1,070 research outputs found
The symmetric, D-invariant and Egorov reductions of the quadrilateral lattice
We present a detailed study of the geometric and algebraic properties of the
multidimensional quadrilateral lattice (a lattice whose elementary
quadrilaterals are planar; the discrete analogue of a conjugate net) and of its
basic reductions. To make this study, we introduce the notions of forward and
backward data, which allow us to give a geometric meaning to the tau-function
of the lattice, defined as the potential connecting these data. Together with
the known circular lattice (a lattice whose elementary quadrilaterals can be
inscribed in circles; the discrete analogue of an orthogonal conjugate net) we
introduce and study two other basic reductions of the quadrilateral lattice:
the symmetric lattice, for which the forward and backward data coincide, and
the D-invariant lattice, characterized by the invariance of a certain natural
frame along the main diagonal. We finally discuss the Egorov lattice, which is,
at the same time, symmetric, circular and D-invariant. The integrability
properties of all these lattices are established using geometric, algebraic and
analytic means; in particular we present a D-bar formalism to construct large
classes of such lattices. We also discuss quadrilateral hyperplane lattices and
the interplay between quadrilateral point and hyperplane lattices in all the
above reductions.Comment: 48 pages, 6 figures; 1 section added, to appear in J. Geom. & Phy
On the occurrence of gauge-dependent secularities in nonlinear gravitational waves
We study the plane (not necessarily monochromatic) gravitational waves at
nonlinear quadratic order on a flat background in vacuum. We show that, in the
harmonic gauge, the nonlinear waves are unstable. We argue that, at this order,
this instability can not be eliminated by means of a multiscale approach, i.e.
introducing suitable long variables, as it is often the case when secularities
appear in a perturbative scheme. However, this is a non-physical and
gauge-dependent effect that disappears in a suitable system of coordinates. In
facts, we show that in a specific gauge such instability does not occur, and
that it is possible to solve exactly the second order nonlinear equations of
gravitational waves. Incidentally, we note that this gauge coincides with the
one used by Belinski and Zakharov to find exact solitonic solutions of
Einstein's equations, that is to an exactly integrable case, and this fact
makes our second order nonlinear solutions less interesting. However, the
important warning is that one must be aware of the existence of the instability
reported in this paper, when studying nonlinear gravitational waves in the
harmonic gauge
The self-adjoint 5-point and 7-point difference operators, the associated Dirichlet problems, Darboux transformations and Lelieuvre formulas
We present some basic properties of two distinguished discretizations of
elliptic operators: the self-adjoint 5-point and 7-point schemes on a two
dimensional lattice. We first show that they allow to solve Dirichlet boundary
value problems; then we present their Darboux transformations. Finally we
construct their Lelieuvre formulas and we show that, at the level of the normal
vector and in full analogy with their continuous counterparts, the self-adjoint
5-point scheme characterizes a two dimensional quadrilateral lattice (a lattice
whose elementary quadrilaterals are planar), while the self-adjoint 7-point
scheme characterizes a generic 2D lattice.Comment: 20 pages, 6 figures, submitted to Glasgow Mathematical Journal Trust
for Island II proceedind
Cryptanalysis of a One-Time Code-Based Digital Signature Scheme
We consider a one-time digital signature scheme recently proposed by
Persichetti and show that a successful key recovery attack can be mounted with
limited complexity. The attack we propose exploits a single signature
intercepted by the attacker, and relies on a statistical analysis performed
over such a signature, followed by information set decoding. We assess the
attack complexity and show that a full recovery of the secret key can be
performed with a work factor that is far below the claimed security level. The
efficiency of the attack is motivated by the sparsity of the signature, which
leads to a significant information leakage about the secret key.Comment: 5 pages, 1 figur
Liquid FM: Recommending Music through Viscous Democracy
Most modern recommendation systems use the approach of collaborative
filtering: users that are believed to behave alike are used to produce
recommendations. In this work we describe an application (Liquid FM) taking a
completely different approach. Liquid FM is a music recommendation system that
makes the user responsible for the recommended items. Suggestions are the
result of a voting scheme, employing the idea of viscous democracy. Liquid FM
can also be thought of as the first testbed for this voting system. In this
paper we outline the design and architecture of the application, both from the
theoretical and from the implementation viewpoints
Layered Label Propagation: A MultiResolution Coordinate-Free Ordering for Compressing Social Networks
We continue the line of research on graph compression started with WebGraph,
but we move our focus to the compression of social networks in a proper sense
(e.g., LiveJournal): the approaches that have been used for a long time to
compress web graphs rely on a specific ordering of the nodes (lexicographical
URL ordering) whose extension to general social networks is not trivial. In
this paper, we propose a solution that mixes clusterings and orders, and devise
a new algorithm, called Layered Label Propagation, that builds on previous work
on scalable clustering and can be used to reorder very large graphs (billions
of nodes). Our implementation uses overdecomposition to perform aggressively on
multi-core architecture, making it possible to reorder graphs of more than 600
millions nodes in a few hours. Experiments performed on a wide array of web
graphs and social networks show that combining the order produced by the
proposed algorithm with the WebGraph compression framework provides a major
increase in compression with respect to all currently known techniques, both on
web graphs and on social networks. These improvements make it possible to
analyse in main memory significantly larger graphs
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