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Collaborative work in Mathematics with a wiki

UseMod. Later, the founders of the project made their own software called Mediawiki [4] which is currently used. The biggest wiki nowadays is the Wikipedia in English, that on April 1st, 2012, had near four million articles. The Wikipedia is more used and is larger than the famous Encyclopædia Britannica, and many of its articles are more accurate.

The most important features of a wiki are:

•It has an hypertextual structure. This makes collaboration possible. Users can create and edit articles without the necessity of knowing HTML, the language behind web pages. It is not necessary use a web page editor either.

•Social authorship. Anyone can write and edit any article. A typical wiki invites all users to participate, but this is not essential. The process of creating and editing articles is very fast. The articles are very dynamic, so they are changing continuously and they are never considered as closed.

•Change history. Every page of the wiki has a history page attached where one can see the contributing users and the dates of every single editing. Depending on the software, changes can be undone and it is possible to revert a page to a previous state.

Wikis in Education are also used to develop collaborative works. E-learning platforms like Moodle [6] incorporate the possibility of setting up wikis. These can be monitorized by the teacher in order to follow the detailed evolution of its development. It is also possible to use open source software to create wikis, like Mediawiki, the Wikipedia engine.

Teachers also use wikis as an alternative to upload their notes on the Internet. This has a number of advantages:

•No knowledge of the HTML language neither of web page editors is needed to create pages in the wiki.

•It is easy to keep, update and make new contributions.

•It allows to introduce external resources.

•Depending on how the subject is organized, the wiki could be opened to the participation of the students.

2Description of the work

The main objective of the work presented in this text is the development through a wiki of problem solving tasks in Mathematics on a group work basis. Furthermore it is intended to obtain and develop a number of competencies within the European higher education system framework. The activity was carried out the second four-month period of the last academic year in a Numerical Analysis course at the University of Oviedo, in Spain.

P. Alonso, R. Gallego

During the academic year 2008-2009 the students of our course were proposed an activity consisting of making periodically some exercises that had to be given to the teachers after a certain period. The volunteers were arranged in groups of three or four people. The resulting numeric mark of the assessment of the activity, up to a maximum of 2 points, was added to the mark of the theory exams as long as the final mark did not exceed 10 points. A problem we noted was that the real contribution of every student to the work could not be determined.

During the academic year 2009-2010 we carried out the educational activity described in this document. The organization was similar to that of the previous year with the important di erence that now the exercises had to be solved on a wiki. For every exercise a deadline was fixed. Beyond the deadline the permission to edit the corresponding pages of the wiki was revoked.

Since we are dealing with mathematical problems, it is necessary that all sort of mathematical expressions can be introduced in the wiki. The builtin wiki module of the moodle elearning software allows the inclusion of formulae with LATEX syntax [8], but we found it too buggy. Instead, we decided to use the free Mediawiki software, on which the famous Wikipedia relies. The features of Mediawiki can be extended by means of a big number of add-ons made by the user community. Moreover, it is possible to insert natively in the articles mathematical expressions using LATEX syntax.

Although the features of Mediawiki to introduce mathematical formulae are actually satisfactory, we decided to add extra functionality by installing an extension named Wikitex [7]. Wikitex allows to insert in the wiki not only mathematical expressions of arbitrary complexity, but also graphics (made with GNUplot [9]), chess matches, several kinds of diagrams, etc.

We were aware that most of the students had little or no knowledge of LATEX. This fact did not have to prevent the students from doing the work so we o ered two alternatives to make the process easier: an exhaustive LATEX tutorial was made in the wiki itself and also the students could use the Mathtype [10] software that includes an equation editor capable of exporting to the mediawiki format.

In Mediawiki, the main page of an article comes with its corresponding discussion page. This is useful for the group members to organize their work and exchange ideas. Since the wiki can be accessed on the Internet, the students can work in the wiki without being physically in a meeting. They only need to have a computer connected to Internet. This is a clear advantage against other more traditional collaborative work strategies.

The history of the wiki plays an important role. The teacher can know in detail the contribution of each member of the group to the proposed work by looking at the history associated to the relevant pages. This allows to measure the collaboration inside the group. When the students gave a hard copy of their works, this kind of evaluations could not be done objectively. In Mediawiki, the page history allows to compare di erent versions and

Collaborative work in Mathematics with a wiki

undo changes. This way, a page can be reverted to a previous state.

3Assessment process

The assessment process was based on a rubric. We give the details in this section.

3.1Rubric

Once finished, the exercises were checked on printed versions that later were given to the students. We also considered the possibility to make the corrections in the wiki itself, but eventually we gave up the idea since we did not have enough time. The task of checking the exercises in the wiki is certainly very interesting from the students learning point of view and it will be taken into account in future activities. In the wiki, the students could easily check both their own exercises and those from other groups.

For each student, the exercises were marked following the next rubric:

P. Alonso, R. Gallego

Note the existence of an oral test. This was compulsory. Although several students decided not to do the oral test (so they were ruled out in the activity), the percent did not reach the 5% of the total number. Taking advantage of these tests, we made an opinion poll to know the students’ opinion about the activity.

3.2Computation of the final marks of the exercises

We considered two marks for each exercise, namely:

•Neatness and Organization (n1): 0 ≤ n1 ≤ 4

•Explanation and Completion (n2): 0 ≤ n2 ≤ 10

On the other hand, the students had to defend their work in an oral test, after which they got a mark n3, 0 ≤ n3 ≤ 4.

If a group made N exercises of a total number of M, the final mark nf of every member of the group, computed up to 2 points, is given by:

Note that a bad mark in the oral test (n3 = 0 or 1) causes the final mark nf to be decreased. Finally, the mark nf was added to that of the theory exams as long as the resulting number was not greater than 10 points.

4Conclusions

We have confirmed the advantages of working with a wiki against more traditional ways of group work. The teacher can track the students’ progresses any time and the works can be accessed in a single location. Besides, he can know each individual contribution inside the group by means of the history pages. On the other hand, the students can work with just

Collaborative work in Mathematics with a wiki

a computer with Internet connection. The wiki has discussion pages that students can use to organize their work and exchange ideas.

We think that this experience has been really positive from the point of view of both the teacher and the students. Furthermore, this activity made it possible that many students worked with a wiki for the first time. This may stimulate them to write an article for the Wikipedia in the future.

We plan to extend this activity to other subjects in Mathematics. We also intend to do the revision of the exercises in the wiki itself, both by teachers and students.

References

[1]http://www.wikispaces.com

[2]B. Leuf y W. Cunningham. The Wiki way: Collaboration and Sharing on the Internet. Addison-Wesley, 2001.

[3]http://en.wikipedia.org/wiki/WikiWikiWeb

[4]http://www.mediawiki.org

[5]http://www.wikipedia.org

[6]http://www.moodle.org

[7]http://wikisophia.org/wiki/Wikitex

[8]http://www.latex-project.org

[9]http://www.gnuplot.info

[10]http://www.dessci.com/en/products/mathtype

Proceedings of the 12th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2012 July, 2-5, 2012.

Aself-adjusting algorithm for solitary wave simulations

I.Alonso-Mallo1 and Nuria Reguera2

1 Department of Applied Mathematics, University of Valladolid, Spain

2 Department of Mathematics and Computation, University of Burgos, Spain

emails: isaias@mac.uva.es, nreguera@ubu.es

Abstract

We introduce a practical algorithm to automate the simulations of solitary wave solutions of some nonlinear dispersive wave equations. The full discretization consists of a spatial discretization with a local basis and an invariant preserving time integrator. The algorithm includes a dynamic cleaning of dispersive tails and an automatic detection of the complete separation of the main pulses.

Key words: Solitary waves, cleaning procedures, conservative methods

MSC 2000: 65M20, 65M99, 35Q53, 76B25

1Introduction

The purpose of this work is to develop a dynamic algorithm for the simulation of solitary waves. This subject requires to pay attention to some numerical problems. In many situations the numerical solution evolves into a main pulse or train of pulses along with some small waves of di erent nature that sometimes form dispersive tails. This is the case, for example, when managing with small perturbations of waves or when studying the interaction of two or more solitary waves. Then the use of a finite computational window along with periodic boundary conditions forces to ‘clean’ the solution: isolating the main pulses, eliminating somehow and sometime the tails and turbulences, and leaving the main pulses alone to evolve.

The simplest case to study is the perturbation of an only solitary wave which evolves into one main pulse along with dispersive tails [2].

Another interesting case in which we focus now is the interaction of two solitary waves. In this case, the computation of the cleaning region is more complicated. Taking into

A self-adjusting algorithm for solitary wave simulations

account the periodic boundary conditions, when the waves are ‘completely separated’ there will be two cleaning intervals, while in other case there will be only one. That is why, it is essential that the algorithm detects automatically when the interaction starts and when the waves that appear after the collision are completely separated. The algorithm that we propose cleans the dispersive tails and turbulences that appear so that they can not alter the new solitary waves that arise after the collision. Moreover, this algorithm allows to carry on the experiment in order to study several consecutive collisions.

Although our algorithm can be used for a wide class of partial di erential equations,

where u = u(x, t) is a real-valued function of the two real independent variables x, t. This equation appear in certain models about the propagation of small-amplitude, nonlinear, dispersive long waves [4, 5]. Solitary wave solutions of (1) are of the form

where the parameter c > 1 represents the velocity of the wave.

2Description of the algorithm

We consider the discretization of the initial boundary value problem for the BBM equation with periodic boundary conditions

The cleaning technique that we propose requires a spatial discretization with local character so that the cleaning of small perturbations does not alter the whole computational window and therefore the main pulses. In this work we are going to consider cubic finite elements although other options are also possible.

On the other hand, in order to choose a suitable time integrator it is important to take into account that the semidiscrete system obtained after the spatial discretization retains a Hamiltonian structure and has as conserved quantities the corresponding discrete versions of the invariants of the original problem. The conservation of these invariants quantities through the numerical integration is a convenient property for a time integrator [1, 3]. Taking this fact into account there are still several possibilities. Between them we have chosen the classical implicit midpoint rule. This symplectic method presents a good behavior with respect to the invariants of our problem while at the same time it is quite easy to implement letting us focus on the implementation of the algorithm.

I. Alonso-Mallo, N. Reguera

2.1Cleaning technique

Let us explain now very briefly the cleaning technique. For simplicity, we consider the case when the initial condition in (3) has evolved into an only main pulse along with some turbulences, although in Section 2.2 it will be used for the case of two pulses. Our cleaning technique calculates in a dynamic way a suitable cleaning region for which a relation between the velocity, the amplitude and the ‘support ’ of the main wave is assumed. In this context the term ‘support ’ associated to a previously fixed tolerance ε will represent an interval where the profile is greater than ε. This relation could be estimated if it is not known in an exact way (see [2] for details).

In order to clean at time tn of the computation, the point xmax,n where the numerical solution attains its maximum absolute value is calculated with the maximum accuracy that the spatial discretization allows. For this, a first estimation xmax,n is done by means of the numerical velocity cn of the main wave. Then, the maximum nodal value xjmax,n is calculated reducing the search, for e ciency, to the nearest nodes to xmax,n. Finally, xmax,n is found by calculating the point where the cubic Hermite piecewise interpolant associated to the numerical solution in the adjacent intervals to xjmax,n reaches its maximum.

Once the point xmax,n has been found, the algorithm computes the supports of the solitary wave with velocity cn associated to two given tolerances ε1 > ε2: (β1,n, β2,n) (γ1,n, γ2,n), both centered at xmax,n. Then, the solution is set equal to zero outside (γ1,n, γ2,n), while in the intervals (γ1,n, β1,n) and (β2,n, γ2,n) a cubic interpolation is implemented in order to obtain a smooth enough numerical approximation.

2.2Algorithm for simulating the collision of two solitary waves

Now we focus on the case of two solitary waves. That is, we assume that the numerical solution consists of two main pulses traveling with di erent velocities. We propose an algorithm for simulating the interaction of both pulses, cleaning in an automatic way the turbulences that appear due to the collision. Moreover, the cleaning technique will let to carry on the experiment in order to study the successive collisions of the waves formed after each interaction, which is possible due to the periodic boundary conditions.

At each time step tn the algorithm is going to consider three intervals associated to each of the two main pulses (we use superscripts to refer to each pulse). Two of them are

the ones associated to the cleaning procedure: (β1(j,n) , β2(j,n) ), (γ1(j,n) , γ2(j,n) ), for j = 1, 2, following with the notation of previous section. If

(γ1(1),n, γ2(1),n) (γ1(2),n, γ2(2),n) =

and therefore the waves are completely separated, there will be two cleaning intervals (we can clean ‘between’ both waves), while in other case, there will be only one.

A self-adjusting algorithm for solitary wave simulations

Notice that in order to apply the cleaning technique in this case, the points x(1)max,n, x(2)max,n where each main pulse attains its amplitude should be calculated. Nevertheless, this can not be done during a short period of time when the collision is taking place, and an estimation should be done instead of computing these points as mentioned in Section 2.1. In order to determine this period of time, two more intervals (α1(j,n) , α2(j,n) ), centered at x(max,nj)

then points x(1)max,n, x(2)max,n will be calculated as explained at Section 2.1, and otherwise an estimation should be done.

Numerical experiments confirm the good properties and usefulness of the proposed algorithm.

Acknowledgements

This research has been supported by MCINN project MTM2011-23417 cofinanced by FEDER funds.

References

[1]I. Alonso-Mallo, A. Duran´ and N. Reguera, Simulation of coherent structures in nonlinear Schroedinger-type equations, J. Comput. Phys. 227 (2010), 8180–8198.

[2]I. Alonso-Mallo, A. Duran´ and N. Reguera, A numerical technique of cleaning in solitary-wave simulations, Proceedings of the 2011 International Conference on Computational and Mathematical Methods in Science and Engineering.

[4] T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Phil Trans R Soc Lond A 272 (1972), 47–78.

[5] D. H. Peregrine, Calculations of the development of an undular bore, J Fluid Mech

25 (1996), 321–330.

Proceedings of the 12th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2012 July, 2-5, 2012.

Hierarchical approaches for multicast based on Euclid’s algorithm

J.A. Alvarez-Bermejo1, N. Antequera2 and J.A. Lopez-Ramos2

1 Department of Computers Architecture and Electronics, University of Almeria

2 Department of Algebra and Analysis, University of Almeria

emails: jaberme@ual.es, nicolas.antequera@gmail.com, jlopez@ual.es

Abstract

We introduce a hierarchical approach for secure multicast where rekeying of groups of users is made through a method based on Euclid’s algorithm for computing GCD. We consider tree arrangements of users and a distributed protocol by groups with group managers that may help both distributing the information and detecting some possible inner attacks.

Key words: Security, Multicast, Euclid’s algorithm,

MSC 2000: 94A60, 68P25

1Introduction

Multicast communications allow a host to simultaneously send information to a set of other hosts, avoiding the establishment of point-to-point connections with all of them. There exist many situations where multicast reveals to be the most suitable way to distribute the information such as pay-per-view IPTV or P2PTV, private multiconferences, or any private service that involves several participants or clients. This has increased the interest in researching on appropriate protocols for secure multicast. Some surveys on this field can be found in [2], [9], or more recently in [12].

In [3] the authors made a computational approach to the problem and introduce a solution based on the Chinese Remainder Theorem, the so-called Secure Lock. However, as shown in [4], computational requirements become quickly huge as the number of user grows. To reduce the number of computations, in [10], a divide-and-conquer extension of Secure Lock is introduced. It combines the well-known Hierarchical Tree Approach, [11] and the