Triangle Norm Research Articles

Aesthetic triangle norms for Japanese adults

Aesthetic triangle norms for Japanese adults

Comparison of Tweed Facial Triangle Norms Among Different Ethnic Groups

Aims: The objectives of the present study were to establish Tweed facial triangle norms in Qatari and to compare the result with Tweed norms and previous reported results in different racial groups.
 Material and method: The sample consisted of 89 lateral cephalometric radiographic radiographs, 75 patients were females and 14 were males with age range 18 to 25 years old (Mean 20.7. +/-2.3).
 Results: The result of the present study showed that the mean values of FMA 33.5 º +/- 6.2 º FMIA 52.8 º +/- 9.5 º angles and IMAP 94.5º +/- 8.8º were significantly higher than Tweed norms. The comparison with Black Brazilian and Bangladeshi indicated extremely statistically significant differences in FMA and IMPA angles whereas in Nepalese, extremely significant difference was observed in FMIA and FMA angles.
 Conclusions: Tweed’s Facial Triangle mean values were established for Qatari. However, Tweed facial triangle norms should be used only as a guide and not as an absolute value. Hence, using specific norms for specific race or ethnic group will help in providing more accurate diagnosis and treatment planning.

Hesitant Fuzzy Linguistic Aggregation Operators Based on the Hamacher t-norm and t-conorm

Hesitant fuzzy linguistic (HFL) term set, as a very flexible tool to represent the judgments of decision makers, has attracted the attention of many researchers. In recent years, some HFL aggregation operators have been developed to aggregate the HFL information. However, most of these operators are proposed based on the Algebraic product and Algebraic sum. In this paper, we presented some HFL aggregation operators to handle HFL information based on Hamacher triangle norms. We first define new operational laws on the HFL element according to Hamacher triangle norms. Then we present a family of HFL Hamacher aggregation operators, including the HFL Hamacher weighted averaging, HFL Hamacher weighted geometric, HFL Hamacher power weighted averaging and HFL Hamacher power weighted geometric operators and their generalized forms. We also investigate some special cases and properties of these operators in detail. Furthermore, we develop two approaches based on the proposed operators to deal with the multi-criteria decision-making problem with HFL information. Finally, a numerical example with regard to choosing a suitable city to release sharing car is provided to illustrate the feasibility of the proposed method, and the advantages of the proposed methods are shown by conducting a sensitivity and comparative analysis.

Intuitionistic [formula omitted]-fuzzy Rough Sets, Intuitionistic [formula omitted]-fuzzy Preorders and Intuitionistic [formula omitted]-fuzzy Topologies

In this paper, by constructing an intuitionistic L-fuzzy triangle norm T and an intuitionistic L-fuzzy implicator I, lower and upper approximations of intuitionistic L-fuzzy sets are defined with respect to an intuitionistic L-fuzzy approximation space. Properties of intuitionistic L-fuzzy approximation operators are then given. This paper is devoted to discussing the relationship between intuitionistic L-fuzzy relations and intuitionistic L-fuzzy topologies. It proves that the set of all lower approximation sets based on a reflexive and transitive intuitionistic L-fuzzy relation consists of an intuitionistic L-fuzzy Alexandrov topology; and conversely, an intuitionistic L-fuzzy Alexandrov topology is just the set of all lower approximation sets under a reflexive and transitive intuitionistic L-fuzzy relation. That is to say, there exists a one-to-one correspondence between the set of all intuitionistic L-fuzzy preorders and the set of all intuitionistic L-fuzzy Alexandrov topologies.

Frank aggregation operators and their application to hesitant fuzzy multiple attribute decision making

In this paper, we investigate multiple attribute decision making (MADM) problems based on Frank triangular norms, in which the attribute values assume the form of hesitant fuzzy information. Firstly, some basic concepts of hesitant fuzzy set (HFS) and the Frank triangle norms are introduced. We develop some hesitant fuzzy aggregation operators based on Frank operations, such as hesitant fuzzy Frank weighted average (HFFWA) operator, hesitant fuzzy Frank ordered weighted averaging (HFFOWA) operator, hesitant fuzzy Frank hybrid averaging (HFFHA) operator, hesitant fuzzy Frank weighted geometric (HFFWG) operator, hesitant fuzzy Frank ordered weighted geometric (HFFOWG) operator, and hesitant fuzzy Frank hybrid geometric (HFFHG) operator. Some essential properties together with their special cases are discussed in detail. Next, a procedure of multiple attribute decision making based on the HFFHWA (or HFFHWG) operator is presented under hesitant fuzzy environment. Finally, a practical example that concerns the human resource selection is provided to illustrate the decision steps of the proposed method. The result demonstrates the practicality and effectiveness of the new method. A comparative analysis is also presented.

The Residual Implication of Interval-Valued Fuzzy Triangle Norm and its Properties

This paper investigates residual implication of interval-valued fuzzy triangle norm and its properties. Residual implication of interval-valued fuzzy triangle norm is first defined. Then the relation between residual implication of interval-valued fuzzy triangle norm and residual implication of triangle norm is established. Finally, some properties of residual implication of interval-valued fuzzy triangle norm are examined.

The stability of the quartic functional equation in various spaces

The purpose of this paper is first to introduce the notation of intuitionistic random normed spaces, and then by virtue of this notation to study the stability of a quartic functional equation in the setting of these spaces under arbitrary triangle norms. Then we prove the stability of above quartic functional equation in non-Archimedean random normed spaces. Furthermore, the interdisciplinary relation among the theory of random spaces, the theory of non-Archimedean spaces, the theory of intuitionistic spaces and the theory of functional equations are also presented in the paper.

Stability of a cubic functional equation in intuitionistic random normed spaces

In this paper, the stability of a cubic functional equation in the setting of intuitionistic random normed spaces is proved. We first introduce the notation of intuitionistic random normed spaces. Then, by virtue of this notation, we study the stability of a cubic functional equation in the setting of these spaces under arbitrary triangle norms. Furthermore, we present the interdisciplinary relation among the theory of random spaces, the theory of intuitionistic spaces, and the theory of functional equations.

On characterization of intuitionistic fuzzy rough sets based on intuitionistic fuzzy implicators

This paper presents a general framework for the study of relation-based ( I , T ) -intuitionistic fuzzy rough sets by using constructive and axiomatic approaches. In the constructive approach, by employing an intuitionistic fuzzy implicator I and an intuitionistic fuzzy triangle norm T , lower and upper approximations of intuitionistic fuzzy sets with respect to an intuitionistic fuzzy approximation space are first defined. Properties of ( I , T ) -intuitionistic fuzzy rough approximation operators are examined. The connections between special types of intuitionistic fuzzy relations and properties of intuitionistic fuzzy approximation operators are established. In the axiomatic approach, an operator-oriented characterization of ( I , T ) -intuitionistic fuzzy rough sets is proposed. Different axiom sets characterizing the essential properties of intuitionistic fuzzy approximation operators associated with various intuitionistic fuzzy relations are explored.

T-regular probabilistic convergence spaces

AbstractA probabilistic convergence structure assigns a probability that a given filter converges to a given element of the space. The role of the t-norm (triangle norm) in the study of regularity of probabilistic convergence spaces is investigated. Given a probabilistic convergence space, there exists a finest T-regular space which is coarser than the given space, and is referred to as the ‘T-regular modification’. Moreover, for each probabilistic convergence space, there is a sequence of spaces, indexed by nonnegative ordinals, whose first term is the given space and whose last term is its T-regular modification. The T-regular modification is illustrated in the example involving ‘convergence with probability λ’ for several t-norms. Suitable function space structures in terms of a given t-norm are also considered.

A fixed point theorem in Menger space

The purpose of this paper is to introduce the concept of weakly compatible maps in Menger space and prove a common fixed point theorem in this space.2010 Mathematics Subject Classification: 54H25.Keywords: Triangle norm; Menger space; Fixed point theorem; Weakly compatible mapsDOI: http://dx.doi.org/ 10.3126/kuset.v7i1.5417 KUSET 2011; 7(1): 9-17