Lattice Squares Research Articles

Covering rectangles by few monotonous polyominoes

A monotonous polyomino is formed by all lattice unit squares met by the graph of some fixed monotonous continuous function $$f:[a,b] \to \mathbb{R}$$ with $$f(k) \notin \mathbb{Z}$$ whenever $$k \in \mathbb{Z}$$ . Our main result says that the least cardinality of a covering of a lattice $$(m \times n)$$ -rectangle by monotonous polyominoes is $$\big\lceil \frac{2}{3}(m+n-\sqrt{m^2+n^2-mn})\big\rceil.$$ The paper is motivated by a problem on arrangements of straight lines on chessboards.

Tromino Tilings with Pegs via Flow Networks

A tromino tiling problem is a packing puzzle where we are given a region of connected lattice squares and we want to decide whether there exists a tiling of the region using trominoes with the shape of an L. In this work we study a slight variation of the tromino tiling problem where some positions of the region have pegs and each tromino comes with a hole that can only be placed on top of the pegs. We present a characterization of this tiling problem with pegs using flow networks and show that (i) there exists a linear-time parsimonious reduction to the maximum-flow problem, and (ii) counting the number of such tilings can be done in linear-time. The proofs of both results contain algorithms that can then be used to decide the tiling of a region with pegs in O(n) time.

A New Simple Proof of the Aztec Diamond Theorem

The Aztec diamond of order n is the union of lattice squares in the plane intersecting the square $$|x|+|y|

Counting Squares to Sum Squares

SummaryPurely combinatorial proofs are given for the sum of squares formula,and the sum of sums of squares formula,More precisely, the following algebraic equivalents are derived,The proofs obtained literally count squares, namely lattice squares whose vertices are in an n × n grid. For the first formula, only lattice squares aligned to the grid are counted; for the second formula, aligned and tilted squares are counted.

The arctic circle boundary and the Airy process

We prove that the, appropriately rescaled, boundary of the north polar region in the Aztec diamond converges to the Airy process. The proof uses certain determinantal point processes given by the extended Krawtchouk kernel. We also prove a version of Propp’s conjecture concerning the structure of the tiling at the center of the Aztec diamond. 1. Introduction and results. Domino tilings of the Aztec diamond were introduced in [8, 9]. Asymptotic properties of random domino tilings of the Aztec diamond have been studied in [5, 12, 15]. In particular, in [12] the existence of the so-called arctic circle was proved. The arctic circle is the asymptotic boundary of the disordered so-called temperate region of the tiling. Outside this boundary the tiling forms a completely regular brick wall pattern. The methods in [12] combined with the results in [13] show that the fluctuations of the point of intersection of the boundary of the temperate region with a line converge to the Tracy–Widom distribution of random matrix theory. In this paper we extend this result to show that the fluctuations of the boundary around the arctic circle converges to the Airy process introduced in [23]. The paper is a continuation of the approach used in [14] and [15], where certain point processes with determinantal correlation functions [24] and the Krawtchouk ensemble, were used. We will use the general techniques developed in [16] and investigate an extended point process which also has determinantal correlation functions given by a kernel, which we call the extended Krawtchouk kernel. The Aztec diamond, An, of order n is the union of all lattice squares [m, m+1]×[l, l+1], m, l 2 Z, that lie inside the region {(x1, y1);|x1|+|y1| �

Local statistics for random domino tilings of the Aztec diamond

We prove an asymptotic formula for the probability that, if one chooses a domino tiling of a large Aztec diamond at random according to the uniform distribution on such tilings, the tiling will contain a domino covering a given pair of adjacent lattice squares. This formula quantifies the effect of the diamond's boundary conditions on the behavior of typical tilings; in addition, it yields a new proof of the arctic circle theorem of Jockusch, Propp, and Shor. Our approach is to use the saddle point method to estimate certain weighted sums of squares of Krawtchouk polynomials (whose relevance to domino tilings is demonstrated elsewhere), and to combine these estimates with some exponential sum bounds to deduce our final result. This approach generalizes straightforwardly to the case in which the probability distribution on the set of tilings incorporates bias favoring horizontal over vertical tiles or vice versa. We also prove a fairly general large deviation estimate for domino tilings of simply-connected planar regions that implies that some of our results on Aztec diamonds apply to many other similar regions as well.

A two-dimensional fast lattice recursive least squares algorithm

This paper is mainly devoted to the derivation of a new two-dimensional fast lattice recursive least squares (2D FLRLS) algorithm. This algorithm updates the filter coefficients in growing-order form with linear computational complexity. After appropriately defining the "order" of 2D data and exploiting the relation with 1D multichannel, "order" recursion relations and shift invariance property are derived. The geometrical approaches of the vector space and the orthogonal projection then can be used for solving this 2D prediction problem. We examine the performances of this new algorithm in comparison with other fast algorithms.

Irreversible multilayer adsorption

AbstractRandom sequential adsorption (RSA) models have been studied [1] due to their relevance to deposition processes on surfaces. The depositing particles are represented by hard‐core extended objects; they are not allowed to overlap. Numerical Monte Carlo studies and analytical considerations are reported for 1D and 2D models of multilayer adsorption processes. Deposition without screening is investigated; in certain models the density may actually increase away from the substrate. Analytical studies of the late stage coverage behavior show the crossover from exponential time dependence for the lattice case of the power law behavior in the continuum deposition. 2D lattice and continuum simulations rule out some “exact” conjectures for the jamming coverage. For the deposition of dimers on a 1D lattice with diffusional relaxation we find that the limiting coverage (100%) is approached according to the ∼1/√t power‐law preceded, for fast diffusion, by the mean‐field crossover regime with the intermediate ∼1/t behavior. In case of k‐mer deposition (k>3) with diffusion the void fraction decreases according to the power‐law t−1/2. In the case of RSA of lattice hard squares in 2D with diffusional relaxation the approach to the full coverage is ∼t−1/2. In case of RSA‐deposition with diffusion of two by two square objects on a 2D square lattice the coverage also approaches 1 according to the power law t−1/2, while on a finite periodic lattice the final state is a frozen random regular grid of domain walls connecting single site defects.

COLLECTIVE EFFECTS IN RANDOM SEQUENTIAL ADSORPTION OF DIFFUSING HARD SQUARES

We study by Monte Carlo computer simulations random sequential adsorption (RSA) with diffusional relaxation, of lattice hard squares in two dimensions. While for RSA without diffusion the coverage approaches its maximum jamming value (large-time fractional coverage) exponentially, added diffusion allows the deposition process to proceed to the full coverage. The approach to the full coverage is consistent with the ~ t−1/2 power law reminiscent of the equilibrium cluster coarsening in models with nonconserved order-parameter dynamics.

Bounds for the efficiency factor of row-column designs

SUMMARY Upper bounds for the average efficiency factor of single block and resolvable row-column designs, based on the first two moments of the canonical efficiency factors, are obtained. In 245 out of the 379 cases considered these bounds are tighter than a bound given previously by Eccleston & McGilchrist (1985). Considerable work has been carried out to find good upper bounds for the efficiency factor of block designs, both resolvable and non-resolvable. For a review of this work see Jarrett (1989); for a more recent bound see Tjur (1990). Apart from a bound by Eccleston & McGilchrist (1985), little attention has been given to row-column designs. The purpose of this paper is to derive upper bounds for row-column designs, using some of the results available for block designs, and to compare these bounds with the one given by Eccleston & McGilchrist (1985). A single block row-column design has v treatments set out in an array of p rows and q columns with each treatment replicated r times, so that vr = pq. Latin and Youden squares are examples of such designs. A resolvable row-column design has v treatments in r replicates, with each replicate set out in an array of p rows and q columns. Each treatment is replicated once in each replicate, so that v = pq. An example of a resolvable row-column design for v = 12, p = 4, q = 3 and r =2 is given in Table 1. Other examples are given by lattice squares. The terminology row-column design will be used to include a design of either type. It will be assumed that p ? q.

Evaluation of real time adaptive noise cancelling algorithms as implemented using a digital signal processor chip

Adaptive lattice noise cancelling algorithms have features that make them appear attractive for the realisation of realtime adaptive noise cancellers with single chip digital signal processors. Two popular adaptive lattice noise cancelling algorithms have been evaluated for this application. Considered are stochastic gradient lattice and least squares lattice algorithms. For the purpose of comparison, the stochastic gradient transversal algorithm is also evaluated. Their performances under a variety of signal and noise excitations are summarised and compared in terms of operating rates and misadjustments.

A simulation study of the nearest neighbour analysis method of papadakis

The relative performances of randomised block, balanced lattice squares and Papadakis nearest neighbour analyses were compared on two simulated fields whose soil heterogeneity profiles were generated, one with a few evenly spaced contours and the other with many unevenly spaced contours. Four levels of random error were generated to simulate different proportions of random error and soil heterogeneity. Dummy treatments, corresponding to 7 x 7 and 11 x 11 balanced lattice squares were applied to the fields. The results from simulated experiments showed an interaction of error mean square (EMS) between size of experiment (7 x 7, 11 x 11) and levels of soil heterogeneity in the lattice analyses, but no such interaction in the Papadakis analyses. The Papadakis EMS decreased as random error decreased but at a rate depending on the map andthe ratio of soil heterogeneity to random error.

Quantitative inheritance studies in sugar-cane. I. Estimation of variance components

Two experiments in quantitative genetics were conducted, one based on a nested design in lattice squares and the other on a factorial design in a balanced lattice. Lattice designs were found to be suitable for genetic experiments if a large number of crosses was involved, but posed some problems in partitioning the sum of squares for treatments. The factorial design was considered preferable to the nested design, although neither design permitted estimation of epistatic variances which, therefore, were assumed to be negligible. Additive genetic variance was found to be more important than dominance genetic variance for most characters. However, most estimates of genetic variance lacked precision in spite of the use of large, precise experiments, which illustrated the difficulty in obtaining estimates of variance components with adequate precision. The validity of assumptions made for these analyses is discussed. The effect of competition was studied and estimates of heritability and degree of genetic determination were determined.

Relative Efficiency of 4×4 Balanced Lattice Squares Compared to Randomized Complete Blocks for Measuring Cotton Variety Lint Yields1

The balanced lattice square experimental design was on the average 36% more efficient than a randomized complete block design based on total lint cotton (Gossypium hirsutum L.) data obtained from four locations in Missouri over 4 years. Efficiency varied with environment but was inconsistent over locations and years.

The Analysis of Variance and Derivation of Standard Errors for Incomplete Data

In a previous paper [10] based on the fundamental paper by Yates [11], the writer has discussed the estimation of missing observations in incomplete data. When the data are completed with these estimates, standard calculations give the correct estimates of treatment effects, etc., and a standard analysis of variance yields the correct residual sum of squares for the estimation of error. However, as Yates [11] observed, other component sums of squares in the standard analysis are incorrect (though perhaps not seriously so). For the randomized blocks design, and for a Latin square with one missing value, Yates derived formulae for adjusting the treatment sum of squares to its correct value. Cornish [2, 3] gave the corresponding results for incomplete block designs, and for designs such as lattice squares, with one missing value. The present paper, following on from the previous paper [10], deals with correcting the standard analysis of variance, and derives a general formula for the necessary corrections. Specific formulae are given which, in particular, provide the necessary correction of the treatment sum of squares when several observations are missing, for the designs with two-way restriction. The standard formulae for determining variances and covariances of treatment comparisons, etc., will also need to be adjusted when the comparisons involve missing values. Tocher [8] gave a general formula for this purpose, which is here extended to cover the singular cases. The principles discussed are illustrated by application to the analysis of a Latin square experiment (numerical example), and in the derivation of an analysis for B.I.B. designs with missing blocks.

Estimation of Missing Values for the Analysis of Incomplete Data

When only a few observations are missing from data that otherwise conform to a planned experimental design, the fitting of a linear model to the data by the principle of least squares is most simply carried out using estimated values for the missing observations, as in this way most effective use is made of the remaining symmetry in the experimental results. A comprehensive account of this procedure was given by Yates [12] in 1933. Firstly, the data are completed by inserting estimates of the missing values, which are determined so that the residual sum of squares for the completed data shall be a minimum. The correct estimates of treatment effects and other parameters in the linear model are then given by the standard formulae for the experimental design, and the correct residual sum of squares for the incomplete data is given by the standard analysis of variance (with the degrees of freedom appropriately reduced). However, the other components of this analysis are only approximations (though usually reasonably good) to the corresponding components of a correct least squares analysis of variance, since the latter involves in principle the fitting of one or more auxiliary models, for which appropriate estimates of the missing values would be necessary. It should also be noted that the usual formulae for standard errors of treatment comparisons need to be modified to take into account the loss in precision due to the missing values. The non-standard aspects of this missing value procedure will be discussed by the present writer in two papers: the present paper, dealing with the estimation of missing values (which enables completion of the first phase of analysis), and a second paper [11] which deals with correction of the analysis of variance and the derivation of standard errors. In connection with incomplete block designs, lattice squares,

Variedades de cana-de-açúcar. II - Série de ensaios realizados no período 1953-1956

No presente trabalho são apresentados os resultados obtidos em seis ensaios regionais de variedades de cana-de-açúcar, realizados de 1953 a 1956, nas usinas Santa Elisa, Tamôio, Pôrto Feliz, Itaiquara, Santa Bárbara e Monte Alegre. A finalidade foi determinar, dentre as variedades estudadas as mais recomendáveis para os diversos tipos de solo, nas diferentes regiões canavieiras do Estado de São Paulo. Foram estudadas 16 variedades. Em cada ensaio foram feitos três cortes (cana-planta, soca e ressoca), sendo estudados, estatisticamente, os resultados da produção de cana e de açúcar provável do primeiro corte e do total dos três cortes. Os resultados obtidos mostraram. que a Co. 290 não mais deve ser cultivada no Estado de São Paulo. A CB. 40/13 apresentou a maior riqueza em açúcar, sendo a de maturação mais precoce, enquanto a CB. 38/30 apresentou baixa riqueza em açúcar. São recomendadas para tôdas as regiões do Estado as variedades CB. 40/69, CB. 41/76 e Co. 419; a CB. 40/13, apesar de sua produtividade e riqueza em açúcar, devido à sua grande sucetibilidade ao mosaico só pode ser cultivada com cuidados especiais. Também são recomendadas, porém para determinadas regiões, as variedades: CB. 40/19, CB. 38/22 e CB. 41/70. Outras variedades podem ser cultivadas com piores resultados, em algumas regiões. Não devem ser cultivadas no Estado as variedades N:Co. 352, CB. 41/35 e CB. 38/30, esta pela baixa riqueza em açúcar que apresenta.

Ensaios de variedades de cana de açúcar. I - série de ensaios realizados no período de 1951 a 1954

No presente trabalho são apresentados os resultados obtidos em quatro ensaios de variedades de cana de açúcar, realizados de 1951 a 1954, nas usinas Junqueira, Porto Feliz e Itaiquara e na Fazenda Pau-a-pique. O objetivo foi determinar, dentre as variedades estudadas, as mais recomendáveis para os diversos tipos de solo e clima do Estado de São Paulo. Estudaram-se 16 variedades, sendo cinco indianas, Co.290, Co.331, Co.413, Co.419 e Co.421, quatro americanas, CP. 27/139, C.P.29/137, C.P.29/291 e C.P.34/120, uma argentina, Tuc. 2645, uma javaneza, P.O.J.2961 e cinco brasileiras, sendo três do Instituto Agronômico de Campinas, I.A.C.34/373, I.A.C.34/536 e I.A.C.34/553, e duas da Estação Experimental de Campos, no Estado do Rio, C.B.36/14 e C.B.38/1. Em cada ensaio foram feitos três cortes, cana planta, soca e ressoca, sendo analisados os resultados de produção de cana e de açúcar provável do primeiro corte e do total dos três cortes. Os resultados obtidos mostraram que podem ser recomendadas para todo o Estado de São Paulo as variedades Co.419 e C.B.36/14 juntamente com a Co.290, que é a variedade mais cultivada. Co.421 e Co.413 revelaram maior exigência quanto a solo e clima, especialmente esta última, C.P.34/120 e Co.331, apesar de produzirem bem como a Co.290, não são recomendadas por serem sucetíveis ao carvão, o mesmo acontecendo com a C.P.29/291, que também apresentou produções satisfatórias.

Recent developments in the design of field experiments. IV. Lattice squares with split plots

The testing of a large number of varieties or treatments can generally be most conveniently made by the use of the quasi-factorial designs devised by Yates. The value of such designs is enhanced by the possibility of introducing further treatments on parts of the plots. The present paper describes a lattice square trial testing 25 organic treatments (actually 22 different treatments with a control represented three times) in which all combinations of nitrogen, phosphate and potash were also tested by splitting the plots and confounding the three-factor interaction with whole plots, the total number of split-plots being 300. Both the design and analysis are comparatively simple and straight-forward, and will serve as an example of the use of split-plot confounding in most types of quasi-factorial designs.

Lattice squares

A new method of analysing the results of experiments arranged in lattice (quasi-Latin) square designs is described. By this method the information contained in the inter-row and inter-column comparisons is recovered. The procedure followed is to estimate the actual accuracy of the inter-row and inter-column comparisons relative to intra-row and column comparisons, and assign weights proportional to these relative accuracies.The methods of computation by which this procedure can be carried out expeditiously are described in detail, and are illustrated by two experiments carried out at Rothamsted and Woburn. The gains in accuracy obtained in these two experiments (as compared with ordinary randomized blocks) were approximately 30% and 60% respectively, allowing for losses of information due to inaccuracies of weighting.