A Perturbation Analysis of an Interaction between Long and Short Surface Waves

The interaction of finite-amplitude long gravity waves with a small-amplitude packet of short capillary waves is studied by a multiple-scale method based on the invariance of the perturbation expansion under certain translations. The result of the analysis is a set of equations coupling the complex amplitude of the packet of short waves with the long-wave velocity potential and surface elevation. The short wave is described by a Ginzburg-Landau equation with coefficients that depend on properties of the long wave. The long-wave potential and surface elevation satisfy the usual free-surface conditions augmented by forcing terms representing effects of the short waves. The derivation removes some of the restrictions imposed in earlier studies.     

Non-recursive functions,knots “with thick ropes,” and self-clenching “thick” hyperspheres

We introduce an approach to certain geometric variational problems based on the use of the algorithmic unrecognizability of the n-dimensional sphere for n ≥ 5. Sometimes this approach allows one to prove the existence of infinitely many solutions of a considered variational problem. This recursion-theoretic approach is applied in this paper to a class of functionals on the space of C^1.1-smooth hypersurfaces diffeomorphic to Sⁿ in Rⁿ⁺¹, where n is any fixed number ≥ 5. The simplest of these functionals k_v is defined by the formula k_v(Σⁿ) = (vol(Σⁿ))^1/n/r(Σⁿ), where r(Σⁿ) denotes the radius of injectivity of the normal exponential map for Σⁿ ? Rⁿ+l. We prove the existence of an infinite set of distinct locally minimal values of k_v on the space of C^1.1-smooth topological hyperspheres in Rⁿ⁺¹ for any n ≥ 5. The functional k_v naturally arises when one attempts to generalize knot theory in order to deal with embeddings and isotopies of “thick” circles and, more generally, “thick” spheres into Euclidean spaces. We introduce the notion of knot “with thick rope” types. The theory of knot “with thick rope” types turns out to be quite different from the classical knot theory because of the following result: There exists an infinite set of non-trivial knot “with thick rope” types in codimension one for every dimension greater than or equal to five.     

Addendum to “A definable nonstandard enlargement”

?o?'s theorem for bounded D ‐ultrapowers, D being the ultrafilter introduced by Kanovei and Shelah [4], is established. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A “Generalized Trace Formula” for Bell numbers

The nth Bell number B_n is the number of ways to partition a set of n elements into nonempty subsets. We generalize the “trace formula” of Barsky and Benzaghou [1], which asserts that for an odd prime p and an appropriate constant τ_p, the relation B_n=-Tr(^n-1-τ_p)B_τp holds in , where is a root of and is the trace form. We deduce some new interesting congruences for the Bell numbers, generalizing miscellaneous well-known results including those of Radoux [4].     

A “Coprolitic Vision” for Earth Science Education

William Buckland (1784–1846) first identified and scientifically studied coprolites in the early 1820s. Although some of his contemporaries did not look favorably upon him or his research, Buckland's early experiments advanced paleoecology and taphonomy. Because our informal presentations with coprolites resulted in students' spirited reactions, we investigated whether coprolite introduction, accompanied with its history of science, had potential for meaningful learning in K‐12 Earth Science classrooms. Practicing Earth Science teachers (N = 28) enrolled in an online paleontology course researched coprolites, identified potential student interest, and designed coprolite activities for their individual classrooms. Resulting projects were diverse and creative, and incorporated investigations into fossilization processes, paleoenvironments, food chains, and geologic time. In anonymous surveys, teachers indicated that their students' interest in coprolites is high. We propose inclusion of coprolites and their history in Earth Science classrooms as a portal to hook students' interest and as springboard to additional scientific topics.