scholarly journals Zariski Geometries and Quantum Mechanics

2021 ◽  
Author(s):  
Milan Zanussi
Keyword(s):  
Quantum Mechanics ◽  
Algebraic Geometry ◽  
Model Theory ◽  
Topological Structure ◽  
Classification Theorem ◽  
Zariski Topology ◽  
Specific Class ◽  
Mathematical Structures ◽  
Topological Structures ◽  
Structure Setting

Model theory is the study of mathematical structures in terms of the logical relationships they define between their constituent objects. The logical relationships defined by these structures can be used to define topologies on the underlying sets. These topological structures will serve as a generalization of the notion of the Zariski topology from classical algebraic geometry. We will adapt properties and theorems from classical algebraic geometry to our topological structure setting. We will isolate a specific class of structures, called Zariski geometries, and demonstrate the main classification theorem of such structures. We will construct some Zariski structures where the classification fails by adding some noncommuting structure to a classical one. Finally we survey an application of these nonclassical Zariski structures to computation of formulas in quantum mechanics using a method of structural approximation developed by Boris Zilber.

scholarly journals Notes on Quasiminimality and Excellence

2004 ◽  
Vol 10 (3) ◽  
pp. 334-366 ◽  
Author(s):  
John T. Baldwin
Keyword(s):  
Algebraic Geometry ◽  
Model Theory ◽  
Special Kind ◽  
Order Logic ◽  
First Order Logic ◽  
Structure Theory ◽  
Mathematical Structures ◽  
First Order ◽  
Elementary Class ◽  
Abstract Elementary Class

AbstractThis paper ties together much of the model theory of the last 50 years. Shelah's attempts to generalize the Morley theorem beyond first order logic led to the notion of excellence, which is a key to the structure theory of uncountable models. The notion of Abstract Elementary Class arose naturally in attempting to prove the categoricity theorem for Lω1,ω(Q). More recently, Zilber has attempted to identify canonical mathematical structures as those whose theory (in an appropriate logic) is categorical in all powers. Zilber's trichotomy conjecture for first order categorical structures was refuted by Hrushovski, by the introducion of a special kind of Abstract Elementary Class. Zilber uses a powerful and essentailly infinitary variant on these techniques to investigate complex exponentiation. This not only demonstrates the relevance of Shelah's model theoretic investigations to mainstream mathematics but produces new results and conjectures in algebraic geometry.

scholarly journals Poisson algebras via model theory and differential-algebraic geometry

2017 ◽  
Vol 19 (7) ◽  
pp. 2019-2049 ◽  
Author(s):  
Jason Bell ◽  
Stéphane Launois ◽  
Omar León Sánchez ◽  
Rahim Moosa
Keyword(s):  
Algebraic Geometry ◽  
Model Theory ◽  
Poisson Algebras

scholarly journals Topological Sensitivity in the Recognition of Disoriented Figures

i-Perception ◽  
2018 ◽  
Vol 9 (6) ◽  
pp. 204166951880971 ◽  
Author(s):  
Fumio Kanbe
Keyword(s):  
Structural Properties ◽  
Topological Structure ◽  
Topological Property ◽  
Topological Sensitivity ◽  
Topological Structures ◽  
Invariant Features

A previous study by the author found that discrimination latencies for figure pairs with the same topological structure (isomorphic pairs) were longer than for pairs with different topological structures (nonisomorphic pairs). These results suggest that topological sensitivity occurs during figure recognition. However, sameness was judged in terms of both shape and orientation. Using this criterion, faster discrimination of nonisomorphic pairs may have arisen from the detection of differences in the corresponding locations of the paired figures, which is not a topological property. The current study examined whether topological sensitivity occurs even when identity judgments are based on the sameness of shapes, irrespective of their orientation, where the sameness of location is not ensured. The current results suggested the involvement of topological sensitivity, indicating that processing of structural properties (invariant features) of a figure may be prioritized over processing of superficial features, such as location, length, and angles, in figure recognition.

Ideas from Zariski Topology in the Study of Cubical Homology

2007 ◽  
Vol 59 (5) ◽  
pp. 1008-1028
Author(s):  
Tomasz Kaczynski ◽  
Marian Mrozek ◽  
Anik Trahan
Keyword(s):  
Algebraic Geometry ◽  
Dynamical Systems ◽  
Digital Imaging ◽  
Zariski Topology ◽  
Euclidean Topology ◽  
Separation Axioms ◽  
Cubical Sets ◽  
Noetherian Property ◽  
Cubical Homology

AbstractCubical sets and their homology have been used in dynamical systems as well as in digital imaging. We take a fresh look at this topic, following Zariski ideas from algebraic geometry. The cubical topology is defined to be a topology in ℝd in which a set is closed if and only if it is cubical. This concept is a convenient frame for describing a variety of important features of cubical sets. Separation axioms which, in general, are not satisfied here, characterize exactly those pairs of points which we want to distinguish. The noetherian property guarantees the correctness of the algorithms. Moreover, maps between cubical sets which are continuous and closed with respect to the cubical topology are precisely those for whom the homology map can be defined and computed without grid subdivisions. A combinatorial version of the Vietoris–Begle theorem is derived. This theorem plays the central role in an algorithm computing homology of maps which are continuous with respect to the Euclidean topology.

scholarly journals On Topological Structures of Fuzzy Parametrized Soft Sets

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Serkan Atmaca ◽  
İdris Zorlutuna
Keyword(s):  
Topological Structure ◽  
Topological Spaces ◽  
Soft Sets ◽  
Topological Structures ◽  
Basic Properties

We introduce the topological structure of fuzzy parametrized soft sets and fuzzy parametrized soft mappings. We define the notion of quasi-coincidence for fuzzy parametrized soft sets and investigated its basic properties. We study the closure, interior, base, continuity, and compactness and properties of these concepts in fuzzy parametrized soft topological spaces.

scholarly journals Computational Algebraic Geometry and Quantum Mechanics: An Initiative toward Post-Contemporary Quantum Chemistry

2019 ◽  
Author(s):  
Ichio Kikuchi ◽  
Akihito Kikuchi
Keyword(s):  
Electronic Structure ◽  
Quantum Mechanics ◽  
Algebraic Geometry ◽  
Quantum Chemistry ◽  
Symbolic Computation ◽  
Algebraic Approach ◽  
Polynomial Systems ◽  
Orbital Theory ◽  
Computational Algebraic Geometry ◽  
Future Direction

A new framework in quantum chemistry has been proposed recently (``An approach to first principles electronic structure calculation by symbolic-numeric computation'' by A. Kikuchi). It is based on the modern technique of computational algebraic geometry, viz. the symbolic computation of polynomial systems. Although this framework belongs to molecular orbital theory, it fully adopts the symbolic method. The analytic integrals in the secular equations are approximated by the polynomials. The indeterminate variables of polynomials represent the wave-functions and other parameters for the optimization, such as atomic positions and contraction coefficients of atomic orbitals. Then the symbolic computation digests and decomposes the polynomials into a tame form of the set of equations, to which numerical computations are easily applied. The key technique is Gr\"obner basis theory, by which one can investigate the electronic structure by unraveling the entangled relations of the involved variables. In this article, at first, we demonstrate the featured result of this new theory. Next, we expound the mathematical basics concerning computational algebraic geometry, which are necessitated in our study. We will see how highly abstract ideas of polynomial algebra would be applied to the solution of the definite problems in quantum mechanics. We solve simple problems in ``quantum chemistry in algebraic variety'' by means of algebraic approach. Finally, we review several topics related to polynomial computation, whereby we shall have an outlook for the future direction of the research.

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