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We consider a discrete system of $n$ simple indistinguishable devices, called \emph{agents}, forming a \emph{connected} shape $S_I$ on a two-dimensional square grid. Agents are equipped with a linear-strength mechanism, called a \emph{line move}, by which an agent can push a whole line of consecutive agents in one of the four directions in a single time-step. We study the problem of transforming an initial shape $S_I$ into a given target shape $S_F$ via a finite sequence of line moves in a distributed model, where each agent can observe the states of nearby agents in a Moore neighbourhood. Our main contribution is the first distributed connectivity-preserving transformation that exploits line moves within a total of $O(n \log_2 n)$ moves, which is asymptotically equivalent to that of the best-known centralised transformations. The algorithm solves the \emph{line formation problem} that allows agents to form a final straight line $S_L$, starting from any shape $ S_I $, whose \emph{associated graph} contains a Hamiltonian path., Comment: 31 pages, 18 figures

This Letter presents a novel ultra-wide-band polarization-selective reflectarray with true-time delay. The reflectarray can be perceived as a combination of polarization grids and Vivaldi antennas, so is characterized by large bandwidth, low cross polarization, and polarization selectivity. A reflectarray of elements arranged in a 41 41 square grid operating from 6 to 32 GHz is designed, fabricated, and tested. The feed is provided by four pyramidal horn antennas operating at the working frequency (C, X, Ku/K, and Ka band). In measurements, the reflectarray shows stable radiation patterns from 6 to 32 GHz (more than 5:1 frequency range) and has a maximum aperture efficiency of 40%. It also has the low cross polarization below 28 dB across the entire band and has the characteristic of polarization selectivity.

Fully trapped survey designs are widely used to delimit adventive pests populations that can be detected using traps and lures. Delimitation includes verifying the presence of the pest and determining its spatial extent. The size and shape of the survey design and the density of traps can vary; however, resulting variation in detecting efficiency is often unknown. We used a trapping network simulation model with diffusion-based insect movement to investigate delimiting survey trapping design performance for fully trapped and some modified designs. Simulations included randomized outbreak locations in a core area and a duration of 30 d. We assessed impacts of insect dispersal ability, grid size and shape, and trap attractiveness and density on survey performance, measured as mean probability of capturing individual pests [p(capture)]. Most published grids are square, but circles performed equally well and are more efficient. Over different grid sizes, p(capture) increased for insects with greater dispersal ability but was generally unresponsive to size because most captures occurred in central areas. For low dispersing insects, the likelihood of egress was approximately zero with a 3.2-km square grid, whereas an 11.3-km grid was needed to contain highly vagile insects. Trap attractiveness affected p(capture) more strongly than density: lower densities of poorly attractive traps may underperform expectations. Variable density designs demonstrated potential for cost savings but highlighted that resource-intensive outer bands are critical to boundary determination. Results suggesting that many grids are oversized need empirical verification, whereas other principles, such as using circular shapes, are readily adoptable now.

We present a novel method to study two-dimensional rotation-symmetric number-conserving multi-state cellular automata with the von Neumann neighborhood with radius one. This method enables a succinct and easy enumeration in all cases examined so far in literature, i.e., cellular automata with at most five states. Moreover, it allows to find all such cellular automata with six and seven states, while so far, even enumerating six-state rules was beyond the reach of computing machines. Such enumeration allows us to revisit some unresolved questions in the field. Furthermore, we give some rough estimates of the asymptotic growth of the number of such cellular automata with n states, as n tends to infinity. The results are obtained for finite square grids with periodic boundary conditions, but they are also valid in the case of the infinite square grid.

Let G be a plane graph, and let S be the union of all edges and vertices of G , equipped with the standard Euclidean topology of the plane. A path between points x , y ∈ S is a minimal, with respect to inclusion, connected subset of S containing x and y (here x and y are not necessarily endpoints of an edge). The distance between points x , y ∈ S , denoted as d ( x , y ) , is the length of a shortest path between x and y in S . An l -covering of G is a finite set of points X ⊆ S such that for every point y ∈ S , there is a point x ∈ X , such that d ( x , y ) ≤ l . The l -covering number of G is the minimum size of an l -covering of G . Let α ( m , n ) be the l -covering number of the grid graph on m n vertices embedded in the plane so that the edges are either horizontal or vertical straight lines of length l . In this note we prove that α ( m , n ) = m n − 1 2 for m n > 1 .

In this study, we report the synthesis and crystal structures of coordination polymers employing tetrabromobenzenedicarboxylate (Br4bdc2−) and pyrazine (pyz). Uncoordinated pyz molecules are stabil...

With over 200 wind farm projects underway in 33 states in America, now more than ever, innovation and precision are necessary in wind farm design. The Wind Farm Layout Optimization Problem (WFLOP) calls for optimally positioning turbines within a wind farm so that a particular objective function is optimized in the presence of wake effect. To make the WFLOP tractable, many solution methods begin by modeling the wind farm as a n x n square grid where the centers of each of the n 2 cells serve as potential locations for wind turbines. After making this modeling assumption, there are 2 n 2 potential layouts to consider, and so heuristic algorithms are often employed in order to search the solution space and generate a near-optimal layout. In this paper, we propose a local, continuous refinement technique that seeks to improve the layouts generated by these heuristic algorithms. In particular, we consider the objective function Levelized Cost of Energy (LCOE) and capture wake using the simple, yet effective, Jensen Model. Given the anemometer data for two potential wind farm sites, we begin by generating initial layouts using a specific heuristic algorithm (the Distributed Genetic Algorithm) and then refine these layouts using our proposed continuous, deterministic refinement scheme. We compare the performance of this deterministic refinement technique to another deterministic refinement technique in the space: a Heuristic Hill-climbing approach. Results suggest that our refinement technique outperforms the compared refinement technique by generating layouts that increase overall energy production, consequently reducing the LCOE. To our knowledge, this is the first paper to compare refinement techniques in wind farm layout optimization, and in doing so, we employ a simulation framework that examines the energy production of the generated wind farm layouts for 10 min intervals.

A dominating set of a graph is a subset D of the vertices such that every vertex not in D is adjacent to some vertex of D. In this paper, we introduce several variants of dominating sets in the square grid, periodic square grid and cylindric square grid by considering translation symmetry. We provide their exact enumerations in terms of domination polynomials. We also analyze the asymptotic behavior of the growth rates of their cardinality.

Automata walking on graphs are a mathematical formalization of autonomous mobile agents with limited memory operating in discrete environments. Under this model a broad area of studies of the behaviour of automata in labyrinths arose and intensively developing last decades (a labyrinth is an embedded directed graph of special form). Research in this regard received a wide range of applications, for example, in the problems of image analysis and navigation of mobile robots. Automata operating in labyrinths can distinguish directions, that is, they have a compass. This paper deals with the problem of constructing square grid graph vertex labelling thanks to which a finite automaton without a compass can walk on graph in any arbitrary direction. The automaton looking over neighbourhood of the current vertex and may travel to some neighbouring vertex selected by its label. In this paper, we propose a minimal deterministic traversable vertex labelling that satisfies the required property. A labelling is said to be deterministic if all vertices in closed neighbourhood of every vertex have different labels. In previous works we have proved that minimal deterministic traversable vertex labelling of square grid graph uses labels of five different types. In this paper we prove that a collective of one automaton and three pebbles can construct this labelling on initially unlabelled infinite square grid graph. We consider pebbles as automata of the simplest form, whose positions are completely determined by the remaining automata of the collective.

Given an n × n grid of dots, where n > 1, when is it possible to connect all of the dots to give a polygon with n2 sides? One is not allowed to have two consecutive segments be collinear, since tha...

We consider a class of power of two choice based assignment policies for allocating users to servers, where both users and servers are located on a two-dimensional Euclidean plane. In this framework, we investigate the inherent tradeoff between the communication cost, and load balancing performance of different allocation policies. To this end, we first design and evaluate a Spatial Power of two (sPOT) policy in which each user is allocated to the least loaded server among its two geographically nearest servers sequentially. When servers are placed on a two-dimensional square grid, sPOT maps to the classical Power of two (POT) policy on the Delaunay graph associated with the Voronoi tessellation of the set of servers. We show that the associated Delaunay graph is 4-regular and provide expressions for asymptotic maximum load using results from the literature. For uniform placement of servers, we map sPOT to a classical balls and bins allocation policy with bins corresponding to the Voronoi regions associated with the second order Voronoi diagram of the set of servers. We provide expressions for the lower bound on the asymptotic expected maximum load on the servers and prove that sPOT does not achieve POT load balancing benefits. However, experimental results suggest the efficacy of sPOT with respect to expected communication cost. Finally, we propose two non-uniform server sampling based POT policies that achieve the best of both the performance metrics. Experimental results validate the effectiveness of our proposed policies.

Circle Grid Analysis (CGA) and Square Grid Analysis (SGA) are commonly used methods to measure the strain in sheet metal forming operations. In these methods, small diameter circles or squares are ...

Convolutional neural networks (CNNs) can effectively handle grid-structured data but not dynamic skeletons, which are usually expressed as graph structures. In this study, we first propose a skeleton-based square grid (SSG) for transforming dynamic skeletons into three-dimensional (3D) grid-structured data so that CNNs can be applied to such data. Each SSG contains a joint-based square grid (JSG) and a rigid-based square grid (RSG) based on intrinsic and extrinsic dependencies of various body parts, respectively. Next, to enhance the ability of deep features to capture the correlations among 3D grid-structured data, a two-stream 3D CNN is constructed to learn spatiotemporal features using the JSG and RSG sequences. Finally, we introduce a soft attention model that selectively focuses on the informative body parts in the skeleton sequences. We validate our model in terms of action recognition using three datasets: NTU RGB+D, Kinetics Motion, and SBU Kinect Interaction datasets. Our experimental results demonstrate the effectiveness of the proposed approach as well as its superior performance when compared with those of state-of-the-art methods.

The notion of a 12-representable graph was introduced by Jones et al.. This notion generalizes the notions of the much studied permutation graphs and co-interval graphs. It is known that any 12-representable graph is a comparability graph, and also that a tree is 12-representable if and only if it is a double caterpillar. Moreover, Jones et al.\ initiated the study of 12-representability of induced subgraphs of a grid graph, and asked whether it is possible to characterize such graphs. This question in is meant to be about induced subgraphs of a grid graph that consist of squares, which we call square grid graphs. However, an induced subgraph in a grid graph does not have to contain entire squares, and we call such graphs line grid graphs. In this paper we answer the question of Jones et al.\ by providing a complete characterization of $12$-representable square grid graphs in terms of forbidden induced subgraphs. Moreover, we conjecture such a characterization for the line grid graphs and give a number of results towards solving this challenging conjecture. Our results are a major step in the direction of characterization of all 12-representable graphs since beyond our characterization, we also discuss relations between graph labelings and 12-representability, one of the key open questions in the area., Comment: To appear in Discussiones Mathematicae Graph Theory

An ultra-wideband phased array antenna using interconnected elements arranged in triangular lattices is investigated. The basic unit cell of the array is a dual orthogonal crossed ring structure. Each element is interconnected to its neighbour using capacitive loading. The ring structure on a planar surface produces broad frequency bandwidth with wide scan angles. Unlike other three-dimensional elements such as Vivaldi antennas, the interconnected crossed ring elements can be readily configured on a triangular grid. As is well known from narrow band arrays, a triangular grid allows reduction in the total number of elements needed for a given performance compared to a square grid. The array presented here applies the triangular grid to a broadband planar structure to reduce the total number of elements required by 13% while the electromagnetic behaviour of the array is essentially maintained. An array prototype of the design has been manufactured and tested. The performance is compared to the associated array with a regular square lattice. The proposed triangular grid structure is shown to be an effective solution for applications where a minimum number of elements is required for large-scale broadband dual-polarised antenna arrays.

Chamfer (or weighted) distances are popular digital distances used in various grids. They are based on the weights assigned to steps to various neighborhoods. In the triangular grid there are three usually used neighbor relations, consequently, chamfer distances based on three weights are used. A chamfer (or digital) disk of a grid is the set of the pixels which have distance from the origin that is not more than a given finite bound called radius. These disks are well known and well characterized on the square grid. Using the two basic (i.e., the cityblock and the chessboard) neighbors, the convex hull of a disk is always an octagon (maybe degenerated). Recently, these disks have been defined on the triangular grid; their shapes have a great variability even with the traditional three type of neighbors, but their complete characterization is still missing. Chamfer balls are convex hulls of integer points that lie in polytopes defined by linear inequalities, and thus can be computed through a linear integer programming approach. Generally, the integer hull of a polyhedral set is the convex hull of the integer points of the set. In most of the cases, for example when the set is bounded, the integer hull is a polyhedral set, as well. The integer hull can be determined in an iterative way by Chvatal cuts. In this paper, sides of the chamfer disks are determined by the inequalities with their Chvatal rank 1. The most popular coordinate system of the triangular grid uses three coordinates. By giving conditions depending only a coordinate, the embedding hexagons of the shapes are obtained. These individual bounds are described completely by Chvatal cuts. They also give the complete description of some disks. Further inequalities having Chvatal rank 1 are also discussed.

The partition function of the exactly solvable four-vertex model on a square grid in the presence of a linearly growing external field is calculated. Namely, we study a system when the external field is applied to one of the columns of the grid. The model is considered both for the fixed and periodic boundary conditions.

Molecular tessellations are known in solid state systems and their formation is often induced or supported by a periodic surface lattice. Here we discover a complex tessellation on the 10 nm length scale, spontaneously formed in the highly dynamic liquid crystalline state. It is composed of overlapping dodecagonal supertiles combining prismatic cells with triangular and square cross sections. This complex honeycomb occurs between a triangular honeycomb at high and a square at low temperature, being opposite to the sequence expected for a thermal expansion of the side chains in the prismatic cells. Formation of the supertiles is supported by the segregation of alkyl chains with different length. The emergent behaviour of this complex soft matter structure is demonstrated, and intriguing connections between self-assembly on surfaces, in liquid crystals, and in block copolymers are drawn. Moreover, the tessellation represents a close approximant of the elusive columnar liquid quasicrystal with dodecagonal symmetry. Bolaamphiphiles can form liquid crystalline phases with unusual honeycomb-like structures. Here a periodic dodecagonal supertiled structure is observed in liquid crystals, emerging at the transition from triangular to square tiling patterns.

Polarization switching in lead-free (K0.40Na0.60)NbO3 (KNN) single crystals was studied by switching spectroscopy piezoresponse force microscopy (SS-PFM). Acquisition of multiple hysteresis loops on a closely spaced square grid enables polarization switching parameters to be mapped in real space. Piezoresponse amplitude and phase hysteresis loops show collective symmetric/asymmetric characteristics, affording information regarding the switching behavior of different domains. As such, the out-of-plane polarization states of the domains, including amplitudes and phases can be determined. Our results could contribute to a further understanding of the relationships between polarization switching and polarization vectors at the nanoscale, and provide a feasible method to correlate the polarization hysteresis loops in a domain under an electric field with the polarization vector states.

A pebbling move on a graph removes two pebbles from a vertex and adds one pebble to an adjacent vertex. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using pebbling moves. The optimal pebbling number $$\pi _{{{\,\mathrm{opt}\,}}}$$πopt is the smallest number m needed to guarantee a pebble distribution of m pebbles from which any vertex is reachable. The optimal pebbling number of the square grid graph $$P_n\square P_m$$Pn□Pm was investigated in several papers (Bunde et al. in J Graph Theory 57(3):215–238, 2008; Xue and Yerger in Graphs Combin 32(3):1229–1247, 2016; Győri et al. in Period Polytech Electr Eng Comput Sci 61(2):217–223 2017). In this paper, we present a new method using some recent ideas to give a lower bound on $$\pi _{{{\,\mathrm{opt}\,}}}$$πopt. We apply this technique to prove that $$\pi _{{{\,\mathrm{opt}\,}}}(P_n\square P_m)\ge \frac{2}{13}nm$$πopt(Pn□Pm)≥213nm. Our method also gives a new proof for $$\pi _{{{\,\mathrm{opt}\,}}}(P_n)=\pi _{{{\,\mathrm{opt}\,}}}(C_n)=\left\lceil \frac{2n}{3}\right\rceil$$πopt(Pn)=πopt(Cn)=2n3.

Structured search strategies have been well studied, but consequences of frequently made assumptions can lead to “optimum” recommendations that are far from optimum in that they ignore risks of failure. Common assumptions are many targets that have average sizes and shapes, and are randomly orientated. The effects of common assumptions are examined for parallel and grid line searches. Equations are provided for estimating probabilities of hitting elliptical targets with parallel and square grid lines along with new simplified equations when orientations are random. In a large portion of searches, the probability of missing the largest, most valuable target, dangerous ordnance, or key target is critical to the decision-maker. The risk of missing a target is frequently central to the decision. Where the target is elliptical in shape and has unknown orientation, square grid line search should be considered over line search. Square grid line search has a slightly lower expected probability of detecting an elliptical target with one or more hits than an equal-cost parallel line search, but the probability of missing regardless of orientation in square grid search is significantly lower. Where two or more hits are required, the square grid has a significantly higher probability of hitting and a slightly lower standard deviation than an equal-cost line search. For elliptical targets, the use of a square grid significantly reduces the chances of search failure.

We consider the ill-posed problem of localizing (finding the position of) the discontinuity lines of a function of two variables that is smooth outside the discontinuity lines and has a discontinuity of the first kind at each point of such lines. A uniform square grid with step τ is considered, and it is assumed that the mean values of a perturbed function over squares with side τ are known at each node of the grid. The perturbed function approximates the exact function in the space L2(ℝ2). The perturbation level δ is known. To solve the problem under consideration, we design and study global discrete algorithms that are based on averaging procedures and approximate the discontinuity lines by a set of points of a uniform grid. The main result of the paper is the development of an approach to the global study of localization algorithms. We formulate conditions for the exact function, thus defining a class of correctness. Within this class, we perform a theoretical study of the proposed algorithms, introduce the characteristics to be estimated (the concept of approximating a set of discontinuity lines by a set of points of a uniform grid), and develop methods for deriving the estimates. To achieve this goal, we use a simplified statement: the discontinuity lines are straight line segments, and the proposed localization algorithm has the simplest thinning block. It is established that the localization error of the algorithm has order O(δ). Estimates of other important parameters characterizing the localization algorithm are given.

According to the circle-packing theorem, the packing efficiency of a hexagonal lattice is higher than an equivalent square tessellation. Consequently, in several contexts, hexagonally sampled images compared to their Cartesian counterparts are better at preserving information content. In this paper, novel mapping techniques alongside the wavelet compression scheme are presented for hexagonal images. Specifically, we introduce two tree-based coding schemes, referred to as SBHex (spirally-mapped branch-coding for hexagonal images) and BBHex (breadth-first block-coding for hexagonal images). Both of these coding schemes respect the geometry of the hexagonal lattice and yield better compression results. Our empirical results show that the proposed algorithms for hexagonal images produce better reconstruction quality at low bits per pixel representations compared to the tree-based coding counterparts for the Cartesian grid., Accepted Manuscript

Various patterns and figures are used widely in image processing including tests of various algorithms, compressing images and also creating/displaying them in computer games. In this paper binary image generation is studied on the triangular grid. On the one hand, the triangular grid has better symmetric properties than the square grid, while on the other hand, the number of closest neighbors of a pixel is less than it is on the square grid. In this way, cellular automata based on the closest neighbors are simpler than similar automata on the square grid, but the generated pictures may be more sophisticated. In our binary cellular automata the state (color) of the closest three neighbors and the pixel's own state determine the next state; we use life-like deterministic cellular automata. In our novel approach we combine two different automata (rules) such that we use them alternately for the picture generation. Various patterns, including highly symmetric mandala type patterns, as well as, airplanes, trees etc. are shown as examples. Some general ideas and hints are also given.

A widely used skeletonization technique is thinning, which is an iterative layer-by-layer erosion in a topology preserving way. In the conventional implementation of thinning algorithms, the deletability of all border pixels in the actual picture is to be investigated. That is why we introduced the concept of 1-attempt thinning. In the case of a 1-attempt algorithm, if a border pixel is not deletable in an iteration step, it cannot be deleted in the remaining thinning phases. This paper presents a computationally efficient implementation scheme for 1-attempt 2-subfield parallel thinning; proves that a 2-subfield parallel thinning algorithm (acting on the conventional 2D square grid) is 1-attempt; shows that 1-attempt property is useful and algorithms fulfilling this property can be implemented with a remarkable speed up.

Conway's Game of Life is a zero-player game played on an infinite square grid, where each cell can be in either of two binary states. The interesting aspect of this game arises when we observe how each cell of this grid interacts with its neighbors over time. There has been much public interest in this game, and several variants have become popular. Similar Games of Life can exist on hexagonal, triangular, and other tiled grids. Games of Life have also been devised in 3 dimensions. In this work, a novel Game of Life is proposed on a Pythagorean tessellation with a unique set of rules, which imitates the reality of Life more accurately. Several interesting life forms in this universe are also illustrated.

The Finite difference method (FDM) is widely used in many applications such as the calculation of electric potential distribution in transmission lines where boundary conditions are known. For the two-dimensional (2D) Laplace Equation (∇ 2 ϕ = 0), the common solution using the FDM is to use a 5-point grid in which the potential at the center point is expressed as the average of the surrounding four potentials. The Nearest Neighbor Finite difference method (NNFDM) is an extension of the FDM which uses the nearest 9 points of a mesh square grid. This work shows when a square mesh grid is defined by few points it is more accurate for the NNFDM as opposed to the FDM. In the case of a rectangular box with a 100V source on one edge the NNFDM showed to be 33 times more accurate than the FDM when compared to the analytical solution and adding no total time in executing the solution.

Let $m_G$ denote the number of perfect matchings of the graph $G$. We introduce a number of combinatorial tools for determining the parity of $m_G$ and giving a lower bound on the power of 2 dividing $m_G$. In particular, we introduce certain vertex sets called channels, which correspond to elements in the kernel of the adjacency matrix of $G$ modulo $2$. A result of Lov\'asz states that the existence of a nontrivial channel is equivalent to $m_G$ being even. We give a new combinatorial proof of this result and strengthen it by showing that the number of channels gives a lower bound on the power of $2$ dividing $m_G$ when $G$ is planar. We describe a number of local graph operations which preserve the number of channels. We also establish a surprising connection between 2-divisibility of $m_G$ and dynamical systems by showing an equivalency between channels and billiard paths. We exploit this relationship to show that $2^{\frac{\gcd(m+1,n+1)-1}{2}}$ divides the number of domino tilings of the $m\times n$ rectangle. We also use billiard paths to give a fast algorithm for counting channels (and hence determining the parity of the number of domino tilings) in simply connected regions of the square grid., Comment: 45 pages, 38 figures

In recent years, computer simulations are playing a fundamental role in unveiling some of the most intriguing features of prime numbers. In this work, we define an algorithm for a deterministic walk through a two-dimensional grid that we refer to as Prime Walk. The walk is constructed from a sequence of steps dictated by and dependent on the sequence of last digits of the primes. Despite the apparent randomness of this generating sequence, the resulting structure -- both in 2d and 3d -- created by the algorithm presents remarkable properties and regularities in its pattern that we proceed to analyze in detail., Comment: 7 pages, 7 figures

A series of donor-acceptor (D-A) naphthalene-viologen-based cyclophanes of different shapes, sizes, and symmetries have been synthesized and characterized. Solution optical studies on these cyclophanes reveal the existence of photoinduced intramolecular charge transfer (CT) at 465 nm from naphthalene (D) to viologen (A) units, resulting in a conformational change in the viologen units and the emergence of an emission at 540 nm. The D-A cyclophanes with box-like and hexagon-like shapes offer an opportunity to control the arrangement within 2D layers where D-A interactions direct the superstructures. While a box-like 2,6-disubstituted naphthalene-based tetracationic cyclophane does not form square tiling patterns, a truncated hexagon-like congener self-assembles to form a hexagonal superstructure which, in turn, adopts a hexagonal tiling pattern. Tessellation of the more rigid and highly symmetrical 2,7-disubstituted naphthalene-based cyclophanes leads to the formation of 2D square and honeycomb tiling patterns with the box-like and hexagon-like cyclophanes, respectively. Co-crystallization of the box-like cyclophanes with tetrathiafulvalene (TTF) results in the formation of D-A CT interactions between TTF and viologen units, leading to tubular superstructures. Co-crystallization of the hexagon-like cyclophane with TTF generates well-ordered and uniform tubular superstructures in which the TTF-viologen CT interactions and naphthalene-naphthalene [π···π] interactions propagate with 2D topology. In the solid state, the TTF-cyclophane co-crystals are paramagnetic and display dual intra- and intermolecular CT behavior at ∼470 and ∼1000 nm, respectively, offering multi-responsive materials with potential pathways for electron transport.

The exactly solvable four-vertex model on a square grid with the fixed boundary conditions in a presence of a special external field is considered. Namely, we study a system in a linear field acting on the central column of the grid. The partition function of the model is calculated by the quantum inverse scattering method. The answer is written in determinantal form.

In this paper, we consider two kinds of matchstick puzzles. In both puzzles, matchsticks can be placed only on the edges of a square grid. The first problem is to make a given placement of matchsticks by moving the designated number of matchsticks. We show an algorithm to solve this problem using string matching with errors that is faster than exhaustive search. The second problem is to make the designated number of squares by moving the designated number of matchsticks. We show that this problem is NP-complete by reduction from circuit-SAT. We also show that the problem can be solved in polynomial time if the width of the grid is bounded by a constant.

Background/Objectives: A study was conducted to observe the self-similarity pattern exhibited within the vegetational field due to their nutritional sharing channels. Methods/Statistical Analysis: A square grid matrix was chosen in the cultivated fields to draw the sample data of the biological as well as environmental variables. Grid matrix was further divided into one hundred and forty -four subareas of 1.2 by 1.2-meter square. Data were analyzed based on a Box-counting method to determine the fractal dimension in each class. Findings: Highest dimension was found to be 2.48 in class one which consists of 144 small grids. Whereas, the lowest dimension found to be 1.03 in class 4. The pattern of the fractal dimension of various classes seems to be self-similar within the vegetational field. Application/Improvements: Vegetational fields share several materials through the nutritional channels within the grid matrix. We observed the pattern in different scales using fractal dimension and found to be the self-similar pattern in all scales. Keywords: Box-Counting Method, Fractal Dimension, Self-Similar Pattern

An implementation of smoothing splines is proposed to reduce orientation noise in electron backscatter diffraction (EBSD) data, and subsequently estimate more accurate geometrically necessary dislocation (GND) densities. The local linear adaptation of smoothing splines (LLASS) filter has two advantages over classical implementations of smoothing splines: (1) it allows for an intuitive calibration of the fitting versus smoothing trade-off and (2) it can be applied directly and in the same manner to both square and hexagonal grids, and to 2D as well as to 3D EBSD data sets. Furthermore, the LLASS filter calculates the filtered orientation gradient, which is actually at the core of the method and which is subsequently used to calculate the GND density. The LLASS filter is applied on a simulated low-misorientation-angle boundary corrupted by artificial orientation noise (on a square grid), and on experimental EBSD data of a compressed Ni-base superalloy (acquired on a square grid) and of a dual austenitic/martensitic steel (acquired on an hexagonal grid). The LLASS filter leads to lower GND density values as compared to raw EBSD data sets, as a result of orientation noise being reduced, while preserving true GND structures. In addition, the results are compared with those of filters available in theMTEXtoolbox.

In vielen Alltagsszenarien zur Orientierung und Navigation sind Vorstellungen möglichst exakter Distanzen unverzichtbar. Die Aneignung von Distanzwissen ist ein prominentes Beispiel der Kartennutzung. Durch die kartographische Kommunikation mit graphischen Zeichen kann die Wahrnehmung und das Lernen von Distanzen in Karten Verzerrungsmechanismen mit unterschiedlichen Einfuss- bzw. Wirkungsgraden unterliegen. Um dies zu verhindern oder zumindest zu verringern, empfehlt die Grundlagenforschung der Kartographie und Raumkognitionsforschung bspw. die Verwendung graphischer Referenzhilfen. Jüngere Studien zeigen etwa, dass die graphische Betonung des Straßennetzes oder die Verwendung von zusätzlich in Karten eingetragenen Gitterlinien Verzerrungen im räumlichen Gedächtnis reduzieren können. Solche graphischen Mittel strukturieren bzw. „regionalisieren“ ein Kartenbild durch die Erzeugung raumreferenzierender Flächeneinheiten, die Einfuss auf Wahrnehmung und Gedächtnis haben. Dadurch konnten bereits gedächtnissteigernde Efekte für den Abruf und das Wiedererkennen von Ortspositionen identifziert werden. Dass raumreferenzierende Grafk auch Vorteile in anderen Feldern der Orientierung mit sich bringt, machen nun die Resultate einer kartenexperimentellen Studie zur Distanzschätzung deutlich. Die Nutzung topographischer Karten, in denen zusätzlich quadratische Gitterzellen enthalten waren, verringerte—im Vergleich zu Karten ohne solche Referenzierungshilfen—das Ausmaß der Fehlschätzungen von Strecken zwischen zwei Orten signifkant. In many everyday scenarios of orientation and navigation, it is inevitable to have a highly accurate distance knowledge. The acquisition of distance knowledge is a common example of map use. As cartographic communication usually depends on graphic symbols, the perception and learning of map information can be influenced by distortion mechanisms. To correct such distortions, foundational research in cartography and spatial cognition have made some design suggestions based on superordinate graphical structures. Some recent studies show that salient road networks and, especially, map grids can help to correct spatial distortion tendencies in memory. These graphic elements regionalize a map surface and serve as space-referencing units. Effects were reported in recall and recognition tasks. The present article includes an experimental study based on topographic maps. The results of the study provide evidence that square grid cells improve the performance in estimating longer distances.

The method to efficiently design a sparse near-field focusing sparse array antenna is introduced here. First, the calculation model and analysis method of three array topologies including the square grid, the circular grid, and the regular triangular grid are established. The evaluation criterion is the lowest sparsity under the condition that the focusing performance of the sparse array approaches to that of a referenced focusing array with the element spacing of a half of one wavelength. After comparing the focusing performance of these array antennas with different grids, the circular-grid array is the optimal one and selected to further discuss the influence of different topology parameters. The optimal sparse near-field focusing array should have the constant distance between two adjacent rings and increase three elements compared with the last ring. Finally, the method based on the proposed selection criteria is employed to design a sparse near-field focusing sparse array. The correctness of the method has been validated by the full-wave simulated results.

The two-part article presents a new theory of the geometric methods used by architects to plan Roman theaters. The study is based on a systematic geometric analysis of the archaeological evidence. It showed that the designers did not follow Vitruvian guidelines, but rather a much more complex and flexible geometric system. The first part of the article covers the geometry of the cavea. The second part will present the analysis of the scaenae building and the geometric methods Roman architects could use to plan the size of the theater for the expected capacity of the auditorium. The article will demonstrate that the layout of the cunei in the cavea was designed based on central angles specific to a variety of regular polygons, including those which are geometrically non-constructible, like heptagon or nonagon. The plan of the scaenae was based on a square grid coordinated with the proportional system of the whole project including the location of the axes of the valvae hospitalia. The capacity of the theater was calculated based on the geometric approximation of the value of π. The plasticity of thus defined geometric design methodology, and its various parameters, resonates much better with our knowledge of the ancient theater structures: they are all similar, but each one is unique. The research offers also a new insight into the Roman design practices.

The 3-ary n-cube, denoted as $$ {Q}_n^3 $$ , is an important interconnection network topology proposed for parallel computers, owing to its many desirable properties such as regular and symmetrical structure, and strong scalability, among others. In this paper, we first obtain an exact formula for the minimum wirelength to embed $$ {Q}_n^3 $$ into grids. We then propose a load balancing algorithm for embedding $$ {Q}_n^3 $$ into a square grid with minimum dilation and congestion. Finally, we derive an O(N2) algorithm for embedding $$ {Q}_n^3 $$ into a gird with balanced communication, where N is the number of nodes in $$ {Q}_n^3 $$ . Simulation experiments are performed to verify the total wirelength and evaluate the network cost of our proposed embedding algorithm.

In this paper we show that for integers s ≥ 2 , t ≥ 1 , any co-edge-regular graph which is cospectral with the s -clique extension of the t × t -grid is the s -clique extension of the t × t -grid, if t is large enough. Gavrilyuk and Koolen used a weaker version of this result to show that the Grassmann graph J q ( 2 D , D ) is characterized by its intersection array as a distance-regular graph, if D is large enough.

Auxetic structures attain certain abnormal mechanical attributes such as negative Poisson’s ratios due to the macro level structural forms. Achieving the auxetic responses is often the central focus of the design of any auxetic structural form. However, the stress distribution across the structures and in particular, the stress concentration aspects are often paid little attention. A numerical simulation analysis is conducted in this paper to establish the stress distribution patterns as well as the stress concentration effects based on variations in the filleting around the critical points of the structure. A square grid auxetic structural form is selected and finite element simulation of the stress fields indicated specific roles of the slight geometrical modifications in terms of filleting leading to variations in both peak stresses and the locations of the stress concentration points. Optimum fillet dimensions could be established based on the results of the numerical experiments.

Miquel dynamics was introduced by Ramassamy as a discrete time evolution of square grid circle patterns on the torus. In each time step every second circle in the pattern is replaced with a new one by employing Miquel’s six circle theorem. Inspired by this dynamics we consider the local Miquel move, which changes the combinatorics and geometry of a circle pattern. We prove that the circle centers under Miquel dynamics are Clifford lattices, an integrable system considered by Konopelchenko and Schief. Clifford lattices have the combinatorics of an octahedral lattice, and every octahedron contains six intersection points of Clifford’s four circle configuration. The Clifford move replaces one of these circle intersection points with the opposite one. We establish a new connection between circle patterns and the dimer model: If the distances between circle centers are interpreted as edge weights, the Miquel move preserves probabilities in the sense of urban renewal.

Gambler’s ruin estimates can be viewed as harmonic measure estimates for finite Markov chains which are absorbed (or killed) at boundary points. We relate such estimates to properties of the underlying chain and its Doob transform. Precisely, we show that gambler’s ruin estimates reduce to a good understanding of the Perron–Frobenius eigenfunction and eigenvalue whenever the underlying chain and its Doob transform are Harnack Markov chains. Finite inner-uniform domains (say, in the square grid Zn) provide a large class of examples where these ideas apply and lead to detailed estimates. In general, understanding the behavior of the Perron–Frobenius eigenfunction remains a challenge.

In the projective plane, we consider congruences of straight lines with the combinatorics of the square grid and with all elementary quadrilaterals possessing touching inscribed conics. The inscribed conics of two combinatorially neighbouring quadrilaterals have the same touching point on their common edge-line. We suggest that these nets are a natural projective generalisation of incircular nets. It is shown that these nets are planar Koenigs nets. Moreover, we show that general Koenigs nets are characterised by the existence of a 1-parameter family of touching inscribed conics. It is shown that the lines of any grid of quadrilaterals with touching inscribed conics are tangent to a common conic. These grids can be constructed via polygonal chains that are inscribed in conics. The special case of billiards in conics corresponds to incircular nets., 23 pages, 19 figures

Xylene isomer separation is considered one of the seven separation challenges that changed the world. In addition, the high-energy demand of xylene separation highlights the need for efficient novel adsorbents. Herein, the liquid-phase separation potential of the anion-pillared hybrid material SIFSIX-1-Cu was studied for preferential adsorption of o-xylene and m-xylene over p-xylene, which was inspired by a previous complexation crystallization method for separating m-xylene. We report detailed experimental liquid-phase adsorption experiments, yielding selectivities of 3.0 for o-xylene versus p-xylene and 2.6 for m-xylene versus p-xylene. Our theoretical calculations thus provide a reasonable explanation that the xylene adsorption selectivity is attributed to the C-H⋅⋅⋅F interaction, and the host-guest interaction order agrees with the adsorption priority: o-xylene > m-xylene > p-xylene.

In a rectangular domain, we discuss about an approximation of the first order derivatives for the solution of the mixed boundary value problem. The boundary values on the sides of the rectangle are supposed to have the second order derivatives satisfying the Holder condition. Under these conditions for the approximate values of the first derivatives of the solution of mixed boundary problem on a square grid, as the solution of the constructed difference scheme a uniform error estimation of order O(h) (h is the grid size) is obtained. Numerical experiments are illustrated to support the theoretical results.

In this paper, two stage difference method is presented to solve the Dirichlet problem for the Laplace equation on rectangle. In the first stage, the sum of the pure fourth order derivatives of the required solution is approximated on a square grid. Then, by using the quantities that are determined in the first stage, the system of difference equations which approximates the Dirichlet problem, is computed during the second stage. The difference equations found in the stages are formulated by using the 5−point averaging operator. Due to these facts that, the boundary values are continuous and sixth times differentiable at the edges of the rectangle, the derivatives of them satisfy Holder ¨ condition and at the end, their second and fourth order derivatives meet the matching condition implied by the Laplace equation. We proved that the difference solution of the Dirichlet problem is uniform convergent with the order O(h4), where h denotes the mesh size.

There are various tessellations of the plane. There are three regular ones, each of them using a sole regular tile. The square grid is self-dual, and the two others, the hexagonal and triangular grids are duals of each other. There are eight semi-regular tessellations, they are based on more than one type of tiles. In this paper, we are interested to their dual tessellations. We show a general method to obtain coordinate system to address the tiles of these tessellations. The properties of the coordinate systems used to address the tiles are playing crucial roles. For some of those grids, including the tetrille tiling D(6, 4, 3, 4) (also called deltoidal trihexagonal tiling and it is the dual of the rhombihexadeltille, T(6, 4, 3, 4) tiling), the rhombille tiling, D(6, 3, 6, 3) (that is the dual of the hexadeltille T(6, 3, 6, 3), also known as trihexagonal tiling) and the kisquadrille tiling D(8, 8, 4) (it is also called tetrakis square tiling and it is the dual of the truncated quadrille tiling T(8, 8, 4) which is also known as Khalimsky grid) we give detailed descriptions. Moreover, we are also presenting formulae to compute the digital, i.e., path-based distance based on the length of a/the shortest path(s) through neighbor tiles for these specific grids.