5,528 research outputs found
An extremal problem on crossing vectors
For positive integers and , two vectors and from
are called -crossing if there are two coordinates and
such that and . What is the maximum size of
a family of pairwise -crossing and pairwise non--crossing vectors in
? We state a conjecture that the answer is . We prove
the conjecture for and provide weaker upper bounds for .
Also, for all and , we construct several quite different examples of
families of desired size . This research is motivated by a natural
question concerning the width of the lattice of maximum antichains of a
partially ordered set.Comment: Corrections and improvement
Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents
We use the conformal bootstrap to perform a precision study of the operator
spectrum of the critical 3d Ising model. We conjecture that the 3d Ising
spectrum minimizes the central charge c in the space of unitary solutions to
crossing symmetry. Because extremal solutions to crossing symmetry are uniquely
determined, we are able to precisely reconstruct the first several Z2-even
operator dimensions and their OPE coefficients. We observe that a sharp
transition in the operator spectrum occurs at the 3d Ising dimension
Delta_sigma=0.518154(15), and find strong numerical evidence that operators
decouple from the spectrum as one approaches the 3d Ising point. We compare
this behavior to the analogous situation in 2d, where the disappearance of
operators can be understood in terms of degenerate Virasoro representations.Comment: 55 pages, many figures; v2 - refs and comments added, to appear in a
special issue of J.Stat.Phys. in memory of Kenneth Wilso
Perturbative and Non-Perturbative Aspects of N=8 Supergravity
Some aspects of quantum properties of N=8 supergravity in four dimensions are
discussed for non-practitioners. At perturbative level, they include the Weyl
trace anomaly as well as composite duality anomalies, the latter being relevant
for perturbative finiteness. At non-perturbative level, we briefly review some
facts about extremal black holes, their Bekenstein-Hawking entropy and
attractor flows for single- and two-centered solutions.Comment: 1+18 pages. Contribution to the Proceedings of the International
School of Subnuclear Physics, 48th Course: "What is Known and Unexpected at
LHC" Erice, Italy, Aug 29 - Sep 7 201
Bending branes for DCFT in two dimensions
We consider a holographic dual model for defect conformal field theories
(DCFT) in which we include the backreaction of the defect on the dual geometry.
In particular, we consider a dual gravity system in which a two-dimensional
hypersurface with matter fields, the brane, is embedded into a
three-dimensional asymptotically Anti-de Sitter spacetime. Motivated by recent
proposals for holographic duals of boundary conformal field theories (BCFT), we
assume the geometry of the brane to be determined by Israel junction
conditions. We show that these conditions are intimately related to the energy
conditions for the brane matter fields, and explain how these energy conditions
constrain the possible geometries. This has implications for the holographic
entanglement entropy in particular. Moreover, we give exact analytical
solutions for the case where the matter content of the brane is a perfect
fluid, which in a particular case corresponds to a free massless scalar field.
Finally, we describe how our results may be particularly useful for extending a
recent proposal for a holographic Kondo model.Comment: 35 pages + appendices, 12 figures, v2: added references and a
paragraph on negative tension solutions, v3: updated reference
X-code: MDS array codes with optimal encoding
We present a new class of MDS (maximum distance separable) array codes of size n×n (n a prime number) called X-code. The X-codes are of minimum column distance 3, namely, they can correct either one column error or two column erasures. The key novelty in X-code is that it has a simple geometrical construction which achieves encoding/update optimal complexity, i.e., a change of any single information bit affects exactly two parity bits. The key idea in our constructions is that all parity symbols are placed in rows rather than columns
Holographic Holes and Differential Entropy
Recently, it has been shown by Balasubramanian et al. and Myers et al. that
the Bekenstein-Hawking entropy formula evaluated on certain closed surfaces in
the bulk of a holographic spacetime has an interpretation as the differential
entropy of a particular family of intervals (or strips) in the boundary theory.
We first extend this construction to bulk surfaces which vary in time. We then
give a general proof of the equality between the gravitational entropy and the
differential entropy. This proof applies to a broad class of holographic
backgrounds possessing a generalized planar symmetry and to certain classes of
higher-curvature theories of gravity. To apply this theorem, one can begin with
a bulk surface and determine the appropriate family of boundary intervals by
considering extremal surfaces tangent to the given surface in the bulk.
Alternatively, one can begin with a family of boundary intervals; as we show,
the differential entropy then equals the gravitational entropy of a bulk
surface that emerges from the intersection of the neighboring entanglement
wedges, in a continuum limit.Comment: 62 pages; v2: minor improvements to presentation, references adde
Energy of a knot: variational principles; Mm-energy
Let be the energy of some knot for any from certain class of
functions. The problem is to find knots with extremal values of energy. We
discuss the notion of the locally perturbed knot. The knot circle minimizes
some energies and maximizes some others. So, is there any energy such
that the circle neither maximizes nor minimizes this energy? Recently it was
shown (A.Abrams, J.Cantarella, J.H.G.Fu, M.Ghomu, and R.Howard) that the answer
is positive. We prove that nevertheless the circle is a locally extremal knot,
i.e. the circle satisfies certain variational equations. We also find these
equations. Finally we represent Mm-energy for a knot. The definition of this
energy differs with one regarded above. Nevertheless besides its own properties
Mm-energy has some similar with M\"obius energy properties.Comment: 17 pages, 6 Postscript figure
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