59,444 research outputs found
An interpretation of the Sigma-2 fragment of classical Analysis in System T
We show that it is possible to define a realizability interpretation for the
-fragment of classical Analysis using G\"odel's System T only. This
supplements a previous result of Schwichtenberg regarding bar recursion at
types 0 and 1 by showing how to avoid using bar recursion altogether. Our
result is proved via a conservative extension of System T with an operator for
composable continuations from the theory of programming languages due to Danvy
and Filinski. The fragment of Analysis is therefore essentially constructive,
even in presence of the full Axiom of Choice schema: Weak Church's Rule holds
of it in spite of the fact that it is strong enough to refute the formal
arithmetical version of Church's Thesis
Kripke Models for Classical Logic
We introduce a notion of Kripke model for classical logic for which we
constructively prove soundness and cut-free completeness. We discuss the
novelty of the notion and its potential applications
Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism
This essay examines the philosophical significance of -logic in Zermelo-Fraenkel set theory with choice (ZFC). The duality between coalgebra and algebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The modal profile of -logical validity can then be countenanced within a coalgebraic logic, and -logical validity can be defined via deterministic automata. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal profiles of -logical validity correspond to those of second-order logical consequence, -logical validity is genuinely logical, and thus vindicates a neo-logicist conception of mathematical truth in the set-theoretic multiverse. Second, the foregoing provides a modal-computational account of the interpretation of mathematical vocabulary, adducing in favor of a realist conception of the cumulative hierarchy of sets
Response maxima in time-modulated turbulence: Direct Numerical Simulations
The response of turbulent flow to time-modulated forcing is studied by direct
numerical simulations of the Navier-Stokes equations. The large-scale forcing
is modulated via periodic energy input variations at frequency . The
response is maximal for frequencies in the range of the inverse of the large
eddy turnover time, confirming the mean-field predictions of von der Heydt,
Grossmann and Lohse (Phys. Rev. E 67, 046308 (2003)). In accordance with the
theory the response maximum shows only a small dependence on the Reynolds
number and is also quite insensitive to the particular flow-quantity that is
monitored, e.g., kinetic energy, dissipation-rate, or Taylor-Reynolds number.
At sufficiently high frequencies the amplitude of the kinetic energy response
decreases as . For frequencies beyond the range of maximal response,
a significant change in phase-shift relative to the time-modulated forcing is
observed.Comment: submitted to Europhysics Letters (EPL), 8 pages, 8 Postscript
figures, uses epl.cl
A syntactic approach to continuity of T-definable functionals
We give a new proof of the well-known fact that all functions which are definable in G\"odel's System T are
continuous via a syntactic approach. Differing from the usual syntactic method,
we firstly perform a translation of System T into itself in which natural
numbers are translated to functions . Then we inductively define a continuity predicate on the
translated elements and show that the translation of any term in System T
satisfies the continuity predicate. We obtain the desired result by relating
terms and their translations via a parametrized logical relation. Our
constructions and proofs have been formalized in the Agda proof assistant.
Because Agda is also a programming language, we can execute our proof to
compute moduli of continuity of T-definable functions
A critical layer model for turbulent pipe flow
A model-based description of the scaling and radial location of turbulent
fluctuations in turbulent pipe flow is presented and used to illuminate the
scaling behaviour of the very large scale motions. The model is derived by
treating the nonlinearity in the perturbation equation (involving the Reynolds
stress) as an unknown forcing, yielding a linear relationship between the
velocity field response and this nonlinearity. We do not assume small
perturbations. We examine propagating modes, permitting comparison of our
results to experimental data, and identify the steady component of the velocity
field that varies only in the wall-normal direction as the turbulent mean
profile. The "optimal" forcing shape, that gives the largest velocity response,
is assumed to lead to modes that will be dominant and hence observed in
turbulent pipe flow.
An investigation of the most amplified velocity response at a given
wavenumber-frequency combination reveals critical layer-like behaviour
reminiscent of the neutrally stable solutions of the Orr-Sommerfeld equation in
linearly unstable flow. Two distinct regions in the flow where the influence of
viscosity becomes important can be identified, namely a wall layer that scales
with and a critical layer, where the propagation velocity is equal
to the local mean velocity, that scales with in pipe flow. This
framework appears to be consistent with several scaling results in wall
turbulence and reveals a mechanism by which the effects of viscosity can extend
well beyond the immediate vicinity of the wall.Comment: Submitted to the Journal of Fluid Mechanics and currently under
revie
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