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It is used to prove Kronecker's lemma, which in turn, is used to prove a version of the strong law of large numbers under variance constraints. It may be used to prove Nicomachus's theorem that the sum of the first n {\displaystyle n} cubes equals the square of the sum of the first n {\displaystyle n} positive integers.
In mathematics and other fields, [a] a lemma (pl.: lemmas or lemmata) is a generally minor, proven proposition which is used to prove a larger statement. For that reason, it is also known as a "helping theorem " or an "auxiliary theorem".
The analogue of Fekete's lemma holds for superadditive sequences as well, that is: + +. (The limit then may be positive infinity: consider the sequence = !.) There are extensions of Fekete's lemma that do not require the inequality a n + m ≤ a n + a m {\displaystyle a_{n+m}\leq a_{n}+a_{m}} to hold for all m and n , but only for m and n ...
For many purposes, it is only necessary to know that an expansion for in terms of iterated commutators of and exists; the exact coefficients are often irrelevant. (See, for example, the discussion of the relationship between Lie group and Lie algebra homomorphisms in Section 5.2 of Hall's book, [2] where the precise coefficients play no role in the argument.)
Abel's summation formula can be generalized to the case where is only assumed to be continuous if the integral is interpreted as a Riemann–Stieltjes integral: ∑ x < n ≤ y a n ϕ ( n ) = A ( y ) ϕ ( y ) − A ( x ) ϕ ( x ) − ∫ x y A ( u ) d ϕ ( u ) . {\displaystyle \sum _{x<n\leq y}a_{n}\phi (n)=A(y)\phi (y)-A(x)\phi (x)-\int _{x ...
Layer cake representation. In mathematics, the layer cake representation of a non-negative, real-valued measurable function defined on a measure space (,,) is the formula = (,) (),
This can be shown to be equivalent to the definition from the first Giambelli formula using the Lindström–Gessel–Viennot lemma (as outlined on that page). Schur polynomials can be expressed as linear combinations of monomial symmetric functions m μ with non-negative integer coefficients K λμ called Kostka numbers ,
An application of the second isomorphism theorem identifies projective linear groups: for example, the group on the complex projective line starts with setting = (), the group of invertible 2 × 2 complex matrices, = (), the subgroup of determinant 1 matrices, and the normal subgroup of scalar matrices = {():}, we have = {}, where is ...