Unit Square Research Articles

Electrical properties of the unit square plate

Numerical values of electrostatic properties of the cube are well investigated whereas numerical ones of the square plate are not. In this paper, we update the numerical values of electrostatic properties of the unit square plate, such as the capacitance, and the edge and corner singularity exponents. For the capacitance, we use the refined Brownian dynamics algorithm with “walk-on-planes” (WOP) method (a kind of first-passage algorithm) and for the edge singularity and corner exponents, the sphere last-passage algorithm.

A new approach to the multiple obnoxious facility location problem based on combinatorial and continuous tools

In this paper we address the multiple obnoxious facility location problem. In this problem p facilities need to be spread within the unit square in such a way that they are far enough from each other and that their minimal distance from n communities, with known positions within the unit square, is maximized. The problem has a combinatorial component, related to the key observation made in Drezner (Omega 87:105–116, 2019) about the role played by Voronoi points. We propose a new approach, which exploits both the combinatorial component of the problem and, through continuous local optimizations, also its continuous component. We also propose techniques to limit the impact on computation times of the number n of communities. The approach turns out to be quite competitive and is able to return 24 new best known solutions with respect to the best results reported in Kalczynski (Optim Lett 16:1153–1166, 2022).

Robust convergence result of discontinuous Galerkin stabilization method for two-dimensional reaction–diffusion equation with discontinuous source term

A reaction–diffusion problem with discontinuous source term and Dirichlet's boundary conditions on the unit square is considered in this paper. The proposed problem has been discretized using a combination of standard Galerkin finite element method (FEM) and non-symmetric discontinuous Galerkin finite element method with an interior penalty (NIPG) with bilinear elements. Layer adapted mesh of Shishkin type has been utilized to discretize the domain. Standard Galerkin FEM is applied on the layer part of the domain where the domain is dense enough and NIPG is applied to the outside layer part. By means of special choice of discontinuity-penalization parameters, the scheme is proved to be uniformly convergent of order O ( ε 1 / 4 N − 1 + N − 1 ln ⁡ N ) . Numerical tests are carried out in support of theoretical findings.

Proving chaos for a system of coupled logistic maps: A topological approach.

In the work, we prove the presence of chaotic dynamics, for suitable values of the model parameters, for the discrete-time system, composed of two coupled logistic maps, as formulated in Yousefi et al. [Discrete Dyn. Nat. Soc. 5, 161-177 (2000)], which describes two interdependent economies, characterized by two competitive markets each, with a demand link between them. In particular, we rely on the SAP (Stretching Along the Paths) method, based on a stretching relation for maps defined on sets homeomorphic to the unit square and expanding the paths along one direction. Such technique is topological in nature and allows to establish the existence of a semiconjugacy between the considered dynamical system and the Bernoulli shift, so that the main features related to the chaos of the latter (e.g., the positivity of the topological entropy) are transmitted to the former. In more detail, we apply the SAP method both to the first and to the second iterate of the map associated with our system, and we show how dealing with the second iterate, although being more demanding in terms of computations, allows for a larger freedom in the sign and in the variation range of the linking parameters for which chaos emerges. Moreover, the latter choice guarantees a good agreement with the numerical simulations, which highlight the presence of a chaotic attractor under the conditions derived for the applicability of the SAP technique to the second iterate of the map.

Tightest Arrangements of All Different Nets of a Cube

Summary Since the 1950’s, polyominoes (that is, shapes consisting of edge-connected unit squares) have gained significant interest both as objects of combinatorial study and as classics of recreational mathematics. There is a very natural collection of such shapes, namely the 11 distinct polyhedral nets of the unit cube. An interesting question is: how tightly can the nets fit together? It is equivalent to search for the smallest perimeter a shape built of them can have. In this article, we provide the exact lower bound as an invitation to study analogous problems for different families of polyominoes.

Error Estimates in the Maximum Norm for the Solution of Poisson’s Equation Approximated by the Five-Point Laplacian Using the Discrete Maximum Principle

In this paper, we study error estimates in the maximum norm in the context of solving Poisson’s equation numerically when approximated the Five-Point Laplacian method using the discrete maximum principle. The primary objective is to assess the accuracy of this numerical approach in solving Poisson’s equation and to provide insights into the behavior of error estimates. We focus on the estimates of maximum norm of the discrete functions defined on a grid in a unit square as well as in a square of side s, and estimate errors measured in the maximum norm.

Entropy generation in a partially heated hybrid nanofluid saturated wavy porous cavity

PurposeA porous cavity flow field generates entropy owing to energy and momentum exchange within the fluid and at solid barriers. The heat transport and viscosity effects on fluid and solid walls irreversibly generate entropy. This numerical study aims to investigate convective heat transfer together with entropy generation in a partially heated wavy porous cavity filled with a hybrid nanofluid.Design/methodology/approachThe governing equations are nondimensionalized and the domain is transformed into a unit square. A second-order finite difference method is used to have numerical solutions to nondimensional unknowns such as stream function and temperature. This numerical computation is conducted to explore a wide range of regulating parameters, e.g. hybrid nano-particle volume fraction (σ = 0.1%, 0.33%, 0.75%, 1%, 2%), Rayleigh–Darcy number (Ra = 10, 102, 103), dimensionless length of the heat source (ϵ = 0.25, 0.50,1.0) and amplitude of the wave (a = 0.05, 0.10, 0.15) for a number of undulations (N = 1, 3) per unit length.FindingsA thorough analysis is conducted to analyze the effect of multiple factors such as thermal convective forces, heat source, surface corrugation factors, nanofluid volume fraction and other parameters on entropy generation. The flow and temperature fields are studied through streamlines and isotherms. The average Bejan number suggested that entropy generation is entirely dominated by irreversibility due to heat transport at Ra = 10, and the irreversibility due to the viscosity effect is severe at Ra = 103, but the increment in s augments irreversibility due to the viscosity effect over the heat transport at Ra = 102.Originality/valueTo the best of the authors’ knowledge, this numerical study, for the first time, analyzes the influence of surface corrugation on the entropy generation related to the cooling of a partial heat source by the convection of a hybrid nanofluid.

Covariational reasoning in Bayesian situations

AbstractPrevious studies on Bayesian situations, in which probabilistic information is used to update the probability of a hypothesis, have often focused on the calculation of a posterior probability. We argue that for an in-depth understanding of Bayesian situations, it is (apart from mere calculation) also necessary to be able to evaluate the effect of changes of parameters in the Bayesian situation and the consequences, e.g., for the posterior probability. Thus, by understanding Bayes’ formula as a function, the concept of covariation is introduced as an extension of conventional Bayesian reasoning, and covariational reasoning in Bayesian situations is studied. Prospective teachers (N=173) for primary (N=112) and secondary (N=61) school from two German universities participated in the study and reasoned about covariation in Bayesian situations. In a mixed-methods approach, firstly, the elaborateness of prospective teachers’ covariational reasoning is assessed by analysing the arguments qualitatively, using an adaption of the Structure of Observed Learning Outcome (SOLO) taxonomy. Secondly, the influence of possibly supportive variables on covariational reasoning is analysed quantitatively by checking whether (i) the changed parameter in the Bayesian situation (false-positive rate, true-positive rate or base rate), (ii) the visualisation depicting the Bayesian situation (double-tree vs. unit square) or (iii) the calculation (correct or incorrect) influences the SOLO level. The results show that among these three variables, only the changed parameter seems to influence the covariational reasoning. Implications are discussed.

Minimum weight Euclidean [formula omitted]-spanners

Given a set S of n points in the plane and a parameter ɛ>0, a Euclidean (1+ɛ)-spanner is a geometric graph G=(S,E) that contains, for all p,q∈S, a pq-path of weight at most (1+ɛ)‖pq‖. We show that the minimum weight of a Euclidean (1+ɛ)-spanner for n points in the unit square [0,1]2 is O(ɛ−3/2n), and this bound is the best possible. The upper bound is based on a new spanner algorithm in the plane. It improves upon the baseline O(ɛ−2n), obtained by combining a tight bound for the weight of a Euclidean minimum spanning tree (MST) on n points in [0,1]2, and a tight bound for the lightness of Euclidean (1+ɛ)-spanners, which is the ratio of the spanner weight to the weight of the MST. Our result generalizes to Euclidean d-space for every constant dimension d∈N: The minimum weight of a Euclidean (1+ɛ)-spanner for n points in the unit cube [0,1]d is Od(ɛ(1−d2)/dn(d−1)/d), and this bound is the best possible.For the n×n section of the integer lattice in the plane, we show that the minimum weight of a Euclidean (1+ɛ)-spanner is between Ω(ɛ−3/4⋅n2) and O(ɛ−1log(ɛ−1)⋅n2). These bounds become Ω(ɛ−3/4⋅n) and O(ɛ−1log(ɛ−1)⋅n) when scaled to a grid of n points in the unit square. In particular, this shows that the integer grid is not an extremal configuration for minimum weight Euclidean (1+ɛ)-spanners.

Packing of non-blocking squares into the unit square

Packing of non-blocking squares into the unit square

Maximum Bipartite Subgraphs of Geometric Intersection Graphs

We study the Maximum Bipartite Subgraph ([Formula: see text]) problem, which is defined as follows. Given a set [Formula: see text] of [Formula: see text] geometric objects in the plane, we want to compute a maximum-size subset [Formula: see text] such that the intersection graph of the objects in [Formula: see text] is bipartite. We first give an [Formula: see text]-time algorithm that computes an almost optimal solution for the problem on circular-arc graphs. We show that the [Formula: see text] problem is [Formula: see text]-hard on geometric graphs for which the maximum independent set is [Formula: see text]-hard (hence, it is [Formula: see text]-hard even on unit squares and unit disks). On the other hand, we give a [Formula: see text] for the problem on unit squares and unit disks. Moreover, we show fast approximation algorithms with small-constant factors for the problem on unit squares, unit disks, and unit-height axis parallel rectangles. Additionally, we prove that the Maximum Triangle-free Subgraph ([Formula: see text]) problem is NP-hard for axis-parallel rectangles. Here the objective is the same as that of the [Formula: see text] except the intersection graph induced by the set [Formula: see text] needs to be triangle-free only (instead of being bipartite).

Quantifying directed dependence via dimension reduction

Studying the multivariate extension of copula correlation yields a dimension reduction principle, which turns out to be strongly related with the ‘simple measure of conditional dependence’ T recently introduced by Azadkia and Chatterjee (2021). In the present paper, we identify and investigate the dependence structure underlying this dimension reduction principle, provide a strongly consistent estimator for it, and demonstrate its broad applicability. For that purpose, we define a bivariate copula capturing the scale-invariant extent of dependence of an endogenous random variable Y on a set of d≥1 exogenous random variables X=(X1,…,Xd), and containing the information whether Y is completely dependent on X, and whether Y and X are independent. The dimension reduction principle becomes apparent insofar as the introduced bivariate copula can be viewed as the distribution function of two random variables Y and Y′ sharing the same conditional distribution and being conditionally independent given X. Evaluating this copula uniformly along the diagonal, i.e., calculating Spearman’s footrule, leads to an unconditional version of Azadkia and Chatterjee’s ‘simple measure of conditional dependence’ T. On the other hand, evaluating this copula uniformly over the unit square, i.e., calculating Spearman’s rho, leads to a distribution-free coefficient of determination (also known as ‘copula correlation’). Several real data examples illustrate the importance of the introduced methodology.

Energy scaling laws for microstructures: from helimagnets to martensites

We consider scalar-valued variational models for pattern formation in helimagnetic compounds and in shape memory alloys. Precisely, we consider a non-convex multi-well bulk energy term on the unit square, which favors four gradients (±α,±β)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\pm }\\,\\alpha ,{\\pm }\\,\\beta )$$\\end{document}, regularized by a singular perturbation in terms of the total variation of the second derivative. We derive scaling laws for the minimal energy in the case of incompatible affine boundary conditions in terms of the singular perturbation parameter as well as the ratio α/β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha /\\beta $$\\end{document} and the incompatibility of the boundary condition. We discuss how well-studied models for martensitic microstructures in shape-memory alloys arise as a limiting case, and relations between the different models in terms of scaling laws. In particular, we show that scaling regimes arise in which an interpolation between the rather different branching-type constructions in the spirit of Kohn and Müller (Commun Pure Appl Math 47(4):405–435, 1994) and Ginster and Zwicknagl (J Nonlinear Sci 33:20, 2023), respectively, occurs. A particular technical difficulty in the lower bounds arises from the fact that the energy scalings involve various logarithmic terms that we capture in matching upper and lower scaling bounds.

38 The Effects of Lysine and Antibiotic Usage on Early Nursery Pig Performance

Abstract Weaning transition, early nursery environment, and the diet is critical for nursery pig performance and health. Nutritional interventions (i.e., low crude protein diets achieved via decreased SID lysine) and water or in-feed antibiotics are often used to aid in this transition in pig flows prone to endemic enteric pathogens. Therefore, the objective of this study was to investigate the effects of high or low dietary lysine curves, with or without antibiotics, in the first two phases of production on early nursery pig performance. A total of 384 weaned nursery pigs (starting body weight of 5.3 ± 1.6 kg) were randomly assigned to split sex pens (8 pigs/pen) over a 21-day study. Using a 2x2 factorial arrangment, pens were fed one of two dietary treatments over phase 1 (P1) and 2 (P2), with or without antibiotics (n = 12 pens/treatment). The treatments consisted of: 1) low lysine curve (1.15% and 1.2% SID Lys, respectively; LLys); 2) high lysine curve (1.45% and 1.375% SID Lys, respectively; HLys); 3) LLys + antibiotics (LLys+); and 4) HLys + antibiotics (HLys+). During P1, LLys+ and HLys+ received water soluble gentamicin for 5 days, while 50 g/ton Mecadox was added to the LLys+ and HLys+ P2 diets. Four days post-placement, pigs were confirmed positive for Rotavirus and hemolytic E. coli. Individual pig body weights and pen feed disappearance were collected on day 7, 14 and 21 to calculate ADG, ADFI, and G:F. Fecal consistency was scored daily by pen. Data were analyzed with pen as the experimental unit and least squares means reported with the fixed effects of lysine, antibiotics, and their interaction. Regardless of dietary treatment or antibiotic usage, LLys, HLys, LLys+, and HLys+ ADG (0.21, 0.22, 0.20, and 0.21 kg/d, respectively) or ADFI (0.30, 0.28, 0.28, and 0.28 kg/d, respectively) were not different over the 21-day period (P > 0.05). No lysine by antibiotic interactions were reported for any variable measured (P > 0.05). Pig removals and mortalities ranged from 1-4% and did not differ between treatments (P > 0.05). No fixed effects were observed in P1 variables. Although growth rates did not differ, G:F was greater in HLys compared with LLys fed pigs in P2 (0.78 vs. 0.73, P = 0.016) and overall (0.75 vs. 0.69, P = 0.031). Antibiotic usage had no effect on P1, P2, or overall performance parameters. In conclusion, feeding pigs on a high lysine curve or utilizing water soluble and in-feed antibiotics shows minimal benefit to young pigs with a natural enteric pathogen burden.

Ortaokul Öğrencilerinin Alan Ölçme Bilgi ve Becerilerinin İncelenmesi

In this study, it is aimed to examine the area measurement knowledge and skills of secondary school students. In the study, in which the descriptive method was used, the data were collected in written form with 8 open-ended questions from 92 seventh grade students. Students' answers to open-ended questions were analyzed with a scoring scale. The research shows that the majority of students who correctly estimate the size of the area measurement units are in the majority, but they have difficulty in transforming the standard area measurement units. Students who can calculate the area as the number of unit squares covering a region are in the majority. It was observed that the students who were successful in measuring the area of the rectangle had low success in measuring the area of the parallelogram, triangle and trapezoid, and it was observed that the students tended to multiply the two sides of the polygon while measuring the area. In addition, it was determined that some students confused the area with the environment.

Concept Formation Processes of Area and its Formula with GeoGebra: Case of Rectangle

This research aims to investigate the construction of the area of rectangle with GeoGebra and how students can formulate it. It focuses on describing the cognitive process involved in constructing this concept and how technology mediates such a process. The research was conducted in a classroom with 5th grade students in a secondary school and was designed qualitatively as a case study. A readiness test was prepared to determine the preliminary knowledge of the students about the concept of area. The teaching process was carried out with the activities based on GeoGebra. Considering the test results, researcher observations, and teacher views, five voluntary participants were selected, and we had three interviews with each of them. The obtained data from the teaching process and interviews were analyzed according to APOS theoretical framework. The results of the research demonstrate that one of the participants could not conceptualize the area as an object as the number of unit squares that cover the area without any space, while two of the participants could not conceptualize the formula of area of rectangle as an object. The students generally had difficulty in encapsulating the concept of the area and its formula. Area conservation and unit concept are important for conceptualizing the concept of area by realizing the covering feature of it, and understanding the dimension relationship is significantly effective in conceptualizing its formula.

Asymptotic Betti Numbers for Hard Squares in the Homological Liquid Regime

Abstract We study configuration spaces $C(n; p, q)$ of $n$ ordered unit squares in a $p$ by $q$ rectangle. Our goal is to estimate the $j$th Betti number for large $n$, $j$, $p$, and $q$. We consider sequences of area-normalized coordinates, where $\left (\frac {n}{pq}, \frac {j}{pq}\right )$ converges as $n$, $j$, $p$, and $q$ approach infinity. For every sequence that converges to a point in the “feasible region” in the $(x,y)$-plane identified in [3], we show that the factorial growth rate of the Betti numbers is the same as the factorial growth rate of $n!$. This implies that (1) the Betti numbers are vastly larger than for the configuration space of $n$ ordered points in the plane, which have the factorial growth rate of $j!$, and (2) every point in the feasible region is eventually in the homological liquid regime.

Generating methods of some classes of uninorms and associated structures

Uninorms are an important class of fuzzy logic connectives that have, so far, neither generating methods nor algebraic structures. In this work, we attempt to propose some novel generating methods of some well established classes of uninorms (with a fixed but arbitrary neutral element e) and investigate the algebraic structures that these generating methods can impose on those specific classes of uninorms. Towards this, we show that the operations point-wise maximum (∨) and the point-wise minimum (∧) on the set of idempotent uninorms are closed and make it a distributive lattice. Also, we show that the operations ∧ and ∨ on some sub-classes of idempotent uninorms with other classes of uninorms generate uninorms. Since the operations ∨ and ∧ on the set of uninorms are not closed in general, we propose some novel generating methods of uninorms that come from various classes such as representable uninorms, pseudo-continuous uninorms, uninorms continuous in the open unit square, uninorms with underlying continuous Archimedean t-norms and t-conorms and for each class of these uninorms, we investigate the algebraic structure imposed on it by the proposed generating methods with the help of results obtained recently in Vemuri et al. (2022) [51].

Planar substitutions to Lebesgue type space-filling curves and relatively dense fractal-like sets in the plane

We present an algorithm for generating curves filling the unit square; i.e. space-filling curves, from any given planar substitution satisfying a mild condition. The proposed algorithm is mimicking construction steps of Lebesgue's curve and is based on linear interpolation. Generated space-filling curves for some known substitutions are elucidated. Some of those substitutions further induce relatively dense fractal-like sets in the plane, whenever some additional assumptions are satisfied.

Irreducible quasi-D-overlap functions: Matrix representation and diagonal generation

As an emerging mathematical model of information aggregation successfully applied in image processing, pattern recognition, decision-making and etc, theoretical research of overlap functions has also continued to develop, especially the research on their related derivative concepts that have emerged in recent years. This work considers so-called irreducible quasi-D-overlap functions defined on grid of unit square, which are one class of discrete aggregation operators originated from overlap functions. The core contents are: (i) it presents the concept of irreducible quasi-D-overlap functions and shows basic properties of them; (ii) it provides equivalent characterization of irreducible quasi-D-overlap functions by means of strong permutation matrixes; (iii) it gives a construction manner for irreducible quasi-D-overlap functions by means of so-called diagonal operators and the greatest and smallest irreducible quasi-D-overlap functions with given diagonal operator are obtained.