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Full Description
Lattice theory extends into virtually every branch of mathematics, ranging from measure theory and convex geometry to probability theory and topology. A more recent development has been the rapid escalation of employing lattice theory for various applications outside the domain of pure mathematics. These applications range from electronic communication theory and gate array devices that implement Boolean logic to artificial intelligence and computer science in general.
Introduction to Lattice Algebra: With Applications in AI, Pattern Recognition, Image Analysis, and Biomimetic Neural Networks lays emphasis on two subjects, the first being lattice algebra and the second the practical applications of that algebra. This textbook is intended to be used for a special topics course in artificial intelligence with a focus on pattern recognition, multispectral image analysis, and biomimetic artificial neural networks. The book is self-contained and - depending on the student's major - can be used for a senior undergraduate level or first-year graduate level course. The book is also an ideal self-study guide for researchers and professionals in the above-mentioned disciplines.
Features
Filled with instructive examples and exercises to help build understanding
Suitable for researchers, professionals and students, both in mathematics and computer science
Contains numerous exercises.
Contents
1. Elements of Algebra. 1.1. Sets, Functions, and Notations. 1.2. Algebraic Structures. 2. Pertinent Properties of R. 2.2. Elementary Properties of Euclidean Spaces. 3. Lattice Theory. 3.1. Historical Background. 3.2. Partial Orders and Lattices. 3.3. Relations with other branches of Mathematics. 4. Lattice Algebra. 4.1. Lattice Semigroups and Lattice Groups. 4.2. Minimax Algebra. 4.3. Minimax Matrix Theory. 4.4. The Geometry of S(X).5. Matrix-Based Lattice Associative Memories. 5.1. Historical Background. 5.2. Associative Memories. 6. Extreme Points of Data Sets. 6.1. Relevant Concepts of Convex Set Theory. 6.2. Affine Subsets of EXT(ß(X)).7. Image Unmixing and Segmentation. 7,1, Spectral Endmembers and Linear Unmixing. 7.2. Aviris Hyperspectral Image Examples. 7.3. Endmembers and Clustering Validation Indexes. 7.4. Color Image Segmentation. 8. Lattice-Based Biomimetic Neural Networks. 8.1. Biomimetics Artificial Neural Networks. 8.2. Lattice Biomimetic Neural Networks. 9. Learning in Biomimetic Neural Networks. 9.1 Learning in Single-Layer LBNNS. 9.2. Multi-Layer Lattice Biomimetic Neural Network. Epilogues. Bibliography.
