On Chebyshev polynomials of matrices

The mth Chebyshev polynomial of a square matrix A is the monic polynomial that minimizes the matrix 2-norm of $p(A)$ over all monic polynomials $p(z)$ of degree m. This polynomial is uniquely defined if m is less than the degree of the minimal polynomial of A. We study general properties of Chebyshev polynomials of matrices, which in some cases turn out to be generalizations of well-known properties of Chebyshev polynomials of compact sets in the complex plane. We also derive explicit formulas of the Chebyshev polynomials of certain classes of matrices, and explore the relation between Chebyshev polynomials of one of these matrix classes and Chebyshev polynomials of lemniscatic regions in the complex plane

The Faber–Manteuffel theorem for linear operators

A short recurrence for orthogonalizing Krylov subspace bases for a matrix A exists if and only if the adjoint of A is a low-degree polynomial in A (i.e., A is normal of low degree). In the area of iterative methods, this result is known as the Faber–Manteuffel theorem [V. Faber and T. Manteuffel, SIAM J. Numer. Anal., 21 (1984), pp. 352–362]. Motivated by the description by J. Liesen and Z. Strakoš, we formulate here this theorem in terms of linear operators on finite dimensional Hilbert spaces and give two new proofs of the necessity part. We have chosen the linear operator rather than the matrix formulation because we found that a matrix-free proof is less technical. Of course, the linear operator result contains the Faber–Manteuffel theorem for matrices

Properties of worst-case GMRES

In the convergence analysis of the GMRES method for a given matrix $A$, one quantity of interest is the largest possible residual norm that can be attained, at a given iteration step $k$, over all unit norm initial vectors. This quantity is called the worst-case GMRES residual norm for $A$ and $k$. We show that the worst-case behavior of GMRES for the matrices $A$ and $A^T$ is the same, and we analyze properties of initial vectors for which the worst-case residual norm is attained. In particular, we prove that such vectors satisfy a certain “cross equality.” We show that the worst-case GMRES polynomial may not be uniquely determined, and we consider the relation between the worst-case and the ideal GMRES approximations, giving new examples in which the inequality between the two quantities is strict at all iteration steps $k\geq 3$. Finally, we give a complete characterization of how the values of the approximation problems change in the context of worst-case and ideal GMRES for a real matrix, when one considers complex (rather than real) polynomials and initial vectors

MINI REVIEW: ASSESSING TECHNICAL SKILLS IN YOUTH ATHLETES USING SPORTS BIOMECHANICAL METHODS

Since technical skills are suggested to play a crucial role in talent identification and development (TID) programs, sports biomechanical assessment methods could gain in importance within this field. This systematic mini review provides a brief overview of the biomechanical approaches used so far to assess technical skills and the respective findings in the context of TID. Our results show that few studies have used biomechanical approaches to identify or develop talented young athletes but those doing so found promising results. On the basis of those studies and given the advancements in technologies, we discuss possible obstacles and the potential of biomechanical assessment methods for motion and technique analysis in the context of talent research

Stereoselective synthesis of γ-hydroxynorvaline through combination of organo- and biocatalysis

An efficient route for the synthesis of all four diastereomers of PMP-protected α-amino-γ-butyrolacton to access γ-hydroxynorvaline was established. The asymmetric key steps comprise an organocatalytic Mannich reaction and an enzymatic ketone reduction. Three reaction steps could be integrated in a one-pot process, using 2-PrOH both as solvent and as reducing agent. The sequential construction of stereogenic centres gave access to each of the four stereoisomers in high yield and with excellent stereocontrol

Cardiac-respiratory self-gated cine ultra-short echo time (UTE) cardiovascular magnetic resonance for assessment of functional cardiac parameters at high magnetic fields

Background: To overcome flow and electrocardiogram-trigger artifacts in cardiovascular magnetic resonance (CMR), we have implemented a cardiac and respiratory self-gated cine ultra-short echo time (UTE) sequence. We have assessed its performance in healthy mice by comparing the results with those obtained with a self-gated cine fast low angle shot (FLASH) sequence and with echocardiography. Methods: 2D self-gated cine UTE (TE/TR = 314 μs/6.2 ms, resolution: 129 × 129 μm, scan time per slice: 5 min 5 sec) and self-gated cine FLASH (TE/TR = 3 ms/6.2 ms, resolution: 129 × 129 μm, scan time per slice: 4 min 49 sec) images were acquired at 9.4 T. Volume of the left and right ventricular (LV, RV) myocardium as well as the end-diastolic and -systolic volume was segmented manually in MR images and myocardial mass, stroke volume (SV), ejection fraction (EF) and cardiac output (CO) were determined. Statistical differences were analyzed by using Student t test and Bland-Altman analyses. Results: Self-gated cine UTE provided high quality images with high contrast-to-noise ratio (CNR) also for the RV myocardium (CNRblood-myocardium = 25.5 ± 7.8). Compared to cine FLASH, susceptibility, motion, and flow artifacts were considerably reduced due to the short TE of 314 μs. The aortic valve was clearly discernible over the entire cardiac cycle. Myocardial mass, SV, EF and CO determined by self-gated UTE were identical to the values measured with self-gated FLASH and showed good agreement to the results obtained by echocardiography. Conclusions: Self-gated UTE allows for robust measurement of cardiac parameters of diagnostic interest. Image quality is superior to self-gated FLASH, rendering the method a powerful alternative for the assessment of cardiac function at high magnetic fields.</p