Symplectic Representation of a Braid Group on 3-Sheeted Covers of the Riemann Sphere

We define Picard cycles on each smooth three-sheeted Galois cover C of the Riemann sphere. The moduli space of all these algebraic curves is a nice Shimura surface, namely a symmetric quotient of the projective plane uniformized by the complex two-dimensional unit ball. We show that all Picard cycles on C form a simple orbit of the Picard modular group of Eisenstein numbers. The proof uses a special surface classification in connection with the uniformization of a classical Picard-Fuchs system. It yields an explicit symplectic representation of the braid groups (coloured or not) of four strings.

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.

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.

On some Finite-Dimensional Representations of Artin Braid Group

Валентин В. Илиев - Авторът изучава някои хомоморфни образи G на групата на Артин на плитките върху n нишки в крайни симетрични групи. Получените пермутационни групи G са разширения на симетричната група върху n букви чрез подходяща абелева група. Разширенията G зависят от един целочислен параметър q ≥ 1 и се разцепват тогава и само тогава, когато 4 не дели q. В случая на нечетно q са намерени всички крайномерни неприводими представяния на G, а те от своя страна генерират безкрайна редица от неприводими представяния на групата на плитките.

The classification and the conjugacy classes of the finite subgroups of the sphere braid groups

Let n >= 3. We classify the finite groups which are realised as subgroups of the sphere braid group B(n)(S(2)). Such groups must be of cohomological period 2 or 4. Depending on the value of n, we show that the following are the maximal finite subgroups of B(n)(S(2)): Z(2(n-1)); the dicyclic groups of order 4n and 4(n - 2); the binary tetrahedral group T*; the binary octahedral group O*; and the binary icosahedral group I(*). We give geometric as well as some explicit algebraic constructions of these groups in B(n)(S(2)) and determine the number of conjugacy classes of such finite subgroups. We also reprove Murasugi`s classification of the torsion elements of B(n)(S(2)) and explain how the finite subgroups of B(n)(S(2)) are related to this classification, as well as to the lower central and derived series of B(n)(S(2)).

Twisted conjugacy classes in symplectic groups, mapping class groups and braid groups (with an appendix written jointly with Francois Dahmani)

We prove that the symplectic group Sp(2n, Z) and the mapping class group Mod(S) of a compact surface S satisfy the R(infinity) property. We also show that B(n)(S), the full braid group on n-strings of a surface S, satisfies the R(infinity) property in the cases where S is either the compact disk D, or the sphere S(2). This means that for any automorphism phi of G, where G is one of the above groups, the number of twisted phi-conjugacy classes is infinite.

METHANE DRY REFORMING OVER Ni SUPPORTED ON PINE SAWDUST ACTIVATED CARBON: EFFECTS OF SUPPORT SURFACE PROPERTIES AND METAL LOADING

The influence of metal loading and support surface functional groups (SFG) on methane dry reforming (MDR) over Ni catalysts supported on pine-sawdust derived activated carbon were studied. Using pine sawdust as the catalyst support precursor, the smallest variety and lowest concentration of SFG led to best Ni dispersion and highest catalytic activity, which increased with Ni loading up to 3 Ni atoms nm-2. At higher Ni loading, the formation of large metal aggregates was observed, consistent with a lower "apparen" surface area and a decrease in catalytic activity. The H2/CO ratio rose with increasing reaction temperature, indicating that increasingly important side reactions were taking place in addition to MDR.

Immersion enthalpy variation of surface-modified mineral activated carbon in lead (II) aqueous solution adsorption: the relation between immersion enthalpy and adsorption capacity

An activated carbon was obtained by chemical activation with phosphoric acid, CM, from a mineral carbon. Afterwards, the carbon was modified with 2 and 5 molL-1, CMox2 and CMox5 nitric acid solutions to increase the surface acid group contents. Immersion enthalpy at pH 4 values and Pb2+ adsorption isotherms were determined by immersing activated carbons in aqueous solution. The surface area values of the adsorbents and total pore volume were approximately 560 m².g-1 and 0.36 cm³g-1, respectively. As regards chemical characteristics, activated carbons had higher acid sites content, 0.92-2.42 meq g-1, than basic sites, 0.63-0.12 meq g-1. pH values were between 7.4 and 4.5 at the point of zero charge, pH PZC. The adsorbed quantity of Pb2+ and the immersion enthalpy in solution of different pH values for CM activated carbon showed that the values are the highest for pH 4, 15.7 mgg-1 and 27.6 Jg-1 respectively. Pb2+ adsorption isotherms and immersion enthalpy were determined for modified activated carbons and the highest values were obtained for the activated carbon that showed the highest content of total acid sites on the surface.

Coincidence Wecken homotopies versus Wecken homotopies relative to a fixed homotopy in one of the maps

We study the 1-parameter Wecken problem versus the restricted Wecken problem, for coincidence free pairs of maps between surfaces. For this we use properties of the function space between two surfaces and of the pure braid group on two strings of a surface. When the target surface is either the 2-sphere or the torus it is known that the two problems are the same. We classify most pairs of homotopy classes of maps according to the answer of the two problems are either the same or different when the target is either projective space or the Klein bottle. Some partial results are given for surfaces of negative Euler characteristic. (C) 2010 Elsevier B.V. All rights reserved.

Surface and Biomechanical Study of Titanium Implants Modified by Laser With and Without Hydroxyapatite Coating, in Rabbits

Surface and biomechanical analysis of titanium implant surfaces modified by laser beam with and without hydroxyapatite. Titanium implants with 3 different surfaces were inserted into the tibias of 30 rabbits: group I (GI) machined surface (control group), group II irradiated with laser (GII), and group III irradiated with laser and hydroxyapatite coating applied-biomimetic method (GIII). Topographical analysis with scanning electron microscopy was made before surgery in the tibia. These rabbits were distributed into 2 periods of observation: 4 and 8 weeks postsurgery, after which biomechanical analysis (removal torque) was conducted. Statistical analysis used the Student-Newman-Keuls method. Surface showed roughness in GII and GIII. Biomechanical analysis demonstrated values with significant differences in GII and GIII. Titanium implants modified by laser irradiation can increase osseointegration during the initial phase.

Influence of surface treatments on the flexural strength of denture base repair

Objective: The purpose of this study was to evaluate the flexural strength of repairs made with autopolymerising acrylic resin after different treatments of joint surfaces.Material and Methods: Fifty rectangular specimens were made with heat-polymerised acrylic resin and 40 were repaired with autopolymerising acrylic resin following joint surface treatments: group 1 (intact specimens), group 2 (chemical treatment: wetting with methyl-methacrylate for 180 s), group 3 (abraded with silicon carbide paper), group 4 (abraded and wetting with methyl-methacrylate for 180 s) and group 5 (without surface treatment). The flexural strength was measured by a three-point bending test using a universal testing machine with a 100 Kgf load cell in the centre of repair at 5 mm/min cross-head speed. All data were analysed using one-way ANOVA and Tukey HSD test for multiple comparisons (p < 0.05).Results: Among repaired specimens, groups 2 and 4 had 66.53 +/- 3.4 and 69.38 +/- 1.8 MPa mean values and were similar. These groups had superior flexural strength than groups 3 and 5 that were similar and had 54.11 +/- 3.4 and 51.24 +/- 2.8 MPa mean values, respectively. Group 1 had a mean value of 108.30 +/- 2.8 MPa being the highest result.Conclusion: It can be concluded that the treatment of the joint surfaces with methyl-methacrylate increases the flexural strength of denture base repairs, although the strength is still lower than that observed for the intact denture base resin. Abrasion with sandpaper was not able to influence the flexural strength of repaired denture bases.

Methane dry reforming over Ni supported on pine sawdust activated carbon: Effects of support surface properties and metal loading

The influence of metal loading and support surface functional groups (SFG) on methane dry reforming (MDR) over Ni catalysts supported on pine-sawdust derived activated carbon were studied. Using pine sawdust as the catalyst support precursor, the smallest variety and lowest concentration of SFG led to best Ni dispersion and highest catalytic activity, which increased with Ni loading up to 3 Ni atoms nm-2. At higher Ni loading, the formation of large metal aggregates was observed, consistent with a lower "apparen" surface area and a decrease in catalytic activity. The H2/CO ratio rose with increasing reaction temperature, indicating that increasingly important side reactions were taking place in addition to MDR.

Braid groups, mapping class groups, and Torelli groups

This thesis discusses subgroups of mapping class groups of particular surfaces. First, we study the Torelli group, that is, the subgroup of the mapping class group that acts trivially on the first homology. We investigate generators of the Torelli group, and we give an algorithm that factorizes elements of the Torelli group into products of particular generators. Furthermore, we investigate normal closures of powers of standard generators of the mapping class group of a punctured sphere. By using the Jones representation, we prove that in most cases these normal closures have infinite index in the mapping class group. We prove a similar result for the hyperelliptic mapping class group, that is, the group that consists of mapping classes that commute with a fixed hyperelliptic involution. As a corollary, we recover an older theorem of Coxeter (with 2 exceptional cases), which states that the normal closure of the m-th power of standard generators of the braid group has infinite index in the braid group. Finally, we study finite index subgroups of braid groups, namely, congruence subgroups of braid groups. We discuss presentations of these groups and we provide a topological interpretation of their generating sets.

Modifying the band gap and optical properties of Germanium nanowires by surface termination

Semiconductor nanowires, based on silicon (Si) or germanium (Ge) are leading candidates for many ICT applications, including next generation transistors, optoelectronics, gas and biosensing and photovoltaics. Key to these applications is the possibility to tune the band gap by changing the diameter of the nanowire. Ge nanowires of different diameter have been studied with H termination, but, using ideas from chemistry, changing the surface terminating group can be used to modulate the band gap. In this paper we apply the generalised gradient approximation of density functional theory (GGA-DFT) and hybrid DFT to study the effect of diameter and surface termination using –H, –NH2 and –OH groups on the band gap of (001), (110) and (111) oriented germanium nanowires. We show that the surface terminating group allows both the magnitude and the nature of the band gap to be changed. We further show that the absorption edge shifts to longer wavelength with the –NH2 and –OH terminations compared to the –H termination and we trace the origin of this effect to valence band modifications upon modifying the nanowire with –NH2 or –OH. These results show that it is possible to tune the band gap of small diameter Ge nanowires over a range of ca. 1.1 eV by simple surface chemistry.