scholarly journals LATTICE POINTS ON THE HOMOGENEOUS CUBIC EQUATION WITH FOUR UNKNOWNS x2 - xy + y2 + 4w2 = 8z3

2020 ◽  
Vol 8 (8) ◽  
Author(s):  
Manju Somanath ◽  
Radhika Das ◽  
V.A Bindu
Keyword(s):  
Three Dimensional ◽  
Lattice Points ◽  
Cubic Equation ◽  
Figurate Number ◽  
Integral Solutions

The Homogeneous cubic equation with four unknowns represented by the equation x2 - xy + y2 + 4w2 = 8z3  is analyzed for its patterns of non zero distinct integral solutions. Here we exhibit four different patterns. In each pattern we can find some interesting relations between the solutions and special numbers like Polygonal number, Three-Dimensional Figurate number, Star number, Rhombic Dodecahedral number etc.

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.

scholarly journals On generation and propagation of tsunamis in a shallow running ocean

1978 ◽  
Vol 1 (3) ◽  
pp. 373-390
Author(s):  
Lokenath Debnath ◽  
Uma Basu
Keyword(s):  
Free Surface ◽  
Three Dimensional ◽  
Relative Magnitude ◽  
Ocean Floor ◽  
Surface Elevation ◽  
Free Surface Elevation ◽  
Surface Disturbance ◽  
Fourier And Laplace Transforms ◽  
The Given ◽  
Integral Solutions

A theory is presented of the generation and propagation of the two and the three dimensional tsunamis in a shallow running ocean due to the action of an arbitrary ocean floor or ocean surface disturbance. Integral solutions for both two and three dimensional problems are obtained by using the generalized Fourier and Laplace transforms. An asymptotic analysis is carried out for the investigation of the principal features of the free surface elevation. It is found that the propagation of the tsunamis depends on the relative magnitude of the given speed of the running ocean and the wave speed of the shallow ocean. When the speed of the running ocean is less than the speed of the shallow ocean wave, both the two and the three dimensional free surface elevation represent the generation and propagation of surface waves which decay asymptotically ast−12for the two dimensional case and ast−1for the three dimensional tsunamis. Several important features of the solution are discussed in some detail. As an application of the general theory, some physically realistic ocean floor disturbances are included in this paper.

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.

scholarly journals Group-theoretical analysis of aperiodic tilings from projections of higher-dimensional latticesBn

2015 ◽  
Vol 71 (2) ◽  
pp. 175-185 ◽  
Author(s):  
Mehmet Koca ◽  
Nazife Ozdes Koca ◽  
Ramazan Koc
Keyword(s):  
Three Dimensional ◽  
Lattice Points ◽  
Icosahedral Symmetry ◽  
The Novel ◽  
Voronoi Cells ◽  
Aperiodic Tilings ◽  
Coxeter Number ◽  
Hypercubic Lattice ◽  
Higher Dimensional ◽  
Coxeter Plane

A group-theoretical discussion on the hypercubic lattice described by the affine Coxeter–Weyl groupWa(Bn) is presented. When the lattice is projected onto the Coxeter plane it is noted that the maximal dihedral subgroupDhofW(Bn) withh= 2nrepresenting the Coxeter number describes theh-fold symmetric aperiodic tilings. Higher-dimensional cubic lattices are explicitly constructed forn= 4, 5, 6. Their rank-3 Coxeter subgroups and maximal dihedral subgroups are identified. It is explicitly shown that when their Voronoi cells are decomposed under the respective rank-3 subgroupsW(A3),W(H2) ×W(A1) andW(H3) one obtains the rhombic dodecahedron, rhombic icosahedron and rhombic triacontahedron, respectively. Projection of the latticeB4onto the Coxeter plane represents a model for quasicrystal structure with eightfold symmetry. TheB5lattice is used to describe both fivefold and tenfold symmetries. The latticeB6can describe aperiodic tilings with 12-fold symmetry as well as a three-dimensional icosahedral symmetry depending on the choice of subspace of projections. The novel structures from the projected sets of lattice points are compatible with the available experimental data.

Symmetry-adapted digital modeling II. The double-helix B-DNA

2016 ◽  
Vol 72 (3) ◽  
pp. 312-323 ◽  
Author(s):  
A. Janner
Keyword(s):  
Dimensional Space ◽  
Double Helix ◽  
Three Dimensional ◽  
Reduction Factor ◽  
Lattice Points ◽  
Three Dimensions ◽  
Coarse Grained ◽  
Digital Modeling ◽  
Dna Strands ◽  
Dna Double Helix

The positions of phosphorus in B-DNA have the remarkable property of occurring (in axial projection) at well defined points in the three-dimensional space of a projected five-dimensional decagonal lattice, subdividing according to the golden mean ratio τ:1:τ [with τ = (1+\sqrt {5})/2] the edges of an enclosing decagon. The corresponding planar integral indicesn1,n2,n3,n4(which are lattice point coordinates) are extended to include the axial indexn5as well, defined for each P position of the double helix with respect to the single decagonal lattice ΛP(aP,cP) withaP= 2.222 Å andcP= 0.676 Å. A finer decagonal lattice Λ(a,c), witha=aP/6 andc=cP, together with a selection of lattice points for each nucleotide with a given indexed P position (so as to define a discrete set in three dimensions) permits the indexing of the atomic positions of the B-DNA d(AGTCAGTCAG) derived by M. J. P. van Dongen. This is done for both DNA strands and the single lattice Λ. Considered first is the sugar–phosphate subsystem, and then each nucleobase guanine, adenine, cytosine and thymine. One gets in this way a digital modeling of d(AGTCAGTCAG) in a one-to-one correspondence between atomic and indexed positions and a maximal deviation of about 0.6 Å (for the value of the lattice parameters given above). It is shown how to get a digital modeling of the B-DNA double helix for any given code. Finally, a short discussion indicates how this procedure can be extended to derive coarse-grained B-DNA models. An example is given with a reduction factor of about 2 in the number of atomic positions. A few remarks about the wider interest of this investigation and possible future developments conclude the paper.

Boundary integral solutions to three-dimensional unconfined Darcy's Flow

1980 ◽  
Vol 16 (4) ◽  
pp. 651-658 ◽  
Author(s):  
Gerard P. Lennon ◽  
Philip L.-F. Liu ◽  
James A. Liggett
Keyword(s):  
Three Dimensional ◽  
Boundary Integral ◽  
Integral Solutions

A GALLERY OF CHUA ATTRACTORS PART V

2007 ◽  
Vol 17 (05) ◽  
pp. 1383-1511 ◽  
Author(s):  
ELEONORA BILOTTA ◽  
GIANPIERO DI BLASI ◽  
FAUSTO STRANGES ◽  
PIETRO PANTANO
Keyword(s):  
Experimental Data ◽  
Time Series ◽  
Characteristic Function ◽  
Parameter Space ◽  
Three Dimensional ◽  
Special Kind ◽  
Cubic Equation ◽  
Different Shapes ◽  
Chua Oscillator ◽  
The Universe

Chua Oscillator and its generalizations display a variety of chaotic behaviors, whose most startling manifestation is the presence of strange attractors. These come in many different shapes and sizes yet share a special kind of beauty. In the work reported in this paper, we explored the universe of these attractors — much of which is still virgin territory. We then recorded the most interesting and significant in a "Gallery of Chua attractors". In papers, published in previous issues of this journal, we showed how different versions of the Oscillator follow different "routes to chaos". Parts II–IV of the Gallery describe the attractors generated by the dimensional and dimensionless forms of the Oscillator as well as attractors generated by circuits with a cubic equation for the characteristic function of the Chua diode. Here, we provide a detailed discussion of single scroll Chua systems and present 248 attractors generated by such systems. For each attractor, we provide a three-dimensional image, time series and FFTs. We go on to describe the techniques used to create the Gallery and the main characteristics of the attractors it includes. We use techniques such as PCA to represent the gallery in parameter space. The same techniques allow us to manipulate the shape of the attractors by enlarging them along their main axes of development. We use Hausdorff distances to compare shapes, and exploit the results to create landscapes in parameter space. Finally, we present experimental data from a single scroll attractor, using inertial ellipsoids and Hausdorff distances to show how the shape of the attractor evolves.

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