Search results for: piecewise linear representation

Authors: Kushakov Kholmurodjon, Muhammadjonov Akbarshox

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This present paper is devoted to a system of second-order nonlinear differential equations with a special right-hand side, exactly, the linear part and a third-order polynomial of a special form. It is shown that for some relations between the parameters, there is a second-order curve in which trajectories leaving the points of this curve remain in the same place. Thus, the curve is invariant with respect to the given system. Moreover, this system is invariant under a non-degenerate linear transformation of variables. The form of this curve, depending on the relations between the parameters and the eigenvalues of the matrix, is proved. All solutions of this system of differential equations are shown analytically.

Keywords: dynamic system, ellipse, hyperbola, Hess system, polar coordinate system

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Authors: Nikolai Krainiukov, Boris Melnikov

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We study the properties of maximal prefix codes. The codes have many applications in computer science, theory of formal languages, data processing and data classification. Practical application is based on the representation of the maximal prefix codes as a sequence of words in a specific order. Our approach to study uses finite state automata (so-called flower automata) for the representation of prefix codes. An important task is the decomposition of prefix codes into prime prefix codes (factors). We discuss the properties of such prefix code decompositions. A linear time algorithm is designed which find the prime decomposition. To verify the correctness of the proposed algorithms, we implemented a system computer algebra GAP.

Keywords: maximal prefix code, regular languages, flower automata, prefix code decomposing

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Authors: Manal Azzidani

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This research consideres the extension of special functions called Positive Linear Operators. the bounded linear operator which defined from normed space to Banach space will extend to the closure of the its domain, And extend identified linear functional on a vector subspace by Hana-Banach theorem which could be generalized to the positive linear operators.

Keywords: extension, positive operator, Riesz space, sublinear function

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Authors: Ang Ling Weay, Mona Masood

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This paper discusses the potential benefits of an interactive multimedia information representation in enhancing students’ critical thinking aligned with history reasoning in learning history between Secondary School students in Malaysia. Two modes of multimedia information representation implemented which are chronological and thematic information representation. A qualitative study of an unstructured interview was conducted among two history teachers, one history education lecturer, two i-think expert and program trainers and five form 4 secondary school students. The interview was to elicit their opinions on the implementation of thinking maps and interactive multimedia information representation in history learning. The key elements of interactive multimedia (e.g. multiple media, user control, interactivity, and use of timelines and concept maps) were then considered to improve the learning process. Findings of the preliminary investigation reveal that the interactive multimedia information representations have the potential benefits to be implemented as instructional resource in enhancing students’ higher order thinking skills (HOTs). This paper concludes by giving suggestions for future work.

Keywords: multimedia information representation, critical thinking, history reasoning, chronological and thematic information representation

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Authors: Samai'la Abdullahi

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A graph is a pair of two set and so that a graph is a pictorial representation of a system using two basic element nodes and edges. A node is represented by a circle (either hallo shade) and edge is represented by a line segment connecting two nodes together. In this paper, we present a circuit network in the concept of graph theory application and also circuit models of graph are represented in logical connection method were we formulate matrix method of adjacency and incidence of matrix and application of truth table.

Keywords: euler circuit and path, graph representation of circuit networks, representation of graph models, representation of circuit network using logical truth table

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Authors: O. Poleshchuk, E. Komarov

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This paper presents a regression model for interval type-2 fuzzy sets based on the least squares estimation technique. Unknown coefficients are assumed to be triangular fuzzy numbers. The basic idea is to determine aggregation intervals for type-1 fuzzy sets, membership functions of whose are low membership function and upper membership function of interval type-2 fuzzy set. These aggregation intervals were called weighted intervals. Low and upper membership functions of input and output interval type-2 fuzzy sets for developed regression models are considered as piecewise linear functions.

Keywords: interval type-2 fuzzy sets, fuzzy regression, weighted interval

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Authors: A. Lashkari, H. Khalafi, H. Khazeminejad, S. Khakshourniya

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In this paper, a numerical model is presented. The model is used to analyze a steady state thermo-hydraulic and reactivity insertion transient in TRR reference cores respectively. The model predictions are compared with the experiments and PARET code results. The model uses the piecewise constant and lumped parameter methods for the coupled point kinetics and thermal-hydraulics modules respectively. The advantages of the piecewise constant method are simplicity, efficiency and accuracy. A main criterion on the applicability range of this model is that the exit coolant temperature remains below the saturation temperature, i.e. no bulk boiling occurs in the core. The calculation values of power and coolant temperature, in steady state and positive reactivity insertion scenario, are in good agreement with the experiment values. However, the model is a useful tool for the transient analysis of most research reactor encountered in practice. The main objective of this work is using simple calculation methods and benchmarking them with experimental data. This model can be used for training proposes.

Keywords: thermal-hydraulic, research reactor, reactivity insertion, numerical modeling

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Authors: Myung Hwan Na, Ho- Chun Song, EunHee Hong

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We widely use normal linear mixed-effects model to analysis data in repeated measurement. In case of detecting heteroscedasticity and the non-normality of the population distribution at the same time, normal linear mixed-effects model can give improper result of analysis. To achieve more robust estimation, we use heavy tailed linear mixed-effects model which gives more exact and reliable analysis conclusion than standard normal linear mixed-effects model.

Keywords: reliability, tires, field data, linear mixed-effects model

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Authors: Fatma Susilawati Mohamad, Azizah Abdul Manaf, Fadhillah Ahmad, Zarina Mohamad, Wan Suryani Wan Awang

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A simple and robust multi-class linear classifier is proposed and implemented. For a pair of classes of the linear boundary, a collection of segments of hyper planes created as perpendicular bisectors of line segments linking centroids of the classes or part of classes. Nearest Neighbor and Linear Discriminant Analysis are compared in the experiments to see the performances of each classifier in discriminating ripeness of oil palm. This paper proposes a multi-class linear classifier using Linear Discriminant Analysis (LDA) for image identification. Result proves that LDA is well capable in separating multi-class features for ripeness identification.

Keywords: multi-class, linear classifier, nearest neighbor, linear discriminant analysis

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Authors: S. H. Nasseri, A. Ebrahimnejad

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Fuzzy set theory has been applied to many fields, such as operations research, control theory, and management sciences. In this paper, we consider two classes of fuzzy linear programming (FLP) problems: Fuzzy number linear programming and linear programming with trapezoidal fuzzy variables problems. We state our recently established results and develop fuzzy primal simplex algorithms for solving these problems. Finally, we give illustrative examples.

Keywords: fuzzy linear programming, fuzzy numbers, duality, sensitivity analysis

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Authors: B. A. A. Paixão, J. V. B. Gomide

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This work aims to analyze the locative structure used by the locative games of the company Niantic. To fulfill this objective, a literature review on the representation and simulation of cities was developed; interviews with Ingress players and playing Ingress. Relating these data, it was possible to deepen the relationship between the virtual and the real to create the simulation of cities and their cultural objects in locative games. Cities representation associates geo-location provided by the Global Positioning System (GPS), with augmented reality and digital image, and provides a new paradigm in the city interaction with its parts and real and virtual world elements, homeomorphic to real world. Bibliographic review of papers related to the representation and simulation study and their application in locative games was carried out and is presented in the present paper. The cities representation and simulation concepts in locative games, and how this setting enables the flow and immersion in urban space, are analyzed. Some examples of games are discussed for this new setting development, which is a mix of real and virtual world. Finally, it was proposed a Locative Structure for electronic games using the concepts of heterotrophic representations and isotropic representations conjoined with immediacy and hypermediacy.

Keywords: cities representation, cities simulation, games simulation, immersion, locative games

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Authors: J. Urban, Á. S. Botella, L. E. Robaina, A. Bárta, P. Souček, P. Císař, Š. Papáček, L. M. Domínguez

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Skin coloration in fish is of great physiological, behavioral and ecological importance and can be considered as an index of animal welfare in aquaculture as well as an important quality factor in the retail value. Currently, in order to compare color in animals fed on different diets, biochemical analysis, and colorimetry of fished, mildly anesthetized or dead body, are very accurate and meaningful measurements. The noninvasive method using digital images of the fish body was developed as a standalone application. This application deals with the computation burden and memory consumption of large input files, optimizing piece wise processing and analysis with the memory/computation time ratio. For the comparison of color distributions of various experiments and different color spaces (RGB, CIE L*a*b*) the comparable semi-equidistant binning of multi channels representation is introduced. It is derived from the knowledge of quantization levels and Freedman-Diaconis rule. The color calibrations and camera responsivity function were necessary part of the measurement process.

Keywords: color distribution, fish skin color, piecewise transformation, object to background segmentation

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Authors: Chetna Jaswal, Nishi Srivastava, Ahammedul Kabeer AP, Puja Prasad

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We are a culture of mass media consumers and cinema as its integral part has high visibility and potential influence on public attitude towards disability which maintains no sociocultural boundaries but experiences substantial social marginalization. Given the lack of awareness and direct experience with disability, on-screen or film representations can give powerful and memorable definitions for the public that can contribute to framing the perception and attitude change. Social representation refers to common ways of thinking, conceiving about and evaluating social reality. It is a product of collective cognition, common sense and thought system. This study aims at analyzing the representations and narratives of disability in Indian cinema and Hollywood with the help of a conceptual understanding of social representation and its theoretical framework.

Keywords: disability, social representation, mainstream cinema, diversity

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Authors: Quy Dang Nguyen, Reza Nakhaie Jazar

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The paper investigates vibration dynamics of tire-road separation for a quarter-car model; this separation model is developed to be close to the real situation considering the tire is able to separate from the ground plane. A set of piecewise linear mathematical models is developed and matches the in-contact and no-contact states to be considered as mother models for further investigations. The bound dynamics are numerically simulated in the time response and phase portraits. The separation analysis may determine which values of suspension parameters can delay and avoid the no-contact phenomenon, which results in improving ride comfort and eliminating the potentially dangerous oscillation. Finally, model verification is carried out in the MSC-ADAMS environment.

Keywords: quarter-car vibrations, tire-road separation, separation analysis, separation dynamics, ride comfort, ADAMS validation

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Authors: Lee Myung-Won, Kwak Keun-Chang

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This paper is concerned with knowledge representation and extraction of fuzzy if-then rules using Interval Type-2 Context-based Fuzzy C-Means clustering (IT2-CFCM) with the aid of fuzzy granulation. This proposed clustering algorithm is based on information granulation in the form of IT2 based Fuzzy C-Means (IT2-FCM) clustering and estimates the cluster centers by preserving the homogeneity between the clustered patterns from the IT2 contexts produced in the output space. Furthermore, we can obtain the automatic knowledge representation in the design of Radial Basis Function Networks (RBFN), Linguistic Model (LM), and Adaptive Neuro-Fuzzy Networks (ANFN) from the numerical input-output data pairs. We shall focus on a design of ANFN in this paper. The experimental results on an estimation problem of energy performance reveal that the proposed method showed a good knowledge representation and performance in comparison with the previous works.

Keywords: IT2-FCM, IT2-CFCM, context-based fuzzy clustering, adaptive neuro-fuzzy network, knowledge representation

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Authors: R. B. Ogunrinde, C. C. Jibunoh

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In this paper, a spectral decomposition method is developed for the direct integration of stiff and nonstiff homogeneous linear (ODE) systems with linear, constant, or zero right hand sides (RHSs). The method does not require iteration but obtains solutions at any random points of t, by direct evaluation, in the interval of integration. All the numerical solutions obtained for the class of systems coincide with the exact theoretical solutions. In particular, solutions of homogeneous linear systems, i.e. with zero RHS, conform to the exact analytical solutions of the systems in terms of t.

Keywords: spectral decomposition, linear RHS, homogeneous linear systems, eigenvalues of the Jacobian

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Authors: Jawad Abuhlail, Hamza Hroub

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Let R be a commutative ring. Representation theory studies the representation of R-modules as (possibly finite) sums of special types of R-submodules. Here we are interested in a class of R-modules between the class of semisimple R-modules and the class of R-modules that can be written as (possibly finite) sums of secondary R-submodules (we know that every simple R-submodule is secondary). We investigate R-modules which can be written as (possibly finite) sums of second R-submodules (we call those modules second representable). Moreover, we investigate the class of (main) second attached prime ideals related to a module with such representation. We provide sufficient conditions for an R-module M to get a (minimal) second representation. We also found the collection of second attached prime ideals for some types of second representable R-modules, in particular within the class of injective R-modules. As we know that every simple R-submodule is second and every second R-submodule is secondary, we can see the importance of the second representable R-module.

Keywords: lifting modules, second attached prime ideals, second representations, secondary representations, semisimple modules, second submodules

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Authors: Revytia Tanera

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This research is a pilot study to examine the rise of selfie trend in dealing with individual self representation and identity building in social media. The symbolic interactionism theory is used as the concept of the desired self image, and Cooley’s looking glass-self concept is used to analyze the mechanical reflection of ourselves; how do people perform their “digital self” in social media. In-depth interviews were conducted in the study with a non-random sample who owns a smartphone with a front camera feature and are active in social media. This research is trying to find out whether the selfie trend brings any influence on identity building on each individual. Through analysis of interview results, it can be concluded that people take selfie photos in order to express themselves and to boost their confidence. This study suggests a follow up and more in depth analysis on identity and self representation from various age groups.

Keywords: self representation, selfie, social media, symbolic interaction, looking glass-self

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Authors: Amanda Julia Binay, Patricia Elise Suarez

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Female representation in media is an important subject in the discussion of gender equality, especially in impactful and influential media like film. As the Filipino film industry continues to grow and evolve, the need for analysis on Filipino female representation on screen is imperative. Additionally, there has been limited research made on female representation in the Philippine film scene. Thus, the paper aims to analyze the presence and evolution of female representation in Philippine cinema and compare the findings with that of American films to see how Filipino filmmakers hold their own against the standards of international movements that call for more and better female representation, especially in Hollywood. The participants selected were Filipino and American films released within the years 1975 to 2020 in five (5) year intervals. Twenty (20) critically acclaimed and highest-grossing Filipino films and twenty (20) critically acclaimed and highest-grossing Hollywood films were then subject to the Bechdel and Peirce tests to obtain statistical measures of their female representation. The findings of the study reveal that the presence of female representation in Philippine film history has been consistent and has continued to grow and evolve throughout the years, with strong female leads with vibrant characteristics and diverse stories. However, analysis of female representation regarding American films has shown an extreme lack thereof with more misogynistic, sexist, and limiting ideals. Thus, the study concludes that the state of female representation in Philippine cinema and film industry holds its own when compared to American cinema and film industry and even outperforms it in many aspects of female representation, such as consistent inclusion and depiction of multi-dimensional female leads and female relationships. Hence, the study implies that women’s consistent presence in Philippine cinema mirrors Filipino women’s prominent role in Philippine society and that American cinema must continue to make efforts to change their portrayals of female characters, leads, and relationships to make them more grounded in reality.

Keywords: female representation, gender studies, feminism, philippine cinema, American cinema, bechdel test, peirce test, comparative analysis

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Authors: Chien-Hung Chen, Yan-Chun Hsing, Yung-Chun Chang

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The Internet has grown into a powerful medium for information dispersion and social interaction that leads to a rapid growth of social media which allows users to easily post their emotions and perspectives regarding certain topics online. Our research aims at using natural language processing and text mining techniques to explore the public emotions expressed on Twitter by analyzing the sentiment behind tweets. In this paper, we propose a composite kernel method that integrates tree kernel with the linear kernel to simultaneously exploit both the tree representation and the distributed emotion keyword representation to analyze the syntactic and content information in tweets. The experiment results demonstrate that our method can effectively detect public emotion of tweets while outperforming the other compared methods.

Keywords: emotion recognition, natural language processing, composite kernel, sentiment analysis, text mining

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Authors: Navdeep Goel, Salvador Gabarda

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Digital images are widely used in computer applications. To store or transmit the uncompressed images requires considerable storage capacity and transmission bandwidth. Image compression is a means to perform transmission or storage of visual data in the most economical way. This paper explains about how images can be encoded to be transmitted in a multiplexing time-frequency domain channel. Multiplexing involves packing signals together whose representations are compact in the working domain. In order to optimize transmission resources each 4x4 pixel block of the image is transformed by a suitable polynomial approximation, into a minimal number of coefficients. Less than 4*4 coefficients in one block spares a significant amount of transmitted information, but some information is lost. Different approximations for image transformation have been evaluated as polynomial representation (Vandermonde matrix), least squares + gradient descent, 1-D Chebyshev polynomials, 2-D Chebyshev polynomials or singular value decomposition (SVD). Results have been compared in terms of nominal compression rate (NCR), compression ratio (CR) and peak signal-to-noise ratio (PSNR) in order to minimize the error function defined as the difference between the original pixel gray levels and the approximated polynomial output. Polynomial coefficients have been later encoded and handled for generating chirps in a target rate of about two chirps per 4*4 pixel block and then submitted to a transmission multiplexing operation in the time-frequency domain.

Keywords: chirp signals, image multiplexing, image transformation, linear canonical transform, polynomial approximation

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Authors: Zainab Abdulkarim

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Literature is the study of life, a source of knowledge. It involves the truth about many things in life. Most of these creative artistes most especially the poets are representatives of the voices of the people. These set of artistes have been the critics to all involved in the development of their nation. This paper will examine how Nigerian Poets goes further not just by writing but by showing the different ways the country has been convoluted. This paper intends to show the power and ability literature has in representation. The power is to represent the important values of life. There is no doubt that literature asserts truth. Through the various poems examined in this paper, Nigerian Poets have proved to portray the realities of the nation.

Keywords: literature, poets, reality, representation

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Authors: Yuan Yan Tang, Lina Yang, Hong Li

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Harmonic model is a very important approximation for the image transform. The harmanic model converts an image into arbitrary shape; however, this mode cannot be described by any fixed functions in mathematics. In fact, it is represented by partial differential equation (PDE) with boundary conditions. Therefore, to develop an efficient method to solve such a PDE is extremely significant in the image transform. In this paper, a novel Integral Equation-Wavelet based method is presented, which consists of three steps: (1) The partial differential equation is converted into boundary integral equation and representation by an indirect method. (2) The boundary integral equation and representation are changed to plane integral equation and representation by boundary measure formula. (3) The plane integral equation and representation are then solved by a method we call wavelet collocation. Our approach has two main advantages, the shape of an image is arbitrary and the program code is independent of the boundary. The performance of our method is evaluated by numerical experiments.

Keywords: harmonic model, partial differential equation (PDE), integral equation, integral representation, boundary measure formula, wavelet collocation

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Authors: Songul Miftakhov

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This study explores political representation and engagement of Eastern and Southeastern Anatolia regions, known to have dense Kurdish population and referred further to as Eastern region, in the Turkish parliament between 1946 and 1980. Traditional local notables had most of the privileges to be represented given their connectedness with political parties. Traditional local notables integrated into right-wing parties considering political and economic aspects. At the same time, they kept control over local political involvement channels. As a result, political representation and presence were monopolized at central, local and civil society levels. One part of Kurdish intellectuals was marginalized from the parliament after addressing issues in Eastern Anatolia and trying to develop solutions apart from the mainstream. Some of them took part in Kurdish oppositional left wing in the 1960s and jounced power of settled notables in 1970s in local administrations or as independent members of the parliament.

Keywords: Kurdish representation, parliament, local nobles, Eastern and Southeastern Anatolia

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Authors: Janis Matvejs

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Visual representation of a city creates unique perspectives that allow to interpret the urban environment and enable to understand a space that is culturally created and territorially organized. Residential complexes are an essential part of cities and cinema is a specific representation form of these areas. There has been very little research done on exploring how these areas are depicted in the Thai movies. The aim of this research is to interpret the discourse of residential areas of Bangkok throughout the 20th and 21st centuries and to examine essential changes in the residential structure. Specific cinematic formal techniques in relation to the urban image were used. The movie review results were compared with changes in Bangkok’s residential development. Movie analysis displayed that residential areas are frequently used in Thai cinematography and they make up an integral part of the urban visual perception.

Keywords: Bangkok, cinema, residential area, representation, visual perception

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Authors: Berceste Gülçin Özdemir

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The representation of women characters in cinema has been discussed for centuries. In cinema where dominant narrative codes prevail and scopophilic views exist over women characters, passive stereotypes of women are observed in the representation of women characters. In films shot from a woman’s point of view in Turkish Cinema and even in the films outside the main stream in which the stories of women characters are told, the fact that women characters are discussed on the basis of feminist film theories triggers the question: ‘Are feminist films produced in Turkish Cinema?’ The spaces that are used in the representation of women characters are observed to be used as spaces that convert characters into passive subjects on the basis of the space factor in the narrative. The representation of women characters in the possible surveillance spaces integrates the characters and compresses them in these spaces. In this study, narrative analysis was used to investigate women characters representation in the surveillance spaces. For the study framework, firstly a case study films are selected, and in the second level, women characters representations in surveillance spaces are argued by narrative analysis using feminist film theories. Two questions are argued with feminist film theories: ‘Why do especially women directors represent their female characters to viewers by representing them in surveillance spaces?’ and ‘Can this type of presentation contribute to the feminist film practice and become important with regard to feminist film theories?’ The representation of women characters in a passive and observed way in surveillance spaces of the narrative reveals the questioning of also the discourses of films outside of the main stream. As films that produce alternative discourses and reveal different cinematic languages, those outside the main stream are expected to bring other points of view also to the representation of women characters in spaces. These questionings are selected as the baseline and Turkish films such as Watch Tower and Mustang, directed by women, were examined. This examination paves the way for discussions regarding the women characters in surveillance spaces. Outcomes can be argued from the viewpoint of representation in the genre by feminist film theories. In the context of feminist film theories and feminist film practice, alternatives should be found that can corporally reveal the existence of women in both the representation of women characters in spaces and in the usage of the space factor.

Keywords: feminist film theory, representation, space, women directors

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Authors: B. Güney, Ç. Teke

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In recent years, rapid and correct decision making is crucial for both people and enterprises. However, uncertainty makes decision-making difficult. Fuzzy logic is used for coping with this situation. Thus, fuzzy linear programming models are developed in order to handle uncertainty in objective function and the constraints. In this study, a problem of a factory in food industry is investigated, required data is obtained and the problem is figured out as a fuzzy linear programming model. The model is solved using Zimmerman approach which is one of the approaches for fuzzy linear programming. As a result, the solution gives the amount of production for each product type in order to gain maximum profit.

Keywords: food industry, fuzzy linear programming, fuzzy logic, linear programming

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Authors: Hazem Al-Mofleh, John Daniels, Joseph McKean

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Within geostatistics research, effective estimation of the variogram points has been examined, particularly in developing robust alternatives. The parametric fit of these variogram points which eventually defines the kriging weights, however, has not received the same attention from a robust perspective. This paper proposes the use of the non-linear Wilcoxon norm over weighted non-linear least squares as a robust variogram fitting alternative. First, we introduce the concept of variogram estimation and fitting. Then, as an alternative to non-linear weighted least squares, we discuss the non-linear Wilcoxon estimator. Next, the robustness properties of the non-linear Wilcoxon are demonstrated using a contaminated spatial data set. Finally, under simulated conditions, increasing levels of contaminated spatial processes have their variograms points estimated and fit. In the fitting of these variogram points, both non-linear Weighted Least Squares and non-linear Wilcoxon fits are examined for efficiency. At all levels of contamination (including 0%), using a robust estimation and robust fitting procedure, the non-weighted Wilcoxon outperforms weighted Least Squares.

Keywords: non-linear wilcoxon, robust estimation, variogram estimation, wilcoxon norm

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Authors: Mehdi Jafari Matehkolaee

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In this paper, we have obtained the adjoint of an arbitrary operator (linear and nonlinear) in Hilbert space by introducing an n-dimensional Riemannian manifold. This general formalism covers every linear operator (non – differential) in Hilbert space. In fact, our approach shows that instead of using the adjoint definition of an operator directly, it can be obtained directly by relying on a suitable generalized space according to the action of the operator in question. For the case of nonlinear operators, we have to change the definition of the linear operator adjoint. But here, we have obtained an adjoint of these operators with respect to the definition of the derivative of the operator. As a matter of fact, we have shown one of the straight applications of the ''Frechet derivative'' in the algebra of the operators.

Keywords: adjoint operator, non-linear operator, differentiable operator, manifold

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Authors: Michela Quadrini

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Chord diagrams occur in mathematics, from the study of RNA to knot theory. They are widely used in theory of knots and links for studying the finite type invariants, whereas in molecular biology one important motivation to study chord diagrams is to deal with the problem of RNA structure prediction. An RNA molecule is a linear polymer, referred to as the backbone, that consists of four types of nucleotides. Each nucleotide is represented by a point, whereas each chord of the diagram stands for one interaction for Watson-Crick base pairs between two nonconsecutive nucleotides. A chord diagram is an oriented circle with a set of n pairs of distinct points, considered up to orientation preserving diffeomorphisms of the circle. A linear chord diagram (LCD) is a special kind of graph obtained cutting the oriented circle of a chord diagram. It consists of a line segment, called its backbone, to which are attached a number of chords with distinct endpoints. There is a natural fattening on any linear chord diagram; the backbone lies on the real axis, while all the chords are in the upper half-plane. Each linear chord diagram has a natural genus of its associated surface. To each chord diagram and linear chord diagram, it is possible to associate the intersection graph. It consists of a graph whose vertices correspond to the chords of the diagram, whereas the chord intersections are represented by a connection between the vertices. Such intersection graph carries a lot of information about the diagram. Our goal is to define an LCD equivalence class in terms of identity of intersection graphs, from which many chord diagram invariants depend. For studying these invariants, we introduce a new representation of Linear Chord Diagrams based on a set of appropriate topological operators that permits to model LCD in terms of the relations among chords. Such set is composed of: crossing, nesting, and concatenations. The crossing operator is able to generate the whole space of linear chord diagrams, and a multiple context free grammar able to uniquely generate each LDC starting from a linear chord diagram adding a chord for each production of the grammar is defined. In other words, it allows to associate a unique algebraic term to each linear chord diagram, while the remaining operators allow to rewrite the term throughout a set of appropriate rewriting rules. Such rules define an LCD equivalence class in terms of the identity of intersection graphs. Starting from a modelled RNA molecule and the linear chord, some authors proposed a topological classification and folding. Our LCD equivalence class could contribute to the RNA folding problem leading to the definition of an algorithm that calculates the free energy of the molecule more accurately respect to the existing ones. Such LCD equivalence class could be useful to obtain a more accurate estimate of link between the crossing number and the topological genus and to study the relation among other invariants.

Keywords: chord diagrams, linear chord diagram, equivalence class, topological language

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