Pattern Reduction in Paper Cutting

A large part of the paper industry involves supplying customers with reels of specified width in specifed quantities. These 'customer reels' must be cut from a set of wider 'jumbo reels', in as economical a way as possible. The first priority is to minimize the waste, i.e. to satisfy the customer demands using as few jumbo reels as possible. This is an example of the one-dimensional cutting stock problem, which has an extensive literature. Greycon have developed cutting stock algorithms which they include in their software packages. Greycon's initial presentation to the Study Group posed several questions, which are listed below, along with (partial) answers arising from the work described in this report. (1) Given a minimum-waste solution, what is the minimum number of patterns required? It is shown in Section 2 that even when all the patterns appearing in minimum-waste solutions are known, determining the minimum number of patterns may be hard. It seems unlikely that one can guarantee to find the minimum number of patterns for large classes of realistic problems with only a few seconds on a PC available. (2) Given an n → n-1 algorithm, will it find an optimal solution to the minimum- pattern problem? There are problems for which n → n - 1 reductions are not possible although a more dramatic reduction is. (3) Is there an efficient n → n-1 algorithm? In light of Question 2, Question 3 should perhaps be rephrased as 'Is there an efficient algorithm to reduce n patterns?' However, if an algorithm guaranteed to find some reduction whenever one existed then it could be applied iteratively to minimize the number of patterns, and we have seen this cannot be done easily. (4) Are there efficient 5 → 4 and 4 → 3 algorithms? (5) Is it worthwhile seeking alternatives to greedy heuristics? In response to Questions 4 and 5, we point to the algorithm described in the report, or variants of it. Such approaches seem capable of catching many higher reductions. (6) Is there a way to find solutions with the smallest possible number of single patterns? The Study Group did not investigate methods tailored specifically to this task, but the algorithm proposed here seems to do reasonably well. It will not increase the number of singleton patterns under any circumstances, and when the number of singletons is high there will be many possible moves that tend to eliminate them. (7) Can a solution be found which reduces the number of knife changes? The algorithm will help to reduce the number of necessary knife changes because it works by bringing patterns closer together, even if this does not proceed fully to a pattern reduction. If two patterns are equal across some of the customer widths, the knives for these reels need not be changed when moving from one to the other

Sub-milliarcsecond Imaging of Quasars and AGN

We have used the VLBA at 15 GHz to image the structure of 132 strong compact AGN and quasars with a resolution better than one milliarcsecond and a dynamic range typically exceeding 1000 to 1. These observations were made as part of a program to investigate the sub-parsec structure of quasars and AGN and to study the changes in their structure with time. Many of the sources included in our study, particularly those located south of +35 degrees, have not been previously imaged with milliarcsecond resolution. Each of the sources has been observed at multiple epochs. In this paper we show images of each of the 132 sources which we have observed. For each source we present data at the epoch which had the best quality data. The milliarcsecond jets generally appear one-sided but two-sided structure is often found in lower luminosity radio galaxies and in high luminosity quasars with gigahertz peaked spectra. Usually the structure is unresolved along the direction perpendicular to the jet, but a few sources have broad plumes. In some low luminosity radio galaxies, the structure appears more symmetric at 2 cm than at long wavelengths. The apparent long wavelength symmetry in these sources is probably due to absorption by intervening material. A few sources contain only a single component with any secondary feature at least a thousand times weaker. We find no obvious correlation of radio morphology and the detection of gamma-ray emission by EGRET.Comment: 19 pages, 3 tables, 3 figures. Figure 2 (132 contour diagrams) is long and is omitted here. Figure 2 may be viewed at http://www.cv.nrao.edu/2cmsurvey/ In press, Astronomical Journal, April 199

Molecular Gas in the Powerful Radio Nucleus of the Ultraluminous Infrared Galaxy PKS 1345+12

Millimeter CO(1-0) interferometry and high resolution, Hubble Space Telescope (HST) 1.1, 1.6, and 2.2 micron imaging of the radio compact galaxy PKS 1345+12 are presented. With an infrared luminosity of 2x10^{12} L_sun, PKS 1345+12 is a prime candidate for studying the link between the ultraluminous infrared galaxy phenomenon and radio galaxies. These new observations probe the molecular gas distribution and obscured nuclear regions of PKS 1345+12 and provide morphological support for the idea that the radio activity in powerful radio galaxies is triggered by the merger of gas rich galaxies. Two nuclei separated by 2" (4.0 kpc) are observed in the near-infrared; the extended southeastern nucleus has colors consistent with reddened starlight, and the compact northwestern nucleus has extremely red colors indicative of an optical quasar with a warm dust component. Further, the molecular gas, 3mm continuum, and radio emission are coincident with the redder nucleus, confirming that the northwestern nucleus is the site of the AGN and that the molecular gas is the likely fuel source.Comment: LaTex, 5 pages with 1 postscript and 1 jpg figure, ApJ Letters, in press (August 20, 1999

The Radio Afterglow From GRB 980519: A Test of the Jet and Circumstellar Models

We present multi-frequency radio observations from the afterglow of GRB 980519 beginning 7.2 hours after the gamma-ray burst and ending 63 days later. The fast decline in the optical and X-ray light curves for this burst has been interpreted either as afterglow emission originating from a collimated outflow -- a jet -- or the result of a blast wave propagating into a medium whose density is shaped by the wind of an evolved massive star. These two models predict divergent behavior for the radio afterglow, and therefore, radio observations are capable, in principle, of discriminating between the two. We show that a wind model describes the subsequent evolution of the radio afterglow rather well. However, we see strong modulation of the light curve, which we interpret as diffractive scintillation. These variations prevent us from decisively rejecting the jet model.Comment: ApJ, submitte

Prognostic significance of the bcl-2 apoptotic family of proteins in primary and recurrent cervical cancer.

bcl-2 is one of a family of genes that control the apoptotic threshold of a cell. bcl-2 protein and its anti-apoptotic homologue, mcl-1, with the pro-apoptotic protein, bax, are thought to function by forming homo- and heterotypic dimers that then control the progression to apoptosis. p53 is also involved as a down-regulator of bcl-2 and a promoter of bax. To determine the effect of these apoptotic mechanisms, we used immunohistochemistry to determine the prognostic significance of the expression of bcl-2, mcl-1, bax and p53 in primary and recurrent cervical cancer. Tissues from 46 patients with primary cervical cancer and 28 women with recurrent carcinoma were stained for bcl-2, mcl-1, bax and p53. Kaplan-Meier survival analysis was performed using the log-rank test for differences between groups. In the primary disease group, positive staining for bcl-2 was associated with a better 5-year survival (bcl-2 +ve, 84% vs bcl-2 -ve, 53%, P = 0.03). Positive staining for p53 was associated with a survival disadvantage (p53 +ve, 4-year survival 38% vs p53 -ve, 4-year survival 78%, P = 0.02). mcl-1 and bax staining were not useful as prognostic indicators in primary disease. No marker was prognostic in recurrent disease. Positive bcl-2 staining defines a group of patients with primary disease with a good prognosis. p53, an activator of the bax promoter, identifies a group with a worse outcome. In recurrent disease, none of the markers reflected prognosis

A Sample of Low Redshift BL Lacs. I. The Radio Data

We present a new sample of 30 nearby (z<0.2) BL Lacs, selected to study the nuclear as well as the large scale properties of low power radio sources. In this first paper, we show and discuss new radio data taken with the VLA (19 objects at 1.4 GHz, either in A or C configuration, or both) as well as with the VLBA (15 sources at 5 GHz). On the kiloparsec scale, all objects exhibit a compact core and a variety of radio morphologies (jets, halos, secondary compact components). On the parsec scale, we find weak cores and a few short, one-sided, jets. From the jet/counter-jet ratio, core dominance, and synchrotron self Compton model we estimate the intrinsic orientation and velocity of the jets. The resulting properties of BL Lacs are similar to those of a parent population composed of FR I radio galaxies.Comment: 46 pages, 12 figures, accepted for publication in Ap