Basic Commutators and Minimal Massey Products

1984 ◽  
Vol 36 (6) ◽  
pp. 1119-1146 ◽  
Author(s):  
Roger Fenn ◽  
Denis Sjerve
Keyword(s):  
Hyperbolic Model ◽  
Two Dimensional ◽  
Massey Product ◽  
Massey Products ◽  
Symmetry Properties ◽  
Cup Products

The purpose of this paper is to continue the investigation into Massey products defined on two dimensional polyhedra, initiated in [13]. It will be shown that for many such spaces there is a hyperbolic model which can be used to study Massey products. More precisely, Massey products may be interpreted as intersections of geodesies in the Poincaré model. These elements are called minimal Massey products and are the analogue of Massey products over a system considered in Porter's paper. They enjoy the property of being uniquely defined (without indeterminacy) and of being multilinear and natural. Minimal products also satisfy symmetry properties generalising the symmetry properties enjoyed by cup products.A device which will be useful in the proof of the main theorem, 7.4, is the introduction of a class of complexes called basic complexes. These generalise the notion of a surface and each one houses a standard copy of a Massey product.

Massey Products and Lower Central Series of Free Groups

1987 ◽  
Vol 39 (2) ◽  
pp. 322-337 ◽  
Author(s):  
Roger Fenn ◽  
Denis Sjerve
Keyword(s):  
Free Group ◽  
Differential Calculus ◽  
Free Groups ◽  
Lower Central Series ◽  
Central Series ◽  
Massey Product ◽  
Massey Products ◽  
One Dimensional ◽  
Image Position

The purpose of this paper is to continue the investigation into the relationships amongst Massey products, lower central series of free groups and the free differential calculus (see [4], [9], [12]). In particular we set forth the notion of a universal Massey product ≪α1, …, αk≫, where the αi are one dimensional cohomology classes. This product is defined with zero indeterminacy, natural and multilinear in its variables.In order to state the results we need some notation. Throughout F will denote the free group on fixed generators x1, …, xn andwill denote the lower central series of F. If I = (i1, …, ik) is a sequence such that 1 ≦ i1, …, ik ≦ n then ∂1 is the iterated Fox derivative and , where is the augmentation. By convention we set ∂1 = identity if I is empty.

Three and Four Centre Oscillators and Their Application in Nuclear Physics

1974 ◽  
Vol 29 (7) ◽  
pp. 1003-1010 ◽  
Author(s):  
Peter Bergmann ◽  
Hans-Joachim Scheefer
Keyword(s):  
Analytical Solution ◽  
Schrödinger Equation ◽  
Nuclear Physics ◽  
Schrodinger Equation ◽  
Numerical Procedure ◽  
Orthogonal Basis ◽  
Two Dimensional ◽  
Schmidt's Method ◽  
Symmetry Properties

The extension of the nuclear two-centre-oscillator to three and four centres is investigated. Some special symmetry-properties are required. In two cases an analytical solution of the Schrödinger equation is possible. A numerical procedure is developed which enables the diagonalization of the Hamiltonian in a non-orthogonal basis without applying Schmidt's method of orthonormalization. This is important for calculations of arbitrary two-dimensional arrangements of the centres.

scholarly journals Realizable response matrices of multi-terminal electrical, acoustic and elastodynamic networks at a given frequency

2008 ◽  
Vol 464 (2092) ◽  
pp. 967-986 ◽  
Author(s):  
Graeme W Milton ◽  
Pierre Seppecher
Keyword(s):  
Three Dimensional ◽  
Complete Characterization ◽  
Two Dimensional ◽  
Response Matrix ◽  
Electrical Networks ◽  
Fixed Frequency ◽  
Symmetry Properties ◽  
Acoustic Networks ◽  
Mathematical Equivalence

We give a complete characterization of the possible response matrices at a fixed frequency of n -terminal electrical networks of inductors, capacitors, resistors and grounds, and of n -terminal discrete linear elastodynamic networks of springs and point masses, both in three-dimensional and two-dimensional cases. Specifically, we construct networks that realize any response matrix that is compatible with the known symmetry properties and thermodynamic constraints of response matrices. Owing to a mathematical equivalence, we also obtain a characterization of the response matrices of discrete acoustic networks.

scholarly journals Similarity in Symmetry Groups of Ornaments as a Measure for Cultural Interactions in Medieval Times

2020 ◽  
Author(s):  
Mehmet Erbudak ◽  
Selim Onat
Keyword(s):  
Systematic Approach ◽  
Symmetry Groups ◽  
Two Dimensional ◽  
Mathematical Methods ◽  
Symmetry Properties ◽  
Art Practices ◽  
Euclidean Distances ◽  
Similarities And Differences ◽  
Wallpaper Groups ◽  
Cultural Interactions

The symmetry properties of an ornament contain information about its civilisation and its interactions with other cultural sources. Two-dimensional periodic ornaments can be strictly classified into a limited set of 17 mathematical symmetry groups, also known as wallpaper groups. The collection of ornaments thus classified for a civilisation is characteristic of the cultural group and serves as a fingerprint to identify that group. If the distribution of wallpaper groups is available for several societies, mathematical methods can be applied to determine similarities and differences between the art practices of these communities. This method allows a systematic approach to the general ornamental practices within a culture and their interactions in the form of similarity of fingerprints. We test the feasibility of the method on examples of medieval Armenians, Byzantium, Seljuks first in Persia and then in Anatolia and among Arabs in the Middle East. For this purpose we present the distribution of the planar ornaments and calculate the Euclidean distances in pairs. We tested to what extend geographical and religious factors could account for the observed similarity of ornamental groups between cultures. The results suggest an intensive interaction between the Seljuk Turks and Arab craftsmen who produced the ornaments. Therefore the cultural interactions are religiously motivated.

SPATIAL, TEMPORAL, AND GLOBAL MODE ENTROPY IN A THALAMO–CORTICAL NETWORK

1993 ◽  
Vol 03 (06) ◽  
pp. 1487-1501 ◽  
Author(s):  
GENE V. WALLENSTEIN
Keyword(s):  
Cortical Neurons ◽  
Space Time ◽  
Time Behavior ◽  
Two Dimensional ◽  
Van Der Pol ◽  
Cortical Cells ◽  
Global Mode ◽  
Time Symmetry ◽  
Symmetry Properties ◽  
Cortical System

We employ an orthogonal decomposition technique with unique space–time symmetry properties to analyze a network of thalamo–cortical oscillators. A small number of thalamic "cells" are used to drive a network of cortical cells structured on a two-dimensional lattice. Two Bonhoeffer–van der Pol (BVP) based models of cortical neurons are compared at the network level in an attempt to reproduce some features of the thalamo–cortical system. It is shown that with the addition of a slowly varying term to the classical two-dimensional BVP model, the network can exhibit both periodic and irregular space–time behavior, along with changes in the temporal coherence and spatial frequency resembling the 8–12 Hz alpha rhythm.

scholarly journals Hyperbolic model for bacterial movement through an orthotropic two-dimensional porous medium

2015 ◽  
Vol 39 (3-4) ◽  
pp. 1050-1062
Author(s):  
R.M. Hernández ◽  
E. Rincón ◽  
R. Herrera ◽  
A.E. Chávez
Keyword(s):  
Porous Medium ◽  
Hyperbolic Model ◽  
Two Dimensional ◽  
Bacterial Movement

Piezoelectricity in Monolayers and Bilayers of Inorganic Two-Dimensional Crystals

2013 ◽  
Vol 1556 ◽  
Author(s):  
Karel-Alexander N. Duerloo ◽  
Mitchell T. Ong ◽  
Evan J. Reed
Keyword(s):  
Density Functional ◽  
Electromechanical Coupling ◽  
Three Dimensional ◽  
Single Layer ◽  
Elastic Model ◽  
Two Dimensional ◽  
Functional Theory ◽  
Symmetry Properties ◽  
Electromechanical Effects ◽  
Out Of Plane

ABSTRACTThe symmetry properties of many inorganic two-dimensional monolayer crystals make them piezoelectric, whereas their three-dimensional parent crystals are not. The emergence of piezoelectricity in the single-layer limit points toward intriguing electromechanical effects and applications in the single- or few-layer regime. We use density functional theory to calculate the piezoelectric coefficients of BN, MoS2, MoSe2, MoTe2, WS2, WSe2 and WTe2. These coefficients are found to be comparable to, and in some cases greater than those of commonly used wurtzite piezoelectrics. The centrosymmetry of a BN bilayer prevents a piezoelectric effect for this structure. However, by developing an elastic model, we find that the bilayer exhibits an unusual electromechanical coupling to the curvature, similar to that of a bimorph. A BN bilayer is found to amplify the constituent monolayers’ in-plane piezoelectric displacements by factors on the order of 103-104 into out-of plane deflections.

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