1. For any instance I1 of π1, I2 = f(I1) is an instance of π2 such that OPTπ2(I2) ≤ OPTπ1(I1). 2. For any feasible solution S2 of I2, S1 = g(I1, S2) (g maps S2 into an instance of I1) we have Costπ1(I1, S1) ≤ Costπ2(I2, S2). Note that OPTπ2(I2) ≤ OPTπ1(I1) ≤ Costπ1(I1, S1) ≤ Costπ2(I2, S2). Therefore, if there is an approximation factor ∆ for π2 then there is an approximation factor ∆ for π1 as...

where Mag(x;y) = 1/| det[Jacη](x)| is the magnification of the lensed image x of y. Suppose that the light source at y moves along a path y(t). Then the shifted center-of-light curve relative to the source’s trajectory is defined by Xcl(t) = xcl(y(t)) − y(t). It is an important fact that locally stable k-plane lensing maps η are generic and each is differentiably equivalent about every critical...

This note is an addendum to Sum theorems for monotone operators and convex functions. In it, we prove some new results on convex functions and monotone operators, and use them to show that several of the constraint qualifications considered in the preceding paper are, in fact, equivalent. Introduction We continue with the notation and the numbering of [4]. For the moment, we shall assume that E...

Equine infectious anemia virus (EIAV) is genetically one of the simplest lentiviruses in that the viral genome encodes only three accessory genes, tat, rev, and S2. Although serological analyses demonstrate the expression of the S2 protein in persistently infected horses, the role of this viral gene remains undefined. We recently reported that the S2 gene is not essential for EIAV replication i...

From now on X denotes a set and S denotes a family of subsets of X. Now we state the propositions: (1) Let us consider sets X1, X2, a family S1 of subsets of X1, and a family S2 of subsets of X2. Then {a×b, where a is an element of S1, b is an element of S2 : a ∈ S1 and b ∈ S2} = {s, where s is a subset of X1 ×X2 : there exist sets a, b such that a ∈ S1 and b ∈ S2 and s = a × b}. Proof: {a × b,...

We study inverse semigroup amalgams [S1, S2;U ], where S1 and S2 are finitely presented inverse semigroups with decidable word problem and U is an inverse semigroup with decidable membership problem in S1 and S2. We use a modified version of Bennett’s work on the structure of Schützenberger graphs of the R-classes of S1 ∗U S2 to state sufficient conditions for the amalgamated free products S1 ∗...