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Algebra ad uso delle scuole medie inferiori / Tommaso Cifarelli
Algebra ad uso delle scuole medie inferiori / Tommaso Cifarelli
Autore Cifarelli, Tommaso
Pubbl/distr/stampa Napoli : Morano, [19..]
Descrizione fisica v. ; 19 cm
Disciplina 512
Soggetto topico Algebra
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione ita
Record Nr. UNISALENTO-991003666319707536
Cifarelli, Tommaso  
Napoli : Morano, [19..]
Materiale a stampa
Lo trovi qui: Univ. del Salento
Opac: Controlla la disponibilità qui
Algebra and applications . 1 Non-associative algebras and categories / / coordinated by Abdenacer Makhlouf
Algebra and applications . 1 Non-associative algebras and categories / / coordinated by Abdenacer Makhlouf
Pubbl/distr/stampa London, England ; ; Hoboken, New Jersey : , : Wiley : , : ISTE, , [2020]
Descrizione fisica 1 online resource (369 pages)
Disciplina 512.9
Soggetto topico Algebra
Soggetto genere / forma Electronic books.
ISBN 1-119-81815-X
1-119-81817-6
1-119-81816-8
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Cover -- Half-Title Page -- Title Page -- Copyright Page -- Contents -- Foreword -- 1 Jordan Superalgebras -- 1.1. Introduction -- 1.2. Tits-Kantor-Koecher construction -- 1.3. Basic examples (classical superalgebras) -- 1.4. Brackets -- 1.5. Cheng-Kac superalgebras -- 1.6. Finite dimensional simple Jordan superalgebras -- 1.6.1. Case F is algebraically closed and char F = 0 -- 1.6.2. Case char F = p > -- 2, the even part J¯0 is semisimple -- 1.6.3. Case char F = p > -- 2, the even part J¯0 is not semisimple -- 1.6.4. Non-unital simple Jordan superalgebras -- 1.7. Finite dimensional representations -- 1.7.1. Superalgebras of rank ≥ 3 -- 1.7.2. Superalgebras of rank ≤ 2 -- 1.8. Jordan superconformal algebras -- 1.9. References -- 2 Composition Algebras -- 2.1. Introduction -- 2.2. Quaternions and octonions -- 2.2.1. Quaternions -- 2.2.2. Rotations in three(and four-) dimensional space -- 2.2.3. Octonions -- 2.3. Unital composition algebras -- 2.3.1. The Cayley-Dickson doubling process and the generalized Hurwitz theorem -- 2.3.2. Isotropic Hurwitz algebras -- 2.4. Symmetric composition algebras -- 2.5. Triality -- 2.6. Concluding remarks -- 2.7. Acknowledgments -- 2.8. References -- 3 Graded-Division Algebras -- 3.1. Introduction -- 3.2. Background on gradings -- 3.2.1. Gradings induced by a group homomorphism -- 3.2.2. Weak isomorphism and equivalence -- 3.2.3. Basic properties of division gradings -- 3.2.4. Graded presentations of associative algebras -- 3.2.5. Tensor products of division gradings -- 3.2.6. Loop construction -- 3.2.7. Another construction of graded-simple algebras -- 3.3. Graded-division algebras over algebraically closed fields -- 3.4. Real graded-division associative algebras -- 3.4.1. Simple graded-division algebras -- 3.4.2. Pauli gradings -- 3.4.3. Commutative case.
3.4.4. Non-commutative graded-division algebras with one-dimensional homogeneous components -- 3.4.5. Equivalence classes of graded-division algebras with one-dimensional homogeneous components -- 3.4.6. Graded-division algebras with non-central two-dimensional identity components -- 3.4.7. Graded-division algebras with four-dimensional identity components -- 3.4.8. Classification of real graded-division algebras, up to isomorphism -- 3.5. Real loop algebras with a non-split centroid -- 3.6. Alternative algebras -- 3.6.1. Cayley-Dickson doubling process -- 3.6.2. Gradings on octonion algebras -- 3.6.3. Graded-simple real alternative algebras -- 3.6.4. Graded-division real alternative algebras -- 3.7. Gradings of fields -- 3.8. References -- 4 Non-associative C*-algebras -- 4.1. Introduction -- 4.2. JB-algebras -- 4.3. The non-associative Vidav-Palmer and Gelfand-Naimark theorems -- 4.4. JB*-triples -- 4.5. Past, present and future of non-associative C*-algebras -- 4.6. Acknowledgments -- 4.7. References -- 5 Structure of H -algebras -- 5.1. Introduction -- 5.2. Preliminaries: aspects of the general theory -- 5.3. Ultraproducts of H -algebras -- 5.4. Quadratic H -algebras -- 5.5. Associative H -algebras -- 5.6. Flexible H -algebras -- 5.7. Non-commutative Jordan H -algebras -- 5.8. Jordan H -algebras -- 5.9. Moufang H -algebras -- 5.10. Lie H -algebras -- 5.11. Topics closely related to Lie H -algebras -- 5.12. Two-graded H -algebras -- 5.13. Other topics: beyond the H -algebras -- 5.14. Acknowledgments -- 5.15. References -- 6 Krichever-Novikov Type Algebras: Definitions and Results -- 6.1. Introduction -- 6.2. The Virasoro algebra and its relatives -- 6.3. The geometric picture -- 6.3.1. The geometric realizations of the Witt algebra -- 6.3.2. Arbitrary genus generalizations -- 6.3.3. Meromorphic forms -- 6.4. Algebraic structures.
6.4.1. Associative structure -- 6.4.2. Lie and Poisson algebra structure -- 6.4.3. The vector field algebra and the Lie derivative -- 6.4.4. The algebra of differential operators -- 6.4.5. Differential operators of all degrees -- 6.4.6. Lie superalgebras of half forms -- 6.4.7. Jordan superalgebra -- 6.4.8. Higher genus current algebras -- 6.4.9. KN-type algebras -- 6.5. Almost-graded structure -- 6.5.1. Definition of almost-gradedness -- 6.5.2. Separating cycle and KN pairing -- 6.5.4. The algebras -- 6.5.5. Triangular decomposition and filtrations -- 6.6. Central extensions -- 6.6.1. Central extensions and cocycles -- 6.6.2. Geometric cocycles -- 6.6.3. Uniqueness and classification of central extensions -- 6.7. Examples and generalizations -- 6.7.1. The genus zero and three-point situation -- 6.7.2. Genus zero multipoint algebras - integrable systems -- 6.7.3. Deformations -- 6.8. Lax operator algebras -- 6.9. Fermionic Fock space -- 6.9.1. Semi-infinite forms and fermionic Fock space representations -- 6.9.2. b - c systems -- 6.10. Sugawara representation -- 6.11. Application to moduli space -- 6.12. Acknowledgments -- 6.13. References -- 7 An Introduction to Pre-Lie Algebras -- 7.1. Introduction -- 7.1.1. Explanation of notions -- 7.1.2. Two fundamental properties -- 7.1.3. Some subclasses -- 7.1.4. Organization of this chapter -- 7.2. Some appearances of pre-Lie algebras -- 7.2.2. Deformation complexes of algebras and right-symmetric algebras -- 7.2.3. Rooted tree algebras: free pre-Lie algebras -- 7.2.4. Complex structures on Lie algebras -- 7.2.5. Symplectic structures on Lie groups and Lie algebras, phase spaces of Lie algebras and Kähler structures -- 7.2.6. Vertex algebras -- 7.3. Some basic results and constructions of pre-Lie algebras -- 7.3.1. Some basic results of pre-Lie algebras.
7.3.2. Constructions of pre-Lie algebras from some known structures -- 7.4. Pre-Lie algebras and CYBE -- 7.4.1. The existence of a compatible pre-Lie algebra on a Lie algebra -- 7.4.2. CYBE: unification of tensor and operator forms -- 7.4.3. Pre-Lie algebras, O-operators and CYBE -- 7.4.4. An algebraic interpretation of "left-symmetry": construction from Lie algebras revisited -- 7.5. A larger framework: Lie analogues of Loday algebras -- 7.5.1. Pre-Lie algebras, dendriform algebras and Loday algebras -- 7.5.2. L-dendriform algebras -- 7.5.3. Lie analogues of Loday algebras -- 7.6. References -- 8 Symplectic, Product and Complex Structures on 3-Lie Algebras -- 8.1. Introduction -- 8.2. Preliminaries -- 8.3. Representations of 3-pre-Lie algebras -- 8.4. Symplectic structures and phase spaces of 3-Lie algebras -- 8.5. Product structures on 3-Lie algebras -- 8.6. Complex structures on 3-Lie algebras -- 8.7. Complex product structures on 3-Lie algebras -- 8.8. Para-Kähler structures on 3-Lie algebras -- 8.9. Pseudo-Kähler structures on 3-Lie algebras -- 8.10. References -- 9 Derived Categories -- 9.1. Introduction -- 9.2. Grothendieck's definition -- 9.3. Verdier's definition -- 9.4. Triangulated structure -- 9.5. Derived functors -- 9.6. Derived Morita theory -- 9.7. Dg categories -- 9.7.1. Dg categories and functors -- 9.7.2. The derived category -- 9.7.3. Derived functors -- 9.7.4. Dg quotients -- 9.7.5. Invariants -- 9.8. References -- List of Authors -- Index -- EULA.
Record Nr. UNINA-9910555016403321
London, England ; ; Hoboken, New Jersey : , : Wiley : , : ISTE, , [2020]
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Algebra and applications . 1 Non-associative algebras and categories / / coordinated by Abdenacer Makhlouf
Algebra and applications . 1 Non-associative algebras and categories / / coordinated by Abdenacer Makhlouf
Pubbl/distr/stampa London, England ; ; Hoboken, New Jersey : , : Wiley : , : ISTE, , [2020]
Descrizione fisica 1 online resource (369 pages)
Disciplina 512.9
Soggetto topico Algebra
ISBN 1-119-81815-X
1-119-81817-6
1-119-81816-8
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Cover -- Half-Title Page -- Title Page -- Copyright Page -- Contents -- Foreword -- 1 Jordan Superalgebras -- 1.1. Introduction -- 1.2. Tits-Kantor-Koecher construction -- 1.3. Basic examples (classical superalgebras) -- 1.4. Brackets -- 1.5. Cheng-Kac superalgebras -- 1.6. Finite dimensional simple Jordan superalgebras -- 1.6.1. Case F is algebraically closed and char F = 0 -- 1.6.2. Case char F = p > -- 2, the even part J¯0 is semisimple -- 1.6.3. Case char F = p > -- 2, the even part J¯0 is not semisimple -- 1.6.4. Non-unital simple Jordan superalgebras -- 1.7. Finite dimensional representations -- 1.7.1. Superalgebras of rank ≥ 3 -- 1.7.2. Superalgebras of rank ≤ 2 -- 1.8. Jordan superconformal algebras -- 1.9. References -- 2 Composition Algebras -- 2.1. Introduction -- 2.2. Quaternions and octonions -- 2.2.1. Quaternions -- 2.2.2. Rotations in three(and four-) dimensional space -- 2.2.3. Octonions -- 2.3. Unital composition algebras -- 2.3.1. The Cayley-Dickson doubling process and the generalized Hurwitz theorem -- 2.3.2. Isotropic Hurwitz algebras -- 2.4. Symmetric composition algebras -- 2.5. Triality -- 2.6. Concluding remarks -- 2.7. Acknowledgments -- 2.8. References -- 3 Graded-Division Algebras -- 3.1. Introduction -- 3.2. Background on gradings -- 3.2.1. Gradings induced by a group homomorphism -- 3.2.2. Weak isomorphism and equivalence -- 3.2.3. Basic properties of division gradings -- 3.2.4. Graded presentations of associative algebras -- 3.2.5. Tensor products of division gradings -- 3.2.6. Loop construction -- 3.2.7. Another construction of graded-simple algebras -- 3.3. Graded-division algebras over algebraically closed fields -- 3.4. Real graded-division associative algebras -- 3.4.1. Simple graded-division algebras -- 3.4.2. Pauli gradings -- 3.4.3. Commutative case.
3.4.4. Non-commutative graded-division algebras with one-dimensional homogeneous components -- 3.4.5. Equivalence classes of graded-division algebras with one-dimensional homogeneous components -- 3.4.6. Graded-division algebras with non-central two-dimensional identity components -- 3.4.7. Graded-division algebras with four-dimensional identity components -- 3.4.8. Classification of real graded-division algebras, up to isomorphism -- 3.5. Real loop algebras with a non-split centroid -- 3.6. Alternative algebras -- 3.6.1. Cayley-Dickson doubling process -- 3.6.2. Gradings on octonion algebras -- 3.6.3. Graded-simple real alternative algebras -- 3.6.4. Graded-division real alternative algebras -- 3.7. Gradings of fields -- 3.8. References -- 4 Non-associative C*-algebras -- 4.1. Introduction -- 4.2. JB-algebras -- 4.3. The non-associative Vidav-Palmer and Gelfand-Naimark theorems -- 4.4. JB*-triples -- 4.5. Past, present and future of non-associative C*-algebras -- 4.6. Acknowledgments -- 4.7. References -- 5 Structure of H -algebras -- 5.1. Introduction -- 5.2. Preliminaries: aspects of the general theory -- 5.3. Ultraproducts of H -algebras -- 5.4. Quadratic H -algebras -- 5.5. Associative H -algebras -- 5.6. Flexible H -algebras -- 5.7. Non-commutative Jordan H -algebras -- 5.8. Jordan H -algebras -- 5.9. Moufang H -algebras -- 5.10. Lie H -algebras -- 5.11. Topics closely related to Lie H -algebras -- 5.12. Two-graded H -algebras -- 5.13. Other topics: beyond the H -algebras -- 5.14. Acknowledgments -- 5.15. References -- 6 Krichever-Novikov Type Algebras: Definitions and Results -- 6.1. Introduction -- 6.2. The Virasoro algebra and its relatives -- 6.3. The geometric picture -- 6.3.1. The geometric realizations of the Witt algebra -- 6.3.2. Arbitrary genus generalizations -- 6.3.3. Meromorphic forms -- 6.4. Algebraic structures.
6.4.1. Associative structure -- 6.4.2. Lie and Poisson algebra structure -- 6.4.3. The vector field algebra and the Lie derivative -- 6.4.4. The algebra of differential operators -- 6.4.5. Differential operators of all degrees -- 6.4.6. Lie superalgebras of half forms -- 6.4.7. Jordan superalgebra -- 6.4.8. Higher genus current algebras -- 6.4.9. KN-type algebras -- 6.5. Almost-graded structure -- 6.5.1. Definition of almost-gradedness -- 6.5.2. Separating cycle and KN pairing -- 6.5.4. The algebras -- 6.5.5. Triangular decomposition and filtrations -- 6.6. Central extensions -- 6.6.1. Central extensions and cocycles -- 6.6.2. Geometric cocycles -- 6.6.3. Uniqueness and classification of central extensions -- 6.7. Examples and generalizations -- 6.7.1. The genus zero and three-point situation -- 6.7.2. Genus zero multipoint algebras - integrable systems -- 6.7.3. Deformations -- 6.8. Lax operator algebras -- 6.9. Fermionic Fock space -- 6.9.1. Semi-infinite forms and fermionic Fock space representations -- 6.9.2. b - c systems -- 6.10. Sugawara representation -- 6.11. Application to moduli space -- 6.12. Acknowledgments -- 6.13. References -- 7 An Introduction to Pre-Lie Algebras -- 7.1. Introduction -- 7.1.1. Explanation of notions -- 7.1.2. Two fundamental properties -- 7.1.3. Some subclasses -- 7.1.4. Organization of this chapter -- 7.2. Some appearances of pre-Lie algebras -- 7.2.2. Deformation complexes of algebras and right-symmetric algebras -- 7.2.3. Rooted tree algebras: free pre-Lie algebras -- 7.2.4. Complex structures on Lie algebras -- 7.2.5. Symplectic structures on Lie groups and Lie algebras, phase spaces of Lie algebras and Kähler structures -- 7.2.6. Vertex algebras -- 7.3. Some basic results and constructions of pre-Lie algebras -- 7.3.1. Some basic results of pre-Lie algebras.
7.3.2. Constructions of pre-Lie algebras from some known structures -- 7.4. Pre-Lie algebras and CYBE -- 7.4.1. The existence of a compatible pre-Lie algebra on a Lie algebra -- 7.4.2. CYBE: unification of tensor and operator forms -- 7.4.3. Pre-Lie algebras, O-operators and CYBE -- 7.4.4. An algebraic interpretation of "left-symmetry": construction from Lie algebras revisited -- 7.5. A larger framework: Lie analogues of Loday algebras -- 7.5.1. Pre-Lie algebras, dendriform algebras and Loday algebras -- 7.5.2. L-dendriform algebras -- 7.5.3. Lie analogues of Loday algebras -- 7.6. References -- 8 Symplectic, Product and Complex Structures on 3-Lie Algebras -- 8.1. Introduction -- 8.2. Preliminaries -- 8.3. Representations of 3-pre-Lie algebras -- 8.4. Symplectic structures and phase spaces of 3-Lie algebras -- 8.5. Product structures on 3-Lie algebras -- 8.6. Complex structures on 3-Lie algebras -- 8.7. Complex product structures on 3-Lie algebras -- 8.8. Para-Kähler structures on 3-Lie algebras -- 8.9. Pseudo-Kähler structures on 3-Lie algebras -- 8.10. References -- 9 Derived Categories -- 9.1. Introduction -- 9.2. Grothendieck's definition -- 9.3. Verdier's definition -- 9.4. Triangulated structure -- 9.5. Derived functors -- 9.6. Derived Morita theory -- 9.7. Dg categories -- 9.7.1. Dg categories and functors -- 9.7.2. The derived category -- 9.7.3. Derived functors -- 9.7.4. Dg quotients -- 9.7.5. Invariants -- 9.8. References -- List of Authors -- Index -- EULA.
Record Nr. UNINA-9910677261803321
London, England ; ; Hoboken, New Jersey : , : Wiley : , : ISTE, , [2020]
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Algebra and applications . 1 Non-associative algebras and categories / / coordinated by Abdenacer Makhlouf
Algebra and applications . 1 Non-associative algebras and categories / / coordinated by Abdenacer Makhlouf
Pubbl/distr/stampa London, England ; ; Hoboken, New Jersey : , : Wiley : , : ISTE, , [2020]
Descrizione fisica 1 online resource (369 pages)
Disciplina 512.9
Soggetto topico Algebra
ISBN 1-119-81815-X
1-119-81817-6
1-119-81816-8
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Cover -- Half-Title Page -- Title Page -- Copyright Page -- Contents -- Foreword -- 1 Jordan Superalgebras -- 1.1. Introduction -- 1.2. Tits-Kantor-Koecher construction -- 1.3. Basic examples (classical superalgebras) -- 1.4. Brackets -- 1.5. Cheng-Kac superalgebras -- 1.6. Finite dimensional simple Jordan superalgebras -- 1.6.1. Case F is algebraically closed and char F = 0 -- 1.6.2. Case char F = p > -- 2, the even part J¯0 is semisimple -- 1.6.3. Case char F = p > -- 2, the even part J¯0 is not semisimple -- 1.6.4. Non-unital simple Jordan superalgebras -- 1.7. Finite dimensional representations -- 1.7.1. Superalgebras of rank ≥ 3 -- 1.7.2. Superalgebras of rank ≤ 2 -- 1.8. Jordan superconformal algebras -- 1.9. References -- 2 Composition Algebras -- 2.1. Introduction -- 2.2. Quaternions and octonions -- 2.2.1. Quaternions -- 2.2.2. Rotations in three(and four-) dimensional space -- 2.2.3. Octonions -- 2.3. Unital composition algebras -- 2.3.1. The Cayley-Dickson doubling process and the generalized Hurwitz theorem -- 2.3.2. Isotropic Hurwitz algebras -- 2.4. Symmetric composition algebras -- 2.5. Triality -- 2.6. Concluding remarks -- 2.7. Acknowledgments -- 2.8. References -- 3 Graded-Division Algebras -- 3.1. Introduction -- 3.2. Background on gradings -- 3.2.1. Gradings induced by a group homomorphism -- 3.2.2. Weak isomorphism and equivalence -- 3.2.3. Basic properties of division gradings -- 3.2.4. Graded presentations of associative algebras -- 3.2.5. Tensor products of division gradings -- 3.2.6. Loop construction -- 3.2.7. Another construction of graded-simple algebras -- 3.3. Graded-division algebras over algebraically closed fields -- 3.4. Real graded-division associative algebras -- 3.4.1. Simple graded-division algebras -- 3.4.2. Pauli gradings -- 3.4.3. Commutative case.
3.4.4. Non-commutative graded-division algebras with one-dimensional homogeneous components -- 3.4.5. Equivalence classes of graded-division algebras with one-dimensional homogeneous components -- 3.4.6. Graded-division algebras with non-central two-dimensional identity components -- 3.4.7. Graded-division algebras with four-dimensional identity components -- 3.4.8. Classification of real graded-division algebras, up to isomorphism -- 3.5. Real loop algebras with a non-split centroid -- 3.6. Alternative algebras -- 3.6.1. Cayley-Dickson doubling process -- 3.6.2. Gradings on octonion algebras -- 3.6.3. Graded-simple real alternative algebras -- 3.6.4. Graded-division real alternative algebras -- 3.7. Gradings of fields -- 3.8. References -- 4 Non-associative C*-algebras -- 4.1. Introduction -- 4.2. JB-algebras -- 4.3. The non-associative Vidav-Palmer and Gelfand-Naimark theorems -- 4.4. JB*-triples -- 4.5. Past, present and future of non-associative C*-algebras -- 4.6. Acknowledgments -- 4.7. References -- 5 Structure of H -algebras -- 5.1. Introduction -- 5.2. Preliminaries: aspects of the general theory -- 5.3. Ultraproducts of H -algebras -- 5.4. Quadratic H -algebras -- 5.5. Associative H -algebras -- 5.6. Flexible H -algebras -- 5.7. Non-commutative Jordan H -algebras -- 5.8. Jordan H -algebras -- 5.9. Moufang H -algebras -- 5.10. Lie H -algebras -- 5.11. Topics closely related to Lie H -algebras -- 5.12. Two-graded H -algebras -- 5.13. Other topics: beyond the H -algebras -- 5.14. Acknowledgments -- 5.15. References -- 6 Krichever-Novikov Type Algebras: Definitions and Results -- 6.1. Introduction -- 6.2. The Virasoro algebra and its relatives -- 6.3. The geometric picture -- 6.3.1. The geometric realizations of the Witt algebra -- 6.3.2. Arbitrary genus generalizations -- 6.3.3. Meromorphic forms -- 6.4. Algebraic structures.
6.4.1. Associative structure -- 6.4.2. Lie and Poisson algebra structure -- 6.4.3. The vector field algebra and the Lie derivative -- 6.4.4. The algebra of differential operators -- 6.4.5. Differential operators of all degrees -- 6.4.6. Lie superalgebras of half forms -- 6.4.7. Jordan superalgebra -- 6.4.8. Higher genus current algebras -- 6.4.9. KN-type algebras -- 6.5. Almost-graded structure -- 6.5.1. Definition of almost-gradedness -- 6.5.2. Separating cycle and KN pairing -- 6.5.4. The algebras -- 6.5.5. Triangular decomposition and filtrations -- 6.6. Central extensions -- 6.6.1. Central extensions and cocycles -- 6.6.2. Geometric cocycles -- 6.6.3. Uniqueness and classification of central extensions -- 6.7. Examples and generalizations -- 6.7.1. The genus zero and three-point situation -- 6.7.2. Genus zero multipoint algebras - integrable systems -- 6.7.3. Deformations -- 6.8. Lax operator algebras -- 6.9. Fermionic Fock space -- 6.9.1. Semi-infinite forms and fermionic Fock space representations -- 6.9.2. b - c systems -- 6.10. Sugawara representation -- 6.11. Application to moduli space -- 6.12. Acknowledgments -- 6.13. References -- 7 An Introduction to Pre-Lie Algebras -- 7.1. Introduction -- 7.1.1. Explanation of notions -- 7.1.2. Two fundamental properties -- 7.1.3. Some subclasses -- 7.1.4. Organization of this chapter -- 7.2. Some appearances of pre-Lie algebras -- 7.2.2. Deformation complexes of algebras and right-symmetric algebras -- 7.2.3. Rooted tree algebras: free pre-Lie algebras -- 7.2.4. Complex structures on Lie algebras -- 7.2.5. Symplectic structures on Lie groups and Lie algebras, phase spaces of Lie algebras and Kähler structures -- 7.2.6. Vertex algebras -- 7.3. Some basic results and constructions of pre-Lie algebras -- 7.3.1. Some basic results of pre-Lie algebras.
7.3.2. Constructions of pre-Lie algebras from some known structures -- 7.4. Pre-Lie algebras and CYBE -- 7.4.1. The existence of a compatible pre-Lie algebra on a Lie algebra -- 7.4.2. CYBE: unification of tensor and operator forms -- 7.4.3. Pre-Lie algebras, O-operators and CYBE -- 7.4.4. An algebraic interpretation of "left-symmetry": construction from Lie algebras revisited -- 7.5. A larger framework: Lie analogues of Loday algebras -- 7.5.1. Pre-Lie algebras, dendriform algebras and Loday algebras -- 7.5.2. L-dendriform algebras -- 7.5.3. Lie analogues of Loday algebras -- 7.6. References -- 8 Symplectic, Product and Complex Structures on 3-Lie Algebras -- 8.1. Introduction -- 8.2. Preliminaries -- 8.3. Representations of 3-pre-Lie algebras -- 8.4. Symplectic structures and phase spaces of 3-Lie algebras -- 8.5. Product structures on 3-Lie algebras -- 8.6. Complex structures on 3-Lie algebras -- 8.7. Complex product structures on 3-Lie algebras -- 8.8. Para-Kähler structures on 3-Lie algebras -- 8.9. Pseudo-Kähler structures on 3-Lie algebras -- 8.10. References -- 9 Derived Categories -- 9.1. Introduction -- 9.2. Grothendieck's definition -- 9.3. Verdier's definition -- 9.4. Triangulated structure -- 9.5. Derived functors -- 9.6. Derived Morita theory -- 9.7. Dg categories -- 9.7.1. Dg categories and functors -- 9.7.2. The derived category -- 9.7.3. Derived functors -- 9.7.4. Dg quotients -- 9.7.5. Invariants -- 9.8. References -- List of Authors -- Index -- EULA.
Record Nr. UNINA-9910820822103321
London, England ; ; Hoboken, New Jersey : , : Wiley : , : ISTE, , [2020]
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Algebra and applications 2 : combinatorial algebra and Hopf algebras / / edited by Abdenacer Makhlouf
Algebra and applications 2 : combinatorial algebra and Hopf algebras / / edited by Abdenacer Makhlouf
Pubbl/distr/stampa Hoboken, New Jersey : , : John Wiley & Sons, Incorporated, , [2021]
Descrizione fisica 1 online resource (336 pages)
Disciplina 512
Soggetto topico Algebra
Soggetto genere / forma Electronic books.
ISBN 1-119-88090-4
1-119-88091-2
1-119-88089-0
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Cover -- Half-Title Page -- Title Page -- Copyright Page -- Contents -- Preface -- 1. Algebraic Background for Numerical Methods, Control Theory and Renormalization -- 1.1. Introduction -- 1.2. Hopf algebras: general properties -- 1.2.1. Algebras -- 1.2.2. Coalgebras -- 1.2.3. Convolution product -- 1.2.4. Bialgebras and Hopf algebras -- 1.2.5. Some simple examples of Hopf algebras -- 1.2.6. Some basic properties of Hopf algebras -- 1.3. Connected Hopf algebras -- 1.3.1. Connected graded bialgebras -- 1.3.2. An example: the Hopf algebra of decorated rooted trees -- 1.3.3. Connected filtered bialgebras -- 1.3.4. The convolution product -- 1.3.5. Characters -- 1.3.6. Group schemes and the Cartier-Milnor-Moore-Quillen theorem -- 1.3.7. Renormalization in connected filtered Hopf algebras -- 1.4. Pre-Lie algebras -- 1.4.1. Definition and general properties -- 1.4.2. The group of formal flows -- 1.4.3. The pre-Lie Poincaré-Birkhoff-Witt theorem -- 1.5. Algebraic operads -- 1.5.1. Manipulating algebraic operations -- 1.5.2. The operad of multi-linear operations -- 1.5.3. A definition for linear operads -- 1.5.4. A few examples of operads -- 1.6. Pre-Lie algebras (continued) -- 1.6.1. Pre-Lie algebras and augmented operads -- 1.6.2. A pedestrian approach to free pre-Lie algebra -- 1.6.3. Right-sided commutative Hopf algebras and the Loday-Ronco theorem -- 1.6.4. Pre-Lie algebras of vector fields -- 1.6.5. B-series, composition and substitution -- 1.7. Other related algebraic structures -- 1.7.1. NAP algebras -- 1.7.2. Novikov algebras -- 1.7.3. Assosymmetric algebras -- 1.7.4. Dendriform algebras -- 1.7.5. Post-Lie algebras -- 1.8. References -- 2. From Iterated Integrals and Chronological Calculus to Hopf and Rota-Baxter Algebras -- 2.1. Introduction -- 2.2. Generalized iterated integrals -- 2.2.1. Permutations and simplices.
2.2.2. Descents, NCSF and the BCH formula -- 2.2.3. Rooted trees and nonlinear differential equations -- 2.2.4. Flows and Hopf algebraic structures -- 2.3. Advances in chronological calculus -- 2.3.1. Chronological calculus and half-shuffles -- 2.3.2. Chronological calculus and pre-Lie products -- 2.3.3. Time-ordered products and enveloping algebras -- 2.3.4. Formal flows and Hopf algebraic structures -- 2.4. Rota-Baxter algebras -- 2.4.1. Origin -- 2.4.2. Definition and examples -- 2.4.3. Related algebraic structures -- 2.4.4. Atkinson's factorization and Bogoliubov's recursion -- 2.4.5. Spitzer's identity: commutative case -- 2.4.6. Free commutative Rota-Baxter algebras -- 2.4.7. Spitzer's identity: noncommutative case -- 2.4.8. Free Rota-Baxter algebras -- 2.5. References -- 3. Noncommutative Symmetric Functions, Lie Series and Descent Algebras -- 3.1. Introduction -- 3.2. Classical symmetric functions -- 3.2.1. Symmetric polynomials -- 3.2.2. The Hopf algebra of symmetric functions -- 3.2.3. The λ-ring notation -- 3.2.4. Symmetric functions and formal power series -- 3.2.5. Duality -- 3.3. Noncommutative symmetric functions -- 3.3.1. Basic definitions -- 3.3.2. Generators and linear bases -- 3.3.3. Duality -- 3.3.4. Solomon's descent algebras -- 3.4. Lie series and Lie idempotents -- 3.4.1. Permutational operators on tensor spaces -- 3.4.2. The Hausdorff series -- 3.4.3. Lie idempotents in the descent algebra -- 3.5. Lie idempotents as noncommutative symmetric functions -- 3.5.1. Noncommutative power-sums -- 3.5.2. The Magnus expansion -- 3.5.3. The continuous BCH expansion -- 3.5.4. Another proof of the Magnus expansion -- 3.5.5. The (1 - q) -transform -- 3.5.6. Hopf algebras enter the scene -- 3.5.7. A one-parameter family of Lie idempotents -- 3.5.8. The iterated q-bracketing and its diagonalization.
3.6. Decompositions of the descent algebras -- 3.6.1. Complete families of minimal orthogonal idempotents -- 3.6.2. Eulerian idempotents -- 3.6.3. Generalized Eulerian idempotents -- 3.7. Decompositions of the tensor algebra -- 3.8. General deformations -- 3.9. Lie quasi-idempotents as Lie polynomials -- 3.9.1. The left derivative -- 3.9.2. Multilinear Lie polynomials -- 3.9.3. Decompositions on other bases -- 3.10. Permutations and free quasi-symmetric functions -- 3.10.1. Free quasi-symmetric functions -- 3.11. Packed words and word quasi-symmetric functions -- 3.12. References -- 4. From Runge-Kutta Methods to Hopf Algebras of Rooted Trees -- 4.1. Numerical integration methods for ordinary differential equations -- 4.1.1. Introduction -- 4.1.2. Runge-Kutta methods -- 4.2. Algebraic theory of Runge-Kutta methods -- 4.2.1. The order conditions of RK methods -- 4.2.2. The independence of order conditions -- 4.2.3. Proof of necessary and sufficient order conditions -- 4.2.4. Composition of RK methods, rooted trees and forests -- 4.2.5. The Butcher group -- 4.2.6. Equivalence classes of RK methods -- 4.2.7. Bibliographical comments -- 4.3. B-series and related formal expansions -- 4.3.1. B-series -- 4.3.2. Backward error analysis, the exponential and the logarithm -- 4.3.3. Series of linear differential operators -- 4.3.4. The Lie algebra of the Butcher group -- 4.3.5. The pre-Lie algebra structure on ɡ -- 4.3.6. Bibliographical comments -- 4.4. Hopf algebras of rooted trees -- 4.4.1. The commutative Hopf algebra of rooted trees -- 4.4.2. The dual algebra Н* and the dual Hopf algebra H° -- 4.4.3. B-series and series of differential operators revisited -- 4.4.4. A universal property of the commutative Hopf algebra of rooted trees -- 4.4.5. The substitution law -- 4.4.6. Bibliographical comments -- 4.5. References.
5. Combinatorial Algebra in Controllability and Optimal Control -- 5.1. Introduction -- 5.1.1. Motivation: idealized examples -- 5.1.2. Controlled dynamical systems -- 5.1.3. Fundamental questions in control -- 5.2. Analytic foundations -- 5.2.1. State-space models and vector fields on manifolds -- 5.2.2. Chronological calculus -- 5.2.3. Piecewise constant controls and the Baker-Campbell-Hausdorff formula -- 5.2.4. Picard iteration and formal series solutions -- 5.2.5. The Chen-Fliess series and abstractions -- 5.3. Controllability and optimality -- 5.3.1. Reachable sets and accessibility -- 5.3.2. Small-time local controllability -- 5.3.3. Nilpotent approximating systems -- 5.3.4. Optimality and the maximum principle -- 5.3.5. Control variations and approximating cones -- 5.4. Product expansions and realizations -- 5.4.1. Variation of parameters and exponential products -- 5.4.2. Computations using Zinbiel products -- 5.4.3. Exponential products and normal forms for nilpotent systems -- 5.4.4. Logarithm of the Chen series -- 5.5. References -- 6. Algebra is Geometry is Algebra - Interactions Between Hopf Algebras, Infinite Dimensional Geometry and Application -- 6.1. The Butcher group and the Connes-Kreimer algebra -- 6.1.1. The Butcher group and B-series from numerical analysis -- 6.1.2. Beyond the Butcher group -- 6.2. Character groups of graded and connected Hopf algebras -- 6.2.1. The exponential and logarithm -- 6.3. Controlled groups of characters -- 6.3.1. Conventions for this section -- 6.3.2. Combinatorial Hopf algebras and the inverse factorial character -- 6.4. Appendix: Calculus in locally convex spaces -- 6.4.1. Cr -Manifolds and Cr -mappings between them -- 6.5. References -- List of Authors -- Index -- EULA.
Record Nr. UNINA-9910555013003321
Hoboken, New Jersey : , : John Wiley & Sons, Incorporated, , [2021]
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Algebra and applications 2 : combinatorial algebra and Hopf algebras / / edited by Abdenacer Makhlouf
Algebra and applications 2 : combinatorial algebra and Hopf algebras / / edited by Abdenacer Makhlouf
Pubbl/distr/stampa Hoboken, New Jersey : , : John Wiley & Sons, Incorporated, , [2021]
Descrizione fisica 1 online resource (336 pages)
Disciplina 512
Soggetto topico Algebra
ISBN 1-119-88090-4
1-119-88091-2
1-119-88089-0
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Cover -- Half-Title Page -- Title Page -- Copyright Page -- Contents -- Preface -- 1. Algebraic Background for Numerical Methods, Control Theory and Renormalization -- 1.1. Introduction -- 1.2. Hopf algebras: general properties -- 1.2.1. Algebras -- 1.2.2. Coalgebras -- 1.2.3. Convolution product -- 1.2.4. Bialgebras and Hopf algebras -- 1.2.5. Some simple examples of Hopf algebras -- 1.2.6. Some basic properties of Hopf algebras -- 1.3. Connected Hopf algebras -- 1.3.1. Connected graded bialgebras -- 1.3.2. An example: the Hopf algebra of decorated rooted trees -- 1.3.3. Connected filtered bialgebras -- 1.3.4. The convolution product -- 1.3.5. Characters -- 1.3.6. Group schemes and the Cartier-Milnor-Moore-Quillen theorem -- 1.3.7. Renormalization in connected filtered Hopf algebras -- 1.4. Pre-Lie algebras -- 1.4.1. Definition and general properties -- 1.4.2. The group of formal flows -- 1.4.3. The pre-Lie Poincaré-Birkhoff-Witt theorem -- 1.5. Algebraic operads -- 1.5.1. Manipulating algebraic operations -- 1.5.2. The operad of multi-linear operations -- 1.5.3. A definition for linear operads -- 1.5.4. A few examples of operads -- 1.6. Pre-Lie algebras (continued) -- 1.6.1. Pre-Lie algebras and augmented operads -- 1.6.2. A pedestrian approach to free pre-Lie algebra -- 1.6.3. Right-sided commutative Hopf algebras and the Loday-Ronco theorem -- 1.6.4. Pre-Lie algebras of vector fields -- 1.6.5. B-series, composition and substitution -- 1.7. Other related algebraic structures -- 1.7.1. NAP algebras -- 1.7.2. Novikov algebras -- 1.7.3. Assosymmetric algebras -- 1.7.4. Dendriform algebras -- 1.7.5. Post-Lie algebras -- 1.8. References -- 2. From Iterated Integrals and Chronological Calculus to Hopf and Rota-Baxter Algebras -- 2.1. Introduction -- 2.2. Generalized iterated integrals -- 2.2.1. Permutations and simplices.
2.2.2. Descents, NCSF and the BCH formula -- 2.2.3. Rooted trees and nonlinear differential equations -- 2.2.4. Flows and Hopf algebraic structures -- 2.3. Advances in chronological calculus -- 2.3.1. Chronological calculus and half-shuffles -- 2.3.2. Chronological calculus and pre-Lie products -- 2.3.3. Time-ordered products and enveloping algebras -- 2.3.4. Formal flows and Hopf algebraic structures -- 2.4. Rota-Baxter algebras -- 2.4.1. Origin -- 2.4.2. Definition and examples -- 2.4.3. Related algebraic structures -- 2.4.4. Atkinson's factorization and Bogoliubov's recursion -- 2.4.5. Spitzer's identity: commutative case -- 2.4.6. Free commutative Rota-Baxter algebras -- 2.4.7. Spitzer's identity: noncommutative case -- 2.4.8. Free Rota-Baxter algebras -- 2.5. References -- 3. Noncommutative Symmetric Functions, Lie Series and Descent Algebras -- 3.1. Introduction -- 3.2. Classical symmetric functions -- 3.2.1. Symmetric polynomials -- 3.2.2. The Hopf algebra of symmetric functions -- 3.2.3. The λ-ring notation -- 3.2.4. Symmetric functions and formal power series -- 3.2.5. Duality -- 3.3. Noncommutative symmetric functions -- 3.3.1. Basic definitions -- 3.3.2. Generators and linear bases -- 3.3.3. Duality -- 3.3.4. Solomon's descent algebras -- 3.4. Lie series and Lie idempotents -- 3.4.1. Permutational operators on tensor spaces -- 3.4.2. The Hausdorff series -- 3.4.3. Lie idempotents in the descent algebra -- 3.5. Lie idempotents as noncommutative symmetric functions -- 3.5.1. Noncommutative power-sums -- 3.5.2. The Magnus expansion -- 3.5.3. The continuous BCH expansion -- 3.5.4. Another proof of the Magnus expansion -- 3.5.5. The (1 - q) -transform -- 3.5.6. Hopf algebras enter the scene -- 3.5.7. A one-parameter family of Lie idempotents -- 3.5.8. The iterated q-bracketing and its diagonalization.
3.6. Decompositions of the descent algebras -- 3.6.1. Complete families of minimal orthogonal idempotents -- 3.6.2. Eulerian idempotents -- 3.6.3. Generalized Eulerian idempotents -- 3.7. Decompositions of the tensor algebra -- 3.8. General deformations -- 3.9. Lie quasi-idempotents as Lie polynomials -- 3.9.1. The left derivative -- 3.9.2. Multilinear Lie polynomials -- 3.9.3. Decompositions on other bases -- 3.10. Permutations and free quasi-symmetric functions -- 3.10.1. Free quasi-symmetric functions -- 3.11. Packed words and word quasi-symmetric functions -- 3.12. References -- 4. From Runge-Kutta Methods to Hopf Algebras of Rooted Trees -- 4.1. Numerical integration methods for ordinary differential equations -- 4.1.1. Introduction -- 4.1.2. Runge-Kutta methods -- 4.2. Algebraic theory of Runge-Kutta methods -- 4.2.1. The order conditions of RK methods -- 4.2.2. The independence of order conditions -- 4.2.3. Proof of necessary and sufficient order conditions -- 4.2.4. Composition of RK methods, rooted trees and forests -- 4.2.5. The Butcher group -- 4.2.6. Equivalence classes of RK methods -- 4.2.7. Bibliographical comments -- 4.3. B-series and related formal expansions -- 4.3.1. B-series -- 4.3.2. Backward error analysis, the exponential and the logarithm -- 4.3.3. Series of linear differential operators -- 4.3.4. The Lie algebra of the Butcher group -- 4.3.5. The pre-Lie algebra structure on ɡ -- 4.3.6. Bibliographical comments -- 4.4. Hopf algebras of rooted trees -- 4.4.1. The commutative Hopf algebra of rooted trees -- 4.4.2. The dual algebra Н* and the dual Hopf algebra H° -- 4.4.3. B-series and series of differential operators revisited -- 4.4.4. A universal property of the commutative Hopf algebra of rooted trees -- 4.4.5. The substitution law -- 4.4.6. Bibliographical comments -- 4.5. References.
5. Combinatorial Algebra in Controllability and Optimal Control -- 5.1. Introduction -- 5.1.1. Motivation: idealized examples -- 5.1.2. Controlled dynamical systems -- 5.1.3. Fundamental questions in control -- 5.2. Analytic foundations -- 5.2.1. State-space models and vector fields on manifolds -- 5.2.2. Chronological calculus -- 5.2.3. Piecewise constant controls and the Baker-Campbell-Hausdorff formula -- 5.2.4. Picard iteration and formal series solutions -- 5.2.5. The Chen-Fliess series and abstractions -- 5.3. Controllability and optimality -- 5.3.1. Reachable sets and accessibility -- 5.3.2. Small-time local controllability -- 5.3.3. Nilpotent approximating systems -- 5.3.4. Optimality and the maximum principle -- 5.3.5. Control variations and approximating cones -- 5.4. Product expansions and realizations -- 5.4.1. Variation of parameters and exponential products -- 5.4.2. Computations using Zinbiel products -- 5.4.3. Exponential products and normal forms for nilpotent systems -- 5.4.4. Logarithm of the Chen series -- 5.5. References -- 6. Algebra is Geometry is Algebra - Interactions Between Hopf Algebras, Infinite Dimensional Geometry and Application -- 6.1. The Butcher group and the Connes-Kreimer algebra -- 6.1.1. The Butcher group and B-series from numerical analysis -- 6.1.2. Beyond the Butcher group -- 6.2. Character groups of graded and connected Hopf algebras -- 6.2.1. The exponential and logarithm -- 6.3. Controlled groups of characters -- 6.3.1. Conventions for this section -- 6.3.2. Combinatorial Hopf algebras and the inverse factorial character -- 6.4. Appendix: Calculus in locally convex spaces -- 6.4.1. Cr -Manifolds and Cr -mappings between them -- 6.5. References -- List of Authors -- Index -- EULA.
Record Nr. UNINA-9910677690703321
Hoboken, New Jersey : , : John Wiley & Sons, Incorporated, , [2021]
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Algebra and applications 2 : combinatorial algebra and Hopf algebras / / edited by Abdenacer Makhlouf
Algebra and applications 2 : combinatorial algebra and Hopf algebras / / edited by Abdenacer Makhlouf
Pubbl/distr/stampa Hoboken, New Jersey : , : John Wiley & Sons, Incorporated, , [2021]
Descrizione fisica 1 online resource (336 pages)
Disciplina 512
Soggetto topico Algebra
ISBN 1-119-88090-4
1-119-88091-2
1-119-88089-0
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Cover -- Half-Title Page -- Title Page -- Copyright Page -- Contents -- Preface -- 1. Algebraic Background for Numerical Methods, Control Theory and Renormalization -- 1.1. Introduction -- 1.2. Hopf algebras: general properties -- 1.2.1. Algebras -- 1.2.2. Coalgebras -- 1.2.3. Convolution product -- 1.2.4. Bialgebras and Hopf algebras -- 1.2.5. Some simple examples of Hopf algebras -- 1.2.6. Some basic properties of Hopf algebras -- 1.3. Connected Hopf algebras -- 1.3.1. Connected graded bialgebras -- 1.3.2. An example: the Hopf algebra of decorated rooted trees -- 1.3.3. Connected filtered bialgebras -- 1.3.4. The convolution product -- 1.3.5. Characters -- 1.3.6. Group schemes and the Cartier-Milnor-Moore-Quillen theorem -- 1.3.7. Renormalization in connected filtered Hopf algebras -- 1.4. Pre-Lie algebras -- 1.4.1. Definition and general properties -- 1.4.2. The group of formal flows -- 1.4.3. The pre-Lie Poincaré-Birkhoff-Witt theorem -- 1.5. Algebraic operads -- 1.5.1. Manipulating algebraic operations -- 1.5.2. The operad of multi-linear operations -- 1.5.3. A definition for linear operads -- 1.5.4. A few examples of operads -- 1.6. Pre-Lie algebras (continued) -- 1.6.1. Pre-Lie algebras and augmented operads -- 1.6.2. A pedestrian approach to free pre-Lie algebra -- 1.6.3. Right-sided commutative Hopf algebras and the Loday-Ronco theorem -- 1.6.4. Pre-Lie algebras of vector fields -- 1.6.5. B-series, composition and substitution -- 1.7. Other related algebraic structures -- 1.7.1. NAP algebras -- 1.7.2. Novikov algebras -- 1.7.3. Assosymmetric algebras -- 1.7.4. Dendriform algebras -- 1.7.5. Post-Lie algebras -- 1.8. References -- 2. From Iterated Integrals and Chronological Calculus to Hopf and Rota-Baxter Algebras -- 2.1. Introduction -- 2.2. Generalized iterated integrals -- 2.2.1. Permutations and simplices.
2.2.2. Descents, NCSF and the BCH formula -- 2.2.3. Rooted trees and nonlinear differential equations -- 2.2.4. Flows and Hopf algebraic structures -- 2.3. Advances in chronological calculus -- 2.3.1. Chronological calculus and half-shuffles -- 2.3.2. Chronological calculus and pre-Lie products -- 2.3.3. Time-ordered products and enveloping algebras -- 2.3.4. Formal flows and Hopf algebraic structures -- 2.4. Rota-Baxter algebras -- 2.4.1. Origin -- 2.4.2. Definition and examples -- 2.4.3. Related algebraic structures -- 2.4.4. Atkinson's factorization and Bogoliubov's recursion -- 2.4.5. Spitzer's identity: commutative case -- 2.4.6. Free commutative Rota-Baxter algebras -- 2.4.7. Spitzer's identity: noncommutative case -- 2.4.8. Free Rota-Baxter algebras -- 2.5. References -- 3. Noncommutative Symmetric Functions, Lie Series and Descent Algebras -- 3.1. Introduction -- 3.2. Classical symmetric functions -- 3.2.1. Symmetric polynomials -- 3.2.2. The Hopf algebra of symmetric functions -- 3.2.3. The λ-ring notation -- 3.2.4. Symmetric functions and formal power series -- 3.2.5. Duality -- 3.3. Noncommutative symmetric functions -- 3.3.1. Basic definitions -- 3.3.2. Generators and linear bases -- 3.3.3. Duality -- 3.3.4. Solomon's descent algebras -- 3.4. Lie series and Lie idempotents -- 3.4.1. Permutational operators on tensor spaces -- 3.4.2. The Hausdorff series -- 3.4.3. Lie idempotents in the descent algebra -- 3.5. Lie idempotents as noncommutative symmetric functions -- 3.5.1. Noncommutative power-sums -- 3.5.2. The Magnus expansion -- 3.5.3. The continuous BCH expansion -- 3.5.4. Another proof of the Magnus expansion -- 3.5.5. The (1 - q) -transform -- 3.5.6. Hopf algebras enter the scene -- 3.5.7. A one-parameter family of Lie idempotents -- 3.5.8. The iterated q-bracketing and its diagonalization.
3.6. Decompositions of the descent algebras -- 3.6.1. Complete families of minimal orthogonal idempotents -- 3.6.2. Eulerian idempotents -- 3.6.3. Generalized Eulerian idempotents -- 3.7. Decompositions of the tensor algebra -- 3.8. General deformations -- 3.9. Lie quasi-idempotents as Lie polynomials -- 3.9.1. The left derivative -- 3.9.2. Multilinear Lie polynomials -- 3.9.3. Decompositions on other bases -- 3.10. Permutations and free quasi-symmetric functions -- 3.10.1. Free quasi-symmetric functions -- 3.11. Packed words and word quasi-symmetric functions -- 3.12. References -- 4. From Runge-Kutta Methods to Hopf Algebras of Rooted Trees -- 4.1. Numerical integration methods for ordinary differential equations -- 4.1.1. Introduction -- 4.1.2. Runge-Kutta methods -- 4.2. Algebraic theory of Runge-Kutta methods -- 4.2.1. The order conditions of RK methods -- 4.2.2. The independence of order conditions -- 4.2.3. Proof of necessary and sufficient order conditions -- 4.2.4. Composition of RK methods, rooted trees and forests -- 4.2.5. The Butcher group -- 4.2.6. Equivalence classes of RK methods -- 4.2.7. Bibliographical comments -- 4.3. B-series and related formal expansions -- 4.3.1. B-series -- 4.3.2. Backward error analysis, the exponential and the logarithm -- 4.3.3. Series of linear differential operators -- 4.3.4. The Lie algebra of the Butcher group -- 4.3.5. The pre-Lie algebra structure on ɡ -- 4.3.6. Bibliographical comments -- 4.4. Hopf algebras of rooted trees -- 4.4.1. The commutative Hopf algebra of rooted trees -- 4.4.2. The dual algebra Н* and the dual Hopf algebra H° -- 4.4.3. B-series and series of differential operators revisited -- 4.4.4. A universal property of the commutative Hopf algebra of rooted trees -- 4.4.5. The substitution law -- 4.4.6. Bibliographical comments -- 4.5. References.
5. Combinatorial Algebra in Controllability and Optimal Control -- 5.1. Introduction -- 5.1.1. Motivation: idealized examples -- 5.1.2. Controlled dynamical systems -- 5.1.3. Fundamental questions in control -- 5.2. Analytic foundations -- 5.2.1. State-space models and vector fields on manifolds -- 5.2.2. Chronological calculus -- 5.2.3. Piecewise constant controls and the Baker-Campbell-Hausdorff formula -- 5.2.4. Picard iteration and formal series solutions -- 5.2.5. The Chen-Fliess series and abstractions -- 5.3. Controllability and optimality -- 5.3.1. Reachable sets and accessibility -- 5.3.2. Small-time local controllability -- 5.3.3. Nilpotent approximating systems -- 5.3.4. Optimality and the maximum principle -- 5.3.5. Control variations and approximating cones -- 5.4. Product expansions and realizations -- 5.4.1. Variation of parameters and exponential products -- 5.4.2. Computations using Zinbiel products -- 5.4.3. Exponential products and normal forms for nilpotent systems -- 5.4.4. Logarithm of the Chen series -- 5.5. References -- 6. Algebra is Geometry is Algebra - Interactions Between Hopf Algebras, Infinite Dimensional Geometry and Application -- 6.1. The Butcher group and the Connes-Kreimer algebra -- 6.1.1. The Butcher group and B-series from numerical analysis -- 6.1.2. Beyond the Butcher group -- 6.2. Character groups of graded and connected Hopf algebras -- 6.2.1. The exponential and logarithm -- 6.3. Controlled groups of characters -- 6.3.1. Conventions for this section -- 6.3.2. Combinatorial Hopf algebras and the inverse factorial character -- 6.4. Appendix: Calculus in locally convex spaces -- 6.4.1. Cr -Manifolds and Cr -mappings between them -- 6.5. References -- List of Authors -- Index -- EULA.
Record Nr. UNINA-9910815668803321
Hoboken, New Jersey : , : John Wiley & Sons, Incorporated, , [2021]
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Algebra and computer science : Joint AMS-EMS-SPM Meeting Algebra and Computer Science June 10-13, 2015: Porto, Portugal : Joint Mathematics Meetings Groups, Algorithms, and Cryptography January 10-13, 2015: San Antonio, TX : Joint AMS-Israel Mathematical Union Meeting Applications of Algebra to Cryptography June 16-19, 2014: Tel-Aviv, Israel / / Delaram Kahrobaei, Bren Cavallo, David Garber, editors
Algebra and computer science : Joint AMS-EMS-SPM Meeting Algebra and Computer Science June 10-13, 2015: Porto, Portugal : Joint Mathematics Meetings Groups, Algorithms, and Cryptography January 10-13, 2015: San Antonio, TX : Joint AMS-Israel Mathematical Union Meeting Applications of Algebra to Cryptography June 16-19, 2014: Tel-Aviv, Israel / / Delaram Kahrobaei, Bren Cavallo, David Garber, editors
Pubbl/distr/stampa Providence, Rhode Island : , : American Mathematical Society, , 2016
Descrizione fisica 1 online resource (242 pages) : illustrations
Disciplina 512.0285
Collana Contemporary Mathematics
Soggetto topico Logic, Symbolic and mathematical
Algebra
Computer science - Mathematics
Soggetto genere / forma Electronic books.
ISBN 1-4704-3587-X
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Record Nr. UNINA-9910480731903321
Providence, Rhode Island : , : American Mathematical Society, , 2016
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Algebra and computer science : Joint AMS-EMS-SPM Meeting Algebra and Computer Science June 10-13, 2015: Porto, Portugal : Joint Mathematics Meetings Groups, Algorithms, and Cryptography January 10-13, 2015: San Antonio, TX : Joint AMS-Israel Mathematical Union Meeting Applications of Algebra to Cryptography June 16-19, 2014: Tel-Aviv, Israel / / Delaram Kahrobaei, Bren Cavallo, David Garber, editors
Algebra and computer science : Joint AMS-EMS-SPM Meeting Algebra and Computer Science June 10-13, 2015: Porto, Portugal : Joint Mathematics Meetings Groups, Algorithms, and Cryptography January 10-13, 2015: San Antonio, TX : Joint AMS-Israel Mathematical Union Meeting Applications of Algebra to Cryptography June 16-19, 2014: Tel-Aviv, Israel / / Delaram Kahrobaei, Bren Cavallo, David Garber, editors
Pubbl/distr/stampa Providence, Rhode Island : , : American Mathematical Society, , 2016
Descrizione fisica 1 online resource (242 pages) : illustrations
Disciplina 512.0285
Collana Contemporary Mathematics
Soggetto topico Logic, Symbolic and mathematical
Algebra
Computer science - Mathematics
ISBN 1-4704-3587-X
Classificazione 20-XX68-XX
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Generic properties of subgroups of free groups and finite presentations / Frédérique Bassino, Cyril Nicaud, and Pascal Weil -- A new multi-server scheme for private information retrieval / Chi Sing Chum and Xiaowen Zhang -- On secret sharing protocols / Chi Sing Chum [and 4 others] -- A verifiable secret sharing scheme using non-abelian groups / Maggie E. Habeer -- Non-associative public-key cryptography / Arkadius Kalka -- Non-associative key establishment protocols and their implementation / Arkadius Kalka and Mina Teicher -- Knapsack and subset sum problems in nilpoint, polycyclic, and co-context-free groups / Daniel König, Markus Lohrey, and George Zetzsche -- On the Tits alternative for a class of finitely presented groups with a special focus on symbolic computations / Anja I.S. Moldenhauer, Gerhard Rosenberger, and Kristina Rosenthal -- Geometry of the conjugacy problem in lamplighter groups / Andew Salle -- A logspace solution to the word and conjugacy problem of generalized Baumslag-Solitar groups / Armin Weiss -- Cryptographic hash functions from sequences of lifted Paley graphs / Serena Yuan.
Record Nr. UNINA-9910794869803321
Providence, Rhode Island : , : American Mathematical Society, , 2016
Materiale a stampa
Lo trovi qui: Univ. Federico II
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Algebra and computer science : Joint AMS-EMS-SPM Meeting Algebra and Computer Science June 10-13, 2015: Porto, Portugal : Joint Mathematics Meetings Groups, Algorithms, and Cryptography January 10-13, 2015: San Antonio, TX : Joint AMS-Israel Mathematical Union Meeting Applications of Algebra to Cryptography June 16-19, 2014: Tel-Aviv, Israel / / Delaram Kahrobaei, Bren Cavallo, David Garber, editors
Algebra and computer science : Joint AMS-EMS-SPM Meeting Algebra and Computer Science June 10-13, 2015: Porto, Portugal : Joint Mathematics Meetings Groups, Algorithms, and Cryptography January 10-13, 2015: San Antonio, TX : Joint AMS-Israel Mathematical Union Meeting Applications of Algebra to Cryptography June 16-19, 2014: Tel-Aviv, Israel / / Delaram Kahrobaei, Bren Cavallo, David Garber, editors
Pubbl/distr/stampa Providence, Rhode Island : , : American Mathematical Society, , 2016
Descrizione fisica 1 online resource (242 pages) : illustrations
Disciplina 512.0285
Collana Contemporary Mathematics
Soggetto topico Logic, Symbolic and mathematical
Algebra
Computer science - Mathematics
ISBN 1-4704-3587-X
Classificazione 20-XX68-XX
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Generic properties of subgroups of free groups and finite presentations / Frédérique Bassino, Cyril Nicaud, and Pascal Weil -- A new multi-server scheme for private information retrieval / Chi Sing Chum and Xiaowen Zhang -- On secret sharing protocols / Chi Sing Chum [and 4 others] -- A verifiable secret sharing scheme using non-abelian groups / Maggie E. Habeer -- Non-associative public-key cryptography / Arkadius Kalka -- Non-associative key establishment protocols and their implementation / Arkadius Kalka and Mina Teicher -- Knapsack and subset sum problems in nilpoint, polycyclic, and co-context-free groups / Daniel König, Markus Lohrey, and George Zetzsche -- On the Tits alternative for a class of finitely presented groups with a special focus on symbolic computations / Anja I.S. Moldenhauer, Gerhard Rosenberger, and Kristina Rosenthal -- Geometry of the conjugacy problem in lamplighter groups / Andew Salle -- A logspace solution to the word and conjugacy problem of generalized Baumslag-Solitar groups / Armin Weiss -- Cryptographic hash functions from sequences of lifted Paley graphs / Serena Yuan.
Record Nr. UNINA-9910820796003321
Providence, Rhode Island : , : American Mathematical Society, , 2016
Materiale a stampa
Lo trovi qui: Univ. Federico II
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