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The Löwenheim-Skolem theorem is a fundamental result in model theory which states that if a countable theory has a model, then it has a countable model. Furthermore, it has a ...

The longest increasing (contiguous) subsequence of a given sequence is the subsequence of increasing terms containing the largest number of elements. For example, the longest ...

Throughout abstract algebra, the term "magma" is most often used as a synonym of the more antiquated term "groupoid," referring to a set equipped with a binary operator. The ...

Let there be three polynomials a(x), b(x), and c(x) with no common factors such that a(x)+b(x)=c(x). Then the number of distinct roots of the three polynomials is one or more ...

A function f:X->R is measurable if, for every real number a, the set {x in X:f(x)>a} is measurable. When X=R with Lebesgue measure, or more generally any Borel measure, then ...

The word "median" has several different meanings in mathematics all related to the "middle" of mathematical objects. The statistical median is an order statistic that gives ...

If a compact manifold M has nonnegative Ricci curvature tensor, then its fundamental group has at most polynomial growth. On the other hand, if M has negative curvature, then ...

Module multiplicity is a number associated with every nonzero finitely generated graded module M over a graded ring R for which the Hilbert series is defined. If dim(M)=d, ...

Let g(x_1,...,x_n,y) be a function such that for any x_1, ..., x_n, there is at least one y such that g(x_1,...,x_n,y)=0. Then the mu-operator muy(g(x_1,...,x_n,y)=0) gives ...

The number of multisets of length k on n symbols is sometimes termed "n multichoose k," denoted ((n; k)) by analogy with the binomial coefficient (n; k). n multichoose k is ...