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For any algebraic number x of degree n>2, a rational approximation p/q to x must satisfy |x-p/q|>1/(q^n) for sufficiently large q. Writing r=n leads to the definition of the ...

A field which is complete with respect to a discrete valuation is called a local field if its field of residue classes is finite. The Hasse principle is one of the chief ...

The term "loop" has a number of meanings in mathematics. Most simply, a loop is a closed curve whose initial and final points coincide in a fixed point p known as the ...

Low-dimensional topology usually deals with objects that are two-, three-, or four-dimensional in nature. Properly speaking, low-dimensional topology should be part of ...

In a set X equipped with a binary operation · called a product, the multiplicative identity is an element e such that e·x=x·e=x for all x in X. It can be, for example, the ...

Let two disks of radius r intersect one another perpendicularly and have a diameter in common. If the distance between the centers of the disks is sqrt(2) times their radius, ...

A Pisot number is a positive algebraic integer greater than 1 all of whose conjugate elements have absolute value less than 1. A real quadratic algebraic integer greater than ...

Pogson's ratio is the constant 100^(1/5)=10^(2/5)=2.511886431... (OEIS A189824) appearing in the definition of the astronomical magnitude (brightness) scale. This scale is ...

The ring of integers is the set of integers ..., -2, -1, 0, 1, 2, ..., which form a ring. This ring is commonly denoted Z (doublestruck Z), or sometimes I (doublestruck I). ...

A semialgebraic set is a subset of R^n which is a finite Boolean combination of sets of the form {x^_=(x_1,...,x_n):f(x^_)>0} and {x^_:g(x^_)=0}, where f and g are ...