ABSTRACT:We report on optimal molecular connectivity descriptors for nitrogen atoms in amines for use in structure᎐property correlations. The descriptors represent generalized molecular connectivity indices with adjusted diagonal entries in the adjacency matrices of the corresponding molecular graphs, such that the standard error in a regression for boiling points in a set of amines is minimized. Advantages of the so-optimized descriptors for multivariate regression analysis in structure᎐property᎐ activity studies are discussed.
Let p be a strong type of an algebraically closed tuple over B = acl eq (B) in any theory T . Depending on a ternary relation ⌣ | * satisfying some basic axioms (there is at least one such, namely the trivial independence in T ), the first homology group H * 1 (p) can be introduced, similarly to [3]. We show that there is a canonical surjective homomorphism from the Lascar group over B to H * 1 (p). We also notice that the map factors naturally via a surjection from the 'relativised' Lascar group of the type (which we define in analogy with the Lascar group of the theory) onto the homology group, and we give an explicit description of its kernel. Due to this characterization, it follows that the first homology group of p is independent from the choice of ⌣ | * , and can be written simply as H 1 (p).As consequences, in any T , we show that |H 1 (p)| ≥ 2 ℵ0 unless H 1 (p) is trivial, and we give a criterion for the equality of stp and Lstp of algebraically closed tuples using the notions of the first homology group and a relativised Lascar group.We also argue how any abelian connected compact group can appear as the first homology group of the type of a model.In this paper we study the first homology group of a strong type in any theory.Originally, in [3] and [4], a homology theory only for rosy theories is developed. Namely, given a strong type p in a rosy theory T , the notion of the nth homology group H n (p) depending on thorn-forking independence relation is introduced. Although the homology groups are defined analogously as in singular homology theory in algebraic topology, the (n + 1)th homology group for n > 0 in the rosy theory context has to do with the nth homology group in algebraic topology. For example as in [3], H 2 (p) in stable theories has to do with the fundamental group in topology. 1 2 JAN DOBROWOLSKI, BYUNGHAN KIM, AND JUNGUK LEE theories, H 1 (p) is detecting somewhat endemic properties of p existing only in model theory context.Indeed, in every known rosy example, H n (p) for n ≥ 2 is a profinite abelian group. In [5], it is proved to be so when T is stable under a canonical condition, and conversely, every profinite abelian group can arise in this form. On the other hand, we show in this paper that the first homology groups appear to have distinct features as follows.Let p = tp(a/B) be a strong type over B = acl eq (B) in any theory T . Fix a ternary invariant independence relation ⌣ | * among small sets satisfying finite character, normality, symmetry, transitivity and extension. (There is at least one such relation, by putting A ⌣ | C D for any sets A, C, D.) Then we can analogously define H * 1 (p) depending on ⌣ | * , (which of course is the same as H 1 (p) when ⌣ | * is thorn-independence in rosy T ). In this note, a canonical epimorphism from the Lascar group over B of T to H * 1 (p) is constructed. Indeed, we also introduce the notion of the relativised Lascar group of a type which is proved to be independent from the choice of the monster model of T , and the homomorphism factors th...
An important dividing line in the class of unstable theories is being NSOP 1 , which is more general than being simple. In NSOP 1 theories forking independence may not be as well behaved as in stable or simple theories, so it is replaced by another independence notion, called Kim-independence. We generalise Kim-independence over models in NSOP 1 theories to positive logic -a proper generalisation of full first-order logic where negation is not built in, but can be added as desired. For example, an important application is that we can add hyperimaginary sorts to a positive theory to get another positive theory, preserving NSOP 1 and various other properties. We prove that, in a thick positive NSOP 1 theory, Kim-independence over existentially closed models has all the nice properties that it is known to have in an NSOP 1 theory in full first-order logic. We also provide a Kim-Pillay style theorem, characterising which thick positive theories are NSOP 1 by the existence of a certain independence relation. Furthermore, this independence relation must then be the same as Kim-independence. Thickness is the mild assumption that being an indiscernible sequence is type-definable.In full first-order logic Kim-independence is defined in terms of Morley sequences in global invariant types. These may not exist in thick positive theories. We solve this by working with Morley sequences in global Lascar-invariant types, which do exist in thick positive theories. We also simplify certain tree constructions that were used in the study of Kim-independence in full first-order logic. In particular, we only work with trees of finite height. 65 4. Kim-dividing 70 5. EM-modelling and parallel-Morley sequences 74 6. Symmetry 81 7. Independence theorem 84 8. Transitivity 94 9. Kim-Pillay style theorem 96 10. Examples 101 Acknowledgements 111 References 111 MSC2020: 03C10, 03C45.
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