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If for each positive integer h, the sequence {u_(n+h)-u_n} is uniformly distributed (mod 1), then the sequence {u_n} is uniformly distributed (mod 1) (Montgomery 2001).

Let R be a ring, and let I be an ideal of R. The correspondence A<->A/I is an inclusion preserving bijection between the set of subrings A of R that contain I and the set of ...

If two single-valued continuous functions kappa(s) (curvature) and tau(s) (torsion) are given for s>0, then there exists exactly one space curve, determined except for ...

Let (L,<=) be any complete lattice. Suppose f:L->L is monotone increasing (or isotone), i.e., for all x,y in L, x<=y implies f(x)<=f(y). Then the set of all fixed points of f ...

Let G be a group having normal subgroups H and K with H subset= K. Then K/H⊴G/H and (G/H)/(K/H)=G/K, where N⊴G indicates that N is a normal subgroup of G and G=H indicates ...

Number theory is a vast and fascinating field of mathematics, sometimes called "higher arithmetic," consisting of the study of the properties of whole numbers. Primes and ...

Enclose a sphere in a cylinder and cut out a spherical segment by slicing twice perpendicularly to the cylinder's axis. Then the lateral surface area of the spherical segment ...

Let F_0 and F_1 denote two directly similar figures in the plane, where P_1 in F_1 corresponds to P_0 in F_0 under the given similarity. Let r in (0,1), and define ...

If f is a schlicht function and D(z_0,r) is the open disk of radius r centered at z_0, then f(D(0,1)) superset= D(0,1/4), where superset= denotes a (not necessarily proper) ...

Let five circles with concyclic centers be drawn such that each intersects its neighbors in two points, with one of these intersections lying itself on the circle of centers. ...