Higher-dimensional knotlike topological defects in local non-Abelian topological tensor currents

2006 ◽  
Vol 47 (9) ◽  
pp. 092303
Author(s):  
Yi-shi Duan ◽  
Shao-feng Wu ◽  
Peng-ming Zhang
Keyword(s):  
Topological Defects ◽  
Higher Dimensional ◽  
Topological Tensor

Symmetry

2021 ◽  
pp. 52-64
Author(s):  
Adrian P Sutton
Keyword(s):  
Physical Properties ◽  
Equations Of Motion ◽  
Topological Defects ◽  
Phase Changes ◽  
Broken Symmetry ◽  
Broken Symmetries ◽  
Higher Dimensional ◽  
Continuous Symmetries ◽  
Dimensional Crystal ◽  
Deep Sense

Symmetry arises not only in the invariance of an object to certain operations, but also in invariance of the equations governing motion of particles. Noether’s theorem connects continuous symmetries of equations of motion to conservation laws. The concept of broken symmetry arises in phase changes and topological defects, such as dislocations and disclinations. The principle of symmetry compensation reveals a deep sense in which symmetry is never destroyed – broken symmetries relate variants of an object displaying reduced symmetry. Symmetry plays a fundamental role in characterising the physical properties of crystals through Neumann’s principle. The concept of quasiperiodicity is introduced and it is shown how it is related to periodicity in a higher dimensional crystal.

scholarly journals PHOTON ROCKETS MOVING ARBITRARILY IN ANY DIMENSION

2011 ◽  
Vol 20 (03) ◽  
pp. 335-360 ◽  
Author(s):  
JIŘÍ PODOLSKÝ
Keyword(s):  
General Relativity ◽  
Gravitational Field ◽  
Cosmic Strings ◽  
Topological Defects ◽  
Higher Dimensions ◽  
Circular Trajectory ◽  
Pure Radiation ◽  
Type D ◽  
Higher Dimensional ◽  
Anisotropic Emission

A family of explicit exact solutions of Einstein's equations in four and higher dimensions is studied which describes the gravitational field of an object accelerating due to an anisotropic emission of photons. It is possible to prescribe an arbitrary motion, so that the acceleration of such photon rocket need not be uniform — both its magnitude and direction may vary with time. Except at the location of the rocket the space–times have no curvature singularities, and topological defects like cosmic strings are also absent. Any value of a cosmological constant is allowed. We investigate some particular examples of motion, namely a straight flight and a circular trajectory, and we derive the corresponding radiation patterns and the mass loss of the rockets. We also demonstrate the absence of "gravitational aberration" in such space–times. This interesting member of the higher-dimensional Robinson–Trautman class of pure radiation space–times of algebraic Type D generalizes the class of Kinnersley's solutions that has long been known in four-dimensional general relativity.

A Pseudo Five-Fold Symmetrical Ligand Drives Geometric Frustration in Porous Metal-Organic and Hydrogen Bonded Frameworks

2020 ◽  
Author(s):  
Frederik Haase ◽  
Gavin Craig ◽  
Mickaele Bonneau ◽  
kunihisa sugimoto ◽  
Shuhei Furukawa
Keyword(s):  
Topological Defects ◽  
Porous Metal ◽  
Building Blocks ◽  
Structural Features ◽  
Equilibrium Structure ◽  
Sorption Behavior ◽  
Building Unit ◽  
Geometric Frustration ◽  
Secondary Building Units ◽  
Hydrogen Bonded

Reticular framework materials thrive on designability, but unexpected reaction outcomes are crucial in exploring new structures and functionalities. By combining “incompatible” building blocks, we employed geometric frustration in reticular materials leading to emergent structural features. The combination of a pseudo C<sub>5</sub> symmetrical organic building unit based on a pyrrole core, with a C<sub>4</sub> symmetrical copper paddlewheel synthon led to three distinct frameworks by tuning the synthetic conditions. The frameworks show structural features typical for geometric frustration: self-limiting assembly, internally stressed equilibrium structures and topological defects in the equilibrium structure, which manifested in the formation of a hydrogen bonded framework, distorted and broken secondary building units and dangling functional groups, respectively. The influence of geometric frustration on the CO<sub>2</sub> sorption behavior and the discovery of a new secondary building unit shows geometric frustration can serve as a strategy to obtain highly complex porous frameworks.

A Pseudo Five-Fold Symmetrical Ligand Drives Geometric Frustration in Porous Metal-Organic and Hydrogen Bonded Frameworks

2020 ◽  
Author(s):  
Frederik Haase ◽  
Gavin Craig ◽  
Mickaele Bonneau ◽  
kunihisa sugimoto ◽  
Shuhei Furukawa
Keyword(s):  
Topological Defects ◽  
Porous Metal ◽  
Building Blocks ◽  
Structural Features ◽  
Equilibrium Structure ◽  
Sorption Behavior ◽  
Building Unit ◽  
Geometric Frustration ◽  
Secondary Building Units ◽  
Hydrogen Bonded

Reticular framework materials thrive on designability, but unexpected reaction outcomes are crucial in exploring new structures and functionalities. By combining “incompatible” building blocks, we employed geometric frustration in reticular materials leading to emergent structural features. The combination of a pseudo C<sub>5</sub> symmetrical organic building unit based on a pyrrole core, with a C<sub>4</sub> symmetrical copper paddlewheel synthon led to three distinct frameworks by tuning the synthetic conditions. The frameworks show structural features typical for geometric frustration: self-limiting assembly, internally stressed equilibrium structures and topological defects in the equilibrium structure, which manifested in the formation of a hydrogen bonded framework, distorted and broken secondary building units and dangling functional groups, respectively. The influence of geometric frustration on the CO<sub>2</sub> sorption behavior and the discovery of a new secondary building unit shows geometric frustration can serve as a strategy to obtain highly complex porous frameworks.

Spectroscopy-Based Mapping with Scanning Microwave Impedance Microscopy

2018 ◽  
Author(s):  
Peter De Wolf ◽  
Zhuangqun Huang ◽  
Bede Pittenger
Keyword(s):  
Single Point ◽  
Electrical Characterization ◽  
Principal Component ◽  
High Sensitivity ◽  
Data Cube ◽  
Nanometer Scale ◽  
Learning Approaches ◽  
Data Set ◽  
3D Data ◽  
Higher Dimensional

Abstract Methods are available to measure conductivity, charge, surface potential, carrier density, piezo-electric and other electrical properties with nanometer scale resolution. One of these methods, scanning microwave impedance microscopy (sMIM), has gained interest due to its capability to measure the full impedance (capacitance and resistive part) with high sensitivity and high spatial resolution. This paper introduces a novel data-cube approach that combines sMIM imaging and sMIM point spectroscopy, producing an integrated and complete 3D data set. This approach replaces the subjective approach of guessing locations of interest (for single point spectroscopy) with a big data approach resulting in higher dimensional data that can be sliced along any axis or plane and is conducive to principal component analysis or other machine learning approaches to data reduction. The data-cube approach is also applicable to other AFM-based electrical characterization modes.

HIGHER DIMENSIONAL ROBERTSON WALKER UNIVERSES INTERACTING WITH BRANS-DICKE FIELD

2020 ◽  
Vol 9 (10) ◽  
pp. 8545-8557
Author(s):  
K. P. Singh ◽  
T. A. Singh ◽  
M. Daimary
Keyword(s):  
Higher Dimensional

Celestial Tapestry

2020 ◽  
Author(s):  
Nicholas Mee
Keyword(s):  
Mathematics Curriculum ◽  
Dimensional Space ◽  
Golden Section ◽  
Single Point ◽  
Historical Context ◽  
Computer Art ◽  
Lightning Strike ◽  
Secret Message ◽  
Axiomatic Method ◽  
Higher Dimensional

Celestial Tapestry places mathematics within a vibrant cultural and historical context, highlighting links to the visual arts and design, and broader areas of artistic creativity. Threads are woven together telling of surprising influences that have passed between the arts and mathematics. The story involves many intriguing characters: Gaston Julia, who laid the foundations for fractals and computer art while recovering in hospital after suffering serious injury in the First World War; Charles Howard, Hinton who was imprisoned for bigamy but whose books had a huge influence on twentieth-century art; Michael Scott, the Scottish necromancer who was the dedicatee of Fibonacci’s Book of Calculation, the most important medieval book of mathematics; Richard of Wallingford, the pioneer clockmaker who suffered from leprosy and who never recovered from a lightning strike on his bedchamber; Alicia Stott Boole, the Victorian housewife who amazed mathematicians with her intuition for higher-dimensional space. The book includes more than 200 colour illustrations, puzzles to engage the reader, and many remarkable tales: the secret message in Hans Holbein’s The Ambassadors; the link between Viking runes, a Milanese banking dynasty, and modern sculpture; the connection between astrology, religion, and the Apocalypse; binary numbers and the I Ching. It also explains topics on the school mathematics curriculum: algorithms; arithmetic progressions; combinations and permutations; number sequences; the axiomatic method; geometrical proof; tessellations and polyhedra, as well as many essential topics for arts and humanities students: single-point perspective; fractals; computer art; the golden section; the higher-dimensional inspiration behind modern art.

Symmetries in higher dimensional crystallography

1989 ◽  
Vol 16 (1-4) ◽  
pp. 87-101 ◽  
Author(s):  
A. Janner
Keyword(s):  
Higher Dimensional
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