Open FRW model in Loop Quantum Cosmology

Open FRW model in Loop Quantum Cosmology is under consideration. The left and right invariant vector fields and holonomies along them are studied. It is shown that in the hyperbolic geometry of $k=-1$ it is possible to construct a suitable loop which provides us with quantum scalar constraint originally introduced by Vandersloot. The quantum scalar constraint operator with negative cosmological constant is proved to be essentially self-adjoint.Comment: 12 pages, no figures, late

Optimal Gersgorin-Style Estimation Of The Largest Singular Value, Ii

In estimating the largest singular value in the class of matrices equiradial with a given n-by-n complex matrix A, it was proved that it is attained at one of n(n - 1) sparse nonnegative matrices (see C.R. Johnson, J.M. Penna and T. Szulc, Optimal Gersgorin-style estimation of the largest singular value; Electronic Journal of Linear Algebra Algebra Appl., 25: 48-59, 2011). Next, some circumstances were identified under which the set of possible optimizers of the largest singular value can be further narrowed (see C.R. Johnson, T. Szulc and D. Wojtera-Tyrakowska, Optimal Gersgorin-style estimation of the largest singular value, Electronic Journal of Linear Algebra Algebra Appl., 25: 48-59, 2011). Here the cardinality of the mentioned set for n-by-n matrices is further reduced. It is shown that the largest singular value, in the class of matrices equiradial with a given n-by-n complex matrix, is attained at one of n (n - 1) /2 sparse nonnegative matrices. Finally, an inequality between the spectral radius of a 3-by-3 nonnegative matrix X and the spectral radius of a modification of X is also proposed

Beyond deficiency:Potential benefits of increased intakesof vitamin K for bone and vascular health

Vitamin K is wellknown for its role in the synthesisof a number of blood coagulationfactors.During recent years vitaminK-dependent proteins werediscovered to be of vital importancefor bone and vascular health.Recommendations for dietary vitaminK intake have been made onthe basis of the hepatic requirementsfor the synthesis of bloodcoagulation factors.Accumulatingevidence suggests that the requirementsfor other functions thanblood coagulation may be higher.This paper is the result of a closedworkshop (Paris,November 2002)in which a number of Europeanvitamin K experts reviewed theavailable data and formulated theirstandpoint with respect to recommendeddietary vitamin K intakeand the use of vitamin K-containingsupplements

Bezpieczeństwo a obronność (dylematy ontologiczno-epistemologiczne)

The subject in question presented in the chapter are among others the differences between safety and defence sciences. Taking into consideration former works in both fields, the author demonstrates their origin and the need of the differentiation of the subject of research. While both disciplines are within social sciences, they can use homogeneous methodology, the differentiation of the subjects of their cognition is considered as prerequisite (sine qua non) for the functioning of both disciplines. Such differentiation is the result of consideration and final author’s suggestion. The suggestion demonstrates clearly the differences and describes epistemological-ontological determinants of both disciplinesPrzedmiotem rozpatrywań prezentowanych w rozdziale są przede wszystkim różnice między naukami o bezpieczeństwie i naukami o obronności. Odnosząc się do dotychczasowych prac w obu dziedzinach autor wskazuje ich rodowód i konieczność rozróżnienia przedmiotu badań. O ile obie dyscypliny mieszczące się w dziedzinie nauk społecznych posługiwać się mogą jednolitą metodologią, o tyle rozróżnienie ich przedmiotów poznania jest traktowane jako warunek konieczny funkcjonowania tych dyscyplin. Takie rozróżnienie jest efektem przemyśleń i końcową propozycją autorską. Propozycja ta wyraźnie wskazuje różnice i określa wyznaczniki epistemologiczno–ontologiczne obu dyscypli

Closed FRW model in Loop Quantum Cosmology

The basic idea of the LQC applies to every spatially homogeneous cosmological model, however only the spatially flat (so called $k=0$) case has been understood in detail in the literature thus far. In the closed (so called: k=1) case certain technical difficulties have been the obstacle that stopped the development. In this work the difficulties are overcome, and a new LQC model of the spatially closed, homogeneous, isotropic universe is constructed. The topology of the spacelike section of the universe is assumed to be that of SU(2) or SO(3). Surprisingly, according to the results achieved in this work, the two cases can be distinguished from each other just by the local properties of the quantum geometry of the universe. The quantum hamiltonian operator of the gravitational field takes the form of a difference operator, where the elementary step is the quantum of the 3-volume derived in the flat case by Ashtekar, Pawlowski and Singh. The mathematical properties of the operator are studied: it is essentially self-adjoint, bounded from above by 0, the 0 itself is not an eigenvalue, the eigenvectors form a basis. An estimate on the dimension of the spectral projection on any finite interval is provided.Comment: 19 pages, latex, no figures, high quality, nea

The status of Quantum Geometry in the dynamical sector of Loop Quantum Cosmology

This letter is motivated by the recent papers by Dittrich and Thiemann and, respectively, by Rovelli discussing the status of Quantum Geometry in the dynamical sector of Loop Quantum Gravity. Since the papers consider model examples, we also study the issue in the case of an example, namely on the Loop Quantum Cosmology model of space-isotropic universe. We derive the Rovelli-Thiemann-Ditrich partial observables corresponding to the quantum geometry operators of LQC in both Hilbert spaces: the kinematical one and, respectively, the physical Hilbert space of solutions to the quantum constraints. We find, that Quantum Geometry can be used to characterize the physical solutions, and the operators of quantum geometry preserve many of their kinematical properties.Comment: Latex, 12 page

Loop Quantum Cosmology corrections to inflationary models

In the recent years the quantization methods of Loop Quantum Gravity have been successfully applied to the homogeneous and isotropic Friedmann-Robertson-Walker space-times. The resulting theory, called Loop Quantum Cosmology (LQC), resolves the Big Bang singularity by replacing it with the Big Bounce. We argue that LQC generates also certain corrections to field theoretical inflationary scenarios. These corrections imply that in the LQC the effective sonic horizon becomes infinite at some point after the bounce and that the scale of the inflationary potential implied by the COBE normalisation increases. The evolution of scalar fields immediately after the Bounce becomes modified in an interesting way. We point out that one can use COBE normalisation to establish an upper bound on the quantum of length of LQG.Comment: 16 pages, 1 figure, plain Late