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In this research article we proposed some important applications of Matrix Algebra tools. Role of determinants in fitting a regression line has been the presented here. Moreover, how the matrices are useful in Information theory is also discussed. The importance of Modal Matrix in obtaining the general solution of first order SLDE is shown. Lastly the use of determinants in Analytical Geometry are also discussed. [ABSTRACT FROM AUTHOR]

Spatial reasoning is the ability to see, visualize, and manipulate objects in two or three dimensions and draw conclusions based on limited information. It is a form of logical reasoning that involves spatial structure, orientation, and perception. The use of Dynamic Geometric Software (DGS) to assist students' reasoning in mathematics lessons has been suggested to affect the student's understanding positively. However, to what extent the software helps, the students' reasoning may be limited by their exploration of the software. This study aims to investigate using DGS, in this case, Geogebra Classroom, to help the students' reasoning on the conic section. The activities in the Geogebra Classroom were designed using inductive reasoning construction (Mental Rotation, Spatial Orientation, and Spatial Visualization). The data were obtained from the Analytical Geometry class of the second-year students in the mathematics education study program. The result of this study suggests that the DGS helps the students make sense of the conic section formed from the conic geometrically assisted. However, the developed DGS activities are still unable to clarify the connection between the geometrical conic section illustration and definition by the coefficient of eccentricity. [ABSTRACT FROM AUTHOR]

The problem in this study is the result of students' mathematical communication skills that are still low on analytic geometry. This study aims to analyze students' mathematical communication abilities. The indicators of mathematical communication ability are (1) Written text, namely giving answers using their own language; modeling situations or problems using spoken, written, concrete, graphics, and algebra; explaining and making mathematics questions that have been studied; listening, discussing, and writing about mathematics, making conjectures, compiling arguments and generalizations, (2) Drawing, namely reflecting real objects, pictures, and diagrams into mathematical ideas, (3) Mathematical expression, namely expressing mathematical concepts by stating the events of a day-day in language or symbols. The method in this research is descriptive qualitative. The sample of this study were students of the mathematics education study program at the Tarbiyah and Teacher Training Faculty of UIN Sultan Syarif Kasim Riau in the fourth semester of 31 students. The results of this study are students' mathematical communication skills in analytical geometry courses at a high level of 83.8%, medium 12.9%, and low 3.3%. [ABSTRACT FROM AUTHOR]

The Geometric transformation set on R2 forms a group with function composition as the binary operation. Therefore, group theory can be used to derive properties applied to the set. This study focused on the Geometric transformation property on R2 based on the type of transformation, the existence of proper subgroup and normal subgroup, and the formation of factor group. Correspondingly, this research aims to derive properties that apply to elements or their subsets. Concepts and theories used to discuss and analyze the research problems were concepts and theorems in group theory, analytic geometry of the plane, and linear algebra. Hence, the scope of the set needs to be limited to examine the research problem using an algebraic approach. Furthermore, by applying concepts and theorems relevant to the research problems, we obtained the composition and inverse transformation formulas, the proof of the existence of proper subgroups and normal subgroups, and the proof of the factor groups isomorphic to GL (2, R). [ABSTRACT FROM AUTHOR]

Zapin is a dance that consists of three elements that are complementary to create movement harmony and rhythm that are beautiful to look at, heard and see. In this research, Zapin dance namely Zapin Pat Lipat Dance were analyzed by observing the formation of the dance to construct the floor pattern and to find the existence of some geometry. The existence of geometry in Zapin Pat Lipat Dance is one of uniqueness in relationship between mathematics and dances, which produce a sense of existence and appreciation for our race as well as a respect for the culture of other races. A floor pattern is a trajectory or line that determines the shape of the pattern in the dance art that is displayed on the stage, forming a formation. Floor patterns become very important so that the movement of the dancers in the group looks neat and well and gives a solid theatrical impression. Geometry is divided into several types which are field geometry, building geometry, differential geometry, descriptive geometry, and analytical geometry. By the definition of field geometry, the floor pattern of Zapin Dance appeared in many shapes of polygon and lines specifically two-dimensional shapes. The floor patterns of Zapin Dance are explored in this research in detail step by step to analyze the movement of the dancers. The floor pattern of Zapin Dance is one of the methods used which plays a crucial role to identify every polygon that appeared in the final step. Thus, in this research, the floor pattern, and the existence of geometry in Malay Traditional Dance (Zapin Pat Lipat Dance) are explored and determined. [ABSTRACT FROM AUTHOR]

This study aims to describe (1) the knowledge that students already have, (2) strategies for solving problems of tangent circles, and (3) didactic situations of students on material tangent circles. The methodology in this study is a qualitative research of phenomenology and the subject taken is a third semester student who is taking analytic geometry courses by giving problem solving questions then conducting interviews with three students. The results of this study are students are able to explore knowledge and problem solving strategies by analyzing didactic situation theory which consists of action situations, formulation situations, and institutional situations. In action situations students develop their own solutions to find a given problem solving and provide more specific information. to be able to determine the equation of the tangent to the circle and also the student's response that is predicted to appear when determining the equation of the tangent line where students previously experienced obstacles in completing the procedure. In addition, the student's response is also predicted to determine the equation of the tangent line in other ways. The student formulation situation generalizes the problem from the given problem solving problem and determines the equation of the tangent to the circle. In the validation situation, students understand problem solving obtained from formulation activities, while in institutional situations students can change previously unknown knowledge into new knowledge they have. [ABSTRACT FROM AUTHOR]

This study aims to analyze and correct errors in the mathematical thinking structure of students in solving non-routine problems by reorganizing or defragmenting the thinking structure. This research is qualitative, using a diagnostic design. Data collection techniques use observations, tests, interviews, and documentation. The subjects in this study were three 2nd semester students of the Mathematics Education Study Program at the Muhammadiyah University of Surakarta, divided into high, medium, and low categories. The results of this study show that subjects with a high category can meet all mathematical thinking indicators before defragmentation and after defragmentation. Subjects with medium categories before defragmenting have not completed all indicators of mathematical thinking: only two indicators can be completed, namely problem identification and problem generalization. After one defragmenting process is given problems, the subject of the medium category can meet all indicators. Subjects with low categories can only complete one indicator of mathematical thinking, namely identifying problems in solving pre-defragmenting problems, as evidenced by the results of tests and interviews before defragmenting, after defragmenting three times, subjects with low categories were able to meet all indicators. [ABSTRACT FROM AUTHOR]

Polymers are rarely processed in their natural state. Single-screw extruders, in particular, have to perform increasingly challenging mixing tasks, often in order to save the additional compounding step in the course of economic efficiency. In this course, special mixing geometries are often used to increase mixing performance. Since the mixing part geometry usually acts as a pressure consumer and thus has a significant influence on the output, knowledge of the pressure-throughput behavior is of enormous importance. One of the most frequently used mixer geometries is the so-called faceted mixer or pineapple mixer. However, this has a high number of degrees of freedom, which is advantageous for the design because it allows the conveying or mixing effect to be adjusted. On the other hand, an unfavorable design can quickly lead to enormous pressure losses and, as a consequence, high throughput losses. Since an extensive investigation would be associated with high manufacturing costs, CFD simulations offer enormous potential for analyzing a large number of geometry variants. Due to this fact a large set of variants based on a simulate CCD test plan with 151 trial points is used. Based on these simulations, a pressure throughput model is then presented and compared with existing analytical models. [ABSTRACT FROM AUTHOR]

In online learning during this pandemic, pre-service mathematics teachers' engagement is the key to success in mathematics course, especially in analytic geometry courses. The way to encourage, pre-service mathematics teachers' engagement in analytical geometry lectures is to use a geogebra applet-based didactic design. This study aims to describe, pre-service mathematics teachers' engagement in online lectures using geogebra, namely in analytical geometry and multivariable calculus courses. This research was conducted on 72 pre-service mathematics teachers who enrolled in analytic geometry or multivariate courses. Data were collected using a Math Engagement Scales questionnaire consisting of cognitive, behavioral, social and emotional dimensions. This questionnaire uses a Likert scale. The data were analyzed quantitatively descriptively. The results show that pre-service mathematics teachers' engagement is in the high category. In detail, pre-service mathematics teachers' engagement in the high category in terms of cognitive, behavioral, emotional, and social engagement. These results show the use of geogebra applet-based didactic design can encourage pre-service mathematics teachers' engagement in online lectures. [ABSTRACT FROM AUTHOR]

An aerodynamic study of air flow effect around different NACA 2412 airfoil geometries was established using methods of CFD. The main focus of work is to improve efficient engineering procedures for analyzing the flow of air around modified NACA 2412 airfoils related to the CFD tools. The coordinates of the airfoil geometry as mentioned in the airfoil database of reference [21], and then importing these coordinates to the geometry modeler. Generating mesh and conduct the experimentation using Fluid Flow (Fluent). The simulation was done by solving the governing equations (Continuity, Reynolds Averaging Navier-Stokes and Energy Equation) in 2-D using Fluent analysis at Reynolds number of 1x106. Based on the results presented, forces of lift and drag has been observed to be increased respectively as increasing the angle of attack till reaching stall. The optimum angle of attack was 4o. At this specified angle, the 2412 downward geometry gives the maximum lift to drag ratio (42.534). The critical or stall angle of attack was 16o. After this angle of attack, the lift force produced from the airfoil body falls gradually and generated region is known as stall. The CFD modeling and simulating which is used in this work is suitable in circulating flow around airfoil models. [ABSTRACT FROM AUTHOR]

This study aims to describe the self-efficacy of prospective mathematics teacher students in terms of gender. The design of this research is descriptive research. The subject of this research is a class of second semester students (11 male and 29 female) who are taking analytical geometry lessons in the Mathematics Education Study Program, Department of Mathematics Education at a university in the city of Bandung, Indonesia. The data collection technique in this study was through a questionnaire designed according to self-efficacy indicators and filled out via Google Form. Data analysis using Rasch model assisted by Winstep software version 4.4.6. Based on the questionnaire analysis of student answers using the Rasch model, it is known that none of the male and female students' self-efficacy is negative. [ABSTRACT FROM AUTHOR]

The purpose of this study is to (1) describe the error of a student's mathematical problem-solving ability by applying Newman's Error Analysis (NEA) based metacognitive approach, and (2) to determine the student's mathematical problem-solving thinking trajectory. This type of research is development research using design research conducted in 2 trials, in the pilot experiment (trial 1) of research subjects amounting to 8 students and on teaching experiment (trial 2) of research subjects amounting to 42 students. The object of this research is problem-solving behavior and learning activities for students as the math learning process in the classroom progresses with the application of a metacognitive approach to the topic "Circle". The results of the study found the type of error in (1) decoding is inaccurate in identifying problems, the type of error (2) comprehension is not careful in making algebraic manipulations, the type of error (3) transformation is an error in using counting operations because understanding the problem / solving is not comprehensive, type of error (4) process skill is not careful in making calculations, type of error, and (5) encoding is not doing. Check the results of the calculation so that it is wrong in writing the final result. The mathematical problem-solving trajectory of a circle topic is the identification of circle elements, simplifying the problem by completing the existing image in the circle, creating a new image model, and observing/looking for solutions to related problems. [ABSTRACT FROM AUTHOR]

In the current era of online learning, the integration of technology in task design in mathematics learning is very important. This integration is called digital task design. This study aims to describe and analyze digital task design with the framework of The Theory of Didactical Situation. The digital task design using the GeoGebra applet. The research method used is Didactic Engineering with four stages of preliminary analysis, a priori analysis, implementation and a posteriori analysis. The research subjects are pre-service mathematics teachers who take the Analytical Geometry course. The result of this research is a digital task design with the Theory of Didactical Situation framework using Geogebra-applet. [ABSTRACT FROM AUTHOR]

Task design is a key factor in learning. In today's online learning context, task design is integrated with technology called digital task design. The success of using digital task design is determined by the perception of it. The purpose of this study is to analyze the perception of pre-service mathematics teachers on digital task design used in analytic geometry and multivariable calculus courses. This study involved 72 pre-service mathematics teachers. Data were collected using a questionnaire with a Likert scale. Data were analyzed using descriptive statistics including mean, maximum-minimum score and standard deviation. The results show that the majority of pre-service mathematics teachers have a positive perception of digital task design in the high category. Positive perceptions with high categories appear in the domain of learning effectiveness, depth and breadth of material and enthusiasm in learning. However, pre-service mathematics teachers see that digital task design is a new burden for them in learning. The implication of this research is that the use of digital task design can be considered as an approach in learning mathematics. [ABSTRACT FROM AUTHOR]

Centerless superfinishing with the longitudinal feed is one of the most productive finishing techniques. Therefore, centerless superfinishing is widely used in the bearing industry. The geometrical and kinematic diagrams of superfinishing with longitudinal feed are complex. The surface of the product is formed as an envelope to the points of contact of individual particles of abrasive bars during their oscillation motion. Therefore, calculating the shafts of a superfinishing machine requires the creation of mathematical models based on analytical geometry and envelope theory. The most difficult can be considered the processing of barrel rollers of rolling bearings. The article presents geometric and kinematic models, obtained on the basis of a new approach. According to the proposed approach, in the process of the rollers move along an arc of a circle relative to the center of oscillation of abrasive bars with a toroidal working surface. During machining, the rollers have a point rather than linear contact with the machine shafts. Equations are obtained for calculating the shafts of a superfinishing machine for a given roller size. A numerical algorithm was used to solve the system of equations due to a large number of unknown parameters. Recommendations on the choice of geometric parameters for setting up the centerless superfinishing machine are given. The developed mathematical models have found application at the plants of the European Bearing Company. [ABSTRACT FROM AUTHOR]

The article discusses an approach to research in the field of interpore space analysis, which is primarily geometric in nature. The method of functional-voxel modeling is used as a basic geometric tool, since it allows you to form graphic M-images (model image) of any dimension. For the analytical representation of the geometry and Voronoi diagram of the model, the mathematical apparatus of R-functions (Rvachev functions) is used. A software implementation of the proposed approaches for constructing a Voronoi diagram in three-dimensional space will be presented. [ABSTRACT FROM AUTHOR]

Using methods of force analysis and analytical geometry, theoretical dependencies are obtained that allow determining the location of vibration impact mechanisms on elastic C-shaped struts of cultivator hoes, as well as calculating the length of the lever of the pendulum impact device of such a mechanism. Calculations showed that for the working organ of the cultivator КПЭ-3,8 the height of the vibration impact mechanism on the C-shaped strut of the cultivator hoe is within 0.38-0.43 m, and the length of the lever of the pendulum percussion device is within 0.14-0.22 m. [ABSTRACT FROM AUTHOR]

This study aims to reveal the mathematical thinking process in the construction pattern of visual-spatial mathematical ability mastery in the Analytical geometry course by using Transcript Based Learning Analysis (TBLA). The research method used is descriptive qualitative, with the subjects being students of mathematics education at the U.G.J. level II in the Analytical Geometry course. The data collection method is Transcript Base Lesson Analysis (TBLA). Thisresearch is focused on learning transcripts that are made, visible learning, and observation of learning implementation. Learning activities using zoom meetings are recorded, then the recording will be used as a transcript of the learning dialogueand then analyzed. The essential concepts analyzed include analytical patterns and language functions. Knowledge construction analysis can be viewed based on the dialogue. The limitation of the type of dialogue is potentially relevant tothe construction of knowledge; student dialogue is focused on student speech related to the construction of student knowledge. The results obtained in this study are the students' thinking processes from the five aspects of students' mathematical thinking, namely Conjecturing, Reasoning, and Proving, Abstraction Generalization, and Specialization havebeen carried out by all students, but for the conjecturing aspect, there are still some students who do not write down the detailed steps in answering some questions. The students' mathematical thinking process builds a construction pattern for the mastery of visual-spatial math skills in the geometry of the subject. Overall mastery of the circle material does not experience much difficulty. [ABSTRACT FROM AUTHOR]

The shoe last is the main forming tool used in the footwear production, the shape and size of which determines both the appearance and convenience of shoes during their use. The shape of the shoe last depends on a number of factors, including ergonomic factors, the purpose, the construction of shoes, the materials and technologies used, fashion requirements, and aesthetic preferences of the fashion designer. A shoe last is developed based on anthropometric data of a foot, but it is not a copy of the foot. The foot shape is transformed into the shape of the shoe last according to complex patterns. This article discusses the principles proposed by the authors for the formation of the surface of a shoe last in its design based on the anthropometric data of the foot. Methods of descriptive and analytical geometry, information about the anatomy, and biomechanics of the foot, are used to develop approaches to the formation of the shoe last surface. The shoe last surface refers to complex non-linear surfaces. To set it, it is proposed to use several guide curves, which are the characteristic lines of the shoe last. The approaches considered are used in the developed methods and algorithms for the design of shoe lasts based on the anthropometric data of the foot and can be applied in the development of software for the automated design of shoe lasts. The use of the proposed approaches to the shaping of the last surface makes it possible to raise the process of designing shoe lasts according to foot data to a new level, to make it more natural and scientifically justified. [ABSTRACT FROM AUTHOR]

The main objective of the article is to develop a mathematical model for assessing the environmental and economic effectiveness of forest management taking into account the planning of the transport network on the territory of the forest fund. The objective is set due to the fact that, despite the relevance of the problem of ensuring the effective environmental, socio-economic development of the territories of the forest fund and the existing scientific achievements in this area, both in Russia and abroad, a small amount of research has so far been accumulated in the scientific environment, which considers forest transport systems as one of the main factors that, in modern conditions of management, contributes to the achievement of the goal of sustainable development of forest areas. A comprehensive consideration of the problem of environmental and economic efficiency of planning a transport network on the territory of the forest fund is only indirectly present in the scientific literature and is local in nature. The presented article proposes an approach according to which a quantitative criterion for assessing the effectiveness of forest management is formed as achieving the maximum of all potential income from the use of various types of forest resources and the ecological potential of the territory taking into account the factor of the level of development of the transport network. This approach allows to strike an optimal balance between the economic and environmental indicators of the project for the creation and development of a transport network on the basis of the criterion of integrated environmental and economic efficiency. When solving the problems set forth in the article, methods and means of economic analysis, financial mathematics, economic and mathematical modeling, and analytical geometry have been used. [ABSTRACT FROM AUTHOR]

Analytic geometry in the plane is a very important part of geometry, especially for the students of mechanical engineering study field. They start studying the analytic geometry in the plane within the secondary school study. Although the study of analytic geometry in the plane is important for the future study at university, students often have insufficient knowledge of the above mentioned part of geometry when entering university. The basics, such as the parametric equations of the straight-line, coordinates of the point lying on the straight-line, the parameter of the point etc. cause students major problems. Therefore, the GeoGebra book containing the chapters of analytic geometry in the plane has been created for the students as a learning tool for the repetitive lessons in the first term of the university study. There is also a quick test for checking the acquired knowledge created in the GeoGebra book. [ABSTRACT FROM AUTHOR]

Metaheuristic algorithm is preferred by the many researches to find the minimum weight of truss structures considering size and/or shape design variables which are the truss element’s cross-sectional area and joint coordinates of the geometry of the trusses for the size and shape optimization, respectively. The design of plane and spatial truss structures are optimized by a popular metaheuristic optimization technique named Teaching-Learning-Based Optimization (TLBO). Finite element analyses of structures and optimization process were carried out by the computer program coded in MATLAB visually developed by the authors. Two benchmark problems (37-bar, and 72-bar) taken from literature were optimized and the optimal solutions belong this paper were compared with previous studies. [ABSTRACT FROM AUTHOR]

The efficiency of different hybrid and meta-hybrid DFT exchange-correlation functionals (B3LYP, B3LYP*, B3LYP-D, CAM-B3LYP, CAM-B3LYP-D, HSE06, ωB97XD, M06L, M06, M062X, M11L, M11, MN12-L, MN12-SX, N12 and N12-SX) in describing the adiabatic singlet-triplet energy gap as well as the geometry conformations obtained with singlet and triplet spin states have been investigated in the case of the organometallic complex of Ni-tetrakis-(pentafluorophenyl)porphyrin-phenazopyridine. The results show that proper consideration of the dispersion effects as well as the amount of Hartree-Fock (HF) exchange is very important in order to have a balanced description for the electronic energies and geometry coordinates, otherwise, we can get significantly different results for different XC functionals. By using functionals with moderate HF-exchange it is possible to get a fortuitous cancellation of errors between the static correlation effects and the self-interaction errors. [ABSTRACT FROM AUTHOR]

In this paper we will highlight some properties of fuzzy bounded sets in fuzzy normed linear spaces. More precisely, we will prove that any finite union or sum of fuzzy bounded sets is fuzzy bounded. We show that the closure of a fuzzy bounded set is also fuzzy bounded. We give a characterization for fuzzy bounded sets and finally we obtain that the Cartesian product of two fuzzy bounded sets is also fuzzy bounded. [ABSTRACT FROM AUTHOR]

In this paper we report the bipolar coordinate formulation of Jeffery-Hamel (JH) flow and study vorticity-stream function, bipolar coordinate JH and conventional JH solutions. It is shown that our derived equation can intrinsically describe the flow behavior in curved geometries and it gives improved results as compared to conventional JH theory. [ABSTRACT FROM AUTHOR]

The BBMCSFilter method was developed to solve mixed integer nonlinear programming problems. This kind of problems have integer and continuous variables and they appear very frequently in process engineering problems. The objective of this work is to analyze the performance of the method when the coordinate searches are interrupted in the context of the multistart strategy. From the numerical experiments, we observed a reduction on the number of function evaluations and on the CPU time. [ABSTRACT FROM AUTHOR]

RF sheaths rectification can be the reason for operational limits for Ion Cyclotron Range of Frequencies (ICRF) heating systems via impurity production or excessive heat loads. To simulate this process in realistic geometry, the Self-consistent Sheaths and Waves for Ion Cyclotron Heating (SSWICH) code is a minimal set of coupled equations that computes self-consistently wave propagation and DC plasma biasing. The present version of its wave propagation module only deals with the Slow Wave assumed to be the source of RF sheath oscillations. However the ICRF power coupling to the plasma is due to the fast wave (FW). This paper proposes to replace this one wave equation module by a full wave module in either 2D or 3D as a first step towards integrated modelling of RF sheaths and wave coupling. Since the FW is propagative in the main plasma, Perfectly Matched Layers (PMLs) adapted for plasmas were implemented at the inner side of the simulation domain to absorb outgoing waves and tested numerically with tilted B0 in Cartesian geometry, by either rotating the cold magnetized plasma dielectric tensors in 2D or rotating the coordinate vector basis in 3D. The PML was further formulated in cylindrical coordinates to account for for the toroidal curvature of the plasma. Toroidal curvature itself does not seem to change much the coupling. A detailed 3D geometrical description of Tore Supra and ASDEX Upgrade (AUG) antennas was included in the coupling code. The full antenna structure was introduced, since its toroidal symmetry with respect to the septum plane is broken (FS bars, toroidal phasing, non-symmetrical structure). Reliable convergence has been obtained with the density profile up to the leading edge of antenna limiters. Parallel electric field maps have been obtained as an input for the present version of SSWICH. [ABSTRACT FROM AUTHOR]

Two main parameters rule the performance of an Image Acquisition System, namely, spatial resolution and contrast. For radiographic systems using cone beam arrangements, the farther the source, the better the resolution, but the contrast would diminish due to the lower statistics. A closer source would yield a higher contrast but it would no longer reproduce the attenuation map of the object, as the incoming beam flux would be reduced by unequal large divergences and attenuation factors. This work proposes a procedure to correct these effects when the object is comprised of a hull -or encased in it - possessing a shape capable to be described in analytical geometry terms. Such a description allows the construction of a matrix containing the attenuation factors undergone by the beam from the source until its final destination at each coordinate on the 2D detector. Each matrix element incorporates the attenuation suffered by the beam after its travel through the hull wall, as well as its reduction due to the square of distance to the source and the angle it hits the detector surface. When the pixel intensities of the original image are corrected by these factors, the image contrast, reduced by the overall attenuation in the exposure phase, are recovered, allowing one to see details otherwise concealed due to the low contrast. In order to verify the soundness of this approach, synthetic images of objects of different shapes, such as plates and tubes, incorporating defects and statistical fluctuation, have been generated, recorded for further comparison and afterwards processed to improve their contrast. The developed algorithm which, generates processes and plots the images has been written in Fortran 90 language. As the resulting final images exhibit the expected improvements, it therefore seemed worthwhile to carry out further tests with actual experimental radiographies. [ABSTRACT FROM AUTHOR]

This paper describes approaches to numerical simulation of various classes of close binary stars using Eulerian gas dynamics and cartesian grids. Possible obstacles, advantages and drawbacks of different methods are discussed with respect to a brought-up classification of binary systems from point of view of a numerical method. Results of a 3D numerical simulation of envelopes of T Tauri stars are presented. [ABSTRACT FROM AUTHOR]

The interface damage and the whole damage of the scratch test of NiTi/Q235 steel composites were simulated. The following conclusions were summed up by analyzing the damage model results and comparing with the results of the model whose interface was perfectly bonded. In the scratch test, when loading normal force and tangential displacement the interface and the substrate damages were in the front of the indenter, not under the indenter. When the interface damage happened and expanded the damage close to interface in the substrate happened and expanded to the front and the underside. The debonding happened in the interface where the shear stress was the maximum. When the coating warped, the shear stress curve fluctuated seriously. But the interface shear stress curve of the perfectly bonded model was smoother. And the surface indentation depth curve of the damage model had mutation where the interface warped. [ABSTRACT FROM AUTHOR]

We present analytic solutions to two test problems that can be used to check the hydrodynamic implementation in computer codes designed to calculate the propagation of shocks in spherically convergent geometry. Our analysis is restricted to fluid materials with constant bulk modulus. In the first problem we present the exact initial acceleration and pressure gradient at the outer surface of a sphere subjected to an exponentially decaying pressure of the form P(t) = P0e-αt. We show that finely-zoned hydro-code simulations are in good agreement with our analytic solution. In the second problem we discuss the implosions of incompressible spherical fluid shells and we present the radial pressure profile across the shell thickness. We also discuss a semi-analytic solution to the time-evolution of a nearly spherical shell with arbitrary but small initial 3-dimensional (3-D) perturbations on its inner and outer surfaces. [ABSTRACT FROM AUTHOR]

In this paper we introduce the notion of anti fuzzy M-semigroup, inspired by the theory of M-semigroup and fuzzy M-semigroup. We justify that the intersection of two anti fuzzy M-semigroup is again an anti fuzzy M-semigroup and the union need not be true by means of a counter example. Given two anti fuzzy M-semigroup we define the generalized Cartesian product of two anti fuzzy M-semigroup and provide some interesting results on it. [ABSTRACT FROM AUTHOR]