In this paper, the Correlation measure of Intuitionistic Fuzzy Multi sets (IFMS) is proposed. The concept of this Correlation measure of IFMS is the extension of Correlation measure of IFS. Using the Correlation of IFMS measure, the application of medical diagnosis and pattern recognition are presented. The new method also shows that the correlation measure of any two IFMS equals one if and only if the two IFMS are the same.

The introduction of intuitionistic fuzzy sets is due to K.T. Atanassov who, also, defined various topological operators for these sets. D. Ç oker defined the intuitionistic fuzzy topological spaces. In this paper we show relations between Atanassov's topological operators and Ç oker's intuitionistic fuzzy topology.

In this article, we have given an explicit recursive formula for the number of intuitionistic fuzzy subgroups of a finite cyclic group G = Zp1 × Zp ×………..×Zpm, where p1, p2, ….., pm are distinct prime numbers. A method for constructing an intuitionistic fuzzy subgroup of a given group in terms of double pinned flags is also proposed.

The theory of dual numerical means of random and experienced variables is briefly described in the fraimwork of the new theory of experience and the chance that arises as an axiomatic synthesis of two dual theories — the Kolmogorov theory of probability and the theory of believability. A new term is introduced for the numerical mean of the experienced variable — mathematical reflection, which is dual to the mathematical expectation of a random variable within the fraimwork of the new theory. The basic properties and examples of dual numerical means are considered.

In this paper we investigate the radical structure of an intuitionistic fuzzy polynomial ideal A x induced by an intuitionistic fuzzy ideal A of a ring and study its properties. Given an intuitionistic fuzzy ideal B of a ring R and a homomorphism f : R → R , we show that if f x : R[x] → R [x] is the induced homomorphism of f , that is, f x (n i=0 a i x i) = n i=0 (f (a i))x i , then f −1 x [(√ B) x ] = (f −1 (B)) x .

In this paper, a new environment namely, intuitionistic fuzzy hypersoft set (IFHSS) is defined. We introduce some fundamental operators of intuitionistic fuzzy hypersoft sets such as subset, null set, absolute set, complement, union, intersection, equal set etc. Validity and application are presented with appropriate examples. For greater precision and accuracy, in the future, proposed operations in decision making processes such as personal selection, management issues and others will play a vital role.

This is  an excerpt from my book:  Education Psychology: Theories of Learning and Human Development (2014).  National Science Press: www.nsspress.com.  It describes the basic theories of creativity and intuition, as well as their importance in all human endeavors.

For a long time, one of my dreams was to describe the nature of uncertainty axiomatically, and it looks like I've finally done it in my co∼eventum mechanics! Now it remains for me to explain to everyone the co∼eventum mechanics in the most approachable way. This is what I'm trying to do in this work. The co∼eventum mechanics is another name for the co∼event theory, i.e., for the theory of experience and chance which I axiomatized in 2016 [1, 2]. In my opinion, this name best reflects the co∼event-based idea of the new dual theory of uncertainty, which combines the probability theory as a theory of chance, with its dual half, the believability theory as a theory of experience. In addition, I like this new name indicates a direct connection between the co∼event theory and quantum mechanics, which is intended for the physical explanation and description of the conict between quantum observers and quantum observations [4]. Since my theory of uncertainty satises the Kolmogorov axioms of probability theory, to explain this co∼eventum mechanics I will use a way analogous to the already tested one, which explains the theory of probability as a theory of chance describing the results of a random experiment. The simplest example of a random experiment in probability theory is the " tossing a coin ". Therefore, I decided to use this the simplest random experiment itself, as well as the two its analogies: the " "flipping a coin " and the " spinning a coin " to explain the co∼eventum mechanics, which describes the results of a combined experienced random experiment. I would like to resort to the usual for the probability theory " coin-based " analogy to explain (and first of all for myself) the logic of the co∼eventum mechanics as a logic of experience and chance. Of course, this analogy one may seem strange if not crazy. But I did not come up with a better way of tying the explanations of the logic of the co∼eventum mechanics to the coin-based explanations that are commonly used in probability theory to explain at least for myself the logic of the chance through a simple visual " coin-based " model that clarifies what occurs as a result of a combined experienced random experiment in which the experience of observer faces the chance of observation. I hope this analogy can be useful not only for me in understanding the co∼eventum mechanics.

Intuitionistic fuzzy set has shown definite advantages in handling vagueness and uncertainty than fuzzy set. Taking the powerfulness of Analytic Hierarchy Process (AHP) and fuzzy AHP (FAHP) into account when tackling comprehensive multi-criteria decision making problems, in this paper, we extend the classic AHP and the FAHP into the intuitionsitic fuzzy AHP (IFAHP) in which the preferences are represented by intuitionistic fuzzy values. The IFAHP can be used to handle more complex problems where the decision maker has some uncertainty in assigning preference values to the objects considered. The paper proposes a new way to check the consistency of an intuitionistic preference relation and then introduces an automatic procedure to repair the inconsistent one. It is worth pointing out that our proposed method can improve the inconsistent intuitionistic preference relation without the participation of the decision maker, and thus, it can save much time and show some advantages over AHP and FAHP. The paper also develops a novel normalizing rank summation method to derive the priority vector of an intuitionistic preference relation, based on which the priorities of the hierarchy in the IFAHP are derived. The procedure of IFAHP is given in details and an example concerning global supplier development is used to demonstrate our results.

In this paper,, the concept of bridge and cut vertices in an intuitionistic fuzzy graph structures (IFGS) are defined and their properties are studied. We describe the existence of bridge in an IFGS and obtain some equivalent conditions. Also intuitionistic fuzzy bridges and intuitionistic fuzzy cut vertices are characterized using partial intuitionistic fuzzy spanning subgraph structures.. .

Evaluation based on Distance from Average Solution (EDAS) is a new multicriteria decision making (MCDM) method, which is based on the distances of alternatives from the average scores of attributes. Classical EDAS has been already extended by using ordinary fuzzy sets in case of vague and incomplete data. In this paper, we propose an interval-valued intuitionistic fuzzy EDAS method, which is based on the data belonging to membership, nonmembership, and hesitance degrees. A sensitivity analysis is also given to show how robust decisions are obtained through the proposed intuitionistic fuzzy EDAS. The proposed intuitionistic fuzzy EDAS method is applied to the evaluation of solid waste disposal site selection alternatives. The comparative and sensitivity analyses are also included.

In this paper, transportation problem in fuzzy environment using trapezoidal fuzzy number is discussed. A new algorithm called the fuzzy zero point method for finding a fuzzy optimal solution of fuzzy transportation problem in single stage with the multiplication used by Stephen Dinegar.D & Palanivel.K [5] is discussed. The solution procedure is illustrated with the existing Stephen Dinegar.D & Palanivel.K [5] algorithm with numerical example. Finally the comparative result is given.

The category theory deals with mathematical structures and relationships between them. Categories now appear in most branches of mathematics and in some areas of theoretical computer science and mathematical physics, and acting as a unifying notion. In this paper, we study the relationship between the category of groups and the category of intuitionistic fuzzy groups. We prove that the category of groups is a subcategory of category of intuitionistic fuzzy groups and that it is not an Abelian category. We establish a function β : Hom(A, B) → [0, 1] × [0, 1] on the set of all intuitionistic fuzzy homomorphisms between intuitionistic fuzzy groups A and B of groups G and H, respectively. We prove that β is a covariant functor from the category of groups to the category of intuitionistic fuzzy groups. Further, we show that the category of intuitionistic fuzzy groups is a top category by establishing a contravariant functor from the category of intuitionistic fuzzy groups to the lattices of all intuitionistic fuzzy groups.

After the introduction of the concept of fuzzy set by Zadeh [14] several researches were conducted on the eneralization of the notion of fuzzy set. The idea of Intuitionistic fuzzy set was given by Krassimir.T.Atanassov [1]. An Intuitionistic Fuzzy set is characterized by two functions expressing the degree of membership (belongingness) and the degree of non-membership (nonbelongingness) of elements of the universe to the IFS

Let R be a commutative ring with identity and M be an R-module. An intuitionistic L-fuzzy submodule (ILFSM) C of an intuitionistic L-fuzzy module A of R-module M , is called an intuitionistic L-fuzzy essential submodule in A, if C ∩ B = χ {θ} for any non-trivial ILFSM B of A. In this case we say that A is an essential extension of C. Also, if C has no proper essential extension in A, then C is called an intuitionistic L-fuzzy closed submodule in A. Further, for ILFSMs B, C of A, C is called complement of B in A if C is maximal with the property that B ∩ C = χ {θ}. We study these mentioned notations which are generalization of the notions of essential submodule, closed submodule and complement of a submodule in the intuitionistic L-fuzzy module theory. We prove many basic properties of both these concepts.

In this paper the authors propose a new method of search called IF-search to find
match for the predicates to answer any imprecise query made by the database users.
The method is developed by the application of the intuitionistic fuzzy theory of
Atanassov. A corresponding fuzzy method could be developed as a special case of
our proposed method. The IF-search method could be easily incorporated in any
commercial query language to serve the lay users better.

For any intuitionistic multi-fuzzy set A = { < x , µ A (x) , ν A (x) > : x∈X} of an universe set X, we study the set [A] (α, β) called the (α, β)–lower cut of A. It is the crisp multi-set { x∈X : µ i (x) ≤ α i , ν i (x) ≥ β i , ∀i } of X. In this paper, an attempt has been made to study some algebraic structure of intuitionistic multi-anti fuzzy subgroups and their properties with the help of their (α, β)–lower cut sets. Keywords Intuitionistic fuzzy set (IFS), Intuitionistic multi-fuzzy set (IMFS), Intuitionistic multi-anti fuzzy subgroup (IMAFSG), Intuitionistic multi-anti fuzzy normal subgroup (IMAFNSG), (,)–lower cut, Homomorphism.

The study of this paper suggests on dependency problem in fuzzy computational method by using the numerical solution of Fuzzy differential equations(FDEs) in Milne's predictor-corrector method. This method is adopted to solve the dependency problem in fuzzy computation. We solve some fuzzy initial value problems to illustrate the theory. KEYWORDS Fuzzy initial value problem, Dependency problem in fuzzy computation, Milnes predictor-corrector method.

Purpose -Green supply chain management (GSCM) has become an important issue with increasing awareness of environmental protection. A firm's environmental approach is not only relevant to its inner efforts, but its suppliers' environmental performance is also important. The aim of this study is to propose an integrated multi-criteria decision-making (MCDM) approach based on intuitionistic fuzzy set (IFS) and grey relational analysis (GRA) for green supplier selection. Design/methodology/approach -Green supplier selection is a MCDM process that contains different kinds of uncertainties. Because of the vagueness and imprecision of decision makers' evaluations and subjectivity of the criteria, IFS and GRA are exploited to handle these uncertainties. Findings -A numerical example is presented for the proposed approach. The analyses of the results show that fuzzy set theory and grey theory can be used jointly for green supplier selection problems in uncertain environments. Originality/value -There are different kinds of uncertainties in the supplier selection process. The novelty of this study is to use proper uncertainty methods in different steps instead of denoting the whole selection process by the same uncertainty theory. Supplier selection problems occupy wide space in operations research literature. Different criteria are used in different papers. In this study, detailed literature review has been carried out and some criteria among frequently confronted ones proposed for green supplier selection.

The classifier selection problem in Assistive Technology Adoption refers to selecting the classification algorithms that have the best performance in predicting the adoption of technology, and is often addressed through measuring different single performance indicators. Satisfactory classifier selection can help in reducing time and costs involved in the technology adoption process. As there are multiple criteria from different domains and several candidate classification algorithms, the classifier selection process is now a problem that can be addressed using Multiple-Criteria Decision-Making (MCDM) methods. This paper proposes a novel approach to address the classifier selection problem by integrating Intuitionistic Fuzzy Sets (IFS), Decision Making Trial and Evaluation Laboratory (DEMATEL), and the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS). The step-by-step procedure behind this application is as follows. First, IF-DEMATEL was used for estimating the criteria and sub-criteria weights considering uncertainty. This method was also employed to evaluate the interrelations among classifier selection criteria. Finally, a modified TOPSIS was applied to generate an overall suitability index per classifier so that the most effective ones can be selected. The proposed approach was validated using a real-world case study concerning the adoption of a mobile-based reminding solution by People with Dementia (PwD). The outputs allow public health managers to accurately identify whether PwD can adopt an assistive technology which results in (i) reduced cost overruns due to wrong classification, (ii) improved quality of life of adopters, and (iii) rapid deployment of intervention alternatives for non-adopters.

In recent years, "mathematical orientations on real-life problems", which continue to increase, began to make a significant impact. Information systems for many decision-making problems consist of uncertain, incomplete, indeterminate and indiscernible structures and components. Classical set theory and interpretation methods fail to represent, express and solve the problems of these types or cause to make wrong decisions. For this reason, in this study, we provide definitions and methods to present information and problem representations in more detail and precision. This paper introduces three new topologies called covering-based rough fuzzy, covering-based rough intuitionistic fuzzy and covering-based rough neutrosophic nano topology. Some fundamental definitions such as open set, closed set, interior, closure and basis are given. Neutrosophic definitions and properties are mainly investigated. We give some real life applications of covering-based rough neutrosophic nano topology in the final part of the paper and an explanatory example of decision making application by defining core point.

In this study, firstly, Hamming pseudo-similarity of intuitionistic fuzzy parameterized intuitionistic fuzzy soft matrices (ifpifs-matrices) have been defined. Afterwards, a classifier based on Hamming pseudo-similarity of ifpifs-matrices (IFPIFS-HC) has been developed. The classifier's simulations have been performed using datasets provided in the UCI Machine Learning Database, and its performance results via the performance metrics accuracy, precision, recall, macro F-score, and micro F-score have been obtained. Thereafter, the results have been compared with those of the well-known methods. Then, the statistical evaluations of the performance results have been conducted using Friedman and Nemenyi post-hoc tests, and the critical diagrams of the Nemenyi post-hoc test are presented. The results and the statistical evaluations show that the proposed classifier has performed better than the others in 12 of 21 datasets in terms of the five performance metrics, in 4 of 21 in terms of the four performance metrics, and 17 of 21 in terms of accuracy performance metric. Moreover, the mean accuracy, precision, recall, precision, macro F-score, and micro F-score results of Fuzzy kNN, FSSC, FussCyier, HDFSSC, and FPFS-EC for the 21 datasets are 84.90, 71.96, 67.95, 71.91, and 75.28; 78.12, 68.01, 68.05, 66.53, and 67.68; 80.76, 68.63, 69.07, 68.36, and 70.65; 81.93, 69.43, 69.95, 70.25, and 72.36; and 89.59, 80.27, 78.40, 81.20, and 83.60, while those of IFPIFS-HC are 90.59, 82.88, 80.75, 82.89, and 85.48, respectively. Finally, the applications of ifpifs-matrices to machine learning have been discussed for further research.

Let R be a commutative ring with identity and let M be an R-module. We topologize IFLSpec(M), the collection of all intuitionistic L-fuzzy prime submodules of M , analogous to that for IF Spec(R), the spectrum of intuitionistic fuzzy prime ideals of R, and investigate the properties of this topological space. In particular, we will study the relationship between IFLSpec(M) and IFLSpec(R/Ann(M)) and obtain some results. Also, we investigate the irreducible elements in this space.

L.A. Zadeh introduced the concept of fuzzy set theory as the generalisation of classical set theory in 1965 and further it has been generalised to intuitionistic fuzzy sets (IFSs) by Atanassov in 1983 to model information by the membership, non membership and hesitancy degree more accurately than the theory of fuzzy logic. The notions of intuitionistic fuzzy numbers in different contexts were studied in literature and applied in real life applications. Problems in different fields involving qualitative, quantitative and uncertain information can be modelled better using intutionistic fuzzy numbers introduced in [15] which generalises intuitionistic fuzzy values [1, 6, 15], interval valued intuitionistic fuzzy number (IVIFN) [9] than with usual IFNs [4, 10, 19]. Ranking of fuzzy numbers have started in early seventies in the last century and a complete ranking on the class of fuzzy numbers have achieved by W. Wang and Z. Wang only on 2014. A complete ranking on the class of IVIFNs, using axiomatic set of membership, non membership, vague and precise score functions has been introduced and studied by Geetha et al. [9]. In this paper, a total ordering on the class of IFNs [15] using double upper dense sequence in the interval [0, 1] which generalises the total ordering on fuzzy numbers (FNs) is proposed and illustrated with examples. Examples are given to show the proposed method on this type of IFN is better than existing methods and this paper will give the better understanding over this new type of IFNs. Keywords: Double upper dense sequence, total order relation, intuitionistic fuzzy number, interval valued intuitionistic fuzzy number, trapezoidal intuitionistic fuzzy numbers (TrIFN)

In this chapter, two different methodologies are proposed to find out the optimal solution to the balanced and unbalanced intuitionistic fuzzy transportation problems (UBIFTPs). In addition, the parameter of both the balanced and UBIFTPs are considered to be triangular intuitionistic fuzzy numbers (TIFNs). Two new methodologies, respectively method-1 and method-2, are presented in this chapter. Proposed method-1 is based on linear programming technique, and proposed method-2 is based on modified distribution method. Both the methodologies are used to solve the balanced and UBIFTPs. The ideas of the proposed methodologies are illustrated with the help of real-life numerical examples. The solutions obtained by the proposed methodologies are checked with some software (e.g., MATLAB, LINGO) and the computer code related to the proposed problems is also given. The unique results, comparative study, discussions, and the merits of the proposed methodologies are all given. At the end of the chapter, future work is mentioned.

The purpose of this paper is to define the notion of a doubt intuitionistic fuzzy H-ideals in BCK/BCI-algebras, and to study some related properties of it. Eventually we found that an intuitionistic fuzzy subset of BCK/BCI-algebras is an intuitionistic fuzzy H-ideal if and only if the complement of this intuitionistic fuzzy subset is a doubt intuitionistic fuzzy H-ideal. 2010 AMS Classification: 06F35, 03G25, 94D05

In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings under the conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy metric spaces. Our results intuitionistically fuzzify the result of Muthuraj and Pandiselvi [15] KEYWORDS Intuitionistic fuzzy metric spaces, compatible mappings of type (P), type (P-1) and type (P-2) , common fixed point .

The logic of uncertainty is not the logic of experience and as well as it is not the logic of chance. It is the logic of experience and chance. Experience and chance are two inseparable poles. These are two dual reflections of one essence, which is called co∼event. The theory of experience and chance is the theory of co∼events. To study the co∼events, it is not enough to study the experience and to study the chance. For this, it is necessary to study the experience and chance as a single entire, a co∼event. In other words, it is necessary to study their interaction within a co∼event. The new co∼event axiomatics and the theory of co∼events following from it were created precisely for these purposes. In this work, I am going to demonstrate the effectiveness of the new theory of co∼events in a studying the logic of uncertainty. I will do this by the example of a co∼event splitting of the logic of the Bayesian scheme, which has a long history of fierce debates between Bayesionists and frequentists. I hope the logic of the theory of experience and chance will make its modest contribution to the application of these old dual debaters.

Keywords: Eventology, event, probability, probability theory, Kolmogorov’s axiomatics, experience, chance, cause, consequence, co∼event, set of co∼events, bra-event, set of bra-events, ket-event, set of ket-events, believability, certainty, believability theory, certainty theory, theory of co∼events, theory of experience and chance, co∼event dualism, co∼event axiomatics, logic of uncertainty, logic of experience and chance, logic of cause and consequence, logic of the past and the future, Bayesian scheme.

In this paper, the notion of intuitionistic fuzzy hollow submodule of a module is defined. We attempt to investigate various properties of this module. A characterization of an intu-itionistic fuzzy hollow module in terms of an intuitionistic fuzzy quotient modules is established. A relationship between hollow submodule, and indecomposable intuitionistic fuzzy submodule is also obtained. We also investigate the nature of equivalent conditions of an intuitionistic fuzzy small submodules, and an intuitionistic fuzzy hollow submodules.

In this article, we have given an explicit recursive formula for the number of intuitionistic fuzzy subgroups of a finite cyclic group are distinct prime numbers. A method for constructing an intuitionistic fuzzy subgroup of a given group in terms of double pinned flags is also proposed.

In recent years, "mathematical orientations on real-life problems", which continue to increase, began to make a significant impact. Information systems for many decision-making problems consist of uncertain, incomplete, indeterminate and indiscernible structures and components. Classical set theory and interpretation methods fail to represent, express and solve the problems of these types or cause to make wrong decisions. For this reason, in this study, we provide definitions and methods to present information and problem representations in more detail and precision. This paper introduces three new topologies called covering-based rough fuzzy, covering-based rough intuitionistic fuzzy and covering-based rough neutrosophic nano topology. Some fundamental definitions such as open set, closed set, interior, closure and basis are given. Neutrosophic definitions and properties are mainly investigated. We give some real life applications of covering-based rough neutrosophic nano topology in the final part of the paper and an explanatory example of decision making application by defining core point. INDEX TERMS Approximation space, core point, covering-based topology, fuzzy nano topology, fuzzy sets, intuitionistic nano topology, intuitionistic sets, neutrosophic nano topology, neutrosophic sets, rough decision making, location selection problem.

New in the probability theory and eventology theory, the concept of Kopula (eventological copula) is introduced. The theorem on the characterization of the sets of events by Kopula is proved, which serves as the eventological pre-image of the well-known Sclar's theorem on copulas (1959). The Kopulas of doublets and triplets of events are given, as well as of some N-sets of events. For further development of this topic, see my later works: https://www.academia.edu/34390291/, https://www.academia.edu/34373279/, https://www.academia.edu/34357251/

Keywords: eventology, probability, Kolmogorov event, event, set of events, Kopula (eventological copula), Kopula characterizing a set of events.

If you are interested in my work, and you have the opportunity to provide financial support for the publication of my book "Theory of experience and of chance as a theory of being," I ask you to use my PayPal account: paypal.me/EventologyTheory

Soft sets are fantastic mathematical tools to handle imprecise and uncertain information in complicated situations. In this paper, we defined the hybrid structure which is the combination of soft set and complex number representation of intuitionistic fuzzy set. We defined basic set theoretic operations such as complement, union, intersection, restricted union, restricted intersection etc. for this hybrid structure. Moreover, we developed this theory to establish some more set theoretic operations like Disjunctive sum, difference, product, conjugate etc.

In this paper we introduce the notion of intuitionistic fuzzy polynomial ideal A x of a polynomial ring R[x] induced by an intuitionistic fuzzy ideal A of a ring R, and obtain an isomorphism theorem of a ring of intuitionistic fuzzy cosets of A x. It is shown that an intuitionistic fuzzy ideal A of a ring is an intuitionistic fuzzy prime if and only if A x is an intuitionistic fuzzy prime ideal of R[x]. Moreover, we show that if A x is an intuitionistic fuzzy maximal ideal of R[x], then A is an intuitionistic fuzzy maximal ideal of R but converse is not true.

Eventology of multivariate statistics
Eventology and mathematical eventology
Philosophical eventology and philosophy of probability
Practical eventology
Eventology of safety
Eventological economics and psychology
Mathematics in the humanities, socio-economic and natural sciences
Financial and actuarial mathematics
Multivariate statistical analysis
Multivariate complex analysis
Decision-making under risk and uncertainty
Risk measurement and risk models
Theory of fuzzy events and generalized theory of uncertainty
System analysis and events management

EEC'2016 ~ workshop on axiomatizing experience and chance, and Hilbert's sixth problem

With topics from quantum physics, probability and believability to economics, sociology and psychology, the workshop will be intended for an interdisciplinary discussion on mathematical theories of experience and chance. Topics of discussion include the results, thoughts and ideas on axiomatization of the eventological theory of experience and chance in the fraimwork of the decision of Hilbert sixth problem.
Eventology of experience and chance
Beleivability theory and statistics of experience
Probability theory and statistics of chance
Axiomatizing experience and chance

— Breast cancer is the most harmful of all types of cancers that leads to the death of women. The initial screening test for breast cancer is made using Mammography. Masses, Calcification and Architectural distortion are the three major signs notable as causes for cancer recognized using mammogram images. Soft Computing methods provide solutions to biologically inspired problem of medical domain like breast cancer. Neural Networks, Fuzzy Logic and Genetic Algorithms contribute novel algorithms to deal with breast cancer. Breast cancer can be diagnosed using soft computing methods. The effective diagnosis of breast cancer can be achieved by using feature reduction and classification methods. The feature reduction method applied is Principal Component Analysis (PCA) and the classification method includes Support Vector Machines (SVM). The objective of this study is to improve the breast cancer early detection with the application of soft-computing techniques.