Comparisons were made among four categories of protein flexibility: (1) low-B-factor ordered regions, (2) high-B-factor ordered regions, (3) short disordered regions, and (4) long disordered regions. Amino acid compositions of the four categories were found to be significantly different from each other, with high-B-factor ordered and short disordered regions being the most similar pair. The hig...

An ordered tree with height h is b-balanced if all its leaves have a level ` with h − b ≤ ` ≤ h, where at least one leaf has a level equal to h− b. For large n, we shall compute asymptotic equivalents to the number of all b-balanced ordered trees with n nodes and of all such trees with height h. Furthermore, assuming that all b-balanced ordered trees with n nodes are equally likely, we shall de...

Fuchs [6] called a partially-ordered integral domain, say D, division closed if it has the property that whenever a > 0 and ab > 0, then b > 0. He showed that if D is a lattice-ordered division closed field, then D is totally ordered. In fact, it is known that for a lattice-ordered division ring, the following three conditions are equivalent: a) squares are positive, b) the order is total, and ...

Given a mapping f : A → B from a partially ordered set A into an unstructured set B, we study the problem of defining a suitable partial ordering relation on B such that there exists a mapping g : B → A such that the pair of mappings (f, g) forms an isotone Galois connection between partially ordered sets.

Problem 1(a). The domain of f is N × N, so elements of the domain of f are ordered pairs 〈a, b〉 where a, b ∈ N. The following equivalences hold for every ordered pair 〈a, b〉 ∈ N× N. 〈a, b〉 ∈ f[E] ⇐⇒ f(〈a, b〉) ∈ E ⇐⇒ a ∈ E The first equivalence is an immediate consequence of the definition of the inverse image f[E]. The second equivalence follows from the definition of function f . We thus have ...

Definition. A binary operation * on a non-empty set S is a rule that assigns to each ordered pair of elements of elements of S a uniquely determined element of S. The element assigned to the ordered pair (a, b) with a, b ∈ S is denoted by a * b. Remark. In other words, a binary operarion of a set S is a function * : S×S → S from the Cartesian product S × S to the set S. The only difference is t...

The notion of absorbent ordered filters in implicative semigroupsis introduced, and its fuzzification is considered. Relations among (fuzzy) orderedfilters, (fuzzy) absorbent ordered filters, and (fuzzy) positive implicativeordered filters are stated. The extensionproperty for (fuzzy) absorbent orderedfilters is established. Conditions for (fuzzy) ordered filters to be (fuzzy)absorbent ordered ...