Q-system Completion of 2-functors

A Q-system is a unitary version of a separable Frobenius algebra object in a C*-tensor category or a C*-2-category. We prove that, for C*-2-categories \mcal C and \mcal D, the C*-2-category \textbf{Fun}(\mcal C, \mcal D) of $*$-$2$-functors, $*$-$2$-transformations and $*$-$2$-modifications is Q-system complete, whenever \mcal D is Q-system complete. We use this result to provide a characterisation of Q-system complete categories in terms of $*$-$2$-functors and to prove that the $2$-category of actions of a unitary fusion category \mcal C on C*-algebras is Q-system complete.Comment: 23 page

Bertini theorems revisited

We prove several new Bertini theorems over arbitrary fields and discrete valuation rings.Comment: 37 pages. Substantial revision. Some results were strengthened and new results were adde

Dimension Reduction in Big Data Environment-A Survey

Relational database management system is able to tackle data set which is structured in some way and by means of querying to the system user gets certain answer. But if the data set itself does not lie under any sort of structure, it is generally very tedious job for user to get answer to certain query. This is the new challenge coming out for the last decade to the scientists, researchers, industrialists and this new form of data is termed as big data. Parallel computation not only from the concept of hardware, but different application dependent software is now being developed to tackle this new data set for solving the challenges generally attached with large data set such as data curation, search, querying, storage etc. Information sensing devices, RFID readers, cloud storage now days are making data set to grow in an increasing manner. The goal of big data analytics is to help industry and organizations to take intelligent decisions by analyzing huge number of transactions that remain untouched till today by conventional business intelligent systems. As the size of dataset grows large also with redundancy, software and people need to analyze only useful information for particular application and this newly reduced dataset are useful compare to noisy and large data

Unitary connections on Bratteli diagrams

In this paper, we extend Ocneanu's theory of connections on graphs to define a 2-category whose 0-cells are tracial Bratteli diagrams, and whose 1-cells are generalizations of unitary connections. We show that this 2-category admits an embedding into the 2-category of hyperfinite von Neumann algebras, generalizing fundamental results from subfactor theory to a 2-categorical setting.Comment: 35 page

Efficient data reconfiguration for today's cloud systems

Performance of big data systems largely relies on efficient data reconfiguration techniques. Data reconfiguration operations deal with changing configuration parameters that affect data layout in a system. They could be user-initiated like changing shard key, block size in NoSQL databases, or system-initiated like changing replication in distributed interactive analytics engine. Current data reconfiguration schemes are heuristics at best and often do not scale well as data volume grows. As a result, system performance suffers. In this thesis, we show that {\it data reconfiguration mechanisms can be done in the background by using new optimal or near-optimal algorithms coupling them with performant system designs}. We explore four different data reconfiguration operations affecting three popular types of systems -- storage, real-time analytics and batch analytics. In NoSQL databases (storage), we explore new strategies for changing table-level configuration and for compaction as they improve read/write latencies. In distributed interactive analytics engines, a good replication algorithm can save costs by judiciously using memory that is sufficient to provide the highest throughput and low latency for queries. Finally, in batch processing systems, we explore prefetching and caching strategies that can improve the number of production jobs meeting their SLOs. All these operations happen in the background without affecting the fast path. Our contributions in each of the problems are two-fold -- 1) we model the problem and design algorithms inspired from well-known theoretical abstractions, 2) we design and build a system on top of popular open source systems used in companies today. Finally, using real-life workloads, we evaluate the efficacy of our solutions. Morphus and Parqua provide several 9s of availability while changing table level configuration parameters in databases. By halving memory usage in distributed interactive analytics engine, Getafix reduces cost of deploying the system by 10 million dollars annually and improves query throughput. We are the first to model the problem of compaction and provide formal bounds on their runtime. Finally, NetCachier helps 30\% more production jobs to meet their SLOs compared to existing state-of-the-art

Carbon-bearing silicate melt at deep mantle conditions

Knowledge about the incorporation and role of carbon in silicate magmas is crucial for our understanding of the deep mantle processes. CO bearing silicate melting and its relevance in the upper mantle regime have been extensively explored. Here we report first-principles molecular dynamics simulations of MgSiO melt containing carbon in three distinct oxidation states - CO , CO, and C at conditions relevant for the whole mantle. Our results show that at low pressures up to 15 GPa, the carbon dioxide speciation is dominated by molecular form and carbonate ions. At higher pressures, the dominant species are silicon-polyhedral bound carbonates, tetrahedral coordination, and polymerized di-carbonates. Our results also indicate that CO component remains soluble in the melt at high pressures and the solution is nearly ideal. However, the elemental carbon and CO components show clustering of carbon atoms in the melt at high pressures, hinting towards possible exsolution of carbon from silicate melt at reduced oxygen contents. Although carbon lowers the melt density, the effect is modest at high pressures. Hence, it is likely that silicate melt above and below the mantle transition zone, and atop the core-mantle boundary could efficiently sequester significant amounts of carbon without being gravitationally unstable. 2 3 2