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 8
... - sition . The function F ( M ) , defined as above , can be equivalently written as the N x N matrix N F ( M ) = F ( X ) Z , [ M ] . ( 1.11 ) A few examples : ( 1 ) the exponential N j = 1 8 Some fundamental notions 1.3 Matrix functions.
... - sition . The function F ( M ) , defined as above , can be equivalently written as the N x N matrix N F ( M ) = F ( X ) Z , [ M ] . ( 1.11 ) A few examples : ( 1 ) the exponential N j = 1 8 Some fundamental notions 1.3 Matrix functions.
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... example . Experiments with small matrices show , however , that the formula works , provided the inverse exists . This suggests that the theory can be extended , and indeed it can [ 1 ] . We shall use ( 1.11 ) as the definition of F ( M ) ...
... example . Experiments with small matrices show , however , that the formula works , provided the inverse exists . This suggests that the theory can be extended , and indeed it can [ 1 ] . We shall use ( 1.11 ) as the definition of F ( M ) ...
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... example . Let us list the main properties of the eigenmatrices Zx [ M ] of a nondegenerate matrix M : ( 1 ) the Z's are linearly independent and nonvanishing ; ( 2 ) the basis { Z [ M ] } depends on M , but is the same for every ...
... example . Let us list the main properties of the eigenmatrices Zx [ M ] of a nondegenerate matrix M : ( 1 ) the Z's are linearly independent and nonvanishing ; ( 2 ) the basis { Z [ M ] } depends on M , but is the same for every ...
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... example ] . The projectors themselves will be presented in Section 15.1 . Comment 1.3.2 That functions of matrices are completely determined by their spec- tra is justified on much more general grounds . Matrix algebras are very ...
... example ] . The projectors themselves will be presented in Section 15.1 . Comment 1.3.2 That functions of matrices are completely determined by their spec- tra is justified on much more general grounds . Matrix algebras are very ...
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... example , they proceed to increase the number of operators up to the point at which all degeneracy is removed . In that case , they stop only at a complete set of intercommuting operators , whose eigen- vectors provide the complete ...
... example , they proceed to increase the number of operators up to the point at which all degeneracy is removed . In that case , they stop only at a complete set of intercommuting operators , whose eigen- vectors provide the complete ...
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