Colouring of Trivalent Polyhedra

1965 ◽  
Vol 17 ◽  
pp. 659-664 ◽  
Author(s):  
Anton Kotzig
Keyword(s):  
Convex Polyhedron ◽  
Number Of Faces

By an Euler polyhedron of valence three or a trivalent convex polyhedron in Euclidean 3-space (4) we mean in the present paper an Euler polyhedron in the sense of Steinitz (8, p. 113), such that each vertex is incident with exactly three edges.In the present paper we establish a theorem concerning the colouring of trivalent polyhedra. A specialization of this theorem solves the following problem implicit in Eberhard (1, p. 84): Does there exist a trivalent Euler polyhedron with an odd number of faces such that the number of edges incident with any face is divisible by three?

An Enumeration of the Five Parallelohedra

1961 ◽  
Vol 4 (1) ◽  
pp. 45-47 ◽  
Author(s):  
William Moser
Keyword(s):  
Convex Polyhedron ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Real Affine ◽  
Whole Space ◽  
Number Of Faces ◽  
Three Dimensional Space

A parallelohedron is a convex polyhedron, in real affine three-dimensional space, which can be repeated by translation to fill the whole space without interstices. It has centrally symmetrical faces [4, p. 120] and hence is centrally symmetrical.Let Fi denote the number of faces each having exactly i edges, Vi denote the number of vertices each incident with exactly i edges, E denote the number of edges, n denote the number of sets of parallel edges, F denote the total number of faces, V denote the total number of vertices.

A ten-faced hexangulaconulariid from Cambrian Stage 2 of South China

2021 ◽  
pp. 1-8
Author(s):  
Junfeng Guo ◽  
Jian Han ◽  
Heyo Van Iten ◽  
Zuchen Song ◽  
Yaqin Qiang ◽  
...  
Keyword(s):  
New Genus ◽  
Radial Symmetry ◽  
New Taxon ◽  
Three Gorges ◽  
Substantial Variation ◽  
The Three Gorges ◽  
Apical Half ◽  
Lateral Margins ◽  
Number Of Faces ◽  
Extinct Group

Abstract Hexangulaconulariids (Cambrian stages 1–2) are an extinct group of medusozoan polyps having a biradially symmetrical, fan-shaped periderm that is distinct from those of medusozoan polyps showing three-, four-, five-, or six-fold radial symmetry. Hexangulaconulariids exhibit substantial variation in gross morphology, including variation in the number of faces on each of the two major sides of the periderm. An intermediate taxon of hexangulaconulariids with ten faces (five on each major side) was expected. Here we describe a new hexangulaconulariid, Decimoconularia isofacialis new genus new species from Bed 5 of the Yanjiahe Formation (Cambrian Stage 2) in the Three Gorges area of Hubei Province, China. The new taxon differs from other hexangulaconulariids (Arthrochites, Hexaconularia, and Septuconularia) mainly in possessing a total of ten faces. The two lateral margins are each marked by a ridge in about the apertural half of the periderm and by a collinear furrow in about the apical half, while the five faces on each major side are bounded by a furrow in about the apertural half and by a collinear ridge in about the apical half. Among hexangulaconulariids, Decimoconularia and Septuconularia may be more closely related to each other than either genus is to Arthrochites or Hexaconularia. UUID: http://zoobank.org/ca270a3b-25ee-4d1f-bdeb-91a963370e70

scholarly journals Data-based polyhedron model for optimization of engineering structures involving uncertainties

2021 ◽  
Vol 2 ◽  
Author(s):  
Zhiping Qiu ◽  
Han Wu ◽  
Isaac Elishakoff ◽  
Dongliang Liu
Keyword(s):  
Linear Programming ◽  
Convex Polyhedron ◽  
Linear Optimization ◽  
Optimal Solution ◽  
Composite Plates ◽  
Convex Polyhedra ◽  
Chebyshev Inequality ◽  
Engineering Structures ◽  
Load Bearing ◽  
Polyhedron Model

Abstract This paper studies the data-based polyhedron model and its application in uncertain linear optimization of engineering structures, especially in the absence of information either on probabilistic properties or about membership functions in the fussy sets-based approach, in which situation it is more appropriate to quantify the uncertainties by convex polyhedra. Firstly, we introduce the uncertainty quantification method of the convex polyhedron approach and the model modification method by Chebyshev inequality. Secondly, the characteristics of the optimal solution of convex polyhedron linear programming are investigated. Then the vertex solution of convex polyhedron linear programming is presented and proven. Next, the application of convex polyhedron linear programming in the static load-bearing capacity problem is introduced. Finally, the effectiveness of the vertex solution is verified by an example of the plane truss bearing problem, and the efficiency is verified by a load-bearing problem of stiffened composite plates.

scholarly journals Typical emotional expression in children’s drawings of the human face

2021 ◽  
Author(s):  
Eleonora Cannoni ◽  
Giuliana Pinto ◽  
Anna Silvia Bombi
Keyword(s):  
Emotional Expression ◽  
Visual Cues ◽  
Age Groups ◽  
Emotional Expressions ◽  
Basic Emotions ◽  
Preferential Expression ◽  
Human Figure ◽  
Figure Drawings ◽  
Number Of Faces ◽  
Human Faces

AbstractThis study was aimed at verifying if children introduce emotional expressions in their drawings of human faces, and if a preferential expression exists; we also wanted to verify if children’s pictorial choices change with increasing age. To this end we examined the human figure drawings made by 160 boys and 160 girls, equally divided in 4 age groups: 6–7; 8–9; 10–11; 12–13 years; mean ages (SD in parentheses) were: 83,30 (6,54); 106,14 (7,16) 130,49 (8,26); 155,40 (6,66). Drawings were collected with the Draw-a-Man test instructions, i.e. without mentioning an emotional characterization. In the light of data from previous studies of emotion drawing on request, and the literature about preferred emotional expressions, we expected that an emotion would be portrayed even by the younger participants, and that the preferred emotion would be happiness. We also expected that with the improving ability to keep into account both mouth and eyes appearance, other expressions would be found besides the smiling face. Data were submitted to non-parametric tests to compare the frequencies of expressions (absolute and by age) and the frequencies of visual cues (absolute and by age and expressions). The results confirmed that only a small number of faces were expressionless, and that the most frequent emotion was happiness. However, with increasing age this representation gave way to a variety of basic emotions (sadness, fear, anger, surprise), whose representation may depend from the ability to modify the shapes of both eyes and mouth and changing communicative aims of the child.

scholarly journals Convex Polyhedron Method to Stability of Continuous Systems with Two Additive Time-Varying Delay Components

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Tiejun Li ◽  
Junkang Tian
Keyword(s):  
Stability Criterion ◽  
Convex Polyhedron ◽  
Linear Matrix ◽  
Continuous Systems ◽  
Time Varying ◽  
Time Varying Delay ◽  
Polyhedron Method ◽  
Delay Dependent ◽  
Delay Dependent Stability ◽  
Varying Delay

This paper is concerned with delay-dependent stability for continuous systems with two additive time-varying delay components. By constructing a new class of Lyapunov functional and using a new convex polyhedron method, a new delay-dependent stability criterion is derived in terms of linear matrix inequalities. The obtained stability criterion is less conservative than some existing ones. Finally, numerical examples are given to illustrate the effectiveness of the proposed method.

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