The solutions of the 3rd and 4th Clay Millennium problems

In this treatise I present the solutions of the third Clay Millennium problem in the computational complexity and the fourth Clay Millennium problem in classical fluid dynamics.Comment: arXiv admin note: text overlap with arXiv:1108.1165 by other author

The continuum of the surreal numbers revisited.The surreal numbers defined through transfinite Cauchy fundamental sequences

In this treatise on the theory of the continuum of the surreal numbers of J.H. Conway, is proved ,that the three different techniques and hierarchies of the continuums of the transfinite real numbers of Glayzal A. (1937) defined through transfinite power series , of the surreal numbers of J.H. Conway (1976) defined by Dedekind cuts ,and of the ordinal real numbers of K. E. Kyritsis (1992) defined by fundamental Cauchy transfinite sequences, give by inductive limit or union the same class and continuum of infinite numbers. This is quite remarkable and is the analogue in the transfinite numbers, of that the real numbers can be constructed either as decimal power series, or by Dedekind cuts, or by Cauchy fundamental sequences.Comment: This text of 64 pages is actually 5 consecutive papers published in the same proceedings of the same conferenc

An Axiomatic System for a Physical or Digital but Continuous 3-Dimensional Euclidean Geometry, Without Infinite Many Points

This paper is concerned with finding an axiomatic system, so as to define the 3-dimensional Euclidean space, without utilizing the infinite ,that can imply all the known geometry for practical applied sciences and engineering applications through computers , and for more natural and perfect education of young people innbsp the Euclidean geometric thinking. In other words by utilizing only finite many visible and invisible points and only finite sets, and only real numbers with finite many digits, in the decimal representation. The inspiration comes from the physical matter , rigid, liquid and gaseous, which consists of only finite manynbsp particles in the physical reality. Or from the way that continuity is produced in a computer screen from only finite many invisible pixels . We present such a system of axioms and explain why it is chosen in such a way. The result is obviously not equivalent, in all the details, with the classical Euclidean geometry.nbsp Our main concern is consistency and adequacy but not independence of the axioms between them. It is obvious that within the space of a single paper, we do not attempt to produce all the main theorems of the Euclidean geometry, but present only the axioms

RoboPol: AGN polarimetric monitoring data

Summarization: We present uniformly reprocessed and re-calibrated data from the RoboPol programme of optopolarimetric monitoring of active galactic nuclei (AGNs), covering observations between 2013, when the instrument was commissioned, and 2017. In total, the data set presented in this paper includes 5068 observations of 222 AGN with Dec. > −25○. We describe the current version of the RoboPol pipeline that was used to process and calibrate the entire data set, and we make the data publicly available for use by the astronomical community. Average quantities summarizing optopolarimetric behaviour (average degree of polarization, polarization variability index) are also provided for each source we have observed and for the time interval we have followed it.Presented on: Monthly Notices of the Royal Astronomical Societ

Dynamics of hydration water in gelatin and hyaluronic acid hydrogels

[EN] We employed broadband dielectric spectroscopy (BDS), for the investigation of the water dynamics in partially hydrated hyaluronic acid (HA), and gelatin (Gel), enzymatically crosslinked hydrogels, in the water fraction ranges [Formula: see text]. Our results indicate that at low hydrations ([Formula: see text]), where the dielectric response of the hydrogels is identical during cooling and heating, water plasticizes strongly the polymeric matrix and is organized in clusters giving rise to [Formula: see text]-process, secondary water relaxation and to an additional slower relaxation process. This later process has been found to be related with the dc charge conductivity and can be described in terms of the conduction current relaxation mechanism. At slightly higher hydrations, however, always below the hydration level where ice is formed during cooling, we have recorded in HA hydrogel a strong water dielectric relaxation process, [Formula: see text], which has Arrhenius-like temperature dependence and large time scale resembling relaxation processes recorded in bulk low density amorphous solid water structures. This relaxation process shows a strong-to-fragile transition at [Formula: see text]C and our data suggest that the VTF-like process recorded at [Formula: see text]C is controlled by the same molecular process like long range charge transport. In addition, our data imply that the crossover temperature is related with the onset of structural rearrangements (increase in configurational entropy) of the macromolecules. In partially crystallized hydrogels ([Formula: see text]) HA exhibits at low temperatures the ice dielectric process consistent with the bulk hexagonal ice, whereas Gel hydrogel exhibits as main low temperature process a slow relaxation process that refers to open tetrahedral structures of water similar to low density amorphous ice structures and to bulk cubic ice. Regarding the water secondary relaxation processes, we have shown that the [Formula: see text]-process and the [Formula: see text] process are activated in water hydrogen bond networks with different structures.The support from Ministerio de Economia, Industria y Competitividad (MINECO) through the MAT2016-76039-C4-1-R project (including the FEDER funds) is acknowledged. The CIBER-BBN initiative is funded by the VI National R&D&I Plan 2008-2011, Iniciativa Ingenio 2010, Consolider Program. CIBER actions are financed by the Instituto de Salud Carlos III with assistance from the European Regional Development Fund. 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