Relevant bibliographies by topics / Expander graphs

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers
  6. Reports

Academic literature on the topic 'Expander graphs'

Author: Grafiati
Published: 4 June 2021
Last updated: 6 September 2023

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Journal articles on the topic "Expander graphs"

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CZUMAJ, ARTUR, and CHRISTIAN SOHLER. "Testing Expansion in Bounded-Degree Graphs." Combinatorics, Probability and Computing 19, no. 5-6 (June 9, 2010): 693–709. http://dx.doi.org/10.1017/s096354831000012x.

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We consider the problem of testing expansion in bounded-degree graphs. We focus on the notion of vertex expansion: an α-expander is a graph G = (V, E) in which every subset U ⊆ V of at most |V|/2 vertices has a neighbourhood of size at least α ⋅ |U|. Our main result is that one can distinguish good expanders from graphs that are far from being weak expanders in time $\widetilde{\O}(\sqrt{n})$. We prove that the property-testing algorithm proposed by Goldreich and Ron with appropriately set parameters accepts every α-expander with probability at least $\frac23$ and rejects every graph that is ϵ-far from any α*-expander with probability at least $\frac23$, where $\expand^* \,{=}\, \Theta(\frac{\expand^2}{d^2 \log(n/\epsilon)})$ and d is the maximum degree of the graphs. The algorithm assumes the bounded-degree graphs model with adjacency list graph representation and its running time is $\O(\frac{d^2 \sqrt{n} \log(n/\epsilon)} {\expand^2 \epsilon^3})$.
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Kamber, Amitay. "Lp-Expander Graphs." Israel Journal of Mathematics 234, no. 2 (October 2019): 863–905. http://dx.doi.org/10.1007/s11856-019-1938-7.

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Polak, Monika, and Vasyl Ustimenko. "Algorithms for generation of Ramanujan graphs, other Expanders and related LDPC codes." Annales Universitatis Mariae Curie-Sklodowska, sectio AI – Informatica 15, no. 2 (October 11, 2015): 14. http://dx.doi.org/10.17951/ai.2015.15.2.14-21.

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Expander graphs are highly connected sparse finite graphs. The property of being an expander seems significant in many of these mathematical, computational and physical contexts. For practical applications it is very important to construct expander and Ramanujan graphs with given regularity and order. In general, constructions of the best expander graphs with a given regularity and order is no easy task. In this paper we present algorithms for generation of Ramanujan graphs and other expanders. We describe properties of obtained graphs in comparison to previously known results. We present a method to obtain a new examples of irregular LDPC codes based on described graphs and we briefly describe properties of this codes.
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Cheung, Ho Yee, Lap Chi Lau, and Kai Man Leung. "Graph Connectivities, Network Coding, and Expander Graphs." SIAM Journal on Computing 42, no. 3 (January 2013): 733–51. http://dx.doi.org/10.1137/110844970.

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Angel, D., R. Mary Jeya Jothi, R. Revathi, and A. Raja. "Expander Graphs – A Study." Journal of Physics: Conference Series 1770, no. 1 (March 1, 2021): 012078. http://dx.doi.org/10.1088/1742-6596/1770/1/012078.

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Dutta, Neelav, and David Jensen. "Gonality of expander graphs." Discrete Mathematics 341, no. 9 (September 2018): 2535–43. http://dx.doi.org/10.1016/j.disc.2018.06.012.

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Harrow, A. W. "Quantum expanders from any classical Cayley graph expander." Quantum Information and Computation 8, no. 8&9 (September 2008): 715–21. http://dx.doi.org/10.26421/qic8.8-9-2.

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We give a simple recipe for translating walks on Cayley graphs of a group G into a quantum operation on any irrep of G. Most properties of the classical walk carry over to the quantum operation: degree becomes the number of Kraus operators, the spectral gap becomes the gap of the quantum operation (viewed as a linear map on density matrices), and the quantum operation is efficient whenever the classical walk and the quantum Fourier transform on G are efficient. This means that using classical constant-degree constant-gap families of Cayley expander graphs on e.g. the symmetric group, we can construct efficient families of quantum expanders.
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LEUNG, KA HIN, VINH NGUYEN, and WASIN SO. "NONEXISTENCE OF A CIRCULANT EXPANDER FAMILY." Bulletin of the Australian Mathematical Society 83, no. 1 (November 17, 2010): 87–95. http://dx.doi.org/10.1017/s0004972710001644.

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AbstractThe expansion constant of a simple graph G of order n is defined as where $E(S, \overline {S})$ denotes the set of edges in G between the vertex subset S and its complement $\overline {S}$. An expander family is a sequence {Gi} of d-regular graphs of increasing order such that h(Gi)>ϵ for some fixed ϵ>0. Existence of such families is known in the literature, but explicit construction is nontrivial. A folklore theorem states that there is no expander family of circulant graphs only. In this note, we provide an elementary proof of this fact by first estimating the second largest eigenvalue of a circulant graph, and then employing Cheeger’s inequalities where G is a d-regular graph and λ2(G) denotes the second largest eigenvalue of G. Moreover, the associated equality cases are discussed.
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JOUVE, FLORENT, and JEAN-SÉBASTIEN SERENI. "EXPANDER GRAPHS AND SIEVING IN COMBINATORIAL STRUCTURES." Journal of the Australian Mathematical Society 105, no. 1 (January 11, 2018): 79–102. http://dx.doi.org/10.1017/s1446788717000234.

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We prove a general large-sieve statement in the context of random walks on subgraphs of a given graph. This can be seen as a generalization of previously known results where one performs a random walk on a group enjoying a strong spectral gap property. In such a context the point is to exhibit a strong uniform expansion property for a suitable family of Cayley graphs on quotients. In our combinatorial approach, this is replaced by a result of Alon–Roichman about expanding properties of random Cayley graphs. Applying the general setting we show, for instance, that with high probability (in a strong explicit sense) random coloured subsets of integers contain monochromatic (nonempty) subsets summing to $0$, and that a random colouring of the edges of a complete graph contains a monochromatic triangle.
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Hoory, Shlomo, Nathan Linial, and Avi Wigderson. "Expander graphs and their applications." Bulletin of the American Mathematical Society 43, no. 04 (August 7, 2006): 439–562. http://dx.doi.org/10.1090/s0273-0979-06-01126-8.

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Dissertations / Theses on the topic "Expander graphs"

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Badaoui, Mohamad. "G-graphs and Expander graphs." Thesis, Normandie, 2018. http://www.theses.fr/2018NORMC207/document.

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L’utilisation de l’algèbre pour résoudre des problèmes de graphes a conduit au développement de trois branches : théorie spectrale des graphes, géométrie et combinatoire des groupes et études des invariants de graphes. La notion de graphe d’expansions (invariant de graphes) est relativement récente, elle a été développée afin d’étudier la robustesse des réseaux de télécommunication. Il s’avère que la construction de familles infinies de graphes expanseurs est un problème difficile. Cette thèse traite principalement de la construction de nouvelles familles de tels graphes. Les graphes expanseurs possèdent des nombreuses applications en informatique, notamment dans la construction de certains algorithmes, en théorie de la complexité, sur les marches aléatoires (random walk), etc. En informatique théorique, ils sont utilisés pour construire des familles de codes correcteurs d’erreur. Comme nous l’avons déjà vu les familles d’expanseurs sont difficiles à construire. La plupart des constructions s'appuient sur des techniques algébriques complexes, principalement en utilisant des graphes de Cayley et des produit Zig-Zag. Dans cette thèse, nous présentons une nouvelle méthode de construction de familles infinies d’expanseurs en utilisant les G-graphes. Ceux-ci sont en quelque sorte une généralisation des graphes de Cayley. Plusieurs nouvelles familles infinies d’expanseurs sont construites, notamment la première famille d’expanseurs irréguliers
Applying algebraic and combinatorics techniques to solve graph problems leads to the birthof algebraic and combinatorial graph theory. This thesis deals mainly with a crossroads questbetween the two theories, that is, the problem of constructing infinite families of expandergraphs.From a combinatorial point of view, expander graphs are sparse graphs that have strongconnectivity properties. Expanders constructions have found extensive applications in bothpure and applied mathematics. Although expanders exist in great abundance, yet their explicitconstructions, which are very desirable for applications, are in general a hard task. Mostconstructions use deep algebraic and combinatorial approaches. Following the huge amountof research published in this direction, mainly through Cayley graphs and the Zig-Zagproduct, we choose to investigate this problem from a new perspective; namely by usingG-graphs theory and spectral hypergraph theory as well as some other techniques. G-graphsare like Cayley graphs defined from groups, but they correspond to an alternative construction.The reason that stands behind our choice is first a notable identifiable link between thesetwo classes of graphs that we prove. This relation is employed significantly to get many newresults. Another reason is the general form of G-graphs, that gives us the intuition that theymust have in many cases such as the relatively high connectivity property.The adopted methodology in this thesis leads to the identification of various approaches forconstructing an infinite family of expander graphs. The effectiveness of our techniques isillustrated by presenting new infinite expander families of Cayley and G-graphs on certaingroups. Also, since expanders stand in no single stem of graph theory, this brings us toinvestigate several closely related threads from a new angle. For instance, we obtain newresults concerning the computation of spectra of certain Cayley and G-graphs, and theconstruction of several new infinite classes of integral and Hamiltonian Cayley graphs
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Kahale, Nabil. "Expander graphs." Thesis, Massachusetts Institute of Technology, 1993. http://hdl.handle.net/1721.1/12511.

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Lountzi, Angeliki. "Expander Graphs and Explicit Constructions." Thesis, Uppsala universitet, Algebra och geometri, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-274643.

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Maceli, Peter Lawson. "Deciding st-connectivity in undirected graphs using logarithmic space." Columbus, Ohio : Ohio State University, 2008. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1211753530.

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Werner, Rose-Line. "Concrete constructions of unbalanced bipartite expander graphs and generalized conductors." Zürich : ETH, Eidgenössische Technische Hochschule Zürich, Departement Informatik, Institut für Informationssysteme, 2008. http://e-collection.ethbib.ethz.ch/show?type=dipl&nr=389.

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Barsukov, Alexey. "On dichotomy above Feder and Vardi's logic." Electronic Thesis or Diss., Université Clermont Auvergne (2021-...), 2022. https://tel.archives-ouvertes.fr/tel-04100704.

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On dit d'un sous-ensemble de NP qu'il présente une dichotomie s'il contient des problèmes qui sont soit résolubles en temps polynomial (dans Ptime), soit difficiles (NP-complets). La classe des problèmes de satisfaction de contraintes (CSP) finis est un sous-ensemble bien connu de NP qui présente une telle dichotomie. La classe de complexité NP n'a pas de dichotomie à moins que P = NP. Pour ces deux classes, il existe des logiques qui leur sont associées. -- NP est capturé par la logique Existentielle du second ordre (ESO) par le théorème de Fagin, c'est-à-dire qu'un problème est dans NP si et seulement s'il est exprimable par une formule ESO.-- CSP est un sous-ensemble de la logique de Feder et Vardi, le fragment monotone, monadique et sans inégalités de SNP, lui-même un fragment syntaxique de ESO (MMSNP); et, pour chaque formule de MMSNP, il existe un problème CSP équivalent via des réductions polynomiales.Ceci implique que la logique ESO, tout comme NP, n'a pas de dichotomie, à contraster avec le fait que MMSNP a une dichotomie tout comme CSP. L'objectif principal de cette thèse est d'étudier les propriétés de dichotomie de sous-ensembles de NP qui contiennent strictement CSP ou MMSNP.Feder et Vardi ont prouvé que si nous omettons une des trois propriétés qui définissent MMSNP, à savoir être monotone, monadique ou omettre les inégalités, alors la logique résultante n'a pas de dichotomie. Comme leurs preuves restent parfois sommaires, nous revisitons ces résultats et fournissons des preuves détaillées. Le fragment guardé et monotone de SNP (GMSNP) est une extension connue de MMSNP qui est obtenue en relâchant la restriction "monadique" de MMSNP. Nous définissons de manière similaire une nouvelle logique appelée MMSNP avec des inégalités gardées, en relâchant la restriction d'être "sans inégalités". Nous prouvons qu'elle est strictement plus expressive que MMSNP et qu'elle possède également une dichotomie.Il existe une logique MMSNP₂ qui étend MMSNP de la même manière que MSO₂ étend la logique monadique du second ordre (MSO). On sait que MMSNP₂ est un fragment de GMSNP et que ces deux classes ont toutes deux une dichotomie ou n'en ont pas. Nous revisitons ce résultat et le renforçons en prouvant que, en ce qui concerne le fait d'avoir une dichotomie, sans perte de généralité, on peut considérer seulement les problèmes MMSNP₂ sur des signatures à un élément, au lieu des problèmes GMSNP sur des signatures finies arbitraires.Nous cherchons à prouver l'existence d'une dichotomie pour les MMSNP₂ en construisant en temps polynomial, pour tout problème MMSNP₂, un problème MMSNP équivalent. Nous rencontrons quelques obstacles pour construire une telle équivalence. Cependant, si nous permettons aux formules MMSNP d'être composées d'un nombre dénombrable de conjonctions négatives, nous prouvons qu'une telle équivalence existe. De plus, la formule MMSNP infinie correspondante a la propriété d'être "régulière". Cette propriété de régularité signifie que, dans un certain sens, cette formule est essentiellement finie. Il est connu que les problèmes MMSNP réguliers peuvent être exprimés par CSP sur des modèles oméga-catégoriques. De plus, il existe une caractérisation de la dichotomie algébrique pour les CSP oméga-catégoriques qui décrivent des problèmes MMSNP. Si l'on parvient à étendre cette caractérisation algébrique sur les problèmes réguliers MMSNP, alors notre résultat fournirait une dichotomie algébrique pour MMSNP₂. (...)
A subset of NP is said to have a dichotomy if it contains problem that are either solvable in P-time or NP-complete. The class of finite Constraint Satisfaction Problems (CSP) is a well-known subset of NP that follows such a dichotomy. The complexity class NP does not have a dichotomy unless P = NP. For both of these classes there exist logics that are associated with them. -- NP is captured by Existential Second-Order (ESO) logic by Fagin's theorem, i.e., a problem is in NP if and only if it is expressible by an ESO sentence.-- CSP is a subset of Feder and Vardi's logic, Monotone Monadic Strict NP without inequalities (MMSNP), and for every MMSNP sentence there exists a P-time equivalent CSP problem. This implies that ESO does not have a dichotomy as well as NP, and that MMSNP has a dichotomy as well as CSP. The main objective of this thesis is to study subsets of NP that strictly contain CSP or MMSNP with respect to the dichotomy existence.Feder and Vardi proved that if we omit one of the three properties that define MMSNP, namely being monotone, monadic or omitting inequalities, then the resulting logic does not have a dichotomy. As their proofs remain sketchy at times, we revisit these results and provide detailed proofs. Guarded Monotone Strict NP (GMSNP) is a known extension of MMSNP that is obtained by relaxing the "monadic" restriction of MMSNP. We define similarly a new logic that is called MMSNP with Guarded inequalities, relaxing the restriction of being "without inequalities". We prove that it is strictly more expressive than MMSNP and that it also has a dichotomy.There is a logic MMSNP₂ that extends MMSNP in the same way as MSO₂ extends Monadic Second-Order (MSO) logic. It is known that MMSNP₂ is a fragment of GMSNP and that these two classes either both have a dichotomy or both have not. We revisit this result and strengthen it by proving that, with respect to having a dichotomy, without loss of generality, one can consider only MMSNP₂ problems over one-element signatures, instead of GMSNP problems over arbitrary finite signatures.We seek to prove the existence of a dichotomy for MMSNP₂ by finding, for every MMSNP₂ problem, a P-time equivalent MMSNP problem. We face some obstacles to build such an equivalence. However, if we allow MMSNP sentences to consist of countably many negated conjuncts, then we prove that such an equivalence exists. Moreover, the corresponding infinite MMSNP sentence has a property of being "regular". This regular property means that, in some sense, this sentence is still finite. It is known that regular MMSNP problems can be expressed by CSP on omega-categorical templates. Also, there is an algebraic dichotomy characterisation for omega-categorical CSPs that describe MMSNP problems. If one manages to extend this algebraic characterisation onto regular MMSNP, then our result would provide an algebraic dichotomy for MMSNP₂.Another potential way to prove the existence of a dichotomy for MMSNP₂ is to mimic the proof of Feder and Vardi for MMSNP. That is, by finding a P-time equivalent CSP problem. The most difficult part there is to reduce a given input structure to a structure of sufficiently large girth. For MMSNP and CSP, it is done using expanders, i.e., structures, where the distribution of tuples is close to a uniform distribution. We study this approach with respect to MMSNP₂ and point out the main obstacles. (...)
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Mendoza-Smith, Rodrigo. "Numerical algorithms for the mathematics of information." Thesis, University of Oxford, 2017. http://ora.ox.ac.uk/objects/uuid:451a418b-eca0-454f-8b54-7b6476056969.

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This thesis presents a series of algorithmic innovations in Combinatorial Compressed Sensing and Persistent Homology. The unifying strategy across these contributions is in translating structural patterns in the underlying data into specific algorithmic designs in order to achieve: better guarantees in computational complexity, the ability to operate on more complex data, highly efficient parallelisations, or any combination of these.
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Balelli, Irène. "Fondements mathématiques de la maturation d’affinité des anticorps." Thesis, Sorbonne Paris Cité, 2016. http://www.theses.fr/2016USPCD091/document.

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Le système immunitaire adaptatif est capable de produire une réponse spécifique contre presque tous le pathogènes qui agressent notre organisme. Ceci est dû aux anticorps qui sont des protéines secrétées par les cellules B. Les molécules qui provoquent cette réaction sont appelées antigènes : pendant une réponse immunitaire, les cellules B sont soumises à un processus d’apprentissage afin d’améliorer leur capacité à reconnaitre un antigène donne. Ce processus est appelé maturation d’affinité des anticorps. Nous établissons un cadre mathématique très flexible dans lequel nous définissons et étudions des modelés évolutionnaires simplifies inspirés par la maturation d’affinité des anticorps. Nous identifions les éléments constitutifs fondamentaux de ce mécanisme d’évolution extrêmement rapide et efficace : mutation, division et sélection. En commençant par une analyse rigoureuse du mécanisme de mutation dans le Chapitre 2, nous procédons à l’enrichissement progressif du modelé en ajoutant et analysant le processus de division dans le Chapitre 3 ,puis des pressions sélectives dépendantes de l’affinité dans le Chapitre 4. Notre objectif n’est pas de construire un modèle mathématique très détaillé et exhaustif de la maturation d’affinité des anticorps, mais plutôt d’enquêter sur les interactions entre mutation, division et sélection dans un contexte théorique simplifie. On cherche à comprendre comment les différents paramètres biologiques influencent la fonctionnalité du système, ainsi qu’à estimer les temps caractéristiques de l’exploration de l’espace d’états des traits des cellules B. Au-delà des motivations biologiques de la modélisation de la maturation d’affinité des anticorps, l’analyse de ce processus d’apprentissage nous a amenée à concevoir un modèle mathématique qui peut également s’appliquer à d’autres systèmes d’évolution, mais aussi à l’étude de la propagation de rumeurs ou de virus. Notre travail théorique s’accompagne de nombreuses simulations numériques qui viennent soit l’illustrer soit montrer que certains résultats demeurent extensibles a des situations plus compliquées
The adaptive immune system is able to produce a specific response against almost any pathogen that could penetrate our organism and inflict diseases. This task is assured by the production of antigen-specific antibodies secreted by B-cells. The agents which causes this reaction are called antigens: during an immune response B-cells are submitted to a learning process in order to improve their ability to recognize the immunizing antigen. This process is called antibody affinity maturation. We set a highly flexible mathematical environment in which we define and study simplified mathematical evolutionary models inspired by antibody affinity maturation. We identify the fundamental building blocks of this extremely efficient and rapid evolutionary mechanism: mutation, division and selection. Starting by a rigorous analysis of the mutational mechanism in Chapter 2, we proceed by successively enriching the model by adding and analyzing the division process in Chapter 3 and affinity-dependent selection pressures in Chapter 4. Our aim is not to build a very detailed and comprehensive mathematical model of antibody affinity maturation, but rather to investigate interactions between mutation, division and selection in a simplified theoretical context. We want to understand how the different biological parameters affect the system’s functionality, as well as estimate the typical time-scales of the exploration of the state-space of B-cell traits. Beyond the biological motivations of antibody affinity maturation modeling, the analysis of this learning process leads us to build a mathematical model which could be relevant to model other evolutionary systems, but also gossip or virus propagation. Our method is based on the complementarity between probabilistic tools and numerical simulations
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Vigolo, Federico. "Geometry of actions, expanders and warped cones." Thesis, University of Oxford, 2018. http://ora.ox.ac.uk/objects/uuid:0b094203-6f94-4b3b-826e-c8b1ac6203b8.

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In this thesis we introduce a notion of graphs approximating actions of finitely generated groups on metric and measure spaces. We systematically investigate expansion properties of said graphs and we prove that a sequence of graphs approximating a fixed action ρ forms a family of expanders if and only if ρ is expanding in measure. This enables us to rely on a number of known results to construct numerous new families of expander (and superexpander) graphs. Proceeding in our investigation, we show that the graphs approximating an action are uniformly quasi-isometric to the level sets of the associated warped cone. The existence of such a relation between approximating graphs and warped cones has twofold advantages: on the one hand it implies that warped cones arising from actions that are expanding in measure coarsely contain families of expanders, on the other hand it provides a geometric model for the approximating graphs allowing us to study the geometry of the expander thus obtained. The rest of the work is devoted to the study of the coarse geometry of warped cones (and approximating graphs). We do so in order to prove rigidity results which allow us to prove that our construction is flexible enough to produce a number of non coarsely equivalent new families of expanders. As a by-product, we also show that some of these expanders enjoy some rather peculiar geometric properties, e.g. we can construct expanders that are coarsely simply connected.
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McGuire, Paul. "Composing with an expanded instrumental palette." Thesis, Brunel University, 2015. http://bura.brunel.ac.uk/handle/2438/12690.

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This thesis is comprised of a portfolio of musical compositions with accompanying media and a written commentary. In each of the seven compositions, the timbral palettes of musical instruments have been expanded through unconventional physical manipulation. The written commentary presents, in detail, specific examples of how this has been achieved. Alongside descriptions of the work in question, select aspects of other composers' music that approach a similar aesthetic are also referred to. In addition, the fundamental role technology has played in the creation or realisation of certain pieces is addressed. Also included are descriptions of the various customised notational systems used throughout the portfolio. It is outlined how each of these systems has been constructed in a clear and practical manner and, where possible, has incorporated elements derived from the lingua franca in order to communicate the required information as efficiently as possible to the performers.
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Books on the topic "Expander graphs"

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Krebs, Mike. Expander families and Cayley graphs: A beginner's guide. Oxford: Oxford University Press, 2011.

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Ron, Hubbard L. Expanded grades references. Los Angeles, Calif: Bridge Publications, 1994.

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Weissman, Annie. Expand and enrich reading, grades 3-6. Worthington, OH: Linworth Learning, 2003.

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Tinsley, Kevin. Digital prepress for comic books: Revised, expanded & updated. Brooklyn, NY: Stickman Graphics, 2009.

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Tinsley, Kevin. Digital prepress for comic books: Revised, expanded & updated. Brooklyn, NY: Stickman Graphics, 2009.

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Tinsley, Kevin. Digital prepress for comic books: Revised, expanded & updated. Brooklyn, NY: Stickman Graphics, 2009.

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Barba, Rick. Myst: Official Strategy Guide, Revised and Expanded. Rocklin, Calif: Prima Games, 1995.

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United States. Office of Justice Programs. Office for Victims of Crimes, ed. Helping outreach programs to expand. Washington, DC: U.S. Dept. of Justice, Office of Justice Programs, Office for Victims of Crimes, 2004.

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1966-, Smith Brian Scott, ed. Story stretchers for the primary grades: Activities to expand children's books. Silver Spring, MD: Gryphon House, 2011.

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J, Canady Robert, ed. Story stretchers for the primary grades: Activities to expand children's favorite books. Mt. Rainier, Md: Gryphon House, 1992.

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Book chapters on the topic "Expander graphs"

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Shum, Kenneth, and Ian Blake. "Expander graphs and codes." In Algebraic Coding Theory and Information Theory, 57–68. Providence, Rhode Island: American Mathematical Society, 2005. http://dx.doi.org/10.1090/dimacs/068/03.

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Goldreich, Oded. "Basic Facts about Expander Graphs." In Studies in Complexity and Cryptography. Miscellanea on the Interplay between Randomness and Computation, 451–64. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-22670-0_30.

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Shalom, Yehuda. "Expander Graphs and Amenable Quotients." In Emerging Applications of Number Theory, 571–81. New York, NY: Springer New York, 1999. http://dx.doi.org/10.1007/978-1-4612-1544-8_23.

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Janwa, H. "Good Expander Graphs and Expander Codes: Parameters and Decoding." In Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 119–28. Berlin, Heidelberg: Springer Berlin Heidelberg, 2003. http://dx.doi.org/10.1007/3-540-44828-4_14.

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Mukhopadhyay, Debdeep. "Generating Expander Graphs Using Cellular Automata." In Lecture Notes in Computer Science, 52–62. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-33350-7_6.

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Ganguly, Sumit. "Data Stream Algorithms via Expander Graphs." In Algorithms and Computation, 52–63. Berlin, Heidelberg: Springer Berlin Heidelberg, 2008. http://dx.doi.org/10.1007/978-3-540-92182-0_8.

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Jo, Hyungrok, Shingo Sugiyama, and Yoshinori Yamasaki. "Ramanujan Graphs for Post-Quantum Cryptography." In International Symposium on Mathematics, Quantum Theory, and Cryptography, 231–50. Singapore: Springer Singapore, 2020. http://dx.doi.org/10.1007/978-981-15-5191-8_17.

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Abstract We introduce a cryptographic hash function based on expander graphs, suggested by Charles et al. ’09, as one prominent candidate in post-quantum cryptography. We propose a generalized version of explicit constructions of Ramanujan graphs, which are seen as an optimal structure of expander graphs in a spectral sense, from the previous works of Lubotzky, Phillips, Sarnak ’88 and Chiu ’92. We also describe the relationship between the security of Cayley hash functions and word problems for group theory. We also give a brief comparison of LPS-type graphs and Pizer’s graphs to draw attention to the underlying hard problems in cryptography.
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Applebaum, Benny, and Pavel Raykov. "Fast Pseudorandom Functions Based on Expander Graphs." In Theory of Cryptography, 27–56. Berlin, Heidelberg: Springer Berlin Heidelberg, 2016. http://dx.doi.org/10.1007/978-3-662-53641-4_2.

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Breuillard, Emmanuel. "Expander graphs, property (𝜏) and approximate groups." In Geometric Group Theory, 325–77. Providence, Rhode Island: American Mathematical Society, 2014. http://dx.doi.org/10.1090/pcms/021/10.

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Spielman, Daniel A. "Constructing Error-Correcting Codes from Expander Graphs." In Emerging Applications of Number Theory, 591–600. New York, NY: Springer New York, 1999. http://dx.doi.org/10.1007/978-1-4612-1544-8_25.

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Conference papers on the topic "Expander graphs"

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Cheung, Ho Yee, Lap Chi Lau, and Kai Man Leung. "Graph Connectivities, Network Coding, and Expander Graphs." In 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS). IEEE, 2011. http://dx.doi.org/10.1109/focs.2011.55.

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Peleg, D., and E. Upfal. "Constructing disjoint paths on expander graphs." In the nineteenth annual ACM conference. New York, New York, USA: ACM Press, 1987. http://dx.doi.org/10.1145/28395.28424.

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Kelsen, Pierre. "Fast parallel matching in expander graphs." In the fifth annual ACM symposium. New York, New York, USA: ACM Press, 1993. http://dx.doi.org/10.1145/165231.166107.

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Chapman, Michael, Nati Linial, and Yuval Peled. "Expander Graphs – Both Local and Global." In 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS). IEEE, 2019. http://dx.doi.org/10.1109/focs.2019.00019.

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Chilappagari, Shashi Kiran, and Bane Vasic. "Fault Tolerant Memories Based on Expander Graphs." In 2007 IEEE Information Theory Workshop. IEEE, 2007. http://dx.doi.org/10.1109/itw.2007.4313061.

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Xu, Weiyu, and Babak Hassibi. "Efficient Compressive Sensing with Deterministic Guarantees Using Expander Graphs." In 2007 IEEE Information Theory Workshop. IEEE, 2007. http://dx.doi.org/10.1109/itw.2007.4313110.

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Wu, Zhenghua, Qiang Wang, Yi Shen, and Jie Liu. "Optimized selection of random expander graphs for Compressive Sensing." In 2013 IEEE International Conference on Information and Automation (ICIA). IEEE, 2013. http://dx.doi.org/10.1109/icinfa.2013.6720446.

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Wu, Zhenghua, Qiang Wang, Yi Shen, and Jie Liu. "Zig-zag and replacement product expander graphs for Compressive Sensing." In 2012 IEEE International Instrumentation and Measurement Technology Conference (I2MTC). IEEE, 2012. http://dx.doi.org/10.1109/i2mtc.2012.6229330.

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Broder, Andrei Z., Alan M. Frieze, and Eli Upfal. "Static and dynamic path selection on expander graphs (preliminary version)." In the twenty-ninth annual ACM symposium. New York, New York, USA: ACM Press, 1997. http://dx.doi.org/10.1145/258533.258646.

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Broder, Andrei Z., Alan M. Frieze, and Eli Upfal. "Existence and construction of edge disjoint paths on expander graphs." In the twenty-fourth annual ACM symposium. New York, New York, USA: ACM Press, 1992. http://dx.doi.org/10.1145/129712.129727.

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Reports on the topic "Expander graphs"

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Leung-Gagné, Melanie, Victoria Wang, Hanna Melnick, and Chris Mauerman. How are California school districts planning for universal prekindergarten? Results from a 2022 survey. Learning Policy Institute, April 2023. http://dx.doi.org/10.54300/109.432.

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In 2021, California committed to providing universal prekindergarten (UPK) for all 4-year-olds and income-eligible 3-year-olds by 2025–26. UPK includes several early learning programs, including transitional kindergarten (TK), the California State Preschool Program (CSPP), Head Start, and expanded learning opportunities to provide full-day early learning and care. TK is the only UPK program that is free and universally available as part of California’s public education system. Offered by local education agencies (LEAs), TK currently serves all 4-year-olds who turn 5 between September 2 and December 2 and will expand to all 4-year-olds by 2025–26. The legislature also made new investments in CSPP, a program for income-eligible 3- and 4-year-old children. Funding for CSPP is provided by the state through grants to both LEAs and community-based organizations. This report provides a snapshot of 1,108 LEAs’ initial plans for UPK expansion through the analysis of a survey administered by the California Department of Education in August 2022. Key findings provide insights into LEA plans for service delivery models, facilities and transportation, instruction and assessment, workforce development, school leader development, and technical assistance needs.
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Lougheed, H. D., M. B. McClenaghan, D. Layton-Matthews, and M. I. Leybourne. Indicator minerals in fine-fraction till heavy-mineral concentrates determined by automated mineral analysis: examples from two Canadian polymetallic base-metal deposits. Natural Resources Canada/CMSS/Information Management, 2022. http://dx.doi.org/10.4095/328011.

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Exploration under glacial sediment cover is a necessary part of modern mineral exploration in Canada. Traditional indicator methods use visual examination to identify mineral grains in the 250 to 2000 µm fraction of till heavy-mineral concentrates (HMC). This study tests automated mineralogical methods using scanning electron microscopy to identify indicator minerals in the fine (<250 µm) HMC fraction of till. Automated mineralogy of polished grains from the fine HMC enables rapid data collection (10 000-300 000 grains/sample). Samples collected near two deposits were used to test this method: four from the upper-amphibolite facies Izok Lake volcanogenic massive-sulfide deposit, Nunavut, and five from the Sisson granite-hosted W-Mo deposit, New Brunswick. The less than 250 µm HMC fraction of till samples collected down ice of each deposit contain ore and alteration minerals typical of their deposit type. Sulfide minerals occur mainly as inclusions in oxidation-resistant minerals, including minerals previously identified in each deposit's metamorphic alteration halo, and are found to occur farther down ice than the grains identified visually in the greater than 250 µm HMC fraction. This project's workflow expands the detectable footprint for certain indicator minerals and enhances the information that can be collected from till samples.
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Dubeck, Margaret M., Jonathan M. B. Stern, and Rehemah Nabacwa. Learning to Read in a Local Language in Uganda: Creating Learner Profiles to Track Progress and Guide Instruction Using Early Grade Reading Assessment Results. RTI Press, June 2021. http://dx.doi.org/10.3768/rtipress.2021.op.0068.2106.

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The Early Grade Reading Assessment (EGRA) is used to evaluate studies and monitor projects that address reading skills in low- and middle-income countries. Results are often described solely in terms of a passage-reading subtask, thereby overlooking progress in related skills. Using archival data of cohort samples from Uganda at two time points in three languages (Ganda, Lango, and Runyankore-Rukiga), we explored a methodology that uses passage-reading results to create five learner profiles: Nonreader, Beginner, Instructional, Fluent, and Next-Level Ready. We compared learner profiles with results on other subtasks to identify the skills students would need to develop to progress from one profile to another. We then used regression models to determine whether students’ learner profiles were related to their results on the various subtasks. We found membership in four categories. We also found a shift in the distribution of learner profiles from Grade 1 to Grade 4, which is useful for establishing program effectiveness. The distribution of profiles within grades expanded as students progressed through the early elementary grades. We recommend that those who are discussing EGRA results describe students by profiles and by the numbers that shift from one profile to another over time. Doing so would help describe abilities and instructional needs and would show changes in a meaningful way.
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Fallas, K. M., and W. Matthews. Age dating of a bentonite in the Duo Lake Formation, western Mackenzie Mountains, Northwest Territories. Natural Resources Canada/CMSS/Information Management, 2022. http://dx.doi.org/10.4095/328830.

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In the Misty Creek Embayment of the western Mackenzie Mountains, Duo Lake Formation locally includes minor volcanic deposits associated with Marmot Formation volcanism. A bentonite layer from an outcrop of graptolitic shale found in NTS map area 106-B, in the upper part of the Duo Lake Formation, was sampled for U-Pb zircon dating. Analytical results yielded a dominant population of grains with a concordia age of 439.8 ± 3.0 Ma, interpreted as the age of deposition. Minor inherited zircon populations yielded ages ranging from approximately 1200 to 2850 Ma. Observed graptolites from the same outcrop likely range from Middle Ordovician to Early Silurian and are compatible with the interpreted U-Pb age of the bentonite. Previously known Middle and Late Ordovician volcanic activity in the Misty Creek Embayment is here expanded to include Early Silurian activity, and serves as a proxy for the timing of active extensional tectonism in the basin.
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Letcher, Theodore, Julie Parno, Zoe Courville, Lauren Farnsworth, and Jason Olivier. A generalized photon-tracking approach to simulate spectral snow albedo and transmittance using X-ray microtomography and geometric optics. Engineer Research and Development Center (U.S.), June 2023. http://dx.doi.org/10.21079/11681/47122.

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A majority of snow radiative transfer models (RTMs) treat snow as a collection of idealized grains rather than an organized ice–air matrix. Here we present a generalized multi-layer photon-tracking RTM that simulates light reflectance and transmittance of snow based on X-ray micro- tomography images, treating snow as a coherent 3D structure rather than a collection of grains. The model uses a blended approach to expand ray-tracing techniques applied to sub-1 cm3 snow samples to snowpacks of arbitrary depths. While this framework has many potential applications, this study’s effort is focused on simulating reflectance and transmittance in the visible and near infrared (NIR) through thin snow- packs as this is relevant for surface energy balance and remote sensing applications. We demonstrate that this framework fits well within the context of previous work and capably reproduces many known optical properties of a snow surface, including the dependence of spectral reflectance on the snow specific surface area and incident zenith angle as well as the surface bidirectional reflectance distribution function (BRDF). To evaluate the model, we compare it against reflectance data collected with a spectroradiometer at a field site in east-central Vermont. In this experiment, painted panels were inserted at various depths beneath the snow to emulate thin snow. The model compares remarkably well against the reflectance measured with a spectroradiometer, with an average RMSE of 0.03 in the 400–1600 nm range. Sensitivity simulations using this model indicate that snow transmittance is greatest in the visible wavelengths, limiting light penetration to the top 6 cm of the snowpack for fine-grain snow but increasing to 12 cm for coarse-grain snow. These results suggest that the 5% transmission depth in snow can vary by over 6 cm according to the snow type.
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Stadnyk, Vаlentyna, Pavlo Izhevskiy, Nila Khrushch, Sergii Lysenko, Galyna Sokoliuk, and Tetjana Tomalja. Strategic priorities of innovation and investment development of the Ukraine's economy industrial sector. [б. в.], October 2020. http://dx.doi.org/10.31812/123456789/4471.

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The problem of determining the investment priorities of the national economy development has been actuated. It has been argued that the formation of institutional preferences for activation of industry investment processes should be carried out taking into account the potential ability of each sectoral group enterprises to increase the added value. The scientific and methodical approach for sub-sectors investment attractiveness assessment has been formed on the example of the Ukrainian food industry. It has been recommended to use for this substantiated set of relative performance indexes which are duplicated in aggregate statistical state surveys based on the enterprise’s financial statements. It has been formed the recommendations for the investment priorities of food industry development in Ukraine which are based on the appropriate calculations made by the TOPSIS and CRITIC methods. Methods of economic-statistical and comparative analysis were used for structural and dynamic characteristics of the Ukraine industrial enterprises activities. Given that innovation processes should also cover small and medium-sized industrial enterprises, whose resource opportunities are mostly limited, it is proposed to expand them within the framework of a strategic partnership. Graphic modeling methods have been used to visualize the process of building the business structures resource potential on the basis of their strategic partnership. The influence of the motivational environment on the value of organizational relations within the partnership has been formalized.
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Taheripour, Farzad, and Wally Tyner. Introducing First and Second Generation Biofuels into GTAP Data Base version 7*. GTAP Research Memoranda, February 2011. http://dx.doi.org/10.21642/gtap.rm21.

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The first version of GTAP-BIO Data Base was built based on the GTAP standard data base version 6 which represents the world economy in 2001 (Taheripour et al., 2007). That data base covers global production, consumption, and trade of the first generation of biofuels including ethanol from grains (eth1), ethanol from sugarcane (eth2), and biodiesel (biod) from oilseeds in 2001. Version 7 of GTAP Data Base, which depicts the world economy in 2004, is now published (Narayanan, B.G. and T.L. Walmsley, 2008). However, this standard data base does not include biofuel industries explicitly. The first objective of this research memorandum is to introduce the first generation of biofuels into this new data base. To accomplish this task we will follow Taheripour et al. (2007). The rapid expansion of the first generation of biofuels in the past decades has raised important concerns related to food-fuel competition, land use change, and other economic and environmental issues. These issues have increased interest in the second generation of biofuels which can be produced from cellulosic materials such as dedicated crops, agricultural and forest residues, and waste materials. To examine the economic and environmental consequences of the second generation of biofuels, a CGE model is an appropriate and essential instrument. A data base which presents the first and second generation of biofuels will facilitate research in this field. Hence the second objective of this research memorandum is to expand the space of biofuel alternatives to the second generation. Given that advanced cellulosic biofuels are not yet commercially viable, we used the most up to date information in this area to define the production technologies for these industries.
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Hostetler, Steven, Cathy Whitlock, Bryan Shuman, David Liefert, Charles Wolf Drimal, and Scott Bischke. Greater Yellowstone climate assessment: past, present, and future climate change in greater Yellowstone watersheds. Montana State University, June 2021. http://dx.doi.org/10.15788/gyca2021.

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The Greater Yellowstone Area (GYA) is one of the last remaining large and nearly intact temperate ecosystems on Earth (Reese 1984; NPSa undated). GYA was originally defined in the 1970s as the Greater Yellowstone Ecosystem, which encompassed the minimum range of the grizzly bear (Schullery 1992). The boundary was enlarged through time and now includes about 22 million acres (8.9 million ha) in northwestern Wyoming, south central Montana, and eastern Idaho. Two national parks, five national forests, three wildlife refuges, 20 counties, and state and private lands lie within the GYA boundary. GYA also includes the Wind River Indian Reservation, but the region is the historical home to several Tribal Nations. Federal lands managed by the US Forest Service, the National Park Service, the Bureau of Land Management, and the US Fish and Wildlife Service amount to about 64% (15.5 million acres [6.27 million ha] or 24,200 square miles [62,700 km2]) of the land within the GYA. The federal lands and their associated wildlife, geologic wonders, and recreational opportunities are considered the GYA’s most valuable economic asset. GYA, and especially the national parks, have long been a place for important scientific discoveries, an inspiration for creativity, and an important national and international stage for fundamental discussions about the interactions of humans and nature (e.g., Keiter and Boyce 1991; Pritchard 1999; Schullery 2004; Quammen 2016). Yellowstone National Park, established in 1872 as the world’s first national park, is the heart of the GYA. Grand Teton National Park, created in 1929 and expanded to its present size in 1950, is located south of Yellowstone National Park1 and is dominated by the rugged Teton Range rising from the valley of Jackson Hole. The Gallatin-Custer, Shoshone, Bridger-Teton, Caribou-Targhee, and Beaverhead-Deerlodge national forests encircle the two national parks and include the highest mountain ranges in the region. The National Elk Refuge, Red Rock Lakes National Wildlife Refuge, and Grays Lake National Wildlife Refuge also lie within GYA.
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Miller, Gad, and Jeffrey F. Harper. Pollen fertility and the role of ROS and Ca signaling in heat stress tolerance. United States Department of Agriculture, January 2013. http://dx.doi.org/10.32747/2013.7598150.bard.

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The long-term goal of this research is to understand how pollen cope with stress, and identify genes that can be manipulated in crop plants to improve reproductive success during heat stress. The specific aims were to: 1) Compare heat stress dependent changes in gene expression between wild type pollen, and mutants in which pollen are heat sensitive (cngc16) or heat tolerant (apx2-1). 2) Compare cngc16 and apx2 mutants for differences in heat-stress triggered changes in ROS, cNMP, and Ca²⁺ transients. 3) Expand a mutant screen for pollen with increased or decreased thermo-tolerance. These aims were designed to provide novel and fundamental advances to our understanding of stress tolerance in pollen reproductive development, and enable research aimed at improving crop plants to be more productive under conditions of heat stress. Background: Each year crop yields are severely impacted by a variety of stress conditions, including heat, cold, drought, hypoxia, and salt. Reproductive development in flowering plants is highly sensitive to hot or cold temperatures, with even a single hot day or cold night sometimes being fatal to reproductive success. In many plants, pollen tube development and fertilization is often the weakest link. Current speculation about global climate change is that most agricultural regions will experience more extreme environmental fluctuations. With the human food supply largely dependent on seeds, it is critical that we consider ways to improve stress tolerance during fertilization. The heat stress response (HSR) has been intensively studied in vegetative tissues, but is poorly understood during reproductive development. A general paradigm is that HS is accompanied by increased production of reactive oxygen species (ROS) and induction of ROS-scavenging enzymes to protect cells from excess oxidative damage. The activation of the HSR has been linked to cytosolic Ca²⁺ signals, and transcriptional and translational responses, including the increased expression of heat shock proteins (HSPs) and antioxidative pathways. The focus of the proposed research was on two mutations, which have been discovered in a collaboration between the Harper and Miller labs, that either increase or decrease reproductive stress tolerance in a model plant, Arabidopsis thaliana (i.e., cngc16--cyclic nucleotide gated channel 16, apx2-1--ascorbate peroxidase 2,). Major conclusions, solutions, achievements. Using RNA-seq technology, the expression profiles of cngc16 and apx2 pollen grains were independently compared to wild type under favourable conditions and following HS. In comparison to a wild type HSR, there were 2,776 differences in the transcriptome response in cngc16 pollen, consistent with a model in which this heat-sensitive mutant fails to enact or maintain a normal wild-type HSR. In a comparison with apx2 pollen, there were 900 differences in the HSR. Some portion of these 900 differences might contribute to an improved HSR in apx2 pollen. Twenty-seven and 42 transcription factor changes, in cngc16 and apx2-1, respectively, were identified that could provide unique contributions to a pollen HSR. While we found that the functional HS-dependent reprogramming of the pollen transcriptome requires specific activity of CNGC16, we identified in apx2 specific activation of flavonol-biosynthesis pathway and auxin signalling that support a role in pollen thermotolerance. Results from this study have identified metabolic pathways and candidate genes of potential use in improving HS tolerance in pollen. Additionally, we developed new FACS-based methodology that can quantify the stress response for individual pollen in a high-throughput fashion. This technology is being adapted for biological screening of crop plant’s pollen to identify novel thermotolerance traits. Implications, both scientific and agricultural. This study has provided a reference data on the pollen HSR from a model plant, and supports a model that the HSR in pollen has many differences compared to vegetative cells. This provides an important foundation for understanding and improving the pollen HSR, and therefor contributes to the long-term goal of improving productivity in crop plants subjected to temperature stress conditions. A specific hypothesis that has emerged from this study is that pollen thermotolerance can be improved by increasing flavonol accumulation before or during a stress response. Efforts to test this hypothesis have been initiated, and if successful have the potential for application with major seed crops such as maize and rice.
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Financial Stability Report - Second Semester of 2020. Banco de la República de Colombia, March 2021. http://dx.doi.org/10.32468/rept-estab-fin.sem2.eng-2020.

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The Colombian financial system has not suffered major structural disruptions during these months of deep economic contraction and has continued to carry out its basic functions as usual, thus facilitating the economy's response to extreme conditions. This is the result of the soundness of financial institutions at the beginning of the crisis, which was reflected in high liquidity and capital adequacy indicators as well as in the timely response of various authorities. Banco de la República lowered its policy interest rates 250 points to 1.75%, the lowest level since the creation of the new independent bank in 1991, and provided ample temporary and permanent liquidity in both pesos and foreign currency. The Office of the Financial Superintendent of Colombia, in turn, adopted prudential measures to facilitate changes in the conditions for loans in effect and temporary rules for rating and loan-loss provisions. Finally, the national government expanded the transfers as well as the guaranteed credit programs for the economy. The supply of real credit (i.e. discounting inflation) in the economy is 4% higher today than it was 12 months ago with especially marked growth in the housing (5.6%) and commercial (4.7%) loan portfolios (2.3% in consumer and -0.1% in microloans), but there have been significant changes over time. During the first few months of the quarantine, firms increased their demands for liquidity sharply while consumers reduced theirs. Since then, the growth of credit to firms has tended to slow down, while consumer and housing credit has grown. The financial system has responded satisfactorily to the changes in the respective demands of each group or sector and loans may grow at high rates in 2021 if GDP grows at rates close to 4.6% as the technical staff at the Bank expects; but the forecasts are highly uncertain. After the strict quarantine implemented by authorities in Colombia, the turmoil seen in March and early April, which was evident in the sudden reddening of macroeconomic variables on the risk heatmap in Graph A,[1] and the drop in crude oil and coal prices (note the high volatility registered in market risk for the region on Graph A) the local financial markets stabilized relatively quickly. Banco de la República’s credible and sustained policy response played a decisive role in this stabilization in terms of liquidity provision through a sharp expansion of repo operations (and changes in amounts, terms, counterparties, and eligible instruments), the purchases of public and private debt, and the reduction in bank reserve requirements. In this respect, there is now abundant aggregate liquidity and significant improvements in the liquidity position of investment funds. In this context, the main vulnerability factor for financial stability in the short term is still the high degree of uncertainty surrounding loan quality. First, the future trajectory of the number of people infected and deceased by the virus and the possible need for additional health measures is uncertain. For that reason, there is also uncertainty about the path for economic recovery in the short and medium term. Second, the degree to which the current shock will be reflected in loan quality once the risk materializes in banks’ financial statements is uncertain. For the time being, the credit risk heatmap (Graph B) indicates that non-performing and risky loans have not shown major deterioration, but past experience indicates that periods of sharp economic slowdown eventually tend to coincide with rises in non-performing loans: the calculations included in this report suggest that the impact of the recession on credit quality could be significant in the short term. This is particularly worrying since the profitability of credit establishments has been declining in recent months, and this could affect their ability to provide credit to the real sector of the economy. In order to adopt a forward-looking approach to this vulnerability, this Report presents several stress tests that evaluate the resilience of the liquidity and capital adequacy of credit institutions and investment funds in the event of a hypothetical scenario that seeks to simulate an extreme version of current macroeconomic conditions. The results suggest that even though there could be strong impacts on the credit institutions’ volume of credit and profitability under such scenarios, aggregate indicators of total and core capital adequacy will probably remain at levels that are above the regulatory limits over the horizon of a year. At the same time, the exercises highlight the high capacity of the system's liquidity to face adverse scenarios. In compliance with its constitutional objectives and in coordination with the financial system's security network, Banco de la República will continue to closely monitor the outlook for financial stability at this juncture and will make the decisions that are necessary to ensure the proper functioning of the economy, facilitate the flow of sufficient credit and liquidity resources, and further the smooth operation of the payment systems. Juan José Echavarría Governor
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