A survey on signature-based Gr\"obner basis computations

This paper is a survey on the area of signature-based Gr\"obner basis algorithms that was initiated by Faug\`ere's F5 algorithm in 2002. We explain the general ideas behind the usage of signatures. We show how to classify the various known variants by 3 different orderings. For this we give translations between different notations and show that besides notations many approaches are just the same. Moreover, we give a general description of how the idea of signatures is quite natural when performing the reduction process using linear algebra. This survey shall help to outline this field of active research.Comment: 53 pages, 8 figures, 11 table

A survey on signature-based algorithms for computing Gröbner basis computations

International audienceThis paper is a survey on the area of signature-based Gröbner basis algorithms that was initiated by Faugère's F5 algorithm in 2002. We explain the general ideas behind the usage of signatures. We show how to classify the various known variants by 3 different orderings. For this we give translations between different notations and show that besides notations many approaches are just the same. Moreover, we give a general description of how the idea of signatures is quite natural when performing the reduction process using linear algebra. This survey shall help to outline this field of active research

Vortex Instability and Transient Growth

The dynamics of vortex flow is studied theoretically and numerically. Starting from a local analysis, where the perturbation in the vortex flow is Fourier decomposed in both radial and azimuthal directions, a modified Chebyshev polynomial method is used to discretize the linearized governing operator. The spectrum of the operator is divided into three groups: discrete spectrum, free-stream spectrum and potential spectrum. The first can be unstable while the latter two are always stable but highly non-normal. The non-normality of the spectra is quantitatively investigated by calculating the transient growth via singular value decomposition of the operator. It is observed that there is significant transient energy growth induced by the non-normality of continuous spectra. The non-normality study is then extended to a global analysis, in which the perturbation is decomposed in the radial or azimuthal direction. The governing equations are discretized through a spectral/hp element method and the maximum energy growth is calculated via an Arnoldi method. In the azimuthally-decomposed case, the development of the optimal perturbation drives the vortex to vibrate while in the stream-wise-decomposed case, the transient effects induce a string of bubbles along the axis of the vortex. A further transient growth study is conducted in the context of a co-rotating vortex pair. It is noted that the development of optimal perturbations accelerates the vortex merging process. Finally, the transient growth study is extended to a sensitivity analysis of the vortex flow to inflow perturbations. An augmented Lagrangian function is built to optimize the inflow perturbations which maximize the energy inside the domain over a fixed time interval

Functional programming and graph algorithms

This thesis is an investigation of graph algorithms in the non-strict purely functional language Haskell. Emphasis is placed on the importance of achieving an asymptotic complexity as good as with conventional languages. This is achieved by using the monadic model for including actions on the state. Work on the monadic model was carried out at Glasgow University by Wadler, Peyton Jones, and Launchbury in the early nineties and has opened up many diverse application areas. One area is the ability to express data structures that require sharing. Although graphs are not presented in this style, data structures that graph algorithms use are expressed in this style. Several examples of stateful algorithms are given including union/find for disjoint sets, and the linear time sort binsort. The graph algorithms presented are not new, but are traditional algorithms recast in a functional setting. Examples include strongly connected components, biconnected components, Kruskal's minimum cost spanning tree, and Dijkstra's shortest paths. The presentation is lucid giving more insight than usual. The functional setting allows for complete calculational style correctness proofs - which is demonstrated with many examples. The benefits of using a functional language for expressing graph algorithms are quantified by looking at the issues of execution times, asymptotic complexity, correctness, and clarity, in comparison with traditional approaches. The intention is to be as objective as possible, pointing out both the weaknesses and the strengths of using a functional language

Users\u27 understandings of the virtual economy in social virtual worlds: consumption and entrepreneurship of virtual goods

Social virtual worlds (SVWs) such as Second Life have gained immense popularity during the last decade. Their popularity is reflected in the explosive growth of the in-world economy, which is mainly supported by users’ consumption of virtual goods and entrepreneurial behavior. Users’ economic behavior has increased the size of the virtual economy to reach over $1B in the United States in 2009 (Nicholson 2009). Given that virtual consumption and entrepreneurship have become an important part of everyday virtual life in SVWs, understanding these two types of economic activities is an essential aspect to understanding user behavior in SVWs. This research aims at investigating the meanings of virtual consumption and entrepreneurship for users. The research consists of two independent essays. In the first essay about virtual consumption, I examine users’ understandings of virtual consumption through core-periphery analysis of its social representation. 154 Second Life users participated in the web-based survey for this study. I also identify user goals for virtual consumption by using means-end chain analysis, based on interviews with 93 Second Life users. The second essay on the topic of virtual entrepreneurship examines the collective meanings of virtual entrepreneurship and their relationship with the collective meanings of SVWs. To understand the meanings of virtual entrepreneurship, the core-periphery structure of social representations of virtual entrepreneurship is analyzed, based on interviews with 24 Second Life entrepreneurs. The meanings of virtual entrepreneurship are explained and compared with the central meanings of SVWs, identified from 101 newspaper articles on SVWs from 2005 to 2009. The results show that virtual consumption penetrates extensively into the virtual life of users and make their virtual experience in SVWs much richer; that virtual entrepreneurship is institutionalized in SVWs, and the key meanings of SVWs are well translated into the meanings of virtual entrepreneurship. This research contributes in providing fundamental knowledge about virtual consumption and entrepreneurship and further suggests potential theoretical frameworks for future research. Implications for SVW service providers and producers and sellers of virtual goods are also identified. Another contribution of the research is to demonstrate alternative qualitative research approaches combined with quantitative analyses