Search results for: J. Littlewood

Authors: Wedad Albalawi

Abstract:

The mathematical inequalities have been the core of mathematical study and used in almost all branches of mathematics as well in various areas of science and engineering. The inequalities by Hardy, Littlewood and Polya were the first significant composition of several science. This work presents fundamental ideas, results and techniques and it has had much influence on research in various branches of analysis. Since 1934, various inequalities have been produced and studied in the literature. Furthermore, some inequalities have been formulated by some operators; in 1989, weighted Hardy inequalities have been obtained for integration operators. Then, they obtained weighted estimates for Steklov operators that were used in the solution of the Cauchy problem for the wave equation. They were improved upon in 2011 to include the boundedness of integral operators from the weighted Sobolev space to the weighted Lebesgue space. Some inequalities have been demonstrated and improved using the Hardy–Steklov operator. Recently, a lot of integral inequalities have been improved by differential operators. Hardy inequality has been one of the tools that is used to consider integrity solutions of differential equations. Then dynamic inequalities of Hardy and Coposon have been extended and improved by various integral operators. These inequalities would be interesting to apply in different fields of mathematics (functional spaces, partial differential equations, mathematical modeling). Some inequalities have been appeared involving Copson and Hardy inequalities on time scales to obtain new special version of them. A time scale is defined as a closed subset contains real numbers. Then the inequalities of time scales version have received a lot of attention and has had a major field in both pure and applied mathematics. There are many applications of dynamic equations on time scales to quantum mechanics, electrical engineering, neural networks, heat transfer, combinatorics, and population dynamics. This study focuses on double integrals to obtain new time-scale inequalities of Copson driven by Steklov operator. They will be applied in the solution of the Cauchy problem for the wave equation. The proof can be done by introducing restriction on the operator in several cases. In addition, the obtained inequalities done by using some concepts in time scale version such as time scales calculus, theorem of Fubini and the inequality of H¨older.

Keywords: time scales, inequality of Hardy, inequality of Coposon, Steklov operator

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Authors: Wedad Albalawi

Abstract:

The mathematical inequalities have been the core of mathematical study and used in almost all branches of mathematics as well in various areas of science and engineering. The inequalities by Hardy, Littlewood and Polya were the first significant composition of several science. This work presents fundamental ideas, results and techniques, and it has had much influence on research in various branches of analysis. Since 1934, various inequalities have been produced and studied in the literature. Furthermore, some inequalities have been formulated by some operators; in 1989, weighted Hardy inequalities have been obtained for integration operators. Then, they obtained weighted estimates for Steklov operators that were used in the solution of the Cauchy problem for the wave equation. They were improved upon in 2011 to include the boundedness of integral operators from the weighted Sobolev space to the weighted Lebesgue space. Some inequalities have been demonstrated and improved using the Hardy–Steklov operator. Recently, a lot of integral inequalities have been improved by differential operators. Hardy inequality has been one of the tools that is used to consider integrity solutions of differential equations. Then, dynamic inequalities of Hardy and Coposon have been extended and improved by various integral operators. These inequalities would be interesting to apply in different fields of mathematics (functional spaces, partial differential equations, mathematical modeling). Some inequalities have been appeared involving Copson and Hardy inequalities on time scales to obtain new special version of them. A time scale is an arbitrary nonempty closed subset of the real numbers. Then, the dynamic inequalities on time scales have received a lot of attention in the literature and has become a major field in pure and applied mathematics. There are many applications of dynamic equations on time scales to quantum mechanics, electrical engineering, neural networks, heat transfer, combinatorics, and population dynamics. This study focuses on Hardy and Coposon inequalities, using Steklov operator on time scale in double integrals to obtain special cases of time-scale inequalities of Hardy and Copson on high dimensions. The advantage of this study is that it uses the one-dimensional classical Hardy inequality to obtain higher dimensional on time scale versions that will be applied in the solution of the Cauchy problem for the wave equation. In addition, the obtained inequalities have various applications involving discontinuous domains such as bug populations, phytoremediation of metals, wound healing, maximization problems. The proof can be done by introducing restriction on the operator in several cases. The concepts in time scale version such as time scales calculus will be used that allows to unify and extend many problems from the theories of differential and of difference equations. In addition, using chain rule, and some properties of multiple integrals on time scales, some theorems of Fubini and the inequality of H¨older.

Keywords: time scales, inequality of hardy, inequality of coposon, steklov operator

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Authors: Olivier Delage, Thierry Portafaix, Hassan Bencherif, Guillaume Guimbretiere

Abstract:

Most observation data sequences in geophysics can be interpreted as resulting from the interaction of several physical processes at several time and space scales. As a consequence, measurements time series in geophysics have often characteristics of non-linearity and non-stationarity and thereby exhibit strong fluctuations at all time-scales and require a time-frequency representation to analyze their variability. Empirical Mode Decomposition (EMD) is a relatively new technic as part of a more general signal processing method called the Hilbert-Huang transform. This analysis method turns out to be particularly suitable for non-linear and non-stationary signals and consists in decomposing a signal in an auto adaptive way into a sum of oscillating components named IMFs (Intrinsic Mode Functions), and thereby acts as a bank of bandpass filters. The advantages of the EMD technic are to be entirely data driven and to provide the principal variability modes of the dynamics represented by the original time series. However, the main limiting factor is the frequency resolution that may give rise to the mode mixing phenomenon where the spectral contents of some IMFs overlap each other. To overcome this problem, J. Gilles proposed an alternative entitled “Empirical Wavelet Transform” (EWT) which consists in building from the segmentation of the original signal Fourier spectrum, a bank of filters. The method used is based on the idea utilized in the construction of both Littlewood-Paley and Meyer’s wavelets. The heart of the method lies in the segmentation of the Fourier spectrum based on the local maxima detection in order to obtain a set of non-overlapping segments. Because linked to the Fourier spectrum, the frequency resolution provided by EWT is higher than that provided by EMD and therefore allows to overcome the mode-mixing problem. On the other hand, if the EWT technique is able to detect the frequencies involved in the original time series fluctuations, EWT does not allow to associate the detected frequencies to a specific mode of variability as in the EMD technic. Because EMD is closer to the observation of physical phenomena than EWT, we propose here a new technic called EAWD (Empirical Adaptive Wavelet Decomposition) based on the coupling of the EMD and EWT technics by using the IMFs density spectral content to optimize the segmentation of the Fourier spectrum required by EWT. In this study, EMD and EWT technics are described, then EAWD technic is presented. Comparison of results obtained respectively by EMD, EWT and EAWD technics on time series of ozone total columns recorded at Reunion island over [1978-2019] period is discussed. This study was carried out as part of the SOLSTYCE project dedicated to the characterization and modeling of the underlying dynamics of time series issued from complex systems in atmospheric sciences

Keywords: adaptive filtering, empirical mode decomposition, empirical wavelet transform, filter banks, mode-mixing, non-linear and non-stationary time series, wavelet

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Authors: Matthew Edmunds, Andrew Hunter, Clare Littlewood, Wisha Gul, Gabriel Heppenstall-Harris, Thomas Humphries

Abstract:

Background: Revision courses for medical students undertaking their final examinations are commonplace throughout the UK. Traditionally these take the form of a series of lectures over multiple weeks or a single day of intensive lectures. The COVID-19 pandemic, however, has required medical educators to create new teaching formats to ensure they adhere to social distancing requirements. It has provided an unexpected opportunity to accelerate the development of students proficiency in the use of ‘technology-enabled communication platforms’, as mandated in the 2018 GMC Outcomes of Graduates. Recent advances in technology have made distance learning possible, whilst also providing novel and more engaging learning opportunities for students. Foundation Year 2 doctors at Aintree University Hospital developed an online series of videos to help prepare medical students in the North West and byond for their final medical school examinations. Method: Eight hour-long videos covering the key topics in medicine and surgery were posted on the Peer Learning Liverpool Youtube channel. These videos were created using new technology such as the screen and audio recording platform, Loom. Each video compromised at least 20 single best answer (SBA) questions, in keeping with the format in most medical school finals. Explanations of the answers were provided, and additional important material was covered. Students were able to ask questions by commenting on the videos, with the authors replying as soon as possible. Feedback was collated using an online Google form. Results: An average of 327 people viewed each video, with 113 students filling in the feedback form. 65.5% of respondents were within one month of their final medical school examinations. The average rating for how well prepared the students felt for their finals was 6.21/10 prior to the course and 8.01/10 after the course. A paired t-test demonstrated a mean increase of 1.80 (95% CI 1.66-1.93). Overall, 98.2% said the online format worked well or very well, and 99.1% would recommend the course to a peer. Conclusions: Based on the feedback received, the online revision course was successful both in terms of preparing students for their final examinations, and with regards to how well the online format worked. Free-text qualitative feedback highlighted advantages such as; students could learn at their own pace, revisit key concepts important to them, and practice exam style questions via the case-based format. Limitations identified included inconsistent audiovisual quality, and requests for a live online Q&A session following the conclusion of the course. This course will be relaunched later in the year with increased opportunities for students to access live feedback. The success of this online course has shown the roll that technology can play in medical education. As well as providing novel teaching modes, online learning allows students to access resources that otherwise would not be available locally, and ensure that they do not miss out on teaching that was previously provided face to face, in the current climate of social distancing.

Keywords: COVID-19 pandemic, Medical School, Online learning, Revision course

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Authors: B. Philip, J. Littlewood, R. Radford, N. Evans, T. Whyman, D. P. Jones

Abstract:

UK buildings and energy infrastructures are heavily dependent on natural gas, a large proportion of which is used for domestic space heating. However, approximately half of the gas consumed in the UK is imported. Improving energy security and reducing carbon emissions are major government drivers for reducing gas dependency. In order to do so there needs to be a wholesale shift in the energy provision to householders without impacting on thermal comfort levels, convenience or cost of supply to the end user. Heat pumps are seen as a potential alternative in modern well insulated homes, however, can the same be said of older homes? A large proportion of housing stock in Britain was built prior to 1919. The age of the buildings bears testimony to the quality of construction; however, their thermal performance falls far below the minimum currently set by UK building standards. In recent years significant sums of money have been invested to improve energy efficiency and combat fuel poverty in some of the most deprived areas of Wales. Increasing energy efficiency of older properties remains a significant challenge, which cannot be achieved through insulation and air-tightness interventions alone, particularly when alterations to historically important architectural features of the building are not permitted. This paper investigates the energy demand of pre-1919 dwellings in a former Welsh mining village, the feasibility of meeting that demand using water from the disused mine workings to supply a district heat network and potential barriers to success of the scheme. The use of renewable solar energy generation and storage technologies, both thermal and electrical, to reduce the load and offset increased electricity demand, are considered. A wholistic surveying approach to provide a more accurate assessment of total household heat demand is proposed. Several surveying techniques, including condition surveys, air permeability, heat loss calculations, and thermography were employed to provide a clear picture of energy demand. Additional insulation can bring unforeseen consequences which are detrimental to the fabric of the building, potentially leading to accelerated dilapidation of the asset being ‘protected’. Increasing ventilation should be considered in parallel, to compensate for the associated reduction in uncontrolled infiltration. The effectiveness of thermal performance improvements are demonstrated and the detrimental effects of incorrect material choice and poor installation are highlighted. The findings show estimated heat demand to be in close correlation to household energy bills. Major areas of heat loss were identified such that improvements to building thermal performance could be targeted. The findings demonstrate that the use of heat pumps in older buildings is viable, provided sufficient improvement to thermal performance is possible. Addition of passive solar thermal and photovoltaic generation can help reduce the load and running cost for the householder. The results were used to predict future heat demand following energy efficiency improvements, thereby informing the size of heat pumps required.

Keywords: heat demand, heat pump, renewable energy, retrofit

Procedia PDF Downloads 69