Fuchs–Frobenius theory

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21: 25.17 Physical Applications

… ►See Armitage (1989), Berry and Keating (1998, 1999), Keating (1993, 1999), and Sarnak (1999). … ►Quantum field theory often encounters formally divergent sums that need to be evaluated by a process of regularization: for example, the energy of the electromagnetic vacuum in a confined space (Casimir–Polder effect). …

22: David M. Bressoud

… ►Bressoud has published numerous papers in number theory, combinatorics, and special functions. … 227, in 1980, Factorization and Primality Testing, published by Springer-Verlag in 1989, Second Year Calculus from Celestial Mechanics to Special Relativity, published by Springer-Verlag in 1992, A Radical Approach to Real Analysis, published by the Mathematical Association of America in 1994, with a second edition in 2007, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, published by the Mathematical Association of America and Cambridge University Press in 1999, A Course in Computational Number Theory (with S. Wagon), published by Key College Press in 2000, and A Radical Approach to Lebesgue’s Theory of Integration, published by the Mathematical Association of America and Cambridge University Press in 2007. …

23: Brian D. Sleeman

… ►Sleeman published numerous papers in applied analysis, multiparameter spectral theory, direct and inverse scattering theory, and mathematical medicine. He is author of the book Multiparameter spectral theory in Hilbert space, published by Pitman in 1978, and coauthor (with D. …

24: 21.7 Riemann Surfaces

… ►

§21.7(iii) Frobenius’ Identity

… ►Then for all

\mathbf{z}_{j}\in{\mathbb{C}}^{g}

j=1,2,3,4

, such that

\mathbf{z}_{1}+\mathbf{z}_{2}+\mathbf{z}_{3}+\mathbf{z}_{4}=0

, and for all

\boldsymbol{{\alpha}}_{j}

\boldsymbol{{\beta}}_{j}

\in{\mathbb{R}}^{g}

, such that

\boldsymbol{{\alpha}}_{1}+\boldsymbol{{\alpha}}_{2}+\boldsymbol{{\alpha}}_{3}+% \boldsymbol{{\alpha}}_{4}=0

and

\boldsymbol{{\beta}}_{1}+\boldsymbol{{\beta}}_{2}+\boldsymbol{{\beta}}_{3}+% \boldsymbol{{\beta}}_{4}=0

, we have Frobenius’ identity: …

25: 20.12 Mathematical Applications

… ►

§20.12(i) Number Theory

… ►This ability to uniformize multiply-connected spaces (manifolds), or multi-sheeted functions of a complex variable (Riemann (1899), Rauch and Lebowitz (1973), Siegel (1988)) has led to applications in string theory (Green et al. (1988a, b), Krichever and Novikov (1989)), and also in statistical mechanics (Baxter (1982)). …

26: 8.23 Statistical Applications

… ►In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of

Q\left(a,x\right)

; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319).

27: Ingram Olkin

… ►Olkin’s research covered a broad range of areas, including multivariate analysis, reliability theory, matrix theory, statistical models in the social and behavioral sciences, life distributions, and meta-analysis. His well-known books in the statistics community include Theory of Majorization and its Applications (with A. …

28: 17.16 Mathematical Applications

… ►Many special cases of

q

-series arise in the theory of partitions, a topic treated in §§27.14(i) and 26.9. …

29: 12.16 Mathematical Applications

… ►PCFs are used as basic approximating functions in the theory of contour integrals with a coalescing saddle point and an algebraic singularity, and in the theory of differential equations with two coalescing turning points; see §§2.4(vi) and 2.8(vi). …

30: Peter L. Walker

… ►Walker’s published work has been mainly in real and complex analysis, with excursions into analytic number theory and geometry, the latter in collaboration with Professor Mowaffaq Hajja of the University of Jordan. ►Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004. …