### Special matrices of mathematical physics: stochastic, circulant, and Bell matrices

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Th e basic propertie s of a matrix family are first introduced , some applications made , then more formal aspect s presented , and so on. The thre e type s of matrice s have not much in common, except for th e fact that they appear t o open new ways t o further developments. In each case old ideas appear under new, simpler visages. Kolmogorov's stochastic matrices are introduced as instrument s to describe in a surprisingly simple way dynamical systems of a wide variety of behaviors.

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Circulant s appear as th e bedrock of Fourier analysis and as a most natura l gateway into Quantum Mechanics and Noncommutative Geometry. Bell polynomials, after turning up in Differential Geometry and in th e theory of gases, promise to be instrumental in th e study of chaotic mappings.

Stochastic matrice s govern, through repeated iteration , th e time evo- lution of probability distribution s of which they are independent. Th e example chosen ha s been glass transition , a fascinating phenomenon for which ther e is no complete theory and for whose description they have made important contributions.

More is said on glasses than on other sub- jects , because not much about th e recent advances is found in textbooks. Some physical feeling once got , their spectral propertie s are presented and shown to epitomize all th e wealth of possible evolution courses.

Th e matri - ces finally provide toy models illustrating involved questions like ergodicity, multi-phase systems , dissipation and non-equilibrium. Circulants, once applied t o Quantum Mechanics, lead t o phase spaces whose points are matrices. Spaces formed by such noncommutative points have noncommutative geometries. With a differential calculus on spaces of matrices as a preliminary, quantum groups turn up as algebras of matrice s whose entries are themselves matrices.

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A matrix symplectic structur e comes out in a natura l way and sheds new light on th e underlying textur e of phase space. Bell polynomials, and th e matrice s of which they are th e entries , lead t o closed expressions for many formulas usually presented with ellipses in th e end, a tru e nightmare for algebraic computation. Those expressions in- clude relationships between determinant s and traces , coefficients and root s of polynomials, virial coefficients and configuration integrals , besides Lie algebra invariants of various types , from Casimir operator s to character- istic classes an d characters.

Furthermore , a Bell polynomial is related t o a mapping.

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## Special Matrices of Mathematical Physics: Stochastic, Circulant and Bell Matrices

A matrix formed with th e Bell polynomials of a mapping is a linear representation of that mapping. Function composition is thereby Preface IX reduced t o matrix multiplication. In particular , th e map iteration s describ- ing discrete-time dynamical systems become powers of matrice s and can b e interpolated into continuous versions of th e time evolution. Th e the - ory of real gases provides th e best physical illustrations of th e polynomials and these , in retribution , reveal th e presence of a matrix structur e in th e backstage of th e theory.

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The text is divided into four Parts : one for th e general background and one for each kind of matrix. Up t o references to th e background introduc- tion, th e Part s are tentatively self-contained, which accounts for a certain amount of repetition. Th e only way we break with standard editing is by th e presence of Comments , which con- tain information additional t o th e main text and may b e omitted in a first reading. Xie et al.

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Zhang et al. Applications and comparisons of four time series models in epidemiological surveillance data.

Ablinger et al. Liu et al. Selective recursive kernel learning for online identification of nonlinear systems with NARX form. Matakos et al. Xin et al. Bai et al. Lee et al. 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. Readership: Mathematical physicists, statistical physicists and researchers in the field of combinatorics and graph theory.

Udvalgte sider Titelside. Indhold Some fundamental notions.

## Carleman matrix

Evolving systems. Glass transition. Formal developments. Equilibrium dissipation and ergodicity.