Integer Coordinates Research Articles

Investigation of performances and mechanism of fluoride removal by Fe (III)-loaded ligand exchange cotton cellulose adsorbent

Novel adsorbent, Fe(III)-loaded ligand exchange cotton cellulose adsorbent [Fe(III)LECCA], was used to investigate the adsorption performances and mechanism of fluoride removal from aqueous solutions. The adsorbent was found to adsorb fluoride rapidly and effectively. The fluoride removal was influenced by pH. Adsorption model followed first-order reaction at different temperature, theapparent adsorption activated energyE, was 6.37 kJ·mol−1, and adsorption enthalpy ΔH was 5.35 kJ·mol−1. The adsorption capacity of fluoride on adsorbent was 3.2 mmol·g−1 (dry weight). The maximal integer coordination ratio of fluoride with Fe(III) LECCA was 3∶1. The ligand exchange mechanism of adsorption was elucidated through chemical methods and IR spectral analysis.

On the unlimited growth of a class of homogeneous multitype Markov chains

We consider a homogeneous multitype Markov chain whose states have non-negative integer coordinates, and give criteria for deciding whether or not the chain grows indefinitely with positive probability. The results are applied to study the extinction problem in a general class of controlled multitype branching processes.

A Five-coordinate Metal Center in Co(II)-substituted VanX

In an effort to structurally probe the metal binding site in VanX, electronic absorption, EPR, and extended x-ray absorption fine structure (EXAFS) spectroscopic studies were conducted on Co(II)-substituted VanX. Electronic spectroscopy revealed the presence of Co(II) ligand field transitions that had molar absorptivities of approximately 100 m(-1) cm(-1), which suggests that Co(II) is five-coordinate in Co(II)-substituted VanX. Low temperature EPR spectra of Co(II)-substituted VanX were simulated using spin Hamiltonian parameters of M(S) = |+/-1/2), E/D = 0.14, g(real(x,y)) = 2.37, and g(real(z)) = 2.03. These parameters lead to the prediction that Co(II) in the enzyme is five-coordinate and that there may be at least one solvent-derived ligand. Single scattering fits of EXAFS data indicate that the metal ions in both native Zn(II)-containing and Co(II)-substituted VanX have the same coordination number and that the metal ions are coordinated by 5 nitrogen/oxygen ligands at approximately 2.0 angstroms. These data demonstrate that Co(II) (and Zn(II) from EXAFS studies) is five-coordinate in VanX in contrast to previous crystallographic studies (Bussiere, D. E., Pratt, S. D., Katz, L., Severin, J. M., Holzman, T., and Park, C. H. (1998) Mol. Cell 2, 75-84). These spectroscopic studies also demonstrate that the metal ion in Co(II)-substituted VanX when complexed with a phosphinate analog of substrate D-Ala-D-Ala is also five-coordinate.

Diophantine properties of sets definable in o-minimal structures

Let be an o-minimal expansion of the ordered field of real numbers , and let S be an -definable subset (parameters allowed unless otherwise stated) of ℝn. In this note I investigate questions concerning the distribution of points on S with integer coordinates. My main theorem gives an estimate which, though probably far from best possible, at least shows that the o-minimal assumption does have diophantine consequences. This is, perhaps, surprising in view of the flexibility that we now seem to have in constructing o-minimal expansions of (see, e. g. [7], [8], [9]).

Generalized B-spline subdivision-surface wavelets for geometry compression

We present a new construction of lifted biorthogonal wavelets on surfaces of arbitrary two-manifold topology for compression and multiresolution representation. Our method combines three approaches: subdivision surfaces of arbitrary topology, B-spline wavelets, and the lifting scheme for biorthogonal wavelet construction. The simple building blocks of our wavelet transform are local lifting operations performed on polygonal meshes with subdivision hierarchy. Starting with a coarse, irregular polyhedral base mesh, our transform creates a subdivision hierarchy of meshes converging to a smooth limit surface. At every subdivision level, geometric detail can be expanded from wavelet coefficients and added to the surface. We present wavelet constructions for bilinear, bicubic, and biquintic B-Spline subdivision. While the bilinear and bicubic constructions perform well in numerical experiments, the biquintic construction turns out to be unstable. For lossless compression, our transform can be computed in integer arithmetic, mapping integer coordinates of control points to integer wavelet coefficients. Our approach provides a highly efficient and progressive representation for complex geometries of arbitrary topology.

Graphical representations of DNA as 2-D map

We describe a modification of the compact representation of DNA sequences which transforms the sequence into a 2-D diagram in which the ‘spots’ have integer coordinates. As a result the accompanying numerical characterization of DNA is quite simple and straightforward. This is an important advantage, particularly when considering DNA sequences having thousands of nucleic bases. The approach starts with the compact representation of DNA based on zigzag spiral template used for placing ‘spots’ associated with binary codes of the nucleic acids and subsequent suppression of the underlying zigzag curve. As a result, a 2-D map is formed in which all ‘spots’ have integer coordinates. By using only distances between spots having the same x or the same y coordinate one can construct a ‘map profile’ using integer arithmetic. The approach is illustrated on DNA sequences of the first exon of human β-globin.

Cutting corners with spheres in d-dimensions

A sphere corner cut A ⊂ N 0 d is a set of points with nonnegative integer coordinates which includes the origin and which can be separated from N 0 d⧹ A by a d-dimensional sphere. We show that in a given d-dimensional space there are O( n d+1 (log n) d−1 ) sphere corner cuts consisting of n points.

Cutting Corners by Circles and Spheres

Abstract A circle corner cut A ⊂ No2 is a planar set of points with nonnegative integer coordinates which includes the origin and which can be separated from No2 A by a circle. In this paper we show that there are O(n3 · log n) different circle corner cuts consisting of n points. If a sphere corner cut is defined as a set A ⊂ No3 of points with nonnegative integer coordinates which includes the origin and which can be separated from No3 A by a sphere, then there are O(n4 · (log n)2) different sphere corner cuts consisting of n points.

The Steinhaus tiling problem and the range of certain quadratic forms

We give a short proof of the fact that there are no measurable subsets of Euclidean space (in dimension d 3), which, no matter how translated and rotated, always contain exactly one integer lattice point. In dimension d = 2 (the original Steinhaus problem) the question remains open. x0. The Steinhaus problem Steinhaus (1957) asked if there exists a subset of the plane which, no matter how translated and rotated, always contains exactly one point with integer coordinates. This question remains unanswered. In this paper we deal only with the measurable version of the Steinhaus problem in dimension d 2, in which a measurable subset E of R d is sought with the property that for almost every x2R d and for almost every isometryS :R d !R d (SE +x)\Z d = 1:

Rounding Voronoi diagram

Computational geometry classically assumes real-number arithmetic which does not exist in actual computers. A solution consists in using integer coordinates for data and exact arithmetic for computations. This approach implies that if the results of an algorithm are the input of another, these results must be rounded to match this hypothesis of integer coordinates. In this paper, we treat the case of two-dimensional Voronoi diagrams and are interested in rounding the Voronoi vertices to grid points while interesting properties of the Voronoi diagram are preserved. These properties are the planarity of the embedding and the convexity of the cells. We give a condition on the grid size to ensure that rounding to the nearest grid point preserves the properties. We also present heuristics to round vertices (not to the nearest grid point) and preserve these properties.

Toric surface patches

We define a toric surface patch associated with a convex polygon, which has vertices with integer coordinates. This rational surface patch naturally generalizes classical Bezier surfaces. Several features of toric patches are considered: affine invariance, convex hull property, boundary curves, implicit degree and singular points. The method of subdivision into tensor product surfaces is introduced. Fundamentals of a multidimensional variant of this theory are also developed.

Computer Text Shading Algorithm Based on Perception of Luminance

With raster display systems, images on a monitor show jaggedness, because they are defined by integer coordinates of the monitor. This jaggedness is called an aliasing effect. To reduce this, engineers developed algorithms. One well-known algorithm is supersampling. This is accomplished by sampling at a higher resolution and reducing the sampled data to a lower resolution. We overlaid arrays of a 3times3 matrix on a text (supersampling) and reduced each array to a pixel on a monitor. In this process, engineers determined the level of the pixel luminance-intensity linearly to the number of elements painted in each array. Instead of using such direct, linear transformation, we determined the level of brightness, not the level of the pixel intensity, to produce better shading. Brightness, not luminance, is subjective. We created nine sets of gray levels based on this algorithm. We ran two experiments to choose the optimal gray-level set. In Experiment 1, participants chose the more legible of two letters shaded with different gray-level sets. In Experiment 2, participants counted target-letters in a string of letters as fast as they could. The experimental results did not favor gray-level sets that were close to the traditional linear transformation from the number of painted elements in arrays to pixel luminance-intensity. The best set was actually the third brightest as identified with this procedure using human participants. This perceptual algorithm can be used for any monitor to reduce aliasing effect.

Large size planar discrete velocity models for gas mixtures

We present results for the construction of physically acceptable large size planar discrete velocity models (DVMs), with momenta tiling all integer coordinates of the plane, for binary gas mixtures. We want, with binary collisions, five conservation laws (four for the restriction along one axis): only one for the light mass species with mass m = 1, only one for the heavy mass species with mass M>1, only two for the momenta along the x,y axes and one for the energy. We restrict our study to models with one zero momentum, while the momenta in the plane are different and occupy only integer coordinates. We start with a preliminary simple physical model satisfying the above constraints and add, with geometrical tools, new momenta. As an illustration we construct models, for M = 2,3,4,5, with an arbitrary number of momenta and which occupy all integer coordinates of the plane.

Optimiation Search Algorithm of Allocation Planning for Strip Coils in Hold for Shipment by Using Operational Know-how.

In the allocation planning of steel products in the hold of a ship, because the restriction which should be considered is complex and the problem scale is very large, planning systematization was difficult in the past. In this paper we study a scheduling algorithm of the allocation problem of steel products such as strip coils and sheets in the hold of a ship. Since steel products are usually stowed hierarchically in a hold, this planning becomes a three-dimensional allocation problem. First, we express the stowing-position of the products as three-dimensional integer coordinates, and model them as a combinatorial optimization problem. Next, the weight balance of the ship, loading ratio, and work efficiency of stowing are evaluated quantitatively, and they are optimized by applying the SA (Simulated Annealing) method. To obtain a preferable solution in practical time, we take the stowing know-how which the operators have in the evaluation function, and can make the search process efficient. In addition, to confirm the effectiveness of the proposed algorithm, several case studies have been carried out with actual data. Through the case studies, it has been found that the evaluation function involving the operation know-how term could reduce more unloaded coils assuring qualities of other evaluation items than the best solution obtained by the evaluation function without know-how term.

On odd points and the volume of lattice polyhedra

Denote by ℤ n 3 ,n≥ 2, the lattice consisting of all pointsx in ℝ3 such thatnx belongs to the fundamental lattice ℤ3 of points with integer coordinates. Letl n be the subset of ℤ n 3 consisting of all points whose coordinates are odd multiples of 1/n. The purpose of this paper is to give several new Pick-type formulae for the volume of three-dimensional lattice polyhedra, that is, polyhedra with vertices in ℤ3. Our formulae are in terms of numbers of only thel n-points belonging to a lattice polyhedronP in contrast to already known formulae which employ numbers of all the ℤ n 3 -points inP. On our way to establishing the formulae we show that the number of points froml n belonging to a three-dimensional lattice polyhedronP has some polynomiality properties similar to those of the well-known Ehrhart polynomial expressing the number of points of ℤ n 3 inP. The paper contains also some comments on a problem of finding a volume formula which would employ only the setsl n and which would be applicable to lattice polyhedra in arbitrary dimensions.

Holey Coronas A Solution of the Grünbaum-Shephard Conjecture on Convex Isohedral Tilings

In an article published recently in the section of Unsolved Problems of this MONTHLY, Branko Griunbaum [1] proposes the following conjecture: If S9 is an isohedral tiling of the Euclidean d-dimensional space by copies of a convex tile, then the corona of each tile is topological ball. The same conjecture is discussed as an open question in the article of Geoffrey C. Shephard [3], in which the author proves its stronger version for d = 2. We shall therefore refer to it as the Griinbaum-Shephard Conjecture. A tiling S9' is a collection of closed topological balls (disks, for d = 2) with mutually disjoint interiors (the tiles of S) such that the union of all tiles is the whole space. If all tiles are congruent, then S9 is monohedral, and their common shape is said to be the prototile of Y A tiling is isohedral provided its symmetry group acts transitively on the tiles; it is convex provided each of its tiles is convex. A lattice (of vectors) in R d is the collection of linear combinations with integer coefficients of a basis for R d. A typical example of a lattice is the collection of all vectors with integer coordinates, called the integer lattice (every lattice is affinely equivalent to the integer lattice). A monohedral tiling consisting of translates of the prototile by the vectors of a lattice is called a lattice tiling. Obviously, every lattice tiling is isohedral, while the converse is not true even in the plane. The corona of a tile T of S is the union of all tiles that meet T, including T itself. For example, in the usual (face-to-face) integer-lattice tiling of d-dimensional space by unit cubes, the corona of each tile is a cube of edge-length 3. Obviously, in an isohedral tiling, the coronas of any two tiles are congruent. See Griinbaum and Shephard [2] for a nearly encyclopedic survey of the theory of plane tilings; also see Schulte [4] and Schattschneider and Senechal [5], which deal with tilings in higher dimensional spaces. Grunbaum [1] observes that a confirmation of his conjecture for d = 2 (even without the assumption of convexity) can be established by inspection, since all types of isohedral tilings of the plane are known [2, Chapter 6]. He also conjectures that in the plane even more is true, namely that in every monohedral tiling of the plane the corona of every tile is simple (meaning simply-connected, hence homeomorphic to a disk). Shephard [3] found a non-convex counterexample, but also proved that in convex monohedral plane tilings every corona is simple [3, p. 358].

Free polygon enumeration and the area of an integral polygon

We introduce the notion of free polygons as combinatorial building blocks for convex integral polygons; that is, polygons with vertices having integer coordinates. In this context, an Euler-type formula is derived for the number of integer points in the interior of an integral polygon. This leads in turn to a formula for the area of an integral polygon P via the enumeration of free integral triangles and parallelograms contained inside P.

A fast algorithm for locating supplying center on a lattice

One of the well known location problems is to find a supply point in a plane such that total weighted distance between supply point and a set of demand points is minimized. No known algorithm finds the exact solution. In this paper, we consider a restricted case. The solution domain considered here is confined to lattice points (that is, points with integer coordinates). Given m demand points in a plane and a convex polygon P with n vertices, we propose an algorithm to find a supply point with integer coordinates in convex polygon P such that total weighted distance from the point to these m points is minimized. If P is contained in U × U lattice, the worst case running time of the algorithm is recursively.

Close Approximations of Minimum Rectangular Coverings

We consider the problem of covering arbitrary polygons with rectangles. The rectangles must lie entirely within the polygon. (This requires that the interior angles of the polygon are all greater than or equal to 90 degrees.) We want to cover the polygon with as few rectangles as possible. This problem has an application in fabricating masks for integrated circuits. In this paper we will describe the first polynomial algorithm, guaranteeing an O(log n) approximation factor, provided that the n vertices of the input polygon are given as polynomially bounded integer coordinates. By the same technique we also obtain the first algorithm producing a covering which is within a constant factor of the optimal in exponential time (compared to the doubly-exponential known before).

Strictly-upward drawings of ordered search trees

We prove that any logarithmic binary tree admits a linear-area straight-line strictly-upward planar grid drawing (in short, strictly-upward drawing), that is, a drawing in which (a) each edge is mapped into a single straight-line segment, (b) each node is placed below its parent, (c) no two edges intersect, and (d) each node is mapped into a point with integer coordinates. Informally, a logarithmic tree has the property that the height of any (sufficiently high) subtree is logarithmic with respect to the number of nodes. As a consequence, we have that k-balanced trees, redblack trees, and BB[α]-trees admit linear-area strictly-upward drawings. We then generalize our esults to logarithmic m-ary trees: as an application, we have that B-trees admit linear-area strictly-upward drawings.