11,485 research outputs found
On non-minimal N=4 supermultiplets in 1D and their associated sigma-models
We construct the non-minimal linear representations of the N=4 Extended
Supersymmetry in one-dimension. They act on 8 bosonic and 8 fermionic fields.
Inequivalent representations are specified by the mass-dimension of the fields
and the connectivity of the associated graphs. The oxidation to minimal N=5
linear representations is given. Two types of N=4 sigma-models based on
non-minimal representations are obtained: the resulting off-shell actions are
either manifestly invariant or depend on a constrained prepotential. The
connectivity properties of the graphs play a decisive role in discriminating
inequivalent actions. These results find application in partial breaking of
supersymmetric theories.Comment: 24 pages, 6 figure
Schnyder decompositions for regular plane graphs and application to drawing
Schnyder woods are decompositions of simple triangulations into three
edge-disjoint spanning trees crossing each other in a specific way. In this
article, we define a generalization of Schnyder woods to -angulations (plane
graphs with faces of degree ) for all . A \emph{Schnyder
decomposition} is a set of spanning forests crossing each other in a
specific way, and such that each internal edge is part of exactly of the
spanning forests. We show that a Schnyder decomposition exists if and only if
the girth of the -angulation is . As in the case of Schnyder woods
(), there are alternative formulations in terms of orientations
("fractional" orientations when ) and in terms of corner-labellings.
Moreover, the set of Schnyder decompositions on a fixed -angulation of girth
is a distributive lattice. We also show that the structures dual to
Schnyder decompositions (on -regular plane graphs of mincut rooted at a
vertex ) are decompositions into spanning trees rooted at such
that each edge not incident to is used in opposite directions by two
trees. Additionally, for even values of , we show that a subclass of
Schnyder decompositions, which are called even, enjoy additional properties
that yield a reduced formulation; in the case d=4, these correspond to
well-studied structures on simple quadrangulations (2-orientations and
partitions into 2 spanning trees). In the case d=4, the dual of even Schnyder
decompositions yields (planar) orthogonal and straight-line drawing algorithms.
For a 4-regular plane graph of mincut 4 with vertices plus a marked
vertex , the vertices of are placed on a grid according to a permutation pattern, and in the orthogonal drawing
each of the edges of has exactly one bend. Embedding
also the marked vertex is doable at the cost of two additional rows and
columns and 8 additional bends for the 4 edges incident to . We propose a
further compaction step for the drawing algorithm and show that the obtained
grid-size is strongly concentrated around for a uniformly
random instance with vertices
Constructing and Classifying Fully Irreducible Outer Automorphisms of Free Groups
The main theorem of this document emulates, in the context of Out(F_r)
theory, a mapping class group theorem (by H. Masur and J. Smillie) that
determines precisely which index lists arise from pseudo-Anosov mapping
classes. Since the ideal Whitehead graph gives a finer invariant in the
analogous setting of a fully irreducible outer automorphism, we instead focus
on determining which of the 21 connected, loop-free, 5-vertex graphs are ideal
Whitehead graphs of ageometric, fully irreducible outer automorphisms of the
free group of rank 3. Our main theorem accomplishes this by showing that there
are precisely 18 graphs arising as such. We also give a method for identifying
certain complications called periodic Nielsen paths, prove the existence of
conveniently decomposed representatives of ageometric, fully irreducible outer
automorphisms having connected, loop-free, (2r-1)-vertex ideal Whitehead
graphs, and prove a criterion for identifying representatives of ageometric,
fully irreducible outer automorphisms. The methods we use for constructing
fully irreducible outer automorphisms of free groups, as well as our
identification and decomposition techniques, can be used to extend our main
theorem, as they are valid in any rank. Our methods of proof rely primarily on
Bestvina-Feighn-Handel train track theory and the theory of attracting
laminations
Multiloop functional renormalization group that sums up all parquet diagrams
We present a multiloop flow equation for the four-point vertex in the
functional renormalization group (fRG) framework. The multiloop flow consists
of successive one-loop calculations and sums up all parquet diagrams to
arbitrary order. This provides substantial improvement of fRG computations for
the four-point vertex and, consequently, the self-energy. Using the X-ray-edge
singularity as an example, we show that solving the multiloop fRG flow is
equivalent to solving the (first-order) parquet equations and illustrate this
with numerical results
Dynamic Dominators and Low-High Orders in DAGs
We consider practical algorithms for maintaining the dominator tree and a low-high order in directed acyclic graphs (DAGs) subject to dynamic operations. Let G be a directed graph with a distinguished start vertex s. The dominator tree D of G is a tree rooted at s, such that a vertex v is an ancestor of a vertex w if and only if all paths from s to w in G include v. The dominator tree is a central tool in program optimization and code generation, and has many applications in other diverse areas including constraint programming, circuit testing, biology, and in algorithms for graph connectivity problems. A low-high order of G is a preorder of D that certifies the correctness of D, and has further applications in connectivity and path-determination problems.
We first provide a practical and carefully engineered version of a recent algorithm [ICALP 2017] for maintaining the dominator tree of a DAG through a sequence of edge deletions. The algorithm runs in O(mn) total time and O(m) space, where n is the number of vertices and m is the number of edges before any deletion. In addition, we present a new algorithm that maintains a low-high order of a DAG under edge deletions within the same bounds. Both results extend to the case of reducible graphs (a class that includes DAGs). Furthermore, we present a fully dynamic algorithm for maintaining the dominator tree of a DAG under an intermixed sequence of edge insertions and deletions. Although it does not maintain the O(mn) worst-case bound of the decremental algorithm, our experiments highlight that the fully dynamic algorithm performs very well in practice. Finally, we study the practical efficiency of all our algorithms by conducting an extensive experimental study on real-world and synthetic graphs
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