Abstract. In this paper, we study the geometry of a (nontrivial) 1-based SU rank-1 complete type. We show that if the (localized, resp.) geometry of the type is modular, then the (localized, resp.) geometry is projective over a division ring. However, unlike the stable case, we construct a locally modular type that is not affine. For the general 1-based case, we prove that even if the geometry of the type itself is not projective over a division ring, it is when we consider a 2-fold or 3-fold of the geometry altogether. In particular, it follows that in any ω-categorical, nontrivial, 1-based theory, a vector space over a finite field is interpretable.Geometric stability theory is one of the most important themes in model theory. Originally developing as a pure subject, it has turned out to be the major technical bridge connecting pure model theory and its applications to algebraic geometry and number theory. It mainly focuses on rank-1 types/structures or regular types where a canonical combinatorial geometry can be assigned. In other words, it is primarily concerned with geometric aspects of Shelah's stability theory [19], the study of stable structures. Arguably, the first major achievements of geometric stability theory are Zilber's results from the early 1980s (the translated version is [23]) on a strongly minimal ω-categorical structure. He showed that the geometry assigned to the structure is locally modular, hence, if nontrivial, must either be affine or projective over a finite field. A different proof was discovered independently by Cherlin, Harrington and Lachlan [14]. Refined notions such as 1-basedness, regular types and p-weight have also been introduced. Pillay, in his book [17], makes a complete exposition of the subject. Hrushovski has now shifted the direction of his research towards applications in algebraic geometry and number theory, which has deepened and broadened the subject. It is well known that, using geometric stability theory and in particular Zilber's principle on "Zariski structures", he solved the Mordell-Lang conjecture [12] and other problems in number theory.From the mid 1990s, after the initial papers [15], [16] of Kim and Pillay, simplicity theory, introduced by Shelah [18], has developed rapidly and extensively. Simplicity
Abstract. Under P(4)− -amalgamation, we obtain the canonical hyperdefinable group from the group configuration.The group configuration theorem for stable theories given by Hrushovski [5], which extends Zilber's result for ω-categorical theories [17], plays a central role in producing deep results in geometric stability theory (For a complete exposition of it, see [14]). For example, it is pivotal in the proof of the dichotomy theorem for Zariski' structures (See [9]). It is fair to say the group configuration theorem is one of the foundational theorems in geometric stability theory and its applications to algebraic geometry.The theorem roughly says that one can get the canonical non-trivial type-definable group from the group configuration, a certain geometrical configuration, in stable theories. The complete generalization of the theorem into the context of simple theories seemed unreachable. In their topical paper [1], Ben-Yaacov, Tomasic and Wagner generalize the group configuration theorem by obtaining an invariant group from the group configuration in simple theories. However the group they produce does not completely fit into the first-order context.On the other hand, Kolesnikov in his important thesis [12], categorizes simple theories by strengthening the type-amalgamation property (the independence theorem [10]), along the lines of early suppositions by Shelah [15] and Hrushovski [6]. These works suggest to us the possibility of using higher amalgamation for the group configuration problem. This approach proves successful, and in this paper we succeed in getting the canonical hyperdefinable group from the group configuration under stronger type-amalgamation in simple theories. The element of the group is a hyperimaginary, an equivalence class of a type-definable equivalence relation, and the group operation is type-definable, hence the group belongs to the domain of the standard first-order logic.We assume that the reader is familiar with basics of simplicity theory [16]. Throughout the paper, T is a complete simple theory. We work in a saturated model M of T with hyperimaginaries, and a, b, ... are (possibly infinitary) hyperimaginaries, M, N are small elementary submodels. (Note that tuples from M eq are also hyperimaginaries). As usual,have the same type (Lascar strong type, resp.) over A. We point out that usually bdd(a) denotes the set of all countable hyperimaginaries definable over a [16, 3.1.7]. Here, depending on the context, it can be either a specific sequence which linearly orders the set bdd(a); or, since a sequence of hyperimaginaries is again a hyperimaginary (of a large arity), a fixed hyperimaginary interdefinable with the sequence.
We clarify some arguments concerning Jefimenko's equations, as a way of constructing solutions to Maxwell's equations, for charge and current satisfying the continuity equation.We then isolate a condition on non-radiation in all inertial frames, which is intuitively reasonable for the stability of an atomic system, and prove that the condition is equivalent to the charge and current satisfying certain relations, including the wave equations. Finally, we prove that with these relations, the energy in the electromagnetic field is quantised and displays the properties of the Balmer series.
We prove that if the frame \(S\) is decaying surface non-radiating, in the sense of Definition 1, then if \(\left(\rho,\overline{J}\right)\) is analytic, either \(\rho=0\) and \(\overline{J}=\overline{0}\), or \(S\) is non-radiating, in the sense of [<a href="#1">1</a>]. In particularly, by the result there, the charge and current satisfy certain wave equations in all the frames \(S_{\overline{v}}\) connected to \(S\) by a real velocity vector \(\overline{v}\), with \(|\overline{v}|< c\).
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