Weak Hopf Algebras I: Integral Theory and C^*-structure

We give an introduction to the theory of weak Hopf algebras proposed recently as a coassociative alternative of weak quasi-Hopf algebras. We follow an axiomatic approach keeping as close as possible to the "classical" theory of Hopf algebras. The emphasis is put on the new structure related to the presence of canonical subalgebras A^L and A^R in any weak Hopf algebra A that play the role of non-commutative numbers in many respects. A theory of integrals is developed in which we show how the algebraic properties of A, such as the Frobenius property, or semisimplicity, or innerness of the square of the antipode, are related to the existence of non-degenerate, normalized, or Haar integrals. In case of C^*-weak Hopf algebras we prove the existence of a unique Haar measure h in A and of a canonical grouplike element g in A implementing the square of the antipode and factorizing into left and right algebra elements. Further discussion of the C^*-case will be presented in Part II.Comment: 40 pages, LaTeX, to appear in J. Algebr

Advanced development of Pb-salt semiconductor lasers for the 8.0 to 15.0 micrometer spectral region

The technology was studied for producing Pb-salt diode lasers for the 8-51 micron spectral region suitable for use as local oscillators in a passive Laser Heterodyne Spectrometer (LHS). Consideration was given to long range NASA plans for the utilization of the passive LHS in a space shuttle environment. The general approach was to further develop the method of compositional interdiffusion (CID) recently reported, and used successfully at shorter wavelength. This technology was shown to provide an effective and reproducible method of producing a single-heterostructure (SH) diode of either the heterojunction or single-sided configuration. Performance specifications were exceeded in several devices, with single-ended CW power outputs as high as 0.88 milliwatts in a mode being achieved. The majority of the CID lasers fabricated had CW operating temperatures of over 60K; 30% of them operated CW above the boiling temperature of liquid nitrogen. CW operation above liquid nitrogen temperature was possible for wavelengths as long as 10.3 microns. Operation at 77K is significant with respect to space shuttle operations since its allows considerable simplification of cooling method

Reliability improvements in tunable Pb1-xSnxSe diode lasers

Recent developments in the technology of Pb-salt diode lasers which have led to significant improvements in reliability and lifetime, and to improved operation at very long wavelengths are described. A combination of packaging and contacting-metallurgy improvements has led to diode lasers that are stable both in terms of temperature cycling and shelf-storage time. Lasers cycled over 500 times between 77 K and 300 K have exhibited no measurable changes in either electrical contact resistance or threshold current. Utilizing metallurgical contacting process, both lasers and experimental n-type and p-type bulk materials are shown to have electrical contact resistance values that are stable for shelf storage periods well in excess of one year. Problems and experiments which have led to devices with improved performance stability are discussed. Stable device configurations achieved for material compositions yielding lasers which operate continuously at wavelengths as long as 30.3 micrometers are described

Development of lead salt semiconductor lasers for the 9-17 micron spectral region

Improved diode lasers of Pb sub 1-x Sn sub x Se operating in the 9-17 micrometers spectral region were developed. The performance characteristics of the best lasers exceeded the contract goals of 500 microW/mode at T 30K in the 9-12 micrometers region and 200 microW/mode at T 18K in the 16-17 micrometers region. Increased reliability and device yields resulted from processing improvements which evolved from a series of diagnostic studies. By means of Auger electron spectroscopy, laser shelf storage degradation was shown to be characterized by the presence of In metal on the semiconductor crystal surfaces. Studies of various metal barrier layers between the crystals and the In metal led to the development of an improved metallurgical contacting technology which has resulted in devices with performance stability values exceeding the contract goal of a one year shelf life. Lasers cycled over 500 times between 300K and 77K were also shown to be stable. Studies on improved methods of fabricating striped geometry lasers indicated that good spectral mode characteristics resulted from lasers which stripe widths of 12 and 25 micrometers

Edge effects in 3D dosimetry: characterisation and correction of the non-uniform dose response of PRESAGE®.

Previous work has shown that PRESAGE® can be used successfully to perform 3D dosimetric measurements of complex radiotherapy treatments. However, measurements near the sample edges are known to be difficult to achieve. This is an issue when the doses at air-material interfaces are of interest, for example when investigating the electron return effect (ERE) present in treatments delivered by magnetic resonance (MR)-linac systems. To study this effect, a set of 3.5 cm-diameter cylindrical PRESAGE® samples was uniformly irradiated with multiple dose fractions, using either a conventional linac or an MR-linac. The samples were imaged between fractions using an optical-CT, to read out the corresponding accumulated doses. A calibration between TPS-predicted dose and optical-CT pixel value was determined for individual dosimeters as a function of radial distance from the axis of rotation. This data was used to develop a correction that was applied to four additional samples of PRESAGE® of the same formulation, irradiated with 3D-CRT and IMRT treatment plans, to recover significantly improved 3D measurements of dose. An alternative strategy was also tested, in which the outer surface of the sample was physically removed prior to irradiation. Results show that for the formulation studied here, PRESAGE® samples have a central region that responds uniformly and an edge region of 6-7 mm where there is gradual increase in dosimeter response, rising to an over-response of 24%-36% at the outer boundary. This non-uniform dose response increases in both extent and magnitude over time. Both mitigation strategies investigated were successful. In our four exemplar studies, we show how discrepancies at edges are reduced from 13%-37% of the maximum dose to between 2 and 8%. Quantitative analysis shows that the 3D gamma passing rates rise from 90.4, 69.3, 63.7 and 43.6% to 97.3, 99.9, 96.7 and 98.9% respectively

Dose verification of dynamic MLC-tracked radiotherapy using small PRESAGE (R) 3D dosimeters and a motion phantom

With the increasing complexity of radiotherapy treatments typical 1D and 2D quality assurance (QA) detectors may fail to detect out-of-plane dose discrepancies, in particular in the presence of motion. In this work, small samples of the PRESAGE® 3D radiochromic dosimeter were used in combination with a motion phantom to measure real-time multileaf collimator (MLC)-tracked radiotherapy treatments. A different sample of PRESAGE® was irradiated for each of three different irradiation scenarios: (1) static: static sample, without tracking (2) motion: moving sample, without tracking and (3) tracking: moving sample, with tracking. Our in-house software DynaTrack dynamically moves the linac's MLC leafs based on the target position. The doses delivered to the samples were reconstructed based on the recorded positions of the MLC and phantom during the beam delivery. PRESAGE® samples were imaged with an in-house optical-CT scanner. Comparison between simulated and measured 3D dose showed good agreement for all three irradiation scenarios (static: 99.2%; motion: 99.7%; tracking: 99.3% with a 3%, 2 mm and a 10% threshold local gamma criterion), failing only at the edges of the PRESAGE® samples (~ 6 mm). Given that the dose distributions deposited using the DynaTrack system have been independently verified, this experiment demonstrates the ability of PRESAGE to measure 3D doses correctly in a tracking context. We conclude that this methodology could be used in the future to validate the delivery of dynamic MLC-tracked radiotherapy

Effect of MLC tracking latency on conformal volumetric modulated arc therapy (VMAT) plans in 4D stereotactic lung treatment

AbstractBackground and purposeThe latency of a multileaf collimator (MLC) tracking system used to overcome respiratory motion causes misalignment of the treatment beam with respect to the gross tumour volume, which may result in reduced target coverage. This study investigates the magnitude of this effect.Material and methodsSimulated superior–inferior breathing motion was used to construct histograms of isocentre offset with respect to the gross tumour volume (GTV) for a variety of tracking latencies. Dose distributions for conformal volumetric modulated arc therapy (VMAT) arcs were then calculated at a range of offsets and summed according to these displacement histograms. The results were verified by delivering the plans to a Delta4 phantom on a motion platform.ResultsIn the absence of an internal target margin, a tracking latency of 150ms reduces the GTV D95% by approximately 2%. With a margin of 2mm, the same drop in dose occurs for a tracking latency of 450ms. Lung V13Gy is unaffected by a range of latencies. These results are supported by the phantom measurements.ConclusionsAssuming that internal motion can be modelled by a rigid translation of the patient, MLC tracking of conformal VMAT can be effectively accomplished in the absence of an internal target margin for substantial breathing motion (4s period and 20mm peak–peak amplitude) so long as the system latency is less than 150ms

On orbits of the automorphism group on a complete toric variety

Let X be a complete toric variety and Aut(X) be the automorphism group. We give an explit description of Aut(X)-orbits on X. In particular, we show that Aut(X) acts on X transitively if and only if X is a product of projective spaces.Comment: 10 pages, 4 figure