Formal languages, part theory, and change

A general definition of interpreted formal language is presented. The notion “is a part of" is formally developed and models of the resulting part theory are used as universes of discourse of the formal languages. It is shown that certain Boolean algebras are models of part theory. With this development, the structure imposed upon the universe of discourse by a formal language is characterized by a group of automorphisms of the model of part theory. If the model of part theory is thought of as a static world, the automorphisms become the changes which take place in the world. Using this formalism, we discuss a notion of abstraction and the concept of definability. A Galois connection between the groups characterizing formal languages and a language-like closure over the groups is determined. It is shown that a set theory can be developed within models of part theory such that certain strong formal languages can be said to determine their own set theory. This development is such that for a given formal language whose universe of discourse is a model of part theory, a set theory can be imbedded as a submodel of part theory so that the formal language has parts which are sets as its discursive entities.</p

Combinatorial triality and representation theory

A new subgroup, the endocenter, is defined. The endocenter is a functorial center . The endocenter also facilitates identification of groups associated with quasigroup modules. We use the endocenter to investigate classes of quasigroups whose combinatorial multiplication group is universal, and classes of quasigroups whose combinatorial multiplication group is not universal;There is a strong connection between groups with triality and Moufang loops. We give a partial classification of those Moufang loops whose combinatorial multiplication group is with triality. We completely characterize all groups with triality associated with cyclic groups. We also identify some universal multiplication groups of Moufang loops and determine their triality status;Unfortunately, the class of groups with triality is not a variety. In an attempt to overcome this apparent deficiency, we axiomatize the variety of triality groups , and initiate an algebraic investigation of this (and related) varieties;There are strong geometric connections between Moufang loops and groups with triality. We investigate some of these connections;A new class of groups associated with Moufang loops, but more general than the class of groups with triality, is defined. This is the class of groups with biality. We investigate groups with biality and obtain abstract characterizations of multiplication groups of various classes of inverse property loops

String Theory on Warped AdS_3 and Virasoro Resonances

We investigate aspects of holographic duals to time-like warped AdS_3 space-times--which include G\"odel's universe--in string theory. Using worldsheet techniques similar to those that have been applied to AdS_3 backgrounds, we are able to identify space-time symmetry algebras that act on the dual boundary theory. In particular, we always find at least one Virasoro algebra with computable central charge. Interestingly, there exists a dense set of points in the moduli space of these models in which there is actually a second commuting Virasoro algebra, typically with different central charge than the first. We analyze the supersymmetry of the backgrounds, finding related enhancements, and comment on possible interpretations of these results. We also perform an asymptotic symmetry analysis at the level of supergravity, providing additional support for the worldsheet analysis.Comment: 24 pages + appendice

New Directions in Descriptive Set Theory

I will start with a quick definition of descriptive set theory: It is the study of the structure of definable sets and functions in separable completely metrizable spaces. Such spaces are usually called Polish spaces. Typical examples are R^n, C^n, (separable) Hilbert space and more generally all separable Banach spaces, the Cantor space 2^N, the Baire space N^N, the infinite symmetric group S_∞, the unitary group (of the Hilbert space), the group of measure preserving transformations of the unit interval, etc

Math Ontological Basis of Quasi Fine-Tuning in Ghc Cosmologies

The subject of fine tuning in physics is a long contentious issue especially now as it has hitched a ride on the Multiverse bandwagon. The maths of quadratic forms are predominately featured and relate the physics parameters G h c, which in turn are weighted during the Planck Era(s) determined by relative Planck time clocking. This simplifies the search to these three values as being the important apparent fine-tuned parameters (quasi fine tuning) for determining the gravitational build structures restricted to SM-4D type Universes. Two gravitational coupling constants (dimensionless) are prescribed within the Ghc complex. Both describe the relative rigidity of gravitational physics in the low energy build of our Universe (General Relativity toward endpoint neutron star, black hole formation). A Master vacuum field symmetry relation (Yang-Mills) is presented using both gravitational coupling constants in their respective degenerate domains (electron to neutron) which shows a relative rigid coherent field of parameters from the Codata set showing the interdependency of these values with each other, particularly G,h,c and particle masses. If this is correct then quasi fine-tuning is a symmetry operation. A consensus example aligns the mass-energy value of the charged pi-meson to 139.58066 MeV (in the near flat space) or in the curved metric to 140.05050 MeV. The interdependency of values demands that the gravitational constant’s empirical value to be 6.67354236 x 10-11 m3kg-1s-2 using Codata 2014 values. The Yang-Mills relation has a perfect symmetry (hidden) due to the inclusion of the very weak gravitational charge (Zxx). This is then the weak gravity unification incorporated into the Standard Model. If the Yang-Mills symmetry relation is true then a double copy pion field permeates the observable Universe

Some applications of set theory to algebra

This thesis deals with two topics. In Part I it is shown that if ZFC is consistent, then so is ZF + the order extension principle + there is an abelian group without a divisible hull. The proof is by forcing. In Part II a technique is developed which, in many varieties of algebras, enables the construction for each positive integer not a non-free Xalpha+n -free algebra of cardinality Xalpha+n from a suitable non-free Xalpha-free algebra, when is regular. The algebras constructed turn out to be elementarily equivalent in the language LinfinityXalpha+n to free algebras in the variety. As applications of the technique, it is shown that for any positive integer n there are 2Xn Xn---free algebras which are generated Xn elements, cannot be generated by fewer than this number and are LinfinityXn-equvalent to free algebras in each of the following varieties: any torsion-free variety of groups, all rings with a 1, all commutative rings with a 1, all K-algebras (with K a not-necessarily commutative integral domain), all Lie algebras over a given field. By a different analysis it is shown too that in any variety of nilpotent groups, a lambda-free group of uncountable cardinality lambda is free (respectively, equivalent in Linfinitylambda to a free group) if and only if its abelianisation is, in the abelian part of the variety. Finally, sufficient conditions are given for a X-free group in a variety of groups to be also para free in the variety. The results imply that in the varieties of all groups soluble of length at most k and of all groups polynil potent of given class, if lambda is singular or weakly compact, then a lambda-free group of cardinality lambda is parafree, while if lambda is strongly compact, then a lambda-free group of any cardinality is parafree.<p