1.1. Graphs. Graphs, or abstract collections of nodes and edges, form the main objects studied in the theory of complex networks. These models can be classified according to many different types of topological properties, but the underlying structures are quite simple. The general procedure for modeling with networks is to first identify the objects of interest (nodes), the crucial relations be...

Since the overall quality of Taiwanese university education has been decreasing in recent years, and universities are losing their competitive advantage while facing foreign countries’ education systems and the threat of closing, upgrading innovation performance and improving total quality performance (Inno-Qual performance) so as to enhance overall operational performance have become an urgent...

Exercise 1.2. If A,B ⊂ R,m∗(A) = 0, then m∗(A ∪B) = m∗(B) Proof: m∗(A ∪B) ≤ m∗(A) +m∗(B), and m∗(B) ≤ m∗(A ∪B), hence we have m∗(B) ≤ m∗(A ∪B) ≤ m∗(A) +m∗(B) = m∗(B) ∴ m∗(A ∪B) = m∗(B) Exercise 1.3. Prove E ∈M iff ∀ > 0,∃O ⊂ R open, such that E ⊂ O and m∗(O\E) < Proof: (⇒) O\E = E ∩O implies that m∗(O\E) = m∗(Ec ∩O), but we have m∗(O) = m∗(Ec ∩O) +m∗(E ∩O) So suppose m∗(E) < ∞ ⇒ m∗(Ec ∩ O) = m∗...

I.1 Two short problems: 1. Suppose that (X,A, μ) is a measure space, and X = ∪j=1Xj with Xj ⊂ Xj+1 for all j. Let χj be the indicator (characteristic) function of Xj, and let Mj denote the multiplication operator by χj acting on Z = Lp(X, dμ), 1 ≤ p < ∞. Show that Mj → I, the identity operator, in the strong operator topology, i.e. the weakest topology on bounded linear operators L(Z) in which ...

We describe a roundness classiication procedure , that is, a procedure to determine if the roundness of a planar object I is within some 0 from an ideal circle. The procedure consists of a probing strategy and an evaluation algorithm working in a feedback loop. This approach of combining probing with evaluation is new in computational metrology. For several deenitions of roundness, our procedur...