53,108 research outputs found
Graded cluster algebras
In the cluster algebra literature, the notion of a graded cluster algebra has
been implicit since the origin of the subject. In this work, we wish to bring
this aspect of cluster algebra theory to the foreground and promote its study.
We transfer a definition of Gekhtman, Shapiro and Vainshtein to the algebraic
setting, yielding the notion of a multi-graded cluster algebra. We then study
gradings for finite type cluster algebras without coefficients, giving a full
classification.
Translating the definition suitably again, we obtain a notion of
multi-grading for (generalised) cluster categories. This setting allows us to
prove additional properties of graded cluster algebras in a wider range of
cases. We also obtain interesting combinatorics---namely tropical frieze
patterns---on the Auslander--Reiten quivers of the categories.Comment: 23 pages, 6 figures. v2: Substantially revised with additional
results. New section on graded (generalised) cluster categories. v3: added
Prop. 5.5 on relationship with Grothendieck group of cluster categor
Automorphism groupoids in noncommutative projective geometry
We address a natural question in noncommutative geometry, namely the rigidity
observed in many examples, whereby noncommutative spaces (or equivalently their
coordinate algebras) have very few automorphisms by comparison with their
commutative counterparts.
In the framework of noncommutative projective geometry, we define a groupoid
whose objects are noncommutative projective spaces of a given dimension and
whose morphisms correspond to isomorphisms of these. This groupoid is then a
natural generalization of an automorphism group. Using work of Zhang, we may
translate this structure to the algebraic side, wherein we consider homogeneous
coordinate algebras of noncommutative projective spaces. The morphisms in our
groupoid precisely correspond to the existence of a Zhang twist relating the
two coordinate algebras.
We analyse this automorphism groupoid, showing that in dimension 1 it is
connected, so that every noncommutative is isomorphic to
commutative . For dimension 2 and above, we use the geometry of
the point scheme, as introduced by Artin-Tate-Van den Bergh, to relate
morphisms in our groupoid to certain automorphisms of the point scheme.
We apply our results to two important examples, quantum projective spaces and
Sklyanin algebras. In both cases, we are able to use the geometry of the point
schemes to fully describe the corresponding component of the automorphism
groupoid. This provides a concrete description of the collection of Zhang
twists of these algebras.Comment: 27 pages; v2: minor corrections and additional reference
Holomorphic subgraph reduction of higher-point modular graph forms
Modular graph forms are a class of modular covariant functions which appear
in the genus-one contribution to the low-energy expansion of closed string
scattering amplitudes. Modular graph forms with holomorphic subgraphs enjoy the
simplifying property that they may be reduced to sums of products of modular
graph forms of strictly lower loop order. In the particular case of dihedral
modular graph forms, a closed form expression for this holomorphic subgraph
reduction was obtained previously by D'Hoker and Green. In the current work, we
extend these results to trihedral modular graph forms. Doing so involves the
identification of a modular covariant regularization scheme for certain
conditionally convergent sums over discrete momenta, with some elements of the
sum being excluded. The appropriate regularization scheme is identified for any
number of exclusions, which in principle allows one to perform holomorphic
subgraph reduction of higher-point modular graph forms with arbitrary
holomorphic subgraphs.Comment: 38 pages; v2: publication versio
On cavitation in Elastodynamics
Motivated by the works of Ball (1982) and Pericak-Spector and Spector (1988), we investigate singular solutions of the compressible nonlinear elastodynamics equations.
These singular solutions contain discontinuities in the displacement field and
can be seen as describing fracture or cavitation.
We explore a definition of singular solution via approximating sequences of smooth functions.
We use these approximating sequences to investigate the energy of such solutions, taking into account the energy needed to open a crack or hole.
In particular, we find that the existence of singular solutions and the finiteness of their energy
is strongly related to the behavior of the stress response function for infinite stretching, i.e.
the material has to display a sufficient amount of softening.
In this note we detail our findings in one space dimension
Singular limiting induced from continuum solutions and the problem of dynamic cavitation
In the works of
K.A. Pericak-Spector and S. Spector [Pericak-Spector, Spector 1988, 1997] a class of self-similar
solutions are constructed for the equations of radial isotropic elastodynamics
that describe cavitating solutions. Cavitating solutions decrease the total
mechanical energy and provide a striking example of non-uniqueness of entropy weak solutions
(for polyconvex energies) due to point-singularities at the cavity. To resolve
this paradox, we introduce the concept of singular limiting induced from continuum solution (or slic-solution),
according to which a discontinuous motion is a slic-solution if its averages
form a family of smooth approximate solutions to the problem. It turns out that there is an energetic cost for
creating the cavity, which is captured by the notion of slic-solution but neglected by the
usual entropic weak solutions. Once this cost is accounted for, the total mechanical energy of the
cavitating solution is in fact larger than that of the homogeneously deformed state.
We also apply the notion of slic-solutions to a one-dimensional example describing the onset of fracture,
and to gas dynamics in Langrangean coordinates with Riemann data inducing vacuum in the wave fan
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