Concept# Computer algebra

Summary

In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects. Although computer algebra could be considered a subfield of scientific computing, they are generally considered as distinct fields because scientific computing is usually based on numerical computation with approximate floating point numbers, while symbolic computation emphasizes exact computation with expressions containing variables that have no given value and are manipulated as symbols.
Software applications that perform symbolic calculations are called computer algebra systems, with the term system alluding to the complexity of the main applications that include, at least, a method to represent mathematical data in a computer, a user programming language (usually different from the language used for the imp

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Javad Ebrahimi Boroojeni, Christina Fragouli

We develop new algebraic algorithms for scalar and vector network coding. In vector network coding, the source multicasts information by transmitting vectors of length L, while intermediate nodes process and combine their incoming packets by multiplying them with L x L coding matrices that play a similar role as coding coefficients in scalar coding. We start our work by extending the algebraic framework developed for multicasting over graphs by Koetter and Medard to include operations over matrices; we build on this generalized framework, to provide a new approach for both scalar and vector code design which attempts to minimize the employed field size and employed vector length, while selecting the coding operations. Our algorithms also lead as a special case to network code designs that employ structured matrices.

Ahlswede et al. in the seminal paper [1] have shown that in data transfer over networks, processing the data at the nodes can significantly improve the throughput. As proved by Li et al. in [2], even with a simple type of operation, namely linear operation, the throughput can still be vastly improved. In [3], it is shown that the multicasting problem over networks can be translated to an algebraic question about a polynomial associated to the network called network polynomial. In this thesis, we start from the algorithm of [3] and extend it in several directions. First, we generalize the framework of [3] to include the case where the messages can also be vectors over some fixed finite field. We also show that in contrast to the original algorithm, ours can be used to reduce the field size for the case of sending finite field elements. In both vector network code algorithm and finite field minimization, we translate the network code design problems into an algebraic problem about network polynomials. Because of the importance of the network polynomials, we investigate more properties of them and we study the relationship between these objects and the topological properties of the network. Then, we extend the result of [3] to the deterministic models for wireless relay networks, a very important class of networks that has been introduced in [4] by Avestimehr, Diggavi and Tse. Finally, for another class of networks, called broadcast networks, we introduce the transfer matrix and using its properties, we show that in the absence of public messages, processing the information at the nodes will not improve the throughput.

Javad Ebrahimi Boroojeni, Christina Fragouli

We develop new algebraic algorithms for scalar and vector network coding. In vector network coding, the source multicasts information by transmitting vectors of length L, while intermediate nodes process and combine their incoming packets by multiplying them with L X L coding matrices that play a similar role as coding coefficients in scalar coding. Our algorithms for scalar network jointly optimize the employed field size while selecting the coding coefficients. Similarly, for vector coding, our algorithms optimize the length L while designing the coding matrices. These algorithms apply both for regular network graphs as well as linear deterministic networks.

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