799,392 research outputs found
Computing the Rank Profile Matrix
The row (resp. column) rank profile of a matrix describes the staircase shape
of its row (resp. column) echelon form. In an ISSAC'13 paper, we proposed a
recursive Gaussian elimination that can compute simultaneously the row and
column rank profiles of a matrix as well as those of all of its leading
sub-matrices, in the same time as state of the art Gaussian elimination
algorithms. Here we first study the conditions making a Gaus-sian elimination
algorithm reveal this information. Therefore, we propose the definition of a
new matrix invariant, the rank profile matrix, summarizing all information on
the row and column rank profiles of all the leading sub-matrices. We also
explore the conditions for a Gaussian elimination algorithm to compute all or
part of this invariant, through the corresponding PLUQ decomposition. As a
consequence, we show that the classical iterative CUP decomposition algorithm
can actually be adapted to compute the rank profile matrix. Used, in a Crout
variant, as a base-case to our ISSAC'13 implementation, it delivers a
significant improvement in efficiency. Second, the row (resp. column) echelon
form of a matrix are usually computed via different dedicated triangular
decompositions. We show here that, from some PLUQ decompositions, it is
possible to recover the row and column echelon forms of a matrix and of any of
its leading sub-matrices thanks to an elementary post-processing algorithm
Common Subexpression Elimination in a Lazy Functional Language
Common subexpression elimination is a well-known compiler optimisation that saves time by avoiding the repetition of the same computation. To our knowledge it has not yet been applied to lazy functional programming languages, although there are several advantages. First, the referential transparency of these languages makes the identification of common subexpressions very simple. Second, more common subexpressions can be recognised because they can be of arbitrary type whereas standard common subexpression elimination only shares primitive values. However, because lazy functional languages decouple program structure from data space allocation and control flow, analysing its effects and deciding under which conditions the elimination of a common subexpression is beneficial proves to be quite difficult. We developed and implemented the transformation for the language Haskell by extending the Glasgow Haskell compiler and measured its effectiveness on real-world programs
Generating facets for the cut polytope of a graph by triangular elimination
The cut polytope of a graph arises in many fields. Although much is known
about facets of the cut polytope of the complete graph, very little is known
for general graphs. The study of Bell inequalities in quantum information
science requires knowledge of the facets of the cut polytope of the complete
bipartite graph or, more generally, the complete k-partite graph. Lifting is a
central tool to prove certain inequalities are facet inducing for the cut
polytope. In this paper we introduce a lifting operation, named triangular
elimination, applicable to the cut polytope of a wide range of graphs.
Triangular elimination is a specific combination of zero-lifting and
Fourier-Motzkin elimination using the triangle inequality. We prove sufficient
conditions for the triangular elimination of facet inducing inequalities to be
facet inducing. The proof is based on a variation of the lifting lemma adapted
to general graphs. The result can be used to derive facet inducing inequalities
of the cut polytope of various graphs from those of the complete graph. We also
investigate the symmetry of facet inducing inequalities of the cut polytope of
the complete bipartite graph derived by triangular elimination.Comment: 19 pages, 1 figure; filled details of the proof of Theorem 4, made
many other small change
Reasoning about Unreliable Actions
We analyse the philosopher Davidson's semantics of actions, using a strongly
typed logic with contexts given by sets of partial equations between the
outcomes of actions. This provides a perspicuous and elegant treatment of
reasoning about action, analogous to Reiter's work on artificial intelligence.
We define a sequent calculus for this logic, prove cut elimination, and give a
semantics based on fibrations over partial cartesian categories: we give a
structure theory for such fibrations. The existence of lax comma objects is
necessary for the proof of cut elimination, and we give conditions on the
domain fibration of a partial cartesian category for such comma objects to
exist
Phase shifts in nonresonant coherent excitation
Far-off-resonant pulsed laser fields produce negligible excitation between
two atomic states but may induce considerable phase shifts. The acquired phases
are usually calculated by using the adiabatic-elimination approximation. We
analyze the accuracy of this approximation and derive the conditions for its
applicability to the calculation of the phases. We account for various sources
of imperfections, ranging from higher terms in the adiabatic-elimination
expansion and irreversible population loss to couplings to additional states.
We find that, as far as the phase shifts are concerned, the adiabatic
elimination is accurate only for a very large detuning. We show that the
adiabatic approximation is a far more accurate method for evaluating the phase
shifts, with a vast domain of validity; the accuracy is further enhanced by
superadiabatic corrections, which reduce the error well below .
Moreover, owing to the effect of adiabatic population return, the adiabatic and
superadiabatic approximations allow one to calculate the phase shifts even for
a moderately large detuning, and even when the peak Rabi frequency is larger
than the detuning; in these regimes the adiabatic elimination is completely
inapplicable. We also derive several exact expressions for the phases using
exactly soluble two-state and three-state analytical models.Comment: 10 pages, 7 figure
Partial elimination ideals and secant cones
For any k \in \Nat, we show that the cone of -secant lines of a
closed subscheme over an algebraically closed field
running through a closed point is defined by the
-th partial elimination ideal of with respect to . We use this fact
to give an algorithm for computing secant cones. Also, we show that under
certain conditions partial elimination ideals describe the length of the fibres
of a multiple projection in a way similar to the way they do for simple
projections. Finally, we study some examples illustrating these results,
computed by means of {\sc Singular}.Comment: 18 pages; revised version, to appear in Journal of Algebr
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