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 $c$, the eavesdropper does not know with certainty which agent holds $c$. Here we present a perfectly safe protocol, which does not alter the eavesdropper's perceived probability that any given agent holds $c$. 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 $a$, the number of cards may be chosen so that each of the $m$ agents holds more than $a$ cards and less than $2m^2a$

The intuitionistic temporal logic of dynamical systems

A dynamical system is a pair $(X,f)$, where $X$ is a topological space and $f\colon X\to X$ 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 ${\sf ITL^c}$, and show that it is decidable. We also show that minimality and Poincar\'e recurrence are both expressible in the language of ${\sf ITL^c}$, thus providing a decidable logic expressive enough to reason about non-trivial asymptotic behavior in dynamical systems

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 A∞A_{\infty}-algebras and double Poisson algebras

In this article we prove that there exists an explicit bijection between nice $d$-pre-Calabi-Yau algebras and $d$-double Poisson differential graded algebras, where $d \in \mathbb{Z}$, 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 $d$-double Poisson dg algebras to the partial category of $d$-pre-Calabi-Yau algebras. Finally, we further generalize it to include double $P_{\infty}$-algebras, as introduced by T. Schedler.Comment: 27 pages. All comments are welcome

Trends in Supersymmetric Quantum Mechanics

Along the years, supersymmetric quantum mechanics (SUSY QM) has been used for studying solvable quantum potentials. It is the simplest method to build Hamiltonians with prescribed spectra in the spectral design. The key is to pair two Hamiltonians through a finite order differential operator. Some related subjects can be simply analyzed, as the algebras ruling both Hamiltonians and the associated coherent states. The technique has been applied also to periodic potentials, where the spectra consist of allowed and forbidden energy bands. In addition, a link with non-linear second-order differential equations, and the possibility of generating some solutions, can be explored. Recent applications concern the study of Dirac electrons in graphene placed either in electric or magnetic fields, and the analysis of optical systems whose relevant equations are the same as those of SUSY QM. These issues will be reviewed briefly in this paper, trying to identify the most important subjects explored currently in the literature