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Let E be a compact connected subset of d-dimensional Euclidean space. Gross (1964) and Stadje (1981) proved that there is a unique real number a(E) such that for all x_1, ...

The Mercator projection is a map projection that was widely used for navigation since loxodromes are straight lines (although great circles are curved). The following ...

The pentagonal hexecontahedron is the 60-faced dual polyhedron of the snub dodecahedron A_8 (Holden 1971, p. 55). It is Wenninger dual W_(18). A tetrahedron 10-compound, cube ...

If p and q are distinct odd primes, then the quadratic reciprocity theorem states that the congruences x^2=q (mod p) x^2=p (mod q) (1) are both solvable or both unsolvable ...

A symmetric matrix is a square matrix that satisfies A^(T)=A, (1) where A^(T) denotes the transpose, so a_(ij)=a_(ji). This also implies A^(-1)A^(T)=I, (2) where I is the ...

Given a regular tetrahedron of unit volume, consider the lengths of line segments connecting pairs of points picked at random inside the tetrahedron. The distribution of ...

The constant e is base of the natural logarithm. e is sometimes known as Napier's constant, although its symbol (e) honors Euler. e is the unique number with the property ...

The van der Grinten projection is a map projection given by the transformation x = (1) y = sgn(phi)(pi|PQ-Asqrt((A^2+1)(P^2+A^2)-Q^2)|)/(P^2+A^2), (2) where A = ...

Orthogonal polynomials are classes of polynomials {p_n(x)} defined over a range [a,b] that obey an orthogonality relation int_a^bw(x)p_m(x)p_n(x)dx=delta_(mn)c_n, (1) where ...

An ordinary differential equation of the form y^('')+P(x)y^'+Q(x)y=0. (1) Such an equation has singularities for finite x=x_0 under the following conditions: (a) If either ...