Regular Squares Research Articles

Iteracyjne poprawianie rozwiązań LZNK obliczonych algorytmem równań normalnych

It is shown that iterative refinement (using normal equations and exclusively standard floating point arithmetic with relative precision v) yields almost full accuracy of computed solution to regular linear least squares problem, provided some conditions.

Coherence conditions in flat regular pullbacks

We investigate the behavior of four coherent-like conditions in regular conductor squares. In particular, we find necessary and sufficient conditions in order that a pullback ring be a finite conductor ring, a coherent ring, a generalized GCD ring, or quasi-coherent ring. As an application of these results, we are able to determine exactly when the ring of integer-valued polynomials determined by a finite subset possesses one of the four coherent-like properties.

Regular sparse anti-magic squares with small odd densities

Sparse anti-magic squares are useful in constructing vertex-magic labelings for bipartite graphs. An n×n array based on {0,1,…,nd} is called a sparse anti-magic square of ordernwith densityd (d<n), denoted by SAMS(n,d), if its row-sums, column-sums and two main diagonal sums constitute a set of 2n+2 consecutive integers. A SAMS(n,d) is called regular if there are d positive entries in each row, each column and each main diagonal. In this paper, we investigate the existence of regular sparse anti-magic squares with densities d=3,5 and it is proved that there exists a regular SAMS(n,3) if and only if n≥4 and there exists a regular SAMS(n,5) if and only if n≥6.

On non-periodic 3-Archimedean tilings with 6-fold rotational symmetry

The purpose of this article is to construct a family of uncountably many non-periodic 3-Archimedean tilings with 6-fold rotational symmetry, which admit three types of vertex configurations by regular triangles and squares.

Overlap Dimensions in Cyclic Tessellable Regular Polygons

In this study we study overlap dimensions in cyclic tessellable regular polygons. Overlap difference and area created by tessellable regular polygons inscribed in disks for covering play a significant role in computational geometry and signal interference in telecommunication network design. Regular triangles and squares are no exceptions except for optimality. We propose general formulae for computing the dimensions of a regular polygon inscribed in a disk. The study also leads to formulae for computing the overlap difference of tessellable regular polygons in disk covering. We realize that the cyclic regular hexagon has both optimal covering area and minimal overlap difference of 17.3 and 86.6% reduction over the original 100% disks size, respectively.

Self-assembly of lamella-forming diblock copolymers confined in nanochannels: Effect of confinement geometry**Project supported by the National Natural Science Foundation of China (Grant Nos. 11204215, 51302187, 20990234, 20925414, 21204040, and 91227121), the Natural Science Foundation of Tianjin City, China (Grant Nos. 12JCYBJC32500 and 14JCZDJC32100), the Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT) (Grant No. IRT1257), and the

The self-assembly of symmetric diblock copolymers confined in the channels of variously shaped cross sections (regular triangles, squares, and ellipses) is investigated using a simulated annealing technique. In the bulk, the studied symmetric diblock copolymers form a lamellar structure with period LL. The geometry and surface property of the confining channels have a large effect on the self-assembled structures and the orientation of the lamellar structures. Stacked perpendicular lamellae with period LL are observed for neutral surfaces regardless of the channel shape and size, but each lamella is in the shape of the corresponding channel's cross section. In the case of triangle-shaped cross sections, stacked parallel lamellae are the majority morphologies for weakly selective surfaces, while morphologies including a triangular-prism-shaped B-cylinder and multiple tridentate lamellae are obtained for strongly selective surfaces. In the cases of square-shaped and ellipse-shaped cross sections, concentric lamellae are the signature morphology for strongly selective surfaces, whereas for weakly selective surfaces, stacked parallel lamellae, and several types of folding lamellae are obtained in the case of square-shaped cross sections, and stacked parallel lamellae are the majority morphologies in the case of ellipse-shaped cross sections when the length of the minor axis is commensurate with the bulk lamellar period. The mean-square end-to-end distance, the average contact number between different species and the surface concentration of the A-monomers are computed to elucidate the mechanisms of the formation of the different morphologies. It is found that the resulting morphology is a consequence of competition among the chain stretching, interfacial energy, and surface energy. Our results suggest that the self-assembled morphology and the orientation of lamellae can be manipulated by the shape, the size, and the surface property of the confining channels.

Topological bifurcation of relative equilibria for the Kirchhoff 8-vortex problem

In this article we prove necessary and sufficient conditions for the existence of local and global bifurcations of relative equilibria of the 8-vortex problem. Applying the equivariant bifurcation theory we prove that from the families of regular concentric squares forming relative equilibria, bifurcate relative equilibria whose shape is more complicated.

Topology preservation on the triangular grid

There are exactly three regular planar grids, which are formed by tiling the 2-dimensional Euclidean space with regular triangles, squares, and hexagons. The topology of the square grid is well-und...

On an open problem concerning regular magic squares of odd order

In this paper we give an answer to an open problem posed by M.Z. Lee et al. (2012) [2]. More precisely, we prove that the classical regular magic square of odd order produced by the centroskew S-circulant matrix with the assignment aj=j−1, j=1,2,⋯,(n+1)/2 is always nonsingular. Moreover an explicit formula for computing the eigenvalues of classical regular magic squares of odd order is given.

Spectra of order-4 special magic squares

A new parameterization for order-4 regular (or associative) magic square matrices leads to general formulas for their eigenvalues, eigenvectors, and singular value decomposition. Known transformations extend these results to order-4 pandiagonal and bent-diagonal magic squares. The effect of various transformations on the eigenvalues and singular values of these special magic squares is considered. Numerical examples are presented and numerical values are obtained from simple formulas for the eigenvalues and singular values of each of the 48 natural pandiagonal, regular, and bent-diagonal magic squares of order 4 and their reflections.

A construction of regular magic squares of odd order

A magic square is an n×n array of numbers whose rows, columns, and the two diagonals sum to μ. A regular magic square satisfies the condition that the entries symmetrically placed with respect to the center sum to 2μn. Using circulant matrices we describe a construction of regular classical magic squares that are nonsingular for all odd orders. A similar construction is given that produces regular classical magic squares that are singular for odd composite orders. This paper is an extension of [3].

Regular sparse anti-magic squares with the second maximum density

Sparse anti-magic squares are useful in constructing vertex-magic labelings for bipartite graphs. An n×n array based on {0,1,⋯,nd} is called a sparse anti-magic square of order n with density d (d<n), denoted by SAMS(n,d), if its row-sums, column-sums and two main diagonal sums form a set of 2n+2 consecutive integers. An SAMS(n,d) is called regular if there are d positive entries in each row, each column and each main diagonal. In this paper, we investigate the existence of regular sparse anti-magic squares with the second maximum density, i.e., d=n−2, and it is shown that there exists a regular SAMS(n,n−2) if and only if n≥4.

Phytosociological variability of weeds in soybean field

Understanding spatial distribution of weeds in the crop enables to perform localized herbicide applications, increasing the technical and economic efficiency of operations and reducing environmental impacts. This work aimed to characterize the spatial and phytosociological variability of weeds occurring in soybean commercial field. It was conducted in an agricultural area located at the municipality of Boa Vista das Missões - RS, during the 2010/2011 harvest season. The area, that had been managed under no-tillage with soybean monoculture (summer) for five years, was divided in regular squares of 50 x 50 m (0.25 ha), totalizing 356 points. For species identification, 0.5 x 0.5 m sample squares were used. During the survey, 1,739 individuals were identified, distributed in 19 species of 13 families. The weed species Cardiospermum halicacabum, Digitaria horizontalis, Urochloa plantaginea and Raphanus raphanistrum showed the highest population variation in the area; however, only C. halicacabum, U. plantaginea and R. raphanistrum stood out based on the Importance Index Value (IVI). Localized management strategies considering the spatial variability of weed species placed in the Magnoliopsidas and Liliopsidas group show a high potential for use in soybean crop. The results show that the sampling method through regular grid was capable of characterizing the occurrence, population density and spatial variability of weed species in soybean crop.

Reduction of deformations of the digital terrain model by merging interpolation algorithms

Majority of contemporary spatial information systems allow generation and visualization of the digital terrain model based on GRID type of the regular squares network. The values at nodal points for such a GRID are in most cases computed by one of the many available interpolation methods on the basis of dispersed measurement points. The commonly used solutions allow in most cases application of a selected interpolation method in a global way within the entire analyzed area, which does not secure the same accuracy of surface generated. The paper describes generation of a digital terrain model by applying a combination of interpolation methods. The criterion of method choice was dependent on the dispersion of measurement points around the GRID node. This solution allows successive complementing the resultant set of computed set with values determined with a specific error and as a result improves the accuracy of the model generated.

Periodic billiard trajectories in regular polygons and closed geodesics on regular polyhedra

We consider the relations between the lengths of periodic billiard trajectories in regular triangles, squares, and regular pentagons and those of closed geodesics on the surfaces of regular polyhedra. The cases of regular tetrahedra and octahedra are fully resolved in [3]. The cases of cubes and regular icosahedra are treated below. In the case of regular dodecahedra we can present only preliminary partial results.

The Optimal Placement of Sensors in Square Target Regions with Varying Boundary Length

The sensors have been applied in many different environments to detect the fire or other things. An optimal placement of sensors is needed to guarantee full coverage and to reduce the financial cost. In this paper, a target region with square boundary is studied to find a suitable placement scheme. The length of the square boundary is growing from zero to infinite, while the sensors’ sensing radius is fixed. Different schemes are discussed to fit the varying target region. The layout of the sensors is substituted by the placement of the sensing circle's inscribed polygons, which are regular triangles, squares and regular hexagons. The minimum number of sensors that each scheme needs is calculated. Through the comparison and analysis, 16 cases are summarized. Each case is given an optimal placement scheme and the minimum number of sensors that should be distributed.

Microarray‐based prediction of Parkinson's disease using clinical data as additional response variables

Recent discoveries and developments in the field of genomics have led to the commercialization of novel diagnostic devices for studying disease or estimating therapeutic outcomes in individual patients. With this emerging field, the emphasis is shifting to integration of clinical research into product development. Data acquisition is primary in the initial exploratory phase of product development, and during the process of sample collection and data generation in clinical microarray studies, great amounts of additional information, such as demographic, clinical, and study design variables associated with the data, are often accumulated and made available. Including additional information in classification has been addressed in many different ways. However, in previous studies, the additional information have consistently been treated as extra predictors, which can be a problem for future prediction if such information are not available or collectable for the new samples. We instead propose to adopt a method called canonical partial least squares, which for our purpose, only uses the additional information at the model building stage to stabilize the construction of a classifier for disease status from microarray data. The canonical partial least squares method is compared with regular partial least squares for the classification of Parkinson's disease from gene expression in peripheral blood samples and also through computer simulations. The present study showed that including clinical data in the model building produces simpler and more stable models for prediction of Parkinson's disease from gene expression data.

On properties of special magic square matrices

By treating regular (or associative), pandiagonal, and most-perfect (MP) magic squares as matrices, we find a number of interesting properties and relationships. In addition, we introduce a new class of quasi-regular (QR) magic squares which includes regular and MP magic squares. These four classes of magic squares are called “special”.We prove that QR magic squares have signed pairs of eigenvalues just as do regular magic squares according to a well-known theorem of Mattingly. This leads to the fact that odd powers of QR magic squares are magic squares which also can be established directly from the QR condition. Since all pandiagonal magic squares of order 4 are MP, they are QR. Also, we show that all pandiagonal magic squares of order 5 are QR but higher-order ones may or may not be. In addition, we prove that odd powers of MP magic squares are MP. A simple proof is given of the known result that natural (or classic) pandiagonal and regular magic squares of singly-even order do not exist.We consider the reflection of a regular magic square about its horizontal or vertical centerline and prove that signed pairs of eigenvalues of the reflected square differ from those of the original square by the factor i. A similar result is found for MP magic squares and a subclass of QR magic squares.The paper begins with mathematical definitions of the special magic squares. Then, a number of useful matrix transformations between them are presented. Next, following a brief summary of the spectral analysis of matrices, the spectra of these special magic squares are considered and the results mentioned above are established. A few numerical examples are presented to illustrate our results.

On nonsingular regular magic squares of odd order

Using centroskew matrices, we provide a necessary and sufficient condition for a regular magic square to be nonsingular. Using latin squares and circulant matrices we describe a method of construction of nonsingular regular magic squares of order n where n is an odd prime power.

Analyzing Spatial Autocorrelation for the Hypsometric Integral to Discriminate Neotectonics and Lithologies Using DEMs and GIS

This investigation documents the sensitivity of the hypsometric integral (HI) and its relationship to neotectonics and lithology. We used digital elevation models (DEMs) of 30 and 90 m spatial resolution from the Shuttle Radar Topographic Mission (SRTM) and the Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) to calculate HI values in the Hindu Kush and its vicinity (Northwest Pakistan and Northeast Afghanistan). We used an analysis grid of regular squares of different sizes to calculate maximum, minimum, and mean elevations. The spatial distributions of HI do not show clear spatial patterns and correlation with mean elevation or relief amplitude. We applied spatial pattern analysis using Local Indices of Spatial Autocorrelation (LISA) to measure the degree to which our HI distribution was clustered, dispersed, or randomized. LISA analysis shows that the data are autocorrelated because of high positive z-scores. Hot spots (clusters with high HI values) are consistent with tectonic uplift and show a strong correlation with the different structural domains in the region. Cold spots represent recent sedimentation close to faults and coincide with shallow earthquake clusters in the region. The HI values do not show any correlation with relative topographic position or lithology. Analysis of HI distribution shows that they are robust and independent of digital elevation model (DEM) resolution but are strongly scale dependent. The LISA technique allows the extraction of clusters of the HI that reveal recent tectonic processes; otherwise it is difficult to interpret the high variability of HI values. The scale dependency of the HI may reflect the varying importance of drainage network and hillslope processes.