A graph with an ordered k-partition of the vertices is radial level planar if there is a strictly outward drawing on k concentric levels without crossings. Radial level planarity extends level planarity, where the vertices are placed on k horizontal lines and the edges are routed strictly downwards without crossings. The extension is characterised by rings, which are certain level non-planar bi...

Proposition 1.2. Let (X, d) be a metric space. Let (x)j=k be a sequence of elements of X. Let x, x′ be elements of X. Assume that the sequence (x)j=k converges to x with respect to d. Assume also that the sequence (x)j=k converges to x ′ with respect to d. Then x = x′. Proposition 1.3. Let a < b be real numbers, and let f : [a, b] → R be a function which is both continuous and strictly monotone...

‎a new approximation method for the set of common fixed points of‎ ‎nonexpansive mappings and the set of solutions of systems of‎ ‎variational inequalities is introduced and studied‎. ‎moreover‎, ‎we‎ ‎apply our main result to obtain strong convergence theorem to a‎ ‎common fixed point of a nonexpannsive mapping and solutions of ‎a ‎system of variational inequalities of an inverse strongly mono...

Let K ⊂ R be a convex body containing the origin. A measurable set G ⊂ R with positive Lebesgue measure is said to be uniformly K-dense if, for any fixed r > 0, the measure of G ∩ (x + rK) is constant when x varies on the boundary of G (here, x + rK denotes a translation of a dilation of K). We first prove that G must always be strictly convex and at least C-regular; also, if K is centrally sym...

Lp−Lq–boundedness of the map f → w(x) ∫ b(x) a(x) k(x, y)f(y)v(y)dy is described by different types of criteria expressed in terms of given parameters 0 < p, q < ∞, strictly increasing boundaries a(x) and b(x), locally integrable weight functions v, w and a positive continuous kernel k(x, y) satisfying some growth conditions.

In this paper we prove that for any simple polygon P with n vertices, the sum of the number of strictly internal edges and the number of strictly external visibility edges of P is at least b 3n−1 2 c − 4. The internal visibility graph of a simple polygon P is the graph with vertex set equal to the vertex set of P , in which two vertices are adjacent if the line segment connecting them does not ...

Performing the shape derivative (Sokolowski and Zolesio, 1992) and using the maximum principle, we show that the so-called Quadrature Surfaces free boundary problem QS ( f , k) ⎪⎪⎪⎨⎪⎪⎪⎩ −ΔuΩ = f in Ω uΩ = 0 on ∂Ω |∇uΩ| = k (constant) on ∂Ω. has a solution which contains strictly the support of f if and only if ∫

For a commutative ring k, the homotopy category of commutative Hk-algebras (strictly unital E∞ ring spectra under the Eilenberg-Mac Lane spectrum Hk) is equivalent to the homotopy category of E∞ differential graded k-algebras. The functor from topology to algebra is a CW approximation and cellular chain functor; the inverse equivalence is constructed by Brown’s representability theorem.