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We define a second (higher) homotopy group for digital images. Namely, we construct a functor from digital images to abelian groups, which closely resembles the ordinary second homotopy group from algebraic topology. We illustrate that our approach can be effective by computing this (digital) second homotopy group for a digital 2-sphere. [ABSTRACT FROM AUTHOR]

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A (signed) graph is said to exhibit the strong reciprocal (anti-reciprocal) eigenvalue property (SR) (resp., (-SR)) if for any eigenvalue λ\lambda , it has 1λ\frac{1}{\lambda } (resp.,−1λ-\frac{1}{\lambda }) as an eigenvalue as well, with the same multiplicity. It is well known that the corona of a (signed) graph does have the property -SR, and if the graph has symmetric spectrum, then it also has the property SR. Therefore, it is interesting to identify (signed) graphs which are not corona graphs with the properties SR or -SR. Recently, a few constructions for unsigned graphs with the property -SR have been offered. In this article, we extend such constructions to signed graphs.

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]

We show that for every $\eta \gt 0$ every sufficiently large $n$ -vertex oriented graph $D$ of minimum semidegree exceeding $(1+\eta)\frac k2$ contains every balanced antidirected tree with $k$ edges and bounded maximum degree, if $k\ge \eta n$. In particular, this asymptotically confirms a conjecture of the first author for long antidirected paths and dense digraphs. Further, we show that in the same setting, $D$ contains every $k$ -edge antidirected subdivision of a sufficiently small complete graph, if the paths of the subdivision that have length $1$ or $2$ span a forest. As a special case, we can find all antidirected cycles of length at most $k$. Finally, we address a conjecture of Addario-Berry, Havet, Linhares Sales, Reed, and Thomassé for antidirected trees in digraphs. We show that this conjecture is asymptotically true in $n$ -vertex oriented graphs for all balanced antidirected trees of bounded maximum degree and of size linear in $n$. [ABSTRACT FROM AUTHOR]

The iterated function system (IFS) is important in different fields like image compression. An important feature of such systems is that they can be used to generate fractals. Yet, for the obtained fractals, it is difficult to locally control them to generate new ones with desired structures at specific places. In this paper, we gave an attempt to solve this problem based on a nonuniform multiple function system. For this, we first analyzed the multiple function systems needed in the generation of the final desired fractals. Based on such analysis, the final fractals with desired structures at specific places can be generated using the nonuniform multiple function system. Moreover, these two procedures were summarized into two algorithms for convenience. Examples were also given to illustrate the performance of the nonuniform multiple function system and the two algorithms in this paper.

Food plays a significant role in the environmental impacts of human activities. However, many agro-industrial processes are multi-product systems and their impacts need to be distributed between the different co-products in order to properly address two major issues: (1) prevention of food spoilage and food losses and (2) the eco-design of food systems, from processing up to recommendations for changes in Western diets. As a culturally and nutritionally central component of most human diets, milk is critical because processing is a preservation issue and most dairy products follow from separations, thereby generating co-products. Life Cycle Assessment (LCA) is a reference and standard method that allows quantification of the potential environmental impacts of a manufactured product throughout its life cycle. Application of the method requires foreground information on the system considered, as well as input and output flows that feed and exit the system. This data paper provides data related to the fractionation of milk into cream, casein, lactose and two whey protein ingredients at industrial scale, using up-to-date technologies used in French dairy factories in years 2000–2010s. Cleaning is included. Transcription of these input and output flows into a selection of processes in the Agribalyse 3.0.1 and Ecoinvent 3.8 databases is also provided. Application of the LCA method in its attributional approach leaves methodological choices up to the practitioner, such as subdivision of the system, allocation of the environmental burden where subdivision is not applied or not possible, and aggregation of the impacts. Therefore, this data paper also provides the allocation factors that are necessary to apply mass, dry matter, protein or economic allocation at every separation operation throughout the processing itinerary. Using the characterization method EF 3.0, this data paper provides the potential environmental impacts of the 5 co-products obtained with an initial input of 600 tons of raw milk, i.e., 63 tons of cream, 183 tons of wet casein, 90 tons of lactose, 1.7 ton of dried β-lactoglobulin and 0.3 ton of dried α-lactalbumin. The respective shares of the 5 co-products are calculated for each allocation rule. Finally, this data paper provides the potential environmental impacts for the manufacture of 1 kg of α-lactalbumin enriched ingredient, as the co-product with the longest process itinerary, with details of all intermediate input contributions as well as two possible aggregation rules: by step or by input type. The dataset participates in providing often confidential industrial-scale LCI data to the public. It will be helpful for the eco-design of future itineraries. In particular, it contributes to taking the fate of the co-products into account when using LCA for such eco-design.

Copyright of China Forest Products Industry is the property of China Forest Products Industry Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We prove that for every digraph F $F$ and every assignment of pairs of integers (re,qe)e∈A(F) ${({r}_{e},{q}_{e})}_{e\in A(F)}$ to its arcs there exists an integer N $N$ such that every digraph D $D$ with dichromatic number greater than N $N$ contains a subdivision of F $F$ in which e $e$ is subdivided into a directed path of length congruent to re ${r}_{e}$ modulo qe ${q}_{e}$, for every e∈A(F) $e\in A(F)$. This generalizes to the directed setting the analogous result by Thomassen for undirected graphs, and at the same time yields a novel short proof of his result. [ABSTRACT FROM AUTHOR]

PHT-splines are defined as polynomial splines over hierarchical T-meshes with very efficient local refinement properties. The original PHT-spline basis functions constructed by the truncation mechanism have a decay phenomenon, resulting in numerical instability. The non-decay basis functions are constructed as the B-splines that are defined on the 2 × 2 tensor product meshes associated with basis vertices in Kang et al., but at the cost of losing the partition of unity. In the field of finite element analysis and topology optimization, forming the partition of unity is the default ingredient for constructing basis functions of approximate spaces. In this paper, we will show that the non-decay PHT-spline basis functions proposed by Kang et al. can be appropriately modified to form a partition of unity. Each non-decay basis function is multiplied by a positive weight to form the weighted basis. The weights are solved such that the sum of weighted bases is equal to 1 on the domain. We provide two methods for calculating weights, based on geometric information of basis functions and the subdivision of PHT-splines. Weights are given in the form of explicit formulas and can be efficiently calculated. We also prove that the weights on the admissible hierarchical T-meshes are positive. [ABSTRACT FROM AUTHOR]

An odd graph is a finite graph all of whose vertices have odd degrees. A graph G $G$ is decomposable into k $k$ odd subgraphs if its edge set can be partitioned into k $k$ subsets each of which induces an odd subgraph of G $G$. The minimum value of k $k$ for which such a decomposition of G $G$ exists is the odd chromatic index, χo′(G) ${\chi }_{o}^{^{\prime} }(G)$, introduced by Pyber. For every k≥χo′(G) $k\ge {\chi }_{o}^{^{\prime} }(G)$, the graph G $G$ is said to be odd k $k$‐edge‐colorable. Apart from two particular exceptions, which are, respectively, odd 5‐ and odd 6‐edge‐colorable, the rest of connected loopless graphs are odd 4‐edge‐colorable, and moreover one of the color classes can be reduced to size ≤2 $\le 2$. In addition, it has been conjectured that an odd 4‐edge‐coloring with a color class of size at most 1 is always achievable. Atanasov et al. characterized the class of loopless subcubic graphs in terms of the value χo′(G)≤4 ${\chi }_{o}^{^{\prime} }(G)\le 4$. In this paper, we extend their result to a characterization of all loopless subdivisions of odd graphs in terms of the value of the odd chromatic index. This larger class S ${\mathscr{S}}$ is of a particular interest as it collects all "least instances" of nonodd graphs. As a prelude to our main result, we show that every connected graph G∈S $G\in {\mathscr{S}}$ requiring the maximum number of four colors, becomes odd 3‐edge‐colorable after removing a certain edge. Thus, we provide support for the mentioned conjecture by proving it for all subdivisions of odd graphs. The paper concludes with few problems for possible further work. [ABSTRACT FROM AUTHOR]

Consider a geodesic triangle on a surface of constant curvature and subdivide it recursively into four triangles by joining the midpoints of its edges. We show the existence of a uniform δ > 0 such that, at any step of the subdivision, all the triangle angles lie in the interval (δ , π - δ) . Additionally, we exhibit stabilising behaviours for both angles and lengths as this subdivision progresses. [ABSTRACT FROM AUTHOR]

La documentation foncière avignonnaise de la fin du Moyen Âge révèle, par les différentes manières de le nommer, de le diviser et de le délimiter, de précieuses informations sur la représentation et la perception de l’espace urbain. Depuis le premier inventaire de biens dressé en 1233 à l’initiative du podestat Perceval Doria jusqu’aux comptes des clavaires de la cour temporelle en passant par les terriers seigneuriaux de la seconde moitié du XIVe siècle, l’ensemble des registres est, en effet, progressivement doté d’une dimension spatiale, dont la qualification ne cesse de s’affiner au fil du temps. Et pour cause, les enjeux territoriaux qui se jouent dans la région conduisent les autorités à localiser de plus en plus précisément les biens qui dépendent de leur seigneurie. Les référents spatiaux utilisés constituent des indices de choix pour étudier tant la constitution des domaines directs que l’évolution des subdivisions de l’espace urbain durant les derniers siècles du Moyen Âge.

We exploit the temporal coherence of closed-form animated implicit surfaces by locally re-evaluating an octree-like discretization of the implicit field only as and where is necessary to rigorously maintain a global error invariant over time, thereby saving resources in static or slowly-evolving areas far from the motion where per-frame updates are not necessary. We treat implicit surface rendering as a special case of the continuous constraint satisfaction problem of set inversion, which seeks preimages of arbitrary sets under vector-valued functions. From this perspective, we formalize a temporally-coherent set inversion algorithm that localizes changes in the field by range-bounding its time derivatives using interval arithmetic. We implement our algorithm on the GPU using persistent thread scheduling and apply it to the scalar case of implicit surface and swept volume rendering where we achieve significant speedups in complex scenes with localized deformations like those found in games and modelling applications where interactivity is required and bounded-error approximation is acceptable. [ABSTRACT FROM AUTHOR]

Circulant networks are a very important and widely studied class of graphs due to their interesting and diverse applications in networking, facility location problems, and their symmetric properties. The structure of the graph ensures that it is symmetric about any line that cuts the graph into two equal parts. Due to this symmetric behavior, the resolvability of these graph becomes interning. Subdividing an edge means inserting a new vertex on the edge that divides it into two edges. The subdivision graph G is a graph formed by a series of edge subdivisions. In a graph, a resolving set is a set that uniquely identifies each vertex of the graph by its distance from the other vertices. A metric basis is a resolving set of minimum cardinality, and the number of elements in the metric basis is referred to as the metric dimension. This paper determines the minimum resolving set for the graphs H l [ 1 , k ] constructed from the circulant graph C l [ 1 , k ] by subdividing its edges. We also proved that, for k = 2 , 3 , this graph class has a constant metric dimension. [ABSTRACT FROM AUTHOR]

Graph operations have a major impact in the aspects of theory and empirical literature of the domain. For relating the molecular topology to any real chemical attribute, the conversion of the relevant details embedded into chemical structure to some numeric value becomes so vital which ultimately paves the way for the emergence of topological indices. Topological descriptor acts as an effective graph invariant in chemical graph theory associated with certain molecular structure. Recently, the study on the sombor index is initiated by I.Gutman [16]. In the article, we utilise combinatorial inequalities, including the general sum-connectivity index, the first general zagreb index and few other indices in their formulations, for the determination of bounds for sombor index for the F-sum operation of connected graphs. [ABSTRACT FROM AUTHOR]

Operations in graph theory have a significant influence in the theoretical and application aspect of the domain. Topological indices serve as a crucial component in chemical graph theory linked with some molecular structure. Recently, the study on the new graph product variants is initiated. In the article, the computation of some bounds for atom-bond connectivity index, inverse sum indeg index, geometric-arithmetic index and sombor index of graph operations notably the corona join product, subdivision vertex join product and the subdivision vertex-edge join is carried out. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Xinyang Normal University Natural Science Edition is the property of Journal of Xinyang Normal University Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we discuss about the pair difference cordial labeling behavior of subdivision of wheel and comb graphs. [ABSTRACT FROM AUTHOR]

χ -bounded classes are studied here in the context of star colorings and, more generally, χ p -colorings. This fits to a general scheme of sparsity and leads to natural extensions of the notion of bounded expansion class. In this paper we solve two conjectures related to star coloring (i.e. χ 2) boundedness. One of the conjectures is disproved and in fact we determine which weakening holds true. χ p -boundedness leads to more stability and we give structural characterizations of (strong and weak) χ p -bounded classes. We also generalize a result of Wood relating the chromatic number of a graph to the star chromatic number of its 1-subdivision. As an application of our characterizations, among other things, we show that for every odd integer g > 3 even hole-free graphs G contain at most φ (g , ω (G)) | G | holes of length g. [ABSTRACT FROM AUTHOR]

The quality of a subdivision scheme in the vicinity of a vertex or a face-centre is related to the eigenstructure of the subdivision matrix. When the scheme has the appropriate symmetries, a common technique, based on discrete Fourier transform, builds small complex matrices that ease the numerical analysis of the eigenelements using in particular their Fourier index. But the numerical analysis of the eigenelements remains difficult when matrix entries involve complex numbers and unknowns, for example, in cases where we are tuning a scheme. We present techniques to build similar small matrices, still associated with a Fourier index and whose eigenstructure is simply related to the full matrix, but which are real. They extend the known techniques to schemes which rotate the lattice and with vertices which do not lie topologically on symmetry axes of the studied vicinity of vertex or face centre. Our techniques make it easier to tune these subdivision schemes. We illustrate it with the analysis of the so-called Simplest Scheme at the centre of an n -sided face. Shifted Fourier Transform and vertex substitution (from Q to Q) allow us to build small real blocks whose eigenelements are simply related to those of the initial scheme, even when it rotates the lattice or involves intermediate vertices, like the Simplest Scheme.   • A set of techniques to build real Fourier blocks from a subdivision matrix. • Apply on schemes which rotate the lattice and with vertices outside of symmetry axes. • Illustrated with Peters and Reif's Simplest scheme around an n-sided face. [ABSTRACT FROM AUTHOR]

Subdivision schemes are used to generate smooth curves by iteratively refining an initial control polygon. The simplest such schemes are corner cutting schemes, which specify two distinct points on each edge of the current polygon and connect them to get the refined polygon, thus cutting off the corners of the current polygon. While de Boor (1987) shows that this process always converges to a Lipschitz continuous limit curve, no matter how the points on each edge are chosen, Gregory and Qu (1996) discover that the limit curve is continuously differentiable under certain constraints. We extend these results and show that the limit curve can even be curvature continuous for specific sequences of cut ratios. • Proof that non-uniform corner cutting schemes can generate curvature continuous limit curves. • Corner cutting rules for generating cubic B-splines as limit curves. • Corner cutting rules for generating cubic non-uniform γ -B-splines as limit curves. • Corner cutting rules for generating cubic non-uniform rational γ -B-splines as limit curves. [ABSTRACT FROM AUTHOR]

Many industrial food processes are multi-steps, multi-products systems, and sharing the environmental impacts produced by each step to each product is critical when implementing life cycle assessment (LCA) method. The objective of the study was therefore to investigate subdivision and different allocation rules as means of accounting for branched process itineraries such as the fractionation of milk into cream, casein, whey proteins and lactose as a case study. Depending on the mass, dry matter, protein or economic allocation, single products may or may not bear significant environmental impacts, thereby stressing the need for equally detailed inventories for all co-products. Aggregating the results by step or by input further helps identifying hotspots. Methodological choices in LCA of multifunctional systems are therefore strategic decisions that ultimately affect the eco-design of products, processes and food chains. • Sharing environmental impacts between up to 5 co-products is an issue in food systems. • Subdivision helps accounting for multi-product i.e. branched processing itineraries. • LCA's sensitivity to allocation or aggregation rules helps identifying the hotspots. • Transparent rules are necessary to compare products' environmental footprints. [ABSTRACT FROM AUTHOR]

With the advent of the era of automation, robotic arms have been more and more applied to industrial production and people􀆳s daily life. The use of robotic arms to classify and grasp objects is widely used in important fields such as hazardous material sorting and aerospace exploration. However, there are still some deficiencies in the realization of the classification of objects. For example, most classification systems mainly rely on the combination of robotic arms and vision. This single-vision classification system doesn't work well in low light or objects with similar appearance but belong to different working environments. In order to realize the subdivision of objects with similar appearance and different materials with the help of a robot arm, A depth camera and a pressure-sensitive sensor were used to comprehensively obtain the appearance, material and other information of the object. Then, these information data were transmitted to the dual input convolutional neural network. A robot arm was used to subdivision objects with similar appearance and different materials. The results show that the correct object recognition rate of the system reaches 98. 5% in the actual robot arm test environment. Compared with AlexNet and VGG16, these two traditional single visual classification models are improved by 35. 7% and 24% respectively. It can be seen that the neural network model combining visual and tactile information of objects can complete the task of subdivision of objects. [ABSTRACT FROM AUTHOR]

Globally, rangelands are undergoing rapid social-ecological changes, yet the scale of these changes is rarely measured. Fencing, sedentarization, and land conversion limit access by wildlife and livestock to vital resources such as water and seasonal forage, leading to rangeland degradation. In addition, these changes limit connectivity between wildlife sub-populations, triggering a spiral of decreasing biodiversity and weakening ecosystem function. Moreover, the combination of land privatization, sedentarization and fencing endangers pastoral livelihoods by reducing resilience to drought and diminishing livestock holdings per person. We provide a unique, urgent, and vital snapshot across >30,000 km2 of southern Kenya’s rangeland, covering four ecosystems renowned for their rich megafauna and pastoral people. We document and explore the drivers of extensive fencing (~40,000 km), the proliferation of livestock enclosures (>27,000), and the conversion of rangelands for cultivation (~1,500 km2). Our analyses were based on an open-access database recently synthesized for the region. Fencing is generally more prolific in areas that have been converted from community tenure to private title, especially where land values are raised by agricultural potential and proximity to Kenya’s capital, Nairobi. These factors drive the transfer of land ownership from traditional pastoralists to speculators, eventually resulting in the transformation of rangeland into agricultural, industrial and urban land uses. Space for wildlife (and traditional pastoralism) is limited on private, subdivided land, where livestock enclosures are at their highest density, and where there is less unfenced land and less untransformed land, compared to conservation areas and pastoral commons. Conflicting planning incentives, policies, and economic forces are driving unsustainable and potentially irreversible social-ecological transitions over unprecedented spatial scales. The lesson from southern Kenya is that a range of financial, policy and governance-related interventions are required to allow people and nature to coexist sustainably in African savannas.

The Gale–Nikaido–Debreu lemma plays an important role in establishing the existence of competitive equilibrium. In this paper, we use Sperner's lemma and basic elements of topology to prove the Gale–Nikaido–Debreu lemma. [ABSTRACT FROM AUTHOR]

Aim: The study aimed to analyze the weight and homogeneity of the parts of tablets containing carbamazepine and tablets with trazodone hydrochloride, obtained after subdivision with a kitchen knife. X-ray microtomography was used for homogeneity analysis. Methods: 30 tablets with carbamazepine and 30 tablets with trazodone hydrochloride were analyzed in terms of weight uniformity after subdivision. Then, seven tablets of each type were analyzed using an X-ray microtomography (Phoenix vǀtomeǀx, General Electric). The absorption of X-rays by an object is proportional to its density. In turn, measurement of the density of the analyzed object in a microtomographic image is the grayscale level. Based on the correlation between the grayscale value and the reference density, from the calibration phantom, we were able to determine the density of any area of the tablet's scan. Results: During the subdivision, the weight loss exceeded 3% for two carbamazepine tablets, while for trazodone tablets, none lost more than 3%, which is the limit recommended by Food and Drug Administration (FDA). As to the density of the tablets resulting from the microtomographic analysis, two of the whole tablets containing trazodone hydrochloride had a significantly higher density than the remainder (p < 0.001). Similarly, some differences in density were observed in the analysis of the density of tablets of carbamazepine (p = 0.008). Parts of one of the analyzed tablets with trazodone obtained after subdivision differed in terms of pixel brightness, thus density. On the other hand, the uniform density was observed for parts of the split tablets containing carbamazepine. Conclusions: Parts of the trazodone hydrochloride tablets obtained after subdivision differed in terms of homogeneity and weight. Microtomographic methods may be an interesting and useful method for evaluating the uniformity of compounds in solid dosage forms. [ABSTRACT FROM AUTHOR]

We prove that every graph of minimum degree at least d≥1 $d\ge 1$ contains a subdivision of some maximal 3‐degenerate graph of order d+1 $d+1$. This generalizes the classic results of Dirac (d=3 $d=3$) and Pelikán (d=4 $d=4$). We conjecture that for any planar maximal 3‐degenerate graph H $H$ of order d+1 $d+1$, every graph of minimum degree at least d $d$ contains a subdivision of H $H$. We verify this in the case H $H$ is P63 ${P}_{6}^{3}$ and P73 ${P}_{7}^{3}$. [ABSTRACT FROM AUTHOR]

This paper is devoted to the construction and analysis of some new families of n-point ternary subdivision schemes. Some members of the families were adapted to the presence of discontinuities converging to limit functions without Gibbs oscillations. We present a numerical comparison where we check the theoretical properties. [ABSTRACT FROM AUTHOR]

Context: Neutral landscape models are useful and popular tools for exploring effects of spatial patterns on ecological processes. Most neutral landscape models mimic natural landscape patterns that often consist of curved, complex, and sometimes fractal shapes. However, human-dominated landscapes often have a spatial rectilinear pattern that is highly aligned and dominated by straight lines and right angles. Objectives: As existing rectilinear neutral landscape models lack controls over either the size, position, orientation, and shape of the rectilinear patches, or do not recognise the hierarchical structure of patch formation in human-dominated landscapes, our objective was to create a neutral landscape model capable of meeting these requirements. Methods: We present binary space partitioning as a method that generates hierarchical and rectilinear neutral landscape models. In doing so we explain how to control the size, position, orientation, and shape of the rectilinear patches, as well as generate a tree that records the hierarchical patch structure. Results: Binary space partitioning succeeds in providing a simple, repeatable, process to generate a range of neutral landscape models for human-dominated landscapes. A large variety of landscape patterns can be efficiently produced from only a very small number of parameters. Conclusions: Binary space partitioning based neutral landscape models would be useful in representing many human-dominated landscapes. Their implementation is straightforward and should be easily understood, used, and developed by landscape ecologists. [ABSTRACT FROM AUTHOR]

Non-Uniform Rational B-splines Surfaces can be defined for any degrees and non-uniform knots, but existing subdivision surfaces are either uniform or of a fixed degree. The only existing non-uniform arbitrary degree subdivision is the scheme in Cashman et al. (2009). However, in order to improve the surface quality, the knot insertion strategy in Cashman et al. (2009) has the problem that the limit surface does not change continuously in terms of the perturbation of knot intervals. This paper solves this problem by introducing higher-degree hybrid non-uniform subdivision surfaces (HNUSS), where the first level refinement converts each valence n extraordinary point (EP) into a valence n face (Li et al., 2019). And then, the subdivision scheme can be defined with one step of splitting and several steps of averaging, where most rules are tensor-product of the arbitrary degree B-spline refinement rule with one double knot. We verify that higher-degree HNUSS limit surface is G 1 at the EPs if the knot intervals for the spoke edges of an EP are the same and has a higher order of continuity in other regions. In the absence of multiple knots at EPs, we provide a knot insertion strategy to create a uniform region around an EP. Additionally, numerical experiments show that the limit surface has satisfactory shape quality. • A higher-degrees hybrid non-uniform subdivision surface is introduced through splitting and averaging. • The new subdivision limit surface is proved to be G 1 -continuous at the extraordinary points and has a higher order of continuity in other regions. • The new subdivision limit surface has satisfactory geometric quality for non-uniform parameterization. [ABSTRACT FROM AUTHOR]

• A sub-pixel image subdivision scheme for incremental optical encoders is proposed. • The local gradient interpolation is introduced to locate the period with high precision. • A wide range of displacement measurements is achieved by counting pulse. • The experiment verified the high accuracy of the image grating. In the field of precision measurement, subdividing displacement within the grating period to achieve high precision and high resolution remains a significant challenge. For subdivision, traditional optical linear encoders need complex coding and complicated optical systems, which struggle to meet the demands of nanometer measurement. In this paper, based on the fundamental principles of grating displacement measurement, a high-precision peak positioning method from grating images is proposed for linear displacement measurements with nanometer resolution and submicron accuracy. By utilizing a local gradient interpolation algorithm to optimize curves, it could further improve the accuracy of displacement measurement. Experiments demonstrate that this method achieves a resolution of 2 nm and an accuracy of ± 0.1 μm in a range of 50 mm using a grating with a 20 μm period. This sets the groundwork for subdividing optical linear encoders further, the resolution and accuracy could be further improved by optimizing the subdivision method. [ABSTRACT FROM AUTHOR]

The four-point interpolatory scheme and the cubic B-spline are examples of the most well-known stationary subdivision procedures. They are based on the space of cubic polynomials and have their respective strengths and weaknesses. In this regard, the purpose of this study is to introduce a new type of subdivision scheme that integrates the advantages of both the four-point and the cubic B-spline schemes. The proposed scheme achieves approximation order 'four' and an improved smoothness C 2 , while keeping the same support of the basic limit function as the four-point scheme. Moreover, under some mild conditions, the new scheme has the properties of monotonicity and convexity preservation. Whereas most high-order shape-preserving schemes are non-linear and rather computationally complicated, the proposed scheme is linear and stationary. Several numerical examples are provided to illustrate the efficiency of our subdivision algorithm. [ABSTRACT FROM AUTHOR]

This article presents a modified butterfly subdivision scheme with improved smoothness over regular triangular meshes. The proposed technique is an approximating scheme with a tension parameter. It achieves fourth-order accuracy and generates C 2 limit surfaces for a suitable range of the parameter while maintaining the same support of the original butterfly scheme. To validate the theoretical results, some numerical examples are provided. [ABSTRACT FROM AUTHOR]

Circulant networks are a very important and widely studied class of graphs due to their interesting and diverse applications in networking, facility location problems, and their symmetric properties. The structure of the graph ensures that it is symmetric about any line that cuts the graph into two equal parts. Due to this symmetric behavior, the resolvability of these graph becomes interning. Subdividing an edge means inserting a new vertex on the edge that divides it into two edges. The subdivision graph G is a graph formed by a series of edge subdivisions. In a graph, a resolving set is a set that uniquely identifies each vertex of the graph by its distance from the other vertices. A metric basis is a resolving set of minimum cardinality, and the number of elements in the metric basis is referred to as the metric dimension. This paper determines the minimum resolving set for the graphs Hl[1,k] constructed from the circulant graph Cl[1,k] by subdividing its edges. We also proved that, for k=2,3, this graph class has a constant metric dimension.