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In this paper, we introduce a new classTs, a k [A, B, a, b] of analytic functions by using a newly defined convolution Operator. This class contains many known classes of analytic and univalent functions as special cases. We derived some interesting results including inclusion relationships, a radius problem and sharp coefficient bound for this class.

In this paper,  a  one-dimensional modified Burgers'  equation is considered for different  Reynolds numbers. For very high Reynolds numbers,  the solution possesses a multiscale character in some part of the independent domain and thus can be classified as a  singularly perturbed problem. A numerical scheme that uses a fitted Operator finite difference scheme to solve the spatial derivatives and the implicit Euler scheme for the time derivative is proposed to solve the modified  Burgers'  equation via Rothe's method. It is important to note that the proposed fitted Operator finite difference scheme is based on the midpoint upwind scheme. The stability of the scheme is established and the error associated with each discretisation is estimated. Numerical simulations are carried out to validate the theoretical findings.

Generally, the criteria involved in a decision making problem are interactive or inter-dependent, and therefore aggregating them by the use of traditional Operators which are based on additive measures is not logical. This verifies that we have to implement fuzzy measures for modelling the interaction phenomena among the criteria. On the other hand, based on the recent extension of hesitant fuzzy set, called higher order hesitant fuzzy set (HOHFS) which allows the membership of a given element to be defined in forms of several possible generalized types of fuzzy set, we encourage to propose the higher order hesitant fuzzy (HOHF) Choquet integral Operator. This concept not only considers the importance of the higher order hesitant fuzzy arguments, but also it can reflect the correlations among those arguments. Then, a detailed discussion on the aggregation properties of the HOHF Choquet integral Operator will be presented. To enhance the application of HOHF Choquet integral Operator in decision making, we first assess the appropriate energy policy for the socio-economic development. Then, the efficiency of the proposed HOHF Choquet integral Operator-based technique over a number of exiting techniques is further verified by employing another decision making problem associated with the technique of TODIM (an acronym in Portuguese of Interactive and Multicriteria Decision Making).

A second order finite difference scheme is constructed to solve a singularly perturbed degenerate parabolic convection diffusion problem via Rothe's method. The solution of the problem exhibits a boundary layer on the left side of the spatial domain.  By means of the Crank Nicolson finite difference scheme, the time derivative is discretised to obtain a set of semi-discrete boundary value problems. Using a fitted Operator finite difference scheme based on the midpoint downwind scheme,  the system of boundary value problems are discretized and analysed for convergence. Second order accuracy is established for each discretisation process. Numerical simulations are carried out to validate the theoretical error estimate.

Intelligent agents are considered as significant means towards realizing the semantic web vision. On the Semantic Web, integrating ontologies and rules enables software agents to interoperate between them, however, this leads to a problem, that no studies have focused on effective distributed reasoning for integrating ontologies and rules in multiple knowledge-bases. The methods that have been presented for distributed reasoning not only get a lot of times and memory, but also do not lead to a complete and sound reasoning. In this paper, to solve this problem, we present a distributed reasoning system that deals with the representation of the knowledge-base of order sorted logic. This logic is able to describe the hierarchy of predicates and inheritance of expressions that there are in our natural language. To have a distributed reasoning, our proposed method uses the expansion of rigid and valid-non-rigid properties between knowledge-bases. Furthermore, with considering time and the situation of properties for reasoning, the non-rigid properties have not been ignored, in fact, in their valid time and situation, they are used. With this method, we achieve a complete reasoning and, moreover, the extracted knowledge is completely considered in the knowledge-bases and we have a distributed reasoning with high efficiency and sound without missing any information.

The Schwarz inequality and Jensen’s one are fundamental in a Hilbert space. Regarding a sesquilinear map B(X, Y ) = Y*X as an Operatorvalued inner product, we discuss Operator versions for the above inequalities and give simple conditions that the equalities hold.