scholarly journals Higher-order synchronization on the sphere

2021 ◽  
Author(s):  
Max A Lohe
Keyword(s):  
Dimensional Space ◽  
Relative Effect ◽  
Higher Order ◽  
Coupling Constants ◽  
Coupling Coefficients ◽  
Continuous Transition ◽  
Final Configuration ◽  
Body Forces ◽  
Invariant Potential ◽  
Negative Coupling

Abstract We construct a system of $N$ interacting particles on the unit sphere $S^{d-1}$ in $d$-dimensional space, which has $d$-body interactions only. The equations have a gradient formulation derived from a rotationally-invariant potential of a determinantal form summed over all nodes, with antisymmetric coefficients. For $d=3$, for example, all trajectories lie on the $2$-sphere and the potential is constructed from the triple scalar product summed over all oriented $2$-simplices. We investigate the cases $d=3,4,5$ in detail, and find that the system synchronizes from generic initial values, for both positive and negative coupling coefficients, to a static final configuration in which the particles lie equally spaced on $S^{d-1}$. Completely synchronized configurations also exist, but are unstable under the $d$-body interactions. We compare the relative effect of $2$-body and $d$-body forces by adding the well-studied $2$-body interactions to the potential, and find that higher-order interactions enhance the synchronization of the system, specifically, synchronization to a final configuration consisting of equally spaced particles occurs for all $d$-body and $2$-body coupling constants of any sign, unless the attractive $2$-body forces are sufficiently strong relative to the $d$-body forces. In this case the system completely synchronizes as the $2$-body coupling constant increases through a positive critical value, with either a continuous transition for $d=3$, or discontinuously for $d=5$. Synchronization also occurs if the nodes have distributed natural frequencies of oscillation, provided that the frequencies are not too large in amplitude, even in the presence of repulsive 2-body interactions which by themselves would result in asynchronous behaviour.

scholarly journals Vortex Polymer Optical Fiber with 64 Stable OAM States

Polymers ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2776
Author(s):  
José A. Borda-Hernández ◽  
Claudia M. Serpa-Imbett ◽  
Hugo E. Hernandez Figueroa
Keyword(s):  
Optical Fiber ◽  
Optical Fibers ◽  
Communication Systems ◽  
Low Cost ◽  
Higher Order ◽  
Coupling Coefficients ◽  
Polymer Optical Fiber ◽  
Optical Polymer ◽  
Polymer Optical Fibers ◽  
Low Dispersion

This research introduces a numerical design of an air-core vortex polymer optical fiber in cyclic transparent optical polymer (CYTOP) that propagates 32 orbital angular momentum (OAM) modes, i.e., it may support up to 64 stable OAM-states considering left- and right-handed circular polarizations. This fiber seeks to be an alternative to increase the capacity of short-range optical communication systems multiplexed by modes, in agreement with the high demand of low-cost, insensitive-to-bending and easy-to-handle fibers similar to others optical fibers fabricated in polymers. This novel fiber possesses unique characteristics: a diameter of 50 µm that would allow a high mechanical compatibility with commercially available polymer optical fibers, a difference of effective index between neighbor OAM modes of around 10−4 over a bandwidth from 1 to 1.6 µm, propagation losses of approximately 15 × 10−3 dB/m for all OAM modes, and a very low dispersion for OAM higher order modes (±l = 16) of up to +2.5 ps/km-nm compared with OAM lower order modes at a telecom wavelength of 1.3 µm, in which the CYTOP exhibits a minimal attenuation. The spectra of mutual coupling coefficients between modes are computed considering small bends of up to 3 cm of radius and slight ellipticity in the ring of up to 5%. Results show lower-charge weights for higher order OAM modes.

Unification of Strong-Weak Interactions and Possible Unified Scheme of Four-Interactions

2020 ◽  
Vol 8 (5) ◽  
Author(s):  
Yi-Fang Chang
Keyword(s):  
Particle Physics ◽  
Short Range ◽  
Weak Interactions ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Coupling Constants ◽  
Strong Interactions ◽  
Main Character ◽  
Unified Theories ◽  
Three Dimensional Space

First, various known unified theories of interactions in particle physics are reviewed. Next, strong and weak interactions are all short-range, which should more be unified. Except different action ranges their main character is: strong interactions are attraction each other, and weak interactions are mutual repulsion and derive decay. We propose a possible method on their unification, whose coupling constants are negative and positive, respectively. Further, we propose a figure on the unification of the four basic interactions in three-dimensional space, and search some possible tests and predictions, for example, strong-weak interactions transform each other, some waves may be produced. Finally, based on the simplest unified gauge group GL(6,C) of four-interactions, a possible form of Lagrangian is researched. Some relations and equations of different interactions are discussed.

A COMPARISON BETWEEN THE TWO-DIMENSIONAL AND THREE-DIMENSIONAL LATTICES

2004 ◽  
Vol 18 (25) ◽  
pp. 1301-1309 ◽  
Author(s):  
ANDREI DOLOCAN ◽  
VOICU OCTAVIAN DOLOCAN ◽  
VOICU DOLOCAN
Keyword(s):  
Lattice Constant ◽  
Cohesive Energy ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Coupling Constants ◽  
Two Dimensional ◽  
Coupling Constant ◽  
Dimensional Lattice ◽  
Analytical Formulae ◽  
Three Dimensional Space

By using a new Hamiltonian of interaction we have calculated the interaction energy for two-dimensional and three-dimensional lattices. We present also, approximate analytical formulae and the analytical formulae for the constant of the elastic force. The obtained results show that in the three-dimensional space, the two-dimensional lattice has the lattice constant and the cohesive energy which are smaller than that of the three-dimensional lattice. For appropriate values of the coupling constants, the two-dimensional lattice in a two-dimensional space has both the lattice constant and the cohesive energy, larger than that of the two-dimensional lattice in a three-dimensional space; this means that if there is a two-dimensional space in the Universe, this should be thinner than the three-dimensional space, while the interaction forces should be stronger. On the other hand, if the coupling constant in the two-dimensional lattice in the two-dimensional space is close to zero, the cohesive energy should be comparable with the cohesive energy from three-dimensional space but this two-dimensional space does not emit but absorbs radiation.

Dispersion characteristics of periodic structural systems using higher order beam element dynamics

2019 ◽  
Vol 25 (2) ◽  
pp. 457-474
Author(s):  
M Ayad ◽  
N Karathanasopoulos ◽  
H Reda ◽  
JF Ganghoffer ◽  
H Lakiss
Keyword(s):  
Dynamic Equilibrium ◽  
Dimensional Space ◽  
Higher Order ◽  
Unit Cell ◽  
Square Lattice ◽  
Structural Systems ◽  
Discrete Dynamics ◽  
Two Dimensional ◽  
Dispersion Characteristics ◽  
Order Expansion

In the current work, we elaborate upon a beam mechanics-based discrete dynamics approach for the computation of the dispersion characteristics of periodic structures. Within that scope, we compute the higher order asymptotic expansion of the forces and moments developed within beam structural elements upon dynamic loads. Thereafter, we employ the obtained results to compute the dispersion characteristics of one- and two-dimensional periodic media. In the one-dimensional space, we demonstrate that single unit-cell equilibrium can provide the fundamental low-frequency band diagram structure, which can be approximated by non-dispersive Cauchy media formulations. However, we show that the discrete dynamics method can access the higher frequency modes by considering multiple unit-cell systems for the dynamic equilibrium, frequency ranges that cannot be accessed by simplified formulations. We extend the analysis into two-dimensional space computing with the dispersion attributes of square lattice structures. Thereupon, we demonstrate that the discrete dynamics dispersion results compare well with that obtained using Bloch theorem computations. We show that a high-order expansion of the inner element forces and moments of the structures is required for the higher wave propagation modes to be accurately represented, in contrast to the shear and the longitudinal mode, which can be captured using a lower, fourth-order expansion of its inner dynamic forces and moments. The provided results can serve as a reference analysis for the computation of the dispersion characteristics of periodic structural systems with the use of discrete element dynamics.

scholarly journals Coupled Bending-Bending-Torsion Vibration of a Pre-Twisted Beam with Aerofoil Cross-Section by the Finite Element Method

2003 ◽  
Vol 10 (4) ◽  
pp. 223-230
Author(s):  
Bulent Yardimoglu ◽  
Daniel J. Inman
Keyword(s):  
Finite Element ◽  
Cross Section ◽  
Degrees Of Freedom ◽  
Element Model ◽  
Higher Order ◽  
Coupling Coefficients ◽  
Cross Sectional ◽  
Torsion Vibration ◽  
Proper Order ◽  
Proposed Model

The present study deals with a finite element model for coupled bending-bending-torsion vibration analysis of a pretwisted Timoshenko beam with varying aerofoil cross-section. The element derived in this paper has two nodes, with seven degrees of freedom at each node. The nodal variables are transverse displacements, cross-section rotations and the shear angles in two planes and torsional displacement. The advantage of the present element is the exclusion of unnecessary derivatives of fundamental nodal variables, which were included to obtain invertable square matrix by other researchers, by choosing proper displacement functions and using relationship between cross-sectional rotation and the shear deformation. Element stiffness and mass matrices are developed from strain and kinetic energy expressions by assigning proper order polynomial expressions for cross-section properties and considering higher order coupling coefficients. The correctness of the present model is confirmed by the experimental results available in the literature. Comparison of the proposed model results with those in the literature indicates that a faster convergence is obtained. The results presented also provide some insights in the formulation by clearly indicating that higher order coupling terms have considerable influence on the natural frequencies.

RICCION FROM HIGHER-DIMENSIONAL SPACE–TIME AND ONE-LOOP RENORMALIZATION

2000 ◽  
Vol 15 (18) ◽  
pp. 2917-2932
Author(s):  
S. K. SRIVASTAVA
Keyword(s):  
Dimensional Space ◽  
Mass Scale ◽  
Space Time ◽  
Energy Scale ◽  
Coupling Constants ◽  
Higher Dimensional Space Time ◽  
Higher Derivative ◽  
Higher Dimensional ◽  
Renormalization Group Equations ◽  
Dimensional Space Time

Using higher-derivative gravitational action in (4+D)-dimensional space–time, Lagrangian density of riccion is obtained with the quartic self-interacting potential. It is found that after compactification to four-dimensional space–time the resulting theory for riccions is one-loop multiplicatively renormalizable. Renormalization group equations are solved and its solutions yield many interesting results such as (i) dependence of extra dimensions on the energy mass scale showing that these dimensions increase with the increasing mass scale, (ii) phase transition at 1.76×1016 GeV and (iii) dependence of gravitational and other coupling constants on energy scale. Results also suggest that space–time above 1.76×1016 GeV should be fractal.

scholarly journals Population heterogeneity in mutation rate increases mean fitness and the frequency of higher order mutants

2016 ◽  
Author(s):  
Helen K. Alexander ◽  
Stephanie I. Mayer ◽  
Sebastian Bonhoeffer
Keyword(s):  
Mutation Rate ◽  
Relative Effect ◽  
Relative Increase ◽  
Higher Order ◽  
Population Heterogeneity ◽  
Arbitrary Distribution ◽  
Rate Heterogeneity ◽  
Standing Variation ◽  
A Genome ◽  
Mean Fitness

AbstractMutation rate is a crucial evolutionary parameter that has typically been treated as a constant in population genetic analyses. However, mutation rate is likely to vary among co-existing individuals within a population, due to genetic polymorphisms, heterogeneous environmental influences, and random physiological fluctuations. We explore the consequences of such mutation rate heterogeneity in a model allowing an arbitrary distribution of mutation rate among individuals, either with or without inheritance. We find that variation of mutation rate about the mean results in a higher probability of producing zero or many simultaneous mutations on a genome. Moreover, it increases the frequency of higher order mutants even under ongoing mutation and selection. We gain a quantitative understanding of how this frequency depends on moments of the mutation rate distribution and selection coefficients. In particular, in a two-locus model, heterogeneity leads to a relative increase in double mutant frequency proportional to the squared coefficient of variation of the mutation rate. Relative effect sizes increase with the number of loci. Finally, this clustering of deleterious mutations into fewer individuals results in a higher population mean fitness. Our results imply that mutation rate heterogeneity allows a population to maintain a higher level of adaptedness to its current environment, while simultaneously harboring greater genetic diversity in the standing variation, which could be crucial for future adaptation to a new environment. Our results also have implications for interpreting mutation rate estimates and mutant frequencies in data.

scholarly journals Constructive Geometric Models with Imaginary Objects

2020 ◽  
pp. short27-1-short27-9
Author(s):  
Denis Voloshinov ◽  
Alexandra Solovjeva
Keyword(s):  
Projective Geometry ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Higher Order ◽  
Second Order ◽  
The Other ◽  
Significant Obstacle ◽  
The Relationship ◽  
Planar Drawing ◽  
Three Dimensional Space

The article is devoted to the consideration of a number of theoretical questions of projective geometry related to specifying and displaying imaginary objects, especially, conics. The lack of development of appropriate constructive schemes is a significant obstacle to the study of quadratic images in three-dimensional space and spaces of higher order. The relationship between the two circles, established by the inversion operation with respect to the other two circles, in particular, one of which is imaginary, allows obtain a simple and effective method for indirect setting of imaginary circles in a planar drawing. The application of the collinear transformation to circles with an imaginary radius also makes it possible to obtain unified algorithms for specifying and controlling imaginary conics along with usual real second-order curves. As a result, it allows eliminate exceptional situations that arise while solving problems with quadratic images in spaces of second and higher order.

scholarly journals ELECTROWEAKLY INTERACTING SCALAR AND GAUGE BOSONS, AND LEPTONS, FROM FIELD EQUATIONS ON SPIN (5+1)-DIMENSIONAL SPACE

2005 ◽  
Vol 20 (01) ◽  
pp. 77-93 ◽  
Author(s):  
J. BESPROSVANY
Keyword(s):  
Dimensional Space ◽  
Spin Symmetry ◽  
Coupling Constants ◽  
Tree Level ◽  
Field Equations ◽  
Gauge Groups ◽  
Scalar Particles ◽  
Additional Information ◽  
The Standard Model ◽  
Mass Terms

Unification ideas motivate the formulation of field equations on an extended matrix-spin space. Demanding that the Poincaré symmetry be maintained, one derives scalar symmetries that are associated with flavor and gauge groups. Boson and fermion solutions are obtained with a fixed representation. A field theory can be equivalently written and interpreted in terms of elements of such a space and is similarly constrained. At 5+1 dimensions, one obtains isospin and hypercharge SU (2)L× U (1) symmetries, their vector carriers, two-flavor charged and chargeless leptons, and scalar particles. Mass terms produce breaking of the symmetry to an electromagnetic U (1), a Weinberg's angle with sin 2(θW)=0.25, and additional information on the respective coupling constants. The particles' underlying spin symmetry gives information on their masses; one reproduces the Standard Model ratio MZ/MW, and predicts possible Higgs masses of MH≈114 and MH≈161 GeV, at tree level.

Sign in / Sign up

Export Citation Format

Share Document