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van der Waerden's theorem is a theorem about the existence of arithmetic progressions in sets. The theorem can be stated in four equivalent forms. 1. If N=C_1 union C_2 union ...

Let {f_n} and {a_n} be sequences with f_n>=f_(n+1)>0 for n=1, 2, ..., then |sum_(n=1)^ma_nf_n|<=Af_1, where A=max{|a_1|,|a_1+a_2|,...,|a_1+a_2+...+a_m|}.

An additive function is an arithmetic function such that whenever positive integers a and b are relatively prime, f(ab)=f(a)+f(b). An example of an additive function is ...

A string or word is said to be admissible if that word appears in a given sequence. For example, in the sequence aabaabaabaabaab..., a, aa, baab are all admissible, but bb is ...

Any link can be represented by a closed braid.

A set (usually of letters) from which a subset is drawn. A sequence of letters is called a word, and a set of words is called a code.

The alternating factorial is defined as the sum of consecutive factorials with alternating signs, a(n)=sum_(k=1)^n(-1)^(n-k)k!. (1) They can be given in closed form as ...

The payment of a debt plus accrued interest by regular payments.

An Andreini tessellation is a three-dimensional equivalent of a semiregular tessellation. There are 28 such tessellations.

The angle bisector of an angle in a triangle divides the opposite side in the same ratio as the sides adjacent to the angle.