This is the first book on matrix and kernel transforms preserving positivity structures. It is self-contained and only requires modest prerequisites in analysis and linear algebra. Covering deep results and modern progress, the book succeeds as an introduction for beginners and a comprehensive reference for experts.
This is the first book on matrix and kernel transforms preserving positivity structures. It is self-contained and only requires modest prerequisites in analysis and linear algebra. Covering deep results and modern progress, the book succeeds as an introduction for beginners and a comprehensive reference for experts.Hinweis: Dieser Artikel kann nur an eine deutsche Lieferadresse ausgeliefert werden.
Apoorva Khare is an Associate Professor of Mathematics at IISc Bangalore. Following a PhD from University of Chicago, he worked at Yale University and Stanford University for eight years. He is a Ramanujan Fellow and Swarnajayanti Fellow of DST, India, with previous support from DARPA, the American Institute of Mathematics (via NSF), and ICMS, UK. Khare has been invited as a Plenary speaker at the leading global conferences in matrix theory and combinatorics: ILAS and FPSAC; and Sectional speaker in the quadrennial leading conferences in Asia (AMC2021) and the Americas (MCA2017).
Inhaltsangabe
Part I. Preliminaries, Entrywise Powers Preserving Positivity in Fixed Dimension: 1. The cone of positive semidefinite matrices 2. The Schur product theorem and nonzero lower bounds 3. Totally positive (TP) and totally non-negative (TN) matrices 4. TP matrices - generalized Vandermonde and Hankel moment matrices 5. Entrywise powers preserving positivity in fixed dimension 6. Mid-convex implies continuous, and 2 x 2 preservers 7. Entrywise preservers of positivity on matrices with zero patterns 8. Entrywise powers preserving positivity, monotonicity, superadditivity 9. Loewner convexity and single matrix encoders of preservers 10. Exercises Part II. Entrywise Functions Preserving Positivity in All Dimensions: 11. History - Shoenberg, Rudin, Vasudeva, and metric geometry 12. Loewner's determinant calculation in Horn's thesis 13. The stronger Horn-Loewner theorem, via mollifiers 14. Stronger Vasudeva and Schoenberg theorems, via Bernstein's theorem 15. Proof of stronger Schoenberg Theorem (Part I) - positivity certificates 16. Proof of stronger Schoenberg Theorem (Part II) - real analyticity 17. Proof of stronger Schoenberg Theorem (Part III) - complex analysis 18. Preservers of Loewner positivity on kernels 19. Preservers of Loewner monotonicity and convexity on kernels 20. Functions acting outside forbidden diagonal blocks 21. The Boas-Widder theorem on functions with positive differences 22. Menger's results and Euclidean distance geometry 23. Exercises Part III. Entrywise Polynomials Preserving Positivity in Fixed Dimension: 24. Entrywise polynomial preservers and Horn-Loewner type conditions 25. Polynomial preservers for rank-one matrices, via Schur polynomials 26. First-order approximation and leading term of Schur polynomials 27. Exact quantitative bound - monotonicity of Schur ratios 28. Polynomial preservers on matrices with real or complex entries 29. Cauchy and Littlewood's definitions of Schur polynomials 30. Exercises.
Part I. Preliminaries, Entrywise Powers Preserving Positivity in Fixed Dimension: 1. The cone of positive semidefinite matrices 2. The Schur product theorem and nonzero lower bounds 3. Totally positive (TP) and totally non-negative (TN) matrices 4. TP matrices - generalized Vandermonde and Hankel moment matrices 5. Entrywise powers preserving positivity in fixed dimension 6. Mid-convex implies continuous, and 2 x 2 preservers 7. Entrywise preservers of positivity on matrices with zero patterns 8. Entrywise powers preserving positivity, monotonicity, superadditivity 9. Loewner convexity and single matrix encoders of preservers 10. Exercises Part II. Entrywise Functions Preserving Positivity in All Dimensions: 11. History - Shoenberg, Rudin, Vasudeva, and metric geometry 12. Loewner's determinant calculation in Horn's thesis 13. The stronger Horn-Loewner theorem, via mollifiers 14. Stronger Vasudeva and Schoenberg theorems, via Bernstein's theorem 15. Proof of stronger Schoenberg Theorem (Part I) - positivity certificates 16. Proof of stronger Schoenberg Theorem (Part II) - real analyticity 17. Proof of stronger Schoenberg Theorem (Part III) - complex analysis 18. Preservers of Loewner positivity on kernels 19. Preservers of Loewner monotonicity and convexity on kernels 20. Functions acting outside forbidden diagonal blocks 21. The Boas-Widder theorem on functions with positive differences 22. Menger's results and Euclidean distance geometry 23. Exercises Part III. Entrywise Polynomials Preserving Positivity in Fixed Dimension: 24. Entrywise polynomial preservers and Horn-Loewner type conditions 25. Polynomial preservers for rank-one matrices, via Schur polynomials 26. First-order approximation and leading term of Schur polynomials 27. Exact quantitative bound - monotonicity of Schur ratios 28. Polynomial preservers on matrices with real or complex entries 29. Cauchy and Littlewood's definitions of Schur polynomials 30. Exercises.
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