We obtain a criterion for approximability by embeddings of piecewise linear maps of a circle to the plane, analogous to the one proved by Minc for maps of a segment to the plane. Theorem. Let S be a triangulation of a circle with s vertices. Let f
be a simplicial map of the graph S to the plane. The map f is approximable by embeddings if and only if for each i=0,...,s the i-th derivative of the map f (defined by Minc) neither contains transversal self-intersections nor is the standard winding of degree greater than 1. We deduce from the Minc result the completeness of the van Kampen obstruction to approximability by embeddings of piecewise linear maps of a segment to the plane. We also generalize these criteria to simplicial maps of a graph without vertices of degree >3 to a circle.
For any collection of graphs we find the minimal dimension d such that the product of these graphs is embeddable into the d-dimensional Euclidean space. In particular, we prove that the n-th powers of the Kuratowsky graphs are not embeddable into the
2n-dimensional Euclidean space. This is a solution of a problem of Menger from 1929. The idea of the proof is the reduction to a problem from so-called Ramsey link theory: we show that any embedding of L into the (2n-1)-dimensional sphere, where L is the join of n copies of a 4-point set, has a pair of linked (n-1)-dimensional spheres.
This paper is on homotopy classification of maps of (n+1)-dimensional manifolds into the n-dimensional sphere. For a continuous map f of an (n+1)-manifold into the n-sphere define the degree deg f to be the class dual to f^*[S^n], where [S^n] is the
fundamental class. We present a short and direct proof of the following specific case of the Pontryagin-Steenrod-Wu theorem: Theorem. Let M be a connected orientable closed smooth (n+1)-manifold, n>2. Then the map deg:pi^n(M)to H_1(M;Z) is 1-to-1 (i.e., bijective), if the product w_2(M) x r_2 H_2(M;Z) is nonzero, where r_2 is the mod2 reduction; 2-to-1 (i.e., each element of H_1(M;Z) has exactly 2 preimages) - otherwise. The proof is based on the Pontryagin-Thom construction and a geometric definition of the Stiefel-Whitney classes w_2(M).
In the present popular science paper we determine when a square can be dissected into rectangles similar to a given rectangle. The approach to the question is based on a physical interpretation using electrical networks. Only secondary school background is assumed in the paper.
We develop linear discretization of complex analysis, originally introduced by R. Isaacs, J. Ferrand, R. Duffin, and C. Mercat. We prove convergence of discrete period matrices and discrete Abelian integrals to their continuous counterparts. We also
prove a discrete counterpart of the Riemann--Roch theorem. The proofs use energy estimates inspired by electrical networks.
This paper is on further development of discrete complex analysis introduced by R. Isaacs, J. Ferrand, R. Duffin, and C. Mercat. We consider a graph lying in the complex plane and having quadrilateral faces. A function on the vertices is called discr
ete analytic, if for each face the difference quotients along the two diagonals are equal. We prove that the Dirichlet boundary value problem for the real part of a discrete analytic function has a unique solution. In the case when each face has orthogonal diagonals we prove that this solution uniformly converges to a harmonic function in the scaling limit. This solves a problem of S. Smirnov from 2010. This was proved earlier by R. Courant-K. Friedrichs-H. Lewy and L. Lusternik for square lattices, by D. Chelkak-S. Smirnov and implicitly by P.G. Ciarlet-P.-A. Raviart for rhombic lattices. In particular, our result implies uniform convergence of the finite element method on Delaunay triangulations. This solves a problem of A. Bobenko from 2011. The methodology is based on energy estimates inspired by alternating-current network theory.
