On monoids, $$2$$ 2 -firs, and semifirs

The experimenter asked if documented equivalence class membership would influence the development of shared discriminative stimulus function established through simple successive discrimination training. In Experiment 1, equivalence classes were established with two sets of 9 stimuli. Common stimulus functions were then trained within or across the equivalence classes. Greater acquisition rates of the simple discriminations with stimuli drawn from within the equivalence classes were observed. In Experiment 2, a third stimulus set was added with which no equivalence relations were explicitly trained. The findings of Experiment 1 were replicated, but the Set 3 results were inconsistent across subjects. The outcomes of the two experiments demonstrate that equivalence classes have an effect on other behavioral relations which requires further investigation.

Two experiments investigated possible interactions between existing equivalence relations among stimuli and the acquisition of simple discriminations involving the same stimuli. During the first phase of Experiment 1, two adult humans learned two sets of six conditional relations among arbitrary visual stimuli designed to produce two independent sets of three, three-member equivalence classes (Set 1 and Set 2). Subjects whose performance met accuracy criteria on tests for equivalence relations proceeded to a second phase in which subjects were taught, via reinforcement, to make the same response to a cluster of three stimuli (e.g., response 1 in the presence of A1, B1, and C1). The clusters of three stimuli were initially consistent with documented equivalence relations for both Set 1 and Set 2 stimuli. In the second half of this phase, the contingencies were realigned such that the simple discriminations being established were consistent with documented equivalence relations for Set 1 stimuli but inconsistent for Set 2 stimuli. Results showed that the simple discriminations were acquired faster for Set 1 stimuli than for Set 2 stimuli. Experiment 2 introduced a third set of stimuli as a control set and systematically replicated Experiment 1 with four subjects. This study found that three of four subjects acquired simple discriminations faster when the cluster of stimuli were drawn from within equivalence classes rather than across equivalence classes. The development of equivalence relations with Set 3 stimuli constrained the utility of the control condition in this experiment. The results are best interpreted as suggesting that existing equivalence relations among stimuli can interact with the development of analytic units such as simple discriminations to either facilitate or retard their development.

Let K be a field. We obtain conditions for elements of the form {1 + a i, 1 + b j} to generate a free group of rank 2 in a quaternion algebra over K, including the case where the characteristic of the field K is 2, based on an article by Professors Jairo Gonçalves, Arnaldo Mandel and Mazi Shirvani. These results are then applied to obtain pairs of elements that freely generate a free group in many other classes of division rings, such as total fields of fractions of Ore domains and Malcev-Neumann series rings. With these results, we partially answer a conjecture by A. Lichtman, regarding the existence of non-abelian free groups in the multiplicative group of non-commutative division rings.

In this note we classify the necessary and the sufficient conditions that an index of a superconformal theory in 3 ≤ d ≤ 6 must obey for the theory to have enhanced supersymmetry. We do that by noting that the index distinguishes a superconformal multiplet contribution to the index only up to a certain equivalence class it lies in. We classify the equivalence classes in d = 4 and build a correspondence between mathcal{N}=1 and mathcal{N}>1 equivalence classes. Using this correspondence, we find a set of necessary conditions and a sufficient condition on the d = 4 mathcal{N}=1 index for the theory to have mathcal{N}>1 SUSY. We also find a necessary and sufficient condition on a d = 4 mathcal{N}>1 index to correspond to a theory with mathcal{N}>2 . We then use our results to study some of the d = 4 theories described by Agarwal, Maruyoshi and Song, and find that the theories in question have only mathcal{N}=1 SUSY despite having rational central charges. In d = 3 we classify the equivalence classes, and build a correspondence between mathcal{N}>2 and mathcal{N}>2 equivalence classes. Using this correspondence, we classify all necessary or sufficient conditions on an 1le mathcal{N}le 3 superconformal index in d = 3 to correspond to a theory with higher SUSY, and find a necessary and sufficient condition on an mathcal{N}=4 index to correspond to an mathcal{N}=4 theory. Finally, in d = 6 we find a necessary and sufficient condition for an mathcal{N}=1 index to correspond to an mathcal{N}>2 theory.

The purpose of this study was to examine the transfer of consequential (reinforcement and punishment) functions through equivalence relations. In Experiment 1, 9 subjects acquired three three-member equivalence classes through matching-to-sample training using arbitrary visual forms. Comparison stimuli were then given conditioned reinforcement or punishment functions by pairing them with verbal feedback during a sorting task. For 8 of the 9 subjects, trained consequential functions transferred through their respective equivalence classes without additional training. In Experiment 2, transfer of function was initially tested before equivalence testing per se. Three of 4 subjects showed the transfer without a formal equivalence test. In Experiment 3, 3 subjects were given training that gave rise to six new three-member conditional equivalence classes. For 2 of the subjects, the same stimulus could have either a reinforcement or punishment function on the basis of contextual cues that defined its class membership. Experiment 4 assessed whether equivalence training had established general or specific consequential functions primarily by adding novel stimuli in the transfer test. Subjects treated even novel feedback stimuli in the transfer test as consequences, but the direction of consequential effects depended upon the transfer of specific consequential functions through equivalence relations.

This thesis presents recent developments concerning two open problems related to algebra and discrete geometry. In 1934, Harold S. M. Coxeter introduced Coxeter groups as abstractions of groups generated by reflections in a vector space [Cox34]. The present work lays out how a study of geometric and combinatorial properties of Coxeter groups contributed to the comprehension of the two open problems. Furthermore, this thesis presents some progress in proving the first problem, and an application of the approach introduced in this thesis for the second problem. Open Problem (Lattice for infinite Coxeter groups, Dyer [Dye11]). Is there, for each infinite Coxeter group, a complete ortholattice that contains the weak order? In Chapter 2, we study the asymptotical behaviour of roots of infinite Coxeter groups, which is part of a joint work with Christophe Hohlweg and Vivien Ripoll [HLR13]. In particular, we show that the directions of roots tend to the isotropic cone of the geometric representation of the root system. Moreover, using this framework, this thesis presents a proof that there is a complete ortholattice structure enclosing the weak order of infinite Coxeter groups of rank at most 3. Open Problem (Existence of multiassociahedra, Jonsson [Jon05]). Let k be an integer such that k >= 1. Is there a polytope whose boundary complex corresponds to the simplicial complex of sets of diagonals of a convex polygon not containing k + 1 mutually crossing diagonals? In Chapter 3 we introduce, for any finite Coxeter group and any nonnegative integer k, a spherical subword complex called multi-cluster complex. This family generalizes the concept of multitriangulations of type A and B to arbitrary finite Coxeter groups. For k = 1, this simplicial complex coincides with the finite cluster complex of the given type. We study combinatorial and geometric properties of multi-cluster complexes. In particular, we show that every spherical subword complex is the link of a face of a multi-cluster complex. This work was realized jointly with Cesar Ceballos and Christian Stump [CLS13]. Finally, this approach allows us to exhibit formulas counting the number of common vertices of permutahedra and generalized associahedra for arbitrary finite Coxeter groups and Coxeter elements.

Recent research has demonstrated the transfer of induced mood functions through equivalence relations by means of a musical mood-induction procedure. The research described in this article replicated and extended such work, primarily with the inclusion of a baseline and two types of reversal procedures. First, 16 adult participants were trained and tested for the formation of two three-member equivalence classes (A1-B1-C1 and A2-B2-C2). A musical mood-induction procedure was used to induce happy and sad mood states in the presence of the B stimuli, and a transfer of mood functions test was administered to determine whether the corresponding mood functions had transferred via equivalence to the C stimuli. Participants were then exposed to either Reversal 1, in which only the functions attached to the B stimuli were reversed, or to Reversal 2, in which the structure of the equivalence classes was altered. At all stages throughout the experiment, participants were asked to rate their current mood state by using mood and incentive rating scales. The findings of the current study replicated the results of previous research in demonstrating a clear transfer of mood functions via equivalence relations. This outcome is compared with the existing literature about mood induction and the transfer of mood functions.

A group is said to be locally indicable if each of its non-trivial finitely generated subgroups has the infinite cyclic group as a homomorphic image. Such groups were studied by Higman [-9] in connection with the zero-divisor and unit problems for group rings. More recently, they have arisen I-2, 12] in the study of equations over groups. In [12], I proved a Freiheitsatz for locally indicable groups. This has been proved independently by Brodskii [2] and Short [22]. The present paper arises from an investigation of a question put to me by S.J. Pride whether torsionfree 1-relator groups are locally indicable. The question was raised originally by Baumslag (I-1], Problem 19) and an affirmative solution has recently been announced by Brodski~ [2]. As a consequence, the group algebra RG of a torsion-free 1-relator group G over an integral domain R has no non-trivial zero-divisors, and no non-trivial units (using Higman's results [9]). The first fact was also proved by Lewin and Lewin [16], by embedding RG in a division ring. The second appears to be new, as was pointed out to me by K.A. Brown. A second consequence is that no 1-relator group has a non-trivial finitely generated perfect subgroup, which answers [1], Problem 7 and [10], Question 1. In fact, using the Freiheitsatz, and the tower method described in [12], it is possible to prove the following general version of Brodskii's theorem.

Extension of the category Og and a vanishing theorem for the Ext functor for Kac-Moody algebras

Similarity is an important way of categorizing mathematical objects. In this paper we consider how similarity between undirected graphs, or networks, can be defined. We do this by reducing any undirected graph, G = (V,E) to its irreducible spine, I. First we show that the reduction process, ω, is a unique transformation and thus is a well-defined function. Each irreducible spine, I, thus defines an equivalence class consisting of all graphs {G_i} such that G_i.ω = I. We then show that reduction by ω preserves many key properties. Specifically it preserves shortest path lengths in I, and (under rather general conditions) preserves the location of distance and betweeness centers. In essence, every member graph of the equivalence class will have much the same path structure. Next, an algorithm is presented which, given any irreducible spine I, will generate (randomly) a graph Gk in the equivalence class defined by I, that is, we must have G_k.ω = I. Since I.e.ω = I, e = ω^-1. It is of some importance in network analysis to be able to quantify the notion of network similarity; to be able to say that a network G1 is closer to the network G2 than to G3. A popular method of comparing similarity between two networks G1 and G2 is to calculate the Spearman coefficient based on corresponding node degrees. We do this for 5 random graphs in the equivalence class of our running example. Finally, we show that any irreducible spine consists of a system of cycles. We then note that every spine, I, is topologically equivalent to a cubic graph. Thus individual cubic graphs generate a second, and coarser, equivalence relation on the family of all undirected graphs, which we call its genus.

Free algebras in division rings with an involution

We recall that two finitely presented groups are said to be proper 2 2 -equivalent if they can be realized as fundamental groups of finite 2 2 -dimensional CW-complexes whose universal covers are proper homotopy equivalent. This yields a topological equivalence relation which is coarser than the quasi-isometry relation. For a 1 1 -ended and semistable at infinity group, its proper 2 2 -equivalence class is determined by the pro-isomorphism type of its fundamental pro-group. This paper is intended to be part of a program for a better understanding of the behaviour of proper 2 2 -equivalences with respect to the basic constructions in combinatorial group theory (i.e., amalgamated free products and HNN-extensions). Previous joint work with R. Roy shows some results in the case the amalgamation subgroups are finite. Here we show that if a 1 1 -ended finitely presented group G G splits non-trivially as an amalgamated product G 1 ∗ F G 2 G_1 \ast _F G_2 (or an HNN-extension G 1 ∗ F G_1 \ast _F ) over a finitely generated virtually free group F F where the groups G i G_i have a fundamental pro-group of a special type (namely, semistable and pro-(finitely generated free)) at each end, then so does G G at infinity. As a consequence, it follows from some previous work and recent results by M. Mihalik and S. Shepherd that such a group G G as above is proper 2 2 -equivalent to either Z × Z {\mathbb {Z}} \times {\mathbb {Z}} (and hence G G is a virtually surface group) or F 2 × Z F_2 \times {\mathbb {Z}} , where F 2 F_2 denotes the free group on two generators.

On λ-homomorphic skew braces

Writing certain commutators as products of cubes in free groups

We are given a division ring D with involution (*) and with a *-valuation V such that V(aa*b − baa*) >0 V(aa*b), for all nonzero elements a, b of D. We assume, further, that the associated residue class division ring D V is a (commutative) field with characteristic 0. In this work, we evidence two criteria in order for D to be either a standard quaternion division algebra or else a purely transcendental extension of its center. We apply one of these to answer our open Question 2.4.3 [2].