ADEPT: A Heuristic Program for Proving Theorems of Group Theory

A computer program, named ADEPT (A Distinctly Empirical Prover of Theorems), has been written which proves theorems taken from the abstract theory of groups. Its operation is basically heuristic, incorporating many of the techniques of the human mathematician in a "natural" way. This program has proved almost 100 theorems, as well as serving as a vehicle for testing and evaluating special-purpose heuristics. A detailed description of the program is supplemented by accounts of its performance on a number of theorems, thus providing many insights into the particular problems inherent in the design of a procedure capable of proving a variety of theorems from this domain. Suggestions have been formulated for further efforts along these lines, and comparisons with related work previously reported in the literature have been made.

Topology synthesis of three-legged spherical parallel manipulators employing Lie group theory

Driven by the requirements of the bionic joint or tracking equipment for the spherical parallel manipulators (SPMs) with three rotational degrees-of-freedom (DoFs), this paper carries out the topology synthesis of a class of three-legged SPMs employing Lie group theory. In order to achieve the intersection of the displacement subgroups, the subgroup characteristics and operation principles are defined in this paper. Mainly drawing on the Lie group theory, the topology synthesis procedure of three-legged SPMs including four stages and two functional blocks is proposed, in which the assembly principles of three legs are defined. By introducing the circular track, a novel class of three-legged SPMs is synthesized, which is the important complement to the existing SPMs. Finally, four typical examples are given to demonstrate the finite displacements of the synthesized three-legged SPMs.

On some Density Theorems in Number Theory and Group Theory

Gowers, dans son article sur les matrices quasi-aléatoires, étudie la question, posée par Babai et Sos, de l'existence d'une constante $c>0$ telle que tout groupe fini possède un sous-ensemble sans produit de taille supérieure ou égale a $c|G|$. En prouvant que, pour tout nombre premier $p$ assez grand, le groupe $PSL_2(\mathbb{F}_p)$ (d'ordre noté $n$) ne posséde aucun sous-ensemble sans produit de taille $c n^{8/9}$, il y répond par la négative. Nous allons considérer le probléme dans le cas des groupes compacts finis, et plus particuliérement des groupes profinis $SL_k(\mathbb{Z}_p)$ et $Sp_{2k}(\mathbb{Z}_p)$. La premiére partie de cette thése est dédiée à l'obtention de bornes inférieures et supérieures exponentielles pour la mesure suprémale des ensembles sans produit. La preuve nécessite d'établir préalablement une borne inférieure sur la dimension des représentations non-triviales des groupes finis $SL_k(\mathbb{Z}/(p^n\mathbb{Z}))$ et $Sp_{2k}(\mathbb{Z}/(p^n\mathbb{Z}))$. Notre théoréme prolonge le travail de Landazuri et Seitz, qui considérent le degré minimal des représentations pour les groupes de Chevalley sur les corps finis, tout en offrant une preuve plus simple que la leur. La seconde partie de la thése à trait à la théorie algébrique des nombres. Un polynome monogéne $f$ est un polynome unitaire irréductible à coefficients entiers qui endengre un corps de nombres monogéne. Pour un nombre premier $q$ donné, nous allons montrer, en utilisant le théoréme de densité de Tchebotariov, que la densité des nombres premiers $p$ tels que $t^q -p$ soit monogéne est supérieure ou égale à $(q-1)/q$. Nous allons également démontrer que, quand $q=3$, la densité des nombres premiers $p$ tels que $\mathbb{Q}(\sqrt[3]{p})$ soit non monogéne est supérieure ou égale à $1/9$.

Introduction to Group Theory

Exam questions and solutions for a second year group theory course.

Modelling QoS of a pervasive MVNO/M2M using semi-group theory

This paper aims to develop a mathematical model based on semi-group theory, which allows to improve quality of service (QoS), including the reduction of the carbon path, in a pervasive environment of a Mobile Virtual Network Operator (MVNO). This paper generalise an interrelationship Machine to Machine (M2M) mathematical model, based on semi-group theory. This paper demonstrates that using available technology and with a solid mathematical model, is possible to streamline relationships between building agents, to control pervasive spaces so as to reduce the impact in carbon footprint through the reduction of GHG.

Asymmetrical three-DOFs rotational-translational parallel-kinematics mechanisms based on lie group theory

The paper introduces four families of three-DOFs translational-rotational Parallel-Kinematics Mechanisms (PKMs) as well as the mobility analysis of such families using Lie group theory. Two of these families are mechanisms with one-rotational two-translational degrees of freedom (DOFs) and each of the other two has one-translational two-rotational DOFs. Four novel mechanisms are presented and discussed as representatives of these four families. Although these mechanisms are asymmetric, the components used to realise them are very similar and, hence, there is no great departure from the favourable modularity of parallel-kinematics mechanisms.

Rotations, Precursor to Rotational Spectroscopy, and Group Theory

Rotations are an integral part of the study of rotational spectroscopy, as well as a part of group theory, hence this introduction.

An application of strategic group theory to the UK pharmaceuticals industry

Conducts a strategic group mapping exercise by analysing R&D investment, sales/marketing cost and leadership information pertaining to the pharmaceuticals industry. Explains that strategic group mapping assists companies in identifying their principal competitors, and hence supports strategic decision-making, and shows that, in the pharmaceutical industry, R&D spending, the cost of sales and marketing, i.e. detailing, and technological leadership are mobility barriers to companies moving between sectors. Illustrates, in bubble-chart format, strategic groups in the pharmaceutical industry, plotting detailing-costs against the scale of activity in therapeutic areas. Places companies into 12 groups, and profiles the strategy and market-position similarities of the companies in each group. Concludes with three questions for companies to ask when evaluating their own, and their competitors, strategies and returns, and suggests that strategy mapping can be carried out in other industries, provided mobility barriers are identified.

Strategic group theory:review, examination and application in the UK pharmaceutical industry

Purpose – The purpose of this paper is to consider the current status of strategic group theory in the light of developments over the last three decades. and then to discuss the continuing value of the concept, both to strategic management research and practising managers. Design/methodology/approach – Critical review of the idea of strategic groups together with a practical strategic mapping illustration. Findings – Strategic group theory still provides a useful approach for management research, which allows a detailed appraisal and comparison of company strategies within an industry. Research limitations/ implications – Strategic group research would undoubtedly benefit from more directly comparable, industry-specific studies, with a more careful focus on variable selection and the statistical methods used for validation. Future studies should aim to build sets of industry specific variables that describe strategic choice within that industry. The statistical methods used to identify strategic groupings need to be robust to ensure that strategic groups are not solely an artefact of method. Practical implications – The paper looks specifically at an application of strategic group theory in the UK pharmaceutical industry. The practical benefits of strategic groups as a classification system and of strategic mapping as a strategy development and analysis tool are discussed. Originality/value – The review of strategic group theory alongside alternative taxonomies and application of the concept to the UK pharmaceutical industry.

Information-theoretic Studies and Capacity Bounds: Group Network Codes and Energy Harvesting Communication Systems

Network information theory and channels with memory are two important but difficult frontiers of information theory. In this two-parted dissertation, we study these two areas, each comprising one part. For the first area we study the so-called entropy vectors via finite group theory, and the network codes constructed from finite groups. In particular, we identify the smallest finite group that violates the Ingleton inequality, an inequality respected by all linear network codes, but not satisfied by all entropy vectors. Based on the analysis of this group we generalize it to several families of Ingleton-violating groups, which may be used to design good network codes. Regarding that aspect, we study the network codes constructed with finite groups, and especially show that linear network codes are embedded in the group network codes constructed with these Ingleton-violating families. Furthermore, such codes are strictly more powerful than linear network codes, as they are able to violate the Ingleton inequality while linear network codes cannot. For the second area, we study the impact of memory to the channel capacity through a novel communication system: the energy harvesting channel. Different from traditional communication systems, the transmitter of an energy harvesting channel is powered by an exogenous energy harvesting device and a finite-sized battery. As a consequence, each time the system can only transmit a symbol whose energy consumption is no more than the energy currently available. This new type of power supply introduces an unprecedented input constraint for the channel, which is random, instantaneous, and has memory. Furthermore, naturally, the energy harvesting process is observed causally at the transmitter, but no such information is provided to the receiver. Both of these features pose great challenges for the analysis of the channel capacity. In this work we use techniques from channels with side information, and finite state channels, to obtain lower and upper bounds of the energy harvesting channel. In particular, we study the stationarity and ergodicity conditions of a surrogate channel to compute and optimize the achievable rates for the original channel. In addition, for practical code design of the system we study the pairwise error probabilities of the input sequences.

Co-evaluation as a strategy to improve working group dynamics: an experience in Sport Sciences

The aim of this study was to analyze if the perceptions of students before and after carrying out the work, that is, their perception of different aspects of the functioning of the group, the working skills acquired as well as those they think that need to be improved, varied depending on whether the contribution of the different members of the group was being co-evaluated or not. 144 students of Physical Activity and Sport Sciences participated in this study. In order to analyze the students' perception of group work the adapted questionnaire by Bourne et al. (2001) was used. Results showed that groups which implemented co-evaluation assessed more negatively the experience in general than those which did not. However, co-evaluation groups perceived their competence to work as a team had improved to a greater extent than the groups without co-evaluation, evaluating more positively both the performance and the result of work and increasing their knowledge of the other team members. Using a co-evaluation system seems to generate both a better assessment of the running of the team and the result of its work.

Involutions and free pairs of bicyclic units in integral group rings

If * : G -> G is an involution on the finite group G, then * extends to an involution on the integral group ring Z[G] . In this paper, we consider whether bicyclic units u is an element of Z[G] exist with the property that the group < u, u*> generated by u and u* is free on the two generators. If this occurs, we say that (u, u*)is a free bicyclic pair. It turns out that the existence of u depends strongly upon the structure of G and on the nature of the involution. One positive result here is that if G is a nonabelian group with all Sylow subgroups abelian, then for any involution *, Z[G] contains a free bicyclic pair.