part. In the next three parts, the author successively analyzes types of social linkages. The final step of the paper is the synthesis, in which the strength and the spatial extent of social linkages in Lodz Metropolitan Area are assessed