Algorithmic problems for free-abelian times free groups

We study direct products of free-abelian and free groups with special emphasis on algorithmic problems. After giving natural extensions of standard notions into that family, we find an explicit expression for an arbitrary endomorphism of \ZZ^m \times F_n. These tools are used to solve several algorithmic and decision problems for \ZZ^m \times F_n : the membership problem, the isomorphism problem, the finite index problem, the subgroup and coset intersection problems, the fixed point problem, and the Whitehead problem.Comment: 38 page

The peculiar supernova remnant CTB 80

Deep CCD exposures of the peculiar supernova remnant CTB 80 in the light of major optical lines have been obtained. These images reveal significant shock heated emission in the area of the remnant. The sulfur line image shows emission in the north along the outer boundary of the IRAS and HI shells. The comparison between the [OIII] and [OII] line images further suggest the presence of significant inhomogeneities in the interstellar medium. The flux calibrated images do not indicate the presence of incomplete recombination zones, and we estimate that the densities of the preshock clouds should not exceed a few atoms per cm^3. The area covered by the optical radiation along with the radio emission at 1410 MHz suggest that CTB 80 occupies a larger angular extent than was previously known.Comment: 19 pages, 6 png figures. Submitted revised version to A &

Kronecker powers of tensors and Strassen’s laser method

We answer a question, posed implicitly in [18, §11], [11, Rem. 15.44] and explicitly in [9, Problem 9.8], showing the border rank of the Kronecker square of the little Coppersmith-Winograd tensor is the square of the border rank of the tensor for all q > 2, a negative result for complexity theory. We further show that when q > 4, the analogous result holds for the Kronecker cube. In the positive direction, we enlarge the list of explicit tensors potentially useful for the laser method. We observe that a well-known tensor, the 3×3 determinant polynomial regarded as a tensor, det3 ∈ C9 C9 C9, could potentially be used in the laser method to prove the exponent of matrix multiplication is two. Because of this, we prove new upper bounds on its Waring rank and rank (both 18), border rank and Waring border rank (both 17), which, in addition to being promising for the laser method, are of interest in their own right. We discuss “skew” cousins of the little Coppersmith-Winograd tensor and indicate why they may be useful for the laser method. We establish general results regarding border ranks of Kronecker powers of tensors, and make a detailed study of Kronecker squares of tensors in C3 C3 C

Rank and border rank of Kronecker powers of tensors and Strassen's laser method

We prove that the border rank of the Kronecker square of the little Coppersmith–Winograd tensor Tcw,q is the square of its border rank for q> 2 and that the border rank of its Kronecker cube is the cube of its border rank for q> 4. This answers questions raised implicitly by Coppersmith & Winograd (1990, §11)and explicitly by Bläser (2013, Problem 9.8) and rules out the possibility of proving new upper bounds on the exponent of matrix multiplication using the square or cube of a little Coppersmith–Winograd tensor in this range. In the positive direction, we enlarge the list of explicit tensors potentially useful for Strassen's laser method, introducing a skew-symmetric version of the Coppersmith–Winograd tensor, Tskewcw,q. For q= 2 , the Kronecker square of this tensor coincides with the 3 × 3 determinant polynomial, det 3∈ C9⊗ C9⊗ C9, regarded as a tensor. We show that this tensor could potentially be used to show that the exponent of matrix multiplication is two. We determine new upper bounds for the (Waring) rank and the (Waring) border rank of det 3, exhibiting a strict submultiplicative behaviour for Tskewcw,2 which is promising for the laser method. We establish general results regarding border ranks of Kronecker powers of tensors, and make a detailed study of Kronecker squares of tensors in C3⊗ C3⊗ C3

Spectroscopy and Imaging Performance of the Liquid Xenon Gamma-Ray Imaging Telescope (LXeGRIT)

LXeGRIT is a balloon-borne Compton telescope based on a liquid xenon time projection chamber (LXeTPC) for imaging cosmic \g-rays in the energy band of 0.2-20 MeV. The detector, with 400 cm$^2$ area and 7 cm drift gap, is filled with high purity LXe. Both ionization and scintillation light signals are detected to measure the energy deposits and the three spatial coordinates of individual \g -ray interactions within the sensitive volume. The TPC has been characterized with repeated measurements of its spectral and Compton imaging response to \g -rays from radioactive sources such as \na, \cs, \yt and Am-Be. The detector shows a linear response to \g -rays in the energy range 511 keV -4.4 MeV, with an energy resolution (FWHM) of \Delta E/E=8.8% \: \sqrt{1\MeV /E}. Compton imaging of \yt \g -ray events with two detected interactions is consistent with an angular resolution of $\sim$ 3 degrees (RMS) at 1.8 MeV.Comment: To appear in: Hard X-Ray, Gamma-Ray and Neutron Detector Physics XI, 2000; Proc. SPIE, vol. 4140; K.A. Flanagan & O.H. Siegmund, ed

Accretion-Induced Lithium Line Enhancements in Classical T Tauri Stars: RW Aur

It is widely accepted that much of the stochastic variability of T Tauri stars is due to accretion by a circumstellar disk. The emission line spectrum as well as the excess continuum emission are common probes of this process. In this communication, we present additional probes of the circumstellar environment in the form of resonance lines of low ionization potential elements. Using a set of 14 high resolution echelle observations of the classical T Tauri star (CTTS), RW Aur, taken between 1986 and 1996, we carefully measure the continuum veiling at each epoch by comparing more than 500 absorption lines with those of an appropriate template. This allows us to accurately subtract out the continuum emission and to recover the underlying photospheric spectrum. In doing so, we find that selected photospheric lines are enhanced by the accretion process, namely the resonance lines of LiI and KI. A resonance line of TiI and a low excitation potential line of CaI also show weak enhancements. Simple slab models and computed line bisectors lead us to propose that these line enhancements are markers of cool gas at the beginning of the accretion flow which provides an additional source of line opacity. These results suggest that published values of surface lithium abundances of classical T Tauri stars are likely to be overestimated. This would account for the various reports of surface lithium abundances in excess of meteoritic values among the extreme CTTS. Computing LTE lithium abundances of RW Aur in a low and then high accretion state yields abundances which vary by one order of magnitude. The low accretion state lithium abundance is consistent with theoretical predictions for a star of this age and mass while the high accretion state spectrum yields a super-meteoritic lithium abundance.Comment: 28 pages, 8 figures, accepted by Ap

Quasihomomorphisms from the integers into Hamming metrics

A function f : Z → Qn is a c-quasihomomorphism if the Hamming distance between f(x + y) and f(x) + f(y) is at most c for all x, y ∈ Z. We show that any c-quasihomomorphism has distance at most some constant C(c) to an actual group homomorphism; here C(c) depends only on c and not on n or f. This gives a positive answer to a special case of a question posed by Kazhdan and Ziegler