Using the (generalized) Darboux transformation in the case of the Clifford torus, we construct for all Pythagorean triples (p, q, n) ∈ Z a CP–family of Willmore tori in S with Willmore energy 2nπ.

We invoke the characterization of Pythagorean-hodograph polynomial curves and prove that it is impossible to parameterize any real curves, other than a straight line, by rational functions of its arc length.

This paper studies forward and reverse projections for the Rényi divergence of order α ∈ (0,∞) on α-convex sets. The forward projection on such a set is motivated by some works of Tsallis et al. in statistical physics, and the reverse projection is motivated by robust statistics. In a recent work, van Erven and Harremoës proved a Pythagorean inequality for Rényi divergences on α-convex sets und...

By using a computer program devised by M. Creutz, we were able to determine all Pythagoeran triplets for which z ^ 300. At this point, a distinction must must be made between P-triplets for which x9 y9 and z have no common divisor [the so-called "primitive solutions" of (1)] and P-triplets which are related to the primitive solutions by multiplication by a common integer factor k. So, if Xi9y^9...

Mathematics most famous marginal note is Fermat’s assertion of what is now called Fermat’s Last Theorem. This was made in his edition of Arithmetica by Diophantus, which had recently been translated from Greek to Latin, the common language of science and all forms of learning. Fermat’s note was also in Latin, and translates roughly as follows: “It is impossible for a cube and a cube to give a c...

In this paper, a class of rational spatial curves that have a rational binormal is introduced. Such curves (called PB curves) play an important role in the derivation of rational rotation-minimizing conformal frames. The PB curve construction proposed is based upon the dual curve representation and the Euler-Rodrigues frame obtained from quaternion polynomials. The construction significantly si...