On Capital Dependent Dynamics of Knowledge

We investigate the dynamics of growth models in terms of dynamical system theory. We analyse some forms of knowledge and its influence on economic growth. We assume that the rate of change of knowledge depends on both the rate of change of physical and human capital. First, we study model with constant savings. The model with optimised behaviour of households is also considered. We show that the model where the rate of change of knowledge depends only on the rate of change of physical capital can be reduced to the form of the two-dimensional autonomous dynamical system. All possible evolutional paths and the stability of solutions in the phase space are discussed in details. We obtain that the rate of growth of capital, consumption and output are greater in the case of capital dependent rate of change of knowledge.Comment: 11 pages, 5 figures; presented at 2nd Polish Symposium on Econo- and Sociophysics, Krakow, Poland, 21-22 April 2006; to be published in Acta Phys. Pol.

Phantom cosmology as a simple model with dynamical complexity

We study the Friedmann-Robertson-Walker model with phantom fields modelled in terms of scalar fields. We apply the Ziglin theory of integrability and find that the flat model is nonintegrable. Then we cannot expect to determine simple analytical solutions of the Einstein equations. We demonstrate that there is only a discrete set of parameters where this model is integrable. For comparison we describe the phantoms fields in terms of the barotropic equation of state. It is shown that in the contrast to the phantoms modelled as scalar fields, the dynamics is always integrable and phase portraits are constructed. In this case we find the duality relation.Comment: RevTeX4, 11 pages, 2 figures; rewritten (v.2); new section on complex dynamics in cases with and without spontaneous symmetry breaking (v.3); this new section on numerical analysis of complex dynamics was dropped, upload to match version in PRE (v.4

Vertices cannot be hidden from quantum spatial search for almost all random graphs

In this paper we show that all nodes can be found optimally for almost all random Erd\H{o}s-R\'enyi ${\mathcal G}(n,p)$ graphs using continuous-time quantum spatial search procedure. This works for both adjacency and Laplacian matrices, though under different conditions. The first one requires $p=\omega(\log^8(n)/n)$, while the seconds requires $p\geq(1+\varepsilon)\log (n)/n$, where $\varepsilon>0$. The proof was made by analyzing the convergence of eigenvectors corresponding to outlying eigenvalues in the $\|\cdot\|_\infty$ norm. At the same time for $p<(1-\varepsilon)\log(n)/n$, the property does not hold for any matrix, due to the connectivity issues. Hence, our derivation concerning Laplacian matrix is tight.Comment: 18 pages, 3 figur

Lyapunov function for cosmological dynamical system

We prove the asymptotic global stability of the de Sitter solution in the Friedmann-Robertson-Walker conservative and dissipative cosmology. In the proof we construct a Lyapunov function in an exact form and establish its relationship with the first integral of dynamical system determining evolution of the flat Universe. Our result is that de-Sitter solution is asymptotically stable solution for general form of equation of state p = (ρ, H), where dependence on the Hubble function H means that the effect of dissipation are included

Idea dynamicznego Wszechświata : style myślowe w kosmologii

Patrząc na korzenie współczesnej kosmologii odnajdujemy dwie ważne okoliczności: przejście z kosmologii na podstawie koncepcji czasoprzestrzeni newtonowskiej do kosmologii relatywistycznej i akceptacji idei dynamicznego relatywistycznego wszechświata. Uważamy, że podczas gdy pierwsza jest rewolucją naukową w sensie Kuhna, druga nie ma takiego charakteru. Powodem jest to, że przejście od statycznego Wszechświata Einsteina do Friedmanna dynamicznego Wszechświata odbywa się na fundamentach stworzonych przez ogólną teorię względności. Teoretyczna możliwość dynamicznego wszechświata jest naturalną konsekwencją ogólnej koncepcji względności krzywizny czasoprzestrzeni, ale nie została uznana od samego początku, gdyż Einstein był przekonany, że Wszechświat jest statyczny i nie istnieje rozwiązanie, zgodne z rozszerzającym się wszechświatem. Uważamy, że koncepcja Flecka stylu myślowego jest bardziej adekwatna w rekonstrukcji bardzo skomplikowanego procesu tworzenia się dynamicznego relatywistycznego obrazu Wszechświata (Weltbilt).Looking at the roots of modern cosmology we find two important circumstances: the transition from cosmology based on the Newtonian concept of space-time to relativistic cosmology and the acceptance of the idea of dynamical relativistic Universe. We argue that while the former is a scientific revolution in the Kuhn’s sense, the latter has no such a character. The reason is that the transition from Einstein’s static to Friedmann’s dynamic Universe takes place on foundations set up by general relativity theory. The theoretical possibility of dynamic Universe is a natural consequence of general relativity concept of curvature space-time, but it was not recognized from the very beginning, when Einstein was convinced that the Universe is static and did not admit the solutions of expanding Universe. We argue that Fleck’s conception of thought style is more adequate to reconstruct the very complicated process of the dynamical relativistic picture of the Universe (‘Weltbilt’) formation

Constraints on a Cardassian model from SNIa data - revisited

We discuss some observational constraints resulting only from SNIa observations, imposed on the behavior of the original flat Cardassian model, and its extension with the curvature term included. We test the models using the Perlmutter SN Ia data as well as the new Knop and Tonry samples. We estimate the Cardassian model parameters using the best-fitting procedure and the likelihood method. In the fitting procedure we use the density variables for matter, Cardassian fluid and curvature, and include the errors in redshift measurements. For the Perlmutter sample in the non-flat Cardassian model we obtain the high or normal matter density universe ($\Omega_{m,0} \ge 0.3$), while for the flat Cardassian model we have the high density universe. For sample A in the high density universe we also find the negative values of estimates of $n$ which can be interpreted as the phantom fluid effect. For the likelihood method we get that a nearly flat universe is preferred. We show that, if we assume that the matter density is 0.3, then $n \approx 0$ in the flat Cardassian model, which corresponds to the Perlmutter model with the cosmological constant. Testing with the Knop and Tonry SN Ia samples show no significant differences.Comment: RevTeX4, 14 pages, 8 figures; new supernovae data by Knop (v.2) and Tonry (v.3) were used for the model estimatio

Der älteste Bericht eines polnischen Pilgers über seine Reise ins Heilige Land: das Itinerarium von Jan Amor Tarnowski (1488–1561)

Najstarsza relacja polskiego pielgrzyma z jego podróży do Ziemi Świętej: Itinerarium Jana Amora Tarnowskiego (1488–1561)(streszczenie) W artykule przedstawiono najstarszą znaną relację z pielgrzymki z Polski do Ziemi Świętej autorstwa Jana Amora Tarnowskiego, w oparciu o którą przeanalizowane zostały między innymi przebieg samej pielgrzymki, okoliczności powstania rękopiśmiennej relacji, a także sposób konstruowania w niej przestrzeni Ziemi Świętej. Omówione zostało również miejsce tego utworu w kontekście ówczesnych dyskursów na temat Orientu.</p