Some geometric conjectures in harmonic function theory

We prove some results on the geometry of the level sets of harmonic functions, particularly regarding their ‘oscillation’ and ‘pinching’ properties. These results allow us to tackle three recent conjectures due to De Carli and Hudson (Bull London Math Soc 42:83–95, 2010). Our approach hinges on a combination of local constructions, methods from differential topology and global extension arguments.     

Unsolved problems in number theory

Periodica Mathematica Hungarica - 68 unsolved problems and conjectures in number theory are presented and brie y discussed. The topics covered are: additive representation functions, the...     

Open problems and conjectures

In this section we present some open problems and conjectures about some interesting types of difference equations. Please submit your problems and conjectures with all relevant information to G. Ladas.     

Open problems and conjectures

In this section we present some pen problems and conjectures about some interesting types of difference equations. Please submit your problems and conjectures with all relevant information to G. Ladas.     

Open problems and conjectures

In this section we present some open problems and conjectures about some interesting types of difference equations, Please submit your problems and conjectures with all relevant information to G. Ladas.     

Open problems and conjectures

In this section we present some open problems and conjectures about some interesting types of difference equations. Please submit your problems and conjecutures with all relevant informations to G. Ladas     

Open problems and conjectures

In this section we present some open problems and conjectures about some interesting types of difference equations. Please submit your problems and conjectures with all relavant information to G.Ladas     

Open problems and conjectures

Two conjectures are made about the dynamics of SI and SIR epidemic models     

Some remarks on number theory

This note contains some disconnected minor remarks on number theory.     

Some problems in approximation theory

This dissertation consists of three parts. The first part is devoted to the approximation of a fixed element by a fixed approximating set. In the second part we study approximation by finite sets (Chapter I, Secs. 3–6) and by finite-dimensional subspaces (Chapter II, Secs. 7–15). Selected problems with geometric (Secs. 1–3) and variational (Secs. 4–6) content are considered in the third chapter.The author's summary of dissertation for the degree of Doctor of Physicomathematical Sciences. The dissertation was defended on December 18, 1970, at a session of the Scientific Council of the Mechanics-Mathematics Faculty of Moscow State University. The official opposition was made up of Academician A. N. Kolmogorov, K. I. Babenko, and S. B. Stechkin.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 593–607, May, 1971.     

Ill-posed problems of number theory and tomography

In honor of the 60th Birthday of Academician M. M. Lavrent'ev.     

Some provably hard crossing number problems

This paper presents a connection between the problem of drawing a graph with the minimum number of edge crossings, and the theory of arrangements of pseudolines, a topic well-studied by combinatorialists. In particular, we show that any given arrangement can be forced to occur in every minimum crossing drawing of an appropriate graph. Using some recent results of Goodman, Pollack, and Sturmfels, this yields that there exists no polynomial-time algorithm for producing a straight-line drawing of a graph, which achieves the minimum number of crossings from among all such drawings. While this result has no bearing on the P versus NP question, it is fairly negative with regard to applications. We also study the problem of drawing a graph with polygonal edges, to achieve the (unrestricted) minimum number of crossings. Here we obtain a tight bound on the smallest number of breakpoints which are required in the polygonal lines. This work was partially supported by the Center for Telecommunications Research, Columbia University.