Matrix representation of quasigroupoids and loopoids

The authors determine the lattice of the clone consisting of the projections and those members of Burle's clone that take on the values 0 and 1 only. Even this lattice is considerably more complicated than the well-known Post's lattice of all clones on the two-element set. On the basis of this result the authors can determine all minimal and essentially minimal TC-clones on the three-element set. >

A monomial clone is a clone which is generated by a monomial. For a prime power k we consider monomial clones on the finite field GF(k). There are two monomial clones on a two-element set and six such clones on a three-element set. Ordered by set inclusion, they form lattices. In the general case, we put emphasis on monomial clones generated by unary and binary monomials. We determine unary minimal monomial clones and binary monomial clones of degree k. Furthermore, we list all unary and binary monomial clones on GF(5) and state several properties that hold on GF(7) and GF(11).

A set with a binary operation is a fundamental concept in algebra and one of the most fundamental properties of a binary operation is associativity. In this paper the authors discuss binary operations on a three-element set and show, by an inclusion-exclusion argument, that exactly 113 operations out of the 19,683 existing operations on the set are associative. Moreover these 113 associative operations are accounted for by means of their operation tables.

In this paper, we propose four new general constructions of LCZ/ZCZ sequence sets based on interleaving technique and affine transformations. A larger family of LCZ/ZCZ sequence sets with longer period are generated by these constructions, which are more flexible among the selection of the alphabet size, the period of the sequences and the length of LCZ/ZCZ, compared with those generated by the known constructions. Especially, two families of the newly constructed sequences can achieve or almost achieve the theoretic bound.

Preface. Conventions and Notations. 1 An Introduction To Maple. 1.1 The Commands . 1.2 Programming. 2 Linear Systems of Equations and Matrices. 2.1 Linear Systems of Equations. 2.2 Augmented Matrix of a Linear System and Row Operations. 2.3 Some Matrix Arithmetic. 3 Gauss-Jordan Elimination and Reduced Row Echelon Form. 3.1 Gauss-Jordan Elimination and rref. 3.2 Elementary Matrices. 3.3 Sensitivity of Solutions to Error in the Linear System. 4 Applications of Linear Systems and Matrices. 4.1 Applications of Linear Systems to Geometry. 4.2 Applications of Linear Systems to Curve Fitting. 4.3 Applications of Linear Systems to Economics. 4.4 Applications of Matrix Multiplication to Geometry. 4.5 An Application of Matrix Multiplication to Economics. 5 Determinants, Inverses and Cramer s Rule. 5.1 Determinants and Inverses from the Adjoint Formula. 5.2 Determinants by Expanding Along Any Row or Column . 5.3 Determinants Found by Triangularizing Matrices. 5.4 LU Factorization. 5.5 Inverses from rref. 5.6 Cramer s Rule. 6 Basic Linear Algebra Topics. 6.1 Vectors. 6.2 Dot Product. 6.3 Cross Product. 6.4 Vector Projection. 7 A Few Advanced Linear Algebra Topics. 7.1 Rotations in Space. 7.2 Rolling a Circle Along a Curve. 7.3 The TNB Frame. 8 Independence, Basis and Dimension for Subspaces of Rn. 8.1 Subspaces of Rn. 8.2 Independent and Dependent Sets of Vectors in Rn. 8.3 Basis and Dimension for Subspaces of Rn. 8.4 Vector Projection onto a Subspace of Rn. 8.5 The Gram-Schmidt Orthonormalization Process. 9 Linear Maps from Rn to Rm. 9.1 Basics About Linear Maps. 9.2 The Kernel and Image Subspaces of a Linear Map. 9.3 Composites of Two Linear maps and Inverses. 9.4 Change of Bases for the Matrix Representation of a Linear Map. 10 The Geometry of Linear and Affine Maps. 10.1 The Effect of a Linear Map on Area and Arclength in Two Dimensions. 10.2 The Decomposition of Linear Maps into Rotations, Reflections and Rescalings in R2. 10.3 The Effect of Linear Maps on Volume, Area and Arclength in R3. 10.4 Rotations, Reflections and Rescalings in Three Dimensions. 10.5 Affine Maps. 11 Least Squares Fits and Pseudoinverses. 11.1 Pseudoinverse to a Non-Square Matrix and Almost Solving an Overdetermined Linear System. 11.2 Fits and Pseudoinverses. 11.3 Least Squares Fits and Pseudoinverses. 12 Eigenvalues and Eigenvectors. 12.1 What Are Eigenvalues and Eigenvectors, and Why Do We Need Them? 12.2 Summary of Definitions and Methods for Computing Eigenvalues and Eigenvectors as well as the Exponential of a Matrix. 12.3 Applications of the Diagonalizability of Square Matrices. 12.4 Solving a Square First Order Linear. System of Differential Equations ... 12.5 Basic Facts About Eigenvalues and Eigenvectors, and Diagonalizability. 12.6 The Geometry of the Ellipse Using Eigenvalues and Eigenvectors. 12.7 A Maple Eigen-Procedure. Bibliography. Indices. Keyword Index. Index of Maple Commands and Packages.

This paper presents a novel system for image retrieval and classification based on a robust multi-dimensional view invariant representation and a linear classifier algorithm. Specifically, multi-dimensional null space invariant (NSI) matrix representation has been derived. Moreover, the robustness of null space representation has been investigated with perturbation analysis in terms of the error ratio and SNR. The proposed representation is invariant to affine transformations and preserves the null space matrix. We use principal component null space analysis (PCNSA) of the NSI operator for recognition, indexing, and retrieval of 2D and 3D images. We rely on PCNSA to determine the distance of a query image to the centroid of the class, which is a statistical information vector in the PCNSA algorithm representing the corresponding class of the object. Our results shows that NSI provides a robust and powerful approach to image recognition and classification even when the query image or the stored image has been subjected to unknown affine transformations due to camera motions.

This chapter gives a brief introduction to the mathematics involved in the determination of the subgroups of space groups. The algebraic concepts of vector spaces, the affine space and the affine group are defined and discussed. A section on groups with special emphasis on actions of groups on sets, the Sylow theorems and the isomorphism theorems follows. After the definition of space groups, their maximal subgroups are considered and the theorem of Hermann is derived. It is shown that a maximal subgroup of a space group has a finite index and is a space group again. From the proof that three-dimensional space groups are soluble groups, it follows that the indices of their maximal subgroups are prime powers. Special considerations are devoted to the subgroups of index 2 and 3. Furthermore, a maximal subgroup is an isomorphic subgroup if its index is larger than 4. In addition, more special quantitative results on the numbers and indices of maximal subgroups of space groups are derived. The abstract definitions and theorems are illustrated by several examples and applications. Keywords: Abelian groups; CARAT; Euclidean affine spaces; Euclidean groups; Euclidean metrics; Euclidean vector spaces; Hermann’s theorem; Lagrange’s theorem; Sylow’s theorems; affine groups; affine mappings; affine spaces; automorphism; characteristic subgroups; congruence; factor groups; faithful actions; isomorphism; generators; groups; homomorphism; klassengleiche subgroups; cosets; isomorphic groups; mappings; normalizers; space groups; isomorphism theorems; translation subgroups; maximal subgroups; translationengleiche subgroups; vector spaces; orbits; alternating groups; symmorphic space groups

This chapter gives a brief introduction to the mathematics involved in the determination of the subgroups of space groups. The algebraic concepts of vector spaces, the affine space and the affine group are defined and discussed. A section on groups with special emphasis on actions of groups on sets, the Sylow theorems and the isomorphism theorems follows. After the definition of space groups, their maximal subgroups are considered and the theorem of Hermann is derived. It is shown that a maximal subgroup of a space group has a finite index and is a space group again. From the proof that three-dimensional space groups are soluble groups, it follows that the indices of their maximal subgroups are prime powers. Special considerations are devoted to the subgroups of index 2 and 3. Furthermore, a maximal subgroup is an isomorphic subgroup if its index is larger than 4. In addition, more special quantitative results on the numbers and indices of maximal subgroups of space groups are derived. The abstract definitions and theorems are illustrated by several examples and applications. Keywords: Abelian groups; CARAT; Euclidean affine spaces; Euclidean groups; Euclidean metrics; Euclidean vector spaces; Hermann’s theorem; Lagrange’s theorem; Sylow’s theorems; affine groups; affine mappings; affine spaces; automorphism; characteristic subgroups; congruence; factor groups; faithful actions; isomorphism; generators; groups; homomorphism; klassengleiche subgroups; cosets; isomorphic groups; mappings; normalizers; space groups; isomorphism theorems; translation subgroups; maximal subgroups; translationengleiche subgroups; vector spaces; orbits; alternating groups; symmorphic space groups

This work concerns with the Taylor expansion remainder term evaluation. The integral defining the remainder is standardized in interval first, by using an appropriate affine transformation. Then it is approximated by the utilization of the fluctuation free integration which was developed quite recently. This method approximates the matrix representation of the function to be integrated in terms of the matrix representation of the independent variable. The approximation quality depends on a lot of issues like curvature, smoothness, singularities, and the basis function used in the representation. This work investigates the basis functions containing a common factor of exponential function times an appropriate power of the independent variable. The purpose is to investigate the role of the decaying nature existing in the basis functions on the approximation quality.

This paper considers a lattice represented by integer coordinates in n‐dimensional space (digital lattice). On the digital lattice, an automorphism is constructed by a newly proposed theory and algorithm, to enable the approximation of the arbitrarily given equivolume affine transformation. The equivolume affine transformation is a general affine transformation that satisfies the condition of volume invariance, and is characterized by the determinant of the representation matrix having an absolute value of 1. An equivolume affine transformation in n‐dimensional space can represent any combination of reflection, rotation, equivolume expansion/contraction and skew deformation. These transformations often are employed as fundamental transformations in the handling of geometrical information in computers.This paper discusses first the geometrical characters of equivolume affine transformations. It then defines the fundamental reflection, skew and translations. It is shown that the equivolume affine transformation in n‐dimensional space can be decomposed into the product of (n2‐1) fundamental skew transformations, n fundamental translations and a finite number of fundamental reflections.For the fundamental transformation, one‐to‐one integer approximation is defined, and a systematic method is proposed to evaluate the upper bound of the approximation error. With the proposed error estimation method, an algorithm is proposed to determine the decomposition suppressing the error in the overall system. One advantage of this theory is that it is no longer necessary to preserve the geometrical information before transformation in the computer, which has been commonplace up to now in computer programming.

Boolean functions with good cryptographic characteristics are needed for the design of robust pseudo-random generators for stream ciphers and of S-boxes for block ciphers. Very few general constructions of such cryptographic Boolean functions are known. The main ones correspond to concatenating affine or quadratic functions. We introduce a general construction corresponding to the concatenation of indicators of flats. We show that the functions it permits to design can present very good cryptographic characteristics.

1. Introduction. Image Processing as Picture Analysis. The Advantages of Interactive Graphics. Representative Uses of Computer Graphics. Classification of Applications. Development of Hardware and Software for Computer Graphics. Conceptual Framework for Interactive Graphics. 2. Programming in the Simple Raster Graphics Package (SRGP)/. Drawing with SRGP/. Basic Interaction Handling/. Raster Graphics Features/. Limitations of SRGP/. 3. Basic Raster Graphics Algorithms for Drawing 2d Primitives. Overview. Scan Converting Lines. Scan Converting Circles. Scan Convertiing Ellipses. Filling Rectangles. Fillign Polygons. Filling Ellipse Arcs. Pattern Filling. Thick Primiives. Line Style and Pen Style. Clipping in a Raster World. Clipping Lines. Clipping Circles and Ellipses. Clipping Polygons. Generating Characters. SRGP_copyPixel. Antialiasing. 4. Graphics Hardware. Hardcopy Technologies. Display Technologies. Raster-Scan Display Systems. The Video Controller. Random-Scan Display Processor. Input Devices for Operator Interaction. Image Scanners. 5. Geometrical Transformations. 2D Transformations. Homogeneous Coordinates and Matrix Representation of 2D Transformations. Composition of 2D Transformations. The Window-to-Viewport Transformation. Efficiency. Matrix Representation of 3D Transformations. Composition of 3D Transformations. Transformations as a Change in Coordinate System. 6. Viewing in 3D. Projections. Specifying an Arbitrary 3D View. Examples of 3D Viewing. The Mathematics of Planar Geometric Projections. Implementing Planar Geometric Projections. Coordinate Systems. 7. Object Hierarchy and Simple PHIGS (SPHIGS). Geometric Modeling. Characteristics of Retained-Mode Graphics Packages. Defining and Displaying Structures. Modeling Transformations. Hierarchical Structure Networks. Matrix Composition in Display Traversal. Appearance-Attribute Handling in Hierarchy. Screen Updating and Rendering Modes. Structure Network Editing for Dynamic Effects. Interaction. Additional Output Features. Implementation Issues. Optimizing Display of Hierarchical Models. Limitations of Hierarchical Modeling in PHIGS. Alternative Forms of Hierarchical Modeling. 8. Input Devices, Interaction Techniques, and Interaction Tasks. Interaction Hardware. Basic Interaction Tasks. Composite Interaction Tasks. 9. Dialogue Design. The Form and Content of User-Computer Dialogues. User-Interfaces Styles. Important Design Considerations. Modes and Syntax. Visual Design. The Design Methodology. 10. User Interface Software. Basic Interaction-Handling Models. Windows-Management Systems. Output Handling in Window Systems. Input Handling in Window Systems. Interaction-Technique Toolkits. User-Interface Management Systems. 11. Representing Curves and Surfaces. Polygon Meshes. Parametric Cubic Curves. Parametric Bicubic Surfaces. Quadric Surfaces. 12. Solid Modeling. Representing Solids. Regularized Boolean Set Operations. Primitive Instancing. Sweep Representations. Boundary Representations. Spatial-Partitioning Representations. Constructive Solid Geometry. Comparison of Representations. User Interfaces for Solid Modeling. 13. Achromatic and Colored Light. Achromatic Light. Chromatic Color. Color Models for Raster Graphics. Reproducing Color. Using Color in Computer Graphics. 14. The Quest for Visual Realism. Why Realism? Fundamental Difficulties. Rendering Techniques for Line Drawings. Rendering Techniques for Shaded Images. Improved Object Models. Dynamics. Stereopsis. Improved Displays. Interacting with Our Other Senses. Aliasing and Antialiasing. 15. Visible-Surface Determination. Functions of Two Variables. Techniques for Efficient Visible-Surface Determination. Algorithms for Visible-Line Determination. The z-Buffer Algorithm. List-Priority Algorithms. Scan-Line Algorithms. Area-Subdivision Algorithms. Algorithms for Octrees. Algorithms for Curved Surfaces. Visible-Surface Ray Tracing. 16. Illumination And Shading. Illumination Modeling. Shading Models for Polygons. Surface Detail. Shadows. Transparency. Interobject Reflections. Physically Based Illumination Models. Extended Light Sources. Spectral Sampling. Improving the Camera Model. Global Illumination Algorithms. Recursive Ray Tracing. Radiosity Methods. The Rendering Pipeline. 17. Image Manipulation and Storage. What Is an Image? Filtering. Image Processing. Geometric Transformations of Images. Multipass Transformations. Image Compositing. Mechanisms for Image Storage. Special Effects with Images. Summary. 18. Advanced Raster Graphic Architecture. Simple Raster-Display System. Display-Processor Systems. Standard Graphics Pipeline. Introduction to Multiprocessing. Pipeline Front-End Architecture. Parallel Front-End Architectures. Multiprocessor Rasterization Architectures. Image-Parallel Rasterization. Object-Parallel Rasterization. Hybrid-Parallel Rasterization. Enhanced Display Capabilities. 19. Advanced Geometric and Raster Algorithms. Clipping. Scan-Converting Primitives. Antialiasing. The Special Problems of Text. Filling Algorithms. Making copyPixel Fast. The Shape Data Structure and Shape Algebra. Managing Windows with bitBlt. Page Description Languages. 20. Advanced Modeling Techniques. Extensions of Previous Techniques. Procedural Models. Fractal Models. Grammar-Based Models. Particle Systems. Volume Rendering. Physically Based Modeling. Special Models for Natural and Synthetic Objects. Automating Object Placement. 21. Animation. Conventional and Computer-Assisted Animation. Animation Languages. Methods of Controlling Animation. Basic Rules of Animation. Problems Peculiar to Animation. Appendix: Mathematics for Computer Graphics. Vector Spaces and Affine Spaces. Some Standard Constructions in Vector Spaces. Dot Products and Distances. Matrices. Linear and Affine Transformations. Eigenvalues and Eigenvectors. Newton-Raphson Iteration for Root Finding. Bibliography. Index. 0201848406T04062001

Matrix Representations for Affine and Projective Transformations

The theory of algebra is highly beneficial and used in daily life. Identifying plagiarism is one of many activities that can be solved with the concept of linear algebra. Even today, plagiarism is still a crucial issue to discuss, preferably in academic scope, both in assignment papers and academic theses. Identifying plagiarism aims to provide a document as a vector. Even the basic concept of linear algebra can be applied in plagiarism detection applications. This paper proposed a modeling example of plagiarism checking on a document with a matrix representation and the calculation of angles among subspaces of each compared document. Finally, the results can be used as one of the considerations to determine the similarity index.Keywords: plagiarism, linear algebra, vector, matrix representation

This presentation is devoted to explain the important aspects of the Mathematical Fluctuations Theory. Although its name started to be used basically by us quite recently it is not a completely independent issue from the other existing ones. It uses many well known concepts of linear algebra, especially Hilbert spaces and matrix representations. This paper bears the characteristics of a review article together with the recent developments in the concrete applications and abstract ideas related to this theory. Main emphasis focuses on the definition and meaning of the fluctuation concept which is related to somehow truncation errors. Although all of them are originated from the same idea, the fluctuations in functions, in operators, and in matrix representations are tried to be distinguished from each others. To this end certain simple illustrations are given to easily explain the authors' ideas about these concepts and their utilization in some applications.