Common Fixed Point and Weak** Commuting Mappings

Existence of common fixed points of weak** commuting mappings which satisfies the contractive condition involving pair of mappings in a complete metric space under certain is shown. Key words: commuting mappings, weak ** commuting mapping

Information content of the angular multipoles of redshift-space galaxy bispectrum

Citation: Gagrani, P., & Samushia, L. (2017). Information content of the angular multipoles of redshift-space galaxy bispectrum. Monthly Notices of the Royal Astronomical Society, 467(1), 928-934. doi:10.1093/mnras/stx135The redshift-space bispectrum (three point statistics) of galaxies depends on the expansion rate, the growth rate and the geometry of the Universe, and hence can be used to measure key cosmological parameters. In a homogeneous Universe, the bispectrum is a function of five variables and unlike its two point statistics counterpart - the power spectrum - which is a function of only two variables - is difficult to analyse unless the information is somehow reduced. The most commonly considered reduction schemes rely on computing angular integrals over possible orientations of the bispectrum triangle, thus reducing it to sets of function of only three variables describing the triangle shape. We use Fisher information formalism to study the information loss associated with this angular integration. Without any reduction, the bispectrum alone can deliver constraints on the growth rate parameter f that are better by a factor of 2.5 compared with the power spectrum, for a sample of luminous red galaxies expected from near future galaxy surveys at a redshift of z similar to 0.65 if we consider all the wavenumbers up to k <= 0.2 h Mpc (1). At lower redshifts the improvement could be up to a factor of 3. We find that most of the information is in the azimuthal averages of the first three even multipoles. This suggests that the bispectrum of every configuration can be reduced to just three numbers (instead of a 2D function) without significant loss of cosmologically relevant information

Action Functional Gradient Descent algorithm for estimating escape paths in Stochastic Chemical Reaction Networks

We first derive the Hamilton-Jacobi theory underlying continuous-time Markov processes, and then use the construction to develop a variational algorithm for estimating escape (least improbable or first passage) paths for a generic stochastic chemical reaction network that exhibits multiple fixed points. The design of our algorithm is such that it is independent of the underlying dimensionality of the system, the discretization control parameters are updated towards the continuum limit, and there is an easy-to-calculate measure for the correctness of its solution. We consider several applications of the algorithm and verify them against computationally expensive means such as the shooting method and stochastic simulation. While we employ theoretical techniques from mathematical physics, numerical optimization and chemical reaction network theory, we hope that our work finds practical applications with an inter-disciplinary audience including chemists, biologists, optimal control theorists and game theorists.Comment: 38 pages, 21 figure

Optimal weights for measuring redshift space distortions in multitracer galaxy catalogues

Citation: Pearson, D. W., Samushia, L., & Gagrani, P. (2016). Optimal weights for measuring redshift space distortions in multitracer galaxy catalogues. Monthly Notices of the Royal Astronomical Society, 463(3), 2708-2715. doi:10.1093/mnras/stw2177Since the volume accessible to galaxy surveys is fundamentally limited, it is extremely important to analyse available data in the most optimal fashion. One way of enhancing the cosmological information extracted from the clustering of galaxies is by weighting the galaxy field. The most widely used weighting schemes assign weights to galaxies based on the average local density in the region (FKP weights) and their bias with respect to the dark matter field (PVP weights). They are designed to minimize the fractional variance of the galaxy power-spectrum. We demonstrate that the currently used bias dependent weighting scheme can be further optimized for specific cosmological parameters. We develop a procedure for computing the optimal weights and test them against mock catalogues for which the values of all fitting parameters, as well as the input power-spectrum are known. We show that by applying these weights to the joint power-spectrum of emission line galaxies and luminous red galaxies from the Dark Energy Spectroscopic Instrument survey, the variance in the measured growth rate parameter can be reduced by as much as 36 per cent

Towards Electrically Injected Semiconductor Nanowire Lasers

We are at a tipping point of the next industrial revolution, which will change or day-to-day lives. Increased needs around communication and data storage will require networks that can handle around 175 zettabyte by the end of 2025. Optical interconnect technology is a promising candidate, as it can transmit data at lightning speeds within chips and boards in efficient way. Laser is the backbone of such optical circuits and is often integrated with its electrical counterpart for power inputs. With the advances in fabrication techniques, micron-sized III-V semiconductor lasers are used in data communication, medicine, robotics, green energy, military etc. However, there is still a strong need to make these lasers even smaller and more energy efficient for optical interconnect technology. Nanowires are a suitable candidate for such devices due to their large surface area-to-volume ratio, confinement of photons in two dimensions and ease in integration with other substrates. However, research of III-V semiconductor nanowire Fabry-Perot cavity lasers is still in the early stages. A lot more research need to be done to realise devices, in particular those that are electrically powered, that are manufacturable for practical applications. In this thesis, in-depth theoretical and experimental studies to fabricate nanowire lasers are presented. First, numerical modelling to optimise the dimensions of nanowires are performed to achieve low threshold lasing. Following this, epitaxial growth optimisation is carried out to achieve the desired dimensions of nanowires. Subsequently, fabrication of two different types of devices, single nanowire and array of nanowires is done. For single nanowire devices, an InP p-i-n axial structure is explored. These devices display light emitting diode (LED) characteristics, but unfortunately, they fail at higher injected current before lasing is observed. The potential causes of degradation of devices are high metal absorption, lower gain due to smaller active region and high free carrier absorption. To overcome high metal absorption and lower gain, radial p-n junction structures are investigated and significant improvement in the device performance is observed. However, the devices also fail at higher current injection levels prior to lasing threshold. To overcome high free carrier absorption, transparent conducting oxides (TCOs) are used as a dopant layer forming a heterojunction with InP nanowire arrays as well as a contact layer. TCOs exhibit metal-like conductivity but with a high degree of transparency. Simulations to compute the optimum nanowire dimensions to obtain lasing, such as diameter and length, are carried out. For n-type TCOs, ZnO and SnOx are explored as potential materials, while for p-type TCOs, SnxNiyOz is evaluated. Optical, compositional and electrical properties of the TCOs are investigated at various deposition conditions. Junction properties as well as band alignment at the TCO-InP interface are studied to have insight of carrier injection and transport across the heterojunction. Finally, electroluminescence characteristics are measured and all the devices show promising LED behaviour, but fail to lase due to excessive heating at higher current injection levels due to carrier crowding, non-uniformity and shifting of the recombination region. For ease of integration, cost effectiveness and minimising the shift of the recombination region, flexible nanowire array devices are also demonstrated, which provide LED characteristics, but still fail to lase. Potential reasons and steps towards improvement such as modified fabrication process to mitigate recombination inside the substrate, proper heat sinking and incorporation of quantum wells and quantum dots to enhance gain are investigated. Nevertheless, the knowledge and understanding gained from devices fabricated in this thesis are promising steps towards realising electrically injected III-V semiconductor nanowire lasers

Polyhedral geometry and combinatorics of an autocatalytic ecosystem

Developing a mathematical understanding of autocatalysis in reaction networks has both theoretical and practical implications. We review definitions of autocatalytic networks and prove some properties for minimal autocatalytic subnetworks (MASs). We show that it is possible to classify MASs in equivalence classes, and develop mathematical results about their behavior. We also provide linear-programming algorithms to exhaustively enumerate them and a scheme to visualize their polyhedral geometry and combinatorics. We then define cluster chemical reaction networks, a framework for coarse-graining real chemical reactions with positive integer conservation laws. We find that the size of the list of minimal autocatalytic subnetworks in a maximally connected cluster chemical reaction network with one conservation law grows exponentially in the number of species. We end our discussion with open questions concerning an ecosystem of autocatalytic subnetworks and multidisciplinary opportunities for future investigation.Comment: 36 pages, 17 figures, 7 table