Optimal threshold padlock systems

In 1968, Liu described the problem of securing documents in a shared secret project. In an example, at least six out of eleven participating scientists need to be present to open the lock securing the secret documents. Shamir proposed a mathematical solution to this physical problem in 1979, by designing an efficient k-out-of-n secret sharing scheme based on Lagrange’s interpolation. Liu and Shamir also claimed that the minimal solution using physical locks is clearly impractical and exponential in the number of participants. In this paper we relax some implicit assumptions in their claim and propose an optimal physical solution to the problem of Liu that uses physical padlocks, but the number of padlocks is not greater than the number of participants. Then, we show that no device can do better for k-out-of-n threshold padlock systems as soon as k ⩾ 2 n , which holds true in particular for Liu’s example. More generally, we derive bounds required to implement any threshold system and prove a lower bound of O ( log ( n ) ) padlocks for any threshold larger than 2. For instance we propose an optimal scheme reaching that bound for 2-out-of-n threshold systems and requiring less than 2 log 2 ( n ) padlocks. We also discuss more complex access structures, a wrapping technique, and other sublinear realizations like an algorithm to generate 3-out-of-n systems with 2.5 n padlocks. Finally we give an algorithm building k-out-of-n threshold padlock systems with only O ( log ( n ) k − 1 ) padlocks. Apart from the physical world, our results also show that it is possible to implement secret sharing over small fields.

Scalarization and convergence in unified set optimization

This paper deals with scalarization and stability aspects for a unified set optimization problem. We provide characterizations for a unified preference relation and the corresponding unified minimal solution in terms of a generalized oriented distance function of the sup-inf type. We establish continuity of a function associated with the generalized oriented distance function and provide an existence result for the unified minimal solution. We establish Painlevé-Kuratowski convergence of minimal solutions of a family of scalar problems to the minimal solutions of the unified set optimization problem.

Minimal solutions of fuzzy relation inequalities with addition-min composition1

This paper investigates minimal solutions of fuzzy relation inequalities with addition-min composition. It first shows the conditions that an element is a minimal solution of the inequalities, and presents the conditions that the inequalities have a unique minimal solution. It then proves that every solution of the inequalities has a minimal one and proposes an algorithm to searching for a minimal solution with computational complexity O (n 2) where n is the number of unknown variables of the inequalities. This paper finally describes all minimal solutions of the inequalities.

A variational sheath model for gyrokinetic Vlasov-Poisson equations

We construct a stationary gyrokinetic variational model for sheaths close to the metallic wall of a magnetized plasma, following a physical extremalization principle for the natural energy. By considering a reduced set of parameters we show that our model has a unique minimal solution, and that the resulting electric potential has an infinite number of oscillations as it propagates towards the core of the plasma. We prove this result for the non linear problem and also provide a simpler analysis for a linearized problem, based on the construction of exact solutions. Some numerical illustrations show the well-posedness of the model after numerical discretization. They also exhibit the oscillating behavior.

An Algorithm for Finding a Minimal Solution to Fuzzy Relation Inequalities with Addition-Min Composition

We first show a sufficient and necessary condition that a solution of fuzzy relation inequalities with addition-min composition is a minimal one. We then prove that for every solution of the fuzzy relation inequalities there exists a minimal solution that is less than or equal to the solution in a very different way. We finally give an algorithm to find a minimal solution for a given solution, which is illustrated by numerical examples.

Positive Solutions for Singular Anisotropic (p, q)-Equations

In this paper, we consider a Dirichlet problem driven by an anisotropic (p, q)-differential operator and a parametric reaction having the competing effects of a singular term and of a superlinear perturbation. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter moves. Moreover, we prove the existence of a minimal positive solution and determine the monotonicity and continuity properties of the minimal solution map.

Fast Healthcare Interoperability Resources (FHIR®) Representation of Medication Data Derived from German Procedure Classification Codes (OPS) Using Identification of Medicinal Products (IDMP) Compliant Terminology

Electronic documentation of medication data is one of the biggest challenges associated with digital clinical documentation. Despite its importance, it has not been consistently implemented in German university hospitals. In this paper we describe the approach of the German Medical Informatics Initiative (MII) towards the modelling of a medication core dataset using FHIR® profiles and standard-compliant terminologies. The FHIR profiles for Medication and MedicationStatement were adapted to the core dataset of the MIl. The terminologies to be used were selected based on the criteria of the ISO-standard for the Identification of Medicinal Products (IDMP). For a first use case with a minimal medication dataset, the entries in the medication chapter of the German Procedure Classification (OPS codes) were analyzed and mapped to IDMP-compliant medication terminology. OPS data are available at all German hospitals as they are mandatory for reimbursement purposes. Reimbursement-relevant encounter data containing OPS medication procedures were used to create a FHIR representation based on the FHIR profiles MedicationStatement and Medication. This minimal solution includes – besides the details on patient and start-/end-dates – the active ingredients identified by the IDMP-compliant codes and – if specified in the OPS code – the route of administration and the range of the amount of substance administered to the patient, using the appropriate unit of measurement code. With FHIR, the medication data can be represented in the data integration centers of the MII to provide a standardized format for data analysis across the MII sites.

Lights Out on graphs

We model the Lights Out game on general simple graphs in the framework of linear algebra over the field $$\mathbb{F}_{2}$$ F 2 . Based upon a version of the Fredholm alternative, we introduce a separating invariant of the game, i.e., an initial state can be transformed into a final state if and only if the values of the invariant of both states agree. We also investigate certain states with particularly interesting properties. Apart from the classical version of the game, we propose several variants, in particular a version with more than only two states (light on, light off), where the analysis relies on systems of linear equations over the ring $$\mathbb{Z}_{n}$$ Z n . Although it is easy to find a concrete solution of the Lights Out problem, we show that it is NP-hard to find a minimal solution. We also propose electric circuit diagrams to actually realize the Lights Out game.