Multiples of Lattice Polytopes without Interior Lattice Points

AbstractIn this paper we motivate some new directions of research regarding the lattice width of convex bodies. We show that convex bodies of sufficiently large width contain a unimodular copy of a standard simplex. Following an argument of Eisenbrand and Shmonin, we prove that every lattice polytope contains a minimal generating set of the affine lattice spanned by its lattice points such that the number of generators (and the lattice width of their convex hull) is bounded by a constant which only depends on the dimension. We also discuss relations to recent results on spanning lattice polytopes and how our results could be viewed as the beginning of the study of generalized flatness constants. Regarding symplectic geometry, we point out how the lattice width of a Delzant polytope is related to upper and lower bounds on the Gromov width of its associated symplectic toric manifold. Throughout, we include several open questions.

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Lattice points in lattice polytopes

Mathematika ◽

10.1112/s0025579300014339 ◽

2001 ◽

Vol 48 (1-2) ◽

pp. 15-24 ◽

Cited By ~ 21

Author(s):

Oleg Pikhurko

Keyword(s):

Lattice Points ◽

Lattice Polytopes

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Circles on lattices and Hadamard matrices

Information and Control Systems ◽

10.31799/1684-8853-2019-3-2-9 ◽

2019 ◽

pp. 2-9 ◽

Cited By ~ 1

Author(s):

N. A. Balonin ◽

M. B. Sergeev ◽

J. Seberry ◽

O. I. Sinitsyna

Keyword(s):

Hadamard Matrices ◽

Lattice Points ◽

Practical Significance ◽

Upper And Lower Bounds ◽

Problem Size ◽

Orthogonal Matrices ◽

Video Information ◽

Scientific Schools ◽

Gauss Theorem ◽

Triangular Numbers

Introduction: The Hadamard conjecture about the existence of Hadamard matrices in all orders multiple of 4, and the Gauss problem about the number of points in a circle are among the most important turning points in the development of mathematics. They both stimulated the development of scientific schools around the world with an immense amount of works. There are substantiations that these scientific problems are deeply connected. The number of Gaussian points (Z3 lattice points) on a spheroid, cone, paraboloid or parabola, along with their location, determines the number and types of Hadamard matrices.Purpose: Specification of the upper and lower bounds for the number of Gaussian points (with odd coordinates) on a spheroid depending on the problem size, in order to specify the Gauss theorem (about the solvability of quadratic problems in triangular numbers by projections onto the Liouville plane) with estimates for the case of Hadamard matrices. Methods: The authors, in addition to their previous ideas about proving the Hadamard conjecture on the base of a one-to-one correspondence between orthogonal matrices and Gaussian points, propose one more way, using the properties of generalized circles on Z3 .Results: It is proved that for a spheroid, the lower bound of all Gaussian points with odd coordinates is equal to the equator radius R, the upper limit of the points located above the equator is equal to the length of this equator L=2πR, and the total number of points is limited to 2L. Due to the spheroid symmetry in the sector with positive coordinates (octant), this gives the values of R/8 and L/4. Thus, the number of Gaussian points with odd coordinates does not exceed the border perimeter and is no less than the relative share of the sector in the total volume of the figure.Practical significance: Hadamard matrices associated with lattice points have a direct practical significance for noise-resistant coding, compression and masking of video information.

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SUMMATIONS OVER LATTICE POINTS IN K-SPACE

The Quarterly Journal of Mathematics ◽

10.1093/qmath/os-19.1.238 ◽

1948 ◽

Vol os-19 (1) ◽

pp. 238-248 ◽

Cited By ~ 2

Author(s):

S. BOCHNER ◽

K. CHANDRASEKHARAN

Keyword(s):

Lattice Points

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

Nanomaterials ◽

10.3390/nano11051145 ◽

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.

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