Modelling Biomolecular Structures in Categorical Systems Theory

In the systemic movement there exist numerous approaches to systems, the most profound of which is the theory of functional systems by Anokhin, which remained largely intuitive science until his pioneering works. The basic principles of functional systems are formalized with the help of the convolutional polycategories in the form of categorical systems theory, which embraced the main systemic approaches, including the traditional mathematical theory of systems. Convolutional polycategories can be built using categorical splices that directly model the external and internal parts of systems. For an algebraic biology using the categorical theory of systems in relation to systemic constructions, the main task of which is to predict the properties of organisms from the genome using strict algebraic methods, new categorical methods are proposed that are widely used in categorical systems theory. These methods are based on the theory of categorical splices, with the help of which the behaviour of quantum-mechanical particles is modelled, in particular, within the framework of the proposed representation of molecules, including RNA and DNA, as categorical systems. Thus, new algebraic and categorical methods (associative algebras with identities, PROP, categorical splices) are involved in the analysis of the genome. The paper presents new results on these matters.

Compounds

The major part of this chapter surveys how music expresses the ten complex emotions of wonder, the sublime, nostalgia, hope, pride, shame, jealousy, envy, disgust, and boredom. Building on the categorical theory of Chapter 2, it explores the extent that complex emotions compound basic ones, or whether they constitute essential emotions in themselves. The chapter considers issues such as display rules, the reality of basic emotions, and the relationship of emotions to topic theory. The survey of ten complex emotions includes a rehabilitation of wonder, and negative emotions that are normally considered nonaesthetic, such as jealousy and disgust. As in Chapter 2, each of the ten complex emotions is considered in relation to an analysis of music from the common practice period.

MASS GROWTH OF OBJECTS AND CATEGORICAL ENTROPY

In the pioneering work by Dimitrov–Haiden–Katzarkov–Kontsevich, they introduced various categorical analogies from the classical theory of dynamical systems. In particular, they defined the entropy of an endofunctor on a triangulated category with a split generator. In the connection between the categorical theory and the classical theory, a stability condition on a triangulated category plays the role of a measured foliation so that one can measure the “volume” of objects, called the mass, via the stability condition. The aim of this paper is to establish fundamental properties of the growth rate of mass of objects under the mapping by the endofunctor and to clarify the relationship between it and the entropy. We also show that they coincide under a certain condition.

Categoricity and the natural numbers

This chapter focuses on modelists who want to pin down the isomorphism type of the natural numbers. This aim immediately runs into two technical barriers: the Compactness Theorem and the Löwenheim-Skolem Theorem (the latter is proven in the appendix to this chapter). These results show that no first-order theory with an infinite model can be categorical; all such theories have non-standard models. Other logics, such as second-order logic with its full semantics, are not so expressively limited. Indeed, Dedekind's Categoricity Theorem tells us that all full models of the Peano axioms are isomorphic. However, it is a subtle philosophical question, whether one is entitled to invoke the full semantics for second-order logic — there are at least four distinct attitudes which one can adopt to these categoricity result — but moderate modelists are unable to invoke the full semantics, or indeed any other logic with a categorical theory of arithmetic.

Categorical Properties of Soft Sets

The present study investigates some novel categorical properties of soft sets. By combining categorical theory with soft set theory, a categorical framework of soft set theory is established. It is proved that the categorySFunof soft sets and soft functions has equalizers, finite products, pullbacks, and exponential properties. It is worth mentioning that we find thatSFunis both a topological construct and Cartesian closed. The categorySRelof soft sets andZ-soft set relations is also characterized, which shows the existence of the zero objects, biproducts, additive identities, injective objects, projective objects, injective hulls, and projective covers. Finally, by constructing proper adjoint situations, some intrinsic connections betweenSFunandSRelare established.

Adding linear orders

We address the following question: Can we expand an NIP theory by adding a linear order such that the expansion is still NIP? Easily, if acl(A)=A for all A, then this is true. Otherwise, we give counterexamples. More precisely, there is a totally categorical theory for which every expansion by a linear order has IP. There is also an ω-stable NDOP theory for which every expansion by a linear order interprets pseudofinite arithmetic.