The Jordan forms of AB and BA∗

The relationship between the Jordan forms of the matrix products AB and BA for some given A and B was first described by Harley Flanders in 1951. Their non-zero eigenvalues and non-singular Jordan structures are the same, but their singular Jordan block sizes can differ by 1. We present an elementary proof that owes its simplicity to a novel use of the Weyr characteristic

Notes on Regularized Least Squares

This is a collection of information about regularized least squares (RLS). The facts here are not new results, but we have not seen them usefully collected together before. A key goal of this work is to demonstrate that with RLS, we get certain things for free: if we can solve a single supervised RLS problem, we can search for a good regularization parameter lambda at essentially no additional cost.The discussion in this paper applies to dense regularized least squares, where we work with matrix factorizations of the data or kernel matrix. It is also possible to work with iterative methods such as conjugate gradient, and this is frequently the method of choice for large data sets in high dimensions with very few nonzero dimensions per point, such as text classifciation tasks. The results discussed here do not apply to iterative methods, which have different design tradeoffs.We present the results in greater detail than strictly necessary, erring on the side of showing our work. We hope that this will be useful to people trying to learn more about linear algebra manipulations in the machine learning context

Nonlinear eigenvalue problems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1998.Includes bibliographical references (p. 211-217).by Ross Adams Lippert.Ph.D

Multiscale Computation with Interpolating Wavelets

Multiresolution analyses based upon interpolets, interpolating scaling functions introduced by Deslauriers and Dubuc, are particularly well-suited to physical applications because they allow exact recovery of the multiresolution representation of a function from its sample values on a finite set of points in space. We present a detailed study of the application of wavelet concepts to physical problems expressed in such bases. The manuscript describes algorithms for the associated transforms which, for properly constructed grids of variable resolution, compute correctly without having to introduce extra grid points. We demonstrate that for the application of local homogeneous operators in such bases, the non-standard multiply of Beylkin, Coifman and Rokhlin also proceeds exactly for inhomogeneous grids of appropriate form. To obtain less stringent conditions on the grids, we generalize the non-standard multiply so that communication may proceed between non-adjacent levels. The manuscript concludes with timing comparisons against naive algorithms and an illustration of the scale-independence of the convergence rate of the conjugate gradient solution of Poisson's equation using a simple preconditioning, suggesting that this approach leads to an O(n) solution of this equation.Comment: 33 pages, figures available at http://laisla.mit.edu/muchomas/Papers/nonstand-figs.ps . Updated: (1) figures file (figs.ps) now appear with the posting on the server; (2) references got lost in the last submissio

Looking to Score: The Dissociation of Goal Influence on Eye Movement and Meta-Attentional Allocation in a Complex Dynamic Natural Scene

Several studies have reported that task instructions influence eye-movement behavior during static image observation. In contrast, during dynamic scene observation we show that while the specificity of the goal of a task influences observers’ beliefs about where they look, the goal does not in turn influence eye-movement patterns. In our study observers watched short video clips of a single tennis match and were asked to make subjective judgments about the allocation of visual attention to the items presented in the clip (e.g., ball, players, court lines, and umpire). However, before attending to the clips, observers were either told to simply watch clips (non-specific goal), or they were told to watch the clips with a view to judging which of the two tennis players was awarded the point (specific goal). The results of subjective reports suggest that observers believed that they allocated their attention more to goal-related items (e.g. court lines) if they performed the goal-specific task. However, we did not find the effect of goal specificity on major eye-movement parameters (i.e., saccadic amplitudes, inter-saccadic intervals, and gaze coherence). We conclude that the specificity of a task goal can alter observer’s beliefs about their attention allocation strategy, but such task-driven meta-attentional modulation does not necessarily correlate with eye-movement behavior

Non-Linear Population Firing Rates and Voltage Sensitive Dye Signals in Visual Areas 17 and 18 to Short Duration Stimuli

Visual stimuli of short duration seem to persist longer after the stimulus offset than stimuli of longer duration. This visual persistence must have a physiological explanation. In ferrets exposed to stimuli of different durations we measured the relative changes in the membrane potentials with a voltage sensitive dye and the action potentials of populations of neurons in the upper layers of areas 17 and 18. For durations less than 100 ms, the timing and amplitude of the firing and membrane potentials showed several non-linear effects. The ON response became truncated, the OFF response progressively reduced, and the timing of the OFF responses progressively delayed the shorter the stimulus duration. The offset of the stimulus elicited a sudden and strong negativity in the time derivative of the dye signal. All these non-linearities could be explained by the stimulus offset inducing a sudden inhibition in layers II–III as indicated by the strongly negative time derivative of the dye signal. Despite the non-linear behavior of the layer II–III neurons the sum of the action potentials, integrated from the peak of the ON response to the peak of the OFF response, was almost linearly related to the stimulus duration

Refinement of 1p36 Alterations Not Involving PRDM16 in Myeloid and Lymphoid Malignancies

Fluorescence in situ hybridization was performed to characterize 81 cases of myeloid and lymphoid malignancies with cytogenetic 1p36 alterations not affecting the PRDM16 locus. In total, three subgroups were identified: balanced translocations (N = 27) and telomeric rearrangements (N = 15), both mainly observed in myeloid disorders; and unbalanced non-telomeric rearrangements (N = 39), mainly observed in lymphoid proliferations and frequently associated with a highly complex karyotype. The 1p36 rearrangement was isolated in 12 cases, mainly myeloid disorders. The breakpoints on 1p36 were more widely distributed than previously reported, but with identifiable rare breakpoint cluster regions, such as the TP73 locus. We also found novel partner loci on 1p36 for the known multi-partner genes HMGA2 and RUNX1. We precised the common terminal 1p36 deletion, which has been suggested to have an adverse prognosis, in B-cell lymphomas [follicular lymphomas and diffuse large B-cell lymphomas with t(14;18)(q32;q21) as well as follicular lymphomas without t(14;18)]. Intrachromosomal telomeric repetitive sequences were detected in at least half the cases of telomeric rearrangements. It is unclear how the latter rearrangements occurred and whether they represent oncogenic events or result from chromosomal instability during oncogenesis

Multi-messenger observations of a binary neutron star merger

On 2017 August 17 a binary neutron star coalescence candidate (later designated GW170817) with merger time 12:41:04 UTC was observed through gravitational waves by the Advanced LIGO and Advanced Virgo detectors. The Fermi Gamma-ray Burst Monitor independently detected a gamma-ray burst (GRB 170817A) with a time delay of ~1.7 s with respect to the merger time. From the gravitational-wave signal, the source was initially localized to a sky region of 31 deg2 at a luminosity distance of 40+8-8 Mpc and with component masses consistent with neutron stars. The component masses were later measured to be in the range 0.86 to 2.26 Mo. An extensive observing campaign was launched across the electromagnetic spectrum leading to the discovery of a bright optical transient (SSS17a, now with the IAU identification of AT 2017gfo) in NGC 4993 (at ~40 Mpc) less than 11 hours after the merger by the One- Meter, Two Hemisphere (1M2H) team using the 1 m Swope Telescope. The optical transient was independently detected by multiple teams within an hour. Subsequent observations targeted the object and its environment. Early ultraviolet observations revealed a blue transient that faded within 48 hours. Optical and infrared observations showed a redward evolution over ~10 days. Following early non-detections, X-ray and radio emission were discovered at the transient’s position ~9 and ~16 days, respectively, after the merger. Both the X-ray and radio emission likely arise from a physical process that is distinct from the one that generates the UV/optical/near-infrared emission. No ultra-high-energy gamma-rays and no neutrino candidates consistent with the source were found in follow-up searches. These observations support the hypothesis that GW170817 was produced by the merger of two neutron stars in NGC4993 followed by a short gamma-ray burst (GRB 170817A) and a kilonova/macronova powered by the radioactive decay of r-process nuclei synthesized in the ejecta