2,938 research outputs found
A computer algebra user interface manifesto
Many computer algebra systems have more than 1000 built-in functions, making
expertise difficult. Using mock dialog boxes, this article describes a proposed
interactive general-purpose wizard for organizing optional transformations and
allowing easy fine grain control over the form of the result even by amateurs.
This wizard integrates ideas including:
* flexible subexpression selection;
* complete control over the ordering of variables and commutative operands,
with well-chosen defaults;
* interleaving the choice of successively less main variables with applicable
function choices to provide detailed control without incurring a combinatorial
number of applicable alternatives at any one level;
* quick applicability tests to reduce the listing of inapplicable
transformations;
* using an organizing principle to order the alternatives in a helpful
manner;
* labeling quickly-computed alternatives in dialog boxes with a preview of
their results,
* using ellipsis elisions if necessary or helpful;
* allowing the user to retreat from a sequence of choices to explore other
branches of the tree of alternatives or to return quickly to branches already
visited;
* allowing the user to accumulate more than one of the alternative forms;
* integrating direct manipulation into the wizard; and
* supporting not only the usual input-result pair mode, but also the useful
alternative derivational and in situ replacement modes in a unified window.Comment: 38 pages, 12 figures, to be published in Communications in Computer
Algebr
Artin's primitive root conjecture -a survey -
This is an expanded version of a write-up of a talk given in the fall of 2000
in Oberwolfach. A large part of it is intended to be understandable by
non-number theorists with a mathematical background. The talk covered some of
the history, results and ideas connected with Artin's celebrated primitive root
conjecture dating from 1927. In the update several new results established
after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer
Provably convergent Newton-Raphson methods for recovering primitive variables with applications to physical-constraint-preserving Hermite WENO schemes for relativistic hydrodynamics
The relativistic hydrodynamics (RHD) equations have three crucial intrinsic
physical constraints on the primitive variables: positivity of pressure and
density, and subluminal fluid velocity. However, numerical simulations can
violate these constraints, leading to nonphysical results or even simulation
failure. Designing genuinely physical-constraint-preserving (PCP) schemes is
very difficult, as the primitive variables cannot be explicitly reformulated
using conservative variables due to relativistic effects. In this paper, we
propose three efficient Newton--Raphson (NR) methods for robustly recovering
primitive variables from conservative variables. Importantly, we rigorously
prove that these NR methods are always convergent and PCP, meaning they
preserve the physical constraints throughout the NR iterations. The discovery
of these robust NR methods and their PCP convergence analyses are highly
nontrivial and technical. As an application, we apply the proposed NR methods
to design PCP finite volume Hermite weighted essentially non-oscillatory
(HWENO) schemes for solving the RHD equations. Our PCP HWENO schemes
incorporate high-order HWENO reconstruction, a PCP limiter, and
strong-stability-preserving time discretization. We rigorously prove the PCP
property of the fully discrete schemes using convex decomposition techniques.
Moreover, we suggest the characteristic decomposition with rescaled
eigenvectors and scale-invariant nonlinear weights to enhance the performance
of the HWENO schemes in simulating large-scale RHD problems. Several demanding
numerical tests are conducted to demonstrate the robustness, accuracy, and high
resolution of the proposed PCP HWENO schemes and to validate the efficiency of
our NR methods.Comment: 49 page
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