Quaternionic Dirac free particle

In this paper, we solve the quaternionic Dirac equation [Formula: see text] in the real Hilbert space, and we ascertain that their free particle solutions set comprises eight elements in the case of a massive particle, and a four elements solutions set in the case of a massless particle, a richer situation when compared to the four elements solutions set of the usual complex Dirac equation [Formula: see text]. These free particle solutions were unknown in the previous solutions of anti-Hermitian quaternionic quantum mechanics, and constitute an essential element in order to build a quaternionic quantum field theory [Formula: see text].

K-Theory for Semigroup C*-Algebras and Partial Crossed Products

Using the Baum–Connes conjecture with coefficients, we develop a K-theory formula for reduced C*-algebras of strongly 0-E-unitary inverse semigroups, or equivalently, for a class of reduced partial crossed products. This generalizes and gives a new proof of previous K-theory results of Cuntz, Echterhoff and the author. Our K-theory formula applies to a rich class of C*-algebras which are generated by partial isometries. For instance, as new applications which could not be treated using previous results, we discuss semigroup C*-algebras of Artin monoids, Baumslag-Solitar monoids and one-relator monoids, as well as C*-algebras generated by right regular representations of semigroups of number-theoretic origin, and C*-algebras attached to tilings.

Nonperturbative aspects of spontaneous symmetry breaking

Spontaneous symmetry breaking is studied in the ultralocal limit of a scalar quantum field theory, that is when [Formula: see text] (or infrared limit). In this infrared approximation the theory [Formula: see text] is formally two-dimensional and its Euclidean solutions are instantons. For BPST-like solutions with [Formula: see text], the map between [Formula: see text] in two dimensions and self-dual Yang–Mills theory is carefully discussed.

Generic expansions by a reduct

Consider the expansion [Formula: see text] of a theory [Formula: see text] by a predicate for a submodel of a reduct [Formula: see text] of [Formula: see text]. We present a setup in which this expansion admits a model companion [Formula: see text]. We show that some of the nice features of the theory [Formula: see text] transfer to [Formula: see text]. In particular, we study conditions for which this expansion preserves the [Formula: see text]-ness, the simplicity or the stability of the starting theory [Formula: see text]. We give concrete examples of new [Formula: see text] not simple theories obtained by this process, among them the expansion of a perfect [Formula: see text]-free PAC field of positive characteristic by generic additive subgroups, and the expansion of an algebraically closed field of any characteristic by a generic multiplicative subgroup.

Generic derivations on o-minimal structures

Let [Formula: see text] be a complete, model complete o-minimal theory extending the theory [Formula: see text] of real closed ordered fields in some appropriate language [Formula: see text]. We study derivations [Formula: see text] on models [Formula: see text]. We introduce the notion of a [Formula: see text]-derivation: a derivation which is compatible with the [Formula: see text]-definable [Formula: see text]-functions on [Formula: see text]. We show that the theory of [Formula: see text]-models with a [Formula: see text]-derivation has a model completion [Formula: see text]. The derivation in models [Formula: see text] behaves “generically”, it is wildly discontinuous and its kernel is a dense elementary [Formula: see text]-substructure of [Formula: see text]. If [Formula: see text], then [Formula: see text] is the theory of closed ordered differential fields (CODFs) as introduced by Michael Singer. We are able to recover many of the known facts about CODF in our setting. Among other things, we show that [Formula: see text] has [Formula: see text] as its open core, that [Formula: see text] is distal, and that [Formula: see text] eliminates imaginaries. We also show that the theory of [Formula: see text]-models with finitely many commuting [Formula: see text]-derivations has a model completion.

Influence of Application Readiness,Trust and E-Service Quality Against Purchase Interest in E-Commerce Xin Jabodetabek

The purpose of this research is to know and analyze how much influence the variable Application Readiness (Perceived Ease of Use and Perceived Usefulness), Trust and E-Service Quality to Purchase Interest in E-commerce X in Jabodetabek. This study used a purposive sampling method with a Hair theory formula amounting to 160 respondents. The data analysis technique used is a Structural equation Modeling (SEM) equation technique using the AMOS version 22 application. The results showed that the Application Readiness, trust, and E-Service Quality significantly affect the Purchase Interest, E-Service Quality has the biggest and most significant contribution to online Purchase Interest.

The Theory of Cognitive Consciousness, and Λ (Lambda)

We provide an overview of the theory of cognitive consciousness (TCC), and of [Formula: see text]; the latter provides a means of measuring the amount of cognitive consciousness present in a given cognizer, whether natural or artificial, at a given time, along a number of different dimensions. TCC and [Formula: see text] stand in stark contrast to Tononi’s Integrated information Theory (IIT) and [Formula: see text]. We believe, for reasons we present, that the former pair is superior to the latter. TCC includes a formal axiomatic theory, [Formula: see text], the 12 axioms of which we present and briefly comment upon herein; no such formal theory accompanies IIT/[Formula: see text]. TCC/[Formula: see text] and IIT/[Formula: see text] each offer radically different verdicts as to whether and to what degree AIs of yesterday, today, and tomorrow were/are/will be conscious. Another noteworthy difference between TCC/[Formula: see text] and IIT/[Formula: see text] is that the former enables the measurement of cognitive consciousness in those who have passed on, and in fictional characters; no such enablement is remotely possible for IIT/[Formula: see text]. For instance, we apply [Formula: see text] to measure the cognitive consciousness of: Descartes; and the first fictional detective to be described on Earth (by Edgar Allen Poe), Auguste Dupin. We also apply [Formula: see text] to compute the cognitive consciousness of an artificial agent able to make ethical decisions using the Doctrine of Double Effect.

A descriptive Main Gap Theorem

Answering one of the main questions of [S.-D. Friedman, T. Hyttinen and V. Kulikov, Generalized descriptive set theory and classification theory, Mem. Amer. Math. Soc. 230(1081) (2014) 80, Chap. 7], we show that there is a tight connection between the depth of a classifiable shallow theory [Formula: see text] and the Borel rank of the isomorphism relation [Formula: see text] on its models of size [Formula: see text], for [Formula: see text] any cardinal satisfying [Formula: see text]. This is achieved by establishing a link between said rank and the [Formula: see text]-Scott height of the [Formula: see text]-sized models of [Formula: see text], and yields to the following descriptive set-theoretical analog of Shelah’s Main Gap Theorem: Given a countable complete first-order theory [Formula: see text], either [Formula: see text] is Borel with a countable Borel rank (i.e. very simple, given that the length of the relevant Borel hierarchy is [Formula: see text]), or it is not Borel at all. The dividing line between the two situations is the same as in Shelah’s theorem, namely that of classifiable shallow theories. We also provide a Borel reducibility version of the above theorem, discuss some limitations to the possible (Borel) complexities of [Formula: see text], and provide a characterization of categoricity of [Formula: see text] in terms of the descriptive set-theoretical complexity of [Formula: see text].

BPS spectra and 3-manifold invariants

We provide a physical definition of new homological invariants [Formula: see text] of 3-manifolds (possibly, with knots) labeled by abelian flat connections. The physical system in question involves a 6d fivebrane theory on [Formula: see text] times a 2-disk, [Formula: see text], whose Hilbert space of BPS states plays the role of a basic building block in categorification of various partition functions of 3d [Formula: see text] theory [Formula: see text]: [Formula: see text] half-index, [Formula: see text] superconformal index, and [Formula: see text] topologically twisted index. The first partition function is labeled by a choice of boundary condition and provides a refinement of Chern–Simons (WRT) invariant. A linear combination of them in the unrefined limit gives the analytically continued WRT invariant of [Formula: see text]. The last two can be factorized into the product of half-indices. We show how this works explicitly for many examples, including Lens spaces, circle fibrations over Riemann surfaces, and plumbed 3-manifolds.

Model theory of Steiner triple systems

A Steiner triple system (STS) is a set [Formula: see text] together with a collection [Formula: see text] of subsets of [Formula: see text] of size 3 such that any two elements of [Formula: see text] belong to exactly one element of [Formula: see text]. It is well known that the class of finite STS has a Fraïssé limit [Formula: see text]. Here, we show that the theory [Formula: see text] of [Formula: see text] is the model completion of the theory of STSs. We also prove that [Formula: see text] is not small and it has quantifier elimination, [Formula: see text], [Formula: see text], elimination of hyperimaginaries and weak elimination of imaginaries.