A note on the Osserman conjecture and isotropic covariant derivative of curvature

Let be a Riemannian manifold with the Jacobi operator, which has constant eigenvalues, independent on the unit vector and the point . Osserman conjectured that these manifolds are flat or rank-one locally symmetric spaces (). It is known that for a general pseudo-Riemannian manifold, the Osserman-type conjecture is not true and 4-dimensional Kleinian Jordan-Osserman manifolds are curvature homogeneous. We show that the length of the first covariant derivative of the curvature tensor is isotropic, i.e. . For known examples of 4-dimensional Osserman manifolds of signature we check also that . By the presentation of a class of examples we show that curvature homogeneity and do not imply local homogeneity; in contrast to the situation in the Riemannian geometry, where it is unknown if the Osserman condition implies local homogeneity.

     

$P_*(\kappa)$线性互补问题的Mehrotra型预估-校正算法

Mehrotra-type predictor-corrector algorithm,as one of most efficient interior point methods,has become the backbones of most optimization packages.Salahi et al.proposed a cut strategy based algorithm for linear optimization that enjoyed polynomial complexity and maintained its efficiency in practice.We extend their algorithm to P_*（κ）linear complementarity problems.The way of choosing corrector direction for our algorithm is different from theirs. The new algorithm has been proved to have an ο(（1+4κ）（17+19κ） √（1+2κn）^3/2log[（x⁰）^Ts⁰/ε] worst case iteration complexity bound.An numerical experiment verifies the feasibility of the new algorithm.     

A mixed Hölder and Minkowski inequality

Hölder's inequality states that for any with . In the same situation we prove the following stronger chains of inequalities, where :

A similar result holds for complex valued functions with Re substituting for . We obtain these inequalities from some stronger (though slightly more involved) ones.

     

Unitarily invariant norm and $Q$-norm estimations for the Moore--Penrose inverse of multiplicative perturbations of matrices

Let $B$ be a multiplicative perturbation of $A\in\mathbb{C}^{m\times n}$ given by $B=D_1^* A D_2$, where $D_1\in\mathbb{C}^{m\times m}$ and $D_2\in\mathbb{C}^{n\times n}$ are both nonsingular. New upper bounds for $\Vert B^\dag-A^\dag\Vert_U$ and $\Vert B^\dag-A^\dag\Vert_Q$ are derived, where $A^\dag,B^\dag$ are the Moore-Penrose inverses of $A$ and $B$, and $\Vert \cdot\Vert_U,\Vert \cdot\Vert_Q$ are any unitarily invariant norm and $Q$-norm, respectively. Numerical examples are provided to illustrate the sharpness of the obtained upper bounds.     

On Schwarz type inequalities

We show Schwarz type inequalities and consider their converses. A continuous function is said to be semi-operator monotone on if is operator monotone on . Let be a bounded linear operator on a complex Hilbert space and be the polar decomposition of . Let and for . (1) If a non-zero function is semi-operator monotone on , then for , where . (2) If are semi-operator monotone on , then for . Also, we show converses of these inequalities, which imply that semi-operator monotonicity is necessary.

     

Geodesic $\gamma$-Pre-$E$-Convex Functions on Riemannian Manifolds

In this paper, we generalize geodesic $E$-convex function and define geodesic $\gamma$-pre-$E$-convex and geodesic $\gamma$-$E$-convex functions on Riemannian manifolds. The sufficient condition of equivalence class of geodesic $\gamma$-pre-$E$-convexity and geodesic $\gamma$-$E$-convexity for differentiable function on Riemannian manifolds is studied. We discuss the sufficient condition for $E$-epigraph to be geodesic $E$-convex set. At the end, we establish some optimality results with the aid of geodesic $\gamma$-pre-$E$-convex and geodesic $\gamma$-$E$-convex functions and discuss the mean value inequality for geodesic $\gamma$-pre-$E$-convex function.     

A nullity condition for complex contact metric manifolds

A nullity condition for real contact manifolds is defined by Blair, Koufogiorgos and Papantoniu. Lately, Boeckx classified such manifolds completely. In this paper, a nullity condition for complex contact manifolds is defined as follows: take a complex contact manifold whose vertical space is annihilated by the curvature. Then, apply an $\mathcal{H}$-homothetic deformation. In this way, we get a condition which is invariant under $\mathcal{H}$-homothetic deformations. A complex contact manifold satisfying this condition is called a complex (,)-space. Some curvature properties of complex (,)-spaces are studied and it is shown that, just as in the real case, the curvature tensor of a complex (,)-space is completely determined.     

Commutator length of symplectomorphisms

Each element $x$ of the commutator subgroup $[G, G]$ of a group $G$ can be represented as a product of commutators. The minimal number of factors in such a product is called the commutator length of $x$. The commutator length of $G$ is defined as the supremum of commutator lengths of elements of $[G, G]$. We show that for certain closed symplectic manifolds $(M,\omega)$, including complex projective spaces and Grassmannians, the universal cover $\widetilde{\hbox{\rm Ham}\, (M,\omega)$ of the group of Hamiltonian symplectomorphisms of $(M,\omega)$ has infinite commutator length. In particular, we present explicit examples of elements in $\widetilde{\hbox{\rm Ham}\, (M,\omega)$ that have arbitrarily large commutator length -- the estimate on their commutator length depends on the multiplicative structure of the quantum cohomology of $(M,\omega)$. By a different method we also show that in the case $c_1 (M) = 0$ the group $\widetilde{\hbox{\rm Ham}\, (M,\omega)$ and the universal cover ${\widetilde{\Symp}}_0\, (M,\omega)$ of the identity component of the group of symplectomorphisms of $(M,\omega)$ have infinite commutator length.     

Rearrangement of Hardy-Littlewood maximal functions in Lorentz spaces

For the classical Hardy-Littlewood maximal function , a well known and important estimate due to Herz and Stein gives the equivalence . In the present note, we study the validity of analogous estimates for maximal operators of the form

where denotes the Lorentz space -norm.

     

Function representation of a noncommutative uniform algebra

We construct a Gelfand type representation of a real noncommutative Banach algebra satisfying , for all

     

On Meinardus' examples for the conjugate gradient method

The conjugate gradient (CG) method is widely used to solve a positive definite linear system of order . It is well known that the relative residual of the th approximate solution by CG (with the initial approximation ) is bounded above by

   with

where is 's spectral condition number. In 1963, Meinardus (Numer. Math., 5 (1963), pp. 14-23) gave an example to achieve this bound for but without saying anything about all other . This very example can be used to show that the bound is sharp for any given by constructing examples to attain the bound, but such examples depend on and for them the th residual is exactly zero. Therefore it would be interesting to know if there is any example on which the CG relative residuals are comparable to the bound for all . There are two contributions in this paper:

1. A closed formula for the CG residuals for all on Meinardus' example is obtained, and in particular it implies that the bound is always within a factor of of the actual residuals;
2. A complete characterization of extreme positive linear systems for which the th CG residual achieves the bound is also presented.

     

Interpolation in self-adjoint settings

We study the operator equation , where the operators and are given and the operator is required to lie in some von Neumann algebra. We derive a necessary and sufficient condition for the existence of a solution . The condition is that there must exist a constant so that, for all finite collections of operators in the commutant, and all collections of vectors , we have We also study the equality , in connection with solving the equation where the operator is required to lie in some CSL algebra.

     

Generalizations of Gonçalves' inequality

Let be a polynomial with complex coefficients and roots , ..., , let denote its norm over the unit circle, and let denote Mahler's measure of . Gonçalves' inequality asserts that

We prove that

for , where is an explicit constant, and that

for . We also establish additional lower bounds on the norms of a polynomial in terms of its coefficients.

     

Sharp estimates for the identity minus Hardy operator on the cone of decreasing functions

It is shown that if we restrict the identity minus Hardy operator on the cone of nonnegative decreasing functions in , then we have the sharp estimate

for In other words,

for each and each integer .

It is also shown, via a connection between the operator and Laguerre functions, that

for all .

     

The growth theorem of convex mappings on the unit ball in

Let be an arbitrary norm on . Let be a normalized biholomorphic convex mapping on the unit ball in with respect to the norm . We will give an upper bound of the growth of .

     

笛卡尔乘积图的圈点连通度

图G的圈点连通度,记为κ_c(G),是所有圈点割中最小的数目,其中每个圈点割S满足G-S不连通且至少它的两个分支含圈.这篇文章中给出了两个连通图的笛卡尔乘积的圈点连通度:(1)如果G_1≌K_m且G_2≌K_n,则κ_c(G_1×G_2)=min{3m+n-6,m+3n-6},其中m+n≥8,m≥n+2,或n≥m+2,且κ_c(G_1×G_2)=2m+2n-8,其中m+n≥8,m=n,或n=m+1,或m=n+11;(2)如果G_1≌K_m(m≥3)且G_2■K_n,则min{3m+κ(G_2)-4,m+3κ(G_2)-3,2m+2κ(G_2)-4}≤κ_c(G_1×G_2)≤mκ(G2);(3)如果G_1■K_m,K_(1,m-1)且G_2■K_n,K_(1,n-1),其中m≥4,n≥4,则min{3κ(G_1)+κ(G_2)-1,κ(G_1)+3κ(G_2)-1,2_κ(G_1)+2_κ(G_2)-2}≤κ_c(G_1×G_2)≤min{mκ(G_2),nκ(G_1),2m+2n-8}.     

Effect of a random real on\kappa \to \left( {\kappa ,{\text{ }}\left( {_{\omega _1 }^\alpha } \right)} \right)^2

It is consistent that $\kappa \to (\kappa ,{\text{ }}\left( {_{\omega _1 }^\alpha } \right))^2 $ holds in the random extension.     

Asymptotic behavior of nonexpansive sequences and mean points

Let be a real Banach space with norm and let be a nonexpansive sequence in (i.e., for all ). Let . We deal with the mean point of concerning a Banach limit. We show that if is reflexive and , then and there exists a unique point with such that . This result is applied to obtain the weak and strong convergence of .

     

Further extension of the Heinz-Kato-Furuta inequality

Let be a bounded operator on a Hilbert space and positive definite operators. Kato has shown that if and for all , then where are operator monotone functions defined on such that . Furuta has shown that Let be any continuous operator monotone functions, and set for We will show that is well defined and Moreover, we will extend this result for unbounded closed operators densely defined on

     

Weak norms of random sums

Let denote a sequence of measurable functions on , and let denote the weak norm. It is shown that

where is a sequence of independent random variables taking on values and with equal probability. Moreover, it is shown that

The paper concludes by providing an example indicating that, if , then the estimate

is the best possible.