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The main goal of this research is to achieve fairer assessment of students' knowledge and skills in tests in view of the fact that learners with similar abilities are often assigned different test grades if their achievements are borderline cases between two grades. As an example, a student who has scored 29 points out of 60 in a test will fail, while another one with 30 or 31 points will pass, which can be viewed as unjust. This problem is addressed by using fuzzy logic and different fuzzy functions to determine whether students with borderline test scores are assigned the higher or the lower grade of the two. Furthermore, although changing the grades of individual students, the authors endeavor to preserve the overall score of all test takers statistically unchanged.

We provided a generalization of the existence and uniqueness of fixed points in partially ordered metric spaces for a monotone map. We applied the major results in the investigation of coupled fixed points for ordered pairs of two maps that met various monotone features, which included a mixed monotone property or a total monotone property. To ascertain necessary requirements for the existence and uniqueness of solutions to systems of matrix equations, the results regarding coupled fixed points for ordered pairs of maps were utilized. These results are illustrated with numerical examples. Some of the known results are a consequence of the results we obtained. [ABSTRACT FROM AUTHOR]

In the study we explore an oligopoly market for equilibrium and stability based on statistical data with the help of response functions rather than payoff maximization. To achieve this, we extend the concept of coupled fixed points to triple fixed points. We propose a new model that leads to generalized triple fixed points. We present a possible application of the generalized tripled fixed point model to the study of market equilibrium in an oligopolistic market dominated by three major competitors. The task of maximizing the payout functions of the three players is modified by the concept of generalized tripled fixed points of response functions. The presented model for generalized tripled fixed points of response functions is equivalent to Cournot payoff maximization, provided that the market price function and the three players' cost functions are differentiable. Furthermore, we demonstrate that the contractive condition corresponds to the second-order constraints in payoff maximization. Moreover, the model under consideration is stable in the sense that it ensures the stability of the consecutive production process, as opposed to the payoff maximization model with which the market equilibrium may not be stable. A possible gap in the applications of the classical technique for maximization of the payoff functions is that the price function in the market may not be known, and any approximation of it may lead to the solution of a task different from the one generated by the market. We use empirical data from Bulgaria's beer market to illustrate the created model. The statistical data gives fair information on how the players react without knowing the price function, their cost function, or their aims towards a specific market. We present two models based on the real data and their approximations, respectively. The two models, although different, show similar behavior in terms of time and the stability of the market equilibrium. Thus, the notion of response functions and tripled fixed points seems to present a justified way of modeling market processes in oligopoly markets when searching whether the market has reached equilibrium and if this equilibrium is unique and stable in time [ABSTRACT FROM AUTHOR]

This work acquaints with the program Sam for interactive computer training of students on the theme "Mutual intersecting of pyramids and prisms in axonometry". The program containing three modules--teacher, student and autopilot--allows for briefest time to teach and study the whole variety of the tasks on this theme. A classification of the pierce points of the surrounding edges of the one polyhedron with the other one, based on the mutual position of their auxiliary planes, is made. The solutions of the tasks can be traced in each of the popular axonometric projection (cabinet projection, cavalier perspective, military perspective, increased orthogonal dimetry, increased orthogonal isometry) in their significant stages as well as in each step. Each stage of the solution is accompanied by commentary. The student can manipulate any object of the drawing or can apply on the whole drawing rotation, translation, expansion, shrink, zoom, etc. The student can watch on all sides the rotating 3D image of the composition of the two intersecting polyhedrons and separate the rotating 3D image of their common part. The combination of traditional methods of training in descriptive geometry using the program Sam and the classification of the pierce points enable for a thorough, creative and self study of the students. They are instruments through which the students can not only learn in depth material for a short time, but they can prepare creative course projects such as: improvisations on a task or a selection of tasks from known sources and creating original tasks on a given topic. The program also allows to create dynamic geometric drawings.

We investigate fixed points for p cyclic maps by introducing a new notion of p–cyclic infimum summing maps and a generalized best proximity point for p–cyclic maps. The idea generalizes some results about best proximity points in order to widen the class of sets and maps for which we can ensure the existence and uniqueness of best proximity points. The replacement of the classical notions of best proximity points and distance between the consecutive set arises from the well-known group traveling salesman problem and presents a different approach to solving it. We illustrate the new result with a map that does not satisfy the known results about best proximity maps for p–cyclic maps. [ABSTRACT FROM AUTHOR]

We show the power of the simultaneous usage of GeoGebra and Maple for generalizing and proving of geometry problems. We present a simple school problem, where with the help of the dynamics in GeoGebra new geometric properties are recognize and then we prove them with the help of Maple. We state an open problem for an investigation. We suggest a new construction for GeoGebra that can optimize the construction process in the extended Euclidian plane. [ABSTRACT FROM AUTHOR]

This work acquaints with the program Sam for interactive computer training of students on the theme "Mutual intersecting of pyramids and prisms in axonometry". The program containing three modules - teacher, student and autopilot - allows for briefest time to teach and study the whole variety of the tasks on this theme. A classification of the pierce points of the surrounding edges of the one polyhedron with the other one, based on the mutual position of their auxiliary planes, is made. The solutions of the tasks can be traced in each of the popular axonometric projection (cabinet projection, cavalier perspective, military perspective, increased orthogonal dimetry, increased orthogonal isometry) in their significant stages as well as in each step. Each stage of the solution is accompanied by commentary. The student can manipulate any object of the drawing or can apply on the whole drawing rotation, translation, expansion, shrink, zoom, etc. The student can watch on all sides the rotating 3D image of the composition of the two intersecting polyhedrons and separate the rotating 3D image of their common part. The combination of traditional methods of training in descriptive geometry using the program Sam and the classification of the pierce points enable for a thorough, creative and self study of the students. They are instruments through which the students can not only learn in depth material for a short time, but they can prepare creative course projects such as: improvisations on a task or a selection of tasks from known sources and creating original tasks on a given topic. The program also allows to create dynamic geometric drawings. [ABSTRACT FROM AUTHOR]

We present a new approach to the teaching of Synthetic Geometry in schools and universities with the help of DGS. The introduction of new elements into DGS helps to optimize the teaching process. These new elements are infinite points in the extended Euclidean plane and the "Swap finite & infinite points" function. We give examples of the usage of these new elements in Projectivity, Homology, Conic Sections, Plane Sections of Polyhedra, and in the application of Pappus' Theorem and Desargues' Theorem. These new features increase the benefits of DGS in teaching and learning Geometry. We optimize the education process by saving time involved in drawing, generalizing large groups of problems, stimulating and helping investigations, and forming a creative style of thinking. [ABSTRACT FROM AUTHOR]