Non-empty Cells Research Articles

Developing a New Estimation Approach for Constructing a Flexible Location Model to Address Challenges with Numerous Empty and Non-Empty Cells

Developing a New Estimation Approach for Constructing a Flexible Location Model to Address Challenges with Numerous Empty and Non-Empty Cells

Quiver Grassmannians of Type widetilde {D}_{n}, Part 2: Schubert Decompositions and F-polynomials

Extending the main result of Lorscheid and Weist (2015), in the first part of this paper we show that every quiver Grassmannian of an indecomposable representation of a quiver of type tilde D_{n} has a decomposition into affine spaces. In the case of real root representations of small defect, the non-empty cells are in one-to-one correspondence to certain, so called non-contradictory, subsets of the vertex set of a fixed tree-shaped coefficient quiver. In the second part, we use this characterization to determine the generating functions of the Euler characteristics of the quiver Grassmannians (resp. F-polynomials). Along these lines, we obtain explicit formulae for all cluster variables of cluster algebras coming from quivers of type tilde D_{n}.

A Conceptual Framework in Developing a New Location Model

The original purpose of the location model is to deal with mixed variables discrimination for classification purposes. Due to the problem of empty cells, smoothed location model is introduced. However, the smoothed location model had smoothed all the cells either empty or not, where the smoothing process causing changesto the original information of the non-empty cells. As it is well known that those original informationis a valuable source and important in any study that should be maintained. To address the aforementioned issues, an amalgamationof maximum likelihood and smoothing estimations is introduced to construct a new location model. The amalgamation of both estimations is expected could handle all situations whether the cells are empty or not based on several settings of sample size and number of variables.

Computation of spatial skyline points

The database skyline query (or non-domination query) has a spatial form: Given a set P with n point sites, and a point set S of m locations of interest, a site p∈P is a skyline point if and only if for each q∈P∖{p}, there exists at least one location s∈S that is closer to p than to q. We reduce the problem of determining skyline points to the problem of finding sites that have non-empty cells in an additively weighted Voronoi diagram under a convex distance function. The weights of said Voronoi diagram are derived from the coordinates of the sites of P, while the convex distance function is derived from the set of locations S. In the two-dimensional plane, this reduction gives an O((n+m)log⁡(n+m))-time algorithm to find the skyline points.

Enumerating Partial Latin Rectangles

This paper deals with different computational methods to enumerate the set $\mathrm{PLR}(r,s,n;m)$ of $r \times s$ partial Latin rectangles on $n$ symbols with $m$ non-empty cells. For fixed $r$, $s$, and $n$, we prove that the size of this set is given by a symmetric polynomial of degree $3m$, and we determine the leading terms (the monomials of degree $3m$ through $3m-9$) using inclusion-exclusion. For $m \leqslant 13$, exact formulas for these symmetric polynomials are determined using a chromatic polynomial method. Adapting Sade's method for enumerating Latin squares, we compute the exact size of $\mathrm{PLR}(r,s,n;m)$, for all $r \leqslant s \leqslant n \leqslant 7$, and all $r \leqslant s \leqslant 6$ when $n=8$. Using an algebraic geometry method together with Burnside's Lemma, we enumerate isomorphism, isotopism, and main classes when $r \leqslant s \leqslant n \leqslant 6$. Numerical results have been cross-checked where possible.

A Pascal Triangle Type Calculation for a Particular Infinite Series

In general, finding the sum of an infinite series is not always possible. However, there are infinite series whose sums can be computed. In our earlier paper, we derived a formula for computing the sum of a type of polylogarithm series involving multinomial coefficients. In this paper, we show that the formula leads to an elementary computation for the series $\sum_{k=1}^{\infty}\frac{k^n}{a^k}$ involving numbers obtained by a method similar to Pascal's triangle. We also show that our result is the number of ways of distributing $n$ distinct objects in $n$ or fewer distinct nonempty cells.

Сomputational procedures for thematic processing of space imagery for agricultural resources monitoring (part 2)

The universal fast algorithm of cluster analysis is considered. The proposed algorithm is a grid type, it uses the point density parameter in the grid cell and the ratio between neighborhoods to unite of neighboring dense cells into clusters.The algorithm sequentially calculates for each point the number of the cell to which it belongs, then generates groups of points for each non-empty cell. Then it sequentially unites cells into clusters, starting the process of fusion of the densest cells.The next cell is included in some cluster if at least one cell neighbor already belongs to the cluster. If the neighbors of the cell do not belong to any formed cluster, then the cell forms a new cluster. If the neighbors of the cell belong to several existing clusters, the respective clusters are merged into a new cluster.Combining cells into clusters uniquely determines the distribution of multiple points between the clusters. The user must specify a grid step parameter and a minimum grid cell density for which the cluster joining process is not performed. Low-density cells are considered noise.The algorithm does not require a preliminary task of the number of clusters and information about the nature of the distribution of points in the input set.The proposed algorithm can be used to process large arrays of point data of large spatial resolution. The most promising area of application of the algorithm is the analysis of multispectral satellite images of medium and high resolution in the fields of the analysis of the state of agricultural resources, forest resources and various natural landscapes. The result of clustering the space image data can also be used to create a classifier's training set.

Using a CAS/DGS to Analyze Computationally the Configuration of Planar Bar Linkage Mechanisms Based on Partial Latin Squares

Currently, the study of new isotopism invariants of partial Latin squares constitutes an open and active problem. This paper delves into this topic by analyzing computationally the configuration of planar bar linkage mechanisms for which the array formed by the lengths of their bars constitutes an empty-diagonal symmetric partial Latin square such that each one of its rows and columns has at least two non-empty cells. These assumptions enable one to define a series of algebraic and geometric constraints that can be readily implemented in any Computer Algebra or Dynamic Geometry System. In order to illustrate the different concepts and results introduced throughout the paper, it is explicitly determined and characterized the distribution of planar bar linkage mechanisms based on partial Latin squares of order up to five.

COMBINATORIAL ANALYSIS IN THE SCHEME OF ALLOCATION OF DISTINGUISHABLE PARTICLES INTO INDISTINGUISHABLE CELLS WITH A GIVEN NUMBER OF NON-EMPTY CELLS

The scheme of allocating r distinguishable particles into n indistinguishable cells with k non-empty cells is studied along the directions of enumerative combinatorics. They are the direct enumeration of the outcomes of the scheme with a particular discipline of their numbering, finding their numbers, establishing a one-to-one correspondence of the types of outcomes with their numbers, also known as numbering problem in direct and inverse statements, and modeling of outcomes of the scheme.

Restricted completion of sparse partial Latin squares

AbstractAnn×npartial Latin squarePis calledα-dense if each row and column has at mostαnnon-empty cells and each symbol occurs at mostαntimes inP. Ann×narrayAwhere each cell contains a subset of {1,…,n} is a (βn,βn, βn)-array if each symbol occurs at mostβntimes in each row and column and each cell contains a set of size at mostβn. Combining the notions of completing partial Latin squares and avoiding arrays, we prove that there are constantsα,β> 0 such that, for every positive integern, ifPis anα-densen×npartial Latin square,Ais ann×n (βn, βn, βn)-array, and no cell ofPcontains a symbol that appears in the corresponding cell ofA, then there is a completion ofPthat avoidsA; that is, there is a Latin squareLthat agrees withPon every non-empty cell ofP, and, for eachi,jsatisfying 1 ≤i,j≤n, the symbol in position (i,j) inLdoes not appear in the corresponding cell ofA.

Secondary Polytope and Secondary Power Diagram

An ingenious construction of Gel’fand, Kapranov, and Zelevinsky [5] geometrizes the triangulations of a point configuration, such that all coherent triangulations form a convex polytope, the so-called secondary polytope. The secondary polytope can be treated as a weighted Delaunay triangulation in the space of all possible coherent triangulations. Naturally, it should have a dual diagram. In this work, we explicitly construct the secondary power diagram, which is the power diagram of the space of all possible power diagrams with nonempty boundary cells. Secondary power diagram gives an alternative proof for the classical secondary polytope theorem based on Alexandrov theorem. Furthermore, secondary power diagram theory shows one can transform a nondegenerated coherent triangulation to another nondegenerated coherent triangulation by a sequence of bistellar modifications, such that all the intermediate triangulations are nondegenerated and coherent. As an application of this theory, we propose an algorithm to triangulate a special class of 3d nonconvex polyhedra without using additional vertices. We prove that this algorithm terminates in $$O({{n}^{3}})$$ time.

A computational algebraic geometry approach to analyze pseudo-random sequences based on Latin squares

Latin squares are used as scramblers on symmetric-key algorithms that generate pseudo-random sequences of the same length. The robustness and effectiveness of these algorithms are respectively based on the extremely large key space and the appropriate choice of the Latin square under consideration. It is also known the importance that isomorphism classes of Latin squares have to design an effective algorithm. In order to delve into this last aspect, we improve in this paper the efficiency of the known methods on computational algebraic geometry to enumerate and classify partial Latin squares. Particularly, we introduce the notion of affine algebraic set of a partial Latin square L = (lij) of order n over a field $\mathbb {K}$ as the set of zeros of the binomial ideal $\langle x_{i}x_{j}-x_{l_{ij}}\colon (i,j) \text { is a non-empty cell in} L \rangle \subseteq \mathbb {K}[x_{1},\ldots ,x_{n}]$ . Since isomorphic partial Latin squares give rise to isomorphic affine algebraic sets, every isomorphism invariant of the latter constitutes an isomorphism invariant of the former. In particular, we deal computationally with the problem of deciding whether two given partial Latin squares have either the same or isomorphic affine algebraic sets. To this end, we introduce a new pair of equivalence relations among partial Latin squares: being partial transpose and being partial isotopic.

Completing Partial Latin Squares with One Nonempty Row, Column, and Symbol

Let $r,c,s\in\{1,2,\ldots,n\}$ and let $P$ be a partial latin square of order $n$ in which each nonempty cell lies in row $r$, column $c$, or contains symbol $s$. We show that if $n\notin\{3,4,5\}$ and row $r$, column $c$, and symbol $s$ can be completed in $P$, then a completion of $P$ exists. As a consequence, this proves a conjecture made by Casselgren and Häggkvist. Furthermore, we show exactly when row $r$, column $c$, and symbol $s$ can be completed.

EVALUATING VOXEL ENABLED SCALABLE INTERSECTION OF LARGE POINT CLOUDS

Abstract. Laser scanning has become a well established surveying solution for obtaining 3D geo-spatial information on objects and environment. Nowadays scanners acquire up to millions of points per second which makes point cloud huge. Laser scanning is widely applied from airborne, carborne and stable platforms, resulting in point clouds obtained at different attitudes and with different extents. Working with such different large point clouds makes the determination of their overlapping area necessary but often time consuming. In this paper, a scalable point cloud intersection determination method is presented based on voxels. The method takes two overlapping point clouds as input. It consecutively resamples the input point clouds according to a preset voxel cell size. For all non-empty cells the center of gravity of the points in contains is computed. Consecutively for those centers it is checked if they are in a voxel cell of the other point cloud. The same process is repeated after interchanging the role of the two point clouds. The quality of the results is evaluated by the distance to the pints from the other data set. Also computation time and quality of the results are compared for different voxel cell sizes. The results are demonstrated on determining he intersection between an airborne and carborne laser point clouds and show that the proposed method takes 0.10%, 0.15%, 1.26% and 14.35% of computation time compared the the classic method when using cell sizes of of 10, 8, 5 and 3 meters respectively.

The asymptotic existence of DR $$(v,k,k-1)$$ ( v , k , k - 1 ) -BIBDs

A Kirkman square with index $$\lambda $$?, latinicity $$\mu $$μ, block size $$k$$k, and $$v$$v points, $$KS_k(v;\mu ,\lambda )$$KSk(v?μ,?), is a $$t\times t$$t×t array ($$t=\lambda (v-1)/\mu (k-1)$$t=?(v-1)/μ(k-1) ) defined on a $$v$$v-set $$V$$V such that (1) every point of $$V$$V is contained in precisely $$\mu $$μ cells of each row and column, (2) each cell of the array is either empty or contains a $$k$$k-subset of $$V$$V, and (3) the collection of blocks obtained from the non-empty cells of the array is a $$(v,k,\lambda )$$(v,k,?)-BIBD. For $$\mu = 1$$μ=1, the existence of a $$KS_k(v;\mu ,\lambda )$$KSk(v?μ,?) is equivalent to the existence of a doubly resolvable $$(v,k,\lambda )$$(v,k,?)-BIBD. The asymptotic existence of $$KS_k(v;1,1)$$KSk(v?1,1) was established in 2009. In this paper we establish necessary and sufficient conditions for the asymptotic existence of $$KS_k(v;1,k-1)$$KSk(v?1,k-1) or DR$$(v,k,k-1)$$(v,k,k-1)-BIBDs.

Performance tuning and sparse traversal technique for a cell‐based fetch length algorithm on a GPU

SummaryFor determining distances (fetch lengths) from points to polygons in a two‐dimensional Euclidean plane, cell‐based algorithms provide a simple and effective solution. They divide the input area into a grid of cells that cover the area. The objects are stored into the appropriate cells, and the resulting structure is used for solving the problem. When the input objects are distributed unevenly or the cell size is small, most of the cells may be empty. The representation is then called sparse. In the method proposed in this work, each cell contains information about its distance to the nonempty cells. It is then possible to skip over several empty cells at a time without memory accesses. A cell‐based fetch length algorithm is implemented on a graphics processing unit (GPU). Because control flow divergence reduces its performance, several methods to reduce the divergence are studied. While many of the explicit attempts turn out to be unsuccessful, sorting of the input data and sparse traversal are observed to greatly improve performance: compared with the initial GPU implementation, up to 45‐fold speedup is reached. The speed improvement is greatest when the map is very sparse and the points are given in a random order. Copyright © 2015 John Wiley & Sons, Ltd.

Completing Partial Latin Squares with Blocks of Non-empty Cells

In this paper we develop two methods for completing partial latin squares and prove the following. Let $$A$$A be a partial latin square of order $$nr$$nr in which all non-empty cells occur in at most $$n-1$$n-1$$r\times r$$r×r squares. If $$t_1,\ldots , t_m$$t1,?,tm are positive integers for which $$n\geqslant t_1^2+t_2^2+\cdots +t_m^2+1$$n?t12+t22+?+tm2+1 and if $$A$$A is the union of $$m$$m subsquares each with order $$rt_i$$rti, then $$A$$A can be completed. We additionally show that if $$n\geqslant r+1$$n?r+1 and $$A$$A is the union of $$n$$n identical $$r\times r$$r×r squares with disjoint rows and columns, then $$A$$A can be completed. For smaller values of $$n$$n we show that a completion does not always exist.

LC-Grid: a linear global contact search algorithm for finite element analysis

The contact searching is computationally intensive and its memory requirement is highly demanding; therefore, it is significant to develop an efficient contact search algorithm with less memory required. In this paper, we propose an efficient global contact search algorithm with linear complexity in terms of computational cost and memory requirement for the finite element analysis of contact problems. This algorithm is named LC-Grid (Lei devised the algorithm and Chen implemented it). The contact space is decomposed; thereafter, all contact nodes and segments are firstly mapped onto layers, then onto rows and lastly onto cells. In each mapping level, the linked-list technique is used for the efficient storing and retrieval of contact nodes and segments. The contact detection is performed in each non-empty cell along non-empty rows in each non-empty layer, and moves to the next non-empty layer once a layer is completed. The use of migration strategy makes the algorithm insensitive to mesh size. The properties of this algorithm are investigated and numerically verified to be linearly proportional to the number of contact segments. Besides, the ideal ranges of two significant scale factors of cell size and buffer zone which strongly affect computational efficiency are determined via an illustrative example.

An Evaluation of Information Criteria Use for Correct Cross-Classified Random Effects Model Selection

The authors assessed correct model identification rates of Akaike's information criterion (AIC), corrected criterion (AICC), consistent AIC (CAIC), Hannon and Quinn's information criterion (HQIC), and Bayesian information criterion (BIC) for selecting among cross-classified random effects models. Performance of default values for the 5 indices used by SAS PROC MIXED for estimating a 2-level cross-classified random effects model were compared with modifications to the sample size used in the AICC, CAIC, HQIC, and BIC formulations. The sample sizes explored included the number of level 1 units (N), the average number of classification units (m), and the number of nonempty classification cells (c). The authors also assessed performance of the χ2 diff test for testing the difference in fit between 2 nested cross-classified random effects models. The χ2 diff exhibited a slightly inflated Type I error rate with high power. The modified information criteria performed better than did the default values. Pairing of N with the HQIC, BIC, and CAIC and of m with the AICC worked best. Results and suggestions for future research are discussed.

Symmetries of partial Latin squares

In this paper, we study symmetries (autoparatopisms) of partial Latin squares. Let s(n) be the minimum number of non-empty cells in a partial Latin square of order n with a trivial autoparatopism group. We show 15(6n−7)≤s(n)≤12(3n−3) for all n≥5. We also show that, if G is a finite group, then there exists a partial Latin square whose autoparatopism group is isomorphic to G (as are its autotopism and automorphism groups). Computational methods are also introduced, and are used to study symmetries of partial Latin squares of small orders; the source code has been made available as supplementary material.