The Interaction between Fuzzy Subsets and Groupoids

Emerging trends in soft set theory and related topics.

Distance measures of fuzzy sets have been developed for feature selection and finding redundant features in the fields of decision-making, prediction, and classification problems. Terms commonly used in the definition of fuzzy sets are normal and convex fuzzy sets. This paper extends the general fuzzy set definitions to subnormal and non-convex fuzzy sets that are more precise when implementing uncertain knowledge representations by weighing fuzzy membership functions. A distance measure method for subnormal and non-convex fuzzy sets is proposed for embedded feature selection. Constructing fuzzy membership functions and extracting fuzzy rules play a critical role in fuzzy classification systems. The weighted fuzzy membership functions prevent the combinatorial explosion of fuzzy rules in multiple fuzzy rule-based systems. The proposed method was validated by a comparison with two other methods. Our proposed method demonstrated higher accuracies in training and test, with scores of 97.95% and 93.98%, respectively, compared to the other two methods.

The aim of this paper is to introduce and study new class of fuzzy function called fuzzy semi pre homeomorphism in a fuzzy topological space by utilizing fuzzy semi pre-open sets. Therefore, some of their characterization has been proved; In addition to that we define, study and develop corresponding to new class of fuzzy semi pre homeomorphism in fuzzy topological spaces using this new class of functions.

Almost all practically encountered decision making problems can be treated as fuzzy decision making problems. In Bellman and Zadeh's seminal approach, fuzzy decision making problems are essentially reduced to the maximization of the aggregated objectives and constraints. Thus, the study of the properties of concavity before and after aggregation should be the a very useful approach for solving fuzzy decision making problems. In this paper, the concept of E-convex, which covers a wider class of sets and functions, is extended to fuzzy sets. Supp-E-concave and supp-E-quasiconcave fuzzy sets are first introduced. Then some aggregation rules for supp-E-concave and supp-E-quasiconcave fuzzy sets are given. Finally, some useful composition rules are developed. These aggregation and composition rules should form the basis for further exploration of fuzzy E convex sets in fuzzy decision-making

On (IO,O)-fuzzy rough sets based on overlap functions

Letter: The concept of stratification and future applications

Neural networks are good at representing functions or data transformations. However just as in the case of the biological brain the mathematical description of the data transformation is hidden. In the case of the human brain the transformation, in terms of rules, may be extracted by interviewing the person, In the case of the artificial neural network other approaches have to be utilized. In the case described here a second neural network that represents the transformation in terms of fuzzy rules is trained using gradient descent. The parameters that are learned are the parameters of the fuzzy sets and also the connection weights in [0,1] between the outputs of the membership function units and the final output units. There is an output unit for each rule and consequent membership function. The fuzzy output set with the highest membership value is taken to be the output fuzzy set. The extracted rules are of the form if x 0 is Small or x 0 is Medium and x 1 is Large or x 1 is Medium then y is Large. x 0 and x 1 are inputs and y is the output. The cost measure consists of several terms indicating how close the actual output is to a target output, how close the weights are to 0 and 1, and how close the output of membership values is to a 1 of n vector. The cost measure is a linear combination of these individual terms. By changing the constant multipliers the relative importance of the cost measures can be changed and studied. The method has been tried on randomly generated feedforward neural networks and also on data produced by functions with specific properties. The fizzy network is trained using data produced by the feedforward neural network or the known function. This method can also be used in extracting rules such as control rules implicitly used by a human if input and output data is gathered from the human.

The purpose of this text is to present some new results in the spectral theory of orthogonal polynomials and Schrodinger operators. These results concern perturbations of the free Schrodinger operator and of the free case for orthogonal polynomials on the unit circle (which corresponds to Verblunsky coefficients equal to 0) and the real line (which corresponds to off-diagonal Jacobi coefficients equal to 1 and diagonal Jacobi coefficients equal to 0). The condition central to our results is that of generalized bounded variation. This class consists of finite linear combinations of sequences of rotated bounded variation with an L^1 perturbation. This generalizes both usual bounded variation and expressions of the form lambda(x) cos(phi x + alpha) with lambda(x) of bounded variation (and, in particular, with lambda(x) = x^gamma, Wigner-von Neumann potentials) as well as their finite linear combinations. Assuming generalized bounded variation and an L^p condition (with any finite p) on the perturbation, our results show preservation of absolutely continuous spectrum, absence of singular continuous spectrum, and that embedded pure points in the continuous spectrum can only occur in an explicit finite set.

The major advantage of this proposed work is to investigate roughness of intuitionistic fuzzy subsemigroups (RIFSs) by using soft relations. In this way, two sets of intuitionistic fuzzy (IF) soft subsemigroups, named lower approximation and upper approximation regarding aftersets and foresets, have been introduced. In RIFSs, incomplete and insufficient information is handled in decision-making problems like symptom diagnosis in medical science. In addition, this new technique is more effective as compared to the previous literature because we use intuitionistic fuzzy set (IFS) instead of fuzzy set (FS). Since the FS describes the membership degree only but often in real-world problems, we need the description of nonmembership degree. That is why an IFS is a more useful set due to its nonmembership degree and hesitation degree. The above technique is applied for left (right) ideals, interior ideals, and bi-ideals in the same manner as described for subsemigroups.

A novel algorithm for segmentation of leukocytes in peripheral blood

Now we reach the centerpiece of this volume, which is the theory of bases in Banach spaces. Since every Banach space is a vector space, it has a basis in the ordinary vector space sense, i.e., a set that spans and is linearly independent.However, this definition of basis restricts us to using only finite linear combinations of vectors, while in any normed space it makes sense to deal with infinite series. Restricting to finite linear combinations when working in an infinite-dimensional space is simply too restrictive for most purposes. Moreover, the proof that a vector space basis exists is nonconstructive in general, as it relies on the Axiom of Choice. Hence we need a new notion of basis that is appropriate for infinite-dimensional Banach spaces, and that is the main topic of this chapter.

We study the arithmetic of a semigroup $\mathcal{M}_{\mathcal{P}}$ of functions with operation of multiplication representable in the form $ f(x) = \sum\nolimits_{n = 0}^\infty {{a_n}{\chi_n}(x)\quad \left( {{a_n} \ge 0,\sum\nolimits_{n = 0}^\infty {{a_n} = 1} } \right)} $ , where $$ \left\{ {{\chi_n}} \right\}_{n = 0}^\infty $$ is a system of multiplicative functions that are generalizations of the classical Walsh functions. For the semigroup $\mathcal{M}_{\mathcal{P}}$ , analogs of the well-known Khinchin theorems related to the arithmetic of a semigroup of probability measures in R n are true. We describe the class $I_0(\mathcal{M}_{\mathcal{P}})$ of functions without indivisible or nondegenerate idempotent divisors and construct a class of indecomposable functions that is dense in $\mathcal{M}_{\mathcal{P}}$ in the topology of uniform convergence.

Fat-Shattering and the Learnability of Real-Valued Functions

A robust complexity reduced proportional-integral-derivative (PID)-like fuzzy controllers is designed for a plant with fuzzy linear model. The plant model is described with the expert's linguistic information involved. The linguistic information for the plant model is represented as fuzzy sets. In order to design a robust fuzzy controller for a plant model with fuzzy sets, an approach is developed to implement the best crisp approximation of fuzzy sets into intervals. Then, Kharitonov's Theorem is applied to construct a robust fuzzy controller for the fuzzy uncertain plant with interval model. With the linear combination of input variables as a new input variable, the complexity of the fuzzy mechanism of PID-like fuzzy controller is significantly reduced. The parameters in the robust fuzzy controller are determined to satisfy the stability conditions. The robustness of the designed fuzzy controller is discussed. Also, with the provided definition of relative robustness, the robustness of the complexity reduced fuzzy controller is compared to the classical PID controller for a second-order plant with fuzzy linear model. The simulation results are included to show the effectiveness of the designed PID-like robust fuzzy controller with the complexity reduced fuzzy mechanism.

The exact distribution of the linear combination α X + β Y is derived when X and Y are exponential and gamma random variables distributed independently of each other. A measure of entropy of the linear combination is investigated. We also provide computer programs for generating tabulations of the percentage points associated with the linear combination. The work is motivated by examples in automation, control, fuzzy sets, neurocomputing and other areas of computer science.