Note on the practical significance of the Drazin inverse

The solution of the differential system Bx = Ax + f where A and B are n x n matrices, and A - Lambda B is not a singular pencil, may be expressed in terms of the Drazin inverse. It is shown that there is a simple reduced form for the pencil A - Lambda B which is adequate for the determination of the general solution and that although the Drazin inverse could be determined efficiently from this reduced form it is inadvisable to do so

Amplification of High Harmonics Using Weak Perturbative High Frequency Radiation

The mechanism underlying the substantial amplification of the high-order harmonics q \pm 2K (K integer) upon the addition of a weak seed XUV field of harmonic frequency q\omega to a strong IR field of frequency \omega is analyzed in the framework of the quantum-mechanical Floquet formalism and the semiclassical re-collision model. According to the Floquet analysis, the high-frequency field induces transitions between several Floquet states and leads to the appearance of new dipole cross terms. The semiclassical re-collision model suggests that the origin of the enhancement lies in the time-dependent modulation of the ground electronic state induced by the XUV field.Comment: 8 pages, 2 figure

A quantum Peierls-Nabarro barrier

Kink dynamics in spatially discrete nonlinear Klein-Gordon systems is considered. For special choices of the substrate potential, such systems support continuous translation orbits of static kinks with no (classical) Peierls-Nabarro barrier. It is shown that these kinks experience, nevertheless, a lattice-periodic confining potential, due to purely quantum effects anaolgous to the Casimir effect of quantum field theory. The resulting ``quantum Peierls-Nabarro potential'' may be calculated in the weak coupling approximation by a simple and computationally cheap numerical algorithm, which is applied, for purposes of illustration, to a certain two-parameter family of substrates.Comment: 13 pages LaTeX, 7 figure

Tridiagonal realization of the anti-symmetric Gaussian β\beta-ensemble

The Householder reduction of a member of the anti-symmetric Gaussian unitary ensemble gives an anti-symmetric tridiagonal matrix with all independent elements. The random variables permit the introduction of a positive parameter $\beta$, and the eigenvalue probability density function of the corresponding random matrices can be computed explicitly, as can the distribution of $\{q_i\}$, the first components of the eigenvectors. Three proofs are given. One involves an inductive construction based on bordering of a family of random matrices which are shown to have the same distributions as the anti-symmetric tridiagonal matrices. This proof uses the Dixon-Anderson integral from Selberg integral theory. A second proof involves the explicit computation of the Jacobian for the change of variables between real anti-symmetric tridiagonal matrices, its eigenvalues and $\{q_i\}$. The third proof maps matrices from the anti-symmetric Gaussian $\beta$-ensemble to those realizing particular examples of the Laguerre $\beta$-ensemble. In addition to these proofs, we note some simple properties of the shooting eigenvector and associated Pr\"ufer phases of the random matrices.Comment: 22 pages; replaced with a new version containing orthogonal transformation proof for both cases (Method III

In Defense of American Criminal Justice

The American criminal justice system is on trial. A chorus of commentators-often but not exclusively in the legal academy-has leveled a sharp indictment of criminal process in our country. The indictment charges that large flaws infect nearly every stage of the adjudicatory process. And the prescriptions are equally far-reaching, with calls for abolition of many current practices and an overhaul of the entire system. What is more, the critics issue their condemnations essentially as givens, often claiming that all reasonable people could not help but agree that fair treatment of the accused has been fatally compromised. For these critics, We live in a time of sharply decreasing faith in the criminal justice system. \u27 As a judge with faith in that system, I am dismayed by the relentless insistence that we have it all wrong. Of course the system, like all human institutions, has its share of flaws. But the attacks have overshadowed what is good about the system and crowded out more measured calls for reform. The critics claim that major aspects of American criminal justice work to the detriment of defendants, when actually the reverse is often true. It is time for a more balanced view of our criminal process, which in fact gets a lot of things right. A brief word as to the scope of this Essay. I have focused mainly on the adjudicatory process and on the criminal trial. I have not sought to explore police investigatory procedure on the one hand, or issues of detention and incarceration on the other, except insofar as they bear on the adjudicatory process in some way. They are vast topics in themselves, and the terrain I have covered is large enough. My own reaction to the critics is one of gratitude for their contributions but dismay that they have allowed the pursuit of perfection in criminal justice to become the enemy of the good. Much about American criminal justice is indeed good. The system provides considerable protections for the accused and sets proper limits on the brutality and deceit that human beings can inflict upon each other. Simply put, in calling for an overhaul of our criminal law and procedure, the critics have failed to appreciate the careful balance our criminal justice system strikes between competing rights and values. They have failed to respect the benefits of the system\u27s front-end features-namely, early process and early resolution. Moreover, they have sold short the democratic virtues of our system. The sensible tradeoffs reflected in American criminal justice are worthy of respect, and the system\u27s democratic tilt is deserving of praise. The critics have extended neither. Ultimately, the often harsh tone of their indictment has done an injustice to the system of criminal justice itself

Crystallization and preliminary crystallographic analysis of the DNA gyrase B protein from B-stearothermophilus

DNA gyrase B (GyrB) from B. stearothermophilus has been crystallized in the presence of the non-hydrolyzable ATP analogue, 5'-adenylpl-beta-gamma-imidodiphosphate (ADPNP), by the dialysis method. A complete native data set to 3.7 Angstrom has been collected from crystals which belonged to the cubic space group I23 with unit-cell dimension a = 250.6 Angstrom. Self-rotation function analysis indicates the position of a molecular twofold axis. Low-resolution data sets of a thimerosal and a selenomethionine derivative have also been analysed. The heavy-atom positions are consistent with one dimer in the asymmetric unit

The path-coalescence transition and its applications

We analyse the motion of a system of particles subjected a random force fluctuating in both space and time, and experiencing viscous damping. When the damping exceeds a certain threshold, the system undergoes a phase transition: the particle trajectories coalesce. We analyse this transition by mapping it to a Kramers problem which we solve exactly. In the limit of weak random force we characterise the dynamics by computing the rate at which caustics are crossed, and the statistics of the particle density in the coalescing phase. Last but not least we describe possible realisations of the effect, ranging from trajectories of raindrops on glass surfaces to animal migration patterns.Comment: 4 pages, 3 figures; revised version, as publishe

Gegenbauer-solvable quantum chain model

In an innovative inverse-problem construction the measured, experimental energies $E_1$, $E_2$, ...$E_N$ of a quantum bound-state system are assumed fitted by an N-plet of zeros of a classical orthogonal polynomial $f_N(E)$. We reconstruct the underlying Hamiltonian $H$ (in the most elementary nearest-neighbor-interaction form) and the underlying Hilbert space ${\cal H}$ of states (the rich menu of non-equivalent inner products is offered). The Gegenbauer's ultraspherical polynomials $f_n(x)=C_n^\alpha(x)$ are chosen for the detailed illustration of technicalities.Comment: 29 pp., 1 fi