RETRACTING 1-SIMPLICIAL CHAOTIC GRAPHS WITH DENSITY VARIATION

In this paper we will discuss retraction transformation on chaotic graphs for different cases of density variation, Two types of retraction will be discussed, geometric retraction and chaotic edges retraction, We shall study and discuss the effects of retraction on the shape and density degree of chaotic graphs shown on figures, the adjacent and incidence matrices with be presented. The density character may present many applications in life such as degree of green color in plants, net perturbation resonance, signals in the nerve system and so many, we focus our study on plants, how retraction effects of the degree of green color of plants leaves and shape of the leave its self. The variation of the density character shows the variation of the degree of green color of plant (i.e. Chlorophyll), so we divided in three cases, the first case when the leave of plant is unit and constant everywhere, or varied from level to level or varied even in the same level line of a plant leave, each case will be discussed for two types of retraction and deduce results for both types.

Pozdně paleolitické a mezolitické osídlení Šumavy: možnosti výzkumu, datování a interpretace Late Palaeolithic and Mesolithic Settlement of Šumava: The Possibilities of Research, Dating and Interpretation

The main objective of the article is to present knowledge of newly discovered sites in the mountainous environment of Šumava (Bohemian Forest). The fieldwork in 2011–2019 identified a total of 30 new sites that can be dated to the Late Palaeolithic and Mesolithic periods. The research of prehistoric settlement of the mountainous regions of Bohemia remained a neglected topic for many years. The presence of hunters-gatherers in the mountains in the Mesolithic was documented by isolated finds from the Ore Mountains and a far greater number from Šumava in south Bohemia. This study presents the result of research conducted in three identified locations: in the floodplains of the Roklanský Stream, the Upper Vltava (Moldau) and the Křemelná Stream. The article documents that the network of sites in this space is relatively dense. Although remnants of intact situations are documented in rare cases, the dating of the majority of sites is based solely on lithic assemblages of varying size. Due to the fragmented condition of the obtained material, we discuss the extent to which it is possible to set the survey results in the broader context of central European Mesolithic settlement and present additional thoughts on the density, character and structure of the settlement of central European mountain ranges. Late Palaeolithic, Mesolithic, lithics, mountain settlement, subsistence strategy, Šumava (Bohemian Forest)

A Rigid Hyperfinite Type II1 Factor

We show that it is relatively consistent with Zermelo-Fraenkel set theory with the axiom of choice that there exists a hyperfinite type $\textrm{II}_1$-factor of density character $\aleph _1$ that is not isomorphic to its opposite, does not have any outer automorphisms, and has trivial fundamental group.

Nonseparable Uhf Algebras Ii: Classification

For every uncountable cardinal $\kappa$ there are $2^\kappa$ nonisomorphic simple AF algebras of density character $\kappa$ and $2^\kappa$ nonisomorphic hyperfinite ${\rm II}_1$ factors of density character $\kappa$. These estimates are maximal possible. All C*-algebras that we construct have the same Elliott invariant and Cuntz semigroup as the CAR algebra.

Uncountable dense categoricity in cats

We prove that under reasonable assumptions, every cat (compact abstract theory) is metric, and develop some of the theory of metric cats. We generalise Morley's theorem: if a countable Hausdorff cat T has a unique complete model of density character λ ≥ ω, then it has a unique complete model of density character λ for every λ ≥ ω.

Ground-State Energy of a Weakly Interacting Bose Gas: Calculation Without Regularization

The textbook calculation of the ground-state energy of a dilute gas of Bose particles is examined in detail, and certain mathematical inconsistencies are pointed out. On the basis of this analysis, a refined approach suitable for soft interaction potentials which lend themselves to a low-order Born approximation is developed. This procedure emphasizes the low-density character of the resulting formula for the ground-state energy, and avoids all divergent expressions at intermediate stages of the computation. It is stressed that the standard Bogoliubov approximation, if not augmented by some additional device, leads to an error which manifests itself already in the lowest order of the density. - PACS 03.75.Hh, 05.30.Jp, 03.65.Nk.