66 research outputs found
Errata and Addenda to Mathematical Constants
We humbly and briefly offer corrections and supplements to Mathematical
Constants (2003) and Mathematical Constants II (2019), both published by
Cambridge University Press. Comments are always welcome.Comment: 162 page
Second International Workshop on Harmonic Oscillators
The Second International Workshop on Harmonic Oscillators was held at the Hotel Hacienda Cocoyoc from March 23 to 25, 1994. The Workshop gathered 67 participants; there were 10 invited lecturers, 30 plenary oral presentations, 15 posters, and plenty of discussion divided into the five sessions of this volume. The Organizing Committee was asked by the chairman of several Mexican funding agencies what exactly was meant by harmonic oscillators, and for what purpose the new research could be useful. Harmonic oscillators - as we explained - is a code name for a family of mathematical models based on the theory of Lie algebras and groups, with applications in a growing range of physical theories and technologies: molecular, atomic, nuclear and particle physics; quantum optics and communication theory
Study of the best linear approximation of nonlinear systems with arbitrary inputs
System identification is the art of modelling of a process (physical, biological,
etc.) or to predict its behaviour or output when the environment condition
or parameter changes. One is modelling the input-output relationship of a system,
for example, linking temperature of a greenhouse (output) to the sunlight intensity
(input), power of a car engine (output) with fuel injection rate (input). In linear
systems, changing an input parameter will result in a proportional increase in the
system output. This is not the case in a nonlinear system. Linear system identification
has been extensively studied, more so than nonlinear system identification.
Since most systems are nonlinear to some extent, there is significant interest in this
topic as industrial processes become more and more complex.
In a linear dynamical system, knowing the impulse response function of a
system will allow one to predict the output given any input. For nonlinear systems
this is not the case. If advanced theory is not available, it is possible to approximate
a nonlinear system by a linear one. One tool is the Best Linear Approximation
(Bla), which is an impulse response function of a linear system that minimises the
output differences between its nonlinear counterparts for a given class of input. The
Bla is often the starting point for modelling a nonlinear system. There is extensive
literature on the Bla obtained from input signals with a Gaussian probability
density function (p.d.f.), but there has been very little for other kinds of inputs.
A Bla estimated from Gaussian inputs is useful in decoupling the linear dynamics
from the nonlinearity, and in initialisation of parameterised models. As Gaussian
inputs are not always practical to be introduced as excitations, it is important to
investigate the dependence of the Bla on the amplitude distribution in more detail.
This thesis studies the behaviour of the Bla with regards to other types of signals,
and in particular, binary sequences where a signal takes only two levels. Such an
input is valuable in many practical situations, for example where the input actuator
is a switch or a valve and hence can only be turned either on or off.
While it is known in the literature that the Bla depends on the amplitude
distribution of the input, as far as the author is aware, there is a lack of comprehensive
theoretical study on this topic. In this thesis, the Blas of discrete-time
time-invariant nonlinear systems are studied theoretically for white inputs with an arbitrary amplitude distribution, including Gaussian and binary sequences. In doing
so, the thesis offers answers to fundamental questions of interest to system engineers,
for example: 1) How the amplitude distribution of the input and the system
dynamics affect the Bla? 2) How does one quantify the difference between the
Bla obtained from a Gaussian input and that obtained from an arbitrary input?
3) Is the difference (if any) negligible? 4) What can be done in terms of experiment
design to minimise such difference?
To answer these questions, the theoretical expressions for the Bla have been
developed for both Wiener-Hammerstein (Wh) systems and the more general Volterra
systems. The theory for the Wh case has been verified by simulation and physical
experiments in Chapter 3 and Chapter 6 respectively. It is shown in Chapter 3
that the difference between the Gaussian and non-Gaussian Bla’s depends on the
system memory as well as the higher order moments of the non-Gaussian input.
To quantify this difference, a measure called the Discrepancy Factor—a measure of
relative error, was developed. It has been shown that when the system memory is
short, the discrepancy can be as high as 44.4%, which is not negligible. This justifies
the need for a method to decrease such discrepancy. One method is to design a random
multilevel sequence for Gaussianity with respect to its higher order moments,
and this is discussed in Chapter 5.
When estimating the Bla even in the absence of environment and measurement
noise, the nonlinearity inevitably introduces nonlinear distortions—deviations
from the Bla specific to the realisation of input used. This also explains why more
than one realisation of input and averaging is required to obtain a good estimate of
the Bla. It is observed that with a specific class of pseudorandom binary sequence
(Prbs), called the maximum length binary sequence (Mlbs or the m-sequence), the
nonlinear distortions appear structured in the time domain. Chapter 4 illustrates
a simple and computationally inexpensive method to take advantage this structure
to obtain better estimates of the Bla—by replacing mean averaging by median
averaging.
Lastly, Chapters 7 and 8 document two independent benchmark studies separate
from the main theoretical work of the thesis. The benchmark in Chapter 7 is
concerned with the modelling of an electrical Wh system proposed in a special session
of the 15th International Federation of Automatic Control (Ifac) Symposium on
System Identification (Sysid) 2009 (Schoukens, Suykens & Ljung, 2009). Chapter 8
is concerned with the modelling of a ‘hyperfast’ Peltier cooling system first proposed
in the U.K. Automatic Control Council (Ukacc) International Conference
on Control, 2010 (Control 2010)
Optimal information storage : nonsequential sources and neural channels
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.MIT Institute Archives copy: pages 101-163 bound in reverse order.Includes bibliographical references (p. 141-163).Information storage and retrieval systems are communication systems from the present to the future and fall naturally into the framework of information theory. The goal of information storage is to preserve as much signal fidelity under resource constraints as possible. The information storage theorem delineates average fidelity and average resource values that are achievable and those that are not. Moreover, observable properties of optimal information storage systems and the robustness of optimal systems to parameter mismatch may be determined. In this thesis, we study the physical properties of a neural information storage channel and also the fundamental bounds on the storage of sources that have nonsequential semantics. Experimental investigations have revealed that synapses in the mammalian brain possess unexpected properties. Adopting the optimization approach to biology, we cast the brain as an optimal information storage system and propose a theoretical framework that accounts for many of these physical properties. Based on previous experimental and theoretical work, we use volume as a limited resource and utilize the empirical relationship between volume anrid synaptic weight.(cont.) Our scientific hypotheses are based on maximizing information storage capacity per unit cost. We use properties of the capacity-cost function, e-capacity cost approximations, and measure matching to develop optimization principles. We find that capacity-achieving input distributions not only explain existing experimental measurements but also make non-trivial predictions about the physical structure of the brain. Numerous information storage applications have semantics such that the order of source elements is irrelevant, so the source sequence can be treated as a multiset. We formulate fidelity criteria that consider asymptotically large multisets and give conclusive, but trivialized, results in rate distortion theory. For fidelity criteria that consider fixed-size multisets. we give some conclusive results in high-rate quantization theory, low-rate quantization. and rate distortion theory. We also provide bounds on the rate-distortion function for other nonsequential fidelity criteria problems. System resource consumption can be significantly reduced by recognizing the correct invariance properties and semantics of the information storage task at hand.by Lav R. Varshney.S.M
Bridging Course: Why, How, and First Impressions
The knowledge gap between high school and university level mathematics is a persistent issue that hinders students in their academic career. Freshman Civil Engineering students at the University of Twente, Netherlands struggle with passing entry level Calculus courses. In 2022, the programme introduced a workshop to help students put their prerequisite knowledge to the test; still, many students could not pass these courses. Capitalising on the idea behind this workshop, a fully digital course was introduced in 2023. In this research we dive into the design of the contents of this course. Furthermore, we investigate its impact on student performance with respect to previous years using a qualitative approach: interviews with second year students provide, to this avail, a valuable comparison
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