Many-Sorted Algebras for Deep Learning and Quantum Technology (eBook)

Many-Sorted Algebras for Deep Learning and Quantum Technology presents a precise and rigorous description of basic concepts in Quantum technologies and how they relate to Deep Learning and Quantum Theory. Current merging of Quantum Theory and Deep Learning techniques provides a need for a text that can give readers insight into the algebraic underpinnings of these disciplines. Although analytical, topological, probabilistic, as well as geometrical concepts are employed in many of these areas, algebra exhibits the principal thread. This thread is exposed using Many-Sorted Algebras (MSA). In almost every aspect of Quantum Theory as well as Deep Learning more than one sort or type of object is involved. For instance, in Quantum areas Hilbert spaces require two sorts, while in affine spaces, three sorts are needed. Both a global level and a local level of precise specification is described using MSA. At a local level operation involving neural nets may appear to be very algebraically different than those used in Quantum systems, but at a global level they may be identical. Again, MSA is well equipped to easily detail their equivalence through text as well as visual diagrams. Among the reasons for using MSA is in illustrating this sameness. Author Charles R. Giardina includes hundreds of well-designed examples in the text to illustrate the intriguing concepts in Quantum systems. Along with these examples are numerous visual displays. In particular, the Polyadic Graph shows the types or sorts of objects used in Quantum or Deep Learning. It also illustrates all the inter and intra sort operations needed in describing algebras. In brief, it provides the closure conditions. Throughout the text, all laws or equational identities needed in specifying an algebraic structure are precisely described. - Includes hundreds of well-designed examples to illustrate the intriguing concepts in quantum systems - Provides precise description of all laws or equational identities that are needed in specifying an algebraic structure - Illustrates all the inter and intra sort operations needed in describing algebras

Charles R. Giardina was born in the Bronx, NY, on December 29, 1942. He received the B.S. degree in mathematics from Fairleigh Dickinson University, Rutherford, NJ, and the M.S. degree in mathematics from Carnegie Institute of Technology, Pittsburgh, PA. He also received the M.E.E. degree in 1969, and the Ph.D. degree in mathematics and electrical engineering in 1970 from Stevens Institute of Technology, Hoboken, NJ. Dr. Giardina was Professor of Mathematics, Electrical Engineering, and Computer Science at Fairleigh Dickinson University from 1965 to 1982. From 1982 to 1986, he was a Professor at the Stevens Institute of Technology. From 1986 to 1996, he was a Professor at the College of Staten Island, City University of New York. From 1996, he was with Bell Telephone Laboratories, Whippany, NJ, USA. His research interests include digital signal and image processing, pattern recognition, artificial intelligence, and the constructive theory of functions. Dr. Giardina has authored numerous papers in these areas, and several books including, Mathematical Models for Artificial Intelligence and Autonomous Systems, Prentice Hall; Matrix Structure Image Processing, Prentice Hall; Parallel Digital Signal Processing: A Unified Signal Algebra Approach, Regency; Morphological Methods in Image and Signal Processing, Prentice Hall; Image Processing - Continuous to Discrete: Geometric, Transform, and Statistical Methods, Prentice Hall; and A Unified Signal Algebra Approach to Two-Dimensional Parallel Digital Signal Processing, Chapman and Hall/CRC Press.

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Many-Sorted Algebras for Deep Learning and Quantum Technology presents a precise and rigorous description of basic concepts in Quantum technologies and how they relate to Deep Learning and Quantum Theory. Current merging of Quantum Theory and Deep Learning techniques provides a need for a text that can give readers insight into the algebraic underpinnings of these disciplines. Although analytical, topological, probabilistic, as well as geometrical concepts are employed in many of these areas, algebra exhibits the principal thread. This thread is exposed using Many-Sorted Algebras (MSA). In almost every aspect of Quantum Theory as well as Deep Learning more than one sort or type of object is involved. For instance, in Quantum areas Hilbert spaces require two sorts, while in affine spaces, three sorts are needed. Both a global level and a local level of precise specification is described using MSA. At a local level operation involving neural nets may appear to be very algebraically different than those used in Quantum systems, but at a global level they may be identical. Again, MSA is well equipped to easily detail their equivalence through text as well as visual diagrams. Among the reasons for using MSA is in illustrating this sameness. Author Charles R. Giardina includes hundreds of well-designed examples in the text to illustrate the intriguing concepts in Quantum systems. Along with these examples are numerous visual displays. In particular, the Polyadic Graph shows the types or sorts of objects used in Quantum or Deep Learning. It also illustrates all the inter and intra sort operations needed in describing algebras. In brief, it provides the closure conditions. Throughout the text, all laws or equational identities needed in specifying an algebraic structure are precisely described. - Includes hundreds of well-designed examples to illustrate the intriguing concepts in quantum systems- Provides precise description of all laws or equational identities that are needed in specifying an algebraic structure- Illustrates all the inter and intra sort operations needed in describing algebras