Unit Square Research Articles

Lissajous curves as aerial search patterns

Manned and unmanned systems are prevalent in a wide range of aerial searching applications. For aircraft whose trajectory is not or cannot be planned on-the-fly, optimal deterministic search pattern generation is a critical area of research. Lissajous curves have recently caught attention as excellent candidates for all kinds of aerial search applications, but little fundamental research has been done to understand how best to design Lissajous pattern (LP)s for this use. This paper examines the optimization of these search patterns from analytical, numerical, and data-driven perspectives to establish the state of the field in Lissajous curves for aerial search. From an analytical perspective, it was found that the average expected distance between a Lissajous searcher and a random target on a unit square approaches 0.586 as search time increases. Furthermore, an analytical approximation for the average searcher speed was found to guarantee error of no more than 22.1%. Important outcomes from the numerical optimization of Lissajous search patterns include the development of an intuitive evaluation criterion and the conclusion that irrational frequency ratios near 0.8 typically yield highest performance. Finally, while a robust predictive model for fast pattern optimization is yet out of reach, initial results indicate that such an approach shows promise.

Numerical study of transitions in lid-driven flows in shallow cavities

In this article, three dimensional (3D) lid-driven flow in shallow cavities with a unit square base are studied. The numerical solution of the Navier–Stokes equations modeling incompressible viscous fluid flow in a cavity is obtained via a methodology combining a first order accurate operator-splitting scheme, a L 2 -projection Stokes solver, a wave-like equation treatment of the advection and finite element space approximations. Numerical results of a lid-driven flow in a cubic cavity show a good agreement with those reported in literature. The critical Reynolds numbers (Re cr ) for having flow with increasing of oscillating amplitude (a Hopf bifurcation) in different shallow cavities are obtained and associated oscillating modes are studied.

And subnormal safe quotients for geometrically regular weighted shifts

Geometrically regular weighted shifts (in short, GRWS) are those with weights α(N,D) given by αn(N,D)=pn+Npn+D, where p>1 and (N,D) is fixed in the open unit square (−1,1)×(−1,1). We study here the zone of pairs (M,P) for which the weight α(N,D)α(M,P) gives rise to a moment infinitely divisible (▪) or a subnormal weighted shift, and deduce immediately the analogous results for product weights α(N,D)α(M,P), instead of quotients. Useful tools introduced for this study are a pair of partial orders on the GRWS.

Coherent distributions on the square–extreme points and asymptotics

Abstract Let $\mathcal{C}$ denote the family of all coherent distributions on the unit square $[0,1]^2$ , i.e. all those probability measures $\mu$ for which there exists a random vector $(X,Y)\sim \mu$ , a pair $(\mathcal{G},\mathcal{H})$ of $\sigma$ -fields, and an event E such that $X=\mathbb{P}(E\mid\mathcal{G})$ , $Y=\mathbb{P}(E\mid\mathcal{H})$ almost surely. We examine the set $\mathrm{ext}(\mathcal{C})$ of extreme points of $\mathcal{C}$ and provide its general characterisation. Moreover, we establish several structural properties of finitely-supported elements of $\mathrm{ext}(\mathcal{C})$ . We apply these results to obtain the asymptotic sharp bound $\lim_{\alpha \to \infty}\alpha\cdot(\sup_{(X,Y)\in \mathcal{C}}\mathbb{E}|X-Y|^{\alpha}) = {2}/{\mathrm{e}}$ .

Modeling, Implementing and Evaluation a Decision Support System Used for Choosing the Best HVAC System in The Buildings, Case Study in Iraq

The life cycle cost of a building is affected by the heating, ventilation, and air conditioning (HVAC) system chosen by the Life Cycle Costs (LCC). Quality, constructability, appearance of the structure's interior and exterior, HVAC size and weight, and LCC are some of the criteria influencing the choice. Methods: To monitor a project's progress based on energy savings, standard measures such as cost variance (CV) and schedule variation have used an idea when tracking the performance of intelligent buildings. Also, as described in the article, this research compared the decision-making limits of Building Information Modelling (BIM) and (MCDM). Analysis: The conventional approach cannot reveal information regarding divergence from the expected level of performance. Based on the outcomes of the construction cost variables, the key finding was the observation of 12 efficient elements. Finding and Novelty: According to the R, a building's most valuable features are its (Energy Saving Features, Warranties, Budget, Protect Your Unit, SEER Ratings, and Home Square Footage). The findings of Actual value (AV) and planned value (PV) were significantly different, as noted by the Multi-Criteria Decision Maker (MCDM). The new method also makes it possible to track project costs and timetables more accurately. The paper will characterize the HVAC Decision Support System's architecture (HVACDSS). Also, a case study of action modeling is provided, and the preliminary findings are addressed. Six criteria characteristics are used by the HVACDSS technique by an analysis of building construction conducted using the WEKA mining tool (decision tree).

Investigation of Earthquakes in Turkey with Cluster Analysis

In this study, earthquakes with a magnitude of 3 and above occurring between 36-42 degrees North latitude and 26-45 degrees East longitude, within and around Turkey from 1980 to 2022, were considered. Turkey and its immediate surroundings were divided into 114-unit squares using latitude and longitude coordinates to investigate seismic similarities. The variables including 50130 earthquake depths and magnitudes in our data set were analyzed using the {K-Means Cluster Analysis} method in the SPSS program. Hierarchical clustering method was used to determine the number of clusters and the number of clusters was determined as 5 with the help of the dendrogram obtained. As a result of the cluster analysis, it was observed that 47% of the earthquakes in Turkey were at a depth of 5 to 10 km. It has been observed that 93% of the earthquake intensities occur between 3 and 4 magnitudes. A cluster analysis was conducted to assess the similarities and dissimilarities among earthquakes within these unit squares in terms of their magnitudes and depths. Due to the large sample size and the analysis involving continuous variables, the earthquake data were grouped into 5 clusters using non-hierarchical clustering methods, and the relationships within and between these clusters were observed.

Abstract 3514: A deep learning model for detection and characterization of tertiary lymphoid structures in H&E-stained images from pancreatic ductal adenocarcinoma

Abstract Tertiary lymphoid structures (TLSs) are ectopic aggregates of a number of immune cells in nonlymphoid tissues under chronically inflamed environments and cancer. Emergence evidence suggested that TLSs were found in diverse cancers and TLSs have been referred as an independently predictive biomarker for immunotherapy response, especially immune checkpoint inhibitors. Moreover, the density and mature status of TLSs often positively associated with the favorable outcomes in cancers. Pancreatic ductal adenocarcinoma (PDAC) is one of the leading causes of cancer-related mortality with low survival rate in the whole world. Here, we developed a multi-resolution machine learning model for detection of TLSs from hema-toxylin and eosin (H&E) stained digital pathology slides as a potential clinical biomarker in PDAC. We selected 120 H&E slides (patients) with a total of 1,896 TLSs manually annotated as immature or mature TLSs by two pathologists. We also used immunohistochemistry (IHC) for CD20 and CD23 markers as a reference standard for H&E-based manual annotations. Our model mainly included detection and segmentation of individual TLSs. First, we used a transformer convolutional neural network to build a classification task to detect the presence of immature or mature TLSs. Then the instance segmentation model was applied to quantify the number of lymphocytes per unit square to validate the presence and maturation of TLSs. F1-score, calculated by the harmonic mean of precision (positive predictive value) and recall (sensitivity), was finally used to quantitatively evaluate the performance of model for each H&E slide. In validation dataset with 250 H&E slides (patients), we selected 50 slides to calculate F1-score, suggesting that our machine learning model achieved high F1-scores for detection and classification of immature TLSs (mean = 0.87, range from 0.79 to 0.95), and mature TLSs (mean = 0.92, range from 0.86 to 0.97). We further stratified 370 PDAC patients into three groups with mature TLSs (23%), immature TLSs (16%), and no TLSs (61%), respectively. Combined the clinical information, survival analysis revealed that patients with mature TLS had better survival outcomes than those without TLS (HR = 0.63; 95% CI, 0.44 - 0.90; P = 0.010). We developed a machine learning model that can accurately detect and classify TLSs in PDAC, and also demonstrated the prognosis value of predictive TLSs in H&E slides by automatedly machine learning model. These data highlight the promise of machine learning model for automated identification and quantification of the TLS on H&E slides in PDAC pathology. Citation Format: Chaoxian Zhao, Jidong Jia, Mingxi Lv, Jiayi Feng, Yijun Wang, Yingbin Liu. A deep learning model for detection and characterization of tertiary lymphoid structures in H&E-stained images from pancreatic ductal adenocarcinoma [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2024; Part 1 (Regular Abstracts); 2024 Apr 5-10; San Diego, CA. Philadelphia (PA): AACR; Cancer Res 2024;84(6_Suppl):Abstract nr 3514.

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.