Lattice Points in Three-Dimensional Convex Bodies

2000 ◽  
Vol 212 (1) ◽  
pp. 77-90 ◽  
Author(s):  
Ekkehard Krätzel
Keyword(s):  
Convex Bodies ◽  
Three Dimensional ◽  
Lattice Points

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 Lattice points in large convex bodies, II

1992 ◽  
Vol 62 (3) ◽  
pp. 285-295 ◽  
Author(s):  
Ekkehard Krätzel ◽  
Werner Nowak
Keyword(s):  
Convex Bodies ◽  
Lattice Points

Spatial Models of Sugars and Their Compounds

2021 ◽  
Vol 11 (2) ◽  
pp. 9-22
Author(s):  
Gennadiy Vladimirovich Zhizhin
Keyword(s):  
Convex Bodies ◽  
Three Dimensional ◽  
Spatial Models ◽  
Higher Dimensions ◽  
Three Dimensional Images

The images of saccharide and polysaccharide molecules in spaces of various dimensions are considered. A method has been developed for obtaining simplified three-dimensional images of sugar molecules and their chains based on their images in spaces of higher dimensions. It was found that three-dimensional images of furanose and pyranose molecules fundamentally differ from each other to form convex and, accordingly, non-convex bodies. This leads to fundamental differences in the structure of polysaccharides from these molecules.

Asymptotic Formulae for the Lattice Point Enumerator

1999 ◽  
Vol 51 (2) ◽  
pp. 225-249 ◽  
Author(s):  
U. Betke ◽  
K. Böröczky
Keyword(s):  
Convex Body ◽  
Surface Area ◽  
Positive Curvature ◽  
Convex Bodies ◽  
Mixed Volume ◽  
Lattice Points ◽  
Asymptotic Formulae ◽  
General Convex ◽  
Lattice Surface ◽  
Small Deformations

AbstractLet M be a convex body such that the boundary has positive curvature. Then by a well developed theory dating back to Landau and Hlawka for large λ the number of lattice points in λM is given by G(λM) = V(λM) + O(λd−1−ε(d)) for some positive ε(d). Here we give for general convex bodies the weaker estimatewhere SZd (M) denotes the lattice surface area of M. The term SZd is optimal for all convex bodies and o(λd−1) cannot be improved in general. We prove that the same estimate even holds if we allow small deformations of M.Further we deal with families {Pλ} of convex bodies where the only condition is that the inradius tends to infinity. Here we havewhere the convex body K satisfies some simple condition, V(Pλ; K; 1) is some mixed volume and S(Pλ) is the surface area of Pλ.

Stable Lattices

1953 ◽  
Vol 5 ◽  
pp. 261-270 ◽  
Author(s):  
Harvey Cohn
Keyword(s):  
Convex Body ◽  
Finite Number ◽  
Quadratic Forms ◽  
Convex Bodies ◽  
Lattice Points ◽  
Three Dimensions ◽  
General Convex ◽  
Critical Lattice ◽  
The Absolute

The consideration of relative extrema to correspond to the absolute extremum which is the critical lattice has been going on for some time. As far back as 1873, Korkine and Zolotareff [6] worked with the ellipsoid in hyperspace (i.e., with quadratic forms), and later Minkowski [8] worked with a general convex body in two or three dimensions. They showed how to find critical lattices by selection from among a finite number of relative extrema. They were aided by the long-recognized premise that only a finite number of lattice points can enter into consideration [1] when one deals with lattices “admissible to convex bodies.”

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