1. CMB Online first
 Li, ChiKwong; Tsai, MingCheng

Factoring a quadratic operator as a product of two positive contractions
Let $T$ be a quadratic operator on a complex Hilbert space $H$.
We show that $T$ can be written as a product of two positive
contractions if and only if $T$ is of the form
\begin{equation*}
aI \oplus bI \oplus
\begin{pmatrix} aI & P \cr 0 & bI \cr
\end{pmatrix} \quad \text{on} \quad H_1\oplus H_2\oplus (H_3\oplus
H_3)
\end{equation*}
for some $a, b\in [0,1]$ and strictly positive operator $P$ with
$\P\ \le \sqrt{a}  \sqrt{b}\sqrt{(1a)(1b)}.$ Also, we
give a necessary condition for a bounded linear operator $T$
with operator matrix
$
\big(
\begin{smallmatrix} T_1 & T_3
\\ 0 & T_2\cr
\end{smallmatrix}
\big)
$ on $H\oplus K$ that can be written as a product
of two positive contractions.
Keywords:quadratic operator, positive contraction, spectral theorem Categories:47A60, 47A68, 47A63 

2. CMB 2014 (vol 57 pp. 431)
 Tagami, Keiji

The Rasmussen Invariant, Fourgenus and Threegenus of an Almost Positive Knot Are Equal
An oriented link is positive if it has a link diagram whose crossings are all positive.
An oriented link is almost positive if it is not positive and has a link diagram with exactly one negative crossing.
It is known that the Rasmussen invariant, $4$genus and $3$genus of a positive knot are equal.
In this paper, we prove that the Rasmussen invariant, $4$genus and $3$genus of an almost positive knot are equal.
Moreover, we determine the Rasmussen invariant of an almost positive knot in terms of its almost positive knot diagram.
As corollaries, we prove that all almost positive knots are not homogeneous, and there is no almost positive knot of $4$genus one.
Keywords:almost positive knot, fourgenus, Rasmussen invariant Categories:57M27, 57M25 

3. CMB 2012 (vol 57 pp. 289)
 Ghasemi, Mehdi; Marshall, Murray; Wagner, Sven

Closure of the Cone of Sums of $2d$powers in Certain Weighted $\ell_1$seminorm Topologies
In a paper from 1976, Berg, Christensen and Ressel prove that the
closure of the cone of sums of squares $\sum
\mathbb{R}[\underline{X}]^2$ in the polynomial ring
$\mathbb{R}[\underline{X}] := \mathbb{R}[X_1,\dots,X_n]$ in the
topology induced by the $\ell_1$norm is equal to
$\operatorname{Pos}([1,1]^n)$, the cone consisting of all polynomials
which are nonnegative on the hypercube $[1,1]^n$. The result is
deduced as a corollary of a general result, established in the same
paper, which is valid for any commutative semigroup.
In later work, Berg and Maserick and Berg, Christensen and Ressel
establish an even more general result, for a commutative semigroup
with involution, for the closure of the cone of sums of squares of
symmetric elements in the weighted $\ell_1$seminorm topology
associated to an absolute value.
In the present paper we give a new proof of these results which is
based on Jacobi's representation theorem from 2001. At the same time,
we use Jacobi's representation theorem to extend these results from
sums of squares to sums of $2d$powers, proving, in particular, that
for any integer $d\ge 1$, the closure of the cone of sums of
$2d$powers $\sum \mathbb{R}[\underline{X}]^{2d}$ in
$\mathbb{R}[\underline{X}]$ in the topology induced by the
$\ell_1$norm is equal to $\operatorname{Pos}([1,1]^n)$.
Keywords:positive definite, moments, sums of squares, involutive semigroups Categories:43A35, 44A60, 13J25 

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. 214)
 Wang, DaBin

Positive Solutions of Impulsive Dynamic System on Time Scales
In this paper, some criteria for the existence of positive solutions of a class
of systems of impulsive dynamic equations on time scales are obtained by
using a fixed point theorem in cones.
Keywords:time scale, positive solution, fixed point, impulsive dynamic equation Categories:39A10, 34B15 

6. CMB 2011 (vol 54 pp. 544)
7. CMB 2010 (vol 53 pp. 256)
8. 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 

9. CMB 2007 (vol 50 pp. 356)
 Filippakis, Michael E.; Papageorgiou, Nikolaos S.

Existence of Positive Solutions for Nonlinear Noncoercive Hemivariational Inequalities
In this paper we investigate the existence of positive solutions
for nonlinear elliptic problems driven by the $p$Laplacian with a
nonsmooth potential (hemivariational inequality). Under asymptotic
conditions that make the Euler functional indefinite and
incorporate in our framework the asymptotically linear problems,
using a variational approach based on nonsmooth critical point
theory, we obtain positive smooth solutions. Our analysis also
leads naturally to multiplicity results.
Keywords:$p$Laplacian, locally Lipschitz potential, nonsmooth critical point theory, principal eigenvalue, positive solutions, nonsmooth Mountain Pass Theorem Categories:35J20, 35J60, 35J85 

10. CMB 2004 (vol 47 pp. 73)
 Li, Ma; Dezhong, Chen

Systems of Hermitian Quadratic Forms
In this paper, we give some conditions to judge when a system of
Hermitian quadratic forms has a real linear combination which is
positive definite or positive semidefinite. We also study some
related geometric and topological properties of the moduli space.
Keywords:hermitian quadratic form, positive definite, positive semidefinite Category:15A63 

11. CMB 2004 (vol 47 pp. 22)
 Goto, Yasuhiro

A Note on the Height of the Formal Brauer Group of a $K3$ Surface
Using weighted Delsarte surfaces, we give examples of $K3$ surfaces
in positive characteristic whose formal Brauer groups have height
equal to $5$, $8$ or $9$. These are among the four values of the
height left open in the article of Yui \cite{Y}.
Keywords:formal Brauer groups, $K3$ surfaces in positive, characteristic, weighted Delsarte surfaces Categories:14L05, 14J28 

12. CMB 2003 (vol 46 pp. 216)
 Li, ChiKwong; Rodman, Leiba; Šemrl, Peter

Linear Maps on Selfadjoint Operators Preserving Invertibility, Positive Definiteness, Numerical Range
Let $H$ be a complex Hilbert space, and $\HH$ be the real linear space of
bounded selfadjoint operators on $H$. We study linear maps $\phi\colon \HH
\to \HH$ leaving invariant various properties such as invertibility, positive
definiteness, numerical range, {\it etc}. The maps $\phi$ are not assumed
{\it a priori\/} continuous. It is shown that under an appropriate surjective
or injective assumption $\phi$ has the form $X \mapsto \xi TXT^*$ or $X \mapsto
\xi TX^tT^*$, for a suitable invertible or unitary $T$ and $\xi\in\{1, 1\}$,
where $X^t$ stands for the transpose of $X$ relative to some orthonormal basis.
Examples are given to show that the surjective or injective assumption cannot
be relaxed. The results are extended to complex linear maps on the algebra of
bounded linear operators on $H$. Similar results are proved for the (real)
linear space of (selfadjoint) operators of the form $\alpha I+K$, where $\alpha$
is a scalar and $K$ is compact.
Keywords:linear map, selfadjoint operator, invertible, positive definite, numerical range Categories:47B15, 47B49 

13. CMB 2001 (vol 44 pp. 210)
 Leung, Man Chun

Growth Estimates on Positive Solutions of the Equation $\Delta u+K u^{\frac{n+2}{n2}}=0$ in $\R^n$
We construct unbounded positive $C^2$solutions of the equation
$\Delta u + K u^{(n + 2)/(n  2)} = 0$ in $\R^n$ (equipped
with Euclidean metric $g_o$) such that $K$ is bounded between two
positive numbers in $\R^n$, the conformal metric $g=u^{4/(n2)}g_o$
is complete, and the volume growth of $g$ can be arbitrarily fast
or reasonably slow according to the constructions. By imposing natural
conditions on $u$, we obtain growth estimate on the $L^{2n/(n2)}$norm
of the solution and show that it has slow decay.
Keywords:positive solution, conformal scalar curvature equation, growth estimate Categories:35J60, 58G03 

14. CMB 2000 (vol 43 pp. 343)
15. CMB 1999 (vol 42 pp. 87)
 Kittaneh, Fuad

Some norm inequalities for operators
Let $A_i$, $B_i$ and $X_i$ $(i=1, 2, \dots, n)$ be operators on a
separable Hilbert space. It is shown that if $f$ and $g$ are
nonnegative continuous functions on $[0,\infty)$ which satisfy the
relation $f(t)g(t) =t$ for all $t$ in $[0,\infty)$, then
$$
\Biglvert \,\Bigl\sum^n_{i=1} A^*_i X_i B_i \Bigr^r \,\Bigrvert^2
\leq \Biglvert \Bigl( \sum^n_{i=1} A^*_i f (X^*_i)^2 A_i \Bigr)^r
\Bigrvert \, \Biglvert \Bigl( \sum^n_{i=1} B^*_i g (X_i)^2 B_i
\Bigr)^r \Bigrvert
$$
for every $r>0$ and for every unitarily invariant norm. This result
improves some known CauchySchwarz type inequalities. Norm
inequalities related to the arithmeticgeometric mean inequality and
the classical Heinz inequalities are also obtained.
Keywords:Unitarily invariant norm, positive operator, arithmeticgeometric mean inequality, CauchySchwarz inequality, Heinz inequality Categories:47A30, 47B10, 47B15, 47B20 
