104,980 research outputs found
Structural Properties of Index Coding Capacity Using Fractional Graph Theory
The capacity region of the index coding problem is characterized through the
notion of confusion graph and its fractional chromatic number. Based on this
multiletter characterization, several structural properties of the capacity
region are established, some of which are already noted by Tahmasbi, Shahrasbi,
and Gohari, but proved here with simple and more direct graph-theoretic
arguments. In particular, the capacity region of a given index coding problem
is shown to be simple functionals of the capacity regions of smaller
subproblems when the interaction between the subproblems is none, one-way, or
complete.Comment: 5 pages, to appear in the 2015 IEEE International Symposium on
Information Theory (ISIT
A new property of the Lov\'asz number and duality relations between graph parameters
We show that for any graph , by considering "activation" through the
strong product with another graph , the relation between the independence number and the Lov\'{a}sz number of
can be made arbitrarily tight: Precisely, the inequality
becomes asymptotically an equality for a suitable sequence of ancillary graphs
.
This motivates us to look for other products of graph parameters of and
on the right hand side of the above relation. For instance, a result of
Rosenfeld and Hales states that with the fractional
packing number , and for every there exists that makes the
above an equality; conversely, for every graph there is a that attains
equality.
These findings constitute some sort of duality of graph parameters, mediated
through the independence number, under which and are dual
to each other, and the Lov\'{a}sz number is self-dual. We also show
duality of Schrijver's and Szegedy's variants and
of the Lov\'{a}sz number, and explore analogous notions for the chromatic
number under strong and disjunctive graph products.Comment: 16 pages, submitted to Discrete Applied Mathematics for a special
issue in memory of Levon Khachatrian; v2 has a full proof of the duality
between theta+ and theta- and a new author, some new references, and we
corrected several small errors and typo
Identifying codes in vertex-transitive graphs and strongly regular graphs
We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2ln(vertical bar V vertical bar) + 1 where V is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that reach both bounds. We exhibit infinite families of vertex-transitive graphs with integer and fractional identifying codes of order vertical bar V vertical bar(alpha) with alpha is an element of{1/4, 1/3, 2/5}These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs
Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates
With a view toward fractal spaces, by using a Korevaar-Schoen space approach,
we introduce the class of bounded variation (BV) functions in a general
framework of strongly local Dirichlet spaces with a heat kernel satisfying
sub-Gaussian estimates. Under a weak Bakry-\'Emery curvature type condition,
which is new in this setting, this BV class is identified with a heat semigroup
based Besov class. As a consequence of this identification, properties of BV
functions and associated BV measures are studied in detail. In particular, we
prove co-area formulas, global Sobolev embeddings and isoperimetric
inequalities. It is shown that for nested fractals or their direct products the
BV class we define is dense in . The examples of the unbounded Vicsek set,
unbounded Sierpinski gasket and unbounded Sierpinski carpet are discussed.Comment: The notes arXiv:1806.03428 will be divided in a series of papers.
This is the third paper. v2: Final versio
B-branes and supersymmetric quivers in 2d
We study 2d supersymmetric quiver gauge theories that
describe the low-energy dynamics of D1-branes at Calabi-Yau fourfold (CY)
singularities. On general grounds, the holomorphic sector of these
theories---matter content and (classical) superpotential interactions---should
be fully captured by the topological -model on the CY. By studying a
number of examples, we confirm this expectation and flesh out the dictionary
between B-brane category and supersymmetric quiver: the matter content of the
supersymmetric quiver is encoded in morphisms between B-branes (that is, Ext
groups of coherent sheaves), while the superpotential interactions are encoded
in the algebra satisfied by the morphisms. This provides us with a
derivation of the supersymmetric quiver directly from the CY geometry. We
also suggest a relation between triality of gauge theories
and certain mutations of exceptional collections of sheaves. 0d
supersymmetric quivers, corresponding to D-instantons probing CY
singularities, can be discussed similarly.Comment: 63 pages plus appendix, 21 figures; v2: corrected typos and added
reference. JHEP version; v3: added referenc
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