Special Matrices Of Mathematical Physics: Stochastic, Circulant And Bell MatricesWorld Scientific, 2001 M08 17 - 340 páginas This book expounds three special kinds of matrices that are of physical interest, centering on physical examples. Stochastic matrices describe dynamical systems of many different types, involving (or not) phenomena like transience, dissipation, ergodicity, nonequilibrium, and hypersensitivity to initial conditions. The main characteristic is growth by agglomeration, as in glass formation. Circulants are the building blocks of elementary Fourier analysis and provide a natural gateway to quantum mechanics and noncommutative geometry. Bell polynomials offer closed expressions for many formulas concerning Lie algebra invariants, differential geometry and real gases, and their matrices are instrumental in the study of chaotic mappings. |
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Página 3
... ... , \ n } of all complex values of À for which the characteristic matrix ( AIM ) is not invertible . It is , consequently , the set of the roots of the characteristic polynomial . 3 Chapter 1 Some fundamental notions 1.1 Definitions.
... ... , \ n } of all complex values of À for which the characteristic matrix ( AIM ) is not invertible . It is , consequently , the set of the roots of the characteristic polynomial . 3 Chapter 1 Some fundamental notions 1.1 Definitions.
Página 6
... values are distinct , we can divide Z1 by ( A1 – A2 ) ( A1 – À3 ) and define a new projector ( M – \ 2I ) ( M – λ3I ) -- Z1 ( A1 – A2 ) ( A1 – λ3 ) ( 1.7 ) This is a true eigenprojector . It is an idempotent 6 Some fundamental notions ...
... values are distinct , we can divide Z1 by ( A1 – A2 ) ( A1 – À3 ) and define a new projector ( M – \ 2I ) ( M – λ3I ) -- Z1 ( A1 – A2 ) ( A1 – λ3 ) ( 1.7 ) This is a true eigenprojector . It is an idempotent 6 Some fundamental notions ...
Página 10
... value 1 ; indeed , define the vectors V ( ks ) with components ( V ( ks ) ) j = [ Z ( k ) ] js ; the equation is then Σn [ Z ( k ) ] rn ( V ( ks ) ) n = ( V ( ks ) ) ri ( 5 ) the Z's can be normalized to tr ( Zê ) ( 1.16 ) = 1 for each ...
... value 1 ; indeed , define the vectors V ( ks ) with components ( V ( ks ) ) j = [ Z ( k ) ] js ; the equation is then Σn [ Z ( k ) ] rn ( V ( ks ) ) n = ( V ( ks ) ) ri ( 5 ) the Z's can be normalized to tr ( Zê ) ( 1.16 ) = 1 for each ...
Página 21
... values . It is possible to go from the first kind of description to the second through the snapshots leading to a Poincaré map [ 9 ] , and from the second to the first by the notion of continuous iteration [ 10 ] ( see section 15.2 ) ...
... values . It is possible to go from the first kind of description to the second through the snapshots leading to a Poincaré map [ 9 ] , and from the second to the first by the notion of continuous iteration [ 10 ] ( see section 15.2 ) ...
Página 22
... values of those variables at some time . The system is said to follow a Markov process when its evolution from a given state depends on that state only , and not on its previous history . Stochastic Matrices are deeply related to Markov ...
... values of those variables at some time . The system is said to follow a Markov process when its evolution from a given state depends on that state only , and not on its previous history . Stochastic Matrices are deeply related to Markov ...
Contenido
19 | |
CIRCULANT MATRICES | 79 |
BELL MATRICES | 147 |
Appendix A Formulary | 283 |
Bibliography | 309 |
Index | 315 |
Otras ediciones - Ver todas
Special Matrices of Mathematical Physics: Stochastic, Circulant, and Bell ... Ruben Aldrovandi Vista previa limitada - 2001 |
Special Matrices of Mathematical Physics: Stochastic, Circulant, and Bell ... Ruben Aldrovandi Vista previa limitada - 2001 |
Términos y frases comunes
alphabet basis Bell matrices Bell polynomials braid group canonical partition function characteristic polynomial circulant matrices circulant matrix classical cluster integrals column commutative components condition consequently continuum convolution corresponding cyclic defined derivative detailed balancing diagonal differential discrete distribution dynamical eigenvalues eigenvectors entries ep+1 equation equilibrium evolution example factor fermions finite formalism formula Fourier transformations Fredholm geometry given glass grand canonical partition Hamiltonian Hopf algebras identity imprimitive invariant inverse irreducible iterate leads Lie algebra Markov chain noncommutative notation obtained operator particles permutation phase space Phys physical Poisson bracket powers projectors properties QN(B Quantum Mechanics recursion representation Statistical Mechanics stochastic matrix summation symmetric functions symmetric group symplectic Taylor coefficients theorem theory totally regular unitary values variables vector virial Weyl-Wigner Wigner functions Σ Σ