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1. CMB Online first

Li, Chi-Kwong; Tsai, Ming-Cheng
 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 $\|P\| \le |\sqrt{a} - \sqrt{b}|\sqrt{(1-a)(1-b)}.$ 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 theoremCategories:47A60, 47A68, 47A63

2. CMB 2014 (vol 57 pp. 431)

Tagami, Keiji
 The Rasmussen Invariant, Four-genus and Three-genus 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, four-genus, Rasmussen invariantCategories: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 non-negative 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 semigroupsCategories: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 Sturm-Liouville problem, we obtain new conditions for the boundary value problem to have a positive solution. Our major tools are the Krein-Rutman 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, Sturm-Liouville problemCategories:34B18, 34B24

5. CMB 2011 (vol 55 pp. 214)

Wang, Da-Bin
 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 equationCategories:39A10, 34B15

6. CMB 2011 (vol 54 pp. 544)

Strungaru, Nicolae
 Positive Definite Measures with Discrete Fourier Transform and Pure Point Diffraction In this paper we characterize the positive definite measures with discrete Fourier transform. As an application we provide a characterization of pure point diffraction in locally compact Abelian groups. Keywords:pure point diffraction, positive definite measure, Fourier transform of measuresCategory:43A25

7. CMB 2010 (vol 53 pp. 256)

Fang, Xiaochun; Wang, Lin
 Equivalent Definitions of Infinite Positive Elements in Simple C*-algebras We prove the equivalence of three definitions given by different comparison relations for infiniteness of positive elements in simple $C^*$-algebras. Keywords:Infinite positive element, Comparison relationCategory:46L99

8. CMB 2008 (vol 51 pp. 386)

Lan, K. Q.; Yang, G. C.
 Positive Solutions of the Falkner--Skan Equation Arising in the Boundary Layer Theory The well-known Falkner--Skan equation is one of the most important equations in laminar boundary layer theory and is used to describe the steady two-dimensional 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 Falkner--Skan equation. The equivalence result provides new techniques to study properties and existence of solutions of the Falkner--Skan equation. Keywords:Falkner-Skan equation, boundary layer problems, singular integral equation, positive solutionsCategories: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 TheoremCategories: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 semi-definite. We also study some related geometric and topological properties of the moduli space. Keywords:hermitian quadratic form, positive definite, positive semi-definiteCategory: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 surfacesCategories:14L05, 14J28

12. CMB 2003 (vol 46 pp. 216)

Li, Chi-Kwong; 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 rangeCategories:47B15, 47B49

13. CMB 2001 (vol 44 pp. 210)

Leung, Man Chun
 Growth Estimates on Positive Solutions of the Equation $\Delta u+K u^{\frac{n+2}{n-2}}=0$ 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 $g=u^{4/(n-2)}g_o$ 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 $L^{2n/(n-2)}$-norm of the solution and show that it has slow decay. Keywords:positive solution, conformal scalar curvature equation, growth estimateCategories:35J60, 58G03

14. CMB 2000 (vol 43 pp. 343)

Hughes, Bruce; Taylor, Larry; Williams, Bruce
 Controlled Homeomorphisms Over Nonpositively Curved Manifolds We obtain a homotopy splitting of the forget control map for controlled homeomorphisms over closed manifolds of nonpositive curvature. Keywords:controlled topology, controlled homeomorphism, nonpositive curvature, Novikov conjecturesCategories:57N15, 53C20, 55R65, 57N37

15. CMB 1999 (vol 42 pp. 87)

 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 Cauchy-Schwarz type inequalities. Norm inequalities related to the arithmetic-geometric mean inequality and the classical Heinz inequalities are also obtained. Keywords:Unitarily invariant norm, positive operator, arithmetic-geometric mean inequality, Cauchy-Schwarz inequality, Heinz inequalityCategories:47A30, 47B10, 47B15, 47B20