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Graf addition theorem

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11: Tom M. Apostol
In 1998, the Mathematical Association of America (MAA) awarded him the annual Trevor Evans Award, presented to authors of an exceptional article that is accessible to undergraduates, for his piece entitled “What Is the Most Surprising Result in Mathematics?” (Answer: the prime number theorem). …In addition, he was the co-author of New Horizons in Geometry, published by the MAA, which received the CHOICE “Outstanding Academic Title” award in 2013. …He additionally served as a visiting lecturer for the MAA, and as a member of the MAA Board of Governors. …
12: 23.20 Mathematical Applications
§23.20(ii) Elliptic Curves
It follows from the addition formula (23.10.1) that the points P j = P ( z j ) , j = 1 , 2 , 3 , have zero sum iff z 1 + z 2 + z 3 𝕃 , so that addition of points on the curve C corresponds to addition of parameters z j on the torus / 𝕃 ; see McKean and Moll (1999, §§2.11, 2.14). … The addition law states that to find the sum of two points, take the third intersection with C of the chord joining them (or the tangent if they coincide); then its reflection in the x -axis gives the required sum. The geometric nature of this construction is illustrated in McKean and Moll (1999, §2.14), Koblitz (1993, §§6, 7), and Silverman and Tate (1992, Chapter 1, §§3, 4): each of these references makes a connection with the addition theorem (23.10.1). … K always has the form T × r (Mordell’s Theorem: Silverman and Tate (1992, Chapter 3, §5)); the determination of r , the rank of K , raises questions of great difficulty, many of which are still open. …
13: 14.18 Sums
§14.18(i) Expansion Theorem
§14.18(ii) Addition Theorems
14: 10.60 Sums
§10.60(i) Addition Theorems
15: 19.26 Addition Theorems
§19.26 Addition Theorems
19.26.27 R C ( x 2 , x 2 θ ) = 2 R C ( s 2 , s 2 θ ) , s = x + x 2 θ , θ x 2 or s 2 .
16: 23.10 Addition Theorems and Other Identities
§23.10 Addition Theorems and Other Identities
§23.10(i) Addition Theorems
For further addition-type identities for the σ -function see Lawden (1989, §6.4). …
17: 34.7 Basic Properties: 9 j Symbol
It constitutes an addition theorem for the 9 j symbol. …
18: 22.8 Addition Theorems
§22.8 Addition Theorems
19: 18.18 Sums
§18.18(ii) Addition Theorems
Ultraspherical
Legendre
Laguerre
Hermite
20: 27.15 Chinese Remainder Theorem
§27.15 Chinese Remainder Theorem
The Chinese remainder theorem states that a system of congruences x a 1 ( mod m 1 ) , , x 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. …By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod m 1 ), (mod m 2 ), (mod m 3 ), and (mod m 4 ), respectively. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result ( mod m ) , which is correct to 20 digits. …