hexadecimal number system

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1: 24.1 Special Notation

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Bernoulli Numbers and Polynomials

►The origin of the notation $B_n$, $B_n\left(x\right)$, is not clear. … ►

Euler Numbers and Polynomials

… ►Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. … ►Various systems of notation are summarized in Adrian (1959) and D’Ocagne (1904).

2: 3.1 Arithmetics and Error Measures

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§3.1(i) Floating-Point Arithmetic

►Computer arithmetic is described for the binary based system with base 2; another system that has been used is the hexadecimal system with base 16. … ► … ►Let $x$ be any positive number with … ►Computer algebra systems use exact rational arithmetic with rational numbers $p/q$, where $p$ and $q$ are multi-length integers. …

3: 27.15 Chinese Remainder Theorem

§27.15 Chinese Remainder Theorem

►The Chinese remainder theorem states that a system of congruences $x\equiv a_1(mod{m}_1),\mathrm{\dots },x\equiv a_k(mod{m}_k)$, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod $m$), where $m$ is the product of the moduli. ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …Their product $m$ has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. …

4: 27.2 Functions

… ►where $p_1,p_2,\mathrm{\dots },p_{\nu \left(n\right)}$ are the distinct prime factors of $n$, each exponent $a_r$ is positive, and $\nu \left(n\right)$ is the number of distinct primes dividing $n$. … ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) … ►Such a set is a reduced residue system modulo $n$. … ►

§27.2(ii) Tables

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5: 37.9 Jacobi Polynomials Associated with Root System $A_2$

§37.9 Jacobi Polynomials Associated with Root System $A_2$

… ►For $\alpha >-\frac{5}{6}$ Jacobi polynomials associated with root system $A_2$ are polynomials $P_{m,n}^{\alpha }(z,\overline{z})$ ($m,n\in {\mathbb{N}}_0$) which are defined uniquely, up to constant factors, by … ►These reflections generate a transformation group of the plane $t_1+t_2+t_3=0$ which contains as a subgroup the Weyl group $W$ for root system $A_2$. The group $W$ is isomorphic to the symmetric group $S_3$ and permutes $t_1-t_2,t_2-t_3,t_3-t_1$. … ►The polynomials $P_{m,n}^{\pm \frac{1}{2}}(z,\overline{z})$ are called Chebyshev polynomials of the first and the second kind for root system $A_2$. …

6: 37.8 Jacobi Polynomials Associated with Root System $B{C}_2$

§37.8 Jacobi Polynomials Associated with Root System $B{C}_2$

… ►the Jacobi polynomials associated with root system $B{C}_2$ are symmetric polynomials $p_{k,n}^{\alpha ,\beta ,\gamma }(x,y)$ ($0\le k\le n$) of the form … ►The polynomials $P_{k,n}^{\alpha ,\beta ,\gamma }(u,v)$, where … ►Moreover, the corresponding OPs $P_{k,n}(u,v)$ as in (37.8.11) satisfy for $\gamma =\pm \frac{1}{2}$ the property that ${\{P_{k,n}\}}_{k=0}^n$ has $\frac{1}{2}(n+1)(n+2)$ real common zeros; see Schmid and Xu (1994). … ►on the square, the OPs are related to the Jacobi polynomials for root system $B{C}_2$. …

7: 20.12 Mathematical Applications

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§20.12(i) Number Theory

… ►For applications of Jacobi’s triple product (20.5.9) to Ramanujan’s $\tau \left(n\right)$ function and Euler’s pentagonal numbers see Hardy and Wright (1979, pp. 132–160) and McKean and Moll (1999, pp. 143–145). For an application of a generalization in affine root systems see Macdonald (1972). … ►The space of complex tori $ℂ/(\mathbb{Z}+\tau \mathbb{Z})$ (that is, the set of complex numbers $z$ in which two of these numbers $z_1$ and $z_2$ are regarded as equivalent if there exist integers $m,n$ such that $z_1-z_2=m+\tau n$) is mapped into the projective space $P^3$ via the identification $z\to ({\theta }_1\left(2z\right|\tau ),{\theta }_2\left(2z\right|\tau ),{\theta }_3\left(2z\right|\tau ),{\theta }_4\left(2z\right|\tau ))$. …

8: Mourad E. H. Ismail

… ► Suslov), Kluwer Academic Publishers, 2001; Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics (with F. … ►Ismail serves on several editorial boards including the Cambridge University Press book series Encyclopedia of Mathematics and its Applications, and on the editorial boards of 9 journals including Proceedings of the American Mathematical Society (Integrable Systems and Special Functions Editor); Constructive Approximation; Journal of Approximation Theory; and Integral Transforms and Special Functions. …

9: Morris Newman

… ►Department of Commerce Gold Medal in 1966 for his work on algorithms for solving integral linear systems exactly by using congruence techniques. … ►He served as Associate Editor for Combinatorics and Number Theory for the DLMF project. …

10: Barry I. Schneider

… ►Schneider’s research interests spanned a broad number of areas of theoretical chemistry, atomic and molecular physics, numerical methods and high performance computing. …Schneider served as Chair and Co-Chair of the APS Division of Computational Physics and the Topical Group on Few-Body Systems and Multipartical Dynamics and has been the organizer of a number of conferences and invited sessions in the US and abroad. …