Appendix Proof of Lemma 1: We first provide the complete parameter regions and the corresponding consumer equilibrium outcome: (I) Unpatched On-premises only (0 < v u < 1): (A) δ ≤ 1 − c p (i) p ≤ p s (1 − c p)/δ and α ≤ α B ; (ii) p s (1 − c p)/δ < p ≤ p s /δ and α ≤ α A ; (B) δ > 1 − c p (i) p ≤ 1 − c p − δ + p s and α ≤ α B ; (ii) 1 − c p − δ + p s < p ≤ p s and α ≤ α C ; (iii) p s < p ≤ p s...

Graph-theoretic properties of certain proximity graphs defined on planar point sets are investigated. We first consider some of the most common proximity graphs of the family of the Delaunay graph, and study their number of edges, minimum and maximum degree, clique number, and chromatic number. In the second part of the paper we focus on the higher order versions of some of these graphs and giv...

Plummer and Toft conjectured in 1987 that the vertices of every 3-connected plane graph with maximum face size Δ⋆ can be colored using at most Δ⋆+2 colors such a way no is incident two same color. The conjecture has been proven for Δ⋆=3, Δ⋆=4 Δ⋆≥18. We prove Δ⋆=16 Δ⋆=17.

A vertex-colored graph is rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertexconnected. It was proved that ifG is a graph of order n with minimum degree δ, then rvc(G) < 11n/δ. I...

Abstract. Let δ(P) be the δ-vector of a reflexive polytope P ⊂ R of dimension d and δ(P) the δ-vector of the dual polytope P ⊂ R. In general, δ(P) = δ(P) does not hold. In this paper, we give a higher-dimensional construction of a reflexive polytope whose δ-vector equals the δ-vector of the dual polytope. In particular, we consider the case that the reflexive polytope and the dual polytope are ...

The convergence rate of a Krylov method such as the Generalized Conjugate Resid10 ual (GCR) [6] method, to solve a linear system Au = f , A = (ai j) ∈ Rm×m,u ∈ 11 R m, f ∈ Rm, decreases with increasing condition number κ2(A) = ||A||2||A−1||2 12 of the non singular matrix A. Left preconditioning techniques consist of solving 13 M−1Au=M−1 f such that κ2(M−1A)<<κ2(A). The Additive Schwarz (AS) pre...

The KLOE measurements of φ radiative decays and of the ratio Γ(K S → π + π − (γ))/Γ(K S → π 0 π 0) are discussed. The first measurement aims at understanding the real nature of the scalar mesons a 0 and f 0 , whose possible compositions are q ¯ q, qq ¯ q ¯ q, or K ¯ K molecule. The second measurement is related to the estimate of the phase shift difference δ 0 − δ 2. Previous measurements were ...

Definition 0.1 Let δ be a cardinal. We say that δ is Woodin with 3 iff there is some sequence (a κ : κ < δ) such that a κ ⊂ V κ for every κ < δ and for every A ⊂ V δ the set {κ < δ : A ∩ V κ = a κ ∧ κ is A–strong up to δ} is stationary in δ. Proof. Let us define ((a κ , c κ) : κ < δ) recursively as follows. If ((a κ , c κ) : κ < µ) is defined for some µ < δ, then we let (a µ , c µ) be the least...

Let N be a lattice and P ⊂ N ⊗ Z R a lattice polytope, i.e., the convex hull of finitely many points in N. Ehrhart's theorem asserts that the lattice-point counting function f P (m) := # (mP ∩ N) is a polynomial, and thus 1 + P m≥1 f P (m) t m = δ P (t) (1−t) d+1 for some polynomial δ P (t), the δ-vector of P. Motivated by the Knudsen–Mumford–Waterman Conjecture about the existence of unimodula...