Conservation laws and normal forms of evolution equations

We study local conservation laws for evolution equations in two independent variables. In particular, we present normal forms for the equations admitting one or two low-order conservation laws. Examples include Harry Dym equation, Korteweg-de-Vries-type equations, and Schwarzian KdV equation. It is also shown that for linear evolution equations all their conservation laws are (modulo trivial conserved vectors) at most quadratic in the dependent variable and its derivatives.Comment: 16 page

Hotel a Amposta

L'edifici es situa a Amposta, entre el carrer de Sant Joan i el passeig fluvial, al costat de la plaça de l'Aube. Es tracta d'un solar privilegiat, amb vistes al riu Ebre i molt proper al centre de la vila. En planta baixa es modifica el volum inicialment rectangular, per a donar lloc a un seguit de porxades i passos cap al paseig fluvial, que protegeixen al vianant de la pluja i del sol alhora que el conviden a entrar. L'entrada principal de l'edifici queda vinculada a la plaça de la Pau i permet als vianants travessar cap al passeig fluvial o accedir al punt d'informació turística que queda desvinculat de l'hotel. La bona orientació és contrària a les vistes al riu, i per això es planteja una solució en forma de pinta, amb un seguit de barres paral.leles entre elles i perpendiculars al riu que permeten bones vistes, i alhora disposar de radiació solar directa. A les proximitats de l'emplaçament, en direcció al pont penjant, existeix un gra d'edificació antiga, que en algun cas és de manposteria de pedra. Amb la finalitat d'adaptar-nos al lloc i d'utilitzar els recursos de la zona, un dels materials principals utilitzats per a la formació de les façanes del projecte serà la manposteria de pedra, combinada amb formigó i vidre. S'ha pres com a referència la casa de Herzog&de Meuron, de la casa de pedra situada a la costa de Liguria, que utilitza els recursos del lloc, combinant-los amb l'estructura de formigó a vista

Elementos de matemática

El libro "Elementos de Matemática" es una obra que fué utilizada por muchos científicos e ingenieros españoles en su formación básica en los años 50-70 del pasado siglo XX. Contiene una primera parte dedicada al Álgebra lineal, una segunda parte dedicada al Análisis infinitesimal y una tercera parte con las soluciones a los ejercicios propuestos en las anteriores. Además de por su interés como texto de introducción a la Matemática, se hace público en el archivo digital por su interés históric

Adaptive memory hierarchies for next generation tiled microarchitectures

Les últimes dècades el rendiment dels processadors i de les memòries ha millorat a diferent ritme, limitant el rendiment dels processadors i creant el conegut memory gap. Sol·lucionar aquesta diferència de rendiment és un camp d'investigació d'actualitat i que requereix de noves sol·lucions. Una sol·lució a aquest problema són les memòries “cache”, que permeten reduïr l'impacte d'unes latències de memòria creixents i que conformen la jerarquia de memòria. La majoria de d'organitzacions de les “caches” estan dissenyades per a uniprocessadors o multiprcessadors tradicionals. Avui en dia, però, el creixent nombre de transistors disponible per xip ha permès l'aparició de xips multiprocessador (CMPs). Aquests xips tenen diferents propietats i limitacions i per tant requereixen de jerarquies de memòria específiques per tal de gestionar eficientment els recursos disponibles. En aquesta tesi ens hem centrat en millorar el rendiment i la eficiència energètica de la jerarquia de memòria per CMPs, des de les “caches” fins als controladors de memòria. A la primera part d'aquesta tesi, s'han estudiat organitzacions tradicionals per les “caches” com les privades o compartides i s'ha pogut constatar que, tot i que funcionen bé per a algunes aplicacions, un sistema que s'ajustés dinàmicament seria més eficient. Tècniques com el Cooperative Caching (CC) combinen els avantatges de les dues tècniques però requereixen un mecanisme centralitzat de coherència que té un consum energètic molt elevat. És per això que en aquesta tesi es proposa el Distributed Cooperative Caching (DCC), un mecanisme que proporciona coherència en CMPs i aplica el concepte del cooperative caching de forma distribuïda. Mitjançant l'ús de directoris distribuïts s'obté una sol·lució més escalable i que, a més, disposa d'un mecanisme de marcatge més flexible i eficient energèticament. A la segona part, es demostra que les aplicacions fan diferents usos de la “cache” i que si es realitza una distribució de recursos eficient es poden aprofitar els que estan infrautilitzats. Es proposa l'Elastic Cooperative Caching (ElasticCC), una organització capaç de redistribuïr la memòria “cache” dinàmicament segons els requeriments de cada aplicació. Una de les contribucions més importants d'aquesta tècnica és que la reconfiguració es decideix completament a través del maquinari i que tots els mecanismes utilitzats es basen en estructures distribuïdes, permetent una millor escalabilitat. ElasticCC no només és capaç de reparticionar les “caches” segons els requeriments de cada aplicació, sinó que, a més a més, és capaç d'adaptar-se a les diferents fases d'execució de cada una d'elles. La nostra avaluació també demostra que la reconfiguració dinàmica de l'ElasticCC és tant eficient que gairebé proporciona la mateixa taxa de fallades que una configuració amb el doble de memòria.Finalment, la tesi es centra en l'estudi del comportament de les memòries DRAM i els seus controladors en els CMPs. Es demostra que, tot i que els controladors tradicionals funcionen eficientment per uniprocessadors, en CMPs els diferents patrons d'accés obliguen a repensar com estan dissenyats aquests sistemes. S'han presentat múltiples sol·lucions per CMPs però totes elles es veuen limitades per un compromís entre el rendiment global i l'equitat en l'assignació de recursos. En aquesta tesi es proposen els Thread Row Buffers (TRBs), una zona d'emmagatenament extra a les memòries DRAM que permetria guardar files de dades específiques per a cada aplicació. Aquest mecanisme permet proporcionar un accés equitatiu a la memòria sense perjudicar el seu rendiment global. En resum, en aquesta tesi es presenten noves organitzacions per la jerarquia de memòria dels CMPs centrades en la escalabilitat i adaptativitat als requeriments de les aplicacions. Els resultats presentats demostren que les tècniques proposades proporcionen un millor rendiment i eficiència energètica que les millors tècniques existents fins a l'actualitat.Processor performance and memory performance have improved at different rates during the last decades, limiting processor performance and creating the well known "memory gap". Solving this performance difference is an important research field and new solutions must be proposed in order to have better processors in the future. Several solutions exist, such as caches, that reduce the impact of longer memory accesses and conform the system memory hierarchy. However, most of the existing memory hierarchy organizations were designed for single processors or traditional multiprocessors. Nowadays, the increasing number of available transistors has allowed the apparition of chip multiprocessors, which have different constraints and require new ad-hoc memory systems able to efficiently manage memory resources. Therefore, in this thesis we have focused on improving the performance and energy efficiency of the memory hierarchy of chip multiprocessors, ranging from caches to DRAM memories. In the first part of this thesis we have studied traditional cache organizations such as shared or private caches and we have seen that they behave well only for some applications and that an adaptive system would be desirable. State-of-the-art techniques such as Cooperative Caching (CC) take advantage of the benefits of both worlds. This technique, however, requires the usage of a centralized coherence structure and has a high energy consumption. Therefore we propose the Distributed Cooperative Caching (DCC), a mechanism to provide coherence to chip multiprocessors and apply the concept of cooperative caching in a distributed way. Through the usage of distributed directories we obtain a more scalable solution and, in addition, has a more flexible and energy-efficient tag allocation method. We also show that applications make different uses of cache and that an efficient allocation can take advantage of unused resources. We propose Elastic Cooperative Caching (ElasticCC), an adaptive cache organization able to redistribute cache resources dynamically depending on application requirements. One of the most important contributions of this technique is that adaptivity is fully managed by hardware and that all repartitioning mechanisms are based on distributed structures, allowing a better scalability. ElasticCC not only is able to repartition cache sizes to application requirements, but also is able to dynamically adapt to the different execution phases of each thread. Our experimental evaluation also has shown that the cache partitioning provided by ElasticCC is efficient and is almost able to match the off-chip miss rate of a configuration that doubles the cache space. Finally, we focus in the behavior of DRAM memories and memory controllers in chip multiprocessors. Although traditional memory schedulers work well for uniprocessors, we show that new access patterns advocate for a redesign of some parts of DRAM memories. Several organizations exist for multiprocessor DRAM schedulers, however, all of them must trade-off between memory throughput and fairness. We propose Thread Row Buffers, an extended storage area in DRAM memories able to store a data row for each thread. This mechanism enables a fair memory access scheduling without hurting memory throughput. Overall, in this thesis we present new organizations for the memory hierarchy of chip multiprocessors which focus on the scalability and of the proposed structures and adaptivity to application behavior. Results show that the presented techniques provide a better performance and energy-efficiency than existing state-of-the-art solutions

A forest simulation approach using weighted Voronoi diagrams. An application to Mediterranean fir Abies pinsapo Boiss stands

Aim of study: a) To present a new version of the forest simulator Vorest, an individual-based spatially explicit model that uses weighted Voronoi diagrams to simulate the natural dynamics of forest stands with closed canopies. b) To apply the model to the current dynamics of a Grazalema pinsapo stand to identify the nature of its competition regime and the stagnation risks it is currently facing.Area of study: Sierra del Pinar de Grazalema (S Spain)Material and methods: Two large plots representative of Grazalema pinsapo stands were used to fit and validate the model (plus 6 accesory plots to increase the availability of mortality data). Two inventories were carried out in 1998 and 2007 producing tree size and location data. We developed a forest simulator based on three submodels: growth, competition and mortality. The model was fitted, evaluated and validated for Grazalema plots. The simulation outputs were used to infer the expected evolution of structural diversity of  forest stands.Main results: Vorest has proved to be a good tool for simulating dynamics of natural closed stands. The application to Grazalema pinsapo stands has allowed assessing the nature of the main processes that are driving its development pathway. We have found that the prevailing size-asymmetric competition dominates the self-thinning process in small-sized trees. At the same time, there is an active tree-size differentiation process.Research highlights:Vorest has proved to be a good tool for simulating natural stands with closed canopies.The Grazalema pinsapo stand under consideration is currently undergoing a natural process of differentiation, avoiding long-term stagnation.Keywords: Vorest; stand dynamics; individual-based forest model; spatially explicit forest model; pinsapo

Incorporating prior knowledge into deep neural networks without handcrafted features

Deep learning (DL) is currently the largest area of research within artificial intelligence (AI). This success can largely be attributed to the data-driven nature of the DL algorithms themselves: unlike previous approaches in AI which required handcrafting and significant human intervention, DL models can be implemented and trained with little to no human involvement. The lack of handcrafting, however, can be a two-edged sword. DL algorithms are notorious for producing uninterpretable features, generalising badly to new tasks and relying on extraordinarily large datasets for training. In this thesis, on the assumption that these shortcomings are symptoms of the under-constrained training setup of deep networks, we address the question of how to incorporate knowledge into DL algorithms without reverting to complete handcrafting in order to train more data efficient algorithms. % In this thesis we consider different alternatives to this problem. We start by motivating this line of work with an example of a DL architecture which, inspired by symbolic AI, aims at extracting symbols from a simple environment and using those for quickly learning downstream tasks. Our proof-of-concept model shows that it is possible to address some of the data efficiency issues as well as obtaining more interpretable representations by reasoning at this higher level of abstraction. Our second approach for data-efficiency is based on pre-training: the idea is to pre-train some parts of the DL network on a different, but related, task to first learn useful feature extractors. For our experiments we pre-train the encoder of a reinforcement learning agent on a 3D scene prediction task and then use the features produced by the encoder to train a simulated robot arm on a reaching task. Crucially, unlike previous approaches that could only learn from fixed view-points, we are able to train an agent using observations captured from randomly changing positions around the robot arm, without having to train a separate policy for each observation position. Lastly, we focus on how to build in prior knowledge through the choice of dataset. To this end, instead of training DL models on a single dataset, we train them on a distribution over datasets that captures the space of tasks we are interested in. This training regime produces models that can flexibly adapt to any dataset within the distribution at test time. Crucially they only need a small number of observations in order to adapt their predictions, thus addressing the data-efficiency challenge at test time. We call this family of meta-learning models for few-shot prediction Neural Processes (NPs). In addition to successfully learning how to carry out few-shot regression and classification, NPs produce uncertainty estimates and can generate coherent samples at arbitrary resolutions.Open Acces

Fun with Fonts: Algorithmic Typography

Over the past decade, we have designed six typefaces based on mathematical theorems and open problems, specifically computational geometry. These typefaces expose the general public in a unique way to intriguing results and hard problems in hinged dissections, geometric tours, origami design, computer-aided glass design, physical simulation, and protein folding. In particular, most of these typefaces include puzzle fonts, where reading the intended message requires solving a series of puzzles which illustrate the challenge of the underlying algorithmic problem.Comment: 14 pages, 12 figures. Revised paper with new glass cane font. Original version in Proceedings of the 7th International Conference on Fun with Algorithm

On the invariants of some solvable rigid Lie algebras

We determine fundamental systems of invariants for complex solvable rigid Lie algebras having nonsplit nilradicals of characteristic sequence $(3,1,..,1)$, these algebras being the natural followers of solvable algebras having Heisenberg nilradicals. A special case of this allows us to obtain a criterion to determine the number of functionally independent invariants of rank one subalgebras of (real or complex) solvable Lie algebras. Finally, we give examples of the inverse procedure, obtaining fundamental systems of an algebra starting from rank one subalgebras.Comment: LaTex, 13 pages, 1 tabl