The first aim of this thesis is to introduce a new definition of a-fuzzy normed algebra after that we introduce some examples to illustrate this notion. Then we study the case when the a-fuzzy normed algebra is fuzzy complete in the first step we proved basic properties of this space.. In the second step we proved properties of the resolvent space and spectrum space when the a-fuzzy normed algebra is fuzzy complete. In the third step further properties of fuzzy complete a-fuzzy normed algebra is proved. The second aim of this thesis is to introduce the notion of the C-a-fuzzy normed algebra in the first step we proved basic properties of fuzzy complete C-a-fuzzy normed algebra. In the second step we proved further properties of C-a-fuzzy normed algebra.

Linear algebra, a branch of mathematics, focuses on the study of matrices, vectors, and linear transformations. It serves as a foundational tool in various domains, with image processing being one notable field. In particular, linear algebra holds significant relevance in the realm of watermarking techniques. Algebraic transformations known as matrix decomposition methods are employed to extract essential values based on eigenvalues and eigenvectors. These decomposition methods encompass a range of techniques utilized for feature extraction in image watermarking. This work aims to introduce mathematical developments and modifications for designing digital image zero watermarking algorithms. The thesis approach primarily relies on two algebraic matrix decomposition methods, namely PCA (Principal Component Analysis) and LU decomposition. These methods are utilized to ensure successful watermark extraction and to enhance the algorithm's resistance against common attacks. In the first stage, two types of transforms, namely the IWT (Integer Wavelet Transform) and the DWT (Discrete Wavelet Transform), are employed to construct two distinct zero watermarking techniques, eliminating the need for additional tools. In the second stage, three zero watermarking techniques are devised. The first technique incorporates PCA, while the second technique combines PCA with IWT. The third technique employs both PCA and LU decomposition in conjunction with IWT. These specific methods have been chosen to develop secure and robust zero watermarking algorithms. The experimental results from this thesis reveal the successful and efficient performance of all the presented algorithms. In particular, the zero watermarking algorithms utilizing the PCA method demonstrated superior performance compared to other algorithms utilizing decomposition methods. This conclusion is supported by the analysis of Normalized Correlation (NC) values before and after attacks. The NC values remained consistently at 1 before any attack, indicating the preservation of watermark integrity, and the highest PSNR (Peak Signal-to-Noise Ratio) value reached 37.6659 after attacks. Based on these findings, it can be inferred that algebraic decomposition methods are highly valuable and effective in the realm of digital image watermarking techniques.

The chromatic polynomial is a graph polynomial that is explored in algebraic graph theory. It was invented by George David Birkhoff to examine the four-color problem and it counts the number of graph colorings as a function of the number of colors. For the first type we provide Chromatic polynomials for rooted graphs, trees, and rooted trees, as well as particular methods for their computation, the relationship between chromatic and Tutte polynomials for rooted graphs. We conclude by giving real-life examples and applications of polynomials. Also, define the numeration polynomial as the numeration of the generating function for the graph's chromatic polynomial. Some features of the polynomial were discovered. Chromatic and Ehrhart's polynomials are two important tools for analyzing graphs. They both provide insight into the structure of the graph but in different ways. In graph theory, combinatorics, and other areas, there is active study on the relationship between chromatic and Ehrhart polynomials. We may better comprehend the structure of graphs and how they interact with one another by knowing the relationship between these two polynomials. This can aid in the quicker and more effective resolution of challenging issues. This work explains the connection between these two significant polynomials, provides the theorem proofs, and discusses an application of these works.

The main objectives of this thesis are as follows: This thesis is devoted to proposing an approximate numerical algorithm based on the use of the state parameterization technique in order to find the solution to the optimal control problem (OCP). An explicit formula for special wavelet Chebyshev functions (SWCFs) is constructed. A new formula that expresses the first-order derivative of the SWCF in terms of their original SWCF is established. The development of our suggested numerical algorithms begins with the extraction of a new operational matrix of derivative from this derivative formula.The expansion’s convergence study is performed in detail, and some illustrative examples of OCP are displayed. The proposed algorithm is compared with the exact one. This confirms the accuracy and the high efficiency of the presented algorithm.

The statistical procedures are employed to estimate the parameters of any distribution or model. In this thesis, one of mixed distributions called Exponential-Rayleigh Distribution and denoted by (ERD) was used. Its two scale parameters of this distribution were estimated by performing and deriving two non-Bayesian (classical) estimation methods, ordinary least squares estimation method (OLSEM) and moment estimation method (MEM) which were provided using Newton-Raphson (NR) method. The numerical values of parameters and survival function were calculated with real data. After that, a new procedure was found and applied to study the statistical logic in fuzzy logic that benefits to generate the fuzzy numbers for the estimated parameters of this distribution by using the confidence interval estimation (CIE) with determining the lower and upper boundaries to create two types of proposed membership functions. These types are a linear pentagonal membership function (L-PMF) and a nonlinear pentagonal membership function (NL-PMF), then forming the Ranking function for each type which is approach to get the crisp estimated parameters of ERD. Following that, to know the best values for the parameters, the results were compared depending on the survival function’s estimators before and after the fuzzy with this actual data via utilizing the mean squared error (MSE) technique to manifest which result is the best.

The main purpose of this thesis is divided three aims. The first aim is to Introduce and study the New Modified Chebyshev Polynomials (NMCPs) with some important properties. These properties are employed to construct a general exact formula of their operational matrices of derivative and product. Second is to Introduce and study the Shifted Modified Chebyshev Polynomials (SMCPs). New analytical formula expressing explicitly the derivatives of SMCPs of any degree in terms of SMCPs themselves is contracted and proved and other interesting properties. And the third is to Devote an appropriate iterative method for solving optimal control problems which is grounded on NMCPs together with state parameterization technique for problem defined on [-1,1]. Then new scheme to obtain approximated solutions of optimal control problems is investigated using NSMCPs for approximating problems defined on [0, 1]. The obtained differentiation matrix is utilized in this scheme. The aim of the presented scheme is to transfer the original optimal control problem to optimization problems by approximating the state variable using NMCPs or NSMCPs with unknown coefficients. Then the constraints equation and the objective function are reduced to algebraic equations. Thus, an optimal control problem is converted to an optimization problem. The proposed algorithm is designed to get simultaneously both efficiency and accuracy. The convergence of NSMCPs is also proved in this work. The high accuracy of the obtained results is illustrated with some numerical tests.

Neural networks in artificial intelligence and the algorithms that relied on these networks in the process of detecting faces and facial emotions, which became a problem and a major challenge after analyzing the image using neural networks that lead to the use of a convolutional neural network (CNN) because it is able to improve the recognition of faces and feelings. In this thesis, the role of polynomials in the world of mathematics appears and converting them into separate wavelets that have the characteristics of orthogonal wavelets to be ready to find their work in analyzing the image to reveal the face with its feelings with linking these wavelets and taking their role as a filter to perform the convolution process to form a new neural network more efficient than the network Ordinary convolutional. Built and programmed derived from Chebyshev polynomials first kind to build a new filter that was used to raise noise and compress the image and measure the most important quality standards Mean Square Error (MSE), Peak Signal for Noise Ratio (PSNR) , Bit Per Pixel (BPP) and Compression Ratio (CR) so that the new filter is connected to the convolutional neural network (CNN) to be Chebyshev Wavelet convolutional neural network (CHWCNN) and define the features of the face after connecting the work Alex Net, Where the new technology has proven its efficiency in achieving the best results for calculating the accuracy of the accuracy and in less time. Face Detection and Six Feelings Read: Normal, Happy, Sad, Surprised, Angry, and Fearful with Deep Learning, Wavelet Transformations, and Convolutional Neural Network Development Alex Net and Google Net.Using the new method with DCHWT and CHWCNN the accuracy was obtained 95% and 91%, respectively. The feelings of the students were read while receiving the lecture, and the percentages of feelings were fixed, and the accuracy was 90% and 72%, respectively. The proposed system is also significant because it can complete the process of recognition and face detection in less than 30 ML seconds.

The game theory has been applied to all situations where agents (people or companies) actions are utility-maximizing. The collaborative offshoot of game theory has proven to be a robust tool for creating effective collaboration strategies in a broad range of applications. The waste file is one of the thorny files that hinder development plans in many countries, where it is necessary to highlight and build an integrated system to achieve the maximum possible benefit from it. Therefore, most developed countries have had to establish a modern manufacturing system based on decreasing the participation of fresh input factors and recycling of waste. Since not all waste is recycled in waste recycling plants, there must be cooperation between waste producers to reduce the overall costs of processing non-recyclable waste. Cooperative game theory has been applied to all situations in which producer actions maximize utility, and has shown to be a robust gadget for creating practical collaboration actions in a broad range of applications. Thesis paper, we apply a mathematical model based on cooperative game theory to model cooperation between producers in waste management. Then, we use the chaining interaction index to show the potential cost to the waste producers in the case of cooperation and reduce the overall costs of processing non-recyclable waste during the cooperation between the five producers. Finally, we use the methodology for a case study to address waste management in Al-Mahmudiya factory. This study assumes that all producers divide sanitary landfills or incineration of non-recyclable waste, meaning that for each producer who earns more, there will be a greater contribution to the disposal or treatment of non-recyclable waste. The results of this study will strongly help professionals formulate well-structured strategies for the waste management system of the future.

In this thesis, principally self injective, and strongly principally self injective modules are introduced. Also, the concepts of (strongly) principally self pure submodules as generalizations of self pure submodules. Principally pure submodules are introduced. We prove that any module is principally pure in any principally injective module containing it as a submodule if and only if it is principally injective or it is principally pure in its injective envelope. Certain properties of (quasi) injective, principally injective, absolutely (self) pure, and finitely 푅-injective modules are extended to (strongly) principally self injective modules, and some properties are studied in a way analogous to absolute self neatness. The fundamental features of these concepts and their interrelationships are linked to the conceptions of some rings. Von Neumann regular, left SF-ring, and left principally projective ring are described by these concepts. For example, the homomorphic image of any injective module is (strongly) principally self injective if and only if 푅 is left principally projective ring, and for a commutative ring 푅 of every principally self injective module is flat if and only if 푅 is Von Neumann regular.

Domination is an area of graph theory that has been studied extensively. The numerous applications of domination and modern research on different classes of domination in graphs, created our interest to investigate in this thesis, a special type of graph domination. This type is called “Modern Roman Domination”. This type of domination is determined by the number of dominated vertices relying on its vertex labeling, which is beneficial for any type of networks that requires such characteristics. A labeling function 𝑓: 𝑉(𝐺) → {0,1,2,3} is a modern Roman domination function on the graph G= (V; E) such that every vertex with label 0 is adjacent to two vertices, one of label 2 and the other of label 3. Every vertex with label 1 has a vertex with label 2 or 3 next to it. In this thesis, some properties of this model of graph domination are introduced. Some upper and lower bounds are proved for different graphs, and has been dealt with the modern Roman domination on complement of certain graphs such as path, cycle, complete bipartite, star, wheel, complete, null, fan, double fan. Also, we characterized the corona operation of two certain graphs.

The concept of domination in graphs is powerful and important concept and contains many applications. In addition to that, domination has flourished in recent years, and many modified definitions have appeared that are attributed to domination. In this thesis, new special types of graph domination have been introduced. The first type is titled “rings domination”, where a condition is put on the dominated set, so that each vertex in this set is neighbored at least two vertices within its set. The second type is the “inverse rings domination”. Three other types of domination were also introduced by adding a condition on the rings dominating set; they are called “independent rings domination”, “total rings domination” and “connected rings domination”. Properties for those types have been found in addition some of upper and lower bounds for domination number. Domination numbers have been calculated for those types on some graphs and their complements. The stability of the ring’s domination set has been investigated when deleting a vertex.

The first aim in this thesis is to introduce linear operator of various types. In the first step we study the case when the linear operator is fuzzy closed, then basic properties of this type is investigated and proved. In the second step we take the case when the linear operator is fuzzy compact also basic properties of this type are investigated and proved. The second aim, is to investigate, basic properties of the spectrum of linear operator of various types. At the first step, we investigate the spectrum properties when the linear operator is fuzzy closed then basic properties of the spectrum of this type of linear operator is investigated and proved. In the second step we study the spectrum when the linear operator is fuzzy compact also basic properties of the spectrum of this type of linear operator is investigated and proved. After that a further properties of spectrum and resolvent of the previous types of linear operators investigated and proved. Finally, we applied complex analysis in spectrum theory and resolvent theory to get more properties of spectrum and resolvent for the previous types of linear operators.

The first aim in this thesis is to introduce the notion of algebra fuzzy absolute value then some examples is introduced to illustrate this notion and in order to introduce new type of fuzzy normed space called algebra fuzzy normed space after that some examples is introduced to illustrate this notion. Then some properties of algebra fuzzy absolute value space are proved moreover basic properties of algebra fuzzy normed space also proved. The second aim is to continuo this study by introducing the notions of fuzzy continuous and uniform fuzzy continuous also to proved basic properties of these notions that will be needed later in this thesis here we proved that the algebra fuzzy norm is a fuzzy continuous function this property is very important in our work as well as the study of a finite dimensional algebra fuzzy normed space to prove that some properties that algebra fuzzy normed spaces does not admit it. This position needed to define fuzzy compact then we shall study this type of space and proved some of its basic properties because we need this notion and its properties later in chapter two. Now we are in the position that make us to introduce further properties of algebra fuzzy normed space such as the notion of algebra fuzzy norm of a fuzzy bounded linear operator. The important results that we have here is when U and V are two algebra fuzzy normed spaces as well as if V is fuzzy complete then fb(U, V) is fuzzy complete where fb(U, V) is the set of all linear fuzzy bounded operators from U into V. Moreover when U and V are algebra fuzzy normed spaces with dimension of U is finite we prove that every linear operator from U into V is fuzzy bounded. Here the direction of the study is changed by recalling the definition of algebra fuzzy metric in order to define the algebra fuzzy distance between two sets then we use this concept to define the Hausdorff algebra fuzzy metric from fuzzy compact set to another fuzzy compact set after that basic properties of the Hausdorff algebra fuzzy metric between two fuzzy compact sets are proved. Tإhe important result we got here is if S is a fuzzy complete then AFH(S) is a fuzzy complete. Finally we introduce the notion of algebra fuzzy norm of the quotient space and the important result that proved here if the quotient space 𝑈 𝐷 is fuzzy complete then U is fuzzy complete. Furthermore we introduce the notion pseudo algebra fuzzy normed space in this direction we define a relation ~ on U in order to introduce the space Û of classes û=[u]= [z∈U: z ~ u} and investigate basic properties of this space here the important results is if U is pseudo algebra fuzzy normed space then Û is algebra fuzzy normed space also if U is a fuzzy complete Û is fuzzy complete.

The close connection between mathematics, especially linear algebra and computer science has shown a great impact in the development of several fields, and the most important is image processing. Algebraic methods aroused interest in building digital image watermarking techniques and are used to find the features of the image to hide the watermark. This thesis aims to use the algebraic Hessemberge decomposition method (HDM) for the first time as an algebraic transformation to extract the features of the image without using any popular transformation for building a zero watermarking technique, besides using two additional transformations to improve the results. In addition, the singular value decomposition (SVD) is used to extract the features of the image side by side with the genetic algorithm (GA) to obtain optimal results. In this work, the HDM and SVD are used to build zero watermarking techniques. The discrete wavelet transform (DWT) and the discrete cosine transform (DCT) as additional transformations are used with HDM to improve the results and to increase the robustness of the technique. On the other hand, GA is used with both HDM and SVD to optimize the proposed techniques in the YCbCr space without using any other transformation such that the watermark can be extracted successfully and the algorithms can resist most of the common attacks. The YCbCr space is one of the spaces that converting the RGB images into three components Y, Cb, and Cr. The Y component represents the brightness (luma) of the color or the grayscale image, while Cb represents the blue component and Cr represents the red component. Firstly, the experimental results manifested that all the algorithms in this thesis worked successfully and efficiently. In particular, the optimized zero watermarking technique using HDM and SVD algebraic methods, with GA, are the best according to the values of the NC before and after attacks. In this case, the value of the NC=1 before any attack for both techniques, and after corrupted by any attack, the NC value is ranging between 0.743345 and 1 for both techniques. The higher value of the PSNR is equal to 36.4464 after attacks . Based on these results, it is concluded that the algebraic decomposition methods are very useful and too efficient in the digital image watermarking techniques.

The problem of the face detection process is a great and difficult challenge even today when using traditional methods because of the large data contained in the features of the face image, which takes a large time to analyze, and the addition of artificial intelligence techniques such as neural networks and their traditional algorithms is the other, which requires a great deal of time during image analysis. The long training time required by the use of traditional neural networks and the problems of falling into the trap of local minima. All these matters have led to the recent trend towards the use of convolutional network algorithms (CNN) because of their good ability to explore complex search spaces and their ability to improve the performance of face recognition When the image is subjected to nonlinear effects that include large light contrasts. In this research, emphasis is placed on the use of deep learning methodology for face detection and recognition. This research mainly focused on training neural networks using sufficient amounts of data to achieve the desired training outcomes. A convolutional neural network (CNN) architecture has been proposed for the problem of face recognition and detection, with a solution capable of dealing with facial images that contain contrasts, facial expressions, and varying lighting, and defining the features and parts of the face in detail and accurately. The current research methodology has depended on the following steps in its implementation. First, by using convolutional neural networks and implementing the application of face recognition using convolutional neural networks deep Alxnet model. Second, bypass neural networks and a face recognition application using deep convolutional neural networks googleNet model is implemented. After analyzing the results of the neural networks and applying face recognition, the experimental results obtained show that the proposal to use the convolutional neural network CNN exceeds the use of any other applications, and achieves accuracy of detection and recognition of faces by 99.5%, which represents in the performance criterion the best result obtained Comparisons with previous works. The importance of the proposed system also lies in its ability to complete the process of recognition and face detection in less than 0.03 seconds . More about this source textSource text required for additional translation information.

This thesis is concerned with establishing theoretical background for modified Chebyshev polynomials of the first kind and deriving some new important properties. In addition, an efficient direct method was proposed with state parameterization technique to solve certain types of optimal control problems approximately. Here the parameterization technique utilizes both of the improved Chebyshev polynomials presented. They are used as linearly independent functions to approximate the state variables by a finite length of the modified Chebyshev function series of unknown parameters. A general new formulation for the operation matrix of derivative for modified Chebyshev polynomials of the first kind has been established to facilitate the computation within the presented method. Moreover, the mathematical proofs of the convergence of the proposed method with first kind modified Chebyshev polynomials and shifted modified Chebyshev polynomials have been reached. Some illustrative examples showed that the suggested algorithms exhibit satisfactory results.

Linear algebra is a subfield of mathematics interested in matrices, vectors, and linear transforms. It is a fundamental key to many fields, in particular, the field of image processing. Minutely, linear algebra plays an important role in watermarking techniques. This thesis aims to introduce mathematical developments and modifications for designing digital image watermarking and zero watermarking algorithms depending basically on two algebraic matrix decomposition methods Singular Value Decomposition (SVD) and Generalized Singular Value Decomposition (GSVD) such that the watermark can be extracted successfully and the algorithm can resist most of the common attacks. The wonderings about the benefit of the algebraic decomposition methods in digital image watermarking techniques have been answered in this work. The SVD and GSVD are the algebraic decomposition methods were chosen to build a secure and robust (zero) watermarking algorithms using two types of transforms the Discrete Wavelet Transform (DWT) and the Discrete Cosine Transform (DCT). In general, the experimental results showed that all the algorithms in this thesis worked successfully and efficiently. In particular, the zero watermarking algorithm using SVD method is the best algorithm according to the values of the Normalized Correlation (NC) and the Peak Signal-to-Noise Ratio (PSNR) before and after attacks and according to the number of steps of the algorithm. The values of the NC before and after attacks are equal to 1 . The higher value of the PSNR is equal to 60.7157 after attack (Jpeg compression). The second best one is the zero watermarking algorithm using GSVD and DCT (the diagonal case). The values of the NC before and after attacks are equal 1 and the higher value of the PSNR is equal to 60.7157 after attack (Jpeg compression). Based on these results, we conclude that the algebraic decomposition methods are very useful and very efficient in digital image watermarking techniques.

In this thesis, new wavelets were derived from polynomials of Laguerre, after relying on the parent function in the wavelet field, which is characterized by expansion and contraction responsible for the two influences c,d, and after the construction of new wavelets that became dependent on four effects, namely t, k, u, v. The intention of the new wavelets was demonstrated by through many important theorems that give the function its durability and its ability to be used in the field in which this thesis is concerned. An image processing of wavelet packages and filters suitable for this function were designed after designing a number of new programs to employ these wavelets and work with them. Four samples were used of the color images were analyzed using the new wavelets and the basic criteria for image quality were calculated and the results were compared with the results of the new wavelets, which proves the efficiency of the new wavelets. The color image is a large matrix, which need for wavelets to compress the image and reduce the area exploited by the image information for the purposes of storage and transport via means of transportation and communications. In the fourth chapter, the new wavelets applied in the process of compression, i.e. Image compression was indicated by using the inverse wavelet. The original image is returned without loss of image information. Using MATLAB software, new programs and important algorithms are used to employ the new waves in image compression. The results were discussed and good results obtained by comparing them with the results reached using traditional basic wavelets used by many researchers. The main aim of this study was to highlight the importance of wavelets in computer applications also the importance of mathematics in the fields of engineering, medicine and all sciences. Finally could create many wavelets from different polynomials because wavelets have mathematics.

Keeping information secure is a major challenge for the development of technology. It is quite popular to share data through optical communication channels, that make their security a matter of urgency. Besides, the development of the current cryptosystems is considered weak due to the development of cryptanalysis methods. There is therefore a great demand for new emerging cryptosystems. The reason behind the use of cryptography based on chaos is important because of non-periodic chaotic signals. Chaotic systems are shown to have a promising future, especially in data encryption algorithms due to the common characteristics between the chaos scheme and the encryption method components. To meet the security and complexity requirements, a new hyperchaotic system has been introduced based on the sine map; we called it a 2D-Adjusted Sine Map (2D-ASM). The behavior of the proposed chaotic system is studied through a several tests to prove that it achieves good performance. Some of the tests used in this work were: time-series analysis, fixed point and eigenvalues, its dynamic properties were studied in terms of trajectory, bifurcation diagram, and Lyapunov exponent. The Approximate Entropy (AE) is used to investigate the complexity of the 2D-ASM. After demonstrating its high chaocity and complexity, this multidimensional system is used to generate PRNG to be used in the design of a new data encryption algorithm for secure semiconductor laser communication. The new data encryption algorithm was implemented practically on Arduino, microcontroller boards and the encryption signal is sent as a binary sequence directly through the injection current of the semiconductor laser, showing promising results. Security analysis tests are the most valid evaluations that have shown the security of the proposed encryption algorithm. So, an efficient analysis indicates that the generated sequence is a good choice for many applications.

This thesis pointed to utilize the peristaltic transport for different nonNewtonian fluids with variable viscosity taken into consideration the following physical influences: variable Magnetohydrodynamics (MHD) field, porous space, temperature depending variable viscosity, Hall effect, Joule’s heating, thermal radiative heat, velocity and thermal slip, heat generation, and compliant wall. Two important models of non-Newtonian fluids are concerning in this work the first one is known as Bingham-plastic model, which can be recognized when its shear stress is overreach the so called yield stress “the entrance of stress level”, and the second fluid is known as Ree-Eyring, which is the most important model than the power law fluids. The mathematical expressions for flow basic equations (i.e. the mass conservation, motion, and energy) are formulated to describe the flowing of the above mentioned fluids through designing of different asymmetric channels (tapered, and vertical) by adopting assumption of “low Renolds number and long wavelength approximation”. Perturbation technique is imposed to find the approximated solution for the resulted nonlinear system of high order partial differential equations for small non-Newtonian parameter. Graphically elucidation is established to figure out the impact of various interesting parameters that controlling the pumping characteristics, velocity profile, temperature distribution, heat transfer rate, and trapping phenomena. This exploration is made via using “Mathematica 11.3” package.

Some basic mathematical tools such as convex sets, polytopes and combinatorial topology, are used quite heavily in applied fields such as geometric modeling, meshing, computer vision, medical imaging and robotics. It is intended for a broad audience of mathematically inclined. A polytope plays a central role in different areas of mathematics; therefore we shall take a polytope with one kind of them, which is known as a Zonohedron. The matroid and arithmetic matroid with their properties are also given. The concept of the Platonic solids is introduced with their properties. Multiplicity Tutte polynomial and Ehrhart polynomial for a Zonohedron Z(X) in 2-dimension and 3-dimensions are introduced. Theorem for the relation between the numbers of integral points of a Zonohedron and the set of generating vectors is proved. Combinatorial interpretation of the associated multiplicity Tutte polynomial with different examples is presented to demonstrate our results. The expression ‘duality’ means a mapping between points and plans (or vertices and faces) is introduced, together with the computation of the Ehrhart polynomial for the dual of Zonohedron and platonic solid.

In this work, a novel construction of a secret sharing scheme is proposed and proved to be new theorems and observations depending on some special graph’s properties such as regularity degree and dominating cardinality. In this work the minimum independent dominating set of vertices as an access structure is used to find the key instead of the theories used in previous methods such as H. Sun method. Some algorithms that helps to implement this novel construction are designed as follows: algorithm to find minimum independent dominating set of vertices (MID). algorithms to construct and reconstruct the proposed secret sharing scheme of ranks 2, 3, m≥4. The information rate (ρ) is used as a measure for the efficiency of secret sharing scheme. Therefore, the main aim of this study is to improve information rate values by applying the proposed method to 21 different graphs which have an order between 4 to 24 and regularity degree between 2 to 8, the resulted information rate for proposed perfect secret sharing scheme has an improvement over other methods such as H. Sun and S. Shieh method and Stinson method and we found that: For rank 2: =2/r , while the higher information rate obtained by H. Sun and S. Shieh method is ρ=2/(r+1). For rank 3: ρ=6/((n-r)(n-r-1)+2), while the higher information rate obtained by H. Sun method is ρ=6/((n-1)^2+2). For rank m≥4: ρ=(m-1)!/(((n-r)(n-r-1)+1) ) , while the higher information rate obtained by H. Sun method is ρ=6/((n-1)^2+2). To apply the proposed method to big graphs with a large number of vertices, the proposed algorithms and H. Sun method have been implemented by using VB. NET 2008. In order to satisfy fair comparison under the same environments, the results values agree with the theoretical results, where the experimental result is deduced to prove the efficiency of this method over other previous method in term of information rate and execution time.

In this thesis the definition of standard fuzzy metric space is introduced as a modification of the notion of fuzzy metric space due to Kramosil and Michalek then several properties of this space are studied and discussed after some illustrative examples are given. Further the definition of convergence sequence, Cauchy sequence and F-bounded standard fuzzy metric space are modified. Also the definition of continuous and uniform continuous function are modified , we proved that a mapping f:X→Y is continuous on X if and only if f^(-1)(G) is open in X for all open subset G of Y . A necessary and sufficient condition for a standard fuzzy metric space to be complete is given then completions of standard fuzzy metric space are discussed. We show that unfortunately there exists a standard fuzzy metric space that dose not admit any completion. Also we show that for each completable standard fuzzy metric space there is a unique completion up to F-isometry. The compact standard fuzzy metric spaces and F-totally bounded are defined then we proved that F-totally bounded complete standard fuzzy metric space is compact. Moreover the continuous and uniform continuous functions on a compact space is studied to prove that continuous functions and uniform continuous functions are equivalent on compact standard fuzzy metric spaces. At this end the notion of standard fuzzy pseudo metric space is introduced to prove that the completion of standard fuzzy pseudo metric space is a standard fuzzy metric space. The definition of standard fuzzy quasi metric space is introduced then an internal characterization of those standard fuzzy quasi-metric space that admit a bicompletion is given, we also show that such a bicompletion is unique up to F-isometry. Finally the fuzzy distance between a compact set and a point is defined as an introduction to define the fuzzy distance between two compact sets then a method for constructing a Hausdorff standard fuzzy metric on the set of the nonempty compact subset of a given standard fuzzy metric space is proposed. At this end we proved that if the given standard fuzzy metric space is complete then the Hausdorff standard fuzzy metric space is complete.

This thesis presents a number of algorithms for the approximate solution of variational problems based on first and second Chebyshev wavelets. The convergence of second. Chebyshev wavelets are first discussed. Then some new relations between first and second Chebyshev wavelets are derived. The proposed algorithms to find extremum value of variational problem are based on using Euler-Lagrange equation. To facilitate the computation, a new property is derived called operational matrix of derivative. Using the operational matrices of derivative and integration, for first and second Chebyshev wavelets. The problem is converted to solving a system of algebraic equation. All algorithms are tested on a variety of problems.

The set of all n×n non singular matrices over the field F form a group under the operation of matrix multiplication, This group is called the general linear group of dimension n over the field F, denoted by GL(n,F) . The subgroup from this group is called the special linear group denoted by SL(n,F) . We take n=2 and F=2k where k natural, k>1. Thus we have SL (2,2k). Our work in this thesis is to find the Artin's exponent from the cyclic subgroups of these groups and the character table of it's. Then we have that: a SL(2,2k ) is equal to 2k-1 .

The main purpose of this work is to find the Artin exponent of finite special linear groups from any arbitrary characters of cyclic subgroups of these special linear groups and denoted by: a (SL(2, p)) Where p is any prime such that p ≥ 5, and we found that a (SL (2, p)) is equal to 2.

In this thesis we recall the definition of a fuzzy metric space and the definition of fuzzy normed space then the related concepts are discussed, much attention is paid to the concept of fuzzy completeness. Also we recall the definition of fuzzy inner product space and give some new results after introducing new concepts such as fuzzy convergence; fuzzy Hilbert spaces then we prove that if A ̃ is any fuzzy closed subspace of a fuzzy Hilbert space H then H =(A ) ̃⊕Z ̃ where Z ̃= A ̃^⊥. Finally we introduce fuzzy Hilbert dimension after that we prove that two fuzzy Hilbert spaces H and G both real or complex are isomorphic if and only if they have the same fuzzy Hilbert dimension.

This thesis is concerned with the approximate solutions of finite quadratic optimal control (QOC) problems that are governed by ordinary differential equations which represent the constraints. The proposed method is classified as indirect methods which are usually based on the necessary optimality conditions. Besides necessary conditions, sufficient condition has to be checked to ensure the optimality of the solution. The result by applying these conditions is two points boundary value problem (TPBV). In this work very efficient algorithms are proposed, which are based on applying the idea of spectral method using the Chebyshev polynomials: which include Chebyshev polynomials of the first kind, Chebyshev polynomials of the second kind and shifted Chebyshev polynomials of the first kind. To facilitate the computations, new properties of Chebyshev polynomials are derived. Finally, the proposed algorithms have been applied to several examples.

In this thesis we introduce a new method for hiding information as images then decrypt them to find the message. Many of us lump cryptography with Steganography, it's true that the later is a way to encrypt the information, but still there are differences. These differences will be cleared in this thesis. We suggested a method to hide the information by the self similarity patterns of fractals, using the affine transformations to create the image as the attractor of the iterated function system. The privilege of the IFS is that we can send long messages, as just one image. That is by embedding the information as a text in an image and extracting it. Decoding the attractor by Barnsley's collage method is to find the values of the coefficients that created the image, that in fact are the letters of the message that are hidden in an innocent colored image.

The aim of this study is to develop Lagrange’s method for the solution of linear programming. Throughout this study many of methods have been carried out to find out the optimal solution for the linear programming problems as the simplex method and Lagrange’s multiplications method and the method of normal Kuhn – Tucker‘s conditions. We have studied Lagrange‘s method and throughout several derivations, we came to a conclusion to mathematical forms by which it can be possible to get to the solution without doing long derivations. The form we obtained is much better than normal Lagrange method .thus; we can avoid a lot of derivations and summarize the number of probabilities of eradicating some of the changeable items which had rapidly got to the optimal solution.

This thesis is concerned with the approximate solution of finite linear quadratic optimal control problem that are governed by a system of ordinary differential equation. The proposed method is classified as direct method, which is employed by using special technique to convert the LQOC problem into a quadratic programming problem. It is based on generalized Laguerre polynomials as a basis functions to aproximate the system state variables by a finite length of the basis functions series of unknown parameters. Furthermore, some important formulas which are concerned the generalized Laguerre polynomials are derived and proved as essential in the proposed method. Finally, the proposed algorithm was illustrated by several examples.

The main Theorem in this work is Solomon Theorem which states that the factor group (R(G))/(T(G)) has a finite exponent dividing |G| . The theorem study the relation between ch'QG, the subring of chG generated by the set of rational valued characters of G and ch'QG generated by characters defined by certain families of subgroups of an arbitrary finite group G. This result has several applications including an extension to ch'QG of the Artin's Induction Theorem in which these certain families of subgroups are cyclic. Lam determined that the last positive integer A(G) such that A(G) χ is an integral linear combination of the induced principal characters of cyclic subgroups, for any rational valued character χ of G, A(G) is called the Artin exponent of G. The main objective of this thesis is to find the Artin Exponent of the Special Linear Groups SL(2,2k), k natural, by the aid of Solomon Theorem of Rational Valued Character. It is found that the Artin Exponent, A(G), of G =SL(2, 2k), k natural and k >1, is A(G) =2K-1. and A(SL(2,2k) ) = 2 , when k = 1

This thesis is concerned with the analytic, numerical and approximate solutions of linear Fredholm integral equation of the second kind. The proposed numerical methods are based on the extrapolation procedure coupled with closed Newton-Cotes formulas, while the approximate methods in this work are employed by using variational technique with different types of basis functions including: algebric polynomials, Legendre polynomials and Chebyshev polynomials. In this technique, we may find many functions for every linear Fredholm integral equation of the second kind whose minimum is the solution of the problem. Furthermore, the proposed algorithms have been applied for several examples. We find the proposed methods produce satisfactory results.

The main aim of this thesis is to study and modify some approximate and numerical method to treat linear system of Fredholm integral equations. For the approximate method for solving linear system of Fredholm integral equations including Neumann Series method have been modified to be applicable for solving this system. Also, the existence and uniqueness theorem for the Fredholm integral equation has been generalized to system of Fredholm integral equations. Moreover, five different types of Nystrom or quadrature methods “Trapezoidal, Simpson’s 1/3, Simpson’s 3/8, Bool’s and Weddel’s” have been modified and successfully employed to give numerical solution for this system. Also, the convergence theorem of the Nystrom methods and the error analysis for the single Fredholm integral equation has been generalized to linear system of Fredholm integral equations. This thesis presents five algorithms based on extrapolation method including: Trapezoidal extrapolation, Simpson’s 1/3 extrapolation, Simpson’s 3/8 extrapolation, Bool’s extrapolation and Weddel’s extrapolation to get more accurate results. Finally, at the end of each methods, Algorithms and programs developed and written in MATHLAB (version 6.5).

This thesis presents some approximate solutions to both linear and nonlinear quadratic optimal control problems that are governed by a system of ordinary differential equations which represent the constraints. The proposed method is classified as direct method which is employed by using state vector parameterization (SVP) to convert the quadratic optimal control problems into quadratic programming problem. The state vector parameterization is based on spline polynomials including: B-spline and Catmull-Rom spline as basis functions to approximate the system state variables by a finite length of the basis functions series of unknown parameters. Furthermore, some important formulas concerning the spline polynomials are derived and proved which are essential in the proposed method. Finally, the proposed algorithms have been applied for several examples giving satisfactory results.

This thesis is based on solving both linear and nonlinear two-points oundary value problem TPBVP using some numerical and approximate methods. Newton-Kantorovich method is used to convert the nonlinear TPBVP into linear boundary value problem. The numerical methods are based on finite difference and Numerov difference methods. The mathematical study of convergence and stability of these methods are introduced. Furthermore, the idea of the extrapolation technique coupled with the proposed numerical methods is applied to solve TPBVP. In addition, the weighted residual method including collocation and Garlekin's methods are applied to solve the above problems approximately with two basis functions: power polynomial and Chebyshev polynomials. Finally, some numerical examples show that the proposed methods give satisfactory results.

Image processing can be considered as an essential part of wide range computer application. In the last decade, the wavelet transform has diffused in most digital signal processing applications, where it plays a very important role in image processing analysis. This thesis deals with image coding. Image coding using Huffman or run length code does not use transform. Moreover image coding using threshold or zonal code is using transforms to code images. Two types of transforms Discrete Cosine Transform (DCT) or Discrete Wavelet Transform (DWT) have been used after segmenting the image into blocks, and then Zonal coding algorithm or threshold coding algorithm is used to get the coded image. As result the (DWT) gives better results than (DCT), where the Mean Square Error (MSE) in DWT is less than the MSE in DCT, which gives us a conclusion that using wavelet transform is better than other transforms when it is used for image coding. Also, the threshold coding gives better results compared with zonal coding. The performance of proposed methods has been evaluated by computer simulations using MATLAB 7.0 language and performed on personal computer Pentium_4 with 2.4 GHZ processor, 40 GB hard disc and 256MB main memory.

Let R be a ring not necessarily with an identity element. A wellknown result proved by I.N.Herstein concering derivations in prime rings have been extensively studied by many authors like, M.Bresar, N.M.Shammu and M.Ashraf and N.Rehman. Also, C. Haetinger and M.Ferro extended this result to higher derivations. The main purpose of this work is i. Extend N.M.Shamm's theorem to higher N-derivations by giving the concept of higher N-derivation. A higher Nderivation of a ring R is defined as a family of additive mappings of R into itself , for all u∈U, r,s∈R , n∈N,where U is a Jordan ideal of R. ii. We answer the question of C.Haetinger and W.Cortes whether the result of C.Haetinger and M.Ferro is also true for Jordan generalized triple higher derivations. iii. We introduce the concept of (U,R) derivations and generalized (U,R) derivations. Then we extend Awatar's theorem and we extend this result to higher (U,R) derivations and generalized higher (U,R) derivations by giving corresponding definitions. A well-known result of I.N.Herstein concerning Jordan homomorphism and Jordan triple homomorphism has been extensively extended by M. Bresar. Also R.C. Shaheen extended these results to generalized Jordan homomorphism and generalized Jordan triple homomorphism iv. We introduce the concepts of higher homomorphism, Jordan homomorphism and generalized Jordan triple homomorphism and their generalization and we extend the above results and study these concepts onto 2-torsion free prime ring.

This thesis presents some Expansion methods for solving linear fractional integro-differential equations of both Fredholm and Volterra types. The expansion method is associated with the weighted residue techniques including the following methods: collocation, least squares and Galerkin which have been used to treat the above problems. In addition two orthogonal functions: Chebyshev and Legender as well as two spline functions: B-spline and Catmall-Rom spline have been used as a basis functions to approximate the unknown function within the proposed approximated methods. Some new formulas of fractional derivatives for each Chebyshev, Legender, B-spline and Catmall-Rom spline functions have been derived which are very useful in simplifying the computations in this work. Moreover, the convergence and stability of all approximated methods are investigated. Additionally, the proposed methods have been applied to several examples with satisfactory results and a program for each method is written with the aid of MATLAB version (6.5).

The main aim of this thesis is to study and modify some analytic and numerical methods to treat a system of linear second kind Volterra integral equations. For the analytic treatment two methods for solving single Volterra integral equation including Laplace transform and successive approximation method have been modified to be applicable to solve the system. Runge-Kutta methods including: Runge-Kutta second order, Runge-Kutta third order and Runge-Kutta of fourth order have been modified to give numerical solution to this system and study the convergent of the algorithms of Runge-Kutta methods. In addition, block methods which include: method of two, three and four blocks have been used to find numerical solution to this system. The package “NSSLSKVIE” is constructed to find the numerical solutions to system of linear Volterra integral equations using all the above methods. Finally, at the end of each method, algorithms and programs are developed and written in MATLAB (Version 6.5).

The objective of this thesis is to study and development the solution of linear programming problems with free variables. This thesis studied number of methods with its algorithms to solve the above problems. The thesis studied one of classical methods which expressed any free variable as a difference between two nonnegative variables, then the modified MPS method is studied which expressed any free variables as a function of remaining nonnegative variables . It is found that MPS method is better than classical method, since MPS method reduced the tables number, make the calculations easier and reduced the solution time. Also MPS method used some basic theorems to determine the solution result for some problems from the initial table with out depending on the problem detail solution . This thesis also includes appendix represented the package programs with Visual Basic language (Microsoft Visual Basic 6.0), to get a good forms of display at execution and solving all the problems that may be found in this scope to apply all methods that included in this thesis.