Convex Window Research Articles

On Helly Numbers for Crystals and Cut-And-Project Sets

We prove existence of Helly numbers for crystals and for cut-and-project sets with convex windows. Also we show that for a two-dimensional crystal consisting of 𝑘 copies of a single lattice the Helly number does not exceed 𝑘 + 6.

Maximizing the Benefits of Daylight in Residential Design and Construction: A Review of Modern Homes and Villas

Daylight is an essential element in human life and plays a crucial role in the design of modern homes and villas. This paper explores the importance and application of daylight in residential design and construction. The paper highlights the benefits of daylight in terms of physical and mental health and reduced energy consumption in buildings. It also explores how daylight resources can be fully utilised in modern architecture, in terms of design, materials, construction and technology, to allow the interior spaces of houses to be adequately illuminated while minimising energy consumption. The paper further examines the influence of daylight on residential design, and the use of daylight in villas, including the choice of appropriate orientation and the use of convex windows, balconies and skylights to obtain optimum daylight conditions while meeting the villa owner’s requirements for views, privacy and comfort. The issues that need to be addressed in the use of daylight, such as avoiding excessive sunlight, maintaining appropriate indoor temperatures, and avoiding glare and inadequate lighting, are also explored. Finally, the paper looks to the future of daylight in architecture, including the more efficient use of daylight resources in buildings through technological means such as intelligent design, machine learning and artificial intelligence, to achieve more comfortable, energy efficient and environmentally friendly designs for homes and villas.

Covariate construction of nonconvex windows for spatial point patterns

In some standard applications of spatial point pattern analysis, window selection for spatial point pattern data is complex. Often, the point pattern window is given a priori. Otherwise, the region is chosen using some objective means reflecting a view that the window may be representative of a larger region. The typical approaches used are the smallest rectangular bounding window and convex windows. The chosen window should however cover the true domain of the point process since it defines the domain for point pattern analysis and supports estimation and inference. Choosing too large a window results in spurious estimation and inference in regions of the window where points cannot occur. We propose a new algorithm for selecting the point pattern domain based on spatial covariate information and without the restriction of convexity, allowing for better estimation of the true domain. Amodified kernel smoothed intensity estimate that uses the Euclidean shortest path distance is proposed as validation of the algorithm. The proposed algorithm is applied in the setting of rural villages in Tanzania. As a spatial covariate, remotely sensed elevation data is used. The algorithm is able to detect and filter out high relief areas and steep slopes; observed characteristics that make the occurrence of a household in these regions improbable. Keywords: Covariate, Euclidean shortest path, Nonconvex, Spatial point pattern, Window selection

A limit theorem for persistence diagrams of random filtered complexes built over marked point processes

Random filtered complexes built over marked point processes on Euclidean spaces are considered. Examples of these filtered complexes include a filtration of $\check{\text{C}}$ech complexes of a family of sets with various sizes, growths, and shapes. The law of large numbers for persistence diagrams is established as the size of the convex window observing a marked point process tends to infinity.

Energetic impact of preheating of fresh air in added convex window

During the renovation of an existing building keeping the original window and installing an added convex window is beneficial for the reduction of heating energy demand, furthermore this solution is free of the disposal and/or recycling of demolition waste. The energy need for heating can be further reduced, if the natural ventilation of the premise is through the buffer zone of the added convex window. In this case the fresh air is pre-heated in the buffer zone, increasing the utilised passive solar gain of the building. Measurements were carried out on experimental huts for analysis of this impact. Dynamic simulations of the huts have been carried out as well. In this paper the results will be presented, which proves the energetic effect.

An innovative technique for TPB deposition on convex window photomultiplier tubes

Tetraphenyl-butadiene (TPB) is an organic fluorescent chemical compound generally used as wavelength shifter thanks to its extremely high efficiency to convert ultra-violet photons into visible light. A common method to use TPB with detectors sensitive to visible light, such as photomultiplier tubes (PMTs), is to deposit thin layers on the device window. To obtain effective TPB layers, different procedures can be used. In this work a specific evaporation technique adopted to coat 8 in convex windows photomultiplier tubes is presented. It consists in evaporating TPB by means of a Knudsen cell, which allows to strictly control the process, and in a rotating sample support, which guarantees the uniformity of the deposition. Simulation results and experimental tests demonstrate the effectiveness of this evaporation technique from the point of view of deposition uniformity and light conversion efficiency.

Prevention of summer overheating of buildings with added convex windows

When renovating an existing building, keeping the original window and installing added convex window is beneficial for the reduction of heating energy demand. This solution is free of the disposal and/or recycling of demolition waste. In the summer season as far as the transparent structures of the building are concerned, in general, heat protection has to be used. In the case of the added convex window, it can be provided by the external air curtain mode; the movable shading device is in the buffer zone. This is a well-protected place for it, so that it can be as simple and cheap as an inner shading device. In this article examinations of the efficiency of the external air curtain mode for the heat protection is presented. The summer overheating has been analysed in two experimental huts. It has been proven that simple shading device in added convex window tempers the summer overheating as efficiently as the combination of usual window and external shading device (which is exposed to heavy weather effects and is difficult to maintain).

Effect of window shape on the detection of hyperuniformity via the local number variance

Hyperuniform many-particle systems in d-dimensional space , which includes crystals, quasicrystals, and some exotic disordered systems, are characterized by an anomalous suppression of density fluctuations at large length scales such that the local number variance within a ‘spherical’ observation window grows slower than the window volume. In usual circumstances, this direct-space condition is equivalent to the Fourier-space hyperuniformity condition that the structure factor vanishes as the wavenumber goes to zero. In this paper, we comprehensively study the effect of aspherical window shapes with characteristic size L on the direct-space condition for hyperuniform systems. For lattices, we demonstrate that the variance growth rate can depend on the shape as well as the orientation of the windows, and in some cases, the growth rate can be faster than the window volume (i.e. Ld), which may lead one to falsely conclude that the system is non-hyperuniform solely according to the direct-space condition. We begin by numerically investigating the variance of two-dimensional lattices using ‘superdisk’ windows, whose convex shapes continuously interpolate between circles (p = 1) and squares (), as prescribed by a deformation parameter p, when the superdisk symmetry axis is aligned with the lattice. Subsequently, we analyze the variance for lattices as a function of the window orientation, especially for two-dimensional lattices using square windows (superdisk when ). Based on this analysis, we explain the reason why the variance for d = 2 can grow faster than the window area or even slower than the window perimeter (e.g. like ). We then study the generalized condition of the window orientation, under which the variance can grow as fast as or faster than Ld (window volume), to the case of Bravais lattices and parallelepiped windows in . In the case of isotropic disordered hyperuniform systems, we prove that the large-L asymptotic behavior of the variance is independent of the window shape for convex windows. We conclude that the orientationally-averaged variance, instead of the conventional one using windows with a fixed orientation, can be used to resolve the window-shape dependence of the direct-space hyperuniformity condition. We suggest a new direct-space hyperuniformity condition that is valid for any convex window. The analysis on the window orientations demonstrates an example of physical systems exhibiting commensurate-incommensurate transitions and is closely related to problems in number theory (e.g. Diophantine approximation and Gauss’ circle problem) and discrepancy theory.

Added Convex Windows for Improving the Energy Balance

Added convex window is an additional glazed construction put on the façade from outside which covers the remaining original window in the plane of the façade. It is suitable for energy conscious retrofit and can be applied on new buildings as well. The added construction provides a buffer effect further improved due to the greenhouse effect. Fresh air supply through buffer space decreases the ventilation heat loss of adjacent room. Preliminary calculations, presented in the article prove that considerable energy saving can be achieved even if the façade is oriented towards North. The energy saving is compared with that of the typical retrofit measures. Besides the energy aspect several positive side effects are worth of mentioning such as fabric protection, protection against street-born noise, aesthetic improvement of old ugly-looking buildings.

Estimation of fracture trace length distributions using probability weighted moments and L-moments

The characterization of fracture trace length distributions is an initial and essential step in estimating three-dimensional fracture size distributions. Present challenge mainly lies in the accurate depiction for the distributional nature of trace lengths from various sizes of trace data, especially for small samples. The present paper is an attempt to solve this problem by using probability weighted moments (PWMs) and L-moments. To quantify the statistical property of trace lengths, the PWMs and L-moments of true trace lengths on an infinite surface from the measured trace lengths by an irregular convex window are estimated. A distribution-free method is then developed using the maximum entropy principle with PWMs for estimating the quantile functions of true trace lengths. Since there is no assumption regarding the type of trace length population distribution, the estimation obtained is distribution-free. For practicing engineers, a method using L-moments for estimating the common trace length distributions is also suggested. Examples that are tested showing the present method provides good approximations of the quantile functions and probability density functions of true trace lengths. The method is effective even for problems with outliers or highly skewed trace data, and can be used as a reliable tool for inferences from various sample sizes with good accuracies.

Proofs of the conjectures by Mecke for mixed line-generated tessellations

For a compact and convex window, Mecke described a process of tessellations which arise from cell divisions in discrete time. At each time step, one of the existing cells is selected according to an equally-likely law. Independently, a line is thrown onto the window. If the line hits the selected cell the cell is divided. If the line does not hit the selected cell nothing happens in that time step. With a geometric distribution whose parameter depends on the time, Mecke transformed his construction into a continuous-time model. He put forward two conjectures in which he assumed this continuous-time model to have certain properties with respect to their iteration. These conjectures lead to a third conjecture which states the equivalence of the construction of STIT tessellations and Mecke’s construction under some homogeneity conditions. In the present paper, all three conjectures are proven. A key tool to do that is a property of a continuous-time version of the equally-likely model classified by Cowan.

Intrinsic volumes of the maximal polytope process in higher dimensional STIT tessellations

Stationary and isotropic iteration stable random tessellations are considered, which are constructed by a random process of iterative cell division. The collection of maximal polytopes at a fixed time t within a convex window W ⊂ R d is regarded and formulas for mean values, variances and a characterization of certain covariance measures are proved. The focus is on the case d ≥ 3 , which is different from the planar one, treated separately in Schreiber and Thäle (2010) [12]. Moreover, a limit theorem for suitably rescaled intrinsic volumes is established, leading — in sharp contrast to the situation in the plane — to a non-Gaussian limit.

Inhomogeneous random planar tessellations generated by lines

Random planar tessellations in bounded convex windows are generated by dividing random cells with random lines. It is suggested that the random STIT tessellations of Nagel and Weiss, if restricted to a bounded convex window, can be interpreted as a special case.

Homometric point sets and inverse problems

Abstract The inverse problem of diffraction theory in essence amounts to the reconstruction of the atomic positions of a solid from its diffraction image. From a mathematical perspective, this is a notoriously difficult problem, even in the idealised situation of perfect diffraction from an infinite structure. Here, the problem is analysed via the autocorrelation measure of the underlying point set, where two point sets are called homometric when they share the same autocorrelation. For the class of mathematical quasicrystals within a given cut and project scheme, the homometry problem becomes equivalent to Matheron´s covariogram problem, in the sense of determining the window from its covariogram. Although certain uniqueness results are known for convex windows, interesting examples of distinct homometric model sets already emerge in the plane. The uncertainty level increases in the presence of diffuse scattering. Already in one dimension, a mixed spectrum can be compatible with structures of different entropy. We expand on this example by constructing a family of mixed systems with fixed diffraction image but varying entropy. We also outline how this generalises to higher dimension.

Estimating Mean Fracture Trace Length and Density from Observations in Convex Windows

Although window samples of fracture traces are widely recognized to be length-biased and censored, they are often the best source of data for inferring statistical parameters of fracture sets. This paper presents new estimators for mean fracture trace length and density that correct for the effects of bias and censoring. A stereological estimator of mean trace length is derived for parallel traces in a rectangular sampling window, an end-point estimator of mean trace length is derived for windows with arbitrary convex boundaries and for arbitrary trace length distributions, and an end-point estimator of trace density is derived for windows with arbitrary convex boundaries and for arbitrary trace length and trace orientation distributions. Results for rectangular and circular windows are obtained as special cases of the general solutions for arbitrary convex windows. When applied to circular windows, the end-point estimator of mean trace length is, in addition, independent of the trace orientation distribution. The estimators are easily determined from field data. The end-point estimator of trace density requires knowing only the area of the window and the number of end-points inside the window. The end-point estimator of mean trace length, when applied to circular windows, requires, in addition, the number of end-points outside the window (of those traces that intersect the window) and the stereological estimator of mean trace length requires only the apparent mean trace length and the height of the window.

O( lgN) line clipping algorithm in E2

A new O( lg N) line clipping algorithm in E 2 against a convex window is presented. The main advantage of the presented algorithm is the principal acceleration of the line clipping problem solution. A comparison of the proposed algorithm with others shows a significant improvement in run-time. Experimental results for selected known algorithms are also shown.

An efficient algorithm for line clipping by convex polygon

A new line clipping algorithm against convex window based on a new approach for intersection detection is presented. Theoretical comparisons with Cyrus-Beck's algorithm are shown together with experimental results obtained by simulations. The main advantage of the presented algorithm is the substantial acceleration of the line clipping problem solution and that edges can be oriented clockwise or anti-clockwise.

Simple and efficient 2D and 3D span clipping algorithms

A machine-independent comparison of several 2D and 3D line segment clipping algorithms is presented. The Nicholl-Lee-Nicholl (NLN), Liang-Barsky (LB), an improved implementation of the Cohen-Sutherland (CS), and Sutherland-Hodgman (SH) algorithms are compared for 2D. The LB, CS and SH algorithms are compared for 3D. The trivial acceptance of the Nicholl-Lee-Nicholl (NLN) algorithm and the Newman-Sproull (NS) implementation of the CS algorithm is proved to be nonoptimal. A new optimal trivial acceptance algorithm is presented, which is similar to the MIN/MAX algorithm. The NLN trivial rejection method is proved to be optimal in the worst case, but is shown to be nonoptimal in the average case. The NS implementation of the CS algorithm is improved by incorporating this optimal trivial rejection method. An alternate implementation of the CS algorithm is presented by incorporating the optimal trivial acceptance method. This method is shown to be more efficient than the implementation that incorporates the optimal trivial rejection method if more than 60% of the line segments are completely inside the clipping window for 2D, and more than 62.5% for 3D. The CS implementation is further improved by reducing the number of redundant comparisons, and the amount of floating-point computation per intersection point. The modifications are performed so as to allow easy extension of the 2D algorithm to 3D. Finally, the SH polygon clipping algorithm is reduced to line segment clipping. It is shown to be conceptually simple, compact, efficient, extendible to polygonal convex window shapes, and applicable to pipeline and systolic architectures.

An optimal algorithm to translate a convex polyhedron through a two-dimensional convex window

An optimal algorithm is proposed to determine all the directions for translating (single translation) a convex polyhedron P through a two-dimensional window Q. In this note we show that it is possible to determine all the directions for translating P through Q in optimal O( ¦P¦ + ¦Q¦) time. This improves upon the previous best result of O( ¦P¦ + ¦Q¦) (K. Tang, Comput. Vision Graphics Image Process. 45, 1989, 357–361).

The survival curve under monotone density constraints with applications to two-dimensional line segment processes

Several situations involving survival data are examined, in which the nonparametric maximum likelihood estimator is not the Kaplan-Meier estimator. Cases covered are inclusion of initial observations in field studies, estimation of the length distribution from convex window sampling of line segment processes in two dimensions, and estimation of a survival curve with monotone density. The first two cases are nonstandard because survival and censoring are not independent, and thus form simple tractable variable-sum models (Williams & Lagakos, 1977).