60,846 research outputs found
Ordered Exchange Graphs
The exchange graph of a cluster algebra encodes the combinatorics of
mutations of clusters. Through the recent "categorifications" of cluster
algebras using representation theory one obtains a whole variety of exchange
graphs associated with objects such as a finite-dimensional algebra or a
differential graded algebra concentrated in non-positive degrees. These
constructions often come from variations of the concept of tilting, the
vertices of the exchange graph being torsion pairs, t-structures, silting
objects, support -tilting modules and so on. All these exchange graphs
stemming from representation theory have the additional feature that they are
the Hasse quiver of a partial order which is naturally defined for the objects.
In this sense, the exchange graphs studied in this article can be considered as
a generalization or as a completion of the poset of tilting modules which has
been studied by Happel and Unger. The goal of this article is to axiomatize the
thus obtained structure of an ordered exchange graph, to present the various
constructions of ordered exchange graphs and to relate them among each other.Comment: References updated, and Theorem A.7 adde
Blow-up of a critical Sobolev norm for energy-subcritical and energy-supercritical wave equations
This work concerns the semilinear wave equation in three space dimensions
with a power-like nonlinearity which is greater than cubic, and not quintic
(i.e. not energy-critical). We prove that a scale-invariant Sobolev norm of any
non-scattering solution goes to infinity at the maximal time of existence. This
gives a refinement on known results on energy-subcritical and
energy-supercritical wave equation, with a unified proof.
The proof relies on the channel of energy method, as in arXiv:1204.0031, in
weighted scale-invariant Sobolev spaces which were introduced in
arXiv:1506.00788. These spaces are local, thus adapted to finite speed of
propagation, and related to a conservation law of the linear wave equation. We
also construct the adapted profile decomposition
The Quantification of Tooth Displacement
By using reference points from a single pixel marker placed at the center point of the cuspid teeth and the center point on each of the incisor teeth, a polynomial curve was generated as a native curve for each dental arch studied. The polynomial curve generated from actual tooth position in each arch provides the forensic odontologist with another reference point that is quantifiable. The study represents that individual characteristics, such as tooth displacement, can be quantified in a simple, reliable, and repeatable format
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