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1: 26.11 Integer Partitions: Compositions
c ( n ) denotes the number of compositions of n , and c m ( n ) is the number of compositions into exactly m parts. …
26.11.1 c ( 0 ) = c ( T , 0 ) = 1 .
26.11.6 c ( T , n ) = F n 1 , n 1 .
2: 3.2 Linear Algebra
The sensitivity of the solution vector 𝐱 in (3.2.1) to small perturbations in the matrix 𝐀 and the vector 𝐛 is measured by the condition number
3.2.16 κ ( 𝐀 ) = 𝐀 p 𝐀 1 p ,
3.2.17 𝐱 𝐱 p 𝐱 p κ ( 𝐀 ) 𝐫 p 𝐛 p .
If 𝐀 is nondefective and λ is a simple zero of p n ( λ ) , then the sensitivity of λ to small perturbations in the matrix 𝐀 is measured by the condition number
3.2.20 κ ( λ ) = 1 | 𝐲 T 𝐱 | ,
3: 26.10 Integer Partitions: Other Restrictions
26.10.1 p ( 𝒟 , 0 ) = p ( 𝒟 k , 0 ) = p ( S , 0 ) = 1 .
26.10.6 p ( 𝒟 , n ) = 1 n t = 1 n p ( 𝒟 , n t ) j | t j  odd j ,
26.10.7 ( 1 ) k p ( 𝒟 , n 1 2 ( 3 k 2 ± k ) ) = { ( 1 ) r , n = 3 r 2 ± r , 0 , otherwise ,
26.10.8 ( 1 ) k p ( 𝒟 , n ( 3 k 2 ± k ) ) = { 1 , n = 1 2 ( r 2 ± r ) , 0 , otherwise ,
26.10.12 p ( 𝒟 , n ) = p ( 𝒪 , n ) ,
4: 34.10 Zeros
In a 3 j symbol, if the three angular momenta j 1 , j 2 , j 3 do not satisfy the triangle conditions (34.2.1), or if the projective quantum numbers do not satisfy (34.2.3), then the 3 j symbol is zero. …However, the 3 j and 6 j symbols may vanish for certain combinations of the angular momenta and projective quantum numbers even when the triangle conditions are fulfilled. …
5: 34.2 Definition: 3 j Symbol
They therefore satisfy the triangle conditions …The corresponding projective quantum numbers m 1 , m 2 , m 3 are given by …
See accompanying text
Figure 34.2.1: Angular momenta j r and projective quantum numbers m r , r = 1 , 2 , 3 . Magnify
If either of the conditions (34.2.1) or (34.2.3) is not satisfied, then the 3 j symbol is zero. When both conditions are satisfied the 3 j symbol can be expressed as the finite sum …
6: 26.12 Plane Partitions
Then the number of plane partitions in B ( r , s , t ) is … The number of symmetric plane partitions in B ( r , r , t ) is … The example of a strict shifted plane partition also satisfies the conditions of a descending plane partition. The number of descending plane partitions in B ( r , r , r ) is …
7: 27.13 Functions
If 3 k = q 2 k + r with 0 < r < 2 k , then equality holds in (27.13.2) provided r + q 2 k , a condition that is satisfied with at most a finite number of exceptions. …
8: 5.11 Asymptotic Expansions
5.11.2 ψ ( z ) ln z 1 2 z k = 1 B 2 k 2 k z 2 k .
For the Bernoulli numbers B 2 k , see §24.2(i). With the same conditions, …For explicit formulas for g k in terms of Stirling numbers see Nemes (2013a), and for asymptotic expansions of g k as k see Boyd (1994) and Nemes (2015a). … Next, and again with the same conditions, …
9: 2.10 Sums and Sequences
Sufficient conditions for the validity of this second result are: … First, the conditions can be weakened. …For example, Condition (b) can be replaced by: … Furthermore, (2.10.31) remains valid with the weaker conditionIn Condition (c) we have …
10: 1.10 Functions of a Complex Variable
The last condition means that given ϵ ( > 0 ) there exists a number a 0 [ a , b ) that is independent of z and is such that …