96,267 research outputs found
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
The exact (up to infinitesimals) infinite perimeter of the Koch snowflake and its finite area
The Koch snowflake is one of the first fractals that were mathematically
described. It is interesting because it has an infinite perimeter in the limit
but its limit area is finite. In this paper, a recently proposed computational
methodology allowing one to execute numerical computations with infinities
and infinitesimals is applied to study the Koch snowflake at infinity. Numerical
computations with actual infinite and infinitesimal numbers can be
executed on the Infinity Computer being a new supercomputer patented in
USA and EU. It is revealed in the paper that at infinity the snowflake is not
unique, i.e., different snowflakes can be distinguished for different infinite
numbers of steps executed during the process of their generation. It is then
shown that for any given infinite number n of steps it becomes possible to
calculate the exact infinite number, Nn, of sides of the snowflake, the exact
infinitesimal length, Ln, of each side and the exact infinite perimeter, Pn,
of the Koch snowflake as the result of multiplication of the infinite Nn by
the infinitesimal Ln. It is established that for different infinite n and k the
infinite perimeters Pn and Pk are also different and the difference can be infinite.
It is shown that the finite areas An and Ak of the snowflakes can be
also calculated exactly (up to infinitesimals) for different infinite n and k and
the difference An − Ak results to be infinitesimal. Finally, snowflakes constructed
starting from different initial conditions are also studied and their
quantitative characteristics at infinity are computed
Towards Machine Wald
The past century has seen a steady increase in the need of estimating and
predicting complex systems and making (possibly critical) decisions with
limited information. Although computers have made possible the numerical
evaluation of sophisticated statistical models, these models are still designed
\emph{by humans} because there is currently no known recipe or algorithm for
dividing the design of a statistical model into a sequence of arithmetic
operations. Indeed enabling computers to \emph{think} as \emph{humans} have the
ability to do when faced with uncertainty is challenging in several major ways:
(1) Finding optimal statistical models remains to be formulated as a well posed
problem when information on the system of interest is incomplete and comes in
the form of a complex combination of sample data, partial knowledge of
constitutive relations and a limited description of the distribution of input
random variables. (2) The space of admissible scenarios along with the space of
relevant information, assumptions, and/or beliefs, tend to be infinite
dimensional, whereas calculus on a computer is necessarily discrete and finite.
With this purpose, this paper explores the foundations of a rigorous framework
for the scientific computation of optimal statistical estimators/models and
reviews their connections with Decision Theory, Machine Learning, Bayesian
Inference, Stochastic Optimization, Robust Optimization, Optimal Uncertainty
Quantification and Information Based Complexity.Comment: 37 page
Pruning Algorithms for Pretropisms of Newton Polytopes
Pretropisms are candidates for the leading exponents of Puiseux series that
represent solutions of polynomial systems. To find pretropisms, we propose an
exact gift wrapping algorithm to prune the tree of edges of a tuple of Newton
polytopes. We prefer exact arithmetic not only because of the exact input and
the degrees of the output, but because of the often unpredictable growth of the
coordinates in the face normals, even for polytopes in generic position. We
provide experimental results with our preliminary implementation in Sage that
compare favorably with the pruning method that relies only on cone
intersections.Comment: exact, gift wrapping, Newton polytope, pretropism, tree pruning,
accepted for presentation at Computer Algebra in Scientific Computing, CASC
201
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