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Let p={a_1,a_2,...,a_n} be a permutation. Then i is a permutation ascent if a_i<a_(i+1). For example, the permutation {1,2,3,4} is composed of three ascents, namely {1,2}, ...

Let f(x) be integrable in [-1,1], let (1-x^2)f(x) be of bounded variation in [-1,1], let M^' denote the least upper bound of |f(x)(1-x^2)| in [-1,1], and let V^' denote the ...

The length of the polygonal spiral is found by noting that the ratio of inradius to circumradius of a regular polygon of n sides is r/R=(cot(pi/n))/(csc(pi/n))=cos(pi/n). (1) ...

The William Lowell Putnam Mathematics Competition is an annual mathematics contest open to North American college students administered by the Mathematical Association of ...

The quaternion group is one of the two non-Abelian groups of the five total finite groups of order 8. It is formed by the quaternions +/-1, +/-i, +/-j, and +/-k , denoted Q_8 ...

A Gaussian quadrature-like formula for numerical estimation of integrals. It requires m+1 points and fits all polynomials to degree 2m, so it effectively fits exactly all ...

A right angle is an angle equal to half the angle from one end of a line segment to the other. A right angle is pi/2 radians or 90 degrees. A triangle containing a right ...

There are a number of interesting results related to the tiling of squares. For example, M. Laczkovich has shown that there are exactly three shapes of non-right triangles ...

A triangle tiling is a tiling of the plane by identical triangles. Any triangle tiles the plane (Wells 1991, p. 208). The total number of triangles (including inverted ones) ...

Let a random n×n (0,1)-matrix have entries which are 1 (with probability p) or 0 (with probability q=1-p) and numbers are assigned to the edges of a grid. A b-cluster is an ...