10,122 research outputs found
Robustness and Randomness
Robustness problems of computational geometry algorithms is a topic that has been subject to intensive research efforts from both computer science and mathematics communities. Robustness problems are caused by the lack of precision in computations involving floating-point instead of real numbers. This paper reviews methods dealing with robustness and inaccuracy problems. It discussed approaches based on exact arithmetic, interval arithmetic and probabilistic methods. The paper investigates the possibility to use randomness at certain levels of reasoning to make geometric constructions more robust
Confidence limits: what is the problem? Is there the solution?
This contribution to the debate on confidence limits focuses mostly on the
case of measurements with `open likelihood', in the sense that it is defined in
the text. I will show that, though a prior-free assessment of {\it confidence}
is, in general, not possible, still a search result can be reported in a mostly
unbiased and efficient way, which satisfies some desiderata which I believe are
shared by the people interested in the subject. The simpler case of `closed
likelihood' will also be treated, and I will discuss why a uniform prior on a
sensible quantity is a very reasonable choice for most applications. In both
cases, I think that much clarity will be achieved if we remove from scientific
parlance the misleading expressions `confidence intervals' and `confidence
levels'.Comment: 20 pages, 6 figures, using cernrepp.cls (included). Contribution to
the Workshop on Confidence Limits, CERN, Geneva, 17-18 January 2000. This
paper and related work are also available at
http://www-zeus.roma1.infn.it/~agostini/prob+stat.htm
Recent progress in exact geometric computation
AbstractComputational geometry has produced an impressive wealth of efficient algorithms. The robust implementation of these algorithms remains a major issue. Among the many proposed approaches for solving numerical non-robustness, Exact Geometric Computation (EGC) has emerged as one of the most successful. This survey describes recent progress in EGC research in three key areas: constructive zero bounds, approximate expression evaluation and numerical filters
Symmetry Detection of Rational Space Curves from their Curvature and Torsion
We present a novel, deterministic, and efficient method to detect whether a
given rational space curve is symmetric. By using well-known differential
invariants of space curves, namely the curvature and torsion, the method is
significantly faster, simpler, and more general than an earlier method
addressing a similar problem. To support this claim, we present an analysis of
the arithmetic complexity of the algorithm and timings from an implementation
in Sage.Comment: 25 page
Near NP-Completeness for Detecting p-adic Rational Roots in One Variable
We show that deciding whether a sparse univariate polynomial has a p-adic
rational root can be done in NP for most inputs. We also prove a
polynomial-time upper bound for trinomials with suitably generic p-adic Newton
polygon. We thus improve the best previous complexity upper bound of EXPTIME.
We also prove an unconditional complexity lower bound of NP-hardness with
respect to randomized reductions for general univariate polynomials. The best
previous lower bound assumed an unproved hypothesis on the distribution of
primes in arithmetic progression. We also discuss how our results complement
analogous results over the real numbers.Comment: 8 pages in 2 column format, 1 illustration. Submitted to a conferenc
Bayesian Inference in Processing Experimental Data: Principles and Basic Applications
This report introduces general ideas and some basic methods of the Bayesian
probability theory applied to physics measurements. Our aim is to make the
reader familiar, through examples rather than rigorous formalism, with concepts
such as: model comparison (including the automatic Ockham's Razor filter
provided by the Bayesian approach); parametric inference; quantification of the
uncertainty about the value of physical quantities, also taking into account
systematic effects; role of marginalization; posterior characterization;
predictive distributions; hierarchical modelling and hyperparameters; Gaussian
approximation of the posterior and recovery of conventional methods, especially
maximum likelihood and chi-square fits under well defined conditions; conjugate
priors, transformation invariance and maximum entropy motivated priors; Monte
Carlo estimates of expectation, including a short introduction to Markov Chain
Monte Carlo methods.Comment: 40 pages, 2 figures, invited paper for Reports on Progress in Physic
Enumerative Galois theory for cubics and quartics
We show that there are monic, cubic
polynomials with integer coefficients bounded by in absolute value whose
Galois group is . We also show that the order of magnitude for
quartics is , and that the respective counts for , ,
are , , . Our work establishes
that irreducible non- cubic polynomials are less numerous than reducible
ones, and similarly in the quartic setting: these are the first two solved
cases of a 1936 conjecture made by van der Waerden
Statistical mechanics of the random K-SAT model
The Random K-Satisfiability Problem, consisting in verifying the existence of
an assignment of N Boolean variables that satisfy a set of M=alpha N random
logical clauses containing K variables each, is studied using the replica
symmetric framework of diluted disordered systems. We present an exact
iterative scheme for the replica symmetric functional order parameter together
for the different cases of interest K=2, K>= 3 and K>>1. The calculation of the
number of solutions, which allowed us [Phys. Rev. Lett. 76, 3881 (1996)] to
predict a first order jump at the threshold where the Boolean expressions
become unsatisfiable with probability one, is thoroughly displayed. In the case
K=2, the (rigorously known) critical value (alpha=1) of the number of clauses
per Boolean variable is recovered while for K>=3 we show that the system
exhibits a replica symmetry breaking transition. The annealed approximation is
proven to be exact for large K.Comment: 34 pages + 1 table + 8 fig., submitted to Phys. Rev. E, new section
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