scholarly journals Sum of squares in discrete physical spaces

2008 ◽  
Vol 30 (4) ◽  
pp. 4302.1-4302.5
Author(s):  
R. De Luca
Keyword(s):  
Number Theory ◽  
Radiation Spectrum ◽  
Three Dimensional ◽  
Standing Waves ◽  
General Criterion ◽  
Square Root ◽  
Introductory Courses ◽  
Physical Spaces ◽  
Physical Quantities ◽  
The Continuum

Discretization of three-dimensional physical spaces can induce, on observable physical quantities, effects which are not present in the continuum. Consider, as an example, the problem of the radiation spectrum of a blackbody, studied in introductory courses in quantum mechanics. One sees that the Rayleigh assumption of continuous and uniform frequency distribution of standing waves inside a cubic cavity with perfectly reflecting inner walls can be validated by a heuristic type of reasoning. However, by means of number theory, one sees that there might exist frequencies for which it is not possible to have standing waves inside the cavity. Nevertheless, within the same context, one can argue that a more general criterion can be adopted to validate the hypothesis of continuity of the observables which are expressed as the square root of the sum of three integers of a threedimensional space On x n y n z.

The two-sided lid-driven cavity: experiments on stationary and time-dependent flows

2002 ◽  
Vol 450 ◽  
pp. 67-95 ◽  
Author(s):  
CH. BLOHM ◽  
H. C. KUHLMANN
Keyword(s):  
Three Dimensional ◽  
Standing Waves ◽  
Reynolds Numbers ◽  
Time Dependent ◽  
Flow State ◽  
Incompressible Fluid Flow ◽  
Velocity Measurements ◽  
Driven Cavity ◽  
Rectangular Container ◽  
Hot Film

The incompressible fluid flow in a rectangular container driven by two facing sidewalls which move steadily in anti-parallel directions is investigated experimentally for Reynolds numbers up to 1200. The moving sidewalls are realized by two rotating cylinders of large radii tightly closing the cavity. The distance between the moving walls relative to the height of the cavity (aspect ratio) is Γ = 1.96. Laser-Doppler and hot-film techniques are employed to measure steady and time-dependent vortex flows. Beyond a first threshold robust, steady, three-dimensional cells bifurcate supercritically out of the basic flow state. Through a further instability the cellular flow becomes unstable to oscillations in the form of standing waves with the same wavelength as the underlying cellular flow. If both sidewalls move with the same velocity (symmetrical driving), the oscillatory instability is found to be tricritical. The dependence on two sidewall Reynolds numbers of the ranges of existence of steady and oscillatory cellular flows is explored. Flow symmetries and quantitative velocity measurements are presented for representative cases.

scholarly journals Thermalization of synchrotron radiation from field-aligned currents

1988 ◽  
Vol 6 (3) ◽  
pp. 493-501 ◽  
Author(s):  
William Peter ◽  
Anthony L. Peratt
Keyword(s):  
Magnetic Field ◽  
Synchrotron Radiation ◽  
Energy Density ◽  
Radiation Spectrum ◽  
Axial Magnetic Field ◽  
Three Dimensional ◽  
Synchrotron Emission ◽  
Microwave Background ◽  
Strong Function ◽  
Current Filaments

Three-dimensional plasma simulations of interacting galactic-dimensioned current filaments show bursts of synchroton radiation of energy density 1·2 ×10−13 erg/cm3 which can be compared with the measured cosmic microwave background energy density of 1·5 × 10−13 erg/cm3. However, the synchrotron emission observed in the simulations is not blackbody. In this paper, we analyze the absorption of the synchrotron emission by the current filaments themselves (i.e., self-absorption) in order to investigate the thermalization of the emitted radiation. It is found that a large number of current filaments (>1031) are needed to make the radiation spectrum blackbody up to the observed measured frequency of 100 GHz. The radiation spectrum and the required number of current filaments is a strong function of the axial magnetic field in the filaments.

Indentation of a Transversely Isotropic Thermoporoelastic Half-Space by a Rigid Circular Cylindrical Punch

2017 ◽  
Vol 84 (11) ◽  
Author(s):  
Yilan Huang ◽  
Guozhan Xia ◽  
Weiqiu Chen ◽  
Xiangyu Li
Keyword(s):  
Half Space ◽  
Original Problem ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
The Finite Element Method ◽  
Thermal Fields ◽  
Cylindrical Punch ◽  
The One ◽  
Physical Quantities ◽  
Potential Theory Method

Exact solutions to the three-dimensional (3D) contact problem of a rigid flat-ended circular cylindrical indenter punching onto a transversely isotropic thermoporoelastic half-space are presented. The couplings among the elastic, hydrostatic, and thermal fields are considered, and two different sets of boundary conditions are formulated for two different cases. We use a concise general solution to represent all the field variables in terms of potential functions and transform the original problem to the one that is mathematically expressed by integral (or integro-differential) equations. The potential theory method is extended and applied to exactly solve these integral equations. As a consequence, all the physical quantities of the coupling fields are derived analytically. To validate the analytical solutions, we also simulate the contact behavior by using the finite element method (FEM). An excellent agreement between the analytical predictions and the numerical simulations is obtained. Further attention is also paid to the discussion on the obtained results. The present solutions can be used as a theoretical reference when practically applying microscale image formation techniques such as thermal scanning probe microscopy (SPM) and electrochemical strain microscopy (ESM).

scholarly journals Characteristic Mode Analysis of surface current distributions on metallic structures exposed to HIRF- and DCI-excitations

2020 ◽  
Vol 18 ◽  
pp. 33-41
Author(s):  
Jan Ückerseifer ◽  
Frank Gronwald
Keyword(s):  
Three Dimensional ◽  
Surface Current ◽  
Resonance Region ◽  
Mode Analysis ◽  
Surface Currents ◽  
Characteristic Mode ◽  
Metallic Structures ◽  
Current Distributions ◽  
Characteristic Modes ◽  
Physical Quantities

Abstract. This paper treats Characteristic Mode Analyses of three-dimensional test objects in the context of EMC. Based on computed Characteristic Modes and mode-specific physical quantities, series expansions for HIRF- and DCI-induced surface currents are deduced. The contribution of single Characteristic Modes to surface currents at different test frequencies is analyzed. HIRF- and DCI-excitations are compared with regard to their surface current distributions in their resonance region determined by Characteristic Mode Analysis.

Three-dimensional standing waves on an obliquely flowing film

1988 ◽  
Vol 28 (4) ◽  
pp. 618-624
Author(s):  
S. V. Alekseenko ◽  
S. I. Shtork
Keyword(s):  
Three Dimensional ◽  
Standing Waves

scholarly journals Discrete-to-continuum modelling of weakly interacting incommensurate two-dimensional lattices

2018 ◽  
Vol 474 (2209) ◽  
pp. 20170612 ◽  
Author(s):  
Malena I. Español ◽  
Dmitry Golovaty ◽  
J. Patrick Wilber
Keyword(s):  
Continuum Model ◽  
Domain Walls ◽  
Three Dimensional ◽  
Variational Model ◽  
Dimensional System ◽  
Two Dimensional ◽  
Starting Point ◽  
Continuum Modelling ◽  
Three Dimensional System ◽  
The Continuum

In this paper, we derive a continuum variational model for a two-dimensional deformable lattice of atoms interacting with a two-dimensional rigid lattice. The starting point is a discrete atomistic model for the two lattices which are assumed to have slightly different lattice parameters and, possibly, a small relative rotation. This is a prototypical example of a three-dimensional system consisting of a graphene sheet suspended over a substrate. We use a discrete-to-continuum procedure to obtain the continuum model which recovers both qualitatively and quantitatively the behaviour observed in the corresponding discrete model. The continuum model predicts that the deformable lattice develops a network of domain walls characterized by large shearing, stretching and bending deformation that accommodates the misalignment and/or mismatch between the deformable and rigid lattices. Two integer-valued parameters, which can be identified with the components of a Burgers vector, describe the mismatch between the lattices and determine the geometry and the details of the deformation associated with the domain walls.

Square-root-like higher-order topological states in three-dimensional sonic crystals

2021 ◽  
Author(s):  
Zhiguo Geng ◽  
Huanzhao Lv ◽  
Zhan Xiong ◽  
Yu-Gui Peng ◽  
Zhaojiang Chen ◽  
...  
Keyword(s):  
Three Dimensional ◽  
Surface States ◽  
Higher Order ◽  
Square Root ◽  
Root Lattice ◽  
Third Order ◽  
Topological Materials ◽  
Topological States ◽  
Sonic Crystal ◽  
Sonic Crystals

Abstract The square-root descendants of higher-order topological insulators were proposed recently, whose topological property is inherited from the squared Hamiltonian. Here we present a three-dimensional (3D) square-root-like sonic crystal by stacking the 2D square-root lattice in the normal (z) direction. With the nontrivial intralayer couplings, the opened degeneracy at the K-H direction induces the emergence of multiple acoustic localized modes, i.e., the extended 2D surface states and 1D hinge states, which originate from the square-root nature of the system. The square-root-like higher order topological states can be tunable and designed by optionally removing the cavities at the boundaries. We further propose a third-order topological corner state in the 3D sonic crystal by introducing the staggered interlayer couplings on each square-root layer, which leads to a nontrivial bulk polarization in the z direction. Our work sheds light on the high-dimensional square-root topological materials, and have the potentials in designing advanced functional devices with sound trapping and acoustic sensing.

The waking-to-dreaming continuum and the effects of emotion

2000 ◽  
Vol 23 (6) ◽  
pp. 947-950 ◽  
Author(s):  
Ernest Hartmann
Keyword(s):  
Emotional State ◽  
Three Dimensional ◽  
The Other ◽  
Alternative View ◽  
Mental Functioning ◽  
The Continuum

The three-dimensional “AIM model” proposed by Hobson et al. is imaginative. However, many kinds of data suggest that the “dimensions” are not orthogonal, but closely correlated. An alternative view is presented in which mental functioning is considered as a continuum, or a group of closely linked continua, running from focused waking activity at one end, to dreaming at the other. The effect of emotional state is increasingly evident towards the dreaming end of the continuum.[Hobson et al.; Nielsen; Solms]

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