The survey gives an overview of the achievements in topology of real algebraic varieties in the direction initiated in the early 70th by V. I. Arnold and V. A. Rokhlin. We make an attempt to systematize the principal results in the subject. After an exposition of general tools and results, special attention is paid to surfaces and curves on surfaces.

Veech groups uniformize Teichmüller geodesic curves in Riemann moduli space. Recently, examples of infinitely generated Veech groups have been given. We show that these can even have infinitely many cusps and infinitely many infinite ends. We further show that examples exist for which each direction of an infinite end is the limit of directions of inequivalent infinite ends.

We consider a unit speed curve α in Euclidean n-dimensional space E n and denote the Frenet frame by {V 1 ,. .. , V n }. We say that α is a cylindrical helix if its tangent vector V 1 makes a constant angle with a fixed direction U. In this work we give different characterizations of such curves in terms of their curvatures.

We consider a unit speed curve α in Euclidean four-dimensional space E and denote the Frenet frame by {T,N,B1,B2}. We say that α is a slant helix if its principal normal vector N makes a constant angle with a fixed direction U . In this work we give different characterizations of such curves in terms of their curvatures. MSC: 53C40, 53C50

In this paper, the nonlinear vibration analysis of a thin cylindrical shell made of Functionally Graded Material (FGM) resting on a nonlinear viscoelastic foundation under compressive axial and lateral loads is studied. Nonlinear governing coupled partial differential equations of motions (PDEs) for cylindrical shell are derived using improved Donnell shell theory. The equations of motions (EOM...

Precise finite speed, in the sense of that the domain of influence is a subset of the union of influence curves through the support of the initial data is proved for hyperbolic systems symmetrized by pseudodifferential operators in the spatial variables. From this, uniqueness in the Cauchy problem at spacelike hypersurfaces is derived by a Hölmgren style duality argument. Sharp finite speed is ...

We consider the endomorphism ring computation problem for supersingular elliptic curves, constructive versions of Deuring’s correspondence, and the security of Charles-Goren-Lauter’s cryptographic hash function. We show that constructing Deuring’s correspondence is easy in one direction and equivalent to the endomorphism ring computation problem in the other direction. We also provide a collisi...

For generalized cylinders (GC) defined by contours of discrete curves, we propose two algorithms to generate GC surfaces (1) in polygonal meshes and (2) in cylindrical type of developable surface patches. To solve the contour blending problem of generalized cylinder, the presented algorithms have adopted the algorithms and properties of LIDM (linear interpolation by direction map) that interpol...

Direction and orientation tuning elicited by moving bars, flashing bars and a moving noise field were compared in cells in area 17 of the cat. Fourier analysis of tuning curves (SDO-analysis) was applied to quantify the general sensitivity (S) to visual stimulation, tuning strength to direction(D) and orientation (O), as well as the preferred direction (PD) and orientation (PO). Results from SD...