The problem of two fixed centers: bifurcation diagram for positive energies

We give a comprehensive analysis of the Euler-Jacobi problem of motion in the field of two fixed centers with arbitrary relative strength and for positive values of the energy. These systems represent nontrivial examples of integrable dynamics and are analysed from the point of view of the energy-momentum mapping from the phase space to the space of the integration constants. In this setting, we describe the structure of the scattering trajectories in phase space and derive an explicit description of the bifurcation diagram, i.e., the set of critical value of the energy-momentum map.

Large deviations of Rouse polymer chain: first passage problem

The purpose of this paper is to investigate several analytical methods of solving first passage (FP) problem for the Rouse model, a simplest model of a polymer chain. We show that this problem has to be treated as a multi-dimensional Kramers' problem, which presents rich and unexpected behavior. We first perform direct and forward-flux sampling (FFS) simulations, and measure the mean first-passage time $\tau(z)$ for the free end to reach a certain distance $z$ away from the origin. The results show that the mean FP time is getting faster if the Rouse chain is represented by more beads. Two scaling regimes of $\tau(z)$ are observed, with transition between them varying as a function of chain length. We use these simulations results to test two theoretical approaches. One is a well known asymptotic theory valid in the limit of zero temperature. We show that this limit corresponds to fully extended chain when each chain segment is stretched, which is not particularly realistic. A new theory based on the well known Freidlin-Wentzell theory is proposed, where dynamics is projected onto the minimal action path. The new theory predicts both scaling regimes correctly, but fails to get the correct numerical prefactor in the first regime. Combining our theory with the FFS simulations lead us to a simple analytical expression valid for all extensions and chain lengths. One of the applications of polymer FP problem occurs in the context of branched polymer rheology. In this paper, we consider the arm-retraction mechanism in the tube model, which maps exactly on the model we have solved. The results are compared to the Milner-McLeish theory without constraint release, which is found to overestimate FP time by a factor of 10 or more.

Profits, malversations, restitutions. Les bénéfices des financiers durant la guerre de la Ligue d’Augsbourg et la taxe de Chamillart

Profit, embezzlement, restitution. The role of the traitants in the Nine Years War and Chamillart’s tax on financial benefits The aim of this article is to revisit the question of the financiers in Old Regime France. It starts with an analysis of the discourses about the financiers under the Absolute monarchy that underlines the complexity of their relationship with the government and the public. It then reviews the secondary literature and highlights the existence of competing historical interpretations (functional, political, utilitarian), which raise the question of their overall capacity to account for the role and impact of the financiers at different times. On this ground, the article focuses on a specific group of financiers, the so-called traitants d’affaires extraordinaires, during the Nine Years War. Further to a description of the specific role and scope of the activities of the various financiers responsible for helping the monarchy to raise the funds it needed to pay for its peace and wartime expenditure, the article examines the conditions and profits granted by the king in his contracts with the traitants whose services were hired for the purpose of selling royal offices in the public and advancing the revenue to the Treasury. It also explores the contractual arrangements of the companies established by the financiers to manage their operations as well as the rights and the responsibilities of their various stakeholders. These bases being laid, the article relies on the administrative correspondence relating to the traités during the Nine Years War to address a range of issues, in particular the extent to which these contracts, and other control procedures, were robust enough to deter fraud. The accounts of two traitants’ companies offer an opportunity to analyse and compare the structure of their income and expenditure (including the volume and cost of the promissory notes sold in the public to finance their payments to the Treasury), to explore the strategies of the contractors, to calculate their net profits and further discuss the problem of embezzlement. The article ends with the study of the context and debates which led to the introduction by finance minister Michel Chamillart, in 1700, of a shortfall tax on the financial profits of the gens d’affaires or traitants, the method used to determine its rate (50 % of the net benefits), its distribution among the various stakeholders (including the bailleurs de fonds or backers), and the related procedures. In total, the article argues that the relationship between the monarchy, society and the financiers under the Ancien Regime was not static and, therefore, suggests that the broad question of control and fraud must be examined against changing circumstances. With regard specifically to the Nine Years War, the article concludes that within the constraints of the Absolute monarchy, contractors offered valuable services by raising capital for the benefit of a king who ruled over a country which, at the time, was by far the wealthiest in Europe, and where ministers failed to foresee long wars of attrition and whose financial strategy was limited by the very existence of privilege. Overall, the traités were too costly to be a viable system of war financing. In these conditions, the substantial fortunes made by a handful of very successful traitants suffice to explain that the government easily gave in to public criticism against the wealth of the financiers and felt compelled, when peace resumed, to cancel the advantageous conditions offered in the treaties by taxing financial profits.

Conditioning and preconditioning of the optimal state estimation problem

Optimal state estimation is a method that requires minimising a weighted, nonlinear, least-squares objective function in order to obtain the best estimate of the current state of a dynamical system. Often the minimisation is non-trivial due to the large scale of the problem, the relative sparsity of the observations and the nonlinearity of the objective function. To simplify the problem the solution is often found via a sequence of linearised objective functions. The condition number of the Hessian of the linearised problem is an important indicator of the convergence rate of the minimisation and the expected accuracy of the solution. In the standard formulation the convergence is slow, indicating an ill-conditioned objective function. A transformation to different variables is often used to ameliorate the conditioning of the Hessian by changing, or preconditioning, the Hessian. There is only sparse information in the literature for describing the causes of ill-conditioning of the optimal state estimation problem and explaining the effect of preconditioning on the condition number. This paper derives descriptive theoretical bounds on the condition number of both the unpreconditioned and preconditioned system in order to better understand the conditioning of the problem. We use these bounds to explain why the standard objective function is often ill-conditioned and why a standard preconditioning reduces the condition number. We also use the bounds on the preconditioned Hessian to understand the main factors that affect the conditioning of the system. We illustrate the results with simple numerical experiments.

An optimization problem concerning multiplicative functions

In this paper we study the problem of maximizing a quadratic form 〈Ax,x〉 subject to ‖x‖q=1, where A has matrix entries View the MathML source with i,j|k and q≥1. We investigate when the optimum is achieved at a ‘multiplicative’ point; i.e. where x1xmn=xmxn. This turns out to depend on both f and q, with a marked difference appearing as q varies between 1 and 2. We prove some partial results and conjecture that for f multiplicative such that 0

Noether-type discrete conserved quantities arising from a finite element approximation of a variational problem

In this work, we prove a weak Noether-type Theorem for a class of variational problems that admit broken extremals. We use this result to prove discrete Noether-type conservation laws for a conforming finite element discretisation of a model elliptic problem. In addition, we study how well the finite element scheme satisfies the continuous conservation laws arising from the application of Noether’s first theorem (1918). We summarise extensive numerical tests, illustrating the conservation of the discrete Noether law using the p-Laplacian as an example and derive a geometric-based adaptive algorithm where an appropriate Noether quantity is the goal functional.

Using problem-based learning in a chemistry practical class for pharmacy students and engaging them with feedback

Objective: To introduce a new approach to problem based learning (PBL) used in the context of medicinal chemistry practical class teaching pharmacy students. Design: The described chemistry practical is based on independent studies by small groups of undergraduate students (4-5), who design their own practical work taking relevant professional standards into account. Students are carefully guided by feedback and acquire a set of skills important to their future profession as healthcare professionals. This model has been tailored to the application of PBL in a chemistry practical class setting for a large student cohort (150 students). Assessment: The achievement of learning outcomes is based on the submission of relevant documentation including a certificate of analysis, in addition to peer assessment. Some of the learning outcomes are also assessed in the final written examination at the end of the academic year. Conclusion: The described design of a novel PBL chemistry laboratory course for pharmacy students has been found to be successful. Self-reflective learning and engagement with feedback were encouraged, and students enjoyed the challenging learning experience. Skills that are highly essential for the students’ future careers as healthcare professionals are promoted.

The multiple team formation problem using sociometry

The Team Formation problem (TFP) has become a well-known problem in the OR literature over the last few years. In this problem, the allocation of multiple individuals that match a required set of skills as a group must be chosen to maximise one or several social positive attributes. Speci�cally, the aim of the current research is two-fold. First, two new dimensions of the TFP are added by considering multiple projects and fractions of people's dedication. This new problem is named the Multiple Team Formation Problem (MTFP). Second, an optimization model consisting in a quadratic objective function, linear constraints and integer variables is proposed for the problem. The optimization model is solved by three algorithms: a Constraint Programming approach provided by a commercial solver, a Local Search heuristic and a Variable Neighbourhood Search metaheuristic. These three algorithms constitute the first attempt to solve the MTFP, being a variable neighbourhood local search metaheuristic the most effi�cient in almost all cases. Applications of this problem commonly appear in real-life situations, particularly with the current and ongoing development of social network analysis. Therefore, this work opens multiple paths for future research.