Algebraic Structures of Neutrosophic Triplets, Neutrosophic Duplets, or Neutrosophic Multisets

Neutrosophy (1995) is a new branch of philosophy that studies triads of the form (, , ), where is an entity {i.e. element, concept, idea, theory, logical proposition, etc.}, is the opposite of , while is the neutral (or indeterminate) between them, i.e., neither nor .Based on neutrosophy, the neutrosophic triplets were founded, which have a similar form (x, neut(x), anti(x)), that satisfy several axioms, for each element x in a given set.This collective book presents original research papers by many neutrosophic researchers from around the world, that report on the state-of-the-art and recent advancements of neutrosophic triplets, neutrosophic duplets, neutrosophic multisets and their algebraic structures – that have been defined recently in 2016 but have gained interest from world researchers. Connections between classical algebraic structures and neutrosophic triplet / duplet / multiset structures are also studied. And numerous neutrosophic applications in various fields, such as: multi-criteria decision making, image segmentation, medical diagnosis, fault diagnosis, clustering data, neutrosophic probability, human resource management, strategic planning, forecasting model, multi-granulation, supplier selection problems, typhoon disaster evaluation, skin lesson detection, mining algorithm for big data analysis, etc

Must Science Make Cosmological Assumptions if it is to be Rational?

Cosmological speculation about the ultimate nature of the universe, being necessary for science to be possible at all, must be regarded as a part of scientific knowledge itself, however epistemologically unsound it may be in other respects. The best such speculation available is that the universe is comprehensible in some way or other and, more specifically, in the light of the immense apparent success of modern natural science, that it is physically comprehensible. But both these speculations may be false; in order to take this possibility into account, we need to adopt an hierarchy of increasingly contentless cosmological conjectures until we arrive at the conjecture that the universe is such that it is possible for us to acquire some knowledge of something, a conjecture which we are justified in accepting as knowledge since doing so cannot harm the pursuit of knowledge in any circumstances whatsoever. As a result of adopting such an hierarchy of increasingly contentless cosmological conjectures in this way, we maximize our chances of adopting conjectures that promote the growth of knowledge, and minimize our chances of taking some cosmological assumption for granted that is false and impedes the growth of knowledge. The hope is that as we increase our knowledge about the world we improve (lower level) cosmological assumptions implicit in our methods, and thus in turn improve our methods. As a result of improving our knowledge we improve our knowledge about how to improve knowledge. Science adapts its own nature to what it learns about the nature of the universe, thus increasing its capacity to make progress in knowledge about the world. This aim-oriented empiricist conception of science solves outstanding problems in the philosophy of science such as the problems of induction, simplicity and verisimilitude

Planck 2015 results. XVIII. Background geometry and topology of the Universe

Maps of cosmic microwave background (CMB) temperature and polarization from the 2015 release of Planck data provide the highestquality full-sky view of the surface of last scattering available to date. This enables us to detect possible departures from a globally isotropic cosmology. We present the first searches using CMB polarization for correlations induced by a possible non-trivial topology with a fundamental domain that intersects, or nearly intersects, the last-scattering surface (at comoving distance χrec), both via a direct scan for matched circular patterns at the intersections and by an optimal likelihood calculation for specific topologies. We specialize to flat spaces with cubic toroidal (T3) and slab (T1) topologies, finding that explicit searches for the latter are sensitive to other topologies with antipodal symmetry. These searches yield no detection of a compact topology with a scale below the diameter of the last-scattering surface. The limits on the radius ℛi of the largest sphere inscribed in the fundamental domain (at log-likelihood ratio Δlnℒ > −5 relative to a simply-connected flat Planck best-fit model) are: ℛi > 0.97 χrec for the T3 cubic torus; and ℛi > 0.56 χrec for the T1 slab. The limit for the T3 cubic torus from the matched-circles search is numerically equivalent, ℛi > 0.97 χrec at 99% confidence level from polarization data alone. We also perform a Bayesian search for an anisotropic global Bianchi VIIh geometry. In the non-physical setting, where the Bianchi cosmology is decoupled from the standard cosmology, Planck temperature data favour the inclusion of a Bianchi component with a Bayes factor of at least 2.3 units of log-evidence. However, the cosmological parameters that generate this pattern are in strong disagreement with those found from CMB anisotropy data alone. Fitting the induced polarization pattern for this model to the Planck data requires an amplitude of −0.10 ± 0.04 compared to the value of + 1 if the model were to be correct. In the physically motivated setting, where the Bianchi parameters are coupled and fitted simultaneously with the standard cosmological parameters, we find no evidence for a Bianchi VIIh cosmology and constrain the vorticity of such models to (ω/H)0 < 7.6 × 10-10 (95% CL)