5,265 research outputs found
Blow-up analysis of conformal metrics of the disk with prescribed Gaussian and geodesic curvatures
This paper is concerned with the compactness of metrics of the disk with
prescribed Gaussian and geodesic curvatures. We consider a blowing-up sequence
of metrics and give a precise description of its asymptotic behavior. In
particular, the metrics blow-up at a unique point on the boundary and we are
able to give necessary conditions on its location. It turns out that such
conditions depend locally on the Gaussian curvatures but they depend on the
geodesic curvatures in a nonlocal way. This is a novelty with respect to the
classical Nirenberg problem where the blow-up conditions are local, and this
new aspect is driven by the boundary condition.Comment: 31 pages, 1 figur
Interacting two-state Markov chains on undirected networks
It is shown that irreducible two-state continuous-time Markov chains
interacting on a network in a bilinear fashion have a unique stable steady
state. The proof is elementary and uses the relative entropy function.Comment: 9 pages; minor typos correcte
Zonotopes and four-dimensional superconformal field theories
The a-maximization technique proposed by Intriligator and Wecht allows us to
determine the exact R-charges and scaling dimensions of the chiral operators of
four-dimensional superconformal field theories. The problem of existence and
uniqueness of the solution, however, has not been addressed in general setting.
In this paper, it is shown that the a-function has always a unique critical
point which is also a global maximum for a large class of quiver gauge theories
specified by toric diagrams. Our proof is based on the observation that the
a-function is given by the volume of a three dimensional polytope called
"zonotope", and the uniqueness essentially follows from Brunn-Minkowski
inequality for the volume of convex bodies. We also show a universal upper
bound for the exact R-charges, and the monotonicity of a-function in the sense
that a-function decreases whenever the toric diagram shrinks. The relationship
between a-maximization and volume-minimization is also discussed.Comment: 29 pages, 15 figures, reference added, typos corrected, version
published in JHE
Grilliot's trick in Nonstandard Analysis
The technique known as Grilliot's trick constitutes a template for explicitly
defining the Turing jump functional in terms of a given
effectively discontinuous type two functional. In this paper, we discuss the
standard extensionality trick: a technique similar to Grilliot's trick in
Nonstandard Analysis. This nonstandard trick proceeds by deriving from the
existence of certain nonstandard discontinuous functionals, the Transfer
principle from Nonstandard analysis limited to -formulas; from this
(generally ineffective) implication, we obtain an effective implication
expressing the Turing jump functional in terms of a discontinuous functional
(and no longer involving Nonstandard Analysis). The advantage of our
nonstandard approach is that one obtains effective content without paying
attention to effective content. We also discuss a new class of functionals
which all seem to fall outside the established categories. These functionals
directly derive from the Standard Part axiom of Nonstandard Analysis.Comment: 21 page
Strategic Interaction and Networks
This paper brings a general network analysis to a wide class of economic games. A network, or interaction matrix, tells who directly interacts with whom. A major challenge is determining how network structure shapes overall outcomes. We have a striking result. Equilibrium conditions depend on a single number: the lowest eigenvalue of a network matrix. Combining tools from potential games, optimization, and spectral graph theory, we study games with linear best replies and characterize the Nash and stable equilibria for any graph and for any impact of playersâ actions. When the graph is sufficiently absorptive (as measured by this eigenvalue), there is a unique equilibrium. When it is less absorptive, stable equilibria always involve extreme play where some agents take no actions at all. This paper is the first to show the importance of this measure to social and economic outcomes, and we relate it to different network link patterns.Networks, potential games, lowest eigenvalue, stable equilibria, asymmetric equilibria
- âŠ