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Gcodes over Formal Power Series Rings and Finite Chain RingsIn this work, we define $G$codes over the infinite ring $R_\infty$ as ideals in the group ring $R_\infty G$. We show that the dual of a $G$code is again a $G$code in this setting. We study the projections and lifts of $G$codes over the finite chain rings and over the formal power series rings respectively. We extend known results of constructing $\gamma$adic codes over $R_\infty$ to $\gamma$adic $G$codes over the same ring. We also study $G$codes over principal ideal rings.

GCodes, selfdual GCodes and reversible GCodes over the Ring Bj,kIn this work, we study a new family of rings, Bj,k, whose base field is the finite field Fpr . We study the structure of this family of rings and show that each member of the family is a commutative Frobenius ring. We define a Gray map for the new family of rings, study Gcodes, selfdual Gcodes, and reversible Gcodes over this family. In particular, we show that the projection of a Gcode over Bj,k to a code over Bl,m is also a Gcode and the image under the Gray map of a selfdual Gcode is also a selfdual Gcode when the characteristic of the base field is 2. Moreover, we show that the image of a reversible Gcode under the Gray map is also a reversible G2j+kcode. The Gray images of these codes are shown to have a rich automorphism group which arises from the algebraic structure of the rings and the groups. Finally, we show that quasiG codes, which are the images of Gcodes under the Gray map, are also Gscodes for some s.

Galerkin finite element approximation of a stochastic semilinear fractional subdiffusion with fractionally integrated additive noiseA Galerkin finite element method is applied to approximate the solution of a semilinear stochastic space and time fractional subdiffusion problem with the Caputo fractional derivative of the order $ \alpha \in (0, 1)$, driven by fractionally integrated additive noise. After discussing the existence, uniqueness and regularity results, we approximate the noise with the piecewise constant function in time in order to obtain a regularized stochastic fractional subdiffusion problem. The regularized problem is then approximated by using the finite element method in spatial direction. The mean squared errors are proved based on the sharp estimates of the various MittagLeffler functions involved in the integrals. Numerical experiments are conducted to show that the numerical results are consistent with the theoretical findings.

Galerkin methods for a Schroedingertype equation with a dynamical boundary condition in two dimensionsIn this paper, we consider a twodimensional Schodingertype equation with a dynamical boundary condition. This model describes the longrange sound propagation in naval environments of variable rigid bottom topography. Our choice for a regular enough finite element approximation is motivated by the dynamical condition and therefore, consists of a cubic splines implicit Galerkin method in space. Furthermore, we apply a CrankNicolson time stepping for the evolutionary variable. We prove existence and stability of the semidiscrete and fully discrete solution.

Gastrointestinal Stents: Materials and DesignsOver the last 25 years stents have developed into an established way of restoring luminal patency throughout the gastrointestinal tract. Materials used as well as the construction of these devices have become more and more sophisticated in order to comply better with the complex environment they are inserted. The requirements vary greatly from organ to organ and stent behavior differs significantly between stent constructions. However this is not necessarily understood by many operators, as the choice of devices is now vast and in many cases decisions are made on availability and cost. An increasing challenge in malignant conditions is the improving survival of incurable patients, which now exceeds the traditional life expectancy of a stent by a factor of 2 to 3. Consequently more patients experience failure of their stent and require repeat interventions. This has a poor impact on patients’ quality of life and potentially on their survival. Reintervention is often more difficult, carries the risk of additional complications and presents an additional economic burden to the health systems. This article illustrates current stent designs, their different behavior and their limitations.

A geneticalgorithm approach to simulating human immunodeficiency virus evolution reveals the strong impact of multiply infected cells and recombinationIt has been previously shown that the majority of human immunodeficiency virus type 1 (HIV1)infected splenocytes can harbour multiple, divergent proviruses with a copy number ranging from one to eight. This implies that, besides point mutations, recombination should be considered as an important mechanism in the evolution of HIV within an infected host. To explore in detail the possible contributions of multiinfection and recombination to HIV evolution, the effects of major microscopic parameters of HIV replication (i.e. the pointmutation rate, the crossover number, the recombination rate and the provirus copy number) on macroscopic characteristics (such as the Hamming distance and the abundance of npoint mutants) have been simulated in silico. Simulations predict that multiple provirus copies per infected cell and recombination act in synergy to speed up the development of sequence diversity. Point mutations can be fixed for some time without fitness selection. The time needed for the selection of multiple mutations with increased fitness is highly variable, supporting the view that stochastic processes may contribute substantially to the kinetics of HIV variation in vivo.

Gradientbased optimization method for producing a contoured beam with singlefed reflector antennaA gradientbased optimization method for producing a contoured beam by using a singlefed reflector antenna is presented. First, a quick and accurate pattern approximation formula based on physical optics (PO) is adopted to calculate the gradients of the directivity with respect to reflector's nodal displacements. Because the approximation formula is a linear function of nodal displacements, the gradient can be easily derived. Then, the method of the steepest descent is adopted, and an optimization iteration procedure is proposed. The iteration procedure includes two loops: an inner loop and an outer loop. In the inner loop, the gradient and pattern are calculated by matrix operation, which is very fast by using the precalculated data in the outer loop. In the outer loop, the ideal terms used in the inner loop to calculate the gradient and pattern are updated, and the real pattern is calculated by the PO method. Due to the high approximation accuracy, when the outer loop is performed once, the inner loop can be performed many times, which will save much time because the integration is replaced by matrix operation. In the end, a contoured beam covering the continental United States (CONUS) is designed, and simulation results show the effectiveness of the proposed algorithm.

Graphene Oxide Bulk Modified ScreenPrinted Electrodes Provide Beneficial Electroanalytical Sensing CapabilitiesWe demonstrate a facile methodology for the mass production of graphene oxide (GO) bulk modified screenprinted electrodes (GOSPEs) that are economical, highly reproducible and provide analytically useful outputs. Through fabricating GOSPEs with varying percentage mass incorporations (2.5, 5, 7.5 and 10%) of GO, an electrocatalytic effect towards the chosen electroanalytical probes is observed, that increases with greater GO incorporated compared to bare/ graphite SPEs. The optimum mass ratio of 10% GO to 90% carbon ink displays an electroanalytical signal towards dopamine (DA) and uric acid (UA), which is ca. ×10 greater in magnitude than that achievable at a bare/unmodified graphite SPE. Furthermore, 10% GOSPEs exhibit a competitively low limit of detection (3σ) towards DA at ca. 81 nM, which is superior to that of a bare/unmodified graphite SPE at ca. 780 nM. The improved analytical response is attributed to the large number of oxygenated species inhabiting the edge and defect sites of the GO nanosheets, which are available to exhibit electrocatalytic responses towards innersphere electrochemical analytes. Our reported methodology is simple, scalable, and cost effective for the fabrication of GOSPEs, that display highly competitive LODs, and is of significant interest for use in commercial and medicinal applications

Graphene oxide electrochemistry: the electrochemistry of graphene oxide modified electrodes reveals coverage dependent beneficial electrocatalysisThe modification of electrode surfaces is widely implemented in order to try and improve electron transfer kinetics and surface interactions, most recently using graphene related materials. Currently, the use of ‘as is’ graphene oxide (GO) has been largely overlooked, with the vast majority of researchers choosing to reduce GO to graphene or use it as part of a composite electrode. In this paper, ‘as is’ GO is explored and electrochemically characterized using a range of electrochemical redox probes, namely potassium ferrocyanide(II), N,N,N ,N tetramethylpphenylenediamine (TMPD), dopamine hydrochloride and epinephrine. Furthermore, the electroanalytical efficacy of GO is explored towards the sensing of dopamine hydrochloride and epinephrine via cyclic voltammetry. The electrochemical response of GO is benchmarked against pristine graphene and edge plane/basal plane pyrolytic graphite (EPPG and BPPG respectively) alternatives, where the GO shows an enhanced electrochemical/electroanalytical response. When using GO as an electrode material, the electrochemical response of the analytes studied herein deviate from that expected and exhibit altered electrochemical responses. The oxygenated species encompassing GO strongly influence and dominate the observed voltammetry, which is crucially coverage dependent. GO electrocatalysis is observed, which is attributed to the presence of beneficial oxygenated species dictating the response in specific cases, demonstrating potential for advantageous electroanalysis to be realized. Note however, that crucial coverage based regions are observed at GO modified electrodes, owing to the synergy of edge plane sites and oxygenated species. We report the true beneficial electrochemistry of GO, which has enormous potential to be beneficially used in various electrochemical applications ‘as is’ rather than be simply used as a precursor to making graphene and is truly a fascinating member of the graphene family

Graphite Felt: A New Material for Electroanalysis?Limit of detection is a key property of any sensor. For electrochemical sensors, a common and successful route to decreasing the limit of detection is maximising current density, thus boosting the signal to noise ratio. In quiescent solutions this is achieved by using micro and nano sized electrodes, where decreasing the electrode size increases the mass transport coefficient.12 However, as the electrode size decreases the fabrication technique becomes more complicate and the cost of the electrode often increases. In addition, the magnitude of the current decreases, eventually requiring the need for high specification potentiostats. This presentation will introduce a promising new type of electrode for electroanalysis based on graphite felt – a commonly used electrode material in redox flow batteries.3 The electrode is porous with a large specific surface area, is easy to fabricate (Figure 1) and has an approximate cost of 1 pence (not including the platinum wire, that can be reused hundreds of times).4 Surprisingly, low limits of detection are possible with this electrode, typically 10100 times lower than conventional carbon macroelectrodes. The reasons for this will be explored, along with an explanation of the distinctive voltammetry observed with graphite felt electrodes. Given the low cost, low limit of detection and relatively high currents, graphite felt is a promising material for electroanalysis that warrants further investigation. References: [1] Henstridge, M.C.; Compton, R.G. The Chemical Record, 2012, 12, 63 [2] Dawson, K.; Wahl, A.; O’Riordan, A. J Phys. Chem. C, 2012, 116, 14665. [3] Chakrabarti, M.H.; Brandon, N.P.; Hajimolana, S.A.; Tariq, F.; Yufit, V.; Hashim, M.A.; Hussain, M.A.; Low, C.T.J.; Aravind, P.V. J. Power Sources, 2014, 253, 150. [4] Smith, R.E.G.; Davies, T.J.; Baynes, N.B.; Nichols, R.J. J. Electroanal. Chem. 2015, 747, 29.

Graphite ScreenPrinted Electrodes Applied for the Accurate and Reagentless Sensing of pHA reagentless pH sensor based upon disposable and economical graphite screenprinted electrodes (GSPEs) is demonstrated for the first time. The voltammetric pH sensor utilises GSPEs which are chemically pretreated to form surface immobilised oxygenated species that when their redox behaviour is monitored, give a Nernstian response over a large pH range (113). An excellent experimental correlation is observed between the voltammetric potential and pH over the entire pH range of 113, such a response is not usually expected but rather deviation from linearity is encountered at alkaline pH values; absence of this has previously been attributed to a change in pKa value of surface immobilised groups. This nondeviation, which is observed here in the case of our facile produced reagentless pH sensor and also reported in the literature for pH sensitive compounds immobilized upon carbon electrodes/surfaces,where a linear response is observed over the entire pH range, is explained alternatively for the first time. The performance of the GSPE pH sensor is directly compared with a glass pH probe and applied to the measurement of pH in real samples where an excellent correlation between the two protocols is observed validating the proposed GSPE pH sensor.

Group Codes, Composite Group Codes and Constructions of SelfDual CodesThe main research presented in this thesis is around constructing binary selfdual codes using group rings together with some wellknown code construction methods and the study of group codes and composite group codes over different alphabets. Both these families of codes are generated by the elements that come from group rings. A search for binary selfdual codes with new weight enumerators is an ongoing research area in algebraic coding theory. For this reason, we present a generator matrix in which we employ the idea of a bisymmetric matrix with its entries being the block matrices that come from group rings and give the necessary conditions for this generator matrix to produce a selfdual code over a fi nite commutative Frobenius ring. Together with our generator matrix and some wellknown code construction methods, we find many binary selfdual codes with parameters [68, 34, 12] that have weight enumerators that were not known in the literature before. There is an extensive literature on the study of different families of codes over different alphabets and speci fically finite fi elds and finite commutative rings. The study of codes over rings opens up a new direction for constructing new binary selfdual codes with a rich automorphism group via the algebraic structure of the rings through the Gray maps associated with them. In this thesis, we introduce a new family of rings, study its algebraic structure and show that each member of this family is a commutative Frobenius ring. Moreover, we study group codes over this new family of rings and show that one can obtain codes with a rich automorphism group via the associated Gray map. We extend a well established isomorphism between group rings and the subring of the n x n matrices and show its applications to algebraic coding theory. Our extension enables one to construct many complex n x n matrices over the ring R that are fully de ned by the elements appearing in the first row. This property allows one to build generator matrices with these complex matrices so that the search field is practical in terms of the computational times. We show how these complex matrices are constructed using group rings, study their properties and present many interesting examples of complex matrices over the ring R. Using our extended isomorphism, we de ne a new family of codes which we call the composite group codes or for simplicity, composite Gcodes. We show that these new codes are ideals in the group ring RG and prove that the dual of a composite Gcode is also a composite Gcode. Moreover, we study generator matrices of the form [In  Ω(v)]; where In is the n x n identity matrix and Ω(v) is the composite matrix that comes from the extended isomorphism mentioned earlier. In particular, we show when such generator matrices produce selfdual codes over finite commutative Frobenius rings. Additionally, together with some generator matrices of the type [In  Ω(v)] and the wellknown extension and neighbour methods, we fi nd many new binary selfdual codes with parameters [68, 34, 12]. Lastly in this work, we study composite Gcodes over formal power series rings and finite chain rings. We extend many known results on projections and lifts of codes over these alphabets. We also extend some known results on γadic codes over the infi nite ring R∞

Group Rings, GCodes and Constructions of SelfDual and Formally SelfDual CodesWe describe Gcodes, which are codes that are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a Gcode is also a Gcode. We give constructions of selfdual and formally selfdual codes in this setting and we improve the existing construction given in [13] by showing that one of the conditions given in the theorem is unnecessary and, moreover, it restricts the number of selfdual codes obtained by the construction. We show that several of the standard constructions of selfdual codes are found within our general framework. We prove that our constructed codes must have an automorphism group that contains G as a subgroup. We also prove that a common construction technique for producing selfdual codes cannot produce the putative [72, 36, 16] Type II code. Additionally, we show precisely which groups can be used to construct the extremal Type II codes over length 24 and 48. We define quasiG codes and give a construction of these codes.

Group rings: Units and their applications in selfdual codesThe initial research presented in this thesis is the structure of the unit group of the group ring Cn x D6 over a field of characteristic 3 in terms of cyclic groups, specifically U(F3t(Cn x D6)). There are numerous applications of group rings, such as topology, geometry and algebraic Ktheory, but more recently in coding theory. Following the initial work on establishing the unit group of a group ring, we take a closer look at the use of group rings in algebraic coding theory in order to construct selfdual and extremal selfdual codes. Using a well established isomorphism between a group ring and a ring of matrices, we construct certain selfdual and formally selfdual codes over a finite commutative Frobenius ring. There is an interesting relationships between the Automorphism group of the code produced and the underlying group in the group ring. Building on the theory, we describe all possible group algebras that can be used to construct the wellknown binary extended Golay code. The double circulant construction is a wellknown technique for constructing selfdual codes; combining this with the established isomorphism previously mentioned, we demonstrate a new technique for constructing selfdual codes. New theory states that under certain conditions, these selfdual codes correspond to unitary units in group rings. Currently, using methods discussed, we construct 10 new extremal selfdual codes of length 68. In the search for new extremal selfdual codes, we establish a new technique which considers a double bordered construction. There are certain conditions where this new technique will produce selfdual codes, which are given in the theoretical results. Applying this new construction, we construct numerous new codes to verify the theoretical results; 1 new extremal selfdual code of length 64, 18 new codes of length 68 and 12 new extremal selfdual codes of length 80. Using the well established isomorphism and the common four block construction, we consider a new technique in order to construct selfdual codes of length 68. There are certain conditions, stated in the theoretical results, which allow this construction to yield selfdual codes, and some interesting links between the group ring elements and the construction. From this technique, we construct 32 new extremal selfdual codes of length 68. Lastly, we consider a unique construction as a combination of block circulant matrices and quadratic circulant matrices. Here, we provide theory surrounding this construction and conditions for full effectiveness of the method. Finally, we present the 52 new selfdual codes that result from this method; 1 new selfdual code of length 66 and 51 new selfdual codes of length 68. Note that different weight enumerators are dependant on different values of β. In addition, for codes of length 68, the weight enumerator is also defined in terms of γ, and for codes of length 80, the weight enumerator is also de ned in terms of α.

Halanaytype theory in the context of evolutionary equations with timelagWe consider extensions and modifications of a theory due to Halanay, and the context in which such results may be applied. Our emphasis is on a mathematical framework for Halanaytype analysis of problems with time lag and simulations using discrete versions or numerical formulae. We present selected (linear and nonlinear, discrete and continuous) results of Halanay type that can be used in the study of systems of evolutionary equations with various types of delayed argument, and the relevance and application of our results is illustrated, by reference to delaydifferential equations, difference equations, and methods.

Haptic feedback from human tissues of various stiffness and homogeneity.This work presents methods for haptic modelling of soft and hard tissue with varying stiffness. The model provides visualization of deformation and calculates force feedback during simulated epidural needle insertion. A springmassdamper (SMD) network is configured from magnetic resonance image (MRI) slices of patient’s lumbar region to represent varying stiffness throughout tissue structure. Reaction force is calculated from the SMD network and a haptic device is configured to produce a needle insertion simulation. The user can feel the changing forces as the needle is inserted through tissue layers and ligaments. Methods for calculating the force feedback at various depths of needle insertion are presented. Voxelization is used to fill ligament surface meshes with spring mass damper assemblies for simulated needle insertion into soft and hard tissues. Modelled vertebrae cannot be pierced by the needle. Graphs were produced during simulated needle insertions to compare the applied force to haptic reaction force. Preliminary saline pressure measurements during Tuohy epidural needle insertion are also used as a basis for forces generated in the simulation.

High order algorithms for numerical solution of fractional differential equationsIn this paper, two novel high order numerical algorithms are proposed for solving fractional differential equations where the fractional derivative is considered in the Caputo sense. The total domain is discretized into a set of small subdomains and then the unknown functions are approximated using the piecewise Lagrange interpolation polynomial of degree three and degree four. The detailed error analysis is presented, and it is analytically proven that the proposed algorithms are of orders 4 and 5. The stability of the algorithms is rigorously established and the stability region is also achieved. Numerical examples are provided to check the theoretical results and illustrate the efficiency and applicability of the novel algorithms.

A high order numerical method for solving nonlinear fractional differential equation with nonuniform meshesWe introduce a highorder numerical method for solving nonlinear fractional differential equation with nonuniform meshes. We first transform the fractional nonlinear differential equation into the equivalent Volterra integral equation. Then we approximate the integral by using the quadratic interpolation polynomials. On the first subinterval $[t_{0}, t_{1}]$, we approximate the integral with the quadratic interpolation polynomials defined on the nodes $t_{0}, t_{1}, t_{2}$ and in the other subinterval $[t_{j}, t_{j+1}], j=1, 2, \dots N1$, we approximate the integral with the quadratic interpolation polynomials defined on the nodes $t_{j1}, t_{j}, t_{j+1}$. A highorder numerical method is obtained. Then we apply this numerical method with the nonuniform meshes with the step size $\tau_{j}= t_{j+1} t_{j}= (j+1) \mu$ where $\mu= \frac{2T}{N (N+1)}$. Numerical results show that this method with the nonuniform meshes has the higher convergence order than the standard numerical methods obtained by using the rectangle and the trapzoid rules with the same nonuniform meshes.

High performing AgNW transparent conducting electrodes with a sheet resistance of 2.5 Ω Sq−1 based upon a rolltoroll compatible postprocessing techniqueThe report of transparent and conducting silver nanowires (AgNWs) that produce remarkable electrical performance, surface planarity and environmental stability is given. This research presents an innovative process that relies on three sequential steps, which are rolltoroll (R2R) compatible; thermal embossing, infrared sintering and plasma treatment. This process leads to the demonstration of a conductive film with a sheet resistance of 2.5Ω/sq and high transmittance, thus demonstrating the highest reported figureofmerit in AgNWs to date (FoM = 933). A further benefit of the process is that the surface roughness is substantially reduced compared to traditional AgNW processing techniques. Finally, consideration of the longterm stability is given by developing an accelerated life test process that simultaneously stresses the applied bias and temperature. Regression line fitting shows that a ∼150times improvement in stability is achieved at ‘normal operational conditions’ when compared to traditionally deposited AgNW films. Xray photoelectron spectroscopy (XPS) is used to understand the root cause of the improvement in longterm stability, which is related to reduced chemcial changes in the AgNWs.

High speed CO2 laser surface modification of iron/cobalt codoped boroaluminosilicate glassA preliminary study into the impact of high speed laser processing on the surface of iron and cobalt codoped glass substrates using a 60 W continuous wave (cw) CO2 laser. Two types of processing, termed fillprocessing and lineprocessing, were trialled. In fillprocessed samples the surface roughness of the glass was found to increase linearly with laser power from an Sa value of 20.8 nm–2.1 μm at a processing power of 54 W. With line processing, a more exponentiallike increase was observed with a roughness of 4 μm at 54 W. The change in surface properties of the glass, such as gloss and wettability, have also been measured. The contact angle of water was found to increase after laser processing by up to 64°. The surface gloss was varied between 45 and 100 gloss units (GUs).