scholarly journals Some New Symmetric Equilateral Embeddings of Platonic and Archimedean Polyhedra

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 382
Author(s):  
Kris Coolsaet ◽  
Stan Schein
Keyword(s):  
Computer Algebra ◽  
Degrees Of Freedom ◽  
State Of The Art ◽  
Edge Length ◽  
Additional Constraint ◽  
Icosahedral Symmetry ◽  
Systems Of Equations ◽  
Number Of Solutions ◽  
Tetrahedral Symmetry ◽  
Reasonable Time

The icosahedron and the dodecahedron have the same graph structures as their algebraic conjugates, the great dodecahedron and the great stellated dodecahedron. All four polyhedra are equilateral and have planar faces—thus “EP”—and display icosahedral symmetry. However, the latter two (star polyhedra) are non-convex and “pathological” because of intersecting faces. Approaching the problem analytically, we sought alternate EP-embeddings for Platonic and Archimedean solids. We prove that the number of equations—E edge length equations (enforcing equilaterality) and 2 E − 3 F face (torsion) equations (enforcing planarity)—and of variables ( 3 V − 6 ) are equal. Therefore, solutions of the equations up to equivalence generally leave no degrees of freedom. As a result, in general there is a finite (but very large) number of solutions. Unfortunately, even with state-of-the-art computer algebra, the resulting systems of equations are generally too complicated to completely solve within reasonable time. We therefore added an additional constraint, symmetry, specifically requiring solutions to display (at least) tetrahedral symmetry. We found 77 non-classical embeddings, seven without intersecting faces—two, four and one, respectively, for the (graphs of the) dodecahedron, the icosidodecahedron and the rhombicosidodecahedron.

scholarly journals Generation of equations for shell models of turbulence in the Maple system

2021 ◽  
Vol 254 ◽  
pp. 02006
Author(s):  
Liubov Feshchenko ◽  
Gleb Vodinchar
Keyword(s):  
Computer Algebra ◽  
Power Law ◽  
Computer Algebra System ◽  
Nonlinear Interactions ◽  
Systems Of Equations ◽  
Shell Models ◽  
Quadratic Invariants ◽  
Automated Compilation ◽  
Shell Models Of Turbulence

The paper describes a technology for the automated compilation of equations for shell models of turbulence in the computer algebra system Maple. A general form of equations for the coefficients of nonlinear interactions is given, which will ensure that the required combination of quadratic invariants and power-law solutions is fulfilled in the model. Described the codes for the Maple system allowing to generate and solve systems of equations for the coefficients. The proposed technology allows you to quickly and accurately generate classes of shell models with the desired properties.

scholarly journals High-Dimensional Pixel Entanglement: Efficient Generation and Certification

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 376
Author(s):  
Natalia Herrera Valencia ◽  
Vatshal Srivastav ◽  
Matej Pivoluska ◽  
Marcus Huber ◽  
Nicolai Friis ◽  
...  
Keyword(s):  
Carrying Capacity ◽  
Communication Systems ◽  
Degrees Of Freedom ◽  
State Of The Art ◽  
Quality Measurement ◽  
High Dimensional ◽  
Single Photons ◽  
Spatial Mode ◽  
Classical Communication ◽  
Entanglement Of Formation

Photons offer the potential to carry large amounts of information in their spectral, spatial, and polarisation degrees of freedom. While state-of-the-art classical communication systems routinely aim to maximize this information-carrying capacity via wavelength and spatial-mode division multiplexing, quantum systems based on multi-mode entanglement usually suffer from low state quality, long measurement times, and limited encoding capacity. At the same time, entanglement certification methods often rely on assumptions that compromise security. Here we show the certification of photonic high-dimensional entanglement in the transverse position-momentum degree-of-freedom with a record quality, measurement speed, and entanglement dimensionality, without making any assumptions about the state or channels. Using a tailored macro-pixel basis, precise spatial-mode measurements, and a modified entanglement witness, we demonstrate state fidelities of up to 94.4% in a 19-dimensional state-space, entanglement in up to 55 local dimensions, and an entanglement-of-formation of up to 4 ebits. Furthermore, our measurement times show an improvement of more than two orders of magnitude over previous state-of-the-art demonstrations. Our results pave the way for noise-robust quantum networks that saturate the information-carrying capacity of single photons.

Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results

2009 ◽  
Vol 63 (1) ◽  
Author(s):  
Yuriy A. Rossikhin ◽  
Marina V. Shitikova
Keyword(s):  
Fractional Calculus ◽  
Degrees Of Freedom ◽  
Solid Mechanics ◽  
State Of The Art ◽  
Dynamic Problems ◽  
Engineering Problems ◽  
Multilayered Systems ◽  
Linear And Nonlinear ◽  
Linear And Nonlinear Systems

The present state-of-the-art article is devoted to the analysis of new trends and recent results carried out during the last 10years in the field of fractional calculus application to dynamic problems of solid mechanics. This review involves the papers dealing with study of dynamic behavior of linear and nonlinear 1DOF systems, systems with two and more DOFs, as well as linear and nonlinear systems with an infinite number of degrees of freedom: vibrations of rods, beams, plates, shells, suspension combined systems, and multilayered systems. Impact response of viscoelastic rods and plates is considered as well. The results obtained in the field are critically estimated in the light of the present view of the place and role of the fractional calculus in engineering problems and practice. This articles reviews 337 papers and involves 27 figures.

scholarly journals Some Lessons from the Schwinger Model

1997 ◽  
Vol 12 (06) ◽  
pp. 1091-1099 ◽  
Author(s):  
Gary McCartor
Keyword(s):  
Boundary Conditions ◽  
Degrees Of Freedom ◽  
Chiral Condensate ◽  
Number Of Solutions ◽  
Different Boundary Conditions ◽  
Schwinger Model ◽  
The Way

I shall recall a number of solutions to the Schwinger model in different gauges, having different boundary conditions and using different quantization surfaces. I shall discuss various properties of these solutions emphasizing the degrees of freedom necessary to represent the solution, the way the operator products are defined and the effects these features have on the chiral condensate.

Solving Polynomial Systems for the Kinematic Analysis and Synthesis of Mechanisms and Robot Manipulators

1995 ◽  
Vol 117 (B) ◽  
pp. 71-79 ◽  
Author(s):  
M. Raghavan ◽  
B. Roth
Keyword(s):  
Kinematic Analysis ◽  
State Of The Art ◽  
Polynomial Systems ◽  
The State ◽  
Polynomial Equations ◽  
Systems Of Equations ◽  
Synthesis Of Mechanisms ◽  
Analysis And Synthesis ◽  
Systems Of Polynomial Equations ◽  
And Robotics

Problems in mechanisms analysis and synthesis and robotics lead naturally to systems of polynomial equations. This paper reviews the state of the art in the solution of such systems of equations. Three well-known methods for solving systems of polynomial equations, viz., Dialytic Elimination, Polynomial Continuation, and Grobner bases are reviewed. The methods are illustrated by means of simple examples. We also review important kinematic analysis and synthesis problems and their solutions using these mathematical procedures.

scholarly journals Nonradiating and radiating modes excited by quantum emitters in open epsilon-near-zero cavities

2016 ◽  
Vol 2 (10) ◽  
pp. e1600987 ◽  
Author(s):  
Iñigo Liberal ◽  
Nader Engheta
Keyword(s):  
Degrees Of Freedom ◽  
Quantum Information Processing ◽  
State Of The Art ◽  
Optical Cavity ◽  
Dynamic Control ◽  
External Boundary ◽  
Dielectric Resonators ◽  
Optical Cavities ◽  
Epsilon Near Zero ◽  
Quantum Emitters

Controlling the emission and interaction properties of quantum emitters (QEs) embedded within an optical cavity is a key technique in engineering light-matter interactions at the nanoscale, as well as in the development of quantum information processing. State-of-the-art optical cavities are based on high quality factor photonic crystals and dielectric resonators. However, wealthier responses might be attainable with cavities carved in more exotic materials. We theoretically investigate the emission and interaction properties of QEs embedded in open epsilon-near-zero (ENZ) cavities. Using analytical methods and numerical simulations, we demonstrate that open ENZ cavities present the unique property of supporting nonradiating modes independently of the geometry of the external boundary of the cavity (shape, size, topology, etc.). Moreover, the possibility of switching between radiating and nonradiating modes enables a dynamic control of the emission by, and the interaction between, QEs. These phenomena provide unprecedented degrees of freedom in controlling and trapping fields within optical cavities, as well as in the design of cavity opto- and acoustomechanical systems.

scholarly journals How many tuples of group elements have a given property? With an appendix by Dmitrii V. Trushin

2014 ◽  
Vol 24 (04) ◽  
pp. 413-428 ◽  
Author(s):  
Anton A. Klyachko ◽  
Anna A. Mkrtchyan
Keyword(s):  
Systems Of Equations ◽  
Number Of Solutions ◽  
First Order ◽  
Group Language ◽  
The Matrix

Generalizing Solomon's theorem, Gordon and Rodriguez-Villegas have proven recently that, in any group, the number of solutions to a system of coefficient-free equations is divisible by the order of this group whenever the rank of the matrix composed of the exponent sums of ith unknown in jth equation is less than the number of unknowns. We generalize this result in two directions: first, we consider equations with coefficients, and second, we consider not only systems of equations but also any first-order formulae in the group language (with constants). Our theorem implies some amusing facts; for example, the number of group elements whose squares lie in a given subgroup is divisible by the order of this subgroup.

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