641 research outputs found
Computability of entropy and information in classical Hamiltonian systems
We consider the computability of entropy and information in classical
Hamiltonian systems. We define the information part and total information
capacity part of entropy in classical Hamiltonian systems using relative
information under a computable discrete partition.
Using a recursively enumerable nonrecursive set it is shown that even though
the initial probability distribution, entropy, Hamiltonian and its partial
derivatives are computable under a computable partition, the time evolution of
its information capacity under the original partition can grow faster than any
recursive function. This implies that even though the probability measure and
information are conserved in classical Hamiltonian time evolution we might not
actually compute the information with respect to the original computable
partition
Design and construction of two genetic tools for use in mycoplasmas
The members of the bacterial class Mollicutes, commonly referred to as mycoplasmas have a number of characteristics that make them difficult research subjects. These include complex media requirements, a lack of cell walls, a high genomic AT/GC ratio and frequently a non-standard translational system. In most orders of the Mollicutes, the codon UGA, a stop codon in the universal genetic code, codes for the amino acid tryptophan. As a consequence, standard genetic tools frequently do not function property in mycoplasmas. The focus of the research described here has been on the construction of two tools with wide applicability for future mycoplasmal research. The first is a derivative of transposon Tn4OOl, mini-Tn4OOltet, that can be used to irreversibly insert a variety of markers (including green fluorescent protein) into mycoplasmal chromosomes. The second is a derivative of the popular genetic marker, green fluorescent protein (GFP). The DNA sequence of this construct, designated GFPmyco has been modified to reflect a mycoplasma coding usage pattern. The results of preliminary expression studies have confirmed that mini-Tn4OOltet is a functional mini-transposon in Mycoplasma gallisepticum and that GFPmyco can produce an observable fluorescent product in E. coli. Factors effecting GFP fluorescence and possible improvements in GFPmyco construction and expression are discussed. Both mini-transposons and GFP markers have been extremely useful for both eucaryotic and bacterial studies. It is hoped that these constructs, or their descendants, will be equally useful for the study of mycoplasmas
On the functions generated by the general purpose analog computer
PreprintWe consider the General Purpose Analog Computer (GPAC), introduced by Claude Shannon in 1941 as a mathematical model of Differential Analysers, that is to say as a model of continuous-time analog (mechanical, and later one electronic) machines of that time.
The GPAC generates as output univariate functions (i.e. functions f:R→R). In this paper we extend this model by: (i) allowing multivariate functions (i.e. functions f:Rn→Rm); (ii) introducing a notion of amount of resources (space) needed to generate a function, which allows the stratification of GPAC generable functions into proper subclasses. We also prove that a wide class of (continuous and discontinuous) functions can be uniformly approximated over their full domain.
We prove a few stability properties of this model, mostly stability by arithmetic operations, composition and ODE solving, taking into account the amount of resources needed to perform each operation.
We establish that generable functions are always analytic but that they can nonetheless (uniformly) approximate a wide range of nonanalytic functions. Our model and results extend some of the results from [19] to the multidimensional case, allow one to define classes of functions generated by GPACs which take into account bounded resources, and also strengthen the approximation result from [19] over a compact domain to a uniform approximation result over unbounded domains.info:eu-repo/semantics/acceptedVersio
Computational bounds on polynomial differential equations
In this paper we study from a computational perspective some prop-erties of the solutions of polynomial ordinary di erential equations.
We consider elementary (in the sense of Analysis) discrete-time dynam-ical systems satisfying certain criteria of robustness. We show that those systems can be simulated with elementary and robust continuous-time
dynamical systems which can be expanded into fully polynomial ordinary diferential equations with coe cients in Q[ ]. This sets a computational lower bound on polynomial ODEs since the former class is large enough
to include the dynamics of arbitrary Turing machines.
We also apply the previous methods to show that the problem of de-termining whether the maximal interval of defnition of an initial-value problem defned with polynomial ODEs is bounded or not is in general undecidable, even if the parameters of the system are computable and comparable and if the degree of the corresponding polynomial is at most
56.
Combined with earlier results on the computability of solutions of poly-nomial ODEs, one can conclude that there is from a computational point of view a close connection between these systems and Turing machines
Boundedness of the domain of definition is undecidable for polynomial odes
Consider the initial-value problem with computable parameters
dx
dt = p(t, x)
x(t0) = x0,
where p : Rn+1 ! Rn is a vector of polynomials and (t0, x0) 2 Rn+1.
We show that the problem of determining whether the maximal interval
of definition of this initial-value problem is bounded or not is in general
undecidable
Effective computability of solutions of ordinary differential equations: the thousand monkeys approach
In this note we consider the computability of the solution of the initial-
value problem for ordinary di erential equations with continuous right-
hand side. We present algorithms for the computation of the solution
using the \thousand monkeys" approach, in which we generate all possi-
ble solution tubes, and then check which are valid. In this way, we show
that the solution of a di erential equation de ned by a locally Lipschitz
function is computable even if the function is not e ectively locally Lips-
chitz. We also recover a result of Ruohonen, in which it is shown that if
the solution is unique, then it is computable, even if the right-hand side is
not locally Lipschitz. We also prove that the maximal interval of existence
for the solution must be e ectively enumerable open, and give an example
of a computable locally Lipschitz function which is not e ectively locally
Lipschitz
An extension of Chaitin's halting probability \Omega to a measurement operator in an infinite dimensional quantum system
This paper proposes an extension of Chaitin's halting probability \Omega to a
measurement operator in an infinite dimensional quantum system. Chaitin's
\Omega is defined as the probability that the universal self-delimiting Turing
machine U halts, and plays a central role in the development of algorithmic
information theory. In the theory, there are two equivalent ways to define the
program-size complexity H(s) of a given finite binary string s. In the standard
way, H(s) is defined as the length of the shortest input string for U to output
s. In the other way, the so-called universal probability m is introduced first,
and then H(s) is defined as -log_2 m(s) without reference to the concept of
program-size.
Mathematically, the statistics of outcomes in a quantum measurement are
described by a positive operator-valued measure (POVM) in the most general
setting. Based on the theory of computability structures on a Banach space
developed by Pour-El and Richards, we extend the universal probability to an
analogue of POVM in an infinite dimensional quantum system, called a universal
semi-POVM. We also give another characterization of Chaitin's \Omega numbers by
universal probabilities. Then, based on this characterization, we propose to
define an extension of \Omega as a sum of the POVM elements of a universal
semi-POVM. The validity of this definition is discussed.
In what follows, we introduce an operator version \hat{H}(s) of H(s) in a
Hilbert space of infinite dimension using a universal semi-POVM, and study its
properties.Comment: 24 pages, LaTeX2e, no figures, accepted for publication in
Mathematical Logic Quarterly: The title was slightly changed and a section on
an operator-valued algorithmic information theory was adde
Computability with polynomial differential equations
In this paper, we show that there are Initial Value Problems de ned
with polynomial ordinary di erential equations that can simulate univer-
sal Turing machines in the presence of bounded noise. The polynomial
ODE de ning the IVP is explicitly obtained and the simulation is per-
formed in real time
Effectively Open Real Functions
A function f is continuous iff the PRE-image f^{-1}[V] of any open set V is
open again. Dual to this topological property, f is called OPEN iff the IMAGE
f[U] of any open set U is open again. Several classical Open Mapping Theorems
in Analysis provide a variety of sufficient conditions for openness.
By the Main Theorem of Recursive Analysis, computable real functions are
necessarily continuous. In fact they admit a well-known characterization in
terms of the mapping V+->f^{-1}[V] being EFFECTIVE: Given a list of open
rational balls exhausting V, a Turing Machine can generate a corresponding list
for f^{-1}[V]. Analogously, EFFECTIVE OPENNESS requires the mapping U+->f[U] on
open real subsets to be effective.
By effectivizing classical Open Mapping Theorems as well as from application
of Tarski's Quantifier Elimination, the present work reveals several rich
classes of functions to be effectively open.Comment: added section on semi-algebraic functions; to appear in Proc.
http://cca-net.de/cca200
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