86,377 research outputs found
Integrating a Global Induction Mechanism into a Sequent Calculus
Most interesting proofs in mathematics contain an inductive argument which
requires an extension of the LK-calculus to formalize. The most commonly used
calculi for induction contain a separate rule or axiom which reduces the valid
proof theoretic properties of the calculus. To the best of our knowledge, there
are no such calculi which allow cut-elimination to a normal form with the
subformula property, i.e. every formula occurring in the proof is a subformula
of the end sequent. Proof schemata are a variant of LK-proofs able to simulate
induction by linking proofs together. There exists a schematic normal form
which has comparable proof theoretic behaviour to normal forms with the
subformula property. However, a calculus for the construction of proof schemata
does not exist. In this paper, we introduce a calculus for proof schemata and
prove soundness and completeness with respect to a fragment of the inductive
arguments formalizable in Peano arithmetic.Comment: 16 page
Fragments of Arithmetic and true sentences
By a theorem of R. Kaye, J. Paris and C. Dimitracopoulos, the class
of the ¦n+1–sentences true in the standard model is the only (up to deductive
equivalence) consistent ¦n+1–theory which extends the scheme of induction for
parameter free ¦n+1–formulas. Motivated by this result, we present a systematic
study of extensions of bounded quantifier complexity of fragments of first–order
Peano Arithmetic. Here, we improve that result and show that this property describes
a general phenomenon valid for parameter free schemes. As a consequence,
we obtain results on the quantifier complexity, (non)finite axiomatizability and
relative strength of schemes for ¢n+1–formulas.Junta de AndalucÃa TIC-13
Forward and inverse problems in fundamental and applied magnetohydrodynamics
This Minireview summarizes the recent efforts to solve forward and inverse
problems as they occur in different branches of fundamental and applied
magnetohydrodynamics. As for the forward problem, the main focus is on the
numerical treatment of induction processes, including self-excitation of
magnetic fields in non-spherical domains and/or under the influence of
non-homogeneous material parameters. As an important application of the
developed numerical schemes, the functioning of the von-K\'{a}rm\'{a}n-sodium
(VKS) dynamo experiment is shown to depend crucially on the presence of
soft-iron impellers. As for the inverse problem, the main focus is on the
mathematical background and some first practical applications of the
Contactless Inductive Flow Tomography (CIFT), in which flow induced magnetic
field perturbations are utilized for the reconstruction of the velocity field.
The promises of CIFT for flow field monitoring in the continuous casting of
steel are substantiated by results obtained at a test rig with a low melting
liquid metal. While CIFT is presently restricted to flows with low magnetic
Reynolds numbers, some selected problems of non-linear inverse dynamo theory,
with possible application to geo- and astrophysics, are also discussed.Comment: 15 pages, 4 figures, accepted for publication in European Physical
Journal Special Topic
First-Order Query Evaluation with Cardinality Conditions
We study an extension of first-order logic that allows to express cardinality
conditions in a similar way as SQL's COUNT operator. The corresponding logic
FOC(P) was introduced by Kuske and Schweikardt (LICS'17), who showed that query
evaluation for this logic is fixed-parameter tractable on classes of structures
(or databases) of bounded degree. In the present paper, we first show that the
fixed-parameter tractability of FOC(P) cannot even be generalised to very
simple classes of structures of unbounded degree such as unranked trees or
strings with a linear order relation.
Then we identify a fragment FOC1(P) of FOC(P) which is still sufficiently
strong to express standard applications of SQL's COUNT operator. Our main
result shows that query evaluation for FOC1(P) is fixed-parameter tractable
with almost linear running time on nowhere dense classes of structures. As a
corollary, we also obtain a fixed-parameter tractable algorithm for counting
the number of tuples satisfying a query over nowhere dense classes of
structures
Critical vortex line length near a zigzag of pinning centers
A vortex line passes through as many pinning centers as possible on its way
from one extremety of the superconductor to the other at the expense of
increasing its self-energy. In the framework of the Ginzburg-Landau theory we
study the relative growth in length, with respect to the straight line, of a
vortex near a zigzag of defects. The defects are insulating pinning spheres
that form a three-dimensional cubic array embedded in the superconductor. We
determine the depinning transition beyond which the vortex line no longer
follows the critical zigzag path of defects.Comment: 8 pages, 25 figures with low resolution option, 1 table. To be
published in Eur. Phys. Jour.
Wave-structure interaction for long wave models in the presence of a freely moving body on the bottom
In this paper we address a particular fluid-solid interaction problem in
which the solid object is lying at the bottom of a layer of fluid and moves
under the forces created by waves travelling on the surface of this layer. More
precisely, we consider the water waves problem in a fluid of fixed depth with a
flat bottom topography and with an object lying on the bottom, allowed to move
horizontally under the pressure forces created by the waves. After establishing
the physical setting of the problem, namely the dynamics of the fluid and the
mechanics of the solid motion, as well as analyzing the nature of the coupling,
we examine in detail two particular shallow water regimes: the case of the
(nonlinear) Saint-Venant system, and the (weakly nonlinear) Boussinesq system.
We prove an existence and uniqueness theorem for the coupled system in both
cases. Using the particular structure of the coupling terms we are able to go
beyond the standard scale for the existence time of solutions to the Boussinesq
system with a moving bottom.Comment: 37 pages, 1 imag
The adjoint problem in the presence of a deformed surface: the example of the Rosensweig instability on magnetic fluids
The Rosensweig instability is the phenomenon that above a certain threshold
of a vertical magnetic field peaks appear on the free surface of a horizontal
layer of magnetic fluid. In contrast to almost all classical hydrodynamical
systems, the nonlinearities of the Rosensweig instability are entirely
triggered by the properties of a deformed and a priori unknown surface. The
resulting problems in defining an adjoint operator for such nonlinearities are
illustrated. The implications concerning amplitude equations for pattern
forming systems with a deformed surface are discussed.Comment: 11 pages, 1 figur
- …