Parallel and Distributed Algorithms for a Class of Graph-Related Computational Problems.

There exist at least two models of parallel computing, namely, shared-memory and message-passing. This research addresses problems in both these types of systems, and proposes efficient parallel (Shared-Memory Model) and distributed (message-passing) algorithms for a variety of graph related computational problems. In part I, we design algorithms for three generic problems in distributed systems: set manipulation, network structure recognition and facility placement. We present optimal distributed algorithms for recognizing rectangular-mesh networks. The time and message complexity of our algorithm is linear in the number of nodes in the network. We also lay the foundation for the recognition of 2-reducible, outer-planar and cactus graphs. These algorithms have a message complexity of O(kn), where, k is the number of isolated two degree nodes in the network. We introduce the problem of reliable r-domination and design unified optimal distributed algorithms for the total, reliable and independent r-domination on trees. The time and message complexity of our algorithm is O(n), where n is the number of nodes in the tree. In the domain of set manipulation we design optimal algorithms for determining the intersection of sets in a distributed environment, where each processor is assumed to have its own set. The time and message complexity of our set intersection algorithm is O(mn), where m is the cardinality of the smallest set. In part II of our research we design optimal algorithms for r-domination and efficient parallel algorithms for the p-center problems on trees. We also present an optimal algorithm for computing the maximum independent set on intervals i the EREW-PRAM model. The r-domination problem on trees can now be solved in O(logn)time with O(n/logn) processors using the EREW-PRAM model. A parallel algorithm for range searching is developed using the concept of distributed data structures. We show that O(logn) search time can be effected for a range search on n 3-dimensional points using (2.log\sp2n-14.logn + 8) processors. Our algorithm can easily be generalized for the case of d-dimensional range search. (Abstract shortened with permission of author.)

Reducible M-curves for Le-networks in the totally-nonnegative Grassmannian and KP-II multiline solitons

We associate real and regular algebraic--geometric data to each multi--line soliton solution of Kadomtsev-Petviashvili II (KP) equation. These solutions are known to be parametrized by points of the totally non--negative part of real Grassmannians $Gr^{TNN}(k,n)$. In Ref.[3] we were able to construct real algebraic-geometric data for soliton data in the main cell $Gr^{TP} (k,n)$ only. Here we do not just extend that construction to all points in $Gr^{TNN}(k,n)$, but we also considerably simplify it, since both the reducible rational $M$-curve $\Gamma$ and the real regular KP divisor on $\Gamma$ are directly related to the parametrization of positroid cells in $Gr^{TNN}(k,n)$ via the Le-networks introduced by A. Postnikov in Ref [62]. In particular, the direct relation of our construction to the Le--networks guarantees that the genus of the underlying smooth $M$-curve is minimal and it coincides with the dimension of the positroid cell in $Gr^{TNN}(k,n)$ to which the soliton data belong to. Finally, we apply our construction to soliton data in $Gr^{TP}(2,4)$ and we compare it with that in Ref [3].Comment: 72 pages; several figures. We have decided to split our paper in Arxiv:1801.00208v1 into two parts. This preprint is the fully revised version of the first part of it. In the next version Arxiv:1801.00208 this part will be removed V2: Minor modifications, proof of Theorem 3.1 improve

Categorification and Dynamics in Generalised Braid Groups

Recent developments in the theory of stability conditions and its relation to Teichmuller theory have revealed a deep connection between triangulated categories and surfaces. Motivated by this, we prove a categorical analogue of the Nielsen-Thurston classification theorem for the rank two generalised braid groups by viewing them as (sub)groups of autoequivalences of certain triangulated categories. This can be seen as a (categorical) generalisation of the classification known for the type A braid groups when viewed as mapping class groups of the punctured discs. Firstly, we realise the generalised braid groups as groups of autoequivalences through categorical actions that categorify the corresponding Burau representations. These categorifications are achieved by constructing certain algebra objects in the tensor categories associated to the quantum group sl2, generalising the construction of zigzag algebras used in the categorical actions of simply-laced-type braid groups to include the non-simply-laced-types. By viewing the elements of the generalised braid groups as autoequivalences of triangulated categories, we study their dynamics through mass growth (categorical entropy), as introduced by Dimitrov-Haiden-Katzarkov-Kontsevich. Our classification is then achieved in a similar fashion to Bestvina-Handel's approach to the Nielsen-Thurston classification for mapping class groups. Namely, our classification can be effectively decided through a given algorithm that also computes the mass growth of the group elements. Moreover, it shows that the mass growth of the pseudo-Anosov elements are computable from certain rank two matrices

New Families of pseudo-Anosov Homeomorphisms with Vanishing Sah-Arnoux-Fathi Invariant

Translation surfaces can be viewed as polygons with parallel and equal sides identified. An affine homeomorphism φ from a translation surface to itself is called pseudo-Anosov when its derivative is a constant matrix in SL₂(R) whose trace is larger than 2 in absolute value. In this setting, the eigendirections of this matrix defines the stable and unstable flow on the translation surface. Taking a transversal to the stable flows, the first return map of the flow induces an interval exchange transformation T. The Sah-Arnoux-Fathi invariant of φ is the sum of the wedge product between the lengths of the subintervals of T and their translations. This wedge product does not depend on the choice of transversal. We apply Veech’s construction of pseudo-Anosov homeomorphisms to produce infinite families of pseudo-Anosov maps in the stratum H(2, 2) with vanishing Sah-Arnoux-Fathi invariant, as well as sporadic examples in other strata

Geochemical modelling of the speciation, transport, dispersal and fate of metal contaminants in water systems in the vicinity of tailings storage facilities

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2016.Gold mining of the Witwatersrand Basin reefs has been responsible for the rise of Johannesburg as an economic centre of South Africa. While mining provided a base for business and infrastructure development for the region, it has also generated social and environmental problems for the country. Tailings storage facilities (TSFs), a common sighting around Johannesburg and across the entire basin, have been built to contain the processed waste following extraction of gold from the pyrite containing quartzite ore. When the fine grained waste is exposed to atmospheric conditions, oxidation of remnant sulphides occurs resulting in acidic, metal rich and sulphate rich plumes that enter the environment through surface and groundwater systems. This thesis sought to better understand the release, transport, dispersal and fate of metals emanating from TSFs and their remnant footprints on the Witwatersrand. These metals included aluminium, copper, chromium, iron, manganese, nickel and uranium and are known to be toxic to humans depending on their concentration and speciation. Traditionally, analytical methods have been employed in studies focussing on the characterisation of some of these processes in the region. While these studies have generally conducted quantitative assessment of the extent of pollution, little comprehensive interrogation and fingerprinting of the processes that are influential in determining the potential risk posed by metals has been done. This has largely been due to the shortcomings of analytical methods to determine these. To this end, this research has employed geochemical modelling to complement the traditional analytical methods. The approach to study the release of metals from TSFs involved assessment of the partitioning of metals within tailings and their potential release using batch and sequential extraction methods. Processes of metal release within the tailings were simulated through geochemical modelling (using the PHREEQC and Geochemist’s Workbench codes). The simulations were based on the percolation of rainwater through these layers and the changes in its chemistry along the path. The potential seepage of this plume along the path was then correlated to observed efflorescent mineral crusts that are temporary sinks for metals and are a common feature in the vicinity of the tailings and water bodies such as ponds and streams. The potential impact of the mineral crusts on the water chemistry of receiving water systems following their dissolution was assessed using forward geochemical modelling. The transport of the metals in groundwater was also studied. This involved simulations of the transition in chemistry of a plume from a TSF along an aquifer of known composition. This was based on a 1-D reactive transport model constructed using information from sequential extraction work on the aquifer rock (to identify the key minerals to consider) and site data (mainly flow rates) from previous studies. The processes occurring in the removal of metals from acid mine drainage (AMD) through a permanent sink in the form of a pump-and-treat plant in the Central Goldfield of the basin were simulated using PHREEQC. The findings from the research showed that two different plumes were produced from an abandoned TSF as a result of rainwater percolation, notably a plume produced from the dissolution of secondary salts formed in the oxidised layer and a sulphuric acid rich plume in the unoxidised layer. These differences were apparent in the geochemical composition of the mineral crusts collected on the walls of tailings dumps and from a pond into which the plumes were draining. On dissolution, mineral crusts were found to produce acidic solutions with crusts containing predominantly Fe producing pH values below 3. The simulated dissolution of various types of mineral crusts gave insight into the impact of minerals present in the smallest amount. This showed that the bulk mineralogy as determined by analytical techniques such as PXRD and remote sensing could not be used with confidence to deduce the impact of the mineral crusts on receiving water bodies. The characteristics of surface plumes released from tailings TSF were compared to other water systems in the area around Soweto, with complementary interpretation conducted using chemometric methods. From principal component analysis (PCA), surface water systems were found to form distinct groups largely influenced by mineral solubility, alkalinity and dissolved oxygen content. The 1-D reactive transport simulations involved acidic, metal and sulphate rich water ingressing the aquifer (below the TSF). Several scenarios were modelled including simulations with different dolomite contents; allowing for surface complexation and the presence of cation exchange surfaces. At a point 500 m from the water ingress in the dolomite rich aquifer, Fe and Mn were largely precipitated out (as confirmed by sequential extraction results on the aquifer rock) while the sulphate concentration was reduced by almost half. On the other hand, Ca concentrations were conservative largely because of continuous dissolution of dolomite and precipitation of gypsum along the flow path. The simulations of the high density sludge treatment plant involved forward modelling of the treatment process with the sludge responsible for the removal of trace metals from the incoming acid mine drainage. The model can be of use for cost and process optimisation at the facility. This research has had notable outputs in the form of publications; models on metal release, transport and attenuation; and models on pump-and-treat processes. These will form an important repository of information and for benchmarking any further studies related to AMD.MT201

An Introduction to Geometric Topology

This book provides a self-contained introduction to the topology and geometry of surfaces and three-manifolds. The main goal is to describe Thurston's geometrisation of three-manifolds, proved by Perelman in 2002. The book is divided into three parts: the first is devoted to hyperbolic geometry, the second to surfaces, and the third to three-manifolds. It contains complete proofs of Mostow's rigidity, the thick-thin decomposition, Thurston's classification of the diffeomorphisms of surfaces (via Bonahon's geodesic currents), the prime and JSJ decomposition, the topological and geometric classification of Seifert manifolds, and Thurston's hyperbolic Dehn filling Theorem

Ahlfors circle maps and total reality: from Riemann to Rohlin

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by including now Rohlin's theory (v.2