503,968 research outputs found
Phase transitions in 3D gravity and fractal dimension
We show that for three dimensional gravity with higher genus boundary
conditions, if the theory possesses a sufficiently light scalar, there is a
second order phase transition where the scalar field condenses. This three
dimensional version of the holographic superconducting phase transition occurs
even though the pure gravity solutions are locally AdS. This is in addition
to the first order Hawking-Page-like phase transitions between different
locally AdS handlebodies. This implies that the R\'enyi entropies of
holographic CFTs will undergo phase transitions as the R\'enyi parameter is
varied, as long as the theory possesses a scalar operator which is lighter than
a certain critical dimension. We show that this critical dimension has an
elegant mathematical interpretation as the Hausdorff dimension of the limit set
of a quotient group of AdS, and use this to compute it, analytically near
the boundary of moduli space and numerically in the interior of moduli space.
We compare this to a CFT computation generalizing recent work of Belin, Keller
and Zadeh, bounding the critical dimension using higher genus conformal blocks,
and find a surprisingly good match
Proving and Computing: Applying Automated Reasoning to the Verification of Symbolic Computation Systems
The application of automated reasoning to the formal verification
of symbolic computation systems is motivated by the need of
ensuring the correctness of the results computed by the system, beyond
the classical approach of testing. Formal verification of properties of the
implemented algorithms require not only to formalize the properties of
the algorithm, but also of the underlying (usually rich) mathematical
theory.
We show how we can use ACL2, a first-order interactive theorem
prover, to reason about properties of algorithms that are typically implemented
as part of symbolic computation systems. We emphasize two
aspects. First, how we can override the apparent lack of expressiveness we
have using a first-order approach (at least compared to higher-order logics).
Second, how we can execute the algorithms (efficiently, if possible)
in the same setting where we formally reason about their correctness.
Three examples of formal verification of symbolic computation algorithms
are presented to illustrate the main issues one has to face in this
task: a Gr¨obner basis algorithm, a first-order unification algorithm based
on directed acyclic graphs, and the Eilenberg-Zilber algorithm, one of
the central components of a symbolic computation system in algebraic
topology
Stability analysis of fractional-order systems with randomly time-varying parameters
This paper is concerned with the stability of fractional-order systems with randomly timevarying parameters. Two approaches are provided to check the stability of such systems in mean sense. The first approach is based on suitable Lyapunov functionals to assess the stability, which is of vital importance in the theory of stability. By an example one finds that the stability conditions obtained by the first approach can be tabulated for some special cases. For some complicated linear and nonlinear systems, the stability conditions present computational difficulties. The second alternative approach is based on integral inequalities and ingenious mathematical method. Finally, we also give two examples to demonstrate the feasibility and advantage of the second approach. Compared with the stability conditions obtained by the first approach, the stability conditions obtained by the second one are easily verified by simple computation rather than complicated functional construction. The derived criteria improve the existing related results
Stability of a Cartesian grid projection method for zero Froude number shallow water flows
In this paper a Godunov-type projection method for computing approximate solutions of the zero Froude number (incompressible) shallow water
equations is presented. It is second-order accurate and locally conserves
height (mass) and momentum. To enforce the underlying divergence
constraint on the velocity field, the predicted numerical fluxes,
computed with a standard second order method for hyperbolic conservation
laws, are corrected in two steps. First, a MAC-type projection adjusts
the advective velocity divergence. In a second projection step, additional
momentum flux corrections are computed to obtain new time level cell-centered
velocities, which satisfy another discrete version of the divergence
constraint.
The scheme features an exact and stable second projection. It is obtained
by a Petrov-Galerkin finite element ansatz with piecewise bilinear
trial functions for the unknown incompressible height and piecewise
constant test functions. The stability of the projection is proved
using the theory of generalized mixed finite elements, which goes
back to Nicolaïdes (1982). In order to do so, the validity of
three different inf-sup conditions has to be shown.
Since the zero Froude number shallow water equations have the same
mathematical structure as the incompressible Euler equations of isentropic
gas dynamics, the method can be easily transfered to the computation
of incompressible variable density flow problems
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