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...

Transport properties in random and scale-free (SF) networks are studied by analyzing the betweenness centrality (BC) distribution P(B) in the minimum spanning trees (MSTs) and infinite incipient percolation clusters (IIPCs) of the networks. It is found that P(B) in MSTs scales as P(B)∼B(-δ). The obtained values of δ are classified into two different categories, δ≃1.6 and δ≃2.0. Using the mappin...

We develop a detailed regularity theory of −∆+ b · ∇ in L(R), for a wide class of vector fields. The L-theory allows us to construct associated strong Feller process in C∞(R ). 1. Let Ld be the Lebesgue measure on Rd, Lp = Lp(Rd,Ld) and W 1,p = W 1,p(Rd,Ld) the standard Lebesgue and Sobolev spaces, C0,γ = C0,γ(Rd) the space of Hölder continuous functions (0 < γ < 1), Cb = Cb(R d) the space of b...

Let A be a set of n > n0(ε) positive real numbers, all at least 1 apart. We show that, if δa,b and δ� a,b are arbitrary real numbers satisfying |δa,b| < n 1−ε a , and |δ � a,b| < n1−ε b , we have |{(a + δa,b)(b + δ� a,b) : a, b ∈ A}| + |A + A| � n1+ε/9 log n . – Dedicated to Mel Nathanson and Carl Pomerance on the occasion of their 65th birthdays

The decays ¯ B → ¯ KD and ¯ B → ¯ K ¯ D taking into account final state interactions are discussed. These decays are described by four strong phases δ 0 , δ 1 , ˜ δ 0 , ˜ δ 1 (subscript 0 and 1 refer to I = 0 and I = 1 isospin final states), one weak phase γ and four real amplitudes. Isospin constraints are taken into account. It is argued that strong interaction dynamics gives δ 1 ≈ ˜ δ 1. The...

a f(x) dx (which one would use for instance to compute the work required to move a particle from a to b). For simplicity we shall restrict attention here to functions f : R → R which are continuous on the entire real line (and similarly, when we come to differential forms, we shall only discuss forms which are continuous on the entire domain). We shall also informally use terminology such as “i...

1 Discrepancy principle for the dynamical systems method Abstract Assume that Au = f, (1) is a solvable linear equation in a Hilbert space, ||A|| < ∞, and R(A) is not closed, so problem (1) is ill-posed. Here R(A) is the range of the linear operator A. A DSM (dynamical systems method) for solving (1), consists of solving the following Cauchy problem: ˙ u = −u + (B + (t)) −1 A * f, u(0) = u 0 , ...

1 Special cases of submodular flows We saw last time that orientation of a 2k-edge connected graph into a k-arc connected digraph and the Lucchesi and Younger Theorem were special cases of submodular flows. Other familiar problems can also be phrased as submodular flows. Let C = 2 V \ {∅, V } and let f be identically zero. Then for any U ∈ C, x(δ in (U)) − x(δ out (U) ≤ 0 (1) x(δ in (V \ U)) − ...

In this paper we introduce a hyperbolic (Poincaré-Bergman type) distance δ on the noncommutative open ball [B(H)]1 := n (X1, . . . ,Xn) ∈ B(H) n : ‖X1X ∗ 1 + · · ·+XnX ∗ n‖ 1/2 < 1 o , where B(H) is the algebra of all bounded linear operators on a Hilbert space H. It is proved that δ is invariant under the action of the free holomorphic automorphism group of [B(H)]1, i.e., δ(Ψ(X),Ψ(Y )) = δ(X, ...

The δ-approximate string matching problem, recently introduced in connection with applications to music retrieval, is a generalization of the exact string matching problem for alphabets of integer numbers. In the δ-approximate variant, (exact) matching between any pair of symbols/integers a and b is replaced by the notion of δ-matching =δ, where a =δ b if and only if |a− b| ≤ δ for a given valu...