66,385 research outputs found
Bounds for D-finite closure properties
We provide bounds on the size of operators obtained by algorithms for
executing D-finite closure properties. For operators of small order, we give
bounds on the degree and on the height (bit-size). For higher order operators,
we give degree bounds that are parameterized with respect to the order and
reflect the phenomenon that higher order operators may have lower degrees
(order-degree curves)
Elementary operators and their lengths
Elementary operators on an algebra, which are finite sums of operators , provide a way to study properties of the algebra. In particular, for C*-algberas we consider results that are related to the length of the operator, defined as the minimal number of summands required. We will review some results concerning complete positivity or complete boundedness. Although all elementary operators on a C*-algebra are completely bounded, that is induce uniformly bounded operators on the algebras , the supremum is always attained for , or for smaller in case has special structure. For positivity, there are also results couched in analagous terms, but with different bounds. In recent work with I.~Gogi\\u27c, we have shown that for prime C*-algebras the elementary operators of length (at most) are norm closed, but that for the rather tractable class of homogeneous C*-algebras more subtle considerations are required for closure. For instance fails to have this closure property if is an open set in with , ()
A Class of Logistic Functions for Approximating State-Inclusive Koopman Operators
An outstanding challenge in nonlinear systems theory is identification or
learning of a given nonlinear system's Koopman operator directly from data or
models. Advances in extended dynamic mode decomposition approaches and machine
learning methods have enabled data-driven discovery of Koopman operators, for
both continuous and discrete-time systems. Since Koopman operators are often
infinite-dimensional, they are approximated in practice using
finite-dimensional systems. The fidelity and convergence of a given
finite-dimensional Koopman approximation is a subject of ongoing research. In
this paper we introduce a class of Koopman observable functions that confer an
approximate closure property on their corresponding finite-dimensional
approximations of the Koopman operator. We derive error bounds for the fidelity
of this class of observable functions, as well as identify two key learning
parameters which can be used to tune performance. We illustrate our approach on
two classical nonlinear system models: the Van Der Pol oscillator and the
bistable toggle switch.Comment: 8 page
Certainty Closure: Reliable Constraint Reasoning with Incomplete or Erroneous Data
Constraint Programming (CP) has proved an effective paradigm to model and
solve difficult combinatorial satisfaction and optimisation problems from
disparate domains. Many such problems arising from the commercial world are
permeated by data uncertainty. Existing CP approaches that accommodate
uncertainty are less suited to uncertainty arising due to incomplete and
erroneous data, because they do not build reliable models and solutions
guaranteed to address the user's genuine problem as she perceives it. Other
fields such as reliable computation offer combinations of models and associated
methods to handle these types of uncertain data, but lack an expressive
framework characterising the resolution methodology independently of the model.
We present a unifying framework that extends the CP formalism in both model
and solutions, to tackle ill-defined combinatorial problems with incomplete or
erroneous data. The certainty closure framework brings together modelling and
solving methodologies from different fields into the CP paradigm to provide
reliable and efficient approches for uncertain constraint problems. We
demonstrate the applicability of the framework on a case study in network
diagnosis. We define resolution forms that give generic templates, and their
associated operational semantics, to derive practical solution methods for
reliable solutions.Comment: Revised versio
Probably Safe or Live
This paper presents a formal characterisation of safety and liveness
properties \`a la Alpern and Schneider for fully probabilistic systems. As for
the classical setting, it is established that any (probabilistic tree) property
is equivalent to a conjunction of a safety and liveness property. A simple
algorithm is provided to obtain such property decomposition for flat
probabilistic CTL (PCTL). A safe fragment of PCTL is identified that provides a
sound and complete characterisation of safety properties. For liveness
properties, we provide two PCTL fragments, a sound and a complete one. We show
that safety properties only have finite counterexamples, whereas liveness
properties have none. We compare our characterisation for qualitative
properties with the one for branching time properties by Manolios and Trefler,
and present sound and complete PCTL fragments for characterising the notions of
strong safety and absolute liveness coined by Sistla
Explicit equations and bounds for the Nakai-Nishimura-Dubois-Efroymson dimension theorem
The Nakai-Nishimura-Dubois-Efroymson dimension theorem asserts the following: "let R be an algebraically closed field or a real closed field, let X be an irreducible algebraic subset of Rn and let Y be an algebraic subset of X of codimention s>=2 (not necessarily irreducible). Then, there is an irreducible algebraic subset W of X of codimention 1 containing Y". In this paper, making use of an elementary construction, we improve this result giving explicit polynomial equations for W. Moreover, denoting by R the algebraic closure of R and embedding canonically W into projective space Pn(R), we obtain explicit upper bounds for the degree and the geometric genus of the Zariski closure of W in Pn(R). In future papers, we will use these bounds in the study of morphism space between algebraic varieties over real closed fields
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