Rotationally coherent Lagrangian vortices are formed by tubes of deforming fluid elements that complete equal bulk material rotation relative to the mean rotation of the deforming fluid volume. We show that the initial positions of such tubes coincide with tubular level surfaces of the Lagrangian-averaged vorticity deviation (LAVD), the trajectory integral of the normed difference of the vorticity from its spatial mean. The LAVD-based vortices are objective, i.e. remain unchanged under time-dependent rotations and translations of the coordinate frame. In the limit of vanishing Rossby numbers in geostrophic flows, cyclonic LAVD vortex centres are precisely the observed attractors for light particles. A similar result holds for heavy particles in anticyclonic LAVD vortices. We also establish a relationship between rotationally coherent Lagrangian vortices and their instantaneous Eulerian counterparts. The latter are formed by tubular surfaces of equal material rotation rate, objectively measured by the instantaneous vorticity deviation (IVD). We illustrate the use of the LAVD and the IVD to detect rotationally coherent Lagrangian and Eulerian vortices objectively in several two- and three-dimensional flows.

Defining coherent vortices objectively from the vorticity

Matrix power means and the Karcher mean

We give an inequality relating the operator norm of T and the numerical radii of T and its Aluthge transform. It is a more precise estimate of the numerical radius than Kittaneh’s result [Studia Math. 158 (2003)]. Then we obtain an equivalent condition for the numerical radius to be equal to half the operator norm.

/pdf/on-upper-and-lower-bounds-of-the-numerical-radius-and-an-3wk3m2nde4.pdf

On upper and lower bounds of the numerical radius and an equality condition

In this paper we present a survey of various algorithms for computing matrix geometric means and derive new second-order optimization algorithms to compute the Karcher mean. These new algorithms are constructed using the standard definition of the Riemannian Hessian. The survey includes the ALM list of desired properties for a geometric mean, the analytical expression for the mean of two matrices, algorithms based on the centroid computation in Euclidean (flat) space, and Riemannian optimization techniques to compute the Karcher mean (preceded by a short introduction into differential geometry). A change of metric is considered in the optimization techniques to reduce the complexity of the structures used in these algorithms. Numerical experiments are presented to compare the existing and the newly developed algorithms. We conclude that currently first-order algorithms are best suited for this optimization problem as the size and/or number of the matrices increase. Copyright © 2012, Kent State University.

http://sma.epfl.ch/~anchpcommon/publications/pd_matrix_geometric_mean.pdf

A survey and comparison of contemporary algorithms for computing the matrix geometric mean

We prove several numerical radius inequalities for certain 2 × 2 operator matrices. Among other inequalities, it is shown that if X, Y, Z, and W are bounded linear operators on a Hilbert space, then 

$$w\left( \left[\begin{array}{cc} X & Y \\ Z & W \end{array} \right] \right) \geq \max \left(w(X),w(W),\frac{w(Y+Z)}{2},\frac{w(Y-Z)}{2}\right) $$

and 

$$w\left( \left[\begin{array}{cc}X & Y \\ Z & W\end{array} \right] \right) \leq \max \left( w(X), w(W)\right)+\frac{w(Y+Z)+w(Y-Z)}{2}. $$

As an application of a special case of the second inequality, it is shown that 

$$\frac{\left\Vert X\right\Vert }{2}+\frac{\left\vert \left\Vert\operatorname{Re}{X}\right\Vert -\frac{\left\Vert X\right\Vert}{2} \right\vert }{4}+\frac{ \left\vert \left\Vert \operatorname{Im}{X} \right\Vert -\frac{\left\Vert X\right\Vert}{2} \right\vert }{4} \leq w(X), $$

which is a considerable improvement of the classical inequality ${\frac{ \left\Vert X\right\Vert }{2}\leq w(X)}$ . Here w(·) and || · || are the numerical radius and the usual operator norm, respectively.

Numerical Radius Inequalities for Certain 2 × 2 Operator Matrices

For an operator T E B(H), the Aluthge transformation of T is defined by T = |T| 2 U|T| 1/2. And also for a natural number n, the n-th Aluthge transformation of T is defined by Tn = (T n-1 ) and T 1 = T. In this paper, we shall show lim ∥T n ∥ = r(T), n→oo where r(T) is the spectral radius.

/pdf/an-expression-of-spectral-radius-via-aluthge-transformation-39indnfuz9.pdf

An expression of spectral radius via Aluthge transformation

The iterated Aluthge transforms of a 2-by-2 matrix converge

In this paper, we shall give an operator transform ${\hat T}$ from class A to the class of hyponormal operators. Then we shall show that $\sigma (T) = \sigma (\hat T)$ and $\sigma_a (T) = \sigma_a (\hat T)$ in case T belongs to class A. Next, as an application of ${\hat T},$ we will show that every class A operator has SVEP and property (β).

An Operator Transform from Class A to the Class of Hyponormal Operators and its Application

Let T = U|T| be the polar decomposition of a bounded linear operator T on a Hilbert space. The transformation e T = |T| 1 2U|T| 1 2 is called the Aluthge transformation and e Tn means the n-th Aluthge transformation. In this paper, firstly, we show that e

/pdf/on-the-polar-decomposition-of-the-aluthge-transformation-and-4kk9b51zll.pdf

Takeaki Yamazaki

Papers

On upper and lower bounds of the numerical radius and an equality condition

An expression of spectral radius via Aluthge transformation

The iterated Aluthge transforms of a 2-by-2 matrix converge

An Operator Transform from Class A to the Class of Hyponormal Operators and its Application

On the polar decomposition of the aluthge transformation and related results