188,801 research outputs found
Z-stability in Constructive Analysis
We introduce Z-stability, a notion capturing the intuition that if a function
f maps a metric space into a normed space and if the norm of f(x) is small,
then x is close to a zero of f. Working in Bishop's constructive setting, we
first study pointwise versions of Z-stability and the related notion of good
behaviour for functions. We then present a recursive counterexample to the
classical argument for passing from pointwise Z-stability to a uniform version
on compact metric spaces. In order to effect this passage constructively, we
bring into play the positivity principle, equivalent to Brouwer's fan theorem
for detachable bars, and the limited anti-Specker property, an intuitionistic
counterpart to sequential compactness. The final section deals with connections
between the limited anti-Specker property, positivity properties, and
(potentially) Brouwer's fan theorem for detachable bars
Computing Solution Operators of Boundary-value Problems for Some Linear Hyperbolic Systems of PDEs
We discuss possibilities of application of Numerical Analysis methods to
proving computability, in the sense of the TTE approach, of solution operators
of boundary-value problems for systems of PDEs. We prove computability of the
solution operator for a symmetric hyperbolic system with computable real
coefficients and dissipative boundary conditions, and of the Cauchy problem for
the same system (we also prove computable dependence on the coefficients) in a
cube . Such systems describe a wide variety of physical
processes (e.g. elasticity, acoustics, Maxwell equations). Moreover, many
boundary-value problems for the wave equation also can be reduced to this case,
thus we partially answer a question raised in Weihrauch and Zhong (2002).
Compared with most of other existing methods of proving computability for PDEs,
this method does not require existence of explicit solution formulas and is
thus applicable to a broader class of (systems of) equations.Comment: 31 page
Classical Mathematics for a Constructive World
Interactive theorem provers based on dependent type theory have the
flexibility to support both constructive and classical reasoning. Constructive
reasoning is supported natively by dependent type theory and classical
reasoning is typically supported by adding additional non-constructive axioms.
However, there is another perspective that views constructive logic as an
extension of classical logic. This paper will illustrate how classical
reasoning can be supported in a practical manner inside dependent type theory
without additional axioms. We will see several examples of how classical
results can be applied to constructive mathematics. Finally, we will see how to
extend this perspective from logic to mathematics by representing classical
function spaces using a weak value monad.Comment: v2: Final copy for publicatio
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