Suppose Gamma is a group acting on a set X. An r-labeling phi: X to {1, 2, ..., r} of X is distinguishing (with respect to the action of Gamma) if for any sigma in Gamma, sigma not equal id_X, there exists an element x in X such that phi(x) not equal phi(sigma(x)). The distinguishing number, D_{Gamma}(X), of the action of Gamma on X is the minimum r for which there is an r-labeling which is distinguishing. Given a graph G, the distinguishing number of G, D(G),is defined as D(G) = D_{Aut(G)}(V(G)). This thesis determines the distinguishing numbers of all actions of S_5. As a consequence, we determine all the possible values of the distinguishing numbers of graphs G with Aut(G)=S_5, confirming a conjecture of Albertson and Collins.

Davis and Januszkiewicz introduced in 1981 a family of compact real manifolds, the Quasi-Toric Manifolds, with a group action by a torus, a direct product of circle (T) groups. Their manifolds have an orbit space which is a simple polytope with a distinct isotropy subgroup associated to each face of the polytope, subject to some consistency conditions. They defined a characteristic function which captured the properties of the isotropy subgroups, and showed that their manifolds can be classified by the polytope and characteristic function. They further showed that the cohomology ring of the manifold can be written down directly from properties derived from the polytope and the characteristic function. This work considers the question of how far the circle group T can be replaced by the group of unit quaternions Q in the construction and description of quasi-toric manifolds. Unlike T, the group Q is not commutative, so the actions of Q n on the product H n of the set of quaternions using quaternionic multiplication are studied in detail. Then, in direct analogy to the quasi-toric manifolds, a family of compact real manifolds, the Quoric Manifolds, is introduced which have an action by Q n, and whose orbit space is a polytope. A characteristic functor is defined on the faces of the polytope which captures the properties of the isotropy classes of the orbits of the action. It is shown that quoric manifolds can be classified in a manner similar to the quasi-toric manifolds, by the polytope and characteristic functor. A restricted family, the global quoric manifolds, which satisfy an additional condition are defined. It is shown that an infinite number of polytopes exist in any dimension over which a global quoric manifold can be defined. It is shown that any global quoric manifold can be described as a quotient space of a moment angle complex over the polytope, and that its integral cohomology ring can be calculated, taking a form analagous to that in the quasi-toric case.

Indiana University-Purdue University Indianapolis (IUPUI) / We introduce the quantum toroidal superalgebra E(m|n) associated with the Lie superalgebra gl(m|n) and initiate its study. For each choice of parity "s" of gl(m|n), a corresponding quantum toroidal superalgebra E(s) is defined. To show that all such superalgebras are isomorphic, an action of the toroidal braid group is constructed. The superalgebra E(s) contains two distinguished subalgebras, both isomorphic to the quantum affine superalgebra Uq sl̂(m|n) with parity "s", called vertical and horizontal subalgebras. We show the existence of Miki automorphism of E(s), which exchanges the vertical and horizontal subalgebras. If m and n are different and "s" is standard, we give a construction of level 1 E(m|n)-modules through vertex operators. We also construct an evaluation map from E(m|n)(q1,q2,q3) to the quantum affine algebra Uq gl̂(m|n) at level c=q3^(m-n)/2.

<p> Existing approaches to the analysis of participation in community groups have adopted either a social-psychological view or a structural-political view of the impetus for participation. This paper attempts to integrate these two approaches, through analysis of the nature of the link between the impact of the issue, which serves as a source of conflict (a psychological view) and the organizational characteristics of the community group (a structural view). Using Dahrendorf's model of latent and manifest interests, research propositions are generated, focusing on four sets of factors conditioning the selection of group participation as a response to conflict. These are: psychological factors, technical conditions of organization, social conditions of organization, and political conditions of organization.</p> <p> Results of an empirical application of these propositions, using a questionnaire, show that the impetus for participation in a community group is a two stage process, depending on the existence of two separate sets of conditions: the impact of the issues, which is dependent upon the individual's distance from the source of conflict; and the social organization of the group. Based on analysis and interpretation of these results, hypotheses are generated, and are used to modify and expand Dahrendorf's model, in order to make it more applicable to the analysis of community group participation in locational conflict.</p> / Thesis / Bachelor of Arts (BA)

Group actions in Czech procedural law Summary The thesis deals with an issue of group actions, which is a legal instrument of collective protection of private rights in civil proceedings. The aim of this thesis is to analyze this procedural institute, to point out shortcomings of current legislation and to propose it's acceptable solution. The thesis is divided into three relatively independent sections. The first section focuses on theoretical basis (including a brief outline of the historical development) and defines basic terminology used in the thesis. Then it describes the most general division of the collective enforcement mechanisms into a group action and a representative action. The second section concentrates on legislative schemes of group actions in certain foreign jurisdictions. Specifically, it deals with a legal conception of class action in the legal system of the USA, where it is applied as a kind of a opt-out group proceedings (group members, who does not agree with adjudication of their claims, may opt-out), then it deals with opt-in group proceedings in Sweden (group members can be required to enter the suit individually) and finally it describes a German model proceedings in capital market disputes, which represents a compromise between individual and collective proceedings. The third...

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In this thesis, we classify finite orbits of the action of the pure braid group over a certain large open subset of the SL(2,C) character variety of the Riemann sphere with five boundary components, i.e. Σ5. This problem arises in the context of classifying algebraic solutions of the Garnier system G2, that is the two variable analogue of the famous sixth Painleve equation PVI. The structure of the analytic continuation of these solutions is described in terms of the action of the pure braid group on the fundamental group of Σ5. To deal with this problem, we introduce a system of co-adjoint coordinates on a big open subset of the SL(2,C) character variety of Σ5. Our classifica- tion method is based on the definition of four restrictions of the action of the pure braid group such that they act on some of the co-adjoint coordi- nates of Σ5 as the pure braid group acts on the co-adjoint coordinates of the character variety of the Riemann sphere with four boundary components, i.e. Σ4, for which the classification of all finite orbits is known. In order to avoid redundant elements in our final list, a group of symmetries G of the large open subset is introduced and the final classification is achieved modulo the action of G. We present a final list of 54 finite orbits.

<div> We introduce the quantum toroidal superalgebra E_m|nassociated with the Lie superalgebra gl_m|n and initiate its study. For each choice of parity "s" of gl_m|n, a corresponding quantum toroidal superalgebra E_s is defined. </div><div> </div><div><br></div><div>To show that all such superalgebras are isomorphic, an action of the toroidal braid group is constructed. </div><div><br></div><div>The superalgebra E_s contains two distinguished subalgebras, both isomorphic to the quantum affine superalgebra U_q sl̂_m|n with parity "s", called vertical and horizontal subalgebras. We show the existence of Miki automorphism of E_s, which exchanges the vertical and horizontal subalgebras.</div><div><br></div><div>If <i>m</i> and <i>n</i> are different and "s" is standard, we give a construction of level 1 E_m|n-modules through vertex operators. We also construct an evaluation map from E_m|n(q₁,q₂,q₃) to the quantum affine algebra U_q gl̂_m|n at level c=q₃^(m-n)/2.</div>

In 2002, Giroux showed that every contact structure had a corresponding open book decomposition. This was the converse to a previous construction of Thurston and Winkelnkemper, and made open books a vital tool in the study of contact three-manifolds. We extend these results to contact orbifolds, i.e. spaces that are locally diffeomorphic to the quotient of a contact manifold and a compatible finite group action. This involves adapting some of the main concepts and constructions of three dimensional contact geometry to the orbifold setting.

This thesis presents the results of testing to determine the significance of the group action factor at the 5% offset yield and capacity of single-shear bolted wood connections loaded parallel to grain. The single and multiple-bolt connections tested represent common connection geometries used in wood construction in the United States. The results of both monotonic and cyclic loading of connections are presented. Monotonic test data was used to determine an appropriately scaled CUREE Displacement Controlled Quasi-Static Cyclic Protocol. Overall, one hundred and eighty connections were tested using this cyclic protocol based on data obtained from thirty-three monotonic tests. Tested assemblies had geometric variables that include number of bolts per row, number of rows, bolt diameter, and side member material. In addition, the main and side member material and thickness were designed to produce three of the four major connection yield modes as defined by the 1997 National Design Specification for Wood Construction (AF&PA, 1997). Results from this research address the need for adequate spacing of bolts in a row to control the brittle connection behavior that directly affected the group action factor at capacity. / Master of Science