75,020 research outputs found
Spectral theory for the q-Boson particle system
We develop spectral theory for the generator of the q-Boson (stochastic)
particle system. Our central result is a Plancherel type isomorphism theorem
for this system. This theorem has various implications. It proves the
completeness of the Bethe ansatz for the q-Boson generator and consequently
enables us to solve the Kolmogorov forward and backward equations for general
initial data. Owing to a Markov duality with q-TASEP, this leads to moment
formulas which characterize the fixed time distribution of q-TASEP started from
general initial conditions. The theorem also implies the biorthogonality of the
left and right eigenfunctions.
We consider limits of our q-Boson results to a discrete delta Bose gas
considered previously by van Diejen, as well as to another discrete delta Bose
gas that describes the evolution of moments of the semi-discrete stochastic
heat equation (or equivalently, the O'Connell-Yor semi-discrete directed
polymer partition function). A further limit takes us to the delta Bose gas
which arises in studying moments of the stochastic heat equation /
Kardar-Parisi-Zhang equation.Comment: 63 pages, 5 figure
HoCHC: A Refutationally Complete and Semantically Invariant System of Higher-order Logic Modulo Theories
We present a simple resolution proof system for higher-order constrained Horn
clauses (HoCHC) - a system of higher-order logic modulo theories - and prove
its soundness and refutational completeness w.r.t. the standard semantics. As
corollaries, we obtain the compactness theorem and semi-decidability of HoCHC
for semi-decidable background theories, and we prove that HoCHC satisfies a
canonical model property. Moreover a variant of the well-known translation from
higher-order to 1st-order logic is shown to be sound and complete for HoCHC in
standard semantics. We illustrate how to transfer decidability results for
(fragments of) 1st-order logic modulo theories to our higher-order setting,
using as example the Bernays-Schonfinkel-Ramsey fragment of HoCHC modulo a
restricted form of Linear Integer Arithmetic
A conformal Hopf-Rinow theorem for semi-Riemannian spacetimes
The famous Hopf-Rinow theorem states that a Riemannian manifold is metrically
complete if and only if it is geodesically complete. The compact Clifton-Pohl
torus fails to be geodesically complete, leading many mathematicians and
Wikipedia to conclude that "the theorem does not generalize to Lorentzian
manifolds". Recall now that in their 1931 paper Hopf and Rinow characterized
metric completeness also by properness. Recently, the author and
Garc\'{\i}a-Heveling obtained a Lorentzian completeness-compactness result with
a similar flavor. In this manuscript, we extend our theorem to cone structures
and to a new class of semi-Riemannian manifolds, dubbed
-spacetimes. Moreover, we demonstrate that our result implies, and
hence generalizes, the metric part of the Hopf-Rinow theorem.Comment: 40 page
Finite and infinite support in nominal algebra and logic: nominal completeness theorems for free
By operations on models we show how to relate completeness with respect to
permissive-nominal models to completeness with respect to nominal models with
finite support. Models with finite support are a special case of
permissive-nominal models, so the construction hinges on generating from an
instance of the latter, some instance of the former in which sufficiently many
inequalities are preserved between elements. We do this using an infinite
generalisation of nominal atoms-abstraction.
The results are of interest in their own right, but also, we factor the
mathematics so as to maximise the chances that it could be used off-the-shelf
for other nominal reasoning systems too. Models with infinite support can be
easier to work with, so it is useful to have a semi-automatic theorem to
transfer results from classes of infinitely-supported nominal models to the
more restricted class of models with finite support.
In conclusion, we consider different permissive-nominal syntaxes and nominal
models and discuss how they relate to the results proved here
Completeness of Larch/C++ specifications for black-box reuse
Develops a methodology to verify the completeness of formal specification intended for a black-box reuse. Identifies the algorithm to apply the methodology semi-automatically with the help of Larch Theorem Prover
- …