Unit Square Research Articles

Binomials Arising from Buchberger Algorithm on Polyomino Ideals

A polyomino is a finite set of unit squares joined side by side on the Cartesian plane. Qureshi introduced an ideal constructed from a polyomino which is called "polyomino ideal". In this paper, we study the binomials arising from Buchberger Algorithm on polyomino ideals. We also introduce socket wrench polyominoes and study the Gr\"{o}bner bases of the ideal $I_{\mathcal P}$ and some algebraic properties of $K[\mathcal P]$.

Quasisymmetrically rigid self-similar carpets

Abstract We define balanced self-similar quasi-round carpets and compare the carpet moduli of some path families relating to such a carpet. Then, using some known results on quasiconformal geometry of carpets, we prove that the group of quasisymmetric self-homeomorphisms of every balanced self-similar quasi-round carpet is finite. Furthermore, we prove that some balanced self-similar carpets in the unit square with strong geometric symmetry are quasisymmetrically rigid by using the quasisymmetry of weak tangents of carpets.

Approximation algorithms for minimum ply covering of points with unit squares and unit disks

Given a set P of points and a set U of geometric objects in the Euclidean plane, a minimum ply cover of P with U is a subset of U that covers P and minimizes the number of objects that share a common intersection, called the minimum ply cover number of P with U. Biedl et al. (2021) [9] showed that for both unit squares and unit disks, determining the minimum ply cover number for a set of points is NP-hard. They gave polynomial-time 2-approximation algorithms for the special case when the minimum ply cover number is constant, and asked whether there exists polynomial-time O(1)-approximation algorithms for these problems. In this paper, we settle the question posed by Biedl et al. by providing polynomial-time O(1)-approximation algorithms for the minimum ply cover problem for both unit squares and unit disks.

Rothman diagrams: the geometry of confounding and standardization.

We outline a geometric perspective on causal inference in cohort studies that can help epidemiologists understand the role of standardization in controlling for confounding. For simplicity, we focus on a binary exposure X, a binary outcome D, and a binary confounder C that is not causally affected by X. Rothman diagrams plot the risk of disease in the unexposed on the x-axis and the risk in the exposed on the y-axis. The crude risks define a point in the unit square, and the stratum-specific risks at each level of C define two other points in the unit square. Standardization produces points along the line segment connecting the stratum-specific points. When there is confounding by C, the crude point is off this line segment. The set of all possible crude points is a rectangle with corners at the stratum-specific points and sides parallel to the axes. When there are more than two strata, standardization produces points in the convex hull of the stratum-specific points, and there is confounding if the crude point is outside this convex hull. We illustrate these ideas using data from a study in Newcastle, United Kingdom, in which the causal effect of smoking on 20-year mortality was confounded by age.

Numerical solution of the mixed boundary value problem for the heat equation in two-dimensional domains of complex shape

A finite-difference computational algorithm is proposed for solving a mixed boundary value problem for heat equation given in a two-dimensional domains of complex shape. To solve the problem, generalized curvilinear coordinates are used. The physical domain is mapped to the computational domain (unit square) in the space of generalized coordinates. The original problem is written in curvilinear coordinates and approximated on a uniform grid in the computational domain. The obtained results are mapped on a non-uniform boundary-fitted difference grid in the physical domain. The second-order approximations of mixed Neumann – Dirichlet boundary conditions are constructed. The results of computational experiments are presented. The second order of accuracy of the presented computational algorithm is confirmed.

Anisotropic error analysis of weak Galerkin finite element method for singularly perturbed biharmonic problems

We consider the weak Galerkin finite element approximation of the singularly perturbed biharmonic elliptic problem on a unit square domain with clamped boundary conditions. Shishkin mesh is used for domain discretization as the solution exhibits boundary layers near the domain boundary. Error estimates in the equivalent H2− norm have been established and the uniform convergence of the proposed method has been proved. Numerical examples are presented corroborating our theoretical findings.

The circle packing problem: A theoretical comparison of various convexification techniques

We consider the problem of packing a given number of congruent circles with the maximum radius in a unit square as a mathematical optimization problem. Due to the presence of non-overlapping constraints, this problem is a notoriously difficult nonconvex quadratically constrained optimization problem, which possesses many local optima. We consider several popular convexification techniques, giving rise to linear programming relaxations and semidefinite programming relaxations for the circle packing problem. We compare the strength of these relaxations theoretically, thereby proving the conjectures by Anstreicher. Our results serve as a theoretical justification for the ineffectiveness of existing machinery for convexification of non-overlapping constraints.

On the Formation of Microstructure for Singularly Perturbed Problems with Two, Three, or Four Preferred Gradients

In this manuscript, singularly perturbed energies with 2, 3, or 4 preferred gradients subject to incompatible Dirichlet boundary conditions are studied. This extends results on models for martensitic microstructures in shape memory alloys (N=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N=2$$\\end{document}), a continuum approximation for the J1-J3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$J_1-J_3$$\\end{document}-model for discrete spin systems (N=4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N=4$$\\end{document}), and models for crystalline surfaces with N different facets (general N). On a unit square, scaling laws are proved with respect to two parameters, one measuring the transition cost between different preferred gradients and the other measuring the incompatibility of the set of preferred gradients and the boundary conditions. By a change of coordinates, the latter can also be understood as an incompatibility of a variable domain with a fixed set of preferred gradients. Moreover, it is shown how simple building blocks and covering arguments lead to upper bounds on the energy and solutions to the differential inclusion problem on general Lipschitz domains.

Decomposition and construction of uninorms on the unit interval

By analyzing the behaviour of uninorms on the boundary of the unit square, we present decomposition theorems for uninorms, showing that a conjunctive (resp. disjunctive) uninorm can be decomposed into a disjunctive (resp. conjunctive) uninorm on an upper set (resp. a lower set) and a triangular subnorm (resp. superconorm) on a lower set (resp. an upper set). As an application of the decomposition theorems, we propose several construction methods for uninorms that are not internal on the boundary.

Online class cover problem

In this paper, we study the online class cover problem where a (finite or infinite) family F of geometric objects and a set Pr of red points in Rd are given a prior, and blue points from Rd arrives one after another. Upon the arrival of a blue point, the online algorithm must make an irreversible decision to cover it with objects from F that do not cover any points of Pr. The objective of the problem is to place a minimum number of objects. When F consists of axis-parallel unit squares in R2, we prove that the competitive ratio of any deterministic online algorithm is Ω(log⁡|Pr|), and also propose an O(log⁡|Pr|)-competitive deterministic algorithm for the problem.

On a variant of Flory model

We consider a one-dimensional variant of a recently introduced settlement planning problem in which houses can be built on finite portions of the rectangular integer lattice subject to certain requirements on the amount of insolation they receive. In our model, each house occupies a unit square on a 1×n strip, with the restriction that at least one of the neighboring squares must be free. We are interested mostly in situations in which no further building is possible, i.e. in maximal configurations of houses in the strip. We reinterpret the problem as a problem of restricted packing of vertices in a path graph and then apply the transfer matrix method in order to compute the bivariate generating functions for the sequences enumerating all maximal configurations of a given length with respect to the number of houses. This allows us to determine the asymptotic behavior of the enumerating sequences and to compute some interesting statistics. Along the way, we establish close connections between our maximal configurations and several other types of combinatorial objects, including restricted permutations and walks on certain small oriented graphs. We then generalize our results in several directions by considering multi-story houses, by varying the insolation restrictions, and, finally, by considering strips of width 2 and 3. At the end we comment on several possible directions of future research.

Proposed theoretical value for TSP constant

The traveling salesman problem (TSP) involves determining the shortest length (optimal tour) for a complete circuit through an arbitrary number of points. In the special case that the points to be visited are randomly distributed in the plane within a unit square, and travel costs correspond to the Euclidean distance between points, it is known that the length of the optimal tour divided by the square root of the number of points ([Formula: see text]) asymptotically approaches a constant ([Formula: see text] as [Formula: see text] becomes large. If individual points are randomly drawn from a uniform distribution, the optimal lengths for a sufficiently large number of TSP instances of the same size ([Formula: see text] comprise a normal distribution whose mean approaches [Formula: see text] as [Formula: see text] increases. Moreover, as [Formula: see text] approaches infinity, this distribution gradually narrows such that its standard deviation asymptotically approaches zero. Although the precise value of [Formula: see text] is unknown, published studies indicate its magnitude is approximately 0.712. In this paper, possibly for the first time, precise formulae are proposed for the parameters of the underlying normal distribution and [Formula: see text], based on intuition, mathematical reasoning, and empirical fits to both published and experimentally derived data.

A note on the 2-colored rectilinear crossing number of random point sets in the unit square

Let S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S$$\\end{document} be a set of four points chosen independently, uniformly at random from a square. Join every pair of points of S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S$$\\end{document} with a straight line segment. Color these edges red if they have positive slope and blue, otherwise. We show that the probability that S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S$$\\end{document} defines a pair of crossing edges of the same color is equal to 1/4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1/4$$\\end{document}. This is connected to a recent result of Aichholzer et al. [1] who showed that by 2-colouring the edges of a geometric graph and counting monochromatic crossings instead of crossings, the number of crossings can be more than halved. Our result shows that for the described random drawings, there is a coloring of the edges such that the number of monochromatic crossings is in expectation 12-750\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{1}{2}-\\frac{7}{50}$$\\end{document} of the total number of crossings.

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.