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Helical Symmetry
Published in Mihai V. Putz, New Frontiers in Nanochemistry, 2020
The helical symmetry operation is defined as a simultaneous translation and rotation of a given molecular system (considered as a unit cell) along the same symmetry axis. It can be used as a special periodic boundary condition for one-dimensional infinite long polymer chain. Its advantage is that instead of a long translational cell which contains a molecular fragment which covers to the whole rotation (rotation with 360°) one can use a much smaller unit cell which corresponds for a rotation with a certain angle. This means that, for example, in the case of the DNA polymer chain, instead of a unit cell with ten base pairs one can consider only a single base pair. It can be demonstrated that in a one-dimensional periodic system the symmetry group of a combined symmetry operation is isomorphic with the group of translation and therefore the Hamiltonian of a system is invariant under the applied helical symmetry operation. It can be applied in the case of one-dimensional polymer chains, like single- or double-strand DNA, helical (3–10, α or π helix) proteins or carbon nanotubes.
Structural Description of Materials
Published in Snehanshu Pal, Bankim Chandra Ray, Molecular Dynamics Simulation of Nanostructured Materials, 2020
Snehanshu Pal, Bankim Chandra Ray
There are a multitude of symmetry elements such as reflection, rotation, roto-inversion, and others. For example, a cubic system has twofold, threefold, and fourfold axes, while a hexagonal system possesses a sixfold symmetry. Crystal systems and their symmetry elements are presented in Table 1.3.
The Group Theory
Published in Mikhail G. Brik, Chong-Geng Ma, Theoretical Spectroscopy of Transition Metal and Rare Earth Ions, 2019
Mikhail G. Brik, Chong-Geng Ma
The group theory emerged from several branches of mathematics: the numbers theory, geometry, and algebraic equations. The foundations of group theory were developed by L. Euler (1707–1783), C.F. Gauss (1777–1855), J.-L. Lagrange (1736–1813), N. H. Abel (1802–1829), E. Galois (1811–1832), and many other prominent mathematicians. At the first glance, the group theory may seem to be a very abstract part of mathematics, which is very far from the real world. However, the area of applications of group theory is surprisingly and unexpectedly wide. It can be used to describe the symmetry properties of various objects. The group theory is very important for combinatorics (permutations and combinations of elements from different sets) and cryptography. It is even applied in the musical set theory, since music—if we forget for the moment about its esthetic component—may be considered as a certain combination of different permutations of the musical notes (well, not every combination of the musical notes can make a pleasant melody though). In chemistry the group theory facilitates description of various properties of molecules and symmetry properties of molecular orbitals. In physics importance of group theory follows from the fact that the symmetry properties of a considered physical system are related to the laws of conservation of certain physical quantities.
A novel mathematical model to measure individuals’ perception of the symmetry level of building facades
Published in Architectural Engineering and Design Management, 2020
Yusuf Cihat Aydin, Parham A. Mirzaei
Visual symmetry can simply be defined as the self-similarity of a visual stimulus under transformations. There are different types of symmetries and these can be grouped under three main headings, namely, reflectional, rotational, and translational symmetry (Wagemans, 1997). Visual demonstration of symmetry typologies is given in Figure 1. Reflection symmetry, also known as a mirror, line, bilateral symmetry, is the reflection of a visual stimulus around a line (e.g. vertical (Figure 1(A)), horizontal (Figure 1(B)) or diagonal-oblique (Figure 1(C))), the visual stimulus can have more than one symmetry line (multiple symmetries) (see Figure 1(D)). Rotational symmetry occurs when the visual stimulus is rotated relative to a specific pivot point (centric (Figure 1(E)) or decentralized (Figure 1(F))) without any change in its form. When the visual stimulus is moved or slide without any change in its form, this called translational symmetry (Figure 1(G)).
Emergence of a new symmetry class for Bogoliubov–de Gennes (BdG) Hamiltonians: expanding 10-fold symmetry classes
Published in Phase Transitions, 2020
A symmetry is a transformation that leaves the physical system invariant. These transformations include translation, reflection, rotation, scaling, etc. One of the most important implications of symmetry in physics is the existence of conservation laws. For every global continuous symmetry, there exists an associated conserved quantity [1]. In quantum mechanics, symmetry transformation can be represented on the Hilbert space of physical states by an operator that is either linear and unitary or anti-linear and anti-unitary [2]. Any symmetry operator acts on these states and transforms them to new states. These symmetry operators can be classified as continuous (rotation, translation) and discrete (parity, lattice translations, time reversal). Continuous symmetry transformations give rise to the conservation of probabilities and discrete symmetry transformations give rise to the quantum numbers. Another important implication of symmetry in quantum mechanics is the symmetry on exchanging identical particles [3].
The holographic principle for the differential game of active target defence
Published in International Journal of Control, 2022
Kamal Mammadov, Cheng-Chew Lim, Peng Shi
The holographic principle reveals a fundamental symmetry in simple motion pursuit-evasion games previously unacknowledged. A symmetry, also known as an invariance, is a transformation or mapping of a certain type which preserves some quantity or property. For example, consider a polynomial equation with only real coefficients; if is a root, then the complex conjugate of is also a root. Thus complex conjugation is a symmetry of the roots of a polynomial with real coefficients.