A Massively Parallel Algorithm for the Approximate Calculation of Inverse p-th Roots of Large Sparse Matrices

We present the submatrix method, a highly parallelizable method for the approximate calculation of inverse p-th roots of large sparse symmetric matrices which are required in different scientific applications. We follow the idea of Approximate Computing, allowing imprecision in the final result in order to be able to utilize the sparsity of the input matrix and to allow massively parallel execution. For an n x n matrix, the proposed algorithm allows to distribute the calculations over n nodes with only little communication overhead. The approximate result matrix exhibits the same sparsity pattern as the input matrix, allowing for efficient reuse of allocated data structures. We evaluate the algorithm with respect to the error that it introduces into calculated results, as well as its performance and scalability. We demonstrate that the error is relatively limited for well-conditioned matrices and that results are still valuable for error-resilient applications like preconditioning even for ill-conditioned matrices. We discuss the execution time and scaling of the algorithm on a theoretical level and present a distributed implementation of the algorithm using MPI and OpenMP. We demonstrate the scalability of this implementation by running it on a high-performance compute cluster comprised of 1024 CPU cores, showing a speedup of 665x compared to single-threaded execution

Modelling Load Balancing and Carrier Aggregation in Mobile Networks

In this paper, we study the performance of multicarrier mobile networks. Specifically, we analyze the flow-level performance of two inter-carrier load balancing schemes and the gain engendered by Carrier Aggregation (CA). CA is one of the most important features of HSPA+ and LTE-A networks; it allows devices to be served simultaneously by several carriers. We propose two load balancing schemes, namely Join the Fastest Queue (JFQ) and Volume Balancing (VB), that allow the traffic of CA and non-CA users to be distributed over the aggregated carriers. We then evaluate the performance of these schemes by means of analytical modeling. We show that the proposed schemes achieve quasi-ideal load balancing. We also investigate the impact of mixing traffic of CA and non-CA users in the same cell and show that performance is practically insensitive to the traffic mix.Comment: 8 pages, 6 figures, submitted to WiOpt201

Reexamination of Quantum Bit Commitment: the Possible and the Impossible

Bit commitment protocols whose security is based on the laws of quantum mechanics alone are generally held to be impossible. In this paper we give a strengthened and explicit proof of this result. We extend its scope to a much larger variety of protocols, which may have an arbitrary number of rounds, in which both classical and quantum information is exchanged, and which may include aborts and resets. Moreover, we do not consider the receiver to be bound to a fixed "honest" strategy, so that "anonymous state protocols", which were recently suggested as a possible way to beat the known no-go results are also covered. We show that any concealing protocol allows the sender to find a cheating strategy, which is universal in the sense that it works against any strategy of the receiver. Moreover, if the concealing property holds only approximately, the cheat goes undetected with a high probability, which we explicitly estimate. The proof uses an explicit formalization of general two party protocols, which is applicable to more general situations, and a new estimate about the continuity of the Stinespring dilation of a general quantum channel. The result also provides a natural characterization of protocols that fall outside the standard setting of unlimited available technology, and thus may allow secure bit commitment. We present a new such protocol whose security, perhaps surprisingly, relies on decoherence in the receiver's lab.Comment: v1: 26 pages, 4 eps figures. v2: 31 pages, 5 eps figures; replaced with published version; title changed to comply with puzzling Phys. Rev. regulations; impossibility proof extended to protocols with infinitely many rounds or a continuous communication tree; security proof of decoherence monster protocol expanded; presentation clarifie

Lattice Green Functions: the seven-dimensional face-centred cubic lattice

We present a recursive method to generate the expansion of the lattice Green function of the d-dimensional face-centred cubic (fcc) lattice. We produce a long series for d =7. Then we show (and recall) that, in order to obtain the linear differential equation annihilating such a long power series, the most economic way amounts to producing the non-minimal order differential equations. We use the method to obtain the minimal order linear differential equation of the lattice Green function of the seven-dimensional face-centred cubic (fcc) lattice. We give some properties of this irreducible order-eleven differential equation. We show that the differential Galois group of the corresponding operator is included in $SO(11, \mathbb{C})$. This order-eleven operator is non-trivially homomorphic to its adjoint, and we give a "decomposition" of this order-eleven operator in terms of four order-one self-adjoint operators and one order-seven self-adjoint operator. Furthermore, using the Landau conditions on the integral, we forward the regular singularities of the differential equation of the d-dimensional lattice and show that they are all rational numbers. We evaluate the return probability in random walks in the seven-dimensional fcc lattice. We show that the return probability in the d-dimensional fcc lattice decreases as $d^{-2}$ as the dimension d goes to infinity.Comment: 19 page

Becoming Aware Of Mathematical Gaps In New Curricular Materials: A Resource-Based Analysis Of Teaching Practice

The study featured in this article, with its central focus on resources-in-use, draws upon salient aspects of the documentational approach of didactics. It includes an a priori analysis of the curricular resources being used by a teacher for the first time, followed by detailed in situ observations of the unfolding of her teaching practice involving these resources. The central mathematical problem of the lesson being analyzed deals with families of polynomial functions. The analysis highlights the teacher’s growing awareness of the mathematical gaps in the resources she is using, which we conjecture to be a first step for her in the evolutionary transformation of resource to document, as well as an essential constituent of her ongoing professional development

Inclusion lexicale et proximité sémantique entre termes.

International audienceNous étudions l'influence de l'inclusion lexicale sur la proximité sémantique entre termes. A partir de l'analyse des relations entre termes d'une ressource terminologique existante, lexicalement inclus dans ceux issus d'un corpus, nous formulons des hypothèses des relations engendrées. Ces hypothèses nous permettent de proposer un ordonnancement automatique des variantes des termes trouvées dans le corpus, par probabilité de proximité sémantique décroissante. Les premières expérimentations montrent que la prise en compte d'indices morpho-lexicaux comme la présence de noms composés, de noms propres et le nombre d'éléments ajoutés sont des critères à prendre en compte pour classer les variantes d'un même terme

Covert channel detection using Information Theory

This paper presents an information theory based detection framework for covert channels. We first show that the usual notion of interference does not characterize the notion of deliberate information flow of covert channels. We then show that even an enhanced notion of "iterated multivalued interference" can not capture flows with capacity lower than one bit of information per channel use. We then characterize and compute the capacity of covert channels that use control flows for a class of systems.Comment: In Proceedings SecCo 2010, arXiv:1102.516

Resources, at the core of mathematics teachers' work

International audienceMathematics teachers work with resources in class and out of class. Textbooks, in particular, hold a central place in this material. Nevertheless, the available resources evolve, with an increasing amount of online resources: software, lesson plans, classroom videos etc. This important change led us to propose a study of mathematics teachers documentation. Mathematics teachers select resources, combine them, use them, revise them, amongst others. Teachers' documentation is both this work and its outcome. Teachers' documentation work is central to their professional activity; it influences the professional activity, which evolves along what we call professional geneses. In this conference, I introduce a specific perspective on teachers resources, which enlightens in particular the changes caused by digitalization, and in particular Internet resources

Reciprocal learning in mathematics education: An interactive study between two Canadian and Chinese elementary schools

In this study, the researchers go beyond the back-and-forth debates on the East-West educational paradigms that often arise from comparative studies, and take a reciprocal learning approach to explore in-depth the commonalities and differences in mathematics education between two Canadian and Chinese elementary schools. Research data were collected through direct and indirect interactions between the pair of research schools, including Skype meetings; formal and informal conversations with teachers and administrators; and the sharing/exchange of documents, texts, teaching materials, and resources. Results show that there is a common emphasis on some thematic issues in the teaching and learning of mathematics including the use of manipulatives, multiple solutions to mathematical problems, and parental involvement, but also some differences between the two schools in teachers’ strategies for teaching problem solving, students’ learning tendencies and schools’ supports for Special Needs students. The researchers conclude that the dichotomies of the East-West educational paradigms need to be further, and more deeply re-examined

The French Didactic Tradition in Mathematics

This chapter presents the French didactic tradition. It first describes theemergence and development of this tradition according to four key features (role ofmathematics and mathematicians, role of theories, role of design of teaching andlearning environments, and role of empirical research), and illustrates it through two case studies respectively devoted to research carried out within this traditionon algebra and on line symmetry-reflection. It then questions the influence of thistradition through the contributions of four researchers from Germany, Italy, Mexicoand Tunisia, before ending with a short epilogue