We investigate asphericity of the relative group presentation 〈G, t | atbtctdtet = 1〉 and prove it aspherical provided the subgroup of G generated by {ab−1, bc−1, cd−1, de−1} is neither finite cyclic nor a finite triangle group. We also prove a similar result for the closely related relative group presentation 〈G, s, t | αsβsγt = 1 = δtεtζs−1〉.

In this note, we prove that an aspherical fully residually free Kähler group is Fuchsian. One instances of this shows that cocompact lattices in PU(1, n) for n > 1 cannot be residually free. The proof uses some recent work of Delzant and Gromov.

We give an explicit recursive presentation for Mihailova’s subgroup M(H) of Fn×Fn corresponding to a finite, concise and Peiffer aspherical presentation H = 〈x1, . . . , xn |R1, . . . , Rm〉. This partially answers a question of R.I. Grigorchuk, [8, Problem 4.14]. As a corollary, we construct a finitely generated recursively presented orbit undecidable subgroup of Aut(F3).

Purpose: To assess the quality of vision in pseudophakic patients with a prolate aspherical intraocular lens (IOL) compared to patients with a spherical IOL. Patients and methods: Twenty patients undergoing cataract surgery were divided into two groups according to the type of IOL: ten prolate aspherical IOLs (TECNIS Z9000, AMO, USA) and ten spherical IOLs (911 CeeOn Edge, AMO, USA). The integr...

With the recent increase in size and image quality of optical equipment, there is a strong demand for higher precision aspherical lenses. Generally, an glass lens or its mold formed by grinding then finished polishing. Since shape accuracy deteriorates this polishing process, it necessary to keep amount small. Therefore, desirable optimize conditions reduce surface roughness. In paper, axisymme...