Tiling Of Space Research Articles

Impact of Subsurface Drainage System Design on Nitrate Loss and Crop Production

Subsurface (or tile) drainage offers a valuable solution for enhancing crop productivity in poorly drained soils. However, this practice is also associated with significant nutrient leaching, which can contribute to water quality problems at the regional scale. This research presents the findings from a 4-year tile depth and spacing study in central Illinois that included three drain spacings (12.2, 18.3, and 24.4 m) and two drain depths (0.8 and 1.1 m) implemented in six plots under the corn and soybean rotation system (plots CS-1 and CS-3: 12.2 m spacing and 1.1 m depth, plots CS-2 and CS-4: 24.4 m spacing and 1.1 m depth, and plots CS-5 and CS-6 18.3 m spacing and 0.8 m depth). Our observations indicate that drain flow and NO3-N losses were higher in plots with narrower drain spacings, while plots with wider drain spacing showed reduced drain flow and NO3-N losses. Specifically, plots set up with drain spacings of 18.3 m and 24.4 m showed significant reductions in drain flow compared to plots featuring a 12.2 m drain spacing. Likewise, plots characterized by 18.3 m and 24.4 m drain spacings (except CS-4) showed better NO3-N retention and lower leaching losses than those with 12.2 m spacing (CS-1 and CS-3). Crop yield results over a 3-year period indicated that CS-2 (wider spacing plot) showed the highest productivity, with up to 13.6% higher yield compared to other plots. Furthermore, when comparing plots with the same drainage designs, CS-2 and CS-4 showed 5.1% to 2.6% higher corn yield (3-year average) compared to CS-1 and CS-3, and CS-5 and CS-6, respectively. Overall, a wider drainage system showed the capacity to export lower nutrient levels while concurrently enhancing productivity. These findings represent that optimizing tile drainage systems can effectively reduce nitrate losses while increasing crop productivity.

Geometrical Penrose tilings are characterized by their 1-atlas

Penrose rhombus tilings are tilings of the plane by two decorated rhombi such that the decorations match at the junction between two tiles (like in a jigsaw puzzle). In dynamical terms, they form a tiling space of finite type. If we remove the decorations, we get, by definition, a sofic tiling space that we here call geometrical Penrose tilings. Here, we show how to compute the patterns of a given size which appear in these tilings by three different methods: two based on the substitutive structure of the Penrose tilings and the last on their definition by the cut and projection method. We use this to prove that the geometrical Penrose tilings are characterized by a small set of patterns called vertex-atlas, i.e., they form a tiling space of finite type. Though considered as folklore, no complete proof of this result has been published, to our knowledge.

Soft cells and the geometry of seashells.

A central problem of geometry is the tiling of space with simple structures. The classical solutions, such as triangles, squares, and hexagons in the plane and cubes and other polyhedra in three-dimensional space are built with sharp corners and flat faces. However, many tilings in Nature are characterized by shapes with curved edges, nonflat faces, and few, if any, sharp corners. An important question is then to relate prototypical sharp tilings to softer natural shapes. Here, we solve this problem by introducing a new class of shapes, the soft cells, minimizing the number of sharp corners and filling space as soft tilings. We prove that an infinite class of polyhedral tilings can be smoothly deformed into soft tilings and we construct the soft versions of all Dirichlet-Voronoi cells associated with point lattices in two and three dimensions. Remarkably, these ideal soft shapes, born out of geometry, are found abundantly in nature, from cells to shells.

Chiral Modulations in Non-Heisenberg Models of Non-Centrosymmetric Magnets Near the Ordering Temperatures

The structure of skyrmion and spiral solutions, investigated within the phenomenological Dzyaloshinskii model of chiral magnets near the ordering temperatures, is characterized by the strong interplay between longitudinal and angular order parameters, which may be responsible for experimentally observed precursor effects. Within the precursor regions, additional effects, such as pressure, electric fields, chemical doping, uniaxial strains and/or magnetocrystalline anisotropies, modify the energetic landscape and may even lead to the stability of such exotic phases as a square staggered lattice of half-skyrmions, the internal structure of which employs the concept of the “soft” modulus and contains points with zero modulus value. Here, we additionally alter the stiffness of the magnetization modulus to favor one- and two-dimensional modulated states with large modulations of the order parameter magnitude. The computed phase diagram, which omits any additional effects, exhibits stability pockets with a square half-skyrmion lattice, a hexagonal skyrmion lattice with the magnetization in the center of the cells parallel to the applied magnetic field, and helicoids with propagation transverse to the field, i.e., those phases in which the notion of localized defects is replaced by the picture of a smooth but more complex tiling of space. We note that the results can be adapted to metallic glasses, in which the energy contributions are the same and originate from the inherent frustration in the models, and chiral liquid crystals with a different ratio of elastic constants.

Tiling iterated function systems

This paper presents a detailed symbolic approach to the study of self-similar tilings. It uses properties of addresses associated with graph-directed iterated function systems to establish conjugacy properties of tiling spaces. Tiles may be fractals and the tiled set maybe a complicated unbounded subset of RM.

K-theory of two-dimensional substitution tiling spaces from AF-algebras

Given a two-dimensional substitution tiling space, we show that, under some reasonable assumptions, the K-theory of the groupoid C⁎-algebra of its unstable groupoid can be explicitly reconstructed from the K-theory of the AF-algebras of the substitution rule and its analogue on the 1-skeleton. We prove this by generalizing the calculations done for the chair tiling in [18] using relative K-theory and excision, and packaging the result into an exact sequence purely in topology. From this exact sequence, it appears that one cannot use only ordinary K-theory to compute using the dimension-filtration on the unstable groupoid. Several examples are computed using Sage and the results are compiled in a table.

DRG afferents that mediate physiologic and pathologic mechanosensation from the distal colon

The properties of dorsal root ganglia (DRG) neurons that innervate the distal colon are poorly defined, hindering our understanding of their roles in normal physiology and gastrointestinal (GI) disease. Here, we report genetically defined subsets of colon-innervating DRG neurons with diverse morphologic and physiologic properties. Four colon-innervating DRG neuron populations are mechanosensitive and exhibit distinct force thresholds to colon distension. The highest threshold population, selectively labeled using Bmpr1b genetic tools, is necessary and sufficient for behavioral responses to high colon distension, which is partly mediated by the mechanosensory ion channel Piezo2. This Aδ-HTMR population mediates behavioral over-reactivity to colon distension caused by inflammation in a model of inflammatory bowel disease. Thus, like cutaneous DRG mechanoreceptor populations, colon-innervating mechanoreceptors exhibit distinct anatomical and physiological properties and tile force threshold space, and genetically defined colon-innervating HTMRs mediate pathophysiological responses to colon distension, revealing a target population for therapeutic intervention.

Projective tilings and full-rank perfect codes

A tiling of a vector space S is the pair (U, V) of its subsets such that every vector in S is uniquely represented as the sum of a vector from U and a vector from V. A tiling is connected to a perfect codes if one of the sets, say U, is projective, i.e., the union of one-dimensional subspaces of S. A tiling (U, V) is full-rank if the affine span of each of U, V is S. For finite non-binary vector spaces of dimension at least 6 (at least 10), we construct full-rank tilings (U, V) with projective U (both U and V, respectively). In particular, that construction gives a full-rank ternary 1-perfect code of length 13, solving a known problem. We also discuss the treatment of tilings with projective components as factorizations of projective spaces.

An Upper bound on the growth of Dirichlet tilings of hyperbolic spaces

An Upper bound on the growth of Dirichlet tilings of hyperbolic spaces

Advanced practice-aided tile drain configuration: A solution to achieving environmentally sustainable agricultural production

Alternatives to conventional crop production practices are needed to achieve environmental sustainability. Because tile-drained crop production systems in the Midwestern United States are one of the single largest contributors to environmental degradation, we explored alternative tile drain configurations by changing the tile spacing and depth of tile-drained fields in central Illinois and then by evaluating comprehensively the impacts of different tile drain configurations on crop productivity and environmental sustainability using the Root Zone Water Quality Model 2 (RZWQM2). An analytic hierarchy process (AHP), which is a multicriteria decision analysis (MCDA) tool, was used to evaluate the optimum tile spacing and depth scenario. The model simulations and AHP tested three management schemes designated as “agricultural productivity” (AP), “environmental sustainability” (ES), and “advance in technology” (AT). The schemes were designed based on three criteria: operational cost, crop profit, and environmental control. Analysis showed that deeper tiles and narrower tile spacing resulted in decreases in flow and nitrate loss from runoff, seepage (SP), and lateral flow (LF), and increases in tile flow, total nitrate loss, and crop yields. On the basis of MCDA, the AP management scheme resulted in higher CP and total nitrate loss, whereas opposite outcomes were generated by the ES management scheme. Under the AT management scheme, higher ranked scenarios resulted in higher CP and lower total amounts of nitrate entering streams. Our analysis indicates that the AT scheme can simultaneously enhance crop production and ES through advances in tile drainage water treatment.

Exploiting Anyonic Behavior of Quasicrystals for Topological Quantum Computing

The concrete realization of topological quantum computing using low-dimensional quasiparticles, known as anyons, remains one of the important challenges of quantum computing. A topological quantum computing platform promises to deliver more robust qubits with additional hardware-level protection against errors that could lead to the desired large-scale quantum computation. We propose quasicrystal materials as such a natural platform and show that they exhibit anyonic behavior that can be used for topological quantum computing. Different from anyons, quasicrystals are already implemented in laboratories. In particular, we study the correspondence between the fusion Hilbert spaces of the simplest non-abelian anyon, the Fibonacci anyons, and the tiling spaces of the one-dimensional Fibonacci chain and the two-dimensional Penrose tiling quasicrystals. A concrete encoding on these tiling spaces of topological quantum information processing is also presented by making use of inflation and deflation of such tiling spaces. While we outline the theoretical basis for such a platform, details on the physical implementation remain open.

A Number System with Base

In the present article we explore a way to represent numbers with respect to the base using the set of digits {0, 1, 2}. Although this number system shares several properties with the classical decimal system, it shows some remarkable differences and reveals interesting new features. For instance, it is related to the field of 2-adic numbers, and to a self-affine “fractal” set that gives rise to a tiling of a non-Euclidean space. Moreover, it has relations to Mahler’s -problem and to the Josephus problem.

On the connectivity of spaces of three-dimensional domino tilings

We consider domino tilings of three-dimensional cubiculated manifolds with or without boundary, including subsets of Euclidean space and three-dimensional tori. In particular, we are interested in the connected components of the space of tilings of such regions under local moves. Building on the work of the third and fourth authors we allow two possible local moves, the flip and trit. These moves are considered with respect to two topological invariants, the twist and flux. Our main result proves that, up to refinement, ∙ \bullet \; Two tilings are connected by flips and trits if and only if they have the same flux. ∙ \bullet \; Two tilings are connected by flips alone if and only if they have the same flux and twist.

On the frequency module of the hull of a primitive substitution tiling

Understanding the properties of tilings is of increasing relevance to the study of aperiodic tilings and tiling spaces. This work considers the statistical properties of the hull of a primitive substitution tiling, where the hull is the family of all substitution tilings with respect to the substitution. A method is presented on how to arrive at the frequency module of the hull of a primitive substitution tiling (the minimal {\bb Z}-module, where {\bb Z} is the set of integers) containing the absolute frequency of each of its patches. The method involves deriving the tiling's edge types and vertex stars; in the process, a new substitution is introduced on a reconstructed set of prototiles.

Cohomology groups for spaces of 12-fold tilings

We consider tilings of the plane with 12-fold symmetry obtained by the cut and projection method. We compute their cohomology groups using the techniques introduced in [5]. To do this we completely describe the window, the orbits of lines under the group action and the orbits of 0-singularities. The complete family of generalized 12-fold tilings can be described using 2-parameters and it presents a surprisingly rich cohomological structure. To put this finding into perspective, one should compare our results with the cohomology of the generalized 5-fold tilings (more commonly known as generalized Penrose tilings). In this case the tilings form a 1-parameter family, which fits in simply one of two types of cohomology.

Пространственные геометрические ячейки – квазимногогранники

Tiling of three-dimensional space is a very interesting and not yet fully explored type of tiling. Tiling by convex polyhedra has been partially investigated, for example, works [1, 15, 20] are devoted to tiling by various tetrahedra, once tiling realized by Platonic, Archimedean and Catalan bodies. The use of tiling begins from ancient times, on the plane with the creation of parquet floors and ornaments, in space - with the construction of houses, but even now new and new areas of applications of tiling are opening up, for example, a recent cycle of work on the use of tiling for packaging information [17]. Until now, tiling in space has been considered almost always by faceted bodies. Bodies bounded by compartments of curved surfaces are poorly considered and by themselves, one can recall the osohedra [14], dihedra, oloids, biconuses, sphericon [21], the Steinmetz figure [22], quasipolyhedra bounded by compartments of hyperbolic paraboloids described in [3] the astroid ellipsoid and hyperbolic tetrahedra, cubes, octahedra mentioned in [6], and tiling bodies with bounded curved surfaces was practically not considered, except for the infinite three-dimensional Schwartz surfaces, but they were also considered as surfaces, not as bodies., although, of course, in each such surface, you can select an elementary cell and fill it with a body, resulting in a geometric cell. With this work, we tried to eliminate this gap and described approaches to identifying geometric cells bounded by compartments of curved surfaces. The concept of tightly packed frameworks is formulated and an approach for their identification are described. A graphical algorithm for identifying polyhedra and quasipolyhedra - geometric cells are described.

TEA-SEA: Tiling and scheduling of non-uniform two-level perfectly nested loops using an evolutionary approach

Nowadays, in scientific and computational programs, increasing the execution speed of programs is very important. In implementing scientific programs, loops are dedicating a large part of time to themselves. Therefore, the parallel execution of the iteration loop reduces the runtime of all programs. This paper has been working on the parallelism of two-level perfectly nested loops with non-uniform dependencies and rectangular iteration space, and then their scheduling is based on the scheduling of task graphs. For this purpose, the iteration space of non-uniform loop is partitioned using a tiling transformation, and then the tiled space is modelled as a task, graph for scheduling. In the task graph, vertices expressive tiles and edges expressive data dependencies inter-tiles. It should be noted that the computational load of vertices is not the same, and the vertices corresponding imperfect tiles have a less computational load. The scheduling is done on both types of homogeneous and heterogeneous multiprocessor systems. The tiling of iteration space loop and the scheduling of task graph is arranged in the category of NP-Hard problems that using evolutionary algorithms can be efficient to discover a suitable solution for them. The results of the proposed approach show the performance of this method better than previous methods in the tiling and scheduling of non-uniform loops. According to the examinations performed on uniform loops, this algorithm also performs better than previous algorithms on these types of loops.

Creating semiflows on simplicial complexes from combinatorial vector fields

Combinatorial vector fields on simplicial complexes introduced by Robin Forman constitute a combinatorial analogue of classical flows. They have found numerous and varied applications in recent years. Yet, their formal relationship to classical dynamical systems has been less clear. In this paper we prove that for every combinatorial vector field on a finite simplicial complex X one can construct a semiflow on the underlying polytope X which exhibits the same dynamics. The equivalence of the dynamical behavior is established in the sense of Conley-Morse graphs and uses a tiling of the topological space X which makes it possible to directly construct isolating blocks for all involved isolated invariant sets based purely on the combinatorial information.

A characterization of metric subspaces of full Assouad dimension

We introduce the notion of tiling spaces for metric spaces. The class of tiling spaces contains the Euclidean spaces, the middle-third Cantor set, and various self-similar spaces appearing in fractal geometry. For doubling tiling spaces, we characterize metric subspaces whose Assouad dimension coincides with that of the whole space.

Binomial ideals of domino tilings

In this paper, we consider the set of all domino tilings of a cubiculated region. The primary question we explore is: How can we move from one tiling to another? Tiling spaces can be viewed as spaces of subgraphs of a fixed graph with a fixed degree sequence. Moves to connect such spaces have been explored in algebraic statistics. Thus, we approach this question from an applied algebra viewpoint, making new connections between domino tilings, algebraic statistics, and toric algebra. Using results from toric ideals of graphs, we are able to describe moves that connect the tiling space of a given cubiculated region of any dimension. This is done by studying binomials that arise from two distinct domino tilings of the same region. Additionally, we introduce tiling ideals and flip ideals and use these ideals to restate what it means for a tiling space to be flip connected. Finally, we show that if R is a 2-dimensional simply connected cubiculated region, any binomial arising from two distinct tilings of R can be written in terms of quadratic binomials. As a corollary to our main result, we obtain an alternative proof to the fact that the set of domino tilings of a 2-dimensional simply connected region is connected by flips.