Search Results for ""

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 ...

Flat polygons embedded in three-space can be transformed into a congruent planar polygon as follows. First, translate the starting vertex to (0, 0, 0) by subtracting it from ...

Let a>|b|, and write h(theta)=(acostheta+b)/(2sintheta). (1) Then define P_n(x;a,b) by the generating function f(x,w)=f(costheta,w)=sum_(n=0)^inftyP_n(x;a,b)w^n ...

The conjecture due to Pollock (1850) that every number is the sum of at most five tetrahedral numbers (Dickson 2005, p. 23; incorrectly described as "pyramidal numbers" and ...

The (signed) area of a planar non-self-intersecting polygon with vertices (x_1,y_1), ..., (x_n,y_n) is A=1/2(|x_1 x_2; y_1 y_2|+|x_2 x_3; y_2 y_3|+...+|x_n x_1; y_n y_1|), ...

Consider the sequence {x_n}_(n=0)^infty defined by x_0=1 and x_(n+1)=[3/2x_n], where [z] is the ceiling function. For n=0, 1, ..., the first few terms are 1, 2, 3, 5, 8, 12, ...

Define a power difference prime as a number of the form n^n-(n-1)^(n-1) that is prime. The first few power difference primes then have n=2, 3, 4, 7, 11, 17, 106, 120, 1907, ...

Pythagoras's theorem states that the diagonal d of a square with sides of integral length s cannot be rational. Assume d/s is rational and equal to p/q where p and q are ...

The Q-chromatic polynomial, introduced by Birkhoff and Lewis (1946) and termed the "Q-chromial" by Bari (1974), is an alternate form of the chromatic polynomial pi(x) defined ...

Quinn et al. (2007) investigated a class of N coupled oscillators whose bifurcation phase offset had a conjectured asymptotic behavior of sinphi∼1-c_1/N, with an experimental ...