SYMMETRY GEOMETRY BY PAIRINGS

2018 ◽  
Vol 106 (03) ◽  
pp. 342-360 ◽  
Author(s):  
G. CHIASELOTTI ◽  
T. GENTILE ◽  
F. INFUSINO
Keyword(s):  
Closure Operator ◽  
Local Symmetry ◽  
Global Symmetry ◽  
Symmetry Relation ◽  
Basic Tool ◽  
Mathematical Structures ◽  
Finite Case ◽  
Set Systems ◽  
Power Set ◽  
Abstract Simplicial Complex

In this paper, we introduce asymmetry geometryfor all those mathematical structures which can be characterized by means of a generalization (which we call pairing) of a finite rectangular table. In more detail, let$\unicode[STIX]{x1D6FA}$be a given set. Apairing$\mathfrak{P}$on$\unicode[STIX]{x1D6FA}$is a triple$\mathfrak{P}:=(U,F,\unicode[STIX]{x1D6EC})$, where$U$and$\unicode[STIX]{x1D6EC}$are nonempty sets and$F:U\times \unicode[STIX]{x1D6FA}\rightarrow \unicode[STIX]{x1D6EC}$is a map having domain$U\times \unicode[STIX]{x1D6FA}$and codomain$\unicode[STIX]{x1D6EC}$. Through this notion, we introduce a local symmetry relation on$U$and a global symmetry relation on the power set${\mathcal{P}}(\unicode[STIX]{x1D6FA})$. Based on these two relations, we establish the basic properties of our symmetry geometry induced by$\mathfrak{P}$. The basic tool of our study is a closure operator$M_{\mathfrak{P}}$, by means of which (in the finite case) we can represent any closure operator. We relate the study of such a closure operator to several types of others set operators and set systems which refine the notion of an abstract simplicial complex.

scholarly journals Goodness of Regularity in Dot Patterns: Global Symmetry, Local Symmetry, and Their Interactions

Perception ◽  
10.1068/p5794 ◽  
2007 ◽  
Vol 36 (9) ◽  
pp. 1305-1319 ◽  
Author(s):  
Massimo Nucci ◽  
Johan Wagemans
Keyword(s):  
Mirror Symmetry ◽  
Local Symmetry ◽  
Detection Task ◽  
Global Symmetry ◽  
Different Types ◽  
Operational Models ◽  
Local Mirror Symmetry ◽  
Axes Of Symmetry ◽  
Global And Local ◽  
Short Presentation

Goodness is a classic Gestalt notion defined as salience or perceptual strength of a given pattern. All operational models of goodness have assigned a central role to mirror symmetry but not much attention has been paid to the distinction between global and local mirror symmetry, and their possible interactions. We designed eight different types of dot patterns (all consisting of 80 dots), combining different numbers (0, 1, and 2) and relative orientations (parallel or orthogonal to each other) of local and global axes of symmetry (affecting 50% or 100% of the dots, respectively) at different absolute orientations (vertical and horizontal). Each of 640 trials consisted of a short presentation of a new dot pattern, which subjects had to classify as regular or random. We hypothesised that the overall goodness of patterns is not the simple sum of the amount of regularity present in them but depends on the cooperation and competition between symmetries. The results confirmed our hypothesis, showing that performance in this regularity-detection task did not increase in a linear way when some symmetries were added to other symmetries.

scholarly journals Anomalous symmetries end at the boundary

2021 ◽  
Vol 2021 (9) ◽  
Author(s):  
Ryan Thorngren ◽  
Yifan Wang
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Boundary Conditions ◽  
Global Symmetries ◽  
Local Symmetry ◽  
Global Symmetry ◽  
Quantum Field ◽  
Ward Identities ◽  
Diffeomorphism Symmetry ◽  
Gravitational Anomalies

Abstract A global symmetry of a quantum field theory is said to have an ’t Hooft anomaly if it cannot be promoted to a local symmetry of a gauged theory. In this paper, we show that the anomaly is also an obstruction to defining symmetric boundary conditions. This applies to Lorentz symmetries with gravitational anomalies as well. For theories with perturbative anomalies, we demonstrate the obstruction by analyzing the Wess-Zumino consistency conditions and current Ward identities in the presence of a boundary. We then recast the problem in terms of symmetry defects and find the same conclusions for anomalies of discrete and orientation-reversing global symmetries, up to the conjecture that global gravitational anomalies, which may not be associated with any diffeomorphism symmetry, also forbid the existence of boundary conditions. This conjecture holds for known gravitational anomalies in D ≤ 3 which allows us to conclude the obstruction result for D ≤ 4.

scholarly journals On the Weak Order of Coxeter Groups

2019 ◽  
Vol 71 (2) ◽  
pp. 299-336 ◽  
Author(s):  
Matthew Dyer
Keyword(s):  
Complete Lattice ◽  
Coxeter Groups ◽  
Closure Operator ◽  
Root Systems ◽  
Bruhat Order ◽  
Weak Order ◽  
Maximal Subgroups ◽  
Rank Zero ◽  
Power Set ◽  
Rank One

AbstractThis paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to whether the set of initial sections of reflection orders, ordered by inclusion, forms a complete lattice. Meet and join in weak order are described in terms of a suitable closure operator. Galois connections are defined from the power set of $W$ to itself, under which maximal subgroups of certain groupoids correspond to certain complete meet subsemilattices of weak order. An analogue of weak order for standard parabolic subsets of any rank of the root system is defined, reducing to the usual weak order in rank zero, and having some analogous properties in rank one (and conjecturally in general).

How to Lay Bricks: Local-Global Alignment Predicts Preference for Shifted Tile Patterns

Perception ◽  
2021 ◽  
Vol 50 (12) ◽  
pp. 983-1001
Author(s):  
Jay Friedenberg ◽  
Preston Martin ◽  
Aimen Khurram ◽  
Mackenzie Kvapil
Keyword(s):  
Spatial Scale ◽  
Equilateral Triangle ◽  
Local Symmetry ◽  
Global Alignment ◽  
Global Symmetry ◽  
Geometric Figures ◽  
Emergent Features ◽  
The Aesthetic ◽  
Aesthetic Characteristics

We examine the aesthetic characteristics of row tile patterns defined by repeating strips of polygons. In experiment 1 participants rated the perceived beauty of equilateral triangle, square and rectangular tilings presented at vertical and horizontal orientations. The tiles were shifted by one-fourth increments of a complete row cycle. Shifts that preserved global symmetry were liked the most. Local symmetry by itself did not predict ratings but tilings with a greater number of emergent features did. In a second experiment we presented row tiles using all types of three- and four-sided geometric figures: acute, obtuse, isosceles and right triangles, kites, parallelograms, a rhombus, trapezoid, and trapezium. Once again, local polygon symmetry did not predict responding but measures of correspondence between local and global levels did. In particular, number of aligned polygon symmetry axes and number of aligned polygon sides were significantly and positively correlated with beauty ratings. Preference was greater for more integrated tilings, possibly because they encourage the formation of gestalts and exploration within and across levels of spatial scale.

The consistency of classical set theory relative to a set theory with intu1tionistic logic

1973 ◽  
Vol 38 (2) ◽  
pp. 315-319 ◽  
Author(s):  
Harvey Friedman
Keyword(s):  
Set Theory ◽  
Classical Theory ◽  
Intuitionistic Logic ◽  
Classical Logic ◽  
Predicate Calculus ◽  
Basic Tool ◽  
Axiom Scheme ◽  
Power Set ◽  
Image Position

Let ZF be the usual Zermelo-Fraenkel set theory formulated without identity, and with the collection axiom scheme. Let ZF−-extensionality be obtained from ZF by using intuitionistic logic instead of classical logic, and dropping the axiom of extensionality. We give a syntactic transformation of ZF into ZF−-extensionality.Let CPC, HPC respectively be classical, intuitionistic predicate calculus without identity, whose only homological symbol is ∈. We use the ~ ~-translation, a basic tool in the metatheory of intuitionistic systems, which is defined byThe two fundamental lemmas about this ~ ~ -translation we will use areFor proofs, see Kleene [3, Lemma 43a, Theorem 60d].This - would provide the desired syntactic transformation at least for ZF into ZF− with extensionality, if A− were provable in ZF− for each axiom A of ZF. Unfortunately, the ~ ~-translations of extensionality and power set appear not to be provable in ZF−. We therefore form an auxiliary classical theory S which has no extensionality and has a weakened power set axiom, and show in §2 that the ~ ~-translation of each axiom of Sis provable in ZF−-extensionality. §1 is devoted to the translation of ZF in S.

REMARKS ON THE MODEL WITH LOCAL U(1)V×U(1)A SYMMETRY IN TWO DIMENSIONS

1991 ◽  
Vol 06 (14) ◽  
pp. 1291-1298 ◽  
Author(s):  
YOSHIYUKI WATABIKI
Keyword(s):  
Energy Momentum Tensor ◽  
Central Charge ◽  
Axial Vector ◽  
Local Symmetry ◽  
Momentum Tensor ◽  
Two Dimensions ◽  
Global Symmetry ◽  
Supersymmetric Extension ◽  
Commutator Algebra ◽  
Local Vector

We investigate a 2-dimensional model which possesses a local vector U (1)V and axial vector U (1)A symmetry. We obtain a general form of Lagrangian which possesses this local symmetry. We also investigate the global symmetry aspects of the model. The commutator algebra of the energy-momentum tensor and the currents is derived, and the central charge of the model is calculated. Supersymmetric extension of the model is also studied.

scholarly journals Strongly Connectedness in Closure Space

2014 ◽  
Vol 9 ◽  
pp. 69-71
Author(s):  
U.D. Tapi ◽  
Bhagyashri A. Deole
Keyword(s):  
Topological Space ◽  
Disjoint Union ◽  
Closure Operator ◽  
Closure Space ◽  
Closed Set ◽  
Power Set ◽  
Strongly Connected

A Čech closure space (X, u) is a set X with Čech closure operator u: P(X) → P(X) where P(X) is a power set of X, which satisfies u𝝓=𝝓, A ⊆uA for every A⊆X, u (A⋃B) = uA⋃uB, for all A, B ⊆ X. Many properties which hold in topological space hold in closure space as well. A topological space X is strongly connected if and only if it is not a disjoint union of countably many but more than one closed set. If X is strongly connected, and Ei‟s are nonempty disjoint closed subsets of X, then X≠ E1∪E2∪. We further extend the concept of strongly connectedness in closure space. The aim of this paper is to introduce and study the concept of strongly connectedness in closure space.

Classical Electromagnetism

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Equations Of Motion ◽  
Classical Mechanics ◽  
Classical Field Theory ◽  
Scalar Fields ◽  
Local Symmetry ◽  
Momentum Tensor ◽  
Global Symmetry ◽  
Complex Scalar ◽  
Symmetry Properties ◽  
Time Variations

In this chapter, Noether’s theorem as a classical field theory is presented and the properties of variations are again discussed for fields (i.e. field variations, space variations, time variations, spacetime variations), resulting in the Noether condition. Quasisymmetries and spontaneous symmetry breaking are discussed, as well as local symmetry and global symmetry. Following these definitions, Noether’s first theorem and Noether’s second theorem are developed. The classical Schrödinger field is investigated and the key equations of classical mechanics are summarised into a single Lagrangian. Symmetry properties of the field action and equations of motion are then compared. The chapter discusses the energy–momentum tensor, the Klein–Utiyama theorem, the Liouville equation and the Hamilton–Jacobi equation. It also discusses material science, special orthogonal groups and complex scalar fields.

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