Q-systems, Heaps, Paths and Cluster Positivity

We consider the cluster algebra associated to the $Q$-system for $A_r$ as a tool for relating $Q$-system solutions to all possible sets of initial data. We show that the conserved quantities of the $Q$-system are partition functions for hard particles on particular target graphs with weights, which are determined by the choice of initial data. This allows us to interpret the simplest solutions of the Q-system as generating functions for Viennot's heaps on these target graphs, and equivalently as generating functions of weighted paths on suitable dual target graphs. The generating functions take the form of finite continued fractions. In this setting, the cluster mutations correspond to local rearrangements of the fractions which leave their final value unchanged. Finally, the general solutions of the $Q$-system are interpreted as partition functions for strongly non-intersecting families of lattice paths on target lattices. This expresses all cluster variables as manifestly positive Laurent polynomials of any initial data, thus proving the cluster positivity conjecture for the $A_r$ $Q$-system. We also give an alternative formulation in terms of domino tilings of deformed Aztec diamonds with defects.Comment: 106 pages, 38 figure

Q-system Cluster Algebras, Paths and Total Positivity

In the first part of this paper, we provide a concise review of our method of solution of the Ar Q-systems in terms of the partition function of paths on a weighted graph. In the second part, we show that it is possible to modify the graphs and transfer matrices so as to provide an explicit connection to the theory of planar networks introduced in the context of totally positive matrices by Fomin and Zelevinsky. As an illustration of the further generality of our method, we apply it to give a simple solution for the rank 2 affine cluster algebras studied by Caldero and Zelevinsky

Asymptotics and scaling analysis of 2-dimensional lattice models of vesicles and polymers

PhDThe subject of this thesis is the asymptotic behaviour of generating functions of different combinatorial models of two-dimensional lattice walks and polygons, enumerated with respect to different parameters, such as perimeter, number of steps and area. These models occur in various applications in physics, computer science and biology. In particular, they can be seen as simple models of biological vesicles or polymers. Of particular interest is the singular behaviour of the generating functions around special, so-called multicritical points in their parameter space, which correspond physically to phase transitions. The singular behaviour around the multicritical point is described by a scaling function, alongside a small set of critical exponents. Apart from some non-rigorous heuristics, our asymptotic analysis mainly consists in applying the method of steepest descents to a suitable integral expression for the exact solution for the generating function of a given model. The similar mathematical structure of the exact solutions of the different models allows for a unified treatment. In the saddle point analysis, the multicritical points correspond to points in the parameter space at which several saddle points of the integral kernels coalesce. Generically, two saddle points coalesce, in which case the scaling function is expressible in terms of the Airy function. As we will see, this is the case for Dyck and Schröder paths, directed column-convex polygons and partially directed self-avoiding walks. The result for Dyck paths also allows for the scaling analysis of Bernoulli meanders (also known as ballot paths). We then construct the model of deformed Dyck paths, where three saddle points coalesce in the corresponding integral kernel, thereby leading to an asymptotic expression in terms of a bivariate, generalised Airy integral.Universität Erlangen-Nürnberg Queen Mary Postgraduate Research Fun

Deviations in neural activity and network integration underpinning the co-occurrence of emotion dysregulation and attention-deficit/hyperactivity disorder: Analyses of fMRI task activations and functional brain network topology

The aim of this thesis was to improve our understanding of the relationship between Attention-deficit/hyperactivity disorder (ADHD) and emotion dysregulation and the underlying neural activity. Three research articles examine specific aspects of the relationship between ADHD and emotion dysregulation, namely the perception of emotional stimuli, the association between functional brain topology and emotion dysregulation in different ADHD presentations, and emotion dysregulation-related neurobiological and phenotypical predictors of the course of ADHD. All three articles are based on functional magnetic resonance imaging (fMRI) data. Individuals with ADHD exhibited aberrant amygdala reactivity and ventromedial prefrontal cortex coupling in the perception and processing of emotional face stimuli. Moreover, functional network topology of the right insula was shown to affect emotion dysregulation in ADHD and emotion dysregulation and integration of emotion-related brain networks were shown to affect intraindividual change in ADHD severity throughout late adolescence. In Summary, the thesis provides evidence that neural activity and functional connectivity between brain structures affecting emotion may be related to the co-occurrence of emotion dysregulation and ADHD. ADHD and the common co-occurring emotional problems should not be attributed to single, isolated systems, e.g., for executive functions and cognitive control. The neurobiological roots appear to be complex and heterogeneous, involving the interplay of different brain networks that are at least partly emotion-related

UNDERSTANDING THE CONTRIBUTION OF THE CENTRAL EXTENDED AMYGDALA TO DISPOSITIONAL NEGATIVITY

Dispositional negativity (DN) is a key risk factor for a spectrum of adverse outcomes, including anxiety disorders, depression, and comorbid substance abuse. The central extended amygdala (EAc; an anatomical concept encompassing the bed nucleus of the stria terminalis [BST] and central nucleus of the amygdala [Ce]) is implicated in the development and maintenance of these disorders. However, disorders, like other psychological processes, reflect the coordinated actions of widely distributed networks. Yet, the functional architecture of the human EAc and its relation to individual differences in DN remains poorly understood. We investigated intrinsic functional connectivity (iFC) of the EAc in 185 healthy adults. Whole-brain regression analyses revealed that the BST and Ce show iFC with one another via the sublenticular extended amygdala. While both regions showed significant iFC with the ventromedial prefrontal cortex and with cingulate territories involved in adaptive control of anxiety-related behavior, the BST showed more robust coupling. Contrary to expectations, EAc iFC was not significantly associated with individual differences in DN. These observations provide a novel neurobiological framework for understanding a range of stress-sensitive disorders

運動計画をフィードバックループに含むヒューマノイドロボットの多点接触全身制御のための計算基盤

学位の種別: 課程博士審査委員会委員 : （主査）東京大学教授 中村 仁彦, 東京大学教授 下山 勲, 東京大学教授 稲葉 雅幸, 東京大学教授 國吉 康夫, 東京大学准教授 高野 渉, LAAS-CNRSSenior Researcher LAUMOND Jean-PaulUniversity of Tokyo(東京大学

Mathematical programming with uncertainty and multiple objectives for sustainable development and wildfire management

Mathematical Programming is a well-placed field of Operational Research to tackle problems as diverse as those that arise in Logistics and Disaster Management. The fundamental objective of Mathematical Programming is the selection of an optimal alternative that meets a series of restrictions. The criterion by which the alternatives are evaluated is traditionally only one (for example, minimizing cost), however it is also common for several objectives to want to be considered simultaneously, thus giving rise to the Multi-criteria Decision. If the conditions to be met by an alternative or the evaluation of said alternative depend on random (or unknown) factors, we are in an optimization context under uncertainty. In the first chapters of this thesis the fields of multicriteria decision and optimization with uncertainty are studied, in two applications in the context of sustainable development and disaster management. Optimization with uncertainty is introduced through an application to rural electrification. In rural areas, access to electricity through solar systems installed in consumers' homes is common. These systems have to be repaired when they fail, so the decision of how to size a maintenance network is affected by great uncertainty. A mathematical programming model is developed by treating uncertainty in an unexplained way, the objective of which is to obtain a maintenance network at minimum cost. This model is later used as a tool to obtain simple rules that can predict the cost of maintenance using little information. The model is validated using information from a real program implemented in Morocco. When studying Multicriteria Optimization it is considered a problem in forest fire management. To mitigate the effects of forest fires, it is common to modify forests, with what is known as fuel treatment. Through this practice, consisting of the controlled felling or burning of trees in selected areas, it is achieved that when fires inevitably occur, they are easier to control. Unfortunately, modifying the flora can affect the existing fauna, so it is sensible to look for solutions that improve the situation in the face of a fire but without great detriment to the existing species. In other words, there are several criteria to take into account when optimizing. A mathematical programming model is developed, which suggests which zones to burn and when, taking into account these competing criteria. This model is applied to a series of simulated realistic cases. The following is a theoretical study of the field of Multiobjective Stochastic Programming (MSP), in which problems that simultaneously have various criteria and uncertainty are considered. In this chapter, a new solution concept is developed for MSP problems with risk aversion, its properties are studied and a linear programming model capable of obtaining said solution is formulated. A computational study of the model is also carried out, applying it to a variation of the well-known backpack problem. Finally, the problem of controlled burning is studied again, this time considering the existing uncertainty as it is not possible to know with certainty how many controlled burns can be carried out in a year, due to the limited window of time in which these can be carried out. The problem is solved using the multi-criteria and stochastic methodology with risk aversion developed in the previous chapter. Finally, the resulting model is applied to a real case located in southern Spain