scholarly journals Bijective, Non-Bijective and Semi-Bijective Translations on the Triangular Plane

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 29 ◽  
Author(s):  
Khaled Abuhmaidan ◽  
Benedek Nagy
Keyword(s):  
Triangular Grid ◽  
Discrete Structure ◽  
Neighborhood Structure ◽  
Square Grid ◽  
Original Shape ◽  
Digitally Continuous ◽  
Triangular Tessellation ◽  
Point Lattice

The triangular plane is the plane which is tiled by the regular triangular tessellation. The underlying discrete structure, the triangular grid, is not a point lattice. There are two types of triangle pixels. Their midpoints are assigned to them. By having a real-valued translation of the plane, the midpoints of the triangles may not be mapped to midpoints. This is the same also on the traditional square grid. However, the redigitized result on the square grid always gives a bijection (gridpoints of the square grid are mapped to gridpoints in a bijective way). This property does not necessarily hold on to the triangular plane, i.e., the redigitized translated points may not be mapped to the original points by a bijection. In this paper, we characterize the translation vectors that cause non bijective translations. Moreover, even if a translation by a vector results in a bijection after redigitization, the neighbor pixels of the original pixels may not be mapped to the neighbors of the resulting pixel, i.e., a bijective translation may not be digitally ‘continuous’. We call that type of translation semi-bijective. They are actually bijective but do not keep the neighborhood structure, and therefore, they seemingly destroy the original shape. We call translations strongly bijective if they are bijective and also the neighborhood structure is kept. Characterizations of semi- and strongly bijective translations are also given.

scholarly journals Vector Arithmetic in the Triangular Grid

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 373
Author(s):  
Khaled Abuhmaidan ◽  
Monther Aldwairi ◽  
Benedek Nagy
Keyword(s):  
Coordinate System ◽  
Triangular Grid ◽  
Vector Addition ◽  
Plane Space ◽  
Physical Simulations ◽  
Rectangular Grids ◽  
Cartesian Coordinate ◽  
Zero Sum ◽  
Coordinate Geometry ◽  
Point Lattice

Vector arithmetic is a base of (coordinate) geometry, physics and various other disciplines. The usual method is based on Cartesian coordinate-system which fits both to continuous plane/space and digital rectangular-grids. The triangular grid is also regular, but it is not a point lattice: it is not closed under vector-addition, which gives a challenge. The points of the triangular grid are represented by zero-sum and one-sum coordinate-triplets keeping the symmetry of the grid and reflecting the orientations of the triangles. This system is expanded to the plane using restrictions like, at least one of the coordinates is an integer and the sum of the three coordinates is in the interval [−1,1]. However, the vector arithmetic is still not straightforward; by purely adding two such vectors the result may not fulfill the above conditions. On the other hand, for various applications of digital grids, e.g., in image processing, cartography and physical simulations, one needs to do vector arithmetic. In this paper, we provide formulae that give the sum, difference and scalar product of vectors of the continuous coordinate system. Our work is essential for applications, e.g., to compute discrete rotations or interpolations of images on the triangular grid.

Neighborhood

2018 ◽  
Author(s):  
Emily Talen
Keyword(s):  
21St Century ◽  
Geographic Location ◽  
Social Contact ◽  
Neighborhood Structure ◽  
Urban Experience ◽  
Local Participation ◽  
Personal Influence ◽  
Social Mix ◽  
Traditional Sense ◽  
American Society

This book is written in support of those who believe that neighborhoods should be genuinely relevant in our lives, not as casual descriptors of geographic location but as places that provide an essential context for daily life. “Neighborhood” in its traditional sense—as a localized, place-based, delimited urban area that has some level of personal influence—seems a vanished part of the urban experience. This book explores whether 21st-century neighborhoods can once again provide a sense of caring and local participation and not devolve into enclaves seeking social insularity and separation. That the localized, diverse neighborhood has often failed to materialize requires thorough exploration. While many factors leading to the decline of the traditional neighborhood—e-commerce, suburban exclusivity, internet-based social contact—seem to be beyond anyone’s control, other factors seem more a product of neglect and confusion about neighborhood definition and its place in American society. Debates about the neighborhood have involved questions about social mix, serviceability, self-containment, centeredness, and connectivity within and without. This book works through these debates and proposes their resolution. The historical and global record shows that there are durable, time-tested regularities about neighborhoods. Many places outside of the West were built with neighborhood structure in evidence—long before professionalized, Western urban planning came on the scene. This book explores the compelling case that the American neighborhood can be connected to these traditions, anchored in human nature and regularities of form, and reinstated as something relevant and empowering in 21st-century urban experience.

scholarly journals Comparing Bayesian Spatial Conditional Overdispersion and the Besag–York–Mollié Models: Application to Infant Mortality Rates

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 282
Author(s):  
Mabel Morales-Otero ◽  
Vicente Núñez-Antón
Keyword(s):  
Infant Mortality ◽  
Count Data ◽  
Spatial Dependence ◽  
Poisson Model ◽  
Mortality Rates ◽  
Estimation Methods ◽  
Neighborhood Structure ◽  
Specific Context ◽  
Count Data Models ◽  
Conditional Mean

In this paper, we review overdispersed Bayesian generalized spatial conditional count data models. Their usefulness is illustrated with their application to infant mortality rates from Colombian regions and by comparing them with the widely used Besag–York–Mollié (BYM) models. These overdispersed models assume that excess of dispersion in the data may be partially caused from the possible spatial dependence existing among the different spatial units. Thus, specific regression structures are then proposed both for the conditional mean and for the dispersion parameter in the models, including covariates, as well as an assumed spatial neighborhood structure. We focus on the case of response variables following a Poisson distribution, specifically concentrating on the spatial generalized conditional normal overdispersion Poisson model. Models were fitted by making use of the Markov Chain Monte Carlo (MCMC) and Integrated Nested Laplace Approximation (INLA) algorithms in the specific context of Bayesian estimation methods.

scholarly journals Fractal-induced 2D flexible net undulation

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Michael Joon Seng Goh ◽  
Yeong Shiong Chiew ◽  
Ji Jinn Foo
Keyword(s):  
3D Reconstruction ◽  
Cross Section ◽  
Channel Cross Section ◽  
Time Varying ◽  
Blockage Ratio ◽  
Transient Time ◽  
Cross Sectional ◽  
Velocity Fluctuations ◽  
Square Grid ◽  
Multilength Scale

AbstractA net immersed in fractal-induced turbulence exhibit a transient time-varying deformation. The anisotropic, inhomogeneous square fractal grid (SFG) generated flow interacts with the flexible net to manifest as visible cross-sectional undulations. We hypothesize that the net’s response may provide a surrogate in expressing local turbulent strength. This is analysed as root-mean-squared velocity fluctuations in the net, displaying intensity patterns dependent on the grid conformation and grid-net separation. The net’s fluctuation strength is found to increase closer to the turbulator with higher thickness ratio while presenting stronger fluctuations compared to regular-square-grid (RSG) of equivalent blockage-ratio, σ. Our findings demonstrate a novel application where 3D-reconstruction of submerged nets is used to experimentally contrast the turbulence generated by RSG and multilength scale SFGs across the channel cross-section. The net’s response shows the unique turbulence developed from SFGs can induce 9 × higher average excitation to a net when compared against RSG of similar σ.

Chromous carbonates containing a square-grid layer of {Cr2(CO3)4}n4n− based on a dichromium(ii,ii) paddlewheel core

2021 ◽  
Vol 50 (7) ◽  
pp. 2387-2392
Author(s):  
Zhi-Qiang Dong ◽  
Jian-Hui Yang ◽  
Bin Liu
Keyword(s):  
Magnetic Properties ◽  
Layer Structure ◽  
Square Grid

The structural, spectroscopic and magnetic properties of chromous carbonates with a square-grid layer structure constructed from Cr2(CO3)44− paddlewheel units.

scholarly journals ON A FABRIC OF KISSING CIRCLES

2021 ◽  
pp. 1-10
Author(s):  
VIERA ČERŇANOVÁ
Keyword(s):  
Sufficient Condition ◽  
Arithmetic Sequence ◽  
Square Grid ◽  
The Individual

Abstract Applying circle inversion on a square grid filled with circles, we obtain a configuration that we call a fabric of kissing circles. We focus on the curvature inside the individual components of the fabric, which are two orthogonal frames and two orthogonal families of chains. We show that the curvatures of the frame circles form a doubly infinite arithmetic sequence (bi-sequence), whereas the curvatures in each chain are arranged in a quadratic bi-sequence. We also prove a sufficient condition for the fabric to be integral.

Phonological potentials and the lower vocal tract

2019 ◽  
pp. 1-35 ◽  
Author(s):  
Scott R. Moisik ◽  
Ewa Czaykowska-Higgins ◽  
John H. Esling
Keyword(s):  
Theoretical Approach ◽  
Vocal Tract ◽  
Speech Sound ◽  
Physical Nature ◽  
Discrete Structure ◽  
Complex Interactions

This paper outlines a theoretical approach to speech sound systems based on the notion of phonological potentials: physical ‘pressures’ or biases that give rise to discrete structure and the tendencies associated with this structure that arise from the physical nature of speech sound systems. We apply this approach to a poorly understood area of phonology – phenomena of the lower vocal tract (LVT) – through a schematic that encapsulates the complex interactions among the vocal tract structures responsible for producing LVT sounds. With the framework, we provide an account of a range of LVT phenomena from several languages, illustrating how tonal, phonatory, and vowel qualities interact. Finally, we consider how the idea of phonological potentials extends across various physical domains and might exhibit patterns of alignment across these domains, thereby serving to guide the formation of patterns found in speech sound systems.

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