For any finite n 3 there are two atomic n-dimensional cylindric algebras with the same atom structure, with one representable, the other, not.Hence, the complex algebra of the atom structure of a representable atomic cylindric algebra is not always representable, so that the class RCAn of representable n-dimensional cylindric algebras is not closed under completions. Further, it follows by an argument of Venema that RCAn is not axiomatisable by Sahlqvist equations, and hence nor by equations where (...) negation can only occur in constant terms.Similar results hold for relation algebras. (shrink)

A cylindric algebra atom structure is said to be strongly representable if all atomic cylindric algebras with that atom structure are representable. This is equivalent to saying that the full complex algebra of the atom structure is a representable cylindric algebra. We show that for any finite n >3, the class of all strongly representable n-dimensional cylindric algebra atom structures is not closed under ultraproducts and is therefore not elementary. Our proof is based on the following (...) construction. From an arbitrary undirected, loop-free graph Γ, we construct an n-dimensional atom structure E(Γ), and prove, for infinite Γ, that E(Γ) is a strongly representable cylindric algebra atom structure if and only if the chromatic number of Γ is infinite. A construction of Erdős shows that there are graphs (k < ω)with infinite chromatic number, but having a non-principal ultraproduct $\prod _D\Gamma _k $ whose chromatic number is just two. It follows that $E(\Gamma _k )$ is strongly representable (each k < ω) but $\Pi _D E(\Gamma _k)$ is not. (shrink)

Strong arguments are presented by Hargrave suggesting that the crystallization of visual rhodopsin for high resolution analysis by X-ray crystallography or electron microscopy is feasible. However, the effort needed to achieve this goal will most likely exceed the resources of a single laboratory and a concerted approach to the research is necessary.

We provide a detailed history of the concepts of atomic number and isotopy before the discovery of protons and neutrons that draws attention to the role of evolving interplays of multiple aims and criteria in chemical and physical research. Focusing on research by Frederick Soddy and Ernest Rutherford, we show that, in the context of differentiating disciplinary projects, the adoption of a complex and shifting concept of elemental identity and the ordering role of the periodic table led to a (...) relatively coherent notion of atomic number. Subsequent attention to valency, still neglected in the secondary literature, and to nuclear charge led to a decoupling of the concepts of elemental identity and weight and allowed for a coherent concept of isotopy. This concept received motivation from empirical investigations on the decomposition series of radioelements and their unstable chemical identity. A new model of chemical order was the result of an ongoing collaboration between chemical and physical research projects with evolving aims and standards. After key concepts were considered resolved and their territories were clarified, chemistry and physics resumed autonomous projects, yet remained bound by newly accepted explanatory relations. It is an episode of scientific collaboration and partial integration without simple, wholesale gestalt switches or chemical revolutions. (shrink)

Psychology suffers from the absence of mathematically-formalized primitives. As a result, conceptual and quantitative studies lack an ontological basis that would situate them in the company of natural sciences. The article addresses this problem by describing a minimal psychic structure, expressed in the algebra of quantum theory. The structure is demarcated into categories of emotion and color, renowned as elementary psychological phenomena. This is achieved by means of quantum-theoretic qubit state space, isomorphic to emotion and color experiences both (...) in meaning and math. In particular, colors are mapped to the qubit states through geometric affinity between the HSL-RGB color solids and the Bloch sphere, widely used in physics. The resulting correspondence aligns with the recent model of subjective experience, producing a unified spherical map of emotions and colors. This structure is identified as a semantic atom of natural thinking—a unit of affectively-colored personal meaning, involved in elementary acts of a binary decision. The model contributes to finding a unified ontology of both inert and living Nature, bridging previously disconnected fields of research. In particular, it enables theory-based coordination of emotion, decision, and cybernetic sciences, needed to achieve new levels of practical impact. (shrink)

In the present work we explore the effects that an H impurity produces upon the geometry and electronic structure of the CaTiO 3 crystal considering the cubic and orthorhombic crystallographic lattices of the material. A quantum-chemical method based on the Hartree-Fock formalism and the periodic large-unit-cell model is used throughout the work. The analysis of the results shows that the interstitial H impurity binds to one of the O atoms forming the so-called O-H group. At equilibrium, the O-H distances (...) are found to be equal to 0.89 and 1.04 Å for cubic and orthorhombic lattices respectively. Atomic displacements and relaxation energies are analysed comparing the obtained results in the cubic lattice with those in the orthorhombic lattice. In the cubic phase the computed relaxation energy of vicinity of the O-H group is found to be equal to 1.1 eV and the atomic displacements generally obey the Coulomb law. So, the negatively charged O atoms move outwards from the defective region by about 0.09 Å while the positively charged Ti and Ca atoms move towards the defective region by about 0.05 and 0.01 Å respectively. A similar effect is observed in the orthorhombic lattice of CaTiO 3 doped with an H atom. It is necessary to mention that different O positions in the orthorhombic structure are considered for the O-H bond creation. The computed relaxation energies of the atomic displacements in this structure are found to be equal to 2.3 and 2.1 eV depending on the crystallographic type of the bonding O atom. (shrink)

The goal of this paper is to present a collection of properties of the set of atoms and the set of finite injective tuples of atoms, as well as of the powersets of atoms in the framework of finitely supported structures. Some properties of atoms are obtained by translating classical Zermelo–Fraenkel results into the new framework, but several important properties are specific to finitely supported structures.

We show that if the structural rules are admissible over a set \ of atomic rules, then they are admissible in the sequent calculus obtained by adding the rules in \ to the multisuccedent minimal and intuitionistic \ calculi as well as to the classical one. Two applications to pure logic and to the sequent calculus with equality are presented.