1. Induction proofs, type I: Sum/product formulas: The most common, and the easiest, application of induction is to prove formulas for sums or products of n terms. All of these proofs follow the same pattern. (a) ∑n i=1 i(i+ 1) = n(n+1)(n+2) 3 (b) ∑n i=0 2 i = 2n+1 − 1 (sum of powers of 2) (c) ∑n i=0 r i = 1−r n+1 1−r (r 6= 1) (sum of finite geometric series) (d) ∑n i=0 i!i = (n+ 1)!− 1. Soluti...

We consider a number of boundary value problems involving the p-Laplacian. The model case is −∆pu = V |u|p−2u for u ∈ W 1,p 0 (D) with D a bounded domain in R. We derive necessary conditions for the existence of nontrivial solutions. These conditions usually involve a lower bound for a product of powers of the norm of V , the measure of D, and a sharp Sobolev constant. In most cases, these ineq...

PURPOSE To introduce a novel autorefractor design that is intended to be manufacturable at low cost and evaluate its performance in measuring refractive errors. METHODS We developed a handheld, open-view autorefractor (the "QuickSee" [QS]) that uses a simplified approach to wavefront sensing that forgoes moving parts and expensive components. Adult subjects (n = 41) were recruited to undergo ...

The present paper reconsiders the Newtonian limit of models of modified gravity including higher order terms in the scalar curvature in the gravitational action. This was studied using the Palatini variational principle in [Meng X and Wang P 2004 Gen. Rel. Grav. 36 1947] and [Domínguez A E and Barraco D E 2004 Phys. Rev. D 70 043505] with contradicting results. Here a different approach is used...

R&D is often, not only, the engine that powers innovation in organisations but it creates and sustains their competitive advantage. The front end of innovation—the ‘R’ R&D—is expected to provide initial spark ignite innovation. Design thinking has become one most popular approaches this crucial challenging phase process. In letter, we contrast its shortcomings with a novel paradigm deri...