7,823 research outputs found
Complete Graphs, Hilbert Series, and the Higgs branch of the 4d N=2 SCFT's
The strongly interacting 4d N=2 SCFT's of type are the simplest
examples of models in the class introduced by Cecotti, Neitzke,
and Vafa in arXiv:1006.3435. These systems have a known 3d N=4 mirror only if
divides , where is the Coxeter number. By 4d/2d
correspondence, we show that in this case these systems have a nontrivial
global flavor symmetry group and, therefore, a non-trivial Higgs branch. As an
application of the methods of arXiv:1309.2657, we then compute the refined
Hilbert series of the Coulomb branch of the 3d mirror for the simplest models
in the series. This equals the refined Hilbert series of the Higgs branch of
the SCFT, providing interesting information about the Higgs branch
of these non-lagrangian theories.Comment: 20 page
Topological finiteness properties of monoids. Part 1: Foundations
We initiate the study of higher dimensional topological finiteness properties
of monoids. This is done by developing the theory of monoids acting on CW
complexes. For this we establish the foundations of -equivariant homotopy
theory where is a discrete monoid. For projective -CW complexes we prove
several fundamental results such as the homotopy extension and lifting
property, which we use to prove the -equivariant Whitehead theorems. We
define a left equivariant classifying space as a contractible projective -CW
complex. We prove that such a space is unique up to -homotopy equivalence
and give a canonical model for such a space via the nerve of the right Cayley
graph category of the monoid. The topological finiteness conditions
left- and left geometric dimension are then defined for monoids
in terms of existence of a left equivariant classifying space satisfying
appropriate finiteness properties. We also introduce the bilateral notion of
-equivariant classifying space, proving uniqueness and giving a canonical
model via the nerve of the two-sided Cayley graph category, and we define the
associated finiteness properties bi- and geometric dimension. We
explore the connections between all of the these topological finiteness
properties and several well-studied homological finiteness properties of
monoids which are important in the theory of string rewriting systems,
including , cohomological dimension, and Hochschild
cohomological dimension. We also develop the corresponding theory of
-equivariant collapsing schemes (that is, -equivariant discrete Morse
theory), and among other things apply it to give topological proofs of results
of Anick, Squier and Kobayashi that monoids which admit presentations by
complete rewriting systems are left-, right- and bi-.Comment: 59 pages, 1 figur
Continuous renormalization for fermions and Fermi liquid theory
I derive a Wick ordered continuous renormalization group equation for fermion
systems and show that a determinant bound applies directly to this equation.
This removes factorials in the recursive equation for the Green functions, and
thus improves the combinatorial behaviour. The form of the equation is also
ideal for the investigation of many-fermion systems, where the propagator is
singular on a surface. For these systems, I define a criterion for Fermi liquid
behaviour which applies at positive temperatures. As a first step towards
establishing such behaviour in d ge 2, I prove basic regularity properties of
the interacting Fermi surface to all orders in a skeleton expansion. The proof
is a considerable simplification of previous ones.Comment: LaTeX, 3 eps figure
Curves having one place at infinity and linear systems on rational surfaces
Denoting by the linear system of plane
curves passing through generic points of the projective
plane with multiplicity (or larger) at each , we prove the
Harbourne-Hirschowitz Conjecture for linear systems determined by a wide family of systems of multiplicities
and arbitrary degree . Moreover, we provide an
algorithm for computing a bound of the regularity of an arbitrary system
and we give its exact value when is in the above family.
To do that, we prove an -vanishing theorem for line bundles on surfaces
associated with some pencils ``at infinity''.Comment: This is a revised version of a preprint of 200
Parallel Processing of Large Graphs
More and more large data collections are gathered worldwide in various IT
systems. Many of them possess the networked nature and need to be processed and
analysed as graph structures. Due to their size they require very often usage
of parallel paradigm for efficient computation. Three parallel techniques have
been compared in the paper: MapReduce, its map-side join extension and Bulk
Synchronous Parallel (BSP). They are implemented for two different graph
problems: calculation of single source shortest paths (SSSP) and collective
classification of graph nodes by means of relational influence propagation
(RIP). The methods and algorithms are applied to several network datasets
differing in size and structural profile, originating from three domains:
telecommunication, multimedia and microblog. The results revealed that
iterative graph processing with the BSP implementation always and
significantly, even up to 10 times outperforms MapReduce, especially for
algorithms with many iterations and sparse communication. Also MapReduce
extension based on map-side join usually noticeably presents better efficiency,
although not as much as BSP. Nevertheless, MapReduce still remains the good
alternative for enormous networks, whose data structures do not fit in local
memories.Comment: Preprint submitted to Future Generation Computer System
Deterministic Approximation of Random Walks in Small Space
We give a deterministic, nearly logarithmic-space algorithm that given an undirected graph G, a positive integer r, and a set S of vertices, approximates the conductance of S in the r-step random walk on G to within a factor of 1+epsilon, where epsilon>0 is an arbitrarily small constant. More generally, our algorithm computes an epsilon-spectral approximation to the normalized Laplacian of the r-step walk.
Our algorithm combines the derandomized square graph operation [Eyal Rozenman and Salil Vadhan, 2005], which we recently used for solving Laplacian systems in nearly logarithmic space [Murtagh et al., 2017], with ideas from [Cheng et al., 2015], which gave an algorithm that is time-efficient (while ours is space-efficient) and randomized (while ours is deterministic) for the case of even r (while ours works for all r). Along the way, we provide some new results that generalize technical machinery and yield improvements over previous work. First, we obtain a nearly linear-time randomized algorithm for computing a spectral approximation to the normalized Laplacian for odd r. Second, we define and analyze a generalization of the derandomized square for irregular graphs and for sparsifying the product of two distinct graphs. As part of this generalization, we also give a strongly explicit construction of expander graphs of every size
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