17,492 research outputs found
Perfectly secure data aggregation via shifted projections
We study a general scenario where confidential information is distributed
among a group of agents who wish to share it in such a way that the data
becomes common knowledge among them but an eavesdropper intercepting their
communications would be unable to obtain any of said data. The information is
modelled as a deck of cards dealt among the agents, so that after the
information is exchanged, all of the communicating agents must know the entire
deal, but the eavesdropper must remain ignorant about who holds each card.
Valentin Goranko and the author previously set up this scenario as the secure
aggregation of distributed information problem and constructed weakly safe
protocols, where given any card , the eavesdropper does not know with
certainty which agent holds . Here we present a perfectly safe protocol,
which does not alter the eavesdropper's perceived probability that any given
agent holds . In our protocol, one of the communicating agents holds a
larger portion of the cards than the rest, but we show how for infinitely many
values of , the number of cards may be chosen so that each of the agents
holds more than cards and less than
The intuitionistic temporal logic of dynamical systems
A dynamical system is a pair , where is a topological space and
is continuous. Kremer observed that the language of
propositional linear temporal logic can be interpreted over the class of
dynamical systems, giving rise to a natural intuitionistic temporal logic. We
introduce a variant of Kremer's logic, which we denote , and show
that it is decidable. We also show that minimality and Poincar\'e recurrence
are both expressible in the language of , thus providing a
decidable logic expressive enough to reason about non-trivial asymptotic
behavior in dynamical systems
Supersymmetric Quantum Mechanics and Painlev\'e IV Equation
As it has been proven, the determination of general one-dimensional
Schr\"odinger Hamiltonians having third-order differential ladder operators
requires to solve the Painlev\'e IV equation. In this work, it will be shown
that some specific subsets of the higher-order supersymmetric partners of the
harmonic oscillator possess third-order differential ladder operators. This
allows us to introduce a simple technique for generating solutions of the
Painlev\'e IV equation. Finally, we classify these solutions into three
relevant hierarchies.Comment: Proceedings of the Workshop 'Supersymmetric Quantum Mechanics and
Spectral Design' (July 18-30, 2010, Benasque, Spain
Supersymmetric quantum mechanics and Painleve equations
In these lecture notes we shall study first the supersymmetric quantum
mechanics (SUSY QM), specially when applied to the harmonic and radial
oscillators. In addition, we will define the polynomial Heisenberg algebras
(PHA), and we will study the general systems ruled by them: for zero and first
order we obtain the harmonic and radial oscillators, respectively; for second
and third order PHA the potential is determined by solutions to Painleve IV
(PIV) and Painleve V (PV) equations. Taking advantage of this connection, later
on we will find solutions to PIV and PV equations expressed in terms of
confluent hypergeometric functions. Furthermore, we will classify them into
several solution hierarchies, according to the specific special functions they
are connected with.Comment: 38 pages, 20 figures. Lecture presented at the XLIII Latin American
School of Physics: ELAF 2013 in Mexico Cit
Harmonic Oscillator SUSY Partners and Evolution Loops
Supersymmetric quantum mechanics is a powerful tool for generating exactly
solvable potentials departing from a given initial one. If applied to the
harmonic oscillator, a family of Hamiltonians ruled by polynomial Heisenberg
algebras is obtained. In this paper it will be shown that the SUSY partner
Hamiltonians of the harmonic oscillator can produce evolution loops. The
corresponding geometric phases will be as well studied
Cyclic -algebras and double Poisson algebras
In this article we prove that there exists an explicit bijection between nice
-pre-Calabi-Yau algebras and -double Poisson differential graded
algebras, where , extending a result proved by N. Iyudu and
M. Kontsevich. We also show that this correspondence is functorial in a quite
satisfactory way, giving rise to a (partial) functor from the category of
-double Poisson dg algebras to the partial category of -pre-Calabi-Yau
algebras. Finally, we further generalize it to include double
-algebras, as introduced by T. Schedler.Comment: 27 pages. All comments are welcome
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