A rectangular drawing of a planar graph G is in which vertices are mapped to grid points, edges horizontal and vertical straight-line segments, faces drawn as rectangles. Sometimes this latter constraint relaxed for the outer face. In paper, we study drawings have unit length. We show complexity dichotomy problem deciding existence unit-length drawing, depending on whether face must also be rec...

We classify the set of quadrilaterals that can be inscribed in convex Jordan curves, continuous as well smooth case. This answers a question Makeev special case curves. The difficulty this problem comes from fact standard topological arguments to prove existence solutions do not apply here due lack sufficient symmetry. Instead, proof makes use an area argument Karasev and Tao, which we furtherm...

In the paper we study configurations which can be represented as families of cyclically inscribed n-gons. The most regular of them arise from quasi difference sets distinguished in a product of two cyclic groups, but some other more general techniques which define series of inscribed n-gons are found and studied. We give conditions which assure the existence of certain automorphisms of the defi...

If E is any ellipse inscribed in a convex quadrilateral, D – , then we prove that Area (E) Area(D – ) π 4 , and equality holds if and only if D – is a parallelogram and E is tangent to the sides of D – at the midpoints. We also prove that the foci of the unique ellipse of maximal area inscribed in a parallelogram, D – , lie on the orthogonal least squares line for the vertices of D – . This doe...

We investigate forcing notions with the rectangle refining property, which is stronger than the countable chain condition, and fragments of Martin’s Axiom for such forcing notions. We prove that it is consistent that every forcing notion with the rectangle refining property has precaliber א1 but MAא1 for forcing notions with the rectangle refining property fails.