scholarly journals Informational Measure of Symmetry vs. Voronoi Entropy and Continuous Measure of Entropy of the Penrose Tiling. Part II of the “Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling”

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2146
Author(s):  
Edward Bormashenko ◽  
Irina Legchenkova ◽  
Mark Frenkel ◽  
Nir Shvalb ◽  
Shraga Shoval
Keyword(s):  
Symmetry Operation ◽  
Continuous Measure ◽  
Penrose Tiling ◽  
Measure Of Symmetry ◽  
2D Pattern ◽  
Equilateral Triangles ◽  
The Given ◽  
The Ideal

The notion of the informational measure of symmetry is introduced according to: Hsym(G)=−∑i=1kP(Gi)lnP(Gi), where P(Gi) is the probability of appearance of the symmetry operation Gi within the given 2D pattern. Hsym(G) is interpreted as an averaged uncertainty in the presence of symmetry elements from the group G in the given pattern. The informational measure of symmetry of the “ideal” pattern built of identical equilateral triangles is established as Hsym(D3)= 1.792. The informational measure of symmetry of the random, completely disordered pattern is zero, Hsym=0. The informational measure of symmetry is calculated for the patterns generated by the P3 Penrose tessellation. The informational measure of symmetry does not correlate with either the Voronoi entropy of the studied patterns nor with the continuous measure of symmetry of the patterns. Quantification of the “ordering” in 2D patterns performed solely with the Voronoi entropy is misleading and erroneous.

scholarly journals Informational Measure of Symmetry vs. Voronoi Entropy and Continuous Measure of Entropy of the Penrose Tiling. Part II of the “Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling”.

2021 ◽  
Author(s):  
Edward Bormashenko ◽  
Irina Legchenkova ◽  
Mark Frenkel ◽  
Nir Shvalb ◽  
Shraga Shoval
Keyword(s):  
Symmetry Operation ◽  
Continuous Measure ◽  
Penrose Tiling ◽  
Measure Of Symmetry ◽  
2D Pattern ◽  
Equilateral Triangles ◽  
The Given

The notion of the informational measure of symmetry is introduced according to: HsymG=-i=1kPGilnPGi, where PGi is the probability of appearance of the symmetry operation Gi within the given 2D pattern. HsymG is interpreted as an averaged uncertainty in the presence of symmetry elements from the group G in the given pattern. The informational measure of symmetry of the “ideal” pattern built of identical equilateral triangles is established as HsymD3=1.792. The informational measure of symmetry of the random, completely disordered pattern is zero, Hsym=0. Informational measure of symmetry is calculated for the patterns generated by the P3 Penrose tessellation. Informational measure of symmetry does not correlate neither with the Voronoi entropy of the studied patterns nor with the continuous measure of symmetry of the patterns.

scholarly journals Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling: Part I. Analysis of the Voronoi Diagrams

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1659
Author(s):  
Edward Bormashenko ◽  
Irina Legchenkova ◽  
Mark Frenkel ◽  
Nir Shvalb ◽  
Shraga Shoval
Keyword(s):  
Symmetry Group ◽  
Voronoi Diagrams ◽  
Continuous Measure ◽  
Penrose Tiling ◽  
Continuous Symmetry ◽  
Measure Of Symmetry ◽  
Symmetry Measure ◽  
Seed Points

A continuous measure of symmetry and the Voronoi entropy of 2D patterns representing Voronoi diagrams emerging from the Penrose tiling were calculated. A given Penrose tiling gives rise to a diversity of the Voronoi diagrams when the centers, vertices, and the centers of the edges of the Penrose rhombs are taken as the seed points (or nuclei). Voronoi diagrams keep the initial symmetry group of the Penrose tiling. We demonstrate that the continuous symmetry measure and the Voronoi entropy of the studied sets of points, generated by the Penrose tiling, do not necessarily correlate. Voronoi diagrams emerging from the centers of the edges of the Penrose rhombs, considered nuclei, deny the hypothesis that the continuous measure of symmetry and the Voronoi entropy are always correlated. The Voronoi entropy of this kind of tiling built of asymmetric convex quadrangles equals zero, whereas the continuous measure of symmetry of this pattern is high. Voronoi diagrams generate new types of Penrose tiling, which are different from the classical Penrose tessellation.

Ahiolohiya poetychnoyi zbirky Lazarya Baranovycha Żywoty świętych (1670)

2020 ◽  
pp. 249-258
Author(s):  
Olena Ruda
Keyword(s):  
Family Life ◽  
18Th Century ◽  
The Given ◽  
The Ideal ◽  
Poetry Collection

The purpose of the article is the analysis of hagiology in Lazar Baranovych’s poetry collection entitled Żywoty świętych (1670). This includes the fulfi lment of such tasks: 1) To enumerate the saints mentioned in the poetry collection; 2) To determine to which church/epoch/place of worship or order of sainthood they belong; 3) To determine how full the saints’ details of biography are refl ected in the poetry collection mentioned above; 4) To understand Lazar Baranovych’s view on the topic of diff erent kinds of sainthood clearly; 5) To measure the actuality of his views given the context of the 18th century Ukraine. The results of the research are shared in the given article, showing how exactly Lazar Baranovych defi ned for himself the concept of the sainthood at the fi rst place. They also tell us about his views on the call for monkhood and family life and help us to reconstruct the images of the ideal spiritual shepherd, female Christian etc.

scholarly journals Malmquist Productivity Index by Extended VIKOR Method Using Interval Numbers

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Mohammad Fallah ◽  
Amir Mohajeri ◽  
Esmaeil Najafi
Keyword(s):  
Multicriteria Decision Making ◽  
Malmquist Productivity Index ◽  
Productivity Index ◽  
Ideal Solution ◽  
Vikor Method ◽  
Decision Matrix ◽  
Interval Numbers ◽  
Productivity Rate ◽  
The Given ◽  
The Ideal

The VIKOR method was developed for multicriteria optimization of complex systems. It determines the compromise ranking list and the compromise solution obtained with the given weights. This method focuses on ranking and selecting from a set of alternatives in the presence of conflicting criteria. Here, the VIKOR method is used for two timestandt+1. In order to calculate the progress or regression via Malmquist productivity index, the positive and negative ideals at timestandt+1are calculated first. Then we introduce the multi-criteria ranking index based on the particular measure of “closeness” to the ideal solution and calculate the separation of each alternative from the ideal solution at timestandt+1. Then we use the Malmquist productivity index to calculate the progress or regression of all alternatives. In this paper, productivity of alternatives available in decision matrix with interval numbers and their improvement or deterioration is researched. To achieve this practical goal, use of extended VIKOR is made to calculate Malmquist productivity index for multicriteria decision-making (MCDM) problem with interval numbers, and by applying Malmquist productivity index, productivity rate of growth for alternatives is calculated. Finally, a numerical example illustrates and clarifies the main results developed in this paper.

Finding an Optimal Set of Cutter Radii for 2D Pocket Machining

2004 ◽  
Author(s):  
Irena Nadjakova ◽  
Sara McMains
Keyword(s):  
Upper Bound ◽  
Approximation Ratio ◽  
Pocket Machining ◽  
Ratio Set ◽  
Tool Set ◽  
The Given ◽  
Optimal Set ◽  
Predetermined Number ◽  
Points Of Discontinuity ◽  
The Ideal

We describe an approach to finding an optimal (within a requested approximation ratio) set of cutter radii for machining a given 2-dimensional pocket. We do not assume that there is a pre-determined set of cutter radii to choose from or a predetermined number of cutters to use. Given an initial set of cutters to choose from, we derive an upper bound on the approximation ratio of what is achievable choosing from this set compared to the ideal set. We then use this bound to subdivide the intervals between the given radii until the requested approximation ratio is achieved. We also look at the machinable area as a function of the tool radius. We show that this area is continuous everywhere, except at a certain set of radii determinable by constructing the Voronoi diagram of the pocket. This lets us avoid subdividing the intervals around the points of discontinuity, improving both running time and the size of the output tool set.

Study on the Obstruction Effect for a Large Explosion Shock Wave Tube Used the Ideal Nozzle Theory

2013 ◽  
Vol 378 ◽  
pp. 87-90
Author(s):  
Chao Cheng Wang ◽  
Hui Qi Ren ◽  
Hai Lu Wang
Keyword(s):  
Shock Wave ◽  
Mach Number ◽  
Dynamic Pressure ◽  
Pressure Ratio ◽  
Wave Tube ◽  
Large Explosion ◽  
Fundamental Equations ◽  
The Given ◽  
The Relationship ◽  
The Ideal

This paper presents a calculation on the obstruction effects for the given large explosion shock wave tube using the ideal nozzle the theory. The relationship among Mach number, Mach number ratio, dynamic pressure ratio in the nozzle throat and blocking area ratio are established according to the fundamental equations of one-dimensional steady flow, which can be taken as the reference of blocking limit design.

scholarly journals Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials

2012 ◽  
Vol 148 (6) ◽  
pp. 1833-1866 ◽  
Author(s):  
Giovanni Cerulli Irelli ◽  
Daniel Labardini-Fragoso
Keyword(s):  
Linear Independence ◽  
Cluster Algebra ◽  
Arbitrary System ◽  
Triangulated Surfaces ◽  
Jacobian Algebra ◽  
Quiver With Potential ◽  
Ideal Triangulations ◽  
The Given ◽  
Marked Points ◽  
The Ideal

AbstractTo each tagged triangulation of a surface with marked points and non-empty boundary we associate a quiver with potential in such a way that whenever we apply a flip to a tagged triangulation the Jacobian algebra of the quiver with potential (QP) associated to the resulting tagged triangulation is isomorphic to the Jacobian algebra of the QP obtained by mutating the QP of the original one. Furthermore, we show that any two tagged triangulations are related by a sequence of flips compatible with QP-mutation. We also prove that, for each of the QPs constructed, the ideal of the non-completed path algebra generated by the cyclic derivatives is admissible and the corresponding quotient is isomorphic to the Jacobian algebra. These results, which generalize some of the second author’s previous work for ideal triangulations, are then applied to prove properties of cluster monomials, like linear independence, in the cluster algebra associated to the given surface by Fomin, Shapiro and Thurston (with an arbitrary system of coefficients).

scholarly journals An Efficient Algorithm for 2-Dimensional Pattern Matching Problem

2020 ◽  
Vol 50 (2) ◽  
pp. 295-313
Author(s):  
Sushil Chandra Dimri ◽  
Umesh Kumar Tiwari ◽  
Mangey Ram
Keyword(s):  
Computer Science ◽  
Prime Number ◽  
Pattern Matching ◽  
Efficient Algorithm ◽  
Matching Problem ◽  
Decimal Number ◽  
2D Pattern ◽  
Decimal Numbers ◽  
The Given

AbstractPattern matching is the area of computer science which deals with security and analysis of data. This work proposes two 2D pattern matching algorithms based on two different input domains. The first algorithm is for the case when the given pattern contains only two symbols, that is, binary symbols 0 and 1. The second algorithm is in the case when the given pattern contains decimal numbers, that is, the collection of symbols between 0 and 9. The algorithms proposed in this manuscript convert the given pattern into an equivalent binary or decimal number, correspondingly find the cofactors of the same dimension and convert these cofactors into numbers if a particular cofactor number matches indicate the matching of the pattern. Furthermore, the algorithm is enhanced for decimal numbers. In the case of decimal numbers, each row of the pattern is changed to its decimal equivalent, and then, modulo with a suitable prime number changes the decimal equivalent into a number less than the prime number. If the number mismatched pattern does not exist, the complexity of the proposed algorithm is very low as compared to other traditional algorithms.

Using Spline Functions for the Shape Description of the Surface of Shell Structures

2014 ◽  
Vol 13 (1) ◽  
Author(s):  
Grzegorz Lenda
Keyword(s):  
Point Of View ◽  
Shell Structures ◽  
Shape Description ◽  
Basic Parameters ◽  
Point Set ◽  
The One ◽  
Half Axis ◽  
The Given ◽  
The Ideal

AbstractThe assessment of the cover shape of shell structures makes an important issue both from the point of view of safety, as well as functionality of the construction. The most numerous group among this type of constructions are objects having the shape of a quadric (cooling towers, tanks with gas and liquids, radio-telescope dishes etc.). The material from observation of these objects (point sets), collected during periodic measurements is usually converted into a continuous form in the process of approximation, with the use of the quadric surface. The created models, are then applied in the assessment of the deformation of surface in the given period of time. Such a procedure has, however, some significant limitations. The approximation with the use of quadrics, allows the determination of basic dimensions and location of the construction, however it results in ideal objects, not providing any information on local surface deformations. They can only be defined by comparison of the model with the point set of observations. If the periodic measurements are carried out in independent, separate points, then it will be impossible to define the existing deformations directly. The second problem results from the one-equation character of the ideal approximation model. Real deformations of the object change its basic parameters, inter alia the lengths of half-axis of main quadrics. The third problem appears when the construction is not a quadric; no information on the equation describing its shape is available either. Accepting wrong kind of approximation function, causes the creation of a model of large deviations from the observed points.All the mentioned above inconveniences can be avoided by applying splines to the shape description of the surface of shell structures. The use of the function of this type, however, comes across other types of limitations. This study deals with the above subject, presenting several methods allowing the increase of accuracy and decrease of the time of the modelling with the splines.

TO THE FORMALIZATION OF THE ESTIMATES OF THE PHENOMENON STATE

2007 ◽  
Vol 06 (04) ◽  
pp. 599-610
Author(s):  
M. E. SALUKVADZE ◽  
R. SH. GOGSADZE ◽  
N. I. JIBLADZE
Keyword(s):  
New York ◽  
Optimization Problems ◽  
Characteristic Parameter ◽  
Academic Press ◽  
Metric Tensor ◽  
Ideal State ◽  
Set Up ◽  
The Given ◽  
Vector Valued ◽  
The Ideal

The questions of the formalization of the estimates of the phenomenon state, which can be used in the decision-making multicriterion problems [Vector-Valued Optimization Problems in Control Theory (Academic Press, New York, 1979)] taking into account the metrics of the space of partial criteria are considered. Any phenomenon is described by some system of characteristic parameters, which values are presented by the coordinates of points in the space of states of the given phenomenon. A notion of the "ideal" state determined by optimal values of each characteristic parameter separately and independently of others is used. Estimate of any state is determined by the distance between that state and the "ideal" one. That permits to choose from a multitude of the given states the best one, to which conforms a minimum distance to the "ideal" state. In the general case of the curved space the distances are measured by geodesic lines drawn by using a metric tensor of the given space. The metric tensor components themselves are determined by the solution of the differential equations set up from the condition of minimization of a certain functional.

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