Abstract:
We develop a direct and elementary (calculus-free) exposition of the famous cubic surface of revolution x^3+y^3+z^3-3xyz=1.12 pages. We have added a second elementary proof that the surface is of revolution.

Abstract:
In this paper a mathematical apparatus for determination of plane section of cone and cylinder was formed. By using the descriptive geometric approach the contour lines of these quadrics were determined. The fact that the tangent lines of a circle could be transformed to the tangent lines of an ellipse using affinity was employed. In that way surfaces are represented by contour lines (tangent lines of basic ellipse in oblique projection) and thus they have a realistic view. Intersecting plane α is a plane normal to a frontal plane. For determination of intersecting points of intersecting curve between the plane α and the quadrics, the lock of auxiliary planes, which contain the vertex of quadrics, was used. Each auxiliary plane from the observed lock intersect the surface in two lines which intersect the given plane α in two points. By using a sufficient number of auxiliary planes the intersecting curve as a set of pairs of points for all auxiliary planes is determined and the intersecting curve was drawn by lightening of these pairs of points on the graphical screen.

Abstract:
We study the existence of solutions an H-system for a revolution surface without boundary for H depending on the radius f. Under suitable conditions we prove that the existence of asolution is equivalent to the solvability of a scalar equationN(a)=L/2, where N:𝒜⊂ℝ

Abstract:
We study the existence of solutions an H -system for a revolution surface without boundary for H depending on the radius f . Under suitable conditions we prove that the existence of a solution is equivalent to the solvability of a scalar equation N( a )=L/ 2 , where N: + → is a function depending on H . Moreover, using the method of upper and lower solutions we prove existence results for some particular examples. In particular, applying a diagonal argument we prove the existence of unbounded surfaces with prescribed H .

Abstract:
A fast inhomogeneous plane wave algorithm is developed for the electromagnetic scattering problem from the composite bodies of revolution (BOR). Poggio-Miller-Chang Harrington-Wu (PMCHW) approach is used for the homogeneous dielectric objects, while the electric field integral equation (EFIE) is used for the perfect electric conducting objects. The aggregation and disaggregation factors can be expressed analytically by using the Weyl identity. Compared with the traditional method of moments (MoM), both the memory requirement and CPU time, are reduced for large-scale composite BOR problems. Numerical results are given to demonstrate the validity and the efficiency of the proposed method.

Abstract:
Spinor fields on surfaces of revolution conformally immersed into 3-dimensional space are considered in the framework of the spinor representations of surfaces. It is shown that a linear problem (a 2-dimensional Dirac equation) related with a modified Veselov- Novikov hierarchy in the case of the surface of revolution reduces to a well-known Zakharov-Shabat system. In the case of one-soliton solution an explicit form of the spinor fields is given by means of linear Bargmann potentials and is expressed via the Jost functions of the Zakharov-Shabat system. It is shown also that integrable deformations of the spinor fields on the surface of revolution are defined by a modified Korteweg-de Vries hierarchy.

Abstract:
If $C \subset P^3_k$ is an integral curve and $k$ an algebraically closed field of characteristic 0, it is known that the points of the general plane section $C \cap H$ of $C$ are in uniform position. From this it follows easily that the general minimal curve containing $C\cap H$ is irreducible. If $char k = p > 0$, the points of $C\cap H$ may not be in uniform position. However, we prove that the general minimal curve containing $C\cap H$ is still irreducible.

Abstract:
We derive analytic expressions for the wavefunctions and energy levels in the semiclassical approximation for perturbed integrable systems. We find that some eigenstates of such systems are substantially different from any of the unperturbed states, which requires some sort of a resonant perturbation theory. We utilize the semiclassical surface of section method by Bogomolny that reduces the spatial dimensions of the problem by one. Among the systems considered are the circular billiard with a perturbed boundary, including the short stadium; the perturbed rectangular billiard, including the tilted square and the square in magnetic field; the bouncing ball states in the stadium and slanted stadium; and the whispering gallery modes. The surface of section perturbation theory is compared with the Born-Oppenheimer approximation, which is an alternative way to describe some classes of states in these systems. We discuss the derivation of the trace formulas from Bogomolny's transfer operator for the chaotic, integrable, and almost integrable systems.

Abstract:
Perturbation theory, the quasiclassical approximation and the quantum surface of section method are combined for the first time. This solves the long standing problem of quantizing the resonances and chaotic regions generically appearing in classical perturbation theory. The result is achieved by expanding the `phase' of the wavefunction in powers of the square root of the small parameter. It gives explicit WKB-like wavefunctions even for systems which classically show hard chaos. We also find analytic solutions to some questions raised recently.

Abstract:
Perturbation theory, the quasiclassical approximation and the quantum surface of section method are combined for the first time. This gives a new solution of the the long standing problem of quantizing the resonances generically appearing in classical perturbation theory. Our method is restricted to two dimensions. In that case, however, the results are simpler, more explicit and more easily expressed visually than the results of earlier techniques. The method involves expanding the `phase' of the wavefunction in powers of the square root of the small parameter. It gives explicit WKB-like wavefunctions and energies even for certain systems which classically show hard chaos.