Showing a Fourier transform to a physics student generally produces the same reaction as showing a crucifix to Count Dracula. This may be because the subject tends to be taught by theorists who themselves use Fourier methods to solve otherwise intractable differential equations. The result is often a heavy load of mathematical analysis.

This need not be so. Engineers and practical physicists use Fourier theory in quite another way: to treat experimental data, to extract information from noisy signals, to design electrical filters, to ‘clean’ TV pictures and for many similar practical tasks. The transforms are done digitally and there is a minimum of mathematics involved.

The chief tools of the trade are the theorems in Chapter 2, and an easy familiarity with these is the way to mastery of the subject. In spite of the forest of integration signs throughout the book there is in fact very little integration done and most of that is at high-school level. There are one or two excursions in places to show the breadth of power that the method can give. These are not pursued to any length but are intended to whet the appetite of those who want to follow more theoretical paths.

The book is deliberately incomplete. Many topics are missing and there is no attempt to explain everything: but I have left, here and there, what I hope are tempting clues to stimulate the reader into looking further; and of course, there is a bibliography at the end.

Undergraduate mathematics syllabi vary considerably in their coverage of measure-theoretic probability theory, so beginning graduates often find substantial gaps in their background when attending modules in advanced analysis, stochastic processes and applications. This text seeks to fill some of these gaps concisely. The exercises form an integral part of the text. The material arose from my experience of teaching AIMS students between 2004 and 2007, of which I retain many fond memories. The AIMS series format allows few explorations of byways; and the objective of arriving at a reasonably honest but concise account of the Itô integral decided most of the material. With motivation from elementary probability we discuss measures and integrals, leading via L2-theory and conditional expectation to discrete martingales and an outline proof of the Radon–Nikodym Theorem. The last two chapters introduce Brownian Motion and Itô integrals, with a brief look at martingale calculus. Here proofs of several key results are only sketched briefly or omitted. The Black–Scholes option pricing model provides the main application. None of the results presented are new; any remaining errors are mine.

Three happy, lively AIMS student cohorts suffered my attempts to introduce them to mathematical finance. My sincere thanks go to them, to the AIMS team for their support, to Alan Beardon for his encouragement, to two helpful reviewers, to Marek Capinski for wise advice and above all to my wife Margaret for her patient, loving support.

Introduction

The main purpose of this chapter is to examine certain trouble-prone aspects of experimental work that are often not considered, but if neglected can have major consequences. These include errors caused by the subconscious biases of experimenters, which are often the origin of mistaken conclusions in scientific research. In those areas of research that involve the study of material samples, very serious errors are commonly made because the true compositions of the samples are different from what is believed. This issue is also discussed below. The chapter also looks at the problems that can arise in reproducing the experimental measurements and techniques of other researchers. Several of the points are illustrated by historical examples. Some potential pitfalls in the analysis of data are discussed here, and also in Section 2.2.7.

Knowing apparatus and software

It is important to understand (at least at the conceptual level, if not in all the details) the inner workings of one's apparatus and software, and the theories they are based on. The effort needed to acquire this understanding is often avoided by experimenters. This tendency is encouraged by the ubiquitous presence in laboratories of highly automated commercial instruments, which are easy to treat as black boxes.

Having a conceptual understanding of apparatus and software can help to reduce human errors that might otherwise occur during their use (see page 17). Also, without such an understanding, the limitations and eccentricities of these things may not be readily perceived.

Most scientists who spend a significant amount of time in the laboratory are only too well aware of the amount of lost time, wasted resources, and diminished morale that result from unexpected problems that inevitably arise in research. These reliability problems include things such as sudden leaks in vacuum systems, vibrations in sensitive optics, and bugs in computer software. The purpose of this book is to help those working in the physical sciences and engineering to:

• identify potential sources of unexpected problems in their work,

• reduce the likelihood of such problems, and

• detect and eliminate them if they occur.

Most of the problems discussed herein concern technical matters, as in the above examples. However, a significant part of the book is devoted to human errors and biases, and other similar issues.

In modern research it is common practice to employ a variety of different experimental methods, often in combination. Some – such as electronics, computing, vacuum, and optics – can be considered “core techniques,” which are widely used in many areas in the physical sciences and engineering. These are a major focus of this book. There are numerous specialized techniques used in particular research fields that can be sources of problems, but which cannot be included in a work of this size.

If one aims to cover a large range of subjects in a single volume, the depth at which they can be treated is necessarily limited.

Introduction

Aside from leaks, which are discussed in Chapters 6 and 8, and outgassing, which is discussed in Chapter 6, other kinds of problem often afflict vacuum apparatus. For instance, vacuum pumps are frequently mechanical devices, and can suffer from the reliability difficulties that are common to machines in general. Contamination is often a concern in vacuum systems, and the pumps themselves are frequently sources of contaminants. These often take the form of pump oils and their vapors, and various types of particulate matter. Besides these general difficulties, there are numerous others that are peculiar to each particular type of pump.

Vacuum gauges are generally affected by contamination, and are often degraded by substances released from the pumps. The vibrations produced by some vacuum pumps can, in addition to increasing the likelihood of leak problems, also damage delicate structures inside certain vacuum gauges. (These points are also relevant for mass spectrometers, which are common items in vacuum equipment.)

These issues, and others pertaining to the use of vacuum equipment, are discussed in the following chapter.

Vacuum pump matters

Primary pumps

General issues concerning mechanical primary pumps

Perhaps the most serious problem pertaining to the use of mechanical vacuum pumps concerns, not difficulties with the pumps themselves, but those that they may cause to the rest of the system in the form of contamination – especially by pump oil.

Introduction

Lack of reliability in cryogenic apparatus is an especially troublesome problem. This is partly because of the inherent susceptibility of such apparatus to breakdowns. The difficulties are magnified by the inaccessibility of the low-temperature components while they are in their working environment, which makes troubleshooting difficult, and immediate repair of faults virtually impossible. Because of the long times that are often needed to cool the equipment down and warm it up again, the correction and detection of defective items can be a very tedious process. Some faults that manifest themselves at low temperatures may vanish when the apparatus is warmed to room temperature. Frequently, it is possible to identify and locate these only through the use of laborious cross-checks. The high cost of liquid cryogens, and especially liquid helium, can lead to large expenditures when low-temperature faults appear. For these reasons, particular care is needed in the design, construction, and use of cryogenic equipment to prevent failures from occurring.

Reliability problems in cryogenic apparatus are primarily the result of the following effects.

1. (a) Leaks can occur into evacuated spaces used to provide thermal insulation. Even if such leaks are of negligible size at room temperature, they may become very large at low temperatures, owing to the special properties of gases and liquids under such conditions.

2. (b) Mechanical stresses are produced in the various parts during thermal cycling. These stresses are caused by differences in thermal expansion coefficients and temperature gradients. They can cause tubes to bend, wires and solder joints to break, leaks to form, and moving parts to seize. Fatigue failure (see Section 3.11) is potentially a serious issue. […]

Motivations for statistics

Statistics is the mathematics of measurement and data. The data might come from measuring the height of a friend, the sizes of fish in a lake, flipping coins, the rate electrons are ejected from a metal illuminated by light, or mathematics test scores of students in Iowa. The tools of statistics are needed whenever the data are partly predictable and partly influenced by factors too complex fully to understand. Statistics makes it possible to summarize the meaning of huge numbers of measurements in a compact form, and to explain the significance of the measurements even when chance makes them uncertain. Two of the main reasons to learn statistics are to deal with measurement error and to be able to describe distributed quantities.

Measurement error

Chapter 2 discussed the fact that almost any attempt to measure values obtains numbers that vary from one trial to the next. It provided recipes for dealing with the variation but no explanation for why the recipes work. This chapter returns to the problem and explains the solutions in more detail.

Suppose you are trying to measure the height of your friend Anne with a ruler. You make the measurement a first time and find she is 1.721 m tall. When you repeat the measurement, the meter stick tilts at a slight angle, and you get 1.722 m. You try again, and she slouches slightly, so you get 1.718 m.

Safety contract

This is an overview of ways to reduce the risk of an accident and to minimize the impact of one if it does occur. The most important part of laboratory safety is the prevention of accidents. However, it is also important to understand how to respond to the various kinds of accidents that can occur in a laboratory setting. You are responsible for reviewing the following rules and following them. The most common laboratory safety equipment is listed below and you are responsible for knowing its location in your laboratory and knowing when and how to use it.

Proper attire Proper laboratory attire includes long pants, close-toed shoes, and shirts that cover the full torso. If you work with any chemicals in the laboratory, you must arrive wearing the appropriate clothing. If not, you may be asked to change clothing before engaging in your experiment, especially if you are using chemicals. Always stand – do not sit – when working with chemicals since in the event of a spill, ability to move out of the way is critical.

Cleanliness and safeness of work area You will always be expected to maintain a clean and uncluttered work area. Be sure to clean and return all materials to the appropriate storage space before leaving the lab. If left unclean and/or cluttered, you will be required to return to the lab on your own time and clean up what you left behind.

Introduction

In certain types of optical measurements, such as those making use of interference, the most elusive sources of reliability difficulties are structural vibrations, acoustic disturbances, and temperature variations in the optical path. The first two phenomena are often a problem in many types of sensitive measurement in research, and these issues are discussed in detail in Section 3.5.3. Temperature effects are examined in the following section.

Precision positioning devices in optical systems can cause difficulties due to mechanical instabilities and other problems. These are discussed in Section 8.2.3.

The most general problem in optical work – affecting all apparatus to a greater or lesser degree – is contamination of optical surfaces. At one end of the scale, the problem may be merely scattering of light by dust (a serious issue in the case of, for example, diffraction gratings), which can often be solved by protecting and/or cleaning the surfaces using elementary methods. At the other extreme, when high-powered lasers are involved, contamination of optical surfaces may result in damage, or even destruction, of optical components. The process of cleaning optical surfaces may itself be difficult, depending on the character of the devices involved and the nature of the contaminants. If done frequently, cleaning can easily be more detrimental to the performance of the optics than the presence of contamination. These issues are discussed in more detail in Section 10.6.

Introduction

Leaks in vacuum systems are an age-old and common problem in scientific research. Although they are nuisances in many situations, leaks are generally most troublesome when they appear in ultrahigh vacuum (UHV) systems and cryogenic apparatus. In both cases, even minuscule leaks, which are difficult to detect and can be intermittent, may be sufficient to disable the equipment. The thermal stresses on both types of equipment can cause cracks to form in the vacuum envelopes and joints that can lead to leaks. Turn-around periods for these systems can be very long. In the case of UHV devices, bakeout of the vacuum chambers is usually required, which may take a day or more. In order to use cryogenic apparatus, one must usually cool down and warm up the equipment very slowly. Time scales for this range from about an hour to a day or more, depending on the type of equipment. Hence, the leak testing of these systems can be very time-consuming and nerve-wracking.

In the case of cryogenic systems in particular, the relative fragility of the equipment, and the presence of large thermal and mechanical stresses during the routine operation and handling of these devices, promotes the formation of leaks. In these systems, leaks may occur only when the apparatus is cold, and therefore often inaccessible to direct investigation.