ENUMERATING PRIME LINKS BY A CANONICAL ORDER

2006 ◽  
Vol 15 (02) ◽  
pp. 217-237 ◽  
Author(s):  
AKIO KAWAUCHI ◽  
IKUO TAYAMA
Keyword(s):  
Ordered Set ◽  
Lattice Points ◽  
Integral Lattice ◽  
Canonical Order

The first author defined a well-order in the set of links by embedding it into a canonical well-ordered set of (integral) lattice points. He also gave elementary transformations among lattice points to enumerate the prime links in terms of lattice points under this order. In this paper, we add some new elementary transformations and explain how to enumerate the prime links. We show a table of the first 443 prime links arising from the lattice points of lengths up to 10 under this order. Our argument enables us to find 7 omissions and 1 overlap in Conway's table of prime links of 10 crossings.

scholarly journals On simplices and lattice points

1981 ◽  
Vol 31 (1) ◽  
pp. 15-19 ◽  
Author(s):  
P. R. Scott
Keyword(s):  
Lattice Points ◽  
Integral Lattice ◽  
Maximal Volume

AbstractLet S be a simplex in En which is homothetic to a given simplex S*, which contains no point of the integral lattice in its interior, and which has maximal volume V(S). We conjecture that V(S) > nn/n!, and establish the conjecture for n < 3.

scholarly journals Distribution of integral lattice points in an ellipsoid with a diophantine center

2015 ◽  
Vol 157 ◽  
pp. 468-506
Author(s):  
Jiyoung Han ◽  
Hyunsuk Kang ◽  
Yong-Cheol Kim ◽  
Seonhee Lim
Keyword(s):  
Lattice Points ◽  
Integral Lattice

scholarly journals On the width of a planar convex set containing zero or one lattice points

1993 ◽  
Vol 48 (1) ◽  
pp. 47-53
Author(s):  
Paul R. Scott
Keyword(s):  
Triangular Lattice ◽  
Short Proof ◽  
Convex Set ◽  
Compact Convex ◽  
Lattice Points ◽  
Rectangular Lattice ◽  
Integral Lattice ◽  
Compact Convex Set ◽  
Maximal Width ◽  
Planar Convex

We generalise to a rectangular lattice a known result about the maximal width of a planar compact convex set containing no points of the integral lattice. As a corollary we give a new short proof that the planar compact convex set of greatest width which contains just one point of the triangular lattice is an equilateral triangle.

scholarly journals Lattice points in a convex Set of given width

1976 ◽  
Vol 21 (4) ◽  
pp. 504-507 ◽  
Author(s):  
G. B. Elkington ◽  
J. Hammer
Keyword(s):  
Euclidean Space ◽  
Minimum Distance ◽  
Convex Set ◽  
Lattice Points ◽  
Dimensional Euclidean Space ◽  
Integral Lattice ◽  
Supporting Hyperplanes ◽  
Bounded Convex

Let S be a closed bounded convex set in d-dimensional Euclidean space Ed. The width w(S) of S is the minimum distance between supporting hyperplanes of S, and L(S) is the number of integral lattice points in the interior of S.

Circles on lattices and Hadamard matrices

2019 ◽  
pp. 2-9 ◽  
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.

SUMMATIONS OVER LATTICE POINTS IN K-SPACE

1948 ◽  
Vol os-19 (1) ◽  
pp. 238-248 ◽  
Author(s):  
S. BOCHNER ◽  
K. CHANDRASEKHARAN
Keyword(s):  
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.

scholarly journals Bounds on the Lattice Point Enumerator via Slices and Projections

2021 ◽  
Author(s):  
Ansgar Freyer ◽  
Martin Henk
Keyword(s):  
Convex Body ◽  
Lattice Point ◽  
Geometric Mean ◽  
Discrete Analogue ◽  
Lattice Points ◽  
General Context ◽  
General Bound

AbstractGardner et al. posed the problem to find a discrete analogue of Meyer’s inequality bounding from below the volume of a convex body by the geometric mean of the volumes of its slices with the coordinate hyperplanes. Motivated by this problem, for which we provide a first general bound, we study in a more general context the question of bounding the number of lattice points of a convex body in terms of slices, as well as projections.

Information Retrieval Systems, an Algebraic Approach I1

1981 ◽  
Vol 4 (3) ◽  
pp. 551-603
Author(s):  
Zbigniew Raś
Keyword(s):  
Information Retrieval ◽  
Boolean Algebra ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Algebraic Approach ◽  
Partially Ordered ◽  
Retrieval Systems ◽  
Information Retrieval Systems ◽  
Present Part ◽  
L Systems

This paper is the first of the three parts of work on the information retrieval systems proposed by Salton (see [24]). The system is defined by the notions of a partially ordered set of requests (A, ⩽), the set of objects X and a monotonic retrieval function U : A → 2X. Different conditions imposed on the set A and a function U make it possible to obtain various classes of information retrieval systems. We will investigate systems in which (A, ⩽) is a partially ordered set, a lattice, a pseudo-Boolean algebra and Boolean algebra. In my paper these systems are called partially ordered information retrieval systems (po-systems) lattice information retrieval systems (l-systems); pseudo-Boolean information retrieval systems (pB-systems) and Boolean information retrieval systems (B-systems). The first part concerns po-systems and 1-systems. The second part deals with pB-systems and B-systems. In the third part, systems with a partial access are investigated. The present part discusses the method for construction of a set of attributes. Problems connected with the selectivity and minimalization of a set of attributes are investigated. The characterization and the properties of a set of attributes are given.

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