An exact analytical solution, based on the method of images, is obtained for the description of the electric field between an atomic force microscope (AFM) tip and a thin dielectric polymer film (30 nm thick) spin coated on a conductive substrate. Three different tip shapes are found to produce electrostatic pressure above the plasticity threshold in the polymers up to 50 MPa. It is shown experimentally that a strong nonuniform electric field ͑5 ϫ 10 8 -5ϫ 10 9 V m −1 ͒ between the AFM tip and polymer substrate produces nanodeformations of two different kinds in planar polymer films. Nanostructures (lines and dots) 10-100 nm wide and 0.1-5 nm high are patterned in the polymer films by using two different experimental techniques. The first technique relies on electric breakdown in the film leading to polymer heating above the glass transition point followed by mass transport of softened polymer material towards the AFM tip. The second technique is believed to be associated with plastic deformation of the polymer surface at the nanoscale. In this case the nanostructures are experimentally patterned in the films with no external biasing of the AFM tip, and using only the motion of the tip. This suggests an additional nanomechanical approach for nanolithography in polymer films of arbitrary thickness.

Special class of surfaces in five-dimensional sphere in C 3 is considered. Immersion equations for minimal tori of that class are shown to be reducible to the equation u zz = e u − e −2u which is integrable by means of inverse scattering method. Finite-gap minimal tori are constructed.

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The problem of constructing a perfect Euler cuboid is reduced to a single Diophantine equation of the degree 12.