1. CMB 2011 (vol 55 pp. 176)
 Spirn, Daniel; Wright, J. Douglas

Linear Dispersive Decay Estimates for the 3+1 Dimensional Water Wave Equation with Surface Tension
We consider the linearization of the threedimensional water waves
equation with surface tension about a flat interface. Using
oscillatory integral methods, we prove that solutions of this equation
demonstrate dispersive decay at the somewhat surprising rate of
$t^{5/6}$. This rate is due to competition between surface tension
and gravitation at $O(1)$ wave numbers and is connected to the fact
that, in the presence of surface tension, there is a socalled "slowest
wave". Additionally, we combine our dispersive estimates with $L^2$
type energy bounds to prove a family of Strichartz estimates.
Keywords:oscillatory integrals, water waves, surface tension, Strichartz estimates Categories:76B07, 76B15, 76B45 

2. CMB 2010 (vol 54 pp. 147)
 Nelson, Sam

Generalized Quandle Polynomials
We define a family of generalizations of the twovariable quandle polynomial.
These polynomial invariants generalize in a natural way to eightvariable
polynomial invariants of finite biquandles. We use these polynomials to define
a family of link invariants that further generalize the quandle counting
invariant.
Keywords:finite quandles, finite biquandles, link invariants Categories:57M27, 76D99 

3. CMB 2008 (vol 51 pp. 386)
 Lan, K. Q.; Yang, G. C.

Positive Solutions of the FalknerSkan Equation Arising in the Boundary Layer Theory
The wellknown FalknerSkan equation is one of the most important
equations in laminar boundary layer theory and is used to describe
the steady twodimensional flow of a slightly viscous
incompressible fluid past wedge shaped bodies of angles related to
$\lambda\pi/2$, where $\lambda\in \mathbb R$ is a parameter
involved in the equation. It is known that there exists
$\lambda^{*}<0$ such that the equation with suitable boundary
conditions has at least one positive solution for each $\lambda\ge
\lambda^{*}$ and has no positive solutions for
$\lambda<\lambda^{*}$. The known numerical result shows
$\lambda^{*}=0.1988$. In this paper, $\lambda^{*}\in
[0.4,0.12]$ is proved analytically by establishing a singular
integral equation which is equivalent to the FalknerSkan
equation. The equivalence result
provides new techniques to study properties and existence of solutions of
the FalknerSkan equation.
Keywords:FalknerSkan equation, boundary layer problems, singular integral equation, positive solutions Categories:34B16, 34B18, 34B40, 76D10 

4. CMB 2007 (vol 50 pp. 547)
 Iakovlev, Serguei

Inverse Laplace Transforms Encountered in Hyperbolic Problems of NonStationary FluidStructure Interaction
The paper offers a study of the inverse Laplace
transforms of the functions $I_n(rs)\{sI_n^{'}(s)\}^{1}$ where
$I_n$ is the modified Bessel function of the first kind and $r$ is
a parameter. The present study is a continuation of the author's
previous work %[\textit{Canadian Mathematical Bulletin} 45]
on the
singular behavior of the special case of the functions in
question, $r$=1. The general case of $r \in [0,1]$ is addressed,
and it is shown that the inverse Laplace transforms for such $r$
exhibit significantly more complex behavior than their
predecessors, even though they still only have two different types
of points of discontinuity: singularities and finite
discontinuities. The functions studied originate from
nonstationary fluidstructure interaction, and as such are of
interest to researchers working in the area.
Categories:44A10, 44A20, 33C10, 40A30, 74F10, 76Q05 

5. CMB 2004 (vol 47 pp. 30)
 He, Xinyu

Existence of Leray's SelfSimilar Solutions of the NavierStokes Equations In $\mathcal{D}\subset\mathbb{R}^3$
Leray's selfsimilar solution of the NavierStokes equations is
defined by
$$
u(x,t) = U(y)/\sqrt{2\sigma (t^*t)},
$$
where $y = x/\sqrt{2\sigma (t^*t)}$, $\sigma>0$. Consider the
equation for $U(y)$ in a smooth bounded domain $\mathcal{D}$ of
$\mathbb{R}^3$ with nonzero boundary condition:
\begin{gather*}
\nu \bigtriangleup U + \sigma U +\sigma y \cdot \nabla U + U\cdot
\nabla U + \nabla P = 0,\quad y \in \mathcal{D}, \\
\nabla \cdot U = 0, \quad y \in \mathcal{D}, \\
U = \mathcal{G}(y), \quad y \in \partial \mathcal{D}.
\end{gather*}
We prove an existence theorem for the Dirichlet problem in Sobolev
space $W^{1,2} (\mathcal{D})$. This implies the local existence of
a selfsimilar solution of the NavierStokes equations which blows
up at $t=t^*$ with $t^* < +\infty$, provided the function
$\mathcal{G}(y)$ is permissible.
Categories:76D05, 76B03 
