A Mathematical Approach to Comply with Ethical Constraints in Compassionate Use Treatments

Patients who are seriously ill may ask doctors to treat them with unapproved medication, about which not much is known, or else with known medication in a high dosage. Apart from strict legal constraints such cases may involve difficult ethical questions as e.g. how long a series of treatments of different patients should be continued. Similar questions also arise in less serious situations. A physician trusts that a certain combination of freely available drugs are efficient against a specific disease and tries to help patients and to follow at the same time the primum-non-nocere principle. The objective of this paper is to contribute to the research on such questions in the form of mathematical models. Arguing in a step-to-step approach, we will show that certain sequential optimisation problems comply in a natural way with the true spirit of major ethical principles in medicine. We then suggest protocols and associate algorithms to find optimal, or approximately optimal, treatment strategies. Although the contribution may sometimes be difficult to apply in medical practice, the author thinks that the rational behind the approach offers a valuable alternative for finding decision support and should attract attention.Comment: 16 page

Resource dependent branching processes and the envelope of societies

Since its early beginnings, mankind has put to test many different society forms, and this fact raises a complex of interesting questions. The objective of this paper is to present a general population model which takes essential features of any society into account and which gives interesting answers on the basis of only two natural hypotheses. One is that societies want to survive, the second, that individuals in a society would, in general, like to increase their standard of living. We start by presenting a mathematical model, which may be seen as a particular type of a controlled branching process. All conditions of the model are justified and interpreted. After several preliminary results about societies in general we can show that two society forms should attract particular attention, both from a qualitative and a quantitative point of view. These are the so-called weakest-first society and the strongest-first society. In particular we prove then that these two societies stand out since they form an envelope of all possible societies in a sense we will make precise. This result (the envelopment theorem) is seen as significant because it is paralleled with precise survival criteria for the enveloping societies. Moreover, given that one of the "limiting" societies can be seen as an extreme form of communism, and the other one as being close to an extreme version of capitalism, we conclude that, remarkably, humanity is close to having already tested the limits.Comment: Published in at http://dx.doi.org/10.1214/13-AAP998 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Optimal realization of the transposition maps

We solve the problem of achieving the optimal physical approximation of the transposition for pure states of arbitrary quantum systems for finite and infinite dimensions. A unitary realization is also given for any finite dimension, which provides the optimal quantum cloning map of the ancilla as well.Comment: 10 pages. No figures. Elsart styl

Explaining National Trends in Terrestrial Water Storage

Access to fresh water is critical for human well-being, economic activity and, in some cases, political stability. Data from the Gravity Recovery and Climate Experiment (GRACE) has been used to monitor variability and trends in total water storage. This makes it possible to associate changes in water storage with both climate variability and large scale water management. Recent research has shown that these trends can be associated, globally, with rainfall, irrigation, and climate model predictions. This research indicates a need for further investigation into specific human predictors of trends in terrestrial water storage. This paper presents the first global scale analysis of GRACE trends focused on national scale socio-economic predictors of terrestrial water storage. We show that rainfall, irrigation, agricultural characteristics, and energy practices all contribute to GRACE trends, and the importance of each differs by country and region. Additionally, this work suggests that other factors such as GDP, population density, urbanization, and forest cover do not explain GRACE trends at a national level. Identifying these key predictors aids in understanding trends in water availability and for informing water management policy in a changing climate

The rencontre problem

Let $\left\{X^{1}_k\right\}_{k=1}^{\infty}, \left\{X^{2}_k\right\}_{k=1}^{\infty}, \cdots, \left\{X^{d}_k\right\}_{k=1}^{\infty}$ be $d$ independent sequences of Bernoulli random variables with success-parameters $p_1, p_2, \cdots, p_d$ respectively, where $d \geq 2$ is a positive integer, and $0<p_j<1$ for all $j=1,2,\cdots,d.$ Let \begin{equation*} S^{j}(n) = \sum_{i=1}^{n} X^{j}_{i} = X^{j}_{1} + X^{j}_{2} + \cdots + X^{j}_{n}, \quad n =1,2 , \cdots. \end{equation*} We declare a "rencontre" at time $n$, or, equivalently, say that $n$ is a "rencontre-time," if \begin{equation*} S^{1}(n) = S^{2}(n) = \cdots = S^{d}(n). \end{equation*} We motivate and study the distribution of the first (provided it is finite) rencontre time.Comment: 40 pages, 1 tabl