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.

Hypothesis-driven experiments

Null and alternative hypotheses

The basic outlines of hypothesis-driven research were provided in Section 1.3.1, but when setting out to create an experiment to test a hypothesis, there is much more to consider.

Your experiment should have at least one control or independent variable, and at least one response or dependent variable. The independent variable is something you are sure will change during the course of experimental measurements. The dependent variable is what you will measure, although you may not be sure if it will change. Indeed the point of the experiment may be to determine if it does or does not change. In laboratory experiments it is common to call the independent variable a control variable because you decide upon its value and control it. For example, in an experiment concerning the effect of fertilizer doses on size of tomatoes, you control the concentration of fertilizer given to each plant. In observational research, the independent variable may be something you cannot control. For example, you might conduct a study of tomatoes grown outside and investigate whether they grow faster on hot days than on cool days. In this case, nature will bring you the hot and cold days; you cannot control when they will happen, but you can still study their effects. Experiments are usually better when the investigator controls the independent variable, but this is not always possible.

Introduction

The most annoying problems that are encountered during the use of electronic systems are often intermittent in nature. Electromagnetic interference, corona and arcing in high-voltage circuits, and some other causes of potentially intermittent faults, are covered in this chapter. Other reliability problems in electronic hardware, such the failure of high-power equipment, are also discussed below.

Difficulties involving electrical contacts, connectors and cables are very common in electronic work, and frequently intermittent. These are dealt with in Chapter 12. Problems with mains power disturbances and overheating of equipment are considered in Sections 3.6 and 3.4.1, respectively. Vibrations can be a cause of noise in electronic systems in the form of microphonics, and these are discussed in Section 3.5.3. A general survey of some of the causes of intermittent failures in experimental work is presented in Section 3.3.

With some exceptions, details of the design, construction, and troubleshooting of electronic circuits and equipment items (e.g. self-contained electronic instruments) are not discussed in this chapter. Such information is provided in references listed in the “Further reading” section on page 403.

Electromagnetic interference

Grounding and ground loops

General points

Importance of grounding arrangements

Proper grounding is generally necessary for the reliability of electronic systems. This includes both the topology of ground networks, and the quality of the electrical contacts made between ground conductors. (The latter is discussed in Section 12.2.4.) Failure to provide an adequate ground arrangement often leads to erratic noise problems and system malfunctions.

Introduction

Among the most common problems encountered when dealing with electronic items are those caused by faulty contacts, connectors, wires and cables. For example, bad solder joints are among the most frequent causes of failures in electronic devices in general. In any given electronic system, the most unreliable components will be its connectors and cables. (Perhaps somewhat surprisingly, the number of connector problems that occurred during the Apollo space program was greater than that of all other problems combined.) More specifically, faulty connectors and cables are an important source of trouble in experimental work, in the form of unreliable measurements and wasted time.

Information on this subject is widely dispersed. The purpose of the following chapter is to bring together knowledge about these mundane but troublesome items from some of the various scattered sources.

Some contact, connector, wire, and cable problems are:

1. (a) excessively high or low resistances, or complete open or short circuits,

2. (b) noisy contacts (perhaps in the form of 1/f noise),

3. (c) production and reception of electromagnetic interference (including crosstalk between nearby wires and cables),

4. (d) leakage currents over insulators in high-impedance circuits,

5. (e) noise produced as a result of cable vibrations in high-impedance circuits (“microphonics”),

6. (f) nonlinear I–V characteristics of poor-quality contacts (i.e. they can become non-ohmic – this may lead to interference with sensitive measurements due to rectification of electromagnetic noise in the environment, and also signal distortion),

7. (g) reflections on cables at radio (and particularly microwave) frequencies, owing to impedance mismatches,

8. (h) arcing and corona discharge at contacts, and in cables and connectors, at high voltages,

9. […]

Ingredients of mathematical modeling

Applied mathematics refers to the portion of mathematics most often useful in representing the physical world. Many scientists spend most of their time working on the theories of physics, chemistry, geology, astronomy, or biology; most of their time is spent with mathematics as well. Statistical analysis is a portion of that analysis, but there is much more as well. Another name for this sort of activity is mathematical modeling.

Numbers

The most useful idea in mathematical modeling is also the oldest and most basic. It is the idea of using numbers to represent the world. There are many different sorts of numbers, and it may be useful to review them before proceeding to more complicated operations.

Positive integers These represent increasing quantities of essentially identical objects. The basic idea is simple yet abstract, that one object can be essentially the same as another, while not literally being identical. Even two symbols, X, X, are different if only because they are in different places, but everyone seems to understand what it means to say that they are the same, and there are two of them.

Zero This is a number representing the absence of some quantity. It makes it possible to state that something is not present.

Negative integers These numbers seem to have been conceived in India during our Medieval period, along with the Arabic numerals (brought by Arabs from India to Europe).