Decreasing norm-trace codes

The decreasing norm-trace codes are evaluation codes defined by a set of monomials closed under divisibility and the rational points of the extended norm-trace curve. In particular, the decreasing norm-trace codes contain the one-point algebraic geometry (AG) codes over the extended norm-trace curve. We use Gr\"obner basis theory and find the indicator functions on the rational points of the curve to determine the basic parameters of the decreasing norm-trace codes: length, dimension, and minimum distance. We also obtain their dual codes. We give conditions for a decreasing norm-trace code to be a self-orthogonal or a self-dual code. We provide a linear exact repair scheme to correct single erasures for decreasing norm-trace codes, which applies to higher rate codes than the scheme developed by Jin, Luo, and Xing (IEEE Transactions on Information Theory {\bf 64} (2), 900-908, 2018) when applied to the one-point AG codes over the extended norm-trace curve

Fornax: a Flexible Code for Multiphysics Astrophysical Simulations

This paper describes the design and implementation of our new multi-group, multi-dimensional radiation hydrodynamics (RHD) code Fornax and provides a suite of code tests to validate its application in a wide range of physical regimes. Instead of focusing exclusively on tests of neutrino radiation hydrodynamics relevant to the core-collapse supernova problem for which Fornax is primarily intended, we present here classical and rigorous demonstrations of code performance relevant to a broad range of multi-dimensional hydrodynamic and multi-group radiation hydrodynamic problems. Our code solves the comoving-frame radiation moment equations using the M1 closure, utilizes conservative high-order reconstruction, employs semi-explicit matter and radiation transport via a high-order time stepping scheme, and is suitable for application to a wide range of astrophysical problems. To this end, we first describe the philosophy, algorithms, and methodologies of Fornax and then perform numerous stringent code tests, that collectively and vigorously exercise the code, demonstrate the excellent numerical fidelity with which it captures the many physical effects of radiation hydrodynamics, and show excellent strong scaling well above 100k MPI tasks.Comment: Accepted to the Astrophysical Journal Supplement Series; A few more textual and reference updates; As before, one additional code test include

Size Effect in Fracture of Concrete Specimens and Structures: New Problems and Progress

Presented is a concise summary of recent Northwestern University studies of six new problems. First, the decrease of fracture energy during crack propagation through a boundary layer, documented by Hu and Wittmann, is shown to be captured by a cohesive crack model in which the softening tail slope depends on the distance from the boundary (which causes an apparent size effect on fracture energy and implies that the nonlocal damage model is more fundamental than the cohesive crack model). Second, an improved universal size effect law giving a smooth transition between failures at large cracks (or notches) and at crack initiation is presented. Third, a recent renewed proposal that the nominal strength variation as a function of notch depth be used for measuring fracture energy is critically examined. Fourth, numerical results and a formula describing the size effect of finite-angle notches are presented. Fifth, a new size effect law derivation from dimensional analysis coupled with asymptotic matching is given. Finally, an improved code-type formula for shear capacity of R.C. beams is proposed.

Constructing Linear Encoders with Good Spectra

Linear encoders with good joint spectra are suitable candidates for optimal lossless joint source-channel coding (JSCC), where the joint spectrum is a variant of the input-output complete weight distribution and is considered good if it is close to the average joint spectrum of all linear encoders (of the same coding rate). In spite of their existence, little is known on how to construct such encoders in practice. This paper is devoted to their construction. In particular, two families of linear encoders are presented and proved to have good joint spectra. The first family is derived from Gabidulin codes, a class of maximum-rank-distance codes. The second family is constructed using a serial concatenation of an encoder of a low-density parity-check code (as outer encoder) with a low-density generator matrix encoder (as inner encoder). In addition, criteria for good linear encoders are defined for three coding applications: lossless source coding, channel coding, and lossless JSCC. In the framework of the code-spectrum approach, these three scenarios correspond to the problems of constructing linear encoders with good kernel spectra, good image spectra, and good joint spectra, respectively. Good joint spectra imply both good kernel spectra and good image spectra, and for every linear encoder having a good kernel (resp., image) spectrum, it is proved that there exists a linear encoder not only with the same kernel (resp., image) but also with a good joint spectrum. Thus a good joint spectrum is the most important feature of a linear encoder.Comment: v5.5.5, no. 201408271350, 40 pages, 3 figures, extended version of the paper to be published in IEEE Transactions on Information Theor

Advanced numerical methods for mantle convection models

Numerical modelling of Earth's mantle is a complex, and computationally demanding task due to, amongst others, the broad spectrum of temporal and spatial scales playing a role in mantle flow, large uncertainties in the physical properties of mantle material, with large and localised transitions in viscosity and density. This thesis introduces and analyses a number of numerical techniques that may bring a significant contribution in meeting some of these challenges. Firstly, we introduce a novel time integration scheme for free surface movement in mantle convection models that is more accurate and stable for large time steps. Secondly, we extend the capabilities of anisotropic mesh optimisation, which allows efficient focussing of mesh resolution, to handle cylindrical and spherical shell domains and demonstrate that a significant reduction in the required number of degrees of freedom is possible while maintaing accuracy. Finally, to verify correctness, and evaluate and compare properties of various numerical schemes, we derive an extensive suite of analytical solutions to the Stokes equations governing mantle flow in cylindrical and spherical shell domains, with physically relevant boundary conditions. As a numerical benchmark they also serve to facilitate comparisons of different geodynamical models, and the further development of numerical techniques to improve these.Open Acces