Generalized Convolution Transforms and Toeplitz Plus Hankel Integral Equations

Mathematics Subject Classification: 44A05, 44A35

An Application of Convolution Integral

MSC 2010: 30C45

Global Results for Solution to the Mass-Critical Schrödinger Equation with Convolution Nonlinearity in R^n

Георги Венков, Христо Генев - Разглеждаме един клас от L^2 - критични нелинейни уравнения на Шрьодингер в R^(1+n) с конволюционна нелинейност от тип Хартри. Целта ни е да установим локалното и глобално съществуване на решенията, както и коректност на задачата на Коши в достатъчно малка околност на нулата в пространството L^2 (R^n). Като естествено следствие на глобалните резултати ние доказваме съществуване на оператор на разсейване за малки начални условия.

On the approximation by convolution operators in homogeneous Banach spaces on R^d

AMS Subject Classification 2010: 41A25, 41A35, 41A40, 41A63, 41A65, 42A38, 42A85, 42B10, 42B20

Number Sequences in an Integral Form with a Generalized Convolution Property and Somos-4 Hankel Determinants

MSC 2010: 11B83, 05A19, 33C45

On the Approximation by Convolution Operators in Homogeneous Banach Spaces of Periodic Functions

AMS Subject Classification 2010: 41A25, 41A27, 41A35, 41A36, 41A40, 42Al6, 42A85.

Weighted Sum of Correlated Lognormals: Convolution Integral Solution

Probability density function (pdf) for sum of n correlated lognormal variables is deducted as a special convolution integral. Pdf for weighted sums (where weights can be any real numbers) is also presented. The result for four dimensions was checked by Monte Carlo simulation.

Orbits of semigroups of truncated convolution operators

We prove that a semigroup generated by finitely many truncated convolution operators on $L_p[0, 1]$ with 1 ≤ p < ∞ is non-supercyclic. On the other hand, there is a truncated convolution operator, which possesses irregular vectors.

Convolution of Orbital Measures on Symmetric Spaces of type Cp and Dp

International audience

Wide-angle visual feature matching for outdoor localization

Wide-angle images exhibit significant distortion for which existing scale-space detectors such as the scale-invariant feature transform (SIFT) are inappropriate. The required scale-space images for feature detection are correctly obtained through the convolution of the image, mapped to the sphere, with the spherical Gaussian. A new visual key-point detector, based on this principle, is developed and several computational approaches to the convolution are investigated in both the spatial and frequency domain. In particular, a close approximation is developed that has comparable computation time to conventional SIFT but with improved matching performance. Results are presented for monocular wide-angle outdoor image sequences obtained using fisheye and equiangular catadioptric cameras. We evaluate the overall matching performance (recall versus 1-precision) of these methods compared to conventional SIFT. We also demonstrate the use of the technique for variable frame-rate visual odometry and its application to place recognition.

The hydrodynamic and capture/retention performance of a gross pollutant trap

This research shows that gross pollutant traps (GPTs) continue to play an important role in preventing visible street waste—gross pollutants—from contaminating the environment. The demand for these GPTs calls for stringent quality control and this research provides a foundation to rigorously examine the devices. A novel and comprehensive testing approach to examine a dry sump GPT was developed. The GPT is designed with internal screens to capture gross pollutants—organic matter and anthropogenic litter. This device has not been previously investigated. Apart from the review of GPTs and gross pollutant data, the testing approach includes four additional aspects to this research, which are: field work and an historical overview of street waste/stormwater pollution, calibration of equipment, hydrodynamic studies and gross pollutant capture/retention investigations. This work is the first comprehensive investigation of its kind and provides valuable practical information for the current research and any future work pertaining to the operations of GPTs and management of street waste in the urban environment. Gross pollutant traps—including patented and registered designs developed by industry—have specific internal configurations and hydrodynamic separation characteristics which demand individual testing and performance assessments. Stormwater devices are usually evaluated by environmental protection agencies (EPAs), professional bodies and water research centres. In the USA, the American Society of Civil Engineers (ASCE) and the Environmental Water Resource Institute (EWRI) are examples of professional and research organisations actively involved in these evaluation/verification programs. These programs largely rely on field evaluations alone that are limited in scope, mainly for cost and logistical reasons. In Australia, evaluation/verification programs of new devices in the stormwater industry are not well established. The current limitations in the evaluation methodologies of GPTs have been addressed in this research by establishing a new testing approach. This approach uses a combination of physical and theoretical models to examine in detail the hydrodynamic and capture/retention characteristics of the GPT. The physical model consisted of a 50% scale model GPT rig with screen blockages varying from 0 to 100%. This rig was placed in a 20 m flume and various inlet and outflow operating conditions were modelled on observations made during the field monitoring of GPTs. Due to infrequent cleaning, the retaining screens inside the GPTs were often observed to be blocked with organic matter. Blocked screens can radically change the hydrodynamic and gross pollutant capture/retention characteristics of a GPT as shown from this research. This research involved the use of equipment, such as acoustic Doppler velocimeters (ADVs) and dye concentration (Komori) probes, which were deployed for the first time in a dry sump GPT. Hence, it was necessary to rigorously evaluate the capability and performance of these devices, particularly in the case of the custom made Komori probes, about which little was known. The evaluation revealed that the Komori probes have a frequency response of up to 100 Hz —which is dependent upon fluid velocities—and this was adequate to measure the relevant fluctuations of dye introduced into the GPT flow domain. The outcome of this evaluation resulted in establishing methodologies for the hydrodynamic measurements and gross pollutant capture/retention experiments. The hydrodynamic measurements consisted of point-based acoustic Doppler velocimeter (ADV) measurements, flow field particle image velocimetry (PIV) capture, head loss experiments and computational fluid dynamics (CFD) simulation. The gross pollutant capture/retention experiments included the use of anthropogenic litter components, tracer dye and custom modified artificial gross pollutants. Anthropogenic litter was limited to tin cans, bottle caps and plastic bags, while the artificial pollutants consisted of 40 mm spheres with a range of four buoyancies. The hydrodynamic results led to the definition of global and local flow features. The gross pollutant capture/retention results showed that when the internal retaining screens are fully blocked, the capture/retention performance of the GPT rapidly deteriorates. The overall results showed that the GPT will operate efficiently until at least 70% of the screens are blocked, particularly at high flow rates. This important finding indicates that cleaning operations could be more effectively planned when the GPT capture/retention performance deteriorates. At lower flow rates, the capture/retention performance trends were reversed. There is little difference in the poor capture/retention performance between a fully blocked GPT and a partially filled or empty GPT with 100% screen blockages. The results also revealed that the GPT is designed with an efficient high flow bypass system to avoid upstream blockages. The capture/retention performance of the GPT at medium to high inlet flow rates is close to maximum efficiency (100%). With regard to the design appraisal of the GPT, a raised inlet offers a better capture/retention performance, particularly at lower flow rates. Further design appraisals of the GPT are recommended.

Developments in texture-based vector field visualisation techniques

Vector field visualisation is one of the classic sub-fields of scientific data visualisation. The need for effective visualisation of flow data arises in many scientific domains ranging from medical sciences to aerodynamics. Though there has been much research on the topic, the question of how to communicate flow information effectively in real, practical situations is still largely an unsolved problem. This is particularly true for complex 3D flows. In this presentation we give a brief introduction and background to vector field visualisation and comment on the effectiveness of the most common solutions. We will then give some examples of current development on texture-based techniques, and given practical examples of their use in CFD research and hydrodynamic applications.

Digital signal processing

Signal Processing (SP) is a subject of central importance in engineering and the applied sciences. Signals are information-bearing functions, and SP deals with the analysis and processing of signals (by dedicated systems) to extract or modify information. Signal processing is necessary because signals normally contain information that is not readily usable or understandable, or which might be disturbed by unwanted sources such as noise. Although many signals are non-electrical, it is common to convert them into electrical signals for processing. Most natural signals (such as acoustic and biomedical signals) are continuous functions of time, with these signals being referred to as analog signals. Prior to the onset of digital computers, Analog Signal Processing (ASP) and analog systems were the only tool to deal with analog signals. Although ASP and analog systems are still widely used, Digital Signal Processing (DSP) and digital systems are attracting more attention, due in large part to the significant advantages of digital systems over the analog counterparts. These advantages include superiority in performance,s peed, reliability, efficiency of storage, size and cost. In addition, DSP can solve problems that cannot be solved using ASP, like the spectral analysis of multicomonent signals, adaptive filtering, and operations at very low frequencies. Following the recent developments in engineering which occurred in the 1980's and 1990's, DSP became one of the world's fastest growing industries. Since that time DSP has not only impacted on traditional areas of electrical engineering, but has had far reaching effects on other domains that deal with information such as economics, meteorology, seismology, bioengineering, oceanology, communications, astronomy, radar engineering, control engineering and various other applications. This book is based on the Lecture Notes of Associate Professor Zahir M. Hussain at RMIT University (Melbourne, 2001-2009), the research of Dr. Amin Z. Sadik (at QUT & RMIT, 2005-2008), and the Note of Professor Peter O'Shea at Queensland University of Technology. Part I of the book addresses the representation of analog and digital signals and systems in the time domain and in the frequency domain. The core topics covered are convolution, transforms (Fourier, Laplace, Z. Discrete-time Fourier, and Discrete Fourier), filters, and random signal analysis. There is also a treatment of some important applications of DSP, including signal detection in noise, radar range estimation, banking and financial applications, and audio effects production. Design and implementation of digital systems (such as integrators, differentiators, resonators and oscillators are also considered, along with the design of conventional digital filters. Part I is suitable for an elementary course in DSP. Part II (which is suitable for an advanced signal processing course), considers selected signal processing systems and techniques. Core topics covered are the Hilbert transformer, binary signal transmission, phase-locked loops, sigma-delta modulation, noise shaping, quantization, adaptive filters, and non-stationary signal analysis. Part III presents some selected advanced DSP topics. We hope that this book will contribute to the advancement of engineering education and that it will serve as a general reference book on digital signal processing.