I have 50 pairs of socks of which 30 are black and 35 are cotton. How many pairs of socks are black and cotton? If I call the set of black socks A and the set of cotton socks B, we are looking at |A ∩ B|. It is not hard to see that: |A ∪ B| = |A| + |B| − |A ∩ B| because every element that belongs to both A and B is counted twice by |A|+|B|, so we subtract the size of the intersection once. Ther...

Separation Logic as introduced by Calcagno, O’Hearn, and Yang [7] is the foun-dation of the separation logic framework in HOL4. In the following, this foundation willbe described. 3.2.1 States and Predicates on States As the name suggests, Abstract Separation Logic is an abstract version of separationlogic. It abstracts from both the concrete specification and the concrete programmi...

20.1. Additional technical results on weak convergence Given two metric spaces S1, S2 and a measurable function f : S1 → S2, suppose S1 is equipped with some probability measure P. This induces a probability measure on S2 which is denoted by Pf−1 and is defined by Pf−1(A) = P(f−1(A) for every measurable set A ⊂ S2. Then for any random variable X : S2 → R, its expectation EPf −1 [X] is equal to ...

In this paper, we consider the asymptotic properties of the numbers of spanning trees and Eulerian trials in circulant digraphs and graphs. Let C(p, s1, s2, . . . , sk) be a directed or undirected circulant graph. Let T (C(p, s1, s2, . . . , sk)) and E(C(p, s1, s2, . . . , sk)) be the numbers ∗This work was partially supported by the Natural Science Foundation of China, email: fjzhang@jingxian....

Suppose two agents, S1 and S2, are epistemic peers regarding a proposition p: that is, suppose S1 and S2 are equally competent, equally impartial, and equally able to evaluate and assess the relevant evidence regarding p (we will call such propositions p peer-propositions for S1 and S2). After carefully reflecting on the salient evidence for p, suppose S1 and S2 discover that they disagree abou...

(1) Let P, Q be subsets of E2 T, p1, p2, q1 be points of E2 T, f be a map from I into (E2 T) P, and s1 be a real number. Suppose that P is an arc from p1 to p2 and q1 ∈ P and q1 ∈ Q and f (s1) = q1 and f is a homeomorphism and f (0) = p1 and f (1) = p2 and 0 ≤ s1 and s1 ≤ 1 and for every real number t such that 0 ≤ t and t < s1 holds f (t) / ∈ Q. Let g be a map from I into (E2 T) P and s2 be a ...

Alpha s1-, alpha s2-, beta- and kappa-caseins from Somali camels (Camelus dromedarius) were purified by acid precipitation at pH 4.4, crudely separated into an alpha-CN and a beta-CN fraction and further purified by reversed-phase HPLC. Fragments of tryptic digests were sequenced. Amino acid patterns obtained were used to screen a cDNA library constructed from mRNA from lactating udder tissue. ...

Given two metric spaces S1, S2 and a measurable function f : S1 → S2, sup­ pose S1 is equipped with some probability measure P. This induces a proba­ bility measure on S2 which is denoted by Pf−1 and is defined by Pf−1(A) = P(f−1(A)) for every measurable set A ⊂ S2. Then for any random variable X : S2 → R, its expectation EPf−1 [X] is equal to EP[X(f)]. (Convince your­ self that this is the cas...

This paper considers classical online scheduling problems on uniform machines. We show the tight competitive ratio of LS for any combinations of speeds of three machines. We prove that LS is optimal when s3 ≥ s2 ≥ s1 = 1 and s3 ≥ s2 + s2s3 + s2, where s1, s2, s3 are the speeds of three machines. On the other hand, LS can not be optimal for all combinations of machine speeds, even restricted to ...

Global surface pressure data from 1976 to 1997 from over 7500 land stations and the Comprehensive Ocean– Atmosphere Data Set have been analyzed using harmonic and zonal harmonic methods. It is found that the diurnal pressure oscillation (S1) is comparable to the semidiurnal pressure oscillation (S2) in magnitude over much of the globe except for the low-latitude open oceans, where S2 is about t...