THE LOWER CENTRAL AND DERIVED SERIES OF THE BRAID GROUPS OF THE SPHERE

In this paper, we determine the lower central and derived series for the braid groups of the sphere. We are motivated in part by the study of Fadell-Neuwirth short exact sequences, but the problem is important in its own right. The braid groups of the 2-sphere S(2) were studied by Fadell, Van Buskirk and Gillette during the 1960s, and are of particular interest due to the fact that they have torsion elements (which were characterised by Murasugi). We first prove that for all n epsilon N, the lower central series of the n-string braid group B(n)(S(2)) is constant from the commutator subgroup onwards. We obtain a presentation of Gamma(2)(Bn(S(2))), from which we observe that Gamma(2)(B(4)(S(2))) is a semi-direct product of the quaternion group Q(8) of order 8 by a free group F(2) of rank 2. As for the derived series of Bn(S(2)), we show that for all n >= 5, it is constant from the derived subgroup onwards. The group Bn(S(2)) being finite and soluble for n <= 3, the critical case is n = 4 for which the derived subgroup is the above semi-direct product Q(8) (sic) F(2). By proving a general result concerning the structure of the derived subgroup of a semi-direct product, we are able to determine completely the derived series of B(4)(S(2)) which from (B(4)(S(2)))(4) onwards coincides with that of the free group of rank 2, as well as its successive derived series quotients.

Studies in Fuzzy Commutative Algebra

The main objective of this thesis was to extend some basic concepts and results in module theory in algebra to the fuzzy setting.The concepts like simple module, semisimple module and exact sequences of R-modules form an important area of study in crisp module theory. In this thesis generalising these concepts to the fuzzy setting we have introduced concepts of ‘simple and semisimple L-modules’ and proved some results which include results analogous to those in crisp case. Also we have defined and studied the concept of ‘exact sequences of L-modules’.Further extending the concepts in crisp theory, we have introduced the fuzzy analogues ‘projective and injective L-modules’. We have proved many results in this context. Further we have defined and explored notion of ‘essential L-submodules of an L-module’. Still there are results in crisp theory related to the topics covered in this thesis which are to be investigated in the fuzzy setting. There are a lot of ideas still left in algebra, related to the theory of modules, such as the ‘injective hull of a module’, ‘tensor product of modules’ etc. for which the fuzzy analogues are not defined and explored.

THE LOWER CENTRAL AND DERIVED SERIES OF THE BRAID GROUPS OF THE FINITELY-PUNCTURED SPHERE

Motivated in part by the study of Fadell-Neuwirth short exact sequences, we determine the lower central and derived series for the braid groups of the finitely-punctured sphere. For n >= 1, the class of m-string braid groups B(m)(S(2)\{x(1), ... , x(n)}) of the n-punctured sphere includes the usual Artin braid groups B(m) (for n = 1), those of the annulus, which are Artin groups of type B (for n = 2), and affine Artin groups of type (C) over tilde (for n = 3). We first consider the case n = 1. Motivated by the study of almost periodic solutions of algebraic equations with almost periodic coefficients, Gorin and Lin calculated the commutator subgroup of the Artin braid groups. We extend their results, and show that the lower central series (respectively, derived series) of B(m) is completely determined for all m is an element of N (respectively, for all m not equal 4). In the exceptional case m = 4, we obtain some higher elements of the derived series and its quotients. When n >= 2, we prove that the lower central series (respectively, derived series) of B(m)(S(2)\{x(1), ... , x(n)}) is constant from the commutator subgroup onwards for all m >= 3 (respectively, m >= 5). The case m = 1 is that of the free group of rank n - 1. The case n = 2 is of particular interest notably when m = 2 also. In this case, the commutator subgroup is a free group of infinite rank. We then go on to show that B(2)(S(2)\{x(1), x(2)}) admits various interpretations, as the Baumslag-Solitar group BS(2, 2), or as a one-relator group with non-trivial centre for example. We conclude from this latter fact that B(2)(S(2)\{x(1), x(2)}) is residually nilpotent, and that from the commutator subgroup onwards, its lower central series coincides with that of the free product Z(2) * Z. Further, its lower central series quotients Gamma(i)/Gamma(i+1) are direct sums of copies of Z(2), the number of summands being determined explicitly. In the case m >= 3 and n = 2, we obtain a presentation of the derived subgroup, from which we deduce its Abelianization. Finally, in the case n = 3, we obtain partial results for the derived series, and we prove that the lower central series quotients Gamma(i)/Gamma(i+1) are 2-elementary finitely-generated groups.

The lower central and derived series of the braid groups of the projective plane

In this paper, we determine the lower central and derived series for the braid groups of the projective plane. We are motivated in part by the study of Fadell-Neuwirth short exact sequences, but the problem is interesting in its own right. The n-string braid groups B(n)(RP(2)) of the projective plane RP(2) were originally studied by Van Buskirk during the 1960s. and are of particular interest due to the fact that they have torsion. The group B(1)(RP(2)) (resp. B(2)(RP(2))) is isomorphic to the cyclic group Z(2) of order 2 (resp. the generalised quaternion group of order 16) and hence their lower central and derived series are known. If n > 2, we first prove that the lower central series of B(n)(RP(2)) is constant from the commutator subgroup onwards. We observe that Gamma(2)(B(3)(RP(2))) is isomorphic to (F(3) X Q(8)) X Z(3), where F(k) denotes the free group of rank k, and Q(8) denotes the quaternion group of order 8, and that Gamma(2)(B(4)(RP(2))) is an extension of an index 2 subgroup K of P(4)(RP(2)) by Z(2) circle plus Z(2). As for the derived series of B(n)(RP(2)), we show that for all n >= 5, it is constant from the derived subgroup onwards. The group B(n)(RP(2)) being finite and soluble for n <= 2, the critical cases are n = 3, 4. We are able to determine completely the derived series of B(3)(RP(2)). The subgroups (B(3)(RP(2)))((1)), (B(3)(RP(2)))((2)) and (B(3)(RP(2)))((3)) are isomorphic respectively to (F(3) x Q(8)) x Z(3), F(3) X Q(8) and F(9) X Z(2), and we compute the derived series quotients of these groups. From (B(3)(RP(2)))((4)) onwards, the derived series of B(3)(RP(2)), as well as its successive derived series quotients, coincide with those of F(9). We analyse the derived series of B(4)(RP(2)) and its quotients up to (B(4)(RP(2)))((4)), and we show that (B(4)(RP(2)))((4)) is a semi-direct product of F(129) by F(17). Finally, we give a presentation of Gamma(2)(B(n)(RP(2))). (C) 2011 Elsevier Inc. All rights reserved.

Charges and fluxes in Maxwell theory on compact manifolds with boundary

We investigate the charges and fluxes that can occur in higher-order Abelian gauge theories defined on compact space-time manifolds with boundary. The boundary is necessary to supply a destination to the electric lines of force emanating from brane sources, thus allowing non-zero net electric charges, but it also introduces new types of electric and magnetic flux. The resulting structure of currents, charges, and fluxes is studied and expressed in the language of relative homology and de Rham cohomology and the corresponding abelian groups. These can be organised in terms of a pair of exact sequences related by the Poincare-Lefschetz isomorphism and by a weaker flip symmetry exchanging the ends of the sequences. It is shown how all this structure is brought into play by the imposition of the appropriately generalised Maxwell's equations. The requirement that these equations be integrable restricts the world-volume of a permitted brane (assumed closed) to be homologous to a cycle on the boundary of space-time. All electric charges and magnetic fluxes are quantised and satisfy the Dirac quantisation condition. But through some boundary cycles there may be unquantised electric fluxes associated with quantised magnetic fluxes and so dyonic in nature.

Cálculo de grupos de homotopia dos grupos clássicos

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Homotopias e aplicações

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Class Notes for Math 901/902: Abstract Algebra, Instructor Tom Marley

Topics include: Free groups and presentations; Automorphism groups; Semidirect products; Classification of groups of small order; Normal series: composition, derived, and solvable series; Algebraic field extensions, splitting fields, algebraic closures; Separable algebraic extensions, the Primitive Element Theorem; Inseparability, purely inseparable extensions; Finite fields; Cyclotomic field extensions; Galois theory; Norm and trace maps of an algebraic field extension; Solvability by radicals, Galois' theorem; Transcendence degree; Rings and modules: Examples and basic properties; Exact sequences, split short exact sequences; Free modules, projective modules; Localization of (commutative) rings and modules; The prime spectrum of a ring; Nakayama's lemma; Basic category theory; The Hom functors; Tensor products, adjointness; Left/right Noetherian and Artinian modules; Composition series, the Jordan-Holder Theorem; Semisimple rings; The Artin-Wedderburn Theorem; The Density Theorem; The Jacobson radical; Artinian rings; von Neumann regular rings; Wedderburn's theorem on finite division rings; Group representations, character theory; Integral ring extensions; Burnside's paqb Theorem; Injective modules.

Class Notes for Math 918: Local Cohomology, Instructor Tom Marley

Topics include: Injective Module, Basic Properties of Local Cohomology Modules, Local Cohomology as a Cech Complex, Long exact sequences on Local Cohomology, Arithmetic Rank, Change of Rings Principle, Local Cohomology as a direct limit of Ext modules, Local Duality, Chevelley’s Theorem, Hartshorne- Lichtenbaum Vanishing Theorem, Falting’s Theorem.

On prediction of individual sequences

Sequential randomized prediction of an arbitrary binary sequence isinvestigated. No assumption is made on the mechanism of generating the bit sequence. The goal of the predictor is to minimize its relative loss, i.e., to make (almost) as few mistakes as the best ``expert'' in a fixed, possibly infinite, set of experts. We point out a surprising connection between this prediction problem and empirical process theory. First, in the special case of static (memoryless) experts, we completely characterize the minimax relative loss in terms of the maximum of an associated Rademacher process. Then we show general upper and lower bounds on the minimaxrelative loss in terms of the geometry of the class of experts. As main examples, we determine the exact order of magnitude of the minimax relative loss for the class of autoregressive linear predictors and for the class of Markov experts.

Calcrete profile development in quaternary alluvial sequences, southeast Spain: Implications for using calcretes as a basis for landform chronologies

A detailed study of the morphology and micro-morphology of Quaternary alluvial calcrete profiles from the Sorbas Basin shows that calcretes may be morphologically simple or complex. The 'simple' profiles reflect pedogenesis occurring after alluvial terrace formation and consist of a single pedogenic horizon near the land surface. The 'complex' profiles reflect the occurrence of multiple calcrete events during terrace sediment aggradation and further periods of pedogenesis after terrace formation. These 'complex' calcrete profiles are consequently described as composite profiles. The exact morphology of the composite profiles depends upon: (1) the number of calcrete-forming events occurring during terrace sediment aggradation; (2) the amount of sediment accretion that occurs between each period of calcrete formation; and (3) the degree of pedogenesis after terrace formation. Simple calcrete profiles are most useful in establishing landform chronologies because they represent a single phase of pedogenesis after terrace formation. Composite profiles are more problematic. Pedogenic calcretes that form within them may inherit carbonate from calcrete horizons occurring lower down in the terrace sediments. In addition erosion may lead to the exhumation of older calcretes within the terrace sediment. Calcrete 'inheritance' may make pedogenic horizons appear more mature than they actually are and produce horizons containing carbonate embracing a range of ages. Calcrete exhumation exposes calcrete horizons whose morphology and radiometric ages are wholly unrelated to terrace surface age. Composite profiles are, therefore, only suitable for chronological studies if the pedogenic horizon capping the terrace sequence can be clearly distinguished from earlier calcrete-forming events. Thus, a detailed morphological/micro-morphological study is required before any chronological study is undertaken. This is the only way to establish whether particular calcrete profiles are suitable for dating purposes. Copyright (C) 2003 John Wiley Sons, Ltd.

Whole-body MRI of multiple myeloma: comparison of different MRI sequences in assessment of different growth patterns

PURPOSE: To determine sensitivity, specificity and inter-observer variability of different whole-body MRI (WB-MRI) sequences in patients with multiple myeloma (MM). METHODS AND MATERIALS: WB-MRI using a 1.5T MRI scanner was performed in 23 consecutive patients (13 males, 10 females; mean age 63+/-12 years) with histologically proven MM. All patients were clinically classified according to infiltration (low-grade, n=7; intermediate-grade, n=7; high-grade, n=9) and to the staging system of Durie and Salmon PLUS (stage I, n=12; stage II, n=4; stage III, n=7). The control group consisted of 36 individuals without malignancy (25 males, 11 females; mean age 57+/-13 years). Two observers independently evaluated the following WB-MRI sequences: T1w-TSE (T1), T2w-TIRM (T2), and the combination of both sequences, including a contrast-enhanced T1w-TSE with fat-saturation (T1+/-CE/T2). They had to determine growth patterns (focal and/or diffuse) and the MRI sequence that provided the highest confidence level in depicting the MM lesions. Results were calculated on a per-patient basis. RESULTS: Visual detection of MM was as follows: T1, 65% (sensitivity)/85% (specificity); T2, 76%/81%; T1+/-CE/T2, 67%/88%. Inter-observer variability was as follows: T1, 0.3; T2, 0.55; T1+/-CE/T2, 0.55. Sensitivity improved depending on infiltration grade (T1: 1=60%; 2=36%; 3=83%; T2: 1=70%; 2=71%; 3=89%; T1+/-CE/T2: 1=50%; 2=50%; 3=89%) and clinical stage (T1: 1=58%; 2=63%; 3=79%; T2: 1=58%; 2=88%; 3=100%; T1+/-CE/T2: 1=50%; 2=63%; 3=100%). T2w-TIRM sequences achieved the best reliability in depicting the MM lesions (65% in the mean of both readers). CONCLUSIONS: T2w-TIRM sequences achieved the highest level of sensitivity and best reliability, and thus might be valuable for initial assessment of MM. For an exact staging and grading the examination protocol should encompass unenhanced and enhanced T1w-MRI sequences, in addition to T2w-TIRM.

Direct and Inverse Results on Row Sequences of Hermite-Pad, Approximants

We give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of simultaneous rational interpolants with a bounded number of poles. The conditions are expressed in terms of intrinsic properties of the system of functions used to build the approximants. Exact rates of convergence for these denominators and the simultaneous rational approximants are provided.

In silico detection of control signals: mRNA 3′-end-processing sequences in diverse species

We have investigated mRNA 3′-end-processing signals in each of six eukaryotic species (yeast, rice, arabidopsis, fruitfly, mouse, and human) through the analysis of more than 20,000 3′-expressed sequence tags. The use and conservation of the canonical AAUAAA element vary widely among the six species and are especially weak in plants and yeast. Even in the animal species, the AAUAAA signal does not appear to be as universal as indicated by previous studies. The abundance of single-base variants of AAUAAA correlates with their measured processing efficiencies. As found previously, the plant polyadenylation signals are more similar to those of yeast than to those of animals, with both common content and arrangement of the signal elements. In all species examined, the complete polyadenylation signal appears to consist of an aggregate of multiple elements. In light of these and previous results, we present a broadened concept of 3′-end-processing signals in which no single exact sequence element is universally required for processing. Rather, the total efficiency is a function of all elements and, importantly, an inefficient word in one element can be compensated for by strong words in other elements. These complex patterns indicate that effective tools to identify 3′-end-processing signals will require more than consensus sequence identification.