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In 1966, Zalgaller completed the task of determining all edge-to-edge tilings of the sphere by regular spherical polygons, proving that the only possibilities are the Platonic tilings, the Archimedean tilings, the Johnson tilings, the prism and anti-prism tilings and two tiles sharing their boundaries. We determine all non-edge-to-edge tilings of the sphere by regular spherical polygons of three or more sides, showing there are five continuous families of kaleidoscope tilings, fifteen continuous families of 2-hemisphere tilings, four lunar tilings, five sporadic tilings, five composed tilings and one magic triangle tiling. [ABSTRACT FROM AUTHOR]

Edge-to-edge tilings of the sphere by congruent a2bc-quadrilaterals are classified as 3 classes: (1) A 1-parameter family of quadrilateral subdivisions of the octahedron with 24 tiles, and a flip modification for one special parameter; (2) a 2-parameter family of 2-layer earth map tilings with 2n tiles for each n ≥ 3; (3) a 3-layer earth map tiling with 8n tiles for each n ≥ 2, and two flip modifications for each odd n. The authors also describe the moduli of parameterized tilings and provide the full geometric data for all tilings. [ABSTRACT FROM AUTHOR]

A family of spherical caps of the 2-dimensional unit sphere S 2 is called a totally separable packing in short, a TS-packing if any two spherical caps can be separated by a great circle which is disjoint from the interior of each spherical cap in the packing. The separable Tammes problem asks for the largest density of given number of congruent spherical caps forming a TS-packing in S 2 . We solve this problem up to eight spherical caps and upper bound the density of any TS-packing of congruent spherical caps in terms of their angular radius. Based on this, we show that the centered separable kissing number of unit balls in Euclidean 3-space is 8. Furthermore, we prove bounds for the maximum of the smallest inradius of the cells of the tilings generated by n > 1 great circles in S 2 . Next, we prove dual bounds for TS-coverings of S 2 by congruent spherical caps. Here a covering of S 2 by spherical caps is called a totally separable covering in short, a TS-covering if there exists a tiling generated by finitely many great circles of S 2 such that the cells of the tiling are covered by pairwise distinct spherical caps of the covering. Finally, we extend some of our bounds on TS-coverings to spherical spaces of dimension > 2 . [ABSTRACT FROM AUTHOR]

A tournament is an orientation of a graph. Each edge is a match, directed towards the winner. The score sequence lists the number of wins by each team. In this article, by interpreting score sequences geometrically, we generalize and extend classical theorems of Landau (Bull. Math. Biophys. 15, 143–148 (1953)) and Moon (Pac. J. Math. 13, 1343–1345 (1963)), via the theory of zonotopal tilings. [ABSTRACT FROM AUTHOR]

We apply Diophantine analysis to classify edge-to-edge tilings of the sphere by congruent almost equilateral quadrilaterals (i.e., edge combination a 3 b ). Parallel to a complete classification by Cheung, Luk, and Yan, the method implemented here is more systematic and applicable to other related tiling problems. We also provide detailed geometric data for the tilings. [ABSTRACT FROM AUTHOR]

Computer-generated aesthetic patterns are widely used as design materials in various fields. The most common methods use fractals or dynamical systems as basic tools to create various patterns. To enhance aesthetics and controllability, some researchers have introduced symmetric layouts along with these tools. One popular strategy employs dynamical systems compatible with symmetries that construct functions with the desired symmetries. However, these are typically confined to simple planar symmetries. The other generates symmetrical patterns under the constraints of tilings. Although it is slightly more flexible, it is restricted to small ranges of tilings and lacks textural variations. Thus, we proposed a new approach for generating aesthetic patterns by symmetrizing quasi-regular patterns using general k-uniform tilings. We adopted a unified strategy to construct invariant mappings for k-uniform tilings that can eliminate texture seams across the tiling edges. Furthermore, we constructed three types of symmetries associated with the patterns: dihedral, rotational, and reflection symmetries. The proposed method can be easily implemented using GPU shaders and is highly efficient and suitable for complicated tiling with regular polygons. Experiments demonstrated the advantages of our method over state-of-the-art methods in terms of flexibility in controlling the generation of patterns with various parameters as well as the diversity of textures and styles. [ABSTRACT FROM AUTHOR]

The kisrhombille tiling is the dual tessellation of one of the semi‐regular tessellations. It consists of right‐angled triangle tiles with 12 different orientations. An adequate coordinate system for the tiles of the grid has been defined that allows a formal description of the grid. In this paper, two tiles are considered to be neighbors if they share at least one point in their boundary. Paths are sequences of tiles such that any two consecutive tiles are neighbors. The digital distance is defined as the minimum number of steps in a path between the tiles, and the distance formula is proven through constructing minimum paths. In fact, the distance between triangles is almost twice the hexagonal distance of their embedding hexagons. [ABSTRACT FROM AUTHOR]

In this paper we define the notion of infinite or bounded fibre-like geodesic cylinder in S L 2 R ~ space, develop a method to determine its volume and total surface area. We prove that the common part of the above congruent fibre-like cylinders with the base plane are Euclidean circles and determine their radii. Using the former classified infinite or bounded congruent regular prism tilings with generating groups pq 2 1 we introduce the notions of cylinder packings, coverings and their densities. Moreover, we determine the densest packing, the thinnest covering cylinder arrangements in S L 2 R ~ space, their densities, their connections with the extremal hyperbolic circle arrangements and with the extremal fibre-like cylinder arrangements in H 3 and H 2 × R spaces. We prove that in these three previous Thurston geometries, the densities of the optimal fiber-like cylinder packings are equal and the same is true for optimal coverings. In our work we use the projective model of S L 2 R ~ introduced by Molnár (Beitr Algebra Geom 38(2):261–288, 1997). [ABSTRACT FROM AUTHOR]

A tiling of a topological disc by topological discs is called monohedral if all tiles are congruent. Maltby (J Comb Theory Ser A 66:40–52, 1994) characterized the monohedral tilings of a square by three topological discs. Kurusa et al. (Mediterr J Math 17:156, 2020) characterized the monohedral tilings of a circular disc by three topological discs. The aim of this note is to connect these two results by characterizing the monohedral tilings of any regular n-gon with at most three tiles for any n ≥ 5 . [ABSTRACT FROM AUTHOR]

Given graphs F and G, a perfect F-tiling in G is a collection of vertex-disjoint copies of F in G that together cover all the vertices in G. The study of the minimum degree threshold forcing a perfect F-tiling in a graph G has a long history, culminating in the Kühn--Osthus theorem [D. Kühn and D. Osthus, Combinatorica, 29 (2009), pp. 65-107] which resolves this problem, up to an additive constant, for all graphs F. In this paper we initiate the study of the analogous question for edge-ordered graphs. In particular, we characterize for which edge-ordered graphs F this problem is well-defined. We also apply the absorbing method to asymptotically determine the minimum degree threshold for forcing a perfect P-tiling in an edge-ordered graph, where P is any fixed monotone path. [ABSTRACT FROM AUTHOR]

We study the problem of perfect tiling in the plane and explore the possibility of tiling a rectangle using integral distinct squares. Assume a set of distinguishable squares (or equivalently a set of distinct natural numbers) is given, and one has to decide whether it can tile the plane or a rectangle or not. Previously, it has been proved that tiling the plane is not feasible using a set of odd numbers or an infinite sequence of natural numbers including exactly two odd numbers. The problem is open for different situations in which the number of odd numbers is arbitrary. In addition to providing a solution to this special case, we discuss some open problems to tile the plane and rectangles in this paper. [ABSTRACT FROM AUTHOR]

Quasiperiodic patterns and crystals—having long range order without translational symmetry—have fascinated researchers since their discovery. In this study, we report on new p‐terphenyl‐based T‐shaped facial polyphiles with two alkyl end chains and a glycerol‐based hydrogen‐bonded side group that self‐assemble into an aperiodic columnar liquid quasicrystal with 12‐fold symmetry and its periodic liquid‐crystalline approximants with complex superstructures. All represent honeycombs formed by the self‐assembly of the p‐terphenyls, dividing space into prismatic cells with polygonal cross‐sections. In the perspective of tiling patterns, the presence of unique trapezoidal tiles, consisting of three rigid sides formed by the p‐terphenyls and one shorter, incommensurate, and adjustable side by the alkyl end chains, plays a crucial role for these phases. A delicate temperature‐dependent balance between conformational, entropic and space‐filling effects determines the role of the alkyl chains, either as network nodes or trapezoid walls, thus resulting in the order‐disorder transitions associated with emergence of quasiperiodicity. In‐depth analysis suggests a change from a quasiperiodic tiling involving trapezoids to a modified one with a contribution of trapezoid pair fusion. This work paves the way for understanding quasiperiodicity emergence and develops fundamental concepts for its generation by chemical design of non‐spherical molecules, aggregates, and frameworks based on dynamic reticular chemistry. [ABSTRACT FROM AUTHOR]

In this paper we study uniformly random lozenge tilings of strip domains. Under the assumption that the limiting arctic boundary has at most one cusp, we prove a nearly optimal concentration estimate for the tiling height functions and arctic boundaries on such domains: with overwhelming probability the tiling height function is within n δ of its limit shape, and the tiling arctic boundary is within n 1 / 3 + δ to its limit shape, for arbitrarily small δ > 0 . This concentration result will be used in Aggarwal and Huang (Edge statistics for lozenge tilings of polygons, II: Airy line ensemble, 2021. arXiv:2108.12874) to prove that the edge statistics of simply-connected polygonal domains, subject to a technical assumption on their limit shape, converge to the Airy line ensemble. [ABSTRACT FROM AUTHOR]

The escalating demand for heat dissipation has prompted considerable interest in the manifold micro-pin–fin heat sink (MMPFHS) as a promising solution for high heat flux cooling applications. The two-dimensional topology of this multi-layered heat sink plays a pivotal role in organizing the three-dimensional spatial flow, consequently influencing both hydraulic and thermal performance. This study proposes and comparatively analyzes the only three rhombus-based tiling topologies applicable to MMPFHS, specifically the rhombus topology (RT), triangle hexies topology (THT), and firecracker topology (FT), all of which adhere to the principles of gapless, non-overlapping, and spatially expandable tiling topology. The relations of topological geometry and the hydraulic and thermal performances are explored and compared. And figure of merit (FOM) is also introduced to assess their overall performance. The results indicate that the overall performance of THT is predominant. When DPF = 100 μm, a = 200 μm, Din = 50, Re = 64.87, the temperature non-uniformity on the heating surface of THT is only 0.022 K, the thermal resistance is 9.4 × 10–6 km2 W−1, and the FOM of THT is about 1.14 times of that of RT. RT and FT have a similar performance, and the maximum deviation of FOM for both of them within the conditions studied in this paper is only 0.04. The paper provides a reference for selecting topologies in MMPFHS design. [ABSTRACT FROM AUTHOR]

We consider the dynamics of light rays in triangle tilings where triangles are transparent and adjacent triangles have equal but opposite indices of refraction. We find that the behavior of a trajectory on a triangle tiling is described by an orientation‐reversing three‐interval exchange transformation on the circle, and that the behavior of all the trajectories on a given triangle tiling is described by a polygon exchange transformation. We observe that, for a particular choice of triangle tiling, certain trajectories appear to approach the Rauzy fractal, under rescaling. [ABSTRACT FROM AUTHOR]

Albeverio, Kondratiev, and Röckner have introduced a type of differential geometry, which we call lifted geometry, for the configuration space ΓX of any manifold X. The name comes from the fact that various elements of the geometry of ΓX are constructed via lifting of the corresponding elements of the geometry of X. In this note, we construct a general algebraic framework for lifted geometry which can be applied to various "infinite dimensional spaces" associated to X. In order to define a lifted geometry for a "space", one dose not need any topology or local coordinate system on the space. As example and application, lifted geometry for spaces of Radon measures on X, mappings into X, embedded submanifolds of X, and tilings on X, are considered. The gradient operator in the lifted geometry of Radon measures is considered. Also, the construction of a natural Dirichlet form associated to a random measure is discussed. It is shown that Stokes' Theorem appears as "differentiability" of "boundary operator" in the lifted geometry of spaces of submanifolds. It is shown that (generalized) action functionals associated with Lagrangian densities on X form the algebra of smooth functions in a specific lifted geometry for the path-space of X. [ABSTRACT FROM AUTHOR]

For a k-vertex graph F and an n-vertex graph G, an F-tiling in G is a collection of vertex-disjoint copies of F in G. For r ∈ N, the r-independence number of G, denoted αr(G), is the largest size of a Kr-free set of vertices in G. In this article, we discuss Ramsey–Turán-type theorems for tilings where one is interested in minimum degree and independence number conditions (and the interaction between the two) that guarantee the existence of optimal F-tilings. [ABSTRACT FROM AUTHOR]

The paper offers a thorough study of multiorders and their applications to measure-preserving actions of countable amenable groups. By a multiorder on a countable group, we mean any probability measure ν on the collection O of linear orders of type Z on G, invariant under the natural action of G on such orders. Multiorders exist on any countable amenable group (and only on such groups) and every multiorder has the Følner property, meaning that almost surely the order intervals starting at the unit form a Følner sequence. Every free measure-preserving G-action (X,μ,G) has a multiorder (O,ν,G) as a factor and has the same orbits as the Z-action (X,μ,S), where S is the successor map determined by the multiorder factor. Moreover, the sub-sigma-algebra ΣO ​ associated with the multiorder factor is invariant under S, which makes the corresponding Z-action (O,ν,S) a factor of (X,μ,S). We prove that the entropy of any G-process generated by a finite partition of X, conditional with respect to ΣO ​, is preserved by the orbit equivalence with (X,μ,S). Furthermore, this entropy can be computed in terms of the so-called random past, by a formula analogous to h(μ,T,P)=H(μ,P∣P-) known for Z-actions. The above fact is then applied to prove a variant of a result by Rudolph and Weiss (2000). The original theorem states that orbit equivalence between free actions of countable amenable groups preserves conditional entropy with respect to a sub-sigma-algebra Σ, as soon as the "orbit change" is measurable with respect to Σ. In our variant, we replace the measurability assumption by a simpler one: Σ should be invariant under both actions and the actions on the resulting factor should be free. In conclusion, we provide a characterization of the Pinsker sigma-algebra of any G-process in terms of an appropriately defined remote past arising from a multiorder. The paper has an appendix in which we present an explicit construction of a particularly regular (uniformly Følner) multiorder based on an ordered dynamical tiling system of G. [ABSTRACT FROM AUTHOR]

First, we consider order- n ribbon tilings of an M -by- N rectangle R M , N where M and N are much larger than n. We prove the existence of the growth rate γ n of the number of tilings and show that γ n ≤ (n − 1) ln 2. Then, we study a rectangle R M , N with fixed width M = n , called a strip. We derive lower and upper bounds on the growth rate μ n for strips as ln n − 1 + o (1) ≤ μ n ≤ ln n. Besides, we construct a recursive system which enables us to enumerate the order- n ribbon tilings of a strip for all n ≤ 8 and calculate the corresponding generating functions. [ABSTRACT FROM AUTHOR]

This paper proposes a hybrid design technique for precast panels that incorporates soft-material formwork, computational form-finding, and the Truchet tiling concept. Initial experiments were conducted to investigate how flexible membranes, such as polyester, can create natural tension shapes that rigid molds cannot achieve. Compared to traditional metal and wooden formworks, fabric formwork offers greater sustainability due to its low cost, rapid fabrication, and ease of transport, making it a viable alternative for fabricating Truchet tile-based panels. The panels will be generated using regular polygons with multiple random curves inside, creating infinitely repeatable patterns. To simulate the continuity of the bump deflection, a parametric design script will be developed in conjunction with the concept of medial axis. Ultimately, an efficient and effective method of fabric formwork based on Truchet tiles will serve as a practical reference for future research. [ABSTRACT FROM AUTHOR]

We consider measurable functions f$f$ on R$\mathbb {R}$ that tile simultaneously by two arithmetic progressions αZ$\alpha \mathbb {Z}$ and βZ$\beta \mathbb {Z}$ at respective tiling levels p$p$ and q$q$. We are interested in two main questions: what are the possible values of the tiling levels p,q$p,q$, and what is the least possible measure of the support of f$f$? We obtain sharp results which show that the answers depend on arithmetic properties of α,β$\alpha , \beta$ and p,q$p,q$, and in particular, on whether the numbers α,β$\alpha , \beta$ are rationally independent or not. [ABSTRACT FROM AUTHOR]

Digital geometry is a field in the intersection of discrete mathematics and geometry having various applications including geographical information systems (GIS). In digital spaces, in grids, distances can be defined based on steps in paths in somewhat similarly as in graph theory. However, the grids have more definite structures, thus one may obtain more concrete results, for example, close formulae, than on arbitrary graphs. In this article, the weighted (also called chamfer) distances, and based on them, the distance transform are investigated on the regular triangular grid. Three types of neighborhood relations are used on the grid, and therefore, three weights are used to define a distance function. Natural conditions are used on the weights such as they are positive and a larger step (in the usual and also in the Euclidean sense) cannot have a smaller weight than a smaller one. Some properties of the weighted distances are discussed; for example, they are proven to be metrics. We also give algorithms and formulae that compute the weighted distance of any point pair on a triangular grid. Algorithm for weighted distance transform is provided based on wave‐front propagation. Therefore, these new distance functions are ready for further applications in GIS, in image processing tasks, in computer vision, in graphics, in networking, and also in other applied fields. [ABSTRACT FROM AUTHOR]

In this paper, we propose an integer linear programming model whose solutions are the aperiodic rhythms tiling with a given rhythm A. We show how it can be used to define an iterative algorithm that, given a period n, finds all the rhythms which tile with a given rhythm A and also to efficiently check the necessity of the Coven-Meyerowitz condition (T2). To conclude, we run several experiments to validate the time efficiency of the model. [ABSTRACT FROM AUTHOR]

A d-dimensional configuration c : Z d ⟶ A is a coloring of the d-dimensional infinite grid by elements of a finite alphabet A ⊆ Z . The configuration c has an annihilator if a non-trivial linear combination of finitely many translations of c is the zero configuration. Writing c as a d-variate formal power series, the annihilator is conveniently expressed as a d-variate Laurent polynomial f whose formal product with c is the zero power series. More generally, if the formal product is a strongly periodic configuration, we call the polynomial f a periodizer of c. A common annihilator (periodizer) of a set of configurations is called an annihilator (periodizer, respectively) of the set. In particular, we consider annihilators and periodizers of d-dimensional subshifts, that is, sets of configurations defined by disallowing some local patterns. We show that a (d - 1) -dimensional linear subspace S ⊆ R d is expansive for a subshift if the subshift has a periodizer whose support contains exactly one element of S. As a subshift is known to be finite if all (d - 1) -dimensional subspaces are expansive, we obtain a simple necessary condition on the periodizers that guarantees finiteness of a subshift or, equivalently, strong periodicity of a configuration. We provide examples in terms of tilings of Z d by translations of a single tile. [ABSTRACT FROM AUTHOR]

We study self-affine tiles generated by iterated function systems consisting of affine mappings whose linear parts are defined by different matrices. We obtain an interior theorem for these tiles. We prove a tiling theorem by showing that for such a self-affine tile, there always exists a tiling set. We also obtain a more complete interior theorem for reptiles, which are tiles obtained when the matrices in the iterated function system are similarities. Our results extend some of the classical ones by Lagarias and Wang (Adv. Math. 121(1), 21–49 (1996)), where the IFS maps are defined by a single matrix. [ABSTRACT FROM AUTHOR]

A self-affine tile is a compact set G ⊂ R d that admits a partition (tiling) by parallel shifts of the set M - 1 G , where M is an expanding matrix. We find all self-affine tiles which are polyhedral sets, i.e., unions of finitely many convex polyhedra. It is shown that there exists an infinite family of such polyhedral sets, not affinely equivalent to each other. A special attention is paid to integral self-affine tiles with standard digit sets, when the matrix M and the translation vectors are integer. Applications to the approximation theory and to the functional analysis are discussed. [ABSTRACT FROM AUTHOR]

We study random domino tilings of the Aztec diamond with a biased 2 × 2 periodic weight function and associate a linear flow on an elliptic curve to this model. Our main result is a double integral formula for the correlation kernel, in which the integrand is expressed in terms of this flow. For special choices of parameters the flow is periodic, and this allows us to perform a saddle point analysis for the correlation kernel. In these cases we compute the local correlations in the smooth disordered (or gaseous) region. The special example in which the flow has period six is worked out in more detail, and we show that in that case the boundary of the rough disordered region is an algebraic curve of degree eight. [ABSTRACT FROM AUTHOR]

The celebrated Pólya's conjecture (1954) in spectral geometry states that the eigenvalue counting functions of the Dirichlet and Neumann Laplacian on a bounded Euclidean domain can be estimated from above and below, respectively, by the leading term of Weyl's asymptotics. Pólya's conjecture is known to be true for domains which tile Euclidean space, and, in addition, for some special domains in higher dimensions. In this paper, we prove Pólya's conjecture for the disk, making it the first non-tiling planar domain for which the conjecture is verified. We also confirm Pólya's conjecture for arbitrary planar sectors, and, in the Dirichlet case, for balls of any dimension. Along the way, we develop the known links between the spectral problems in the disk and certain lattice counting problems. A key novel ingredient is the observation, made in recent work of the last named author, that the corresponding eigenvalue and lattice counting functions are related not only asymptotically, but in fact satisfy certain uniform bounds. Our proofs are purely analytic, except for a rigorous computer-assisted argument needed to cover the short interval of values of the spectral parameter in the case of the Neumann problem in the disk. [ABSTRACT FROM AUTHOR]

This paper describes a system which takes user input of a pattern of regular polygons around one vertex and attempts to construct a uniform tiling with the same pattern at every vertex by adding one polygon at a time. The system constructs spherical, planar, or hyperbolic tilings when the sum of the interior angles of the user-specified regular polygons is respectively less than, equal to, or greater than 360 ∘ . Other works have catalogued uniform tilings in tables and/or illustrations. In contrast, this system was developed as an interactive educational tool for people to learn about symmetry and tilings by trial and error through proposing potential vertex patterns and investigating whether they work. Users can watch the rest of the polygons being automatically added one by one with recursive backtracking. When a trial polygon addition is found to violate the conditions of a regular tiling, polygons are removed one by one until a configuration with another compatible choice is found, and that choice is tried next. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In the famous works "Tilings, substitution systems and dynamical systems generated by them" by S. Mozes and "Matching rules and substitution tilings" by Ch. Goodman-Strauss, important properties of tiling on the plane connecting the language of substitution systems and local rules are investigated. In particular, it is proved that with almost any substitution system of tiles, it is possible to link a system of decorations and local rules to the boundaries of tiles in such a way that any tiling of the plane allowed by the rules will belong to the family generated by this substitution system. Geometric constructions based on tiling and geometric complexes of glued squares open up new opportunities for the study and construction of algebraic objects. An important property of tiling that allows them to be associated with algebraic objects is determinicity. For square tiles whose sides are painted in a finite number of colors, this property means that the colors of the two adjacent sides of each tile uniquely determine the colors of the other pair of sides. J. Kari and P. Papasoglu constructed a deterministic aperiodic set. Thus, the analogue of the theorems of Goodman-Strauss and Mozes with an additional determinicity condition is an important statement in the theory of aperiodic mosaics. We prove an analog of this statement for a specific fixed substitution system. But its arbitrariness allows us to hope for the possibility of improving the construction for the general case. This system is important when constructing an infinite finitely presented nilsemigroup, which solves the problem of L. N. Shevrin and M. V. Sapir. The work is also devoted to the construction of a family of geometric complexes and the introduction of coding of vertices and edges on them. For the resulting complex, both properties are fulfilled simultaneously: finiteness of the set of defining local rules and determinicity on paths of length 2 on minimal squares. [ABSTRACT FROM AUTHOR]

We consider the five-vertex model on a regular square lattice of the size L × M with boundary conditions fixed in such a way that configurations of the model are in one-to-one correspondence with the lozenge tilings of the hexagon with a dent. We obtain two determinant representations for the partition function. In the free-fermionic limit, this result implies some summation formulae for Schur functions. [ABSTRACT FROM AUTHOR]

In this paper we study extremal behaviors of the mean to max ratio of the p-torsion function with respect to the geometry of the domain. For p larger than the dimension of the space N, we prove that the upper bound is uniformly below 1, contrary to the case p ∈ (1 , N ] . For p = + ∞ , in two dimensions, we prove that the upper bound is asymptotically attained by a disc from which is removed a network of points consisting on the vertices of a tiling of the plane with regular hexagons of vanishing size. [ABSTRACT FROM AUTHOR]

We show that $ \{0, \varphi+3, -3\varphi+2, -\varphi+\frac{5}{2}\} $ is the set of slopes of nonexpansive directions for a minimal subshift in the Jeandel-Rao Wang shift, where $ \varphi = (1+\sqrt{5})/2 $ is the golden mean. This set is a topological invariant allowing to distinguish the Jeandel-Rao Wang shift from other subshifts. Moreover, we describe the combinatorial structure of the two resolutions of the Conway worms along the nonexpansive directions in terms of irrational rotations of the unit interval. The introduction finishes with pictures of nonperiodic Wang tilings corresponding to what Conway called the cartwheel tiling in the context of Penrose tilings. The article concludes with open questions regarding the description of octopods and essential holes in the Jeandel-Rao Wang shift. [ABSTRACT FROM AUTHOR]

In 1966 Gallai asked the following question: Do all longest paths (cycles) of a connected graph contain a common vertex? After a positive answer to Gallai's question, another question has been raised, Is there any family of graphs without Gallai's property? Menke found one such family, the square lattices. Embedding methods hold the promise to transform not just the way calculations are performed, but to significantly reduce computational costs and often used in quantum mechanics and material sciences. In this paper, we prove the existence of graphs with the empty intersection of their longest cycles as subgraphs of Archimedean lattices. [ABSTRACT FROM AUTHOR]

The article deals with a combination of parallelization on distributed memory of iterative algorithms with skewed tiling and par-allelization on a multicore processor of each supercomputer node. This combination is possible due to the data distribution with overlaps. The justification of the correctness and efficiency of a combination of several methods for accelerating a sequential program is given. The analysis of the dependence of the methods of accelerating the program on the size of the tile is given. The results of numerical experiments of OpenMP-parallelization of the Gauss-Seidel algorithm for the 3D Dirichlet problem are presented, demonstrating the dependence of the acceleration on the height of a tile. [ABSTRACT FROM AUTHOR]

Optimal graph embeddings represent graphs in a lower dimensional space in a way that preserves the structure and properties of the original graph. These techniques have wide applications in fields such as machine learning, data mining, and network analysis. Do we have small (if possible minimal) k -connected graphs with the property that for any j vertices there is a longest path avoiding all of them? This question of Zamfirescu (1972) was the first variant of Gallai's question (1966): Do all longest paths in a connected graph share a common vertex? Several good examples answering Zamfirescu's question are known. In 2001, he asked to investigate the family of geometrical lattices with respect to this property. In 2017, Chang and Yuan proved the existence of such graphs in Archimedean tiling. Here, we prove that the graphs presented by Chang and Yuan are not optimal by constructing such graphs of sufficiently smaller orders. The problem of finding nonconcurrent longest paths in Archimedean tessellations refers to finding paths in a lattice such that the paths do not overlap or intersect with each other and are as long as possible. The complexity of embedding graph is still unknown. This problem can be challenging because it requires finding paths that are both long and do not intersect, which can be difficult due to the constraints of the lattice structure. [ABSTRACT FROM AUTHOR]

Given two graphs H and G, an H-tiling of G is a collection of vertex-disjoint copies of H in G and an H-factor is an H-tiling that covers all vertices of G. K\"uhn and Osthus managed to characterize, up to an additive constant, the minimum degree threshold which forces an H-factor in a host graph G. In this paper we study a similar tiling problem in a system that is locally bounded. An incompatibility system (G,\scrF) consists of a graph G and a family F = {Fv}v∊V(G) over G with Fv... We say that two edges e, et ∊ E(G) are incompatible if\{e,et} ∊ Fv for some v ∊ V (G), and otherwise compatible. A subgraph H of G is compatible if every pair of edges in H are compatible. An incompatibility system (G,F) is Δ-bounded if for any vertex v and any edge e incident with v, there are at most Δ members of Fv containing e. This notion was partly motivated by a concept of transition system introduced by Kotzig [Matematický časopis, 18 (1968), pp. 76--80], and first formulated by Krivelevich, Lee, and Sudakov [Combinatorica, 37 (2017), pp. 697--732], to study the robustness of Hamiltonicity of Dirac graphs. We prove that for any α >0 and any graph H with h vertices, there exists a constant mu >0 such that for any sufficiently large n with n ∊ hN, if G is an n-vertex graph with Δ (G)... and (G,F) is a\mu n-bounded incompatibility system, then there exists a compatible H-factor in G, where the value\chi\ast (H) is either the chromatic number χ (H) or the critical chromatic number χ cr(H) and we provide a dichotomy as in the K\"uhn--Osthus result. Moreover, we give examples H for which there exists an\mu n-bounded incompatibility system (G,F) with n ∊ hN and Δ (G)...n such that G contains no compatible H-factor. Unlike in the previous work of K\"uhn and Osthus on embedding H-factors, our proof uses the lattice-based absorption method. [ABSTRACT FROM AUTHOR]

An h-tiling on a finite simplicial complex is a partition of its geometric realization by maximal simplices deprived of several codimension one faces together with possibly their remaining face of highest codimension. In this last case, the tiles are said to be critical. An h-tiling thus induces a partitioning of its face poset by closed or semi-open intervals. We prove the existence of h-tilings on every finite simplicial complex after finitely many stellar subdivisions at maximal simplices. These tilings are moreover shellable. We also prove that the number of tiles of each type used by a tiling, encoded by its h-vector, is determined by the number of critical tiles of each index it uses, encoded by its critical vector. In the case of closed triangulated manifolds, these vectors satisfy some palindromic property. We finally study the behavior of tilings under any stellar subdivision. [ABSTRACT FROM AUTHOR]

A 3-isogonal tiling is an edge-to-edge tiling by regular polygons having 3 distinct transitivity classes of vertices. We know that there are sixty-one distinct 3-isogonal tilings on the plane. In this article, we discuss and determine the bounds of the vertex orbits of the plane's 3-isogonal lattices on the torus and will show that these bounds are sharp. [ABSTRACT FROM AUTHOR]

A polyomino is a polygonal region with axis-parallel edges and corners of integral coordinates, which may have holes. In this paper, we consider planar tiling and packing problems with polyomino pieces and a polyomino container P. We give polynomial-time algorithms for deciding if P can be tiled with k × k squares for any fixed k which can be part of the input (that is, deciding if P is the union of a set of non-overlapping k ×k squares) and for packing P with a maximum number of non-overlapping and axis-parallel 2 × 1 dominos, allowing rotations by 90 c. As packing is more general than tiling, the latter algorithm can also be used to decide if P can be tiled by 2 × 1 dominos. These are classical problems with important applications in VLSI design, and the related problem of finding a maximum packing of 2 × 2 squares is known to be NP-hard [6]. For our three problems there are known pseudo-polynomial-time algorithms, that is, algorithms with running times polynomial in the area or perimeter of P. However, the standard, compact way to represent a polygon is by listing the coordinates of the corners in binary. We use this representation, and thus present the first polynomial-time algorithms for the problems. Concretely, we give a simple O(n logn)-time algorithm for tiling with squares, where n is the number of corners of P. We then give a more involved algorithm that reduces the problems of packing and tiling with dominos to finding a maximum and perfect matching in a graph with O(n3) vertices. This leads to algorithms with running times O(n3 log3 n log2 log n) and O(n3 log2 n log log n), respectively. [ABSTRACT FROM AUTHOR]

This research employs shape grammars to generate variations of the Truchet tile to characterize a two-dimensional and three-dimensional generative tiling system. [ABSTRACT FROM AUTHOR]

By‐example aperiodic tilings are popular texture synthesis techniques that allow a fast, on‐the‐fly generation of unbounded and non‐periodic textures with an appearance matching an arbitrary input sample called the "exemplar". But by relying on uniform random sampling, these algorithms fail to preserve the autocovariance function, resulting in correlations that do not match the ones in the exemplar. The output can then be perceived as excessively random. In this work, we present a new method which can well preserve the autocovariance function of the exemplar. It consists in fetching contents with an importance sampler taking the explicit autocovariance function as the probability density function (pdf) of the sampler. Our method can be controlled for increasing or decreasing the randomness aspect of the texture. Besides significantly improving synthesis quality for classes of textures characterized by pronounced autocovariance functions, we moreover propose a real‐time tiling and blending scheme that permits the generation of high‐quality textures faster than former algorithms with minimal downsides by reducing the number of texture fetches. [ABSTRACT FROM AUTHOR]

Considerable energy has been devoted to understanding domino tilings: for example, Elkies, Kuperberg, Larsen and Propp proved the Aztec diamond theorem, which states that the number of domino tilings for the Aztec diamond of order n is equal to 2 n (n + 1) / 2 , and the authors recently counted the number of domino tilings for augmented Aztec rectangles and their chains by using Delannoy paths. In this paper, we count domino tilings for two new shapes of regions, bounded augmented Aztec rectangles and Aztec octagons by constructing a bijection between domino tilings for these regions and the associated generalized Motzkin paths. [ABSTRACT FROM AUTHOR]

We investigate Conway's Game of Life played on the Robinson triangle Penrose tiling. In this paper, we classify all four-cell still lifes. [ABSTRACT FROM AUTHOR]