On the eigenvalues in problems with spherical symmetry. Ill

1959 ◽  
Vol 252 (1271) ◽  
pp. 436-444 ◽  
Keyword(s):  
Spherical Symmetry ◽  
General Type ◽  
Three Dimensional ◽  
Lattice Points ◽  
Whole Space ◽  
Dimensional Equation

Let N (λ) denote the number of eigenvalues not exceeding λ of the three-dimensional equation ∇ 2 Ψ + {λ- q ( r )} Ψ = 0 over the whole space. The problem of the behaviour of N (λ) as λ → ∞ is considered in the case where q ( r — r c , c being a constant. It is shown that if c = 4 or 6 N (λ) = a μ 3 + b μ 2 + O (μ 5/3 ), where μ =λ 1/2+1/c and a and b are constants. This result is derived from a theorem due to van der Corput on the lattice-points in a region of a general type. It does not hold in the case c = 2, which is exceptional.

On the eigenvalues in problems with spherical symmetry. II

1959 ◽  
Vol 251 (1264) ◽  
pp. 46-54 ◽  
Keyword(s):  
Asymptotic Distribution ◽  
Spherical Symmetry ◽  
Large Range ◽  
Three Dimensional ◽  
Dimensional Equation

The problem of the asymptotic distribution of the eigenvalues of the three-dimensional equation ∇ 2 Ψ + {λ-q(r)} Ψ = 0 with spherical symmetry, involves that of the eigenvalues of the equations d 2 Ψ/d 2 x + {λ -q(r) - l 2 +l / r 2 } Ψ = 0, where λ is large and l varies over a certain large range 0 ≤ l ≤ L . It is shown that these eigenvalues satisfy equations of the form ƒ R R´ {λ l, n - q(r) - l 2 + l / r 2 } 1/2 dr = (n + 1/2) π +0 {L/l(L-l)}.

scholarly journals Organotrialkoxysilane-Functionalized Prussian Blue Nanoparticles-Mediated Fluorescence Sensing of Arsenic(III)

Nanomaterials ◽  
2021 ◽  
Vol 11 (5) ◽  
pp. 1145
Author(s):  
Prem. C. Pandey ◽  
Shubhangi Shukla ◽  
Roger J. Narayan
Keyword(s):  
Prussian Blue ◽  
Chemical Reduction ◽  
Low Cost ◽  
Three Dimensional ◽  
Heterogeneous Systems ◽  
Lattice Points ◽  
Radiative Energy ◽  
Fluorescence Sensing ◽  
Emission Signal ◽  
Prussian Blue Nanoparticles

Prussian blue nanoparticles (PBN) exhibit selective fluorescence quenching behavior with heavy metal ions; in addition, they possess characteristic oxidant properties both for liquid–liquid and liquid–solid interface catalysis. Here, we propose to study the detection and efficient removal of toxic arsenic(III) species by materializing these dual functions of PBN. A sophisticated PBN-sensitized fluorometric switching system for dosage-dependent detection of As3+ along with PBN-integrated SiO2 platforms as a column adsorbent for biphasic oxidation and elimination of As3+ have been developed. Colloidal PBN were obtained by a facile two-step process involving chemical reduction in the presence of 2-(3,4-epoxycyclohexyl)ethyl trimethoxysilane (EETMSi) and cyclohexanone as reducing agents, while heterogeneous systems were formulated via EETMSi, which triggered in situ growth of PBN inside the three-dimensional framework of silica gel and silica nanoparticles (SiO2). PBN-induced quenching of the emission signal was recorded with an As3+ concentration (0.05–1.6 ppm)-dependent fluorometric titration system, owing to the potential excitation window of PBN (at 480–500 nm), which ultimately restricts the radiative energy transfer. The detection limit for this arrangement is estimated around 0.025 ppm. Furthermore, the mesoporous and macroporous PBN-integrated SiO2 arrangements might act as stationary phase in chromatographic studies to significantly remove As3+. Besides physisorption, significant electron exchange between Fe3+/Fe2+ lattice points and As3+ ions enable complete conversion to less toxic As5+ ions with the repeated influx of mobile phase. PBN-integrated SiO2 matrices were successfully restored after segregating the target ions. This study indicates that PBN and PBN-integrated SiO2 platforms may enable straightforward and low-cost removal of arsenic from contaminated water.

Monte Carlo Simulation of Precipitate Nucleation and Growth: Time Dependent Results

1988 ◽  
Vol 141 ◽  
Author(s):  
James P. Lavine ◽  
Gilbert A. Hawkins
Keyword(s):  
Monte Carlo ◽  
Crystalline Silicon ◽  
Three Dimensional ◽  
Nucleation Site ◽  
Nucleation And Growth ◽  
Lattice Points ◽  
Growth Time ◽  
Decay Kinetics ◽  
Time Step ◽  
Oxygen Atoms

AbstractA three-dimensional Monte Carlo computer program has been developed to study the heterogeneous nucleation and growth of oxide precipitates during the thermal treatment of crystalline silicon. In the simulations, oxygen atoms move on a lattice with randomly selected lattice points serving as nucleation sites. The change in free energy that the oxygen cluster would experience in gaining or losing one oxygen atom is used to govern growth or dissolution of the cluster. All the oxygen atoms undergo a jump or a growth decision during each time step of the anneal. The growth and decay kinetics of each nucleation site display interesting fluctuation phenomena. The time dependence of the cluster size generally differs from the expected 3/2 power law due to the fluctuations in oxygen arrival at and incorporation in a precipitate. Competition between growing sites and coarsening are observed.

Dynamically Feasible Periodic Trajectories for Generic Spatial Three-Degree-of-Freedom Cable-Suspended Parallel Robots1

2018 ◽  
Vol 10 (3) ◽  
Author(s):  
Giovanni Mottola ◽  
Clément Gosselin ◽  
Marco Carricato
Keyword(s):  
Real Time ◽  
General Type ◽  
Three Dimensional ◽  
Initial State ◽  
End Effector ◽  
Periodic Trajectories ◽  
Real Time Applications ◽  
State Of Rest ◽  
Three Degree Of Freedom ◽  
Inertia Forces

Cable-suspended robots may move beyond their static workspace by keeping all cables under tension, thanks to end-effector inertia forces. This may be used to extend the robot capabilities, by choosing suitable dynamical trajectories. In this paper, we consider three-dimensional (3D) elliptical trajectories of a point-mass end effector suspended by three cables from a base of generic geometry. Elliptical trajectories are the most general type of spatial sinusoidal motions. We find a range of admissible frequencies for which said trajectories are feasible; we also show that there is a special frequency, which allows the robot to have arbitrarily large oscillations. The feasibility of these trajectories is verified via algebraic conditions that can be quickly verified, thus being compatible with real-time applications. By generalizing previous studies, we also study the possibility to change the frequency of oscillation: this allows the velocity at which a given ellipse is tracked to be varied, thus providing more latitude in the trajectory definition. We finally study transition trajectories to move the robot from an initial state of rest (within the static workspace) to the elliptical trajectory (and vice versa) or to connect two identical ellipses having different centers.

scholarly journals The Lissajous lens: a three-dimensional absolute optical instrument without spherical symmetry

2015 ◽  
Vol 23 (5) ◽  
pp. 5716 ◽  
Author(s):  
Aaron J. Danner ◽  
H. L. Dao ◽  
Tomáš Tyc
Keyword(s):  
Spherical Symmetry ◽  
Three Dimensional ◽  
Optical Instrument

Special directions in momentum space. II. Hexagonal, tetragonal and trigonal symmetries

2012 ◽  
Vol 45 (6) ◽  
pp. 1254-1260 ◽  
Author(s):  
G. Kontrym-Sznajd ◽  
M. Samsel-Czekała
Keyword(s):  
Brillouin Zone ◽  
Momentum Space ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Cubic Lattices ◽  
Whole Space ◽  
Full Symmetry ◽  
Optimal Set ◽  
Three Dimensional Space

This paper is a continuation of a previous one,Special directions in momentum space. I. Cubic symmetries[Kontrym-Sznajd & Samsel-Czekała (2011).J. Appl. Cryst.44, 1246–1254], where new sets of special directions (SDs), having the full symmetry of the Brillouin zone, were proposed for cubic lattices. In the present paper, such directions are derived for structures with unique six-, four- and threefold axes,i.e.hexagonal, tetragonal and trigonal lattices, for both two- and three-dimensional space. The SDs presented here allow for construction, in the whole space, of anisotropic quantities from the knowledge of such quantities along a limited number of SDs. The task at hand is to determine as many anisotropic components as the number of available sampling directions. Also discussed is a way of dealing with data when the number of anisotropic components is restricted by a non-optimal set of SDs.

Transverse mixing of pollutants in streams: a review

2014 ◽  
Vol 41 (5) ◽  
pp. 472-482 ◽  
Author(s):  
H. Sharma ◽  
Z. Ahmad
Keyword(s):  
Vertical Mixing ◽  
Three Dimensional ◽  
Water Quality Modeling ◽  
Mixing Zone ◽  
Transverse Mixing ◽  
Know How ◽  
Advection Dispersion ◽  
Quality Modeling ◽  
Longitudinal Mixing ◽  
Dimensional Equation

Spilling or release of foreign particles in the flowing water is considered as pollution of water, and due to the inherent property of water to dissolve the substance, the particulate is well mixed in water. To monitor the extent of pollution in a stream it is essential to know how the pollutants mix in the river. It is observed that vertical mixing of pollutants is a very rapid process in the vertical directions and longitudinal mixing occurs very far from source of pollutant, which is generally out of reach of observations. Thus intermediate or transverse mixing zone is considered very important for water quality modeling. This paper is an attempt to summarize the phenomenon behind pollutant transport, reduction of three-dimensional advection–dispersion equation to two-dimensional equation, and factors causing and affecting transverse mixing of pollutants.

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