By Dorothy Buck and Erica Flapan, Dorothy Buck, Erica Flapan

ISBN-10: 0821844660

ISBN-13: 9780821844663

Over the last 20-30 years, knot idea has rekindled its historical ties with biology, chemistry, and physics as a way of constructing extra subtle descriptions of the entanglements and houses of usual phenomena--from strings to natural compounds to DNA. This quantity relies at the 2008 AMS brief direction, functions of Knot conception. the purpose of the fast path and this quantity, whereas no longer overlaying all facets of utilized knot conception, is to supply the reader with a mathematical appetizer, for you to stimulate the mathematical urge for food for extra learn of this intriguing box. No past wisdom of topology, biology, chemistry, or physics is thought. particularly, the 1st 3 chapters of this quantity introduce the reader to knot thought (by Colin Adams), topological chirality and molecular symmetry (by Erica Flapan), and DNA topology (by Dorothy Buck). the second one 1/2 this quantity is targeted on 3 specific purposes of knot idea. Louis Kauffman discusses purposes of knot idea to physics, Nadrian Seeman discusses how topology is utilized in DNA nanotechnology, and Jonathan Simon discusses the statistical and vigorous homes of knots and their relation to molecular biology.

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4], and [1010, p. 65]. 26. ) (Remark: These results are Bernoulli’s inequality. 2. Let x be a nonnegative number, and let α be a real number. If α ∈ [0, 1], then α + xα ≤ 1 + αx, whereas, if either α ≤ 0 or α ≥ 1, then 1 + αx ≤ α + xα. 1. 21. ) (Remark: This result is equivalent to Bernoulli’s inequality. 3. Let x and α be real numbers, assume that either α ≤ 0 or α ≥ 1, and assume that x ∈ [0, 1]. Then, (1 + x)α ≤ 1 + (2α − 1)x. Furthermore, equality holds if and only if either α = 0, α = 1, x = 0, or x = 1.

2) The set with no elements, denoted by ∅, is the empty set. If X = ∅, then X is nonempty. A set cannot have repeated elements. For example, {x, x} = {x}. However, a multiset is a collection of elements that allows for repetition. The multiset consisting of two copies of x is written as {x, x}ms . However, we do not assume that the listed elements x, y of the conventional set {x, y} are distinct. The number of distinct elements of the set S or not-necessarily-distinct elements of the multiset S is the cardinality of S, which is denoted by card(S).

In this case, f (x) = y. 2. Let R be a relation on X. Then, the following terminology is deﬁned: i) R is reﬂexive if, for all x ∈ X, it follows that (x, x) ∈ R. ii) R is symmetric if, for all (x1, x2 ) ∈ R, it follows that (x2 , x1 ) ∈ R. iii) R is transitive if, for all (x1, x2 ) ∈ R and (x2 , x3 ) ∈ R, it follows that (x1, x3 ) ∈ R. iv) R is an equivalence relation if R is reﬂexive, symmetric, and transitive. 3. Let R1 and R2 be relations on X. If R1 and R2 are (reﬂexive, symmetric) relations, then so are R1 ∩ R2 and R1 ∪ R2 .

### Applications of Knot Theory (Proceedings of Symposia in Applied Mathematics) by Dorothy Buck and Erica Flapan, Dorothy Buck, Erica Flapan

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