Erasure-Resilient Property Testing

Property testers form an important class of sublinear algorithms. In the standard property testing model, an algorithm accesses the input function f:D -> R via an oracle. With very few exceptions, all property testers studied in this model rely on the oracle to provide function values at all queried domain points. However, in many realistic situations, the oracle may be unable to reveal the function values at some domain points due to privacy concerns, or when some of the values get erased by mistake or by an adversary. The testers do not learn anything useful about the property by querying those erased points. Moreover, the knowledge of a tester may enable an adversary to erase some of the values so as to increase the query complexity of the tester arbitrarily or, in some cases, make the tester entirely useless. In this work, we initiate a study of property testers that are resilient to the presence of adversarially erased function values. An alpha-erasure-resilient epsilon-tester is given parameters alpha, epsilon in (0,1), along with oracle access to a function f such that at most an alpha fraction of function values have been erased. The tester does not know whether a value is erased until it queries the corresponding domain point. The tester has to accept with high probability if there is a way to assign values to the erased points such that the resulting function satisfies the desired property P. It has to reject with high probability if, for every assignment of values to the erased points, the resulting function has to be changed in at least an epsilon-fraction of the non-erased domain points to satisfy P. We design erasure-resilient property testers for a large class of properties. For some properties, it is possible to obtain erasure-resilient testers by simply using standard testers as a black box. However, there are more challenging properties for which all known testers rely on querying a specific point. If this point is erased, all these testers break. We give efficient erasure-resilient testers for several important classes of such properties of functions including monotonicity, the Lipschitz property, and convexity. Finally, we show a separation between the standard testing and erasure-resilient testing. Specifically, we describe a property that can be epsilon-tested with O(1/epsilon) queries in the standard model, whereas testing it in the erasure-resilient model requires number of queries polynomial in the input size

Influence of reaction conditions on the preparation of Cu" pyrazolate CPs: Synthetic approaches and XRD structural characterization

The preparation of Cu"-pyrazolate CPS has been carried out in different sovents, at room temperature and in solvothermal conditions. Trinuclear triangolar compounds and linear CPS were obtained. Trinuclear compounds were also reacted with bipyridine in solvothermal conditional.The products were charactezized with elemental analysis, IR and XRPD. XRD structures were determined for some new product

Oxamato/Oxamidato-Based Multifunctional Porous Coordination Polymers

El principal objetivo de esta presente Tesis concierne el diseño y la síntesis de materiales multifuncionales, una línea de investigación de creciente interés dentro del campo de la Química de Materiales. Para llevar a cabo dicho objetivo, hemos aprovechado los avances que ha experimentado la química metalosupramolecular en los últimos años. Concretamente pretendemos hacer uso de los métodos de auto-ensamblaje programado basado en las diferentes preferencias que muestran los iones metálicos en sus entornos de coordinación y en la funcionalización de ligandos orgánicos. Para ello, hemos sintetizado ligandos derivados de grupos oxamato y oxamidato, que actuarán como moléculas de partida para la obtención de estructuras de alta dimensionalidad con diversas propiedades, siendo el control de la porosidad la más relevante. Nuestros esfuerzos se centran, por tanto, en la obtención de polímeros de coordinación porosos (PCPs) y en dotarles de propiedades físicas tales como quiralidad, adsorción y separación de gases o propiedades magnéticas, entre otras. La Tesis, a su vez, se divide en dos líneas de investigación cuyos resultados se muestran en las Partes 1 y 2. La Parte 1 se basa en el diseño de una nueva estrategia sintética para obtener materiales porosos quirales de una forma sencilla y efectiva. La aproximación consiste en el uso de metaloligandos enantiopuros que han sido sintetizados a partir de la funcionalización de aminoácidos (alanina, valina, leucina y fenilglicina) con los grupos oxamato y oxamidato. La quiralidad intrínseca de los aminoácidos se transmite de manera efectiva a sus derivados y los diferentes residuos alifáticos y aromáticos juegan un papel fundamental en el proceso de ensamblaje de las estructuras de mayor dimensionalidad. A su vez, la Parte 1 está divida en las Partes 1.A y 1.B que se centran en los derivados de oxamato y oxamidato, respectivamente. Resulta interesante que a partir de ambas familias de proligandos se obtuvieron metaloligandos quirales de estructuras complejas que dieron lugar a una amplia variedad de fascinantes redes tridimensionales quirales. En la Parte 1.A se demuestra el éxito de la estrategia del metaloligando para transmitir la quiralidad de unos sistemas a otros siendo uno de los resultados más sorprendentes y sin precedentes la obtención de un MOF quiral sintetizado a partir del auto-ensamblaje de cadenas quirales con complejos catiónicos actuando como nodos. En la Parte 1.B, de entre todas las familias de PCPs que se han obtenido, destacan especialmente los PCPs derivados de calcio(II). Esta familia representa una plataforma excelente para el estudio de cómo las propiedades de adsorción y separación de gases pueden ser moduladas basándose en la diferente funcionalización de los poros de los PCPs. De esta forma, se consiguió separar metano de otros hidrocarburos de mayor cadena carbonada. La Parte 2 consiste en el uso de algunos métodos post-sintéticos (MPSs) para insertar nuevas propiedades en materiales que ya están formados y así obtener PCPs multifuncionales. En este capítulo de la Tesis se han explorado tres aproximaciones diferentes que se muestran en las Partes 2.A, 2.B, y 2.C. Haciendo uso de un MOF aniónico poroso, cuya estructura 3D presenta contra-cationes distribuidos a lo largo de los canales, hemos llevado a cabo la sustitución de dichos cationes por otros de distinta naturaleza e investigado las propiedades físicas que los nuevos materiales así preparados pueden exhibir. De esta manera, la Parte 2.A muestra como el intercambio de los iones sodio(I) de un PCP preformado por iones litio(I) y potasio(I) da lugar a dos nuevos materiales más robustos que muestran capacidades de adsorción de gases y propiedades magnéticas mejoradas. En la Parte 2.B, hemos encapsulado de forma efectiva un complejo catiónico de hierro(III) en los poros de un PCP a través de un intercambio catiónico con los iones sodio(I). La estabilidad estructural del nuevo material se ve reforzada y las propiedades magnéticas tanto del PCP como del complejo de hierro(III) mejoran. Por último, en la Parte 2.C, llevamos a cabo el método post-sintético más exigente ya que implica el reemplazo no sólo de los contra-cationes sino también el de los iones metálicos que actúan como nodos en la red covalente del PCP. Así, conseguimos la sustitución de la totalidad de los iones magnesio(II) diamagnéticos por iones cobalto(II) y níquel(II) paramagnéticos a través de un proceso de transmetalación de monocristal a monocristal, dando lugar a dos nuevos materiales más robustos que su precursor y que no se pudieron obtener mediante síntesis directa. La obtención dos nuevos materiales que presentan un canje magnético de largo alcance a partir de un material paramagnético resulta la característica más fascinante.The main goal of this Ph.D. Thesis concerns the design and synthesis of multifunctional materials which is one of the most challenging topics for chemists and physicists working together in the multidisciplinary field of Materials Chemistry. In order to do so, we have taken advantage of the new developments of the metallosupramolecular chemistry, in particular the molecular-programmed self-assembly methods that exploit the coordination preferences of the metal ions and the versatility of the tailored ligands. In this sense, we have chosen functionalised oxamato and oxamidato derivatives to build extended architectures which can exhibit interesting features, the control of the porosity being one of them. Our efforts have been devoted to prepare porous coordination polymers (PCPs) and investigate the introduction of new physical properties such as chirality, gas sorption and separation or magnetic properties, among others. In this respect, two separated research lines have been explored whose results are shown in Parts 1 and 2. Part 1 deals with the development of a synthetic strategy to obtain chiral porous materials in an easy and effective manner. It consists of the functionalisation of enantiopure amino acids (alanine, valine, leucine and phenylglycine) whose encoded chiral information is efficiently transmitted to their derivatives and their different aliphatic residues play a non-negligible role in the self-assembling processes of the extended structures. In turn, Part 1 has been divided into Parts 1.A and 1.B, focusing on oxamate- and oxamidate-based compounds, respectively. Interestingly, both families of ligands gave rise to very different metalloligands and consequently, to a wide variety of fascinating chiral 3D frameworks which display interesting properties. In Part 1.A, we demonstrate that our metalloligand strategy represents an effective synthetic route to rationally prepare chiral PCPs, one of the unprecedented and striking result being the synthesis of a rod-like MOF from a preformed chiral 1D SBU. Among the oxamidato-derived PCPs shown in Part 1.B, we report a family of calcium(II)-derived PCPs which serves as an excellent platform to study how the gas sorption and selectivity can be tailored by tuning the electron density of the channels of the PCPs, thus achieving an easy manner to separate, for instance, methane from longer hydrocarbons in natural gas. Part 2 concerns the use of several post-synthetic methods (PSMs) to introduce new physical properties into preformed materials and thus to obtain multifunctional PCPs. In this chapter, we have explored three PSMs which are discussed in Parts 2.A, 2.B and 2.C. Taking advantage of the porous and anionic nature of the preformed PCPs and the resulting presence of counter-balancing cations within their 3D frameworks, we have performed the substitution of such cations and investigated the physical properties that the new materials show. In Part 2.A, we show how the exchange of the sodium(I) ions by lithium(I) and potassium(I) cations affords the derived PCPs which exhibit improved structural stability, gas sorption and magnetic properties. In Part 2.B, we are able to encapsulate a preformed iron(III) cationic complex within the pores of a PCP through cation exchange. This encapsulation results into interesting properties for both, the encapsulated complex and the original PCP. We go a step forward in Part 2.C and explore the substitution of not only the counter-balancing cations but also the metal ions constituting the coordination framework. Hence, we satisfactory exchange the diamagnetic magnesium(II) ions by paramagnetic cations from the first-transition row through transmetallation processes, affording two new magnetic materials which could not be prepared by direct synthesis

Efficient Domain Partitioning for Stencil-based Parallel Operators

Partial Differential Equations (PDEs) are used ubiquitously in modelling natural phenomena. It is generally not possible to obtain an analytical solution and hence they are commonly discretized using schemes such as the Finite Difference Method (FDM) and the Finite Element Method (FEM), converting the continuous PDE to a discrete system of sparse algebraic equations. The solution of this system can be approximated using iterative methods, which are better suited to many sparse systems than direct methods. In this thesis we use the FDM to discretize linear, second order, Elliptic PDEs and consider parallel implementations of standard iterative solvers. The dominant paradigm in this field is distributed memory parallelism which requires the FDM grid to be partitioned across the available computational cores. The orthodox approach to domain partitioning aims to minimize only the communication volume and achieve perfect load-balance on each core. In this work, we re-examine and challenge this traditional method of domain partitioning and show that for well load-balanced problems, minimizing only the communication volume is insufficient for obtaining optimal domain partitions. To this effect we create a high-level, quasi-cache-aware mathematical model that quantifies cache-misses at the sub-domain level and minimizes them to obtain families of high performing domain decompositions. To our knowledge this is the first work that optimizes domain partitioning by analyzing cache misses, establishing a relationship between cache-misses and domain partitioning. To place our model in its true context, we identify and qualitatively examine multiple other factors such as the Least Recently Used policy, Cache Line Utilization and Vectorization, that influence the choice of optimal sub-domain dimensions. Since the convergence rate of point iterative methods, such as Jacobi, for uniform meshes is not acceptable at a high mesh resolution, we extend the model to Parallel Geometric Multigrid (GMG). GMG is a multilevel, iterative, optimal algorithm for numerically solving Elliptic PDEs. Adaptive Mesh Refinement (AMR) is another multilevel technique that allows local refinement of a global mesh based on parameters such as error estimates or geometric importance. We study a massively parallel, multiphysics, multi-resolution AMR framework called BoxLib, and implement and discuss our model on single level and adaptively refined meshes, respectively. We conclude that “close to 2-D” partitions are optimal for stencil-based codes on structured 3-D domains and that it is necessary to optimize for both minimizing cache-misses and communication. We advise that in light of the evolving hardware-software ecosystem, there is an imperative need to re-examine conventional domain partitioning strategies

Monotonicity Testing for Boolean Functions over Graph Products

We establish a directed analogue of Chung and Tetali's isoperimetric inequality for graph products. We use this inequality to obtain new bounds on the query complexity for testing monotonicity of Boolean-valued functions over products of general posets

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum