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38520 results in Cambridge Textbooks

Page 1594 of 1926

  • 1 - Vectors and Kinematics

  • Daniel Kleppner, Massachusetts Institute of Technology, Robert Kolenkow
  • Book:
    An Introduction to Mechanics
    Published online:
    28 May 2018
    Print publication:
    18 November 2013, pp 1-46

  • Summary

    Introduction

    Mechanics is at the heart of physics; its concepts are essential for understanding the world around us and phenomena on scales from atomic to cosmic. Concepts such as momentum, angular momentum, and energy play roles in practically every area of physics. The goal of this book is to help you acquire a deep understanding of the principles of mechanics.

    The reason we start by discussing vectors and kinematics rather than plunging into dynamics is that we want to use these tools freely in discussing physical principles. Rather than interrupt the flow of discussion later, we are taking time now to ensure they are on hand when required.

    Vectors

    The topic of vectors provides a natural introduction to the role of mathematics in physics. By using vector notation, physical laws can often be written in compact and simple form. Modern vector notation was invented by a physicist, Willard Gibbs of Yale University, primarily to simplify the appearance of equations. For example, here is how Newton's second law appears in nineteenth century notation:

    Fx = max

    Fy = may

    Fz = maz .

    In vector notation, one simply writes

    F = ma,

    where the bold face symbols F and a stand for vectors.

    Our principal motivation for introducing vectors is to simplify the form of equations. However, as we shall see in Chapter 14, vectors have a much deeper significance. Vectors are closely related to the fundamental ideas of symmetry and their use can lead to valuable insights into the possible forms of unknown laws.

  • 3 - Forces and Equations of Motion

  • Daniel Kleppner, Massachusetts Institute of Technology, Robert Kolenkow
  • Book:
    An Introduction to Mechanics
    Published online:
    28 May 2018
    Print publication:
    18 November 2013, pp 81-114

  • Summary

    Introduction

    The concept of force is central in Newtonian physics. This chapter describes the gravitational force and the electrostatic force, two of the fundamental forces of nature. We also discuss several phenomenological forces, for example friction. Such forces are commonly encountered in “everyday” physics and are approximately described by empirical equations. Because the concept of force is meaningful only if one knows how to solve problems involving forces, this chapter includes many examples in which Newton's laws are put into practice.

    The problem of calculating motion from known forces frequently occurs in physics. For instance, a physicist who sets out to design a particle accelerator employs the laws of mechanics and knowledge of electric and magnetic forces to calculate how the particles will move in the accelerator. Equally important, however, is the converse process of deducing the physical interaction from observations of the motion, which is how new laws are discovered. The classic example is Newton's deduction of the inverse-square law of gravitation from Kepler's laws of planetary motion. A contemporary example is the effort to elucidate the interactions between elementary particles from high energy scattering experiments at the Large Hadron Collider at CERN in Geneva and at other high energy laboratories.

    Unscrambling experimental observations to find the underlying forces can be complicated. In a facetious mood, the British cosmologist Arthur Eddington once said that force is the mathematical expression we put into the left-hand side of Newton's second law to obtain results that agree with observed motions. Fortunately, force has a more concrete physical reality.

  • 12 - The Special Theory of Relativity

  • Daniel Kleppner, Massachusetts Institute of Technology, Robert Kolenkow
  • Book:
    An Introduction to Mechanics
    Published online:
    28 May 2018
    Print publication:
    18 November 2013, pp 439-476

  • Summary

    Introduction

    In the centuries following publication of the Principia, Newtonian dynamics was accepted whole-heartedly not only because of its enormous success in explaining planetary motion but also in accounting for all motions commonly encountered on the Earth. Physicists and mathematicians (often the same people) created elegant reformulations of Newtonian physics and introduced more powerful analytical and calculational techniques, but the foundations of Newtonian physics were assumed to be unassailable. Then, on June 30 1905, Albert Einstein presented his special theory of relativity in his publication The Electrodynamics of Moving Bodies. The English translation, available on the web, is reprinted from Relativity: The Special and General Theory, Albert Einstein, Methuen, London (1920). The original publication is Zur Elektrodynamik bewegter Körper, Annalen der Physik 17 (1905). Einstein's paper transformed our fundamental view of space, time, and measurement.

    The reason that Newtonian dynamics went unchallenged for over two centuries is that although we now realize that it is only an approximation to the laws of motion, the approximation is superb for motion with speed much less than the speed of light, c ≈ 3 × 108 m/s. Relativistic modifications to observations of a body moving with speed v typically involve a factor of v2/c2 . Most familiar phenomena involve speeds v ≪ c. Even for the high speed of an Earth-orbiting satellite, v2/c2 ≈ 10−10. There is one obvious exception to this generalization about speed: light itself.

  • 10 - Central Force Motion

  • Daniel Kleppner, Massachusetts Institute of Technology, Robert Kolenkow
  • Book:
    An Introduction to Mechanics
    Published online:
    28 May 2018
    Print publication:
    18 November 2013, pp 373-410

  • Summary

    Introduction

    Johannes Kepler was the assistant of the sixteenth-century Danish astronomer Tycho Brahe. They had an ideal combination of talents. Brahe had the ingenuity and skill to measure planetary positions to better than

    0.01°, all made by naked eye because the telescope was not invented until a few years after his death. Kepler had the mathematical genius and fortitude to discover that Brahe's measurements could be fitted by three simple empirical laws. The task was formidable. It took Kepler 18 years of laborious calculation to obtain the following three laws of planetary motion, which he stated early in the seventeenth century:

  • Every planet moves in an ellipse with the Sun at one focus.

  • The radius vector from the Sun to a planet sweeps out equal areas in equal times.

  • The period of revolution T of a planet about the Sun is related to the major axis A of the ellipse by T2 = kA3, where k is the same for all the planets.

    • Kepler's empirical laws went unexplained until the latter half of the seventeenth century, when Newton's fascination with the problem of planetary motion inspired him to formulate his laws of motion and the law of universal gravitation. Using these mathematical laws, Newton explained Kepler's empirical laws, giving an overwhelming argument in favor of the new mechanics and marking the beginning of modern mathematical physics. Planetary motion and the more general problem of motion under a central force continue to play an important role in many branches of physics and turn up in such topics as particle scattering, atomic structure, and space navigation.

  • Preface

  • Daniel Kleppner, Massachusetts Institute of Technology, Robert Kolenkow
  • Book:
    An Introduction to Mechanics
    Published online:
    28 May 2018
    Print publication:
    18 November 2013, pp xi-xiv

  • Summary

    An Introduction to Mechanics grew out of a one-semester course at the Massachusetts Institute of Technology—Physics 8.012—intended for students who seek to understand physics more deeply than the usual freshman level. In the four decades since this text was written physics has moved forward on many fronts but mechanics continues to be a bedrock for concepts such as inertia, momentum, and energy; fluency in the physicist's approach to problem-solving—an underlying theme of this book—remains priceless. The positive comments we have received over the years from students, some of whom are now well advanced in their careers, as well as from faculty at M.I.T. and elsewhere, reassures us that the approach of the text is fundamentally sound. We have received many suggestions from colleagues and we have taken this opportunity to incorporate their ideas and to update some of the discussions.

    We assume that our readers know enough elementary calculus to differentiate and integrate simple polynomials and trigonometric functions. We do not assume any familiarity with differential equations. Our experience is that the principal challenge for most students is not with understanding mathematical concepts but in learning how to apply them to physical problems. This comes with practice and there is no substitute for solving challenging problems. Consequently problem-solving takes high priority. We have provided numerous worked examples to help provide guidance. Where possible we try to tie the examples to interesting physical phenomena but we are unapologetic about totally pedagogical problems.

  • 10 - Asymptotic analysis of queues

  • from II - Performance analysis
  • R. Srikant, University of Illinois, Urbana-Champaign, Lei Ying, Arizona State University
  • Book:
    Communication Networks
    Published online:
    30 January 2019
    Print publication:
    14 November 2013, pp 290-322

  • Summary

    Stationary distributions of queueing systems with general arrivals and service-time distributions are difficult to compute. In this chapter, we will develop techniques to understand the behavior of such systems in certain asymptotic regimes. We consider two such regimes: the heavy-traffic regime and the large-deviations regime. The analysis in the heavy-traffic regime can be thought of as the analog of the central limit theorem for random variables, and the analysis of the large-deviations regime can be thought of as the analog of the Chernoff bound. Traditionally, heavy-traffic analysis is performed by scaling arrival and service processes so that they can be approximated by Brownian motions, which are the stochastic process analogs of Gaussian random variables. Here, we take a different approach: we will show that heavy-traffic results can be obtained by extending the ideas behind the discrete-time Kingman bound presented in Chapter 3. The analysis of the large-deviations regime in this chapter refines the probability of overflow estimates obtained using the Chernoff bound, also in Chapter 3.

    Roughly speaking, heavy traffic refers to a regime in which the mean arrival rate to a queueing system is close to the boundary of the capacity region. We will first use a Lyapunov argument to derive bounds on the moments of the scaled queue length ∈q(t) for the discrete-time G/G/1 queue; where ∈ = μ — λ; μ is the service rate and λ is the arrival rate.

  • 4 - Scheduling in packet switches

  • from I - Network architecture and algorithms
  • R. Srikant, University of Illinois, Urbana-Champaign, Lei Ying, Arizona State University
  • Book:
    Communication Networks
    Published online:
    30 January 2019
    Print publication:
    14 November 2013, pp 86-109

  • Summary

    In Chapter 2, we learned about routing algorithms that determine the sequence of links a packet should traverse to get to its destination. But we did not explain how a router actually moves a packet from one link to another. To understand this process, let us first look at the architecture of a router. Generally speaking, a router has four major components: the input and the output ports, which are interfaces connecting the router to input and output links, respectively, a switch fabric, and a routing processor, as shown in Figure 4.1. The routing processor maintains the routing table and makes routing decisions. The switch fabric is the component that moves packets from one link to another link. In this chapter, we will assume that all packets are of equal size. In reality, packets in the Internet have widely variable sizes. In the switch fabric, packets are divided into equal-sized cells and reassembled at the output, hence our assumption holds.

    Earlier, we implicitly assumed that this switch fabric operates infinitely fast, so packets are moved from input ports to output ports immediately. This allowed us to focus on the buffers at output ports. So, all our discussions so far on buffer overflow probabilities are for output queues since an output buffer is the place where packets “enter” a link. For example, WFQ, introduced in Chapter 3, may be implemented at output port buffers to provide isolation among flows. However, in reality, the switch fabric does not really operate at infinite speed.

  • Preface

  • R. Srikant, University of Illinois, Urbana-Champaign, Lei Ying, Arizona State University
  • Book:
    Communication Networks
    Published online:
    30 January 2019
    Print publication:
    14 November 2013, pp xi-xii

  • Summary

    Why we wrote this book

    Traditionally, analytical techniques for communication networks discussed in textbooks fall into two categories: (i) analysis of network protocols, primarily using queueing theoretic tools, and (ii) algorithms for network provisioning which use tools from optimization theory. Since the mid 1990s, a new viewpoint of the architecture of a communication network has emerged. Network architecture and algorithms are now viewed as slow-time-scale, distributed solutions to a large-scale optimization problem. This approach illustrates that the layered architecture of a communication network is a natural by-product of the desire to design a fair and stable system. On the other hand, queueing theory, stochastic processes, and combinatorics play an important role in designing low-complexity and distributed algorithms that are viewed as operating at fast time scales.

    Our goal in writing this book is to present this modern point of view of network protocol design and analysis to a wide audience. The book provides readers with a comprehensive view of the design of communication networks using a combination of tools from optimization theory, control theory, and stochastic networks, and introduces mathematical tools needed to analyze the performance of communication network protocols.

    Organization of the book

    The book has been organized into two major parts. In the first part of the book, with a few exceptions, we present mathematical techniques only as tools to design algorithms implemented at various layers of a communication network. We start with the transport layer, and then consider algorithms at the link layer and the medium access layer, and finally present a unified view of all these layers along with the network layer.

  • 1 - Introduction

  • R. Srikant, University of Illinois, Urbana-Champaign, Lei Ying, Arizona State University
  • Book:
    Communication Networks
    Published online:
    30 January 2019
    Print publication:
    14 November 2013, pp 1-4

  • Summary

    A communication network is an interconnection of devices designed to carry information from various sources to their respective destinations. To execute this task of carrying information, a number of protocols (algorithms) have to be developed to convert the information to bits and transport these bits reliably over the network. The first part of this book deals with the development of mathematical models which will be used to design the protocols used by communication networks. To understand the scope of the book, it is useful first to understand the architecture of a communication network.

    The sources (also called end hosts) that generate information (also called data) first convert the data into bits (0s and 1s) which are then collected into groups called packets. We will not discuss the process of converting data into packets in this book, but simply assume that the data are generated in the form of packets. Let us consider the problem of sending a stream of packets from a source S to destination D, and assume for the moment that there are no other entities (such as other sources or destinations or intermediate nodes) in the network. The source and destination must be connected by some communication medium, such as a coaxial cable, telephone wire, or optical fiber, or they have to communicate in a wireless fashion. In either case, we can imagine that S and D are connected by a communication link, although the link is virtual in the case of wireless communication.

  • 3 - Links: statistical multiplexing and queues

  • from I - Network architecture and algorithms
  • R. Srikant, University of Illinois, Urbana-Champaign, Lei Ying, Arizona State University
  • Book:
    Communication Networks
    Published online:
    30 January 2019
    Print publication:
    14 November 2013, pp 49-85

  • Summary

    In Chapter 2, we assumed that the transmission rates xr are positive, and we derived fair and stable resource allocation algorithms. In reality, since data are transmitted in the form of packets, the rates xr are converted to discrete window sizes, which results in bursty (non-smooth) arrival rates at the links in the network. In addition, many flows in the Internet are very short (consisting of only a few packets), for which the convergence analysis in the previous chapter does not apply. Further, there may also be flows which are not congestion controlled. Because of these deviations, the number of incoming packets at a link varies over time and may exceed the link capacity occasionally even if the mean arrival rate is less than the link capacity. So buffers are needed to absorb bursty arrivals and to reduce packet losses. To understand the effect of bursty arrivals and the role of buffering in communication networks, in this chapter we model packet arrivals at links as random processes and study the packet level performance at a link using discrete-time queueing theory. This chapter is devoted to answering the following questions.

  • • How large should the buffer size be to store bursty packet arrivals temporarily before transmission over a link?

  • • What is the relationship between buffer overflow probabilities, delays, and the burstiness of the arrival processes?

  • • How do we provide isolation among flows so that each flow is guaranteed a minimum rate at a link, independent of the burstiness of the other flows sharing the link?

  • 5 - Scheduling in wireless networks

  • from I - Network architecture and algorithms
  • R. Srikant, University of Illinois, Urbana-Champaign, Lei Ying, Arizona State University
  • Book:
    Communication Networks
    Published online:
    30 January 2019
    Print publication:
    14 November 2013, pp 110-141

  • Summary

    So far we have looked at resource allocation algorithms in networks with wireline links. In this chapter, we consider networks with wireless components. The major difference between wireless and wireline networks is that in wireless networks links contend for a common resource, namely the wireless spectrum. As a result, we have to design Medium Access Control (MAC) algorithms to decide which links access the wireless medium at each time instant. As we will see, wireless MAC algorithms have features similar to scheduling algorithms for high-speed switches, which were studied in Chapter 4. However, there are some differences: wireless networks are subject to time-varying link quality due to channel fluctuations, also known as channel fading; and transmissions in wireless networks may interfere with each other, so transmissions have to be scheduled to avoid interference. In addition, some wireless networks do not have a central coordinator to perform scheduling, so scheduling decisions have to be taken independently by each link. In this chapter, we will address the following issues specific to wireless networks.

  • • Does channel-state information play a critical role in scheduling in wireless networks?

  • • What is the capacity region of a cellular network, and what scheduling algorithm can be used to achieve the full capacity region?

  • • What is the capacity region of an ad hoc wireless network, what scheduling algorithm can be used to support the capacity region, and can the algorithm be implemented in a distributed fashion?

Page 1594 of 1926