1. CMB 2016 (vol 59 pp. 542)
 Jiang, Yongxin; Wang, Wei; Feng, Zhaosheng

Spatial Homogenization of Stochastic Wave Equation with Large Interaction
A dynamical approximation of a stochastic wave
equation with large interaction is derived.
A random invariant manifold is discussed. By a key linear transformation,
the random invariant manifold is shown to be close to the random
invariant manifold
of a secondorder stochastic ordinary differential equation.
Keywords:stochastic wave equation, homogeneous system, approximation, random invariant manifold, Neumann boundary condition Categories:60F10, 60H15, 35Q55 

2. CMB 2015 (vol 59 pp. 303)
3. CMB 2014 (vol 58 pp. 9)
 Chavan, Sameer

Irreducible Tuples Without the Boundary Property
We examine spectral behavior of irreducible tuples which do not
admit boundary property. In particular, we prove under some mild
assumption that the spectral radius of such an $m$tuple $(T_1,
\dots, T_m)$ must be the operator norm of $T^*_1T_1 + \cdots +
T^*_mT_m$. We use this simple observation to ensure boundary
property for an irreducible, essentially normal joint $q$isometry provided it
is not a joint isometry.
We further exhibit a family of
reproducing Hilbert $\mathbb{C}[z_1, \dots, z_m]$modules (of which
the DruryArveson Hilbert module is a prototype) with the property that any
two nested unitarily equivalent submodules are indeed equal.
Keywords:boundary representations, subnormal, joint pisometry Categories:47A13, 46E22 

4. CMB 2011 (vol 56 pp. 102)
 Kong, Qingkai; Wang, Min

Eigenvalue Approach to Even Order System Periodic Boundary Value Problems
We study an even order system boundary value problem with
periodic boundary conditions. By establishing
the existence of a positive eigenvalue of an associated linear system
SturmLiouville problem, we obtain new conditions for the boundary
value problem to have a positive solution. Our major tools are the
KreinRutman theorem for linear spectra and the fixed point index theory
for compact operators.
Keywords:Green's function, high order system boundary value problems, positive solutions, SturmLiouville problem Categories:34B18, 34B24 

5. CMB 2011 (vol 55 pp. 285)
 Eloe, Paul W.; Henderson, Johnny; Khan, Rahmat Ali

Uniqueness Implies Existence and Uniqueness Conditions for a Class of $(k+j)$Point Boundary Value Problems for $n$th Order Differential Equations
For the $n$th order nonlinear differential equation, $y^{(n)} = f(x, y, y',
\dots, y^{(n1)})$, we consider uniqueness implies uniqueness and existence
results for solutions satisfying certain $(k+j)$point
boundary conditions for $1\le j \le n1$ and $1\leq k \leq nj$. We
define $(k;j)$point unique solvability in analogy to $k$point
disconjugacy and we show that $(nj_{0};j_{0})$point
unique solvability implies $(k;j)$point unique solvability for $1\le j \le
j_{0}$, and $1\leq k \leq nj$. This result is
analogous to
$n$point disconjugacy implies $k$point disconjugacy for $2\le k\le
n1$.
Keywords:boundary value problem, uniqueness, existence, unique solvability, nonlinear interpolation Categories:34B15, 34B10, 65D05 

6. CMB 2010 (vol 53 pp. 674)
 Kristály, Alexandru; Papageorgiou, Nikolaos S.; Varga, Csaba

Multiple Solutions for a Class of Neumann Elliptic Problems on Compact Riemannian Manifolds with Boundary
We study a semilinear elliptic problem on a compact Riemannian
manifold with boundary, subject to an inhomogeneous Neumann
boundary condition. Under various hypotheses on the nonlinear
terms, depending on their behaviour in the origin and infinity, we
prove multiplicity of solutions by using variational arguments.
Keywords:Riemannian manifold with boundary, Neumann problem, sublinearity at infinity, multiple solutions Categories:58J05, 35P30 

7. CMB 2010 (vol 53 pp. 475)
 Jankowski, Tadeusz

Nonlinear Multipoint Boundary Value Problems for Second Order Differential Equations
In this paper we shall discuss nonlinear multipoint boundary value problems for second order differential equations when deviating arguments depend on the unknown solution. Sufficient conditions under which such problems have extremal and quasisolutions are given. The problem of when a unique solution exists is also investigated. To obtain existence results, a monotone iterative technique is used. Two examples are added to verify theoretical results.
Keywords:second order differential equations, deviated arguments, nonlinear boundary conditions, extremal solutions, quasisolutions, unique solution Categories:34A45, 34K10 

8. CMB 2009 (vol 52 pp. 555)
 Hirata, Kentaro

Boundary Behavior of Solutions of the Helmholtz Equation
This paper is concerned with the boundary behavior of solutions of
the Helmholtz equation in $\mathbb{R}^\di$.
In particular, we give a Littlewoodtype theorem to show that
the approach region introduced by Kor\'anyi and Taylor (1983) is best possible.
Keywords:boundary behavior, Helmholtz equation Categories:31B25, 35J05 

9. 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 

10. CMB 2008 (vol 51 pp. 140)
 Rossi, Julio D.

First Variations of the Best Sobolev Trace Constant with Respect to the Domain
In this paper we study the best constant of the Sobolev trace
embedding $H^{1}(\Omega)\to L^{2}(\partial\Omega)$, where $\Omega$
is a bounded smooth domain in $\RR^N$. We find a formula for the
first variation of the best constant with respect to the domain.
As a consequence, we prove that the ball is a critical domain when
we consider deformations that preserve volume.
Keywords:nonlinear boundary conditions, Sobolev trace embedding Categories:35J65, 35B33 

11. CMB 2007 (vol 50 pp. 579)
 Kot, Piotr

$p$Radial Exceptional Sets and Conformal Mappings
For $p>0$ and for a given set $E$ of type $G_{\delta}$ in the boundary
of the unit disc $\partial\mathbb D$ we construct a holomorphic function
$f\in\mathbb O(\mathbb D)$ such that
\[
\int_{\mathbb D\setminus[0,1]E}ft^{p}\,d\mathfrak{L}^{2}<\infty\]
and\[
E=E^{p}(f)=\Bigl\{ z\in\partial\mathbb D:\int_{0}^{1}f(tz)^{p}\,dt=\infty\Bigr\} .\]
In particular if a set $E$ has a measure equal to zero, then a function
$f$ is constructed as integrable with power $p$ on the unit disc $\mathbb D$.
Keywords:boundary behaviour of holomorphic functions, exceptional sets Categories:30B30, 30E25 

12. CMB 2005 (vol 48 pp. 580)
 Kot, Piotr

Exceptional Sets in Hartogs Domains
Assume that $\Omega$ is a Hartogs domain in $\mathbb{C}^{1+n}$,
defined as $\Omega=\{(z,w)\in\mathbb{C}^{1+n}:z<\mu(w),w\in H\}$, where $H$ is an open set in
$\mathbb{C}^{n}$ and $\mu$ is a continuous function with positive values in $H$ such that $\ln\mu$
is a strongly plurisubharmonic function in $H$. Let $\Omega_{w}=\Omega\cap(\mathbb{C}\times\{w\})$.
For a given set $E$ contained in $H$ of the type $G_{\delta}$ we construct a holomorphic function
$f\in\mathbb{O}(\Omega)$ such that
\[
E=\Bigl\{ w\in\mathbb{C}^{n}:\int_{\Omega_{w}}f(\cdot\,,w)^{2}\,d\mathfrak{L}^{2}=\infty\Bigr\}.
\]
Keywords:boundary behaviour of holomorphic functions,, exceptional sets Category:30B30 

13. CMB 1997 (vol 40 pp. 395)
 Boudhraa, Zineddine

$D$spaces and resolution
A space $X$ is a $D$space if, for every neighborhood assignment $f$
there is a closed discrete set $D$ such that $\bigcup{f(D)}=X$. In this
paper we give some necessary conditions and some sufficient conditions
for a resolution of a topological space to be a $D$space. In particular,
if a space $X$ is resolved at each $x\in X$ into a $D$space $Y_x$ by
continuous mappings $f_x\colon X\{{x}\} \rightarrow Y_x$, then the
resolution is a $D$space if and only if $\bigcup{\{{x}\}}\times \Bd(Y_x)$
is a $D$space.
Keywords:$D$space, neighborhood assignment, resolution, boundary Categories:54D20, 54B99, 54D10, 54D30 
