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# Fine particle magnetism

Publisher: Atlantic Publishers and Distributors in New Delhi

Written in English

## Subjects:

• Particles (Nuclear physics),
• Magnetic fields.

## Edition Notes

Includes bibliographical references (p. [177]-183).

The Physical Object ID Numbers Statement Bibek Bandyopadhyay. Pagination vi, 183 p. : Number of Pages 183 Open Library OL19249065M ISBN 10 8126901772

The first characterization of pre‐ and postshock magnetic properties for sized Fe particles and the first characterization of the transformation remanent magnetization (TMRM) associated with the face‐centered‐cubic (fcc) to body‐centered‐cubic (bcc) transformation in fine particle . @article{osti_, title = {RESEARCH ON FINE PARTICLE MAGNETIC MATERIALS}, author = {Schuele, W.J.}, abstractNote = {Ferromagnetic and ferrimagnetic fine particles were prepared by suitably heat treating coprecipitated oxalates of the systems Co-Fe, Ni-Fe, and Cu-Fe in a wide range of compositions. Decomposition of the oxalates in hydrogen produced fine particle metals or alloys, .   This book intends to provide the reader with the recent progress of studies of fine particle plasmas from the viewpoints of wide and interdisciplinary directions, such as self-organized fine particles, Coulomb crystal formation, behaviors of fine particles, their stability, and syntheses of nano-sized particles in reactive plasmas. In September , the UK Nanomagnetism Network and the London Centre for Nanotechnology hosted the 5th International Conference on Fine Particle Magnetism. The objective of the meeting, as in Rome (), Bangor (), Barcelona () and Pittsburgh (), was to explore latest developments in the fundamentals and applications of nanoscale magnetic clusters, particles and grains. This.

A finite force acts on a charged particle if it enters a uniform magnetic field in a direction with finite angle with the field. If two charged particles of masses m 1 and m 2 and charges q 1 and q 2 are projected in a uniform magnetic field with same constant velocity in a direction perpendicular to the field then the ratio of their radii (R 1. The 11th International Conference on Fine Particle Magnetism ICFPM will be held in Yokohama, Japan. Dates: 30 May (Mo) – 3 Jun (Fr), Last Conference. Gijón, a beautiful city at the north coast of Spain, was the venue for ICFPM In theory, magnetic particle inspection (MPI) is a relatively simple concept. It can be considered as a combination of two nondestructive testing methods: magnetic flux leakage testing and visual testing. Consider the case of a bar magnet. It has a magnetic field in and around the magnet. Any place that a magnetic line of force exits or enters. fine-particle magnetic oxides in response to a magnetic field is of fundamental importancet. This is partly due to the ubiquity of fine-particle magnetic oxides in naturally occurring forms, and partly due to the special properties of finely subdivided materials which lead to their widespread utilization in manufactured forms.

Experimentally observed dynamic behavior, such as a particle circulation under magnetic field, a bow shock formation in an upper stream of an obstacle, etc., will be reviewed. Dust particles confined in a cylindrical glass tube show a dynamic circulation when strong magnetic field is applied from the bottom of the tube using a permanent magnet. The circulation consists of two kinds . Artisan Photo Books. We distinguish ourselves from other online photo books by giving premium finishing for all our products, using % recycled artisan quality paper, signature binding for flatter pages, finest materials, and hand made artisan finishing. Books starts with 40 pages by default, and you can add extra pages up to pages. Figure A negatively charged particle moves in the plane of the paper in a region where the magnetic field is perpendicular to the paper (represented by the small × × ’s—like the tails of arrows). The magnetic force is perpendicular to the velocity, so velocity changes in direction but not magnitude. The result is uniform circular motion. In , ICFPM was called 3 rd EuroConference on Magnetic Properties of Fine Particles and their Relevance to Materials Science. It was held in Barcelona from 19 th to 22 nd October.. The co-chairs of this edition were X. Batlle and A. Labarta from the Group of Magnetism and Transport Properties, Department of Fundamental Physics, Faculty of Physics, University of Barcelona.

## Recent

Purchase Magnetic Properties of Fine Particles - 1st Edition. Print Book & E-Book. ISBNBook Edition: 1. In fine magnetic particle systems, all of these parameters may be quite different from their values when the particles are consolidated into a large single entity.

For an assembly of weakly-interacting nanometer-sized clusters of magnetic spins, as occurs in nanocomposite materials, it was predicted [, ] that the magnetocaloric effect may be. Recent advances in fine-particle magnetism are reviewed from an experimental point of view, especially with reference to surface magnetism and also to magnetism in low dimensions.

Emphasis is placed on intrinsic properties as compared with the behavior of bulk by: Magnetism of Fine Particles Hysteresis Loops Magnetic Domains Energetics of a Ferrimagnetic Particle S f MtiAi t Fe 3 O 4 ources o Magnetic Anisotropy Hysteresis in Single domain and Multidomain particles Domain Calculations and Micromagnetism Néel.

Abstract. Magnetic fine particles find a large number of applications, perhaps most notably as particulate recording media. In addition, the ideas of fine particle magnetism are applicable in a number of areas as diverse as geomagnetism and by: The book contains a fairly detailed though, of course, incomplete reference list, which may help to satisfy the reader who wishes to investigate anyone question in its entirity.

The book opens with a review of the well-known aspects of magnetism of the elementary particle that. Magnetic Particle Imaging: A Novel SPIO Nanoparticle Imaging Technique (Springer Proceedings in Physics Book ) - Kindle edition by Buzug, Thorsten M., Borgert, Jörn.

Download it once and read it on your Kindle device, PC, phones or tablets. Use features like bookmarks, note taking and highlighting while reading Magnetic Particle Imaging: A Novel SPIO Nanoparticle Imaging Manufacturer: Springer. Personnel Training Publications: Magnetic Particle Testing (MT), Classroom Training Book Second Edition (eBook) Contains up-to-date information on many aspects of magnetic particle testing.

Organized to follow the body of knowledge outlined in CP () for Levels I and II. Content includes theories of magnetism, demagnetization techniques. Part of the NanoScience and Technology book series (NANO) Abstract.

Magnetic nanoparticles are important physical systems, relevant for many applications and, in many cases, the starting point of fundamental developments in Fine particle magnetism book. E.P. Wohlfarth, Fine particle magnetism, in Magnetic Properties of Low-Dimensional Systems, ed.

by L. At what level. I recommend Jackson, “Classical Electrodynamics” — it'll last a lifetime. Ebook Title: Fine Particle Magnetism - Read Fine Particle Magnetism PDF on your Android, iPhone, iPad or PC directly, the following PDF file is submitted in 23 Jul,Ebook ID PDF-6FPM4.

Download full version PDF for Fine Particle Magnetism using the link below. Pub Date: August Bibcode: RvGeoSM Keywords: Geomagnetism and Paleomagnetism: Rock and mineral magnetism; Geomagnetism and.

In contrast, the magnetic force on a charge particle is orthogonal to the magnetic field vector, and depends on the velocity of the particle. The right hand rule can be used to determine the direction of the force.

An electric field may do work on a charged particle, while a magnetic field does no work. Fine Structure The Lamb Shift Hyperfine Structure Positronium Quarkonium Light Quark Mesons Baryons Baryon Masses and Magnetic Moments References and Notes Problems 6 The Feynman Calculus CONTENTS 81 Lifetimes and Cross Sections   Contributor; The force on a charged particle in an electric and a magnetic field is $\textbf{F} = q(\textbf{E} +\textbf{v} \times \textbf{B})$.

As an example, let us investigate the motion of a charged particle in uniform electric and magnetic fields that are at right angles to each other. In magnetism: B= F /q v sin θ Units of B: Tesla (T) 1 Gauss = T Magnetic fields (B fields) point towards south poles and away from north poles.

Force on a moving charged particle when it enters a magnetic field. The particle experiences a force that is evident because the charge is. During the years since this book was first published in there have very few developments in the technology of magnetic particle inspection apart from improvements in instrumentation which has made the measurement of peak values of time varying currents practicable.

The major changes have arisen from health and safety and environmental Reviews: 1. In the final region, there is only a uniform magnetic field, and so the charged particles move in circular arcs with radii proportional to particle mass.

The paths also depend on charge q q size 12{q} {}, but since q q size 12{q} {} is in multiples of electron charges, it is easy to determine and to discriminate between ions in different. Electricity and magnetism are 2 waves on the same field.

each photon (as a model of a single event of electromagnetism on a path from a to b) has a polarization, a rotation/orientation along its movement direction with an electric wave and a magnetic wave relative to its two waves on the same field are in a 90° angle to each other for the same photon, coplanar with the line that.

Journals & Books; Help; Journal of Magnetism and Magnetic Materials. Supports open access. CiteScore. Impact Factor. Submit your article. Articles & Issues Proceedings of the 4th International Conference on Fine Particle Magnetism (ICFPM) (ICFPM) 14 August • Pittsburgh, PA, USA.

Robert D. Shull and Dieter Weller. Volume Magnetism is a class of physical phenomena that are mediated by magnetic fields. Electric currents and the magnetic moments of elementary particles give rise to a magnetic field, which acts on other currents and magnetic moments. Magnetism is one aspect of the combined phenomenon of most familiar effects occur in ferromagnetic materials, which are strongly.

Books by Robert G. Brown Physics Textbooks • Introductory Physics I and II A lecture note style textbook series intended to support the teaching of introductory physics, with calculus, at a level suitable for Duke undergraduates.

Theory of Magnetism. This note explains the following topics: The Bohr-van Leeuwen theorem, The electron spin and magnetic moment, ipole-dipole interaction, Magnetism of free atoms and ions, Magnetic ions in crystals, Exchange interactions between local spins, The Heisenberg model, Mean-field theory for magnetic insulators, The paramagnetic phase of magnetic insulators, Excitations in the.

During the years since this book was first published in there have very few developments in the technology of magnetic particle inspection apart from improvements in instrumentation which has made the measurement of peak values of time varying currents practicable. The major changes have arisen from health and safety and environmental concerns.5/5(2).

Oprah’s Book Club Magnetic Properties of Fine Particles. J.L. Dormann. $;$; Publisher Description. There have not been many changes in the use or the understanding of the concept of superparamagnetism since Jacobs and Bean [1] reviewed it almost 30 years ago. The measuring techniques have improved during that time, and have become.

Personnel Training Publications: Magnetic Particle Testing (MT), Classroom Training Book Second Edition. Contains up-to-date information on many aspects of magnetic particle testing.

Organized to follow the body of knowledge outlined in CP () for Levels I and II. Content includes theories of magnetism, demagnetization techniques.

where a and b represent the endpoints of the wire. As an example, consider a curved wire carrying a current I in a uniform magnetic field B G, as shown in Figure Figure A curved wire carrying a current I. Using Eq. (), the magnetic force on the wire is given by.

Question and answer For Magnetic Particle Testing Method Levels I, II, and III for download. Magnetic Particle Testing Method booklet contains suggested questions and answers in the Magnetic Particle Testing Method for use in conjunction with Recommended Personnel Qualification and Certification in Nondestructive Testing.


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# How do you find the vertex given y=x^2 -4?

If you are into formulas, you can remember $\left(y - k\right) = {\left(x - h\right)}^{2}$, where the vertex is $\left(h , k\right)$. This formula is easily derivable, and you can probably find a proof for it in Wikipedia.

format: plain


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# zbMATH — the first resource for mathematics

Topological and differentiable structures of the complement of an arrangement of hyperplanes. (English) Zbl 0795.57012
Greene, Robert (ed.) et al., Differential geometry. Part 2: Geometry in mathematical physics and related topics. Proceedings of a summer research institute, held at the University of California, Los Angeles, CA, USA, July 8-28, 1990. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 54, Part 2, 337-357 (1993).
This article is a brief survey of selected topics in the homotopy theory of hyperplane arrangements. Let $${\mathcal A}$$ denote a finite collection of homogeneous hyperplanes in $$C^ n$$. The main object of interest in this theory is the homotopy type of the complement $$M=C^ n - \bigcup {\mathcal A}$$, especially in its dependence on the intersection lattice $$L({\mathcal A}) = \{X= \bigcap {\mathcal B} \mid {\mathcal B} \subseteq {\mathcal A}\}$$.
The authors begin with a discussion of the Orlik-Solomon presentation of the cohomology algebra $$H^*(M)$$ in terms of $$L({\mathcal A})$$. Next they focus on the higher homotopy groups, giving a precise description, due to Hattori, of the homotopy type of $$\pi_ i (M)$$, $$i \geq 2$$, when $${\mathcal A}$$ is generic. Supersolvable lattices and fiber-type arrangements are introduced, with primary emphasis on a formula of Kohno, Randell and the reviewer relating the lower central series of $$\pi_ 1 (M)$$ to $$H^*(M)$$. An approach to this formula using Sullivan’s theory of minimal models is described in some detail. Finally the authors present their own partial solution to the homotopy-type conjecture, that $$L({\mathcal A})$$ determines the homotopy type of $$M$$. In the case $$n=3$$ they give a graphical condition under which this conjecture holds. Most proofs are omitted. The paper closes with two nontrivial examples.
For the entire collection see [Zbl 0773.00023].

##### MSC:
 57R55 Differentiable structures in differential topology 14B05 Singularities in algebraic geometry 05B35 Combinatorial aspects of matroids and geometric lattices 52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)


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IEEE Transactions on Cybernetics

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Publication Year: 2016, Page(s):C1 - 1
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• A Study on Estimation of Joint Force Through Isometric Index Finger Abduction With the Help of SEMG Peaks for Biomedical Applications

Publication Year: 2016, Page(s):2 - 8
Cited by:  Papers (5)
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We propose a new method to estimate joint force using a biomechanical muscle model and peaks of surface electromyography (SEMG). The SEMG measurement was carried out from the first dorsal interosseous muscle during isometric index finger abduction. The SEMG peaks were used as the input of the biomechanical muscle model which is a transfer function to generate the force. The force estimation perfor... View full abstract»

• Neural Controller Design-Based Adaptive Control for Nonlinear MIMO Systems With Unknown Hysteresis Inputs

Publication Year: 2016, Page(s):9 - 19
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This paper studies an adaptive neural control for nonlinear multiple-input multiple-output systems in interconnected form. The studied systems are composed of ${N}$ subsystems in pure feedback structure and the interconnection terms are contained in every equation of each subsystem. Moreover, the studied systems consider the effects of Prandtl-Ishlinskii (PI) hysteresis model. It is for the first ... View full abstract»

• Fast Sound Source Localization Using Two-Level Search Space Clustering

Publication Year: 2016, Page(s):20 - 26
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Steered response power phase transform (SRP-PHAT) is a method that is widely used for robust sound source localization (SSL). However, since SRP-PHAT searches over a large number of candidate locations, it is too slow to run in real-time for large-scale microphone array systems. In this paper, we propose a robust two-level search space clustering method to speed-up SRP-PHAT-based SSL. The proposed... View full abstract»

• Spectral Multimodal Hashing and Its Application to Multimedia Retrieval

Publication Year: 2016, Page(s):27 - 38
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In recent years, multimedia retrieval has sparked much research interest in the multimedia, pattern recognition, and data mining communities. Although some attempts have been made along this direction, performing fast multimodal search at very large scale still remains a major challenge in the area. While hashing-based methods have recently achieved promising successes in speeding-up large-scale s... View full abstract»

• No-Reference Image Blur Assessment Based on Discrete Orthogonal Moments

Publication Year: 2016, Page(s):39 - 50
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Blur is a key determinant in the perception of image quality. Generally, blur causes spread of edges, which leads to shape changes in images. Discrete orthogonal moments have been widely studied as effective shape descriptors. Intuitively, blur can be represented using discrete moments since noticeable blur affects the magnitudes of moments of an image. With this consideration, this paper presents... View full abstract»

• Robust Visual Tracking via Exclusive Context Modeling

Publication Year: 2016, Page(s):51 - 63
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In this paper, we formulate particle filter-based object tracking as an exclusive sparse learning problem that exploits contextual information. To achieve this goal, we propose the context-aware exclusive sparse tracker (CEST) to model particle appearances as linear combinations of dictionary templates that are updated dynamically. Learning the representation of each particle is formulated as an e... View full abstract»

• Ensemble and Arithmetic Recombination-Based Speciation Differential Evolution for Multimodal Optimization

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Multimodal optimization problems consists of multiple equal or comparable spatially distributed solutions. Niching and clustering differential evolution (DE) techniques have been demonstrated to be highly effective for solving such problems. The key challenge in the speciation niching technique is to balance between local solution exploitation and global exploration. Our proposal enhances explorat... View full abstract»

• Consensus Control With Failure—Wait or Abandon?

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This paper introduces and solves a decision-making problem under the context of consensus control with failure. We study an optimal consensus control problem in which n autonomous agents try to arrive at a target at the same time. One of the agents suddenly fails and the rest n - 1 agents can either wait or abandon the failed agent. If they wait, they must slow down and delay the consensus time. I... View full abstract»

• Adaptive Neural Control of a Class of Output-Constrained Nonaffine Systems

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In this paper, we present a novel tracking controller for a class of uncertain nonaffine systems with time-varying asymmetric output constraints. Firstly, the original nonaffine constrained (in the sense of the output signal) control system is transformed into a output-feedback control problem of an unconstrained affine system in normal form. As a result, stabilization of the transformed system is... View full abstract»

• Multiobjective Optimization of Linear Cooperative Spectrum Sensing: Pareto Solutions and Refinement

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In linear cooperative spectrum sensing, the weights of secondary users and detection threshold should be optimally chosen to minimize missed detection probability and to maximize secondary network throughput. Since these two objectives are not completely compatible, we study this problem from the viewpoint of multiple-objective optimization. We aim to obtain a set of evenly distributed Pareto solu... View full abstract»

• Evidential Matrix Metrics as Distances Between Meta-Data Dependent Bodies of Evidence

Publication Year: 2016, Page(s):109 - 122
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As part of the theory of belief functions, we address the problem of appraising the similarity between bodies of evidence in a relevant way using metrics. Such metrics are called evidential distances and must be computed from mathematical objects depicting the information inside bodies of evidence. Specialization matrices are such objects and, therefore, an evidential distance can be obtained by c... View full abstract»

• Pervasive Sound Sensing: A Weakly Supervised Training Approach

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Modern smartphones present an ideal device for pervasive sensing of human behavior. Microphones have the potential to reveal key information about a person's behavior. However, they have been utilized to a significantly lesser extent than other smartphone sensors in the context of human behavior sensing. We postulate that, in order for microphones to be useful in behavior sensing applications, the... View full abstract»

• A Gesture Recognition System for Detecting Behavioral Patterns of ADHD

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We present an application of gesture recognition using an extension of dynamic time warping (DTW) to recognize behavioral patterns of attention deficit hyperactivity disorder (ADHD). We propose an extension of DTW using one-class classifiers in order to be able to encode the variability of a gesture category, and thus, perform an alignment between a gesture sample and a gesture class. We model the... View full abstract»

• Consensus of Linear Multi-Agent Systems by Distributed Event-Triggered Strategy

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This paper studies the consensus problem of multi-agent systems with general linear dynamics. We propose a novel event-triggered control scheme with some desirable features, namely, distributed, asynchronous, and independent. It is shown that consensus of the controlled multi-agent system can be reached asymptotically. The feasibility of the event-triggered strategy is further verified by the excl... View full abstract»

• Learning Spatio-Temporal Representations for Action Recognition: A Genetic Programming Approach

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Extracting discriminative and robust features from video sequences is the first and most critical step in human action recognition. In this paper, instead of using handcrafted features, we automatically learn spatio-temporal motion features for action recognition. This is achieved via an evolutionary method, i.e., genetic programming (GP), which evolves the motion feature descriptor on a populatio... View full abstract»

• FLS-Based Adaptive Synchronization Control of Complex Dynamical Networks With Nonlinear Couplings and State-Dependent Uncertainties

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This paper is concerned with the problem of synchronization control of complex dynamical networks (CDN) subject to nonlinear couplings and uncertainties. An fuzzy logical system-based adaptive distributed controller is designed to achieve the synchronization. The asymptotic convergence of synchronization errors is analyzed by combining algebraic graph theory and Lyapunov theory. In contrast to the... View full abstract»

• Cluster Prototypes and Fuzzy Memberships Jointly Leveraged Cross-Domain Maximum Entropy Clustering

Publication Year: 2016, Page(s):181 - 193
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The classical maximum entropy clustering (MEC) algorithm usually cannot achieve satisfactory results in the situations where the data is insufficient, incomplete, or distorted. To address this problem, inspired by transfer learning, the specific cluster prototypes and fuzzy memberships jointly leveraged (CPM-JL) framework for cross-domain MEC (CDMEC) is firstly devised in this paper, and then the ... View full abstract»

• Robust Extreme Learning Machine With its Application to Indoor Positioning

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The increasing demands of location-based services have spurred the rapid development of indoor positioning system and indoor localization system interchangeably (IPSs). However, the performance of IPSs suffers from noisy measurements. In this paper, two kinds of robust extreme learning machines (RELMs), corresponding to the close-to-mean constraint, and the small-residual constraint, have been pro... View full abstract»

• Learning Flexible Graph-Based Semi-Supervised Embedding

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This paper introduces a graph-based semi-supervised embedding method as well as its kernelized version for generic classification and recognition tasks. The aim is to combine the merits of flexible manifold embedding and nonlinear graph-based embedding for semi-supervised learning. The proposed linear method will be flexible since it estimates a nonlinear manifold that is the closest one to a line... View full abstract»

• Self-Adaptive Differential Evolution Algorithm With Zoning Evolution of Control Parameters and Adaptive Mutation Strategies

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The performance of the differential evolution (DE) algorithm is significantly affected by the choice of mutation strategies and control parameters. Maintaining the search capability of various control parameter combinations throughout the entire evolution process is also a key issue. A self-adaptive DE algorithm with zoning evolution of control parameters and adaptive mutation strategies is propos... View full abstract»

• Semi-Supervised Nonnegative Matrix Factorization via Constraint Propagation

Publication Year: 2016, Page(s):233 - 244
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As is well known, nonnegative matrix factorization (NMF) is a popular nonnegative dimensionality reduction method which has been widely used in computer vision, document clustering, and image analysis. However, traditional NMF is an unsupervised learning mode which cannot fully utilize the priori or supervised information. To this end, semi-supervised NMF methods have been proposed by incorporatin... View full abstract»

• MOD* Lite: An Incremental Path Planning Algorithm Taking Care of Multiple Objectives

Publication Year: 2016, Page(s):245 - 257
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The need for determining a path from an initial location to a target one is a crucial task in many applications, such as virtual simulations, robotics, and computer games. Almost all of the existing algorithms are designed to find optimal or suboptimal solutions considering only a single objective, namely path length. However, in many real life application path length is not the sole criteria for ... View full abstract»

• Weakly Supervised Human Fixations Prediction

Publication Year: 2016, Page(s):258 - 269
Cited by:  Papers (14)
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Automatically predicting human eye fixations is a useful technique that can facilitate many multimedia applications, e.g., image retrieval, action recognition, and photo retargeting. Conventional approaches are frustrated by two drawbacks. First, psychophysical experiments show that an object-level interpretation of scenes influences eye movements significantly. Most of the existing saliency model... View full abstract»

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The scope of the IEEE Transactions on Cybernetics includes computational approaches to the field of cybernetics.

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Editor-in-Chief
Prof. Jun Wang
Dept. of Computer Science
City University of Hong Kong
Kowloon Tong, Kowloon, Hong Kong
Tel: +852 34429701
Email: jwang.cs@cityu.edu.hk


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# Connexions

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Textbook by: Lawrence Baggett. E-mail the author

# A Little Topology

Module by: Lawrence Baggett. E-mail the author

Summary: We now investigate some properties that subsets of RR and CC may possess. We will define “closed sets,” “open sets,” and “limit points” of sets. These notions are the rudimentary notions of what is called topology. As in earlier definitions, these topological ones will be enlightening when we come to continuity.

We now investigate some properties that subsets of RR and CC may possess. We will define “closed sets,” “open sets,” and “limit points” of sets. These notions are the rudimentary notions of what is called topology. As in earlier definitions, these topological ones will be enlightening when we come to continuity.

Definition 1:

Let SS be a subset of C.C. A complex number xx is called a limit point of SS if there exists a sequence {xn}{xn} of elements of SS such that x=limxn.x=limxn.

A set SCSC is called closed if every limit point of SS belongs to S.S.

Every limit point of a set of real numbers is a real number. Closed intervals [a,b][a,b] are examples of closed sets in R,R, while open intervals and half-open intervals may not be closed sets. Similarly, closed disks Br(c)¯Br(c)¯ of radius rr around a point cc in CC, and closed neighborhoods N¯r(S)N¯r(S) of radius rr around a set SC,SC, are closed sets, while the open disks or open neighborhoods are not closed sets. As a first example of a limit point of a set, we give the following exercise.

## Exercise 1

Let SS be a nonempty bounded set of real numbers, and let M=supS.M=supS. Prove that there exists a sequence {an}{an} of elements of SS such that M=liman.M=liman. That is, prove that the supremum of a bounded set of real numbers is a limit point of that set. State and prove an analogous result for infs.

HINT: Use (Reference), and let ϵϵ run through the numbers 1/n.1/n.

## Exercise 2

1. Suppose SS is a set of real numbers, and that z=a+biCz=a+biC with b0.b0. Show that zz is not a limit point of S.S. That is, every limit point of a set of real numbers is a real number. HINT: Suppose false; write a+bi=limxn,a+bi=limxn, and make use of the positive number |b|.|b|.
2. Let cc be a complex number, and let S=B¯r(c)S=B¯r(c) be the set of all zCzC for which |z-c|r.|z-c|r. Show that SS is a closed subset of CC. HINT: Use part (b) of (Reference).
3. Show that the open disk Br(0)Br(0) is not a closed set in CC by finding a limit point of Br(0)Br(0) that is not in Br(0).Br(0).
4. State and prove results analogous to parts b and c for intervals in R.R.
5. Show that every element xx of a set SS is a limit point of S.S.
6. Let SS be a subset of CC, and let xx be a complex number. Show that xx is not a limit point of SS if and only if there exists a positive number ϵϵ such that if |y-x|<ϵ,|y-x|<ϵ, then yy is not in S.S. That is, SBϵ(x)=.SBϵ(x)=. HINT: To prove the “ only if” part, argue by contradiction, and use the sequence {1/n}{1/n} as ϵϵ's.
7. Let {an}{an} be a sequence of complex numbers, and let SS be the set of all the anan's. What is the difference between a cluster point of the sequence {an}{an} and a limit point of the set S?S?
8. (h) Prove that the cluster set of a sequence is a closed set. HINT: Use parts (e) and (f).

## Exercise 3

1. Show that the set QQ of all rational numbers is not a closed set. Show also that the set of all irrational numbers is not a closed set.
2. Show that if SS is a closed subset of RR that contains Q,Q, then SS must equal all of R.R.

Here is another version of the Bolzano-Weierstrass Theorem, this time stated in terms of closed sets rather than bounded sequences.

## Theorem 1

Let SS be a bounded and closed subset of CC. Then every sequence {xn}{xn} of elements of SS has a subsequence that converges to an element of S.S.

### Proof

Let {xn}{xn} be a sequence in S.S. Since SS is bounded, we know by (Reference) that there exists a subsequence {xnk}{xnk} of {xn}{xn} that converges to some number x.x. Since each xnkxnk belongs to S,S, it follows that xx is a limit point of S.S. Finally, because SS is a closed subset of C,C, it then follows that xS.xS.

We have defined the concept of a closed set. Now let's give the definition of an open set.

Definition 2:

Let SS be a subset of C.C. A point xSxS is called an interior point of SS if there exists an ϵ>0ϵ>0 such that the open disk Bϵ(x)Bϵ(x) of radius ϵϵ around xx is entirely contained in S.S. The set of all interior points of SS is denoted by S0S0 and we call S0S0 the interior of S.S.

A subset SS of CC is called an open subset of CC if every point of SS is an interior point of S;S; i.e., if S=S0.S=S0.

Analogously, let SS be a subset of R.R. A point xSxS is called an interior point of SS if there exists an ϵ>0ϵ>0 such that the open interval (x-ϵ,x+ϵ)(x-ϵ,x+ϵ) is entirely contained in S.S. Again, we denote the set of all interior points of SS by S0S0 and call S0S0 the interior of S.S.

A subset SS of RR is called an open subset of RR if every point of SS is an interior point of S;S; i.e., if S=S0.S=S0.

## Exercise 4

1. Prove that an open interval (a,b)(a,b) in RR is an open subset of RR; i.e., show that every point of (a,b)(a,b) is an interior point of (a,b).(a,b).
2. Prove that any disk Br(c)Br(c) is an open subset of C.C. Show also that the punctured diskBr'(c)Br'(c) is an open set, where Br'(c)={z:0<|z-c|<r},Br'(c)={z:0<|z-c|<r}, i.e., evrything in the disk Br(c)Br(c) except the central point c.c.
3. Prove that the neighborhood Nr(S)Nr(S) of radius rr around a set SS is an open subset of C.C.
4. Prove that no nonempty subset of RR is an open subset of C.C.
5. (e) Prove that the set QQ of all rational numbers is not an open subset of R.R. We have seen in part (a) of Exercise 3 that QQ is not a closed set. Consequently it is an example of a set that is neither open nor closed. Show that the set of all irrational numbers is neither open nor closed.

We give next a useful application of the Bolzano-Weierstrass Theorem, or more precisely an application of Theorem 1. This also provides some insight into the structure of open sets.

## Theorem 2

Let SS be a closed and bounded subset of C,C, and suppose SS is a subset of an open set U.U. Then there exists an r>0r>0 such that the neighborhood Nr(S)Nr(S) is contained in U.U. That is, every open set containing a closed and bounded set SS actually contains a neighborhood of S.S.

### Proof

If SS is just a singleton {x},{x}, then this theorem is asserting nothing more than the fact that xx is in the interior of U,U, which it is if UU is an open set. However, when SS is an infinite set, then the result is more subtle. We argue by contradiction. Thus, suppose there is no such r>0r>0 for which Nr(S)U.Nr(S)U. then for each positive integer nn there must be a point xnxn that is not in U,U, and a corresponding point ynS,ynS, such that |xn-yn|<1/n.|xn-yn|<1/n. Otherwise, the number r=1/nr=1/n would satisfy the claim of the theorem. Now, because the ynyn's all belong to S,S, we know from Theorem 1 that a subsequence {ynk}{ynk} of the sequence {yn}{yn} must converge to a number yS.yS. Next, we see that

| x n k - y | | x n k - y n k | + | y n k - y | , < 1 n k + | y n k - y | , | x n k - y | | x n k - y n k | + | y n k - y | , < 1 n k + | y n k - y | ,
(1)

and this quantity tends to 0. Hence, the subsequence {xnk}{xnk} of the sequence {xn}{xn} also converges to y.y.

Finally, because yy belongs to SS and hence to the open set U,U, we know that there must exist an ϵ>0ϵ>0 such that the entire disk Bϵ(y)U.Bϵ(y)U. Then, since the subsequence {xnk}{xnk} converges to y,y, there must exist anknk such that |xnk-y|<ϵ,|xnk-y|<ϵ, implying that xnkBϵ(y),xnkBϵ(y), and hence belongs to U.U. But this is our contradiction, because all of the xnxn's were not in U.U. So, the theorem is proved.

We give next a result that clarifies to some extent the connection between open sets and closed sets. Always remember that there are sets that are neither open nor closed, and just because a set is not open does not mean that it is closed.

## Theorem 3

A subset SS of CC (RR) is open if and only if its complement S˜=CSS˜=CS (RSRS) is closed.

### Proof

First, assume that SS is open, and let us show that S˜S˜ is closed. Suppose not. We will derive a contradiction. Suppose then that there is a sequence {xn}{xn} of elements of S˜S˜ that converges to a number xx that is not in S˜;S˜; i.e., xx is an element of S.S. Since every element of SS is an interior point of S,S, there must exist an ϵ>0ϵ>0 such that the entire disk Bϵ(x)Bϵ(x) (or interval (x-ϵ,x+ϵ)(x-ϵ,x+ϵ)) is a subset of S.S. Now, since x=limxn,x=limxn, there must exist anNN such that |xn-x|<ϵ|xn-x|<ϵ for every nN.nN. In particular, |xN-x|<ϵ;|xN-x|<ϵ; i.e., xNxN belongs to Bϵ(x)Bϵ(x) (or (x-ϵ,x+ϵ)(x-ϵ,x+ϵ)). This implies that xNS.xNS. But xNS˜,xNS˜, and this is a contradiction. Hence, if SS is open, then S˜S˜ is closed.

Conversely, assume that S˜S˜ is closed, and let us show that SS must be open. Again we argue by contradiction. Thus, assuming that SS is not open, there must exist a point xSxS that is not an interior point of S.S. Hence, for every ϵ>0ϵ>0 the disk Bϵ(x)Bϵ(x) (or interval (x-ϵ,x+ϵ)(x-ϵ,x+ϵ)) is not entirely contained in S.S. So, for each positive integer n,n, there must exist a point xnxn such that |xn-x|<1/n|xn-x|<1/n and xnS.xnS. It follows then that x=limxn,x=limxn, and that each xnS˜.xnS˜. Since S˜S˜ is a closed set, we must have that xS˜.xS˜. But xS,xS, and we have arrived at the desired contradiction. Hence, if S˜S˜ is closed, then SS is open, and the theorem is proved.

The theorem below, the famous Heine-Borel Theorem, gives an equivalent and different description of closed and bounded sets. This description is in terms of open sets, whereas the original definitions were interms of limit points. Any time we can find two very different descriptions of the same phenomenon, we have found something useful.

Definition 3:

Let SS be a subset of CC (respectively R).R). By an open cover of SS we mean a sequence {Un}{Un} of open subsets of CC (respectively RR) such that SUn;SUn; i.e., for every xSxSthere exists an nn such that xUn.xUn.

A subset SS of CC (respectively RR) is called compact, or is said to satisfy the Heine-Borel property, if every open cover of SS has a finite subcover. That is, if {Un}{Un} is an open cover of S,S, then there exists an integer NN such that Sn=1NUn.Sn=1NUn. In other words, only a finite number of the open sets are necessary to cover S.S.

REMARK The definition we have given here for a set being compact is a little less general from the one found in books on topology. We have restricted the notion of an open cover to be a sequence of open sets, while in the general setting an open cover is just a collection of open sets. The distinction between a sequence of open sets and a collection of open sets is genuine in general topology, but it can be disregarded in the case of the topological spaces RR and C.C.

## Theorem 4: Heine-Borel Theorem

A subset SS of CC (respectively RR) is compact if and only if it is a closed and bounded set.

### Proof

We prove this theorem for subsets SS of C,C, and leave the proof for subsets of RR to the exercises.

Suppose first that SCSC is compact, i.e., satisfies the Heine-Borel property. For each positive integer n,n, define UnUn to be the open set Bn(0).Bn(0). Then SUn,SUn, because C=Un.C=Un. Hence, by the Heine-Borel property, there must exist an NN such that Sn=1NUn.Sn=1NUn. But then SBN(0),SBN(0), implying that SS is bounded. Indeed, |x|N|x|N for all xS.xS.

Next, still assuming that SS is compact, we will show that SS is closed by showing that S˜S˜ is open. Thus, let xx be an element of S˜.S˜. For each positive integer n,n, define UnUn to be the complement of the closed set B1/n(x)¯.B1/n(x)¯. Then each UnUn is an open set by Theorem 2.12, and we claim that {Un}{Un} is an open cover of S.S. Indeed, if yS,yS, then yx,yx, and |y-x|>0.|y-x|>0. Choose an nn so that 1/n<|y-x|.1/n<|y-x|. Then yB1/n(x)¯,yB1/n(x)¯, implying that yUn.yUn. This proves our claim that {Un}{Un} is an open cover of S.S. Now, by the Heine-Borel property, there exists an NN such that Sn=1NUn.Sn=1NUn. But this implies that for every zSzS we must have |z-x|1/N,|z-x|1/N, and this implies that the disk B1/N(x)B1/N(x) is entirely contained in S˜.S˜. Therefore, every element xx of S˜S˜ is an interior point of S˜.S˜. So, S˜S˜ is open, whence SS is closed. This finishes the proof that compact sets are necessarily closed and bounded.

Conversely, assume that SS is both closed and bounded. We must show that SS satisfies the Heine-Borel property. Suppose not. Then, there exists an open cover {Un}{Un} that has no finite subcover. So, for each positive integer nn there must exist an element xnSxnS for which xnk=1nUk.xnk=1nUk. Otherwise, there would be a finite subcover. By Theorem 1, there exists a subsequence {xnj}{xnj} of {xn}{xn} that converges to an element xx of S.S. Now, because {Un}{Un} is an open cover of S,S, there must exist an NN such that xUN.xUN. Because UNUN is open, there exists an ϵ>0ϵ>0 so that the entire disk Bϵ(x)Bϵ(x) is contained in UN.UN. Since x=limxnj,x=limxnj, there exists a JJ so that |xnj-x|<ϵ|xnj-x|<ϵ if jJ.jJ. Therefore, if jJ,jJ, then xnjUN.xnjUN. But the sequence {nj}{nj} is strictly increasing, so that there exists a j'Jj'J such that nj'>N,nj'>N, and by the choice of the point xnj',xnj', we know that xnj'k=1NUk.xnj'k=1NUk. We have arrived at a contradiction, and so the second half of the theorem is proved.

## Exercise 5

1. Prove that the union ABAB of two open sets is open and the intersection ABAB is also open.
2. Prove that the union ABAB of two closed sets is closed and the intersection ABAB is also closed. HINT: Use Theorem 3 and the set equations AB˜=A˜B˜,AB˜=A˜B˜, and AB˜=A˜B˜.AB˜=A˜B˜. These set equations are known as Demorgan's Laws.
3. Prove that the union ABAB of two bounded sets is bounded and the intersection ABAB is also bounded.
4. Prove that the union ABAB of two compact sets is compact and the intersection ABAB is also compact.
5. Prove that the intersection of a compact set and a closed set is compact.
6. Suppose SS is a compact set in CC and rr is a positive real number. Prove that the closed neighborhood N¯r(S)N¯r(S) of radius rr around SS is compact. HINT: To see that this set is closed, show that its coplement is open.

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text:
# Adiabatic approximation in the derivation of the speed of sound

• B
The speed of sound in a gas at temperature T is given to be ## v=\sqrt{\frac{\gamma RT}{M}}##, where ##\gamma## is the adiabatic exponent, R is the gas constant and M is the molar mass of the gas. In deriving this expression, we assumed that the compression and expansion processes were so fast that there isn't enough time for heat transfer to take place, and therefore that the processes are approximately adiabatic (Laplace's correction to Newton's formula). We therefore used ##PV^\gamma = constant##. However the relation ##PV^\gamma = constant## is valid only for a quasi-static adiabatic process, and cannot be used for very fast processes. The quasi-static assumption is made in the derivation of adiabatic process relations. So how are we allowed to use it in the speed of sound derivation? The very reason we assumed the process was adiabatic is because it is very fast in the first place. So how is it consistent to assume it is quasi-static as well?

Last edited:

## Answers and Replies

Related Classical Physics News on Phys.org
Mister T
Science Advisor
Gold Member
The process has to be slow enough that it's quasi-static, but fast enough that negligible heat is transferred.

The process has to be slow enough that it's quasi-static, but fast enough that negligible heat is transferred.
How do we know that this case satisfies that condition? Is there a practical limit below which a process can be reasonably assumed to be quasistatic? I have read that we can make the quasistatic approximation for expansion and compression processes, if the boundaries are moving much slower than the speed of sound in the gas. This makes perfect sense because then there would be sufficient time for the boundary disturbance to propagate through the rest of the gas and equalize for each small part of the process. However, I don't see how I can use that here, since the very derivation is for speed of sound in the gas.

Mister T
Science Advisor
Gold Member
How do we know that this case satisfies that condition?
It's an assumption used in the derivation. If the result of the derivation matches what we observe then that gives us confidence that the assumption is justified.

Terry Bing
FWIW, my text derives the basic relation (##c = \sqrt{\frac{\Delta p}{\Delta \rho}}##) using mass and momentum equations, then justifies the use of the isentropic relation (##\frac{p}{\rho ^k} = const##) by referring to experimental results which "indicate that the relationship between pressure and density across a sound wave is nearly isentropic."

So, rather than being an assumption in the derivation, it is an experimental finding that is applied to the general result.

vanhees71
It's an assumption used in the derivation. If the result of the derivation matches what we observe then that gives us confidence that the assumption is justified.
Thanks. I get that it matches the observed speed of sound. I am just curious as to why? Was there a way we could have guessed beforehand that it would work?

vanhees71
Science Advisor
Gold Member
2019 Award
You can derive hydrodynamics from the Boltzmann-transport equation. The ideal hydrodynamics equation, the Euler equation of an ideal fluid, comes out when you consider the fluid being always in local thermal equilibrium, i.e., at any time the phase-space distribution function is that of an ideal gas with temperature and chemical potential being functions of time and position. This approximation is valid if the typical changes of these macroscopic quantities are slow in time and with not too large gradients in space. This implies that no heat is transferred and no dissipation occurs, i.e., the thermodynamic process is adiabatic, i.e., entropy doesn't change. That's because the collision term of the Boltzmann equation then vanishes.

On top of this ideal-hydro approximation you can consider deviations from local thermal equilibrium using a gradient expansion or the method of moments. This leads to the calucation of transport coefficients like heat conductivity and shear and bulk viscosity. In the lowest order of the gradient expansion you obtain the Navier-Stokes equation.


text:
# Constructing a particular Toeplitz matrix with a certain rule

I want to construct a Toeplitz matrix in Mathematica. Usually it is done in Mathematica using the following command:

ToeplitzMatrix[{a,b,...},{a,y,...}]

I want to have the first and the second list like {a,b,...} and {a,y,...} above to each contain N elements for a given number N (say 5 or 6; I want to keep the option of changing it). I want a=0. From b onwards I want to have (-1)^n nx (starting with n=1) and from y onwards -(-1)^n nx alternatively.

I am quite new to all the powers of Mathematica. So some help will be greatly appreciated.

tf[nn_] := With[{n = Range[0, nn]}, ToeplitzMatrix[(-1)^n n x, -(-1)^n n x]]

For the question in the comment:

tf2[nn_] :=
With[{n = Range[0, nn]},
Module[{expr = (-1)^n Cot@(n x)}, expr[[1]] = 0; ToeplitzMatrix[expr, -expr]]]

tf3[nn_] :=
With[{n = Range@nn},
With[{expr = Join[{0}, (-1)^n Cot@(n x)]}, ToeplitzMatrix[expr, -expr]]]

• Sorry, but it says: – Student Jan 28 '16 at 14:39
• "Objects of unequal length in {1,-1,1,-1,1}\ {0,1,2,3,4}\ {[Pi]/12,\ [Pi]/6,[Pi]/4,[Pi]/3,(5\[Pi])/12,[Pi]/2,(7\[Pi])/12,(2\[Pi])/3,\ (3\[Pi])/4,(5\[Pi])/6,(11\[Pi])/12,[Pi],(13\[Pi])/12,(7\[Pi])/6,\ (5\[Pi])/4,(4\[Pi])/3,(17\[Pi])/12,(3\[Pi])/2,(19\[Pi])/12,(5\[\ Pi])/3,(7\[Pi])/4,(11\[Pi])/6,(23\[Pi])/12,2\ [Pi]} cannot be \ combined" – Student Jan 28 '16 at 14:39
• Also, here the first element 0 is obtained just by putting 0 for n in (-1)^n nx. But what if I have 0 for the diagonals and say (-1)^n cot(nx), so that I really need to put a second rule for the diagonals? – Student Jan 28 '16 at 14:41
• @Student You forgot to Clear[x] – xzczd Jan 28 '16 at 14:41
• Sorry, it is working for what you said. But for the other one I mentioned, I can't do e.g. tf[nn_] := With[{n = Range[0, nn]}, ToeplitzMatrix[{0,(-1)^n Cot[n x]}, {0,-(-1)^n Cot[n x]}]]. – Student Jan 28 '16 at 14:47


text:
## Method For Determining Relationship Between Binomial Coefficients

We can prove many relationships between the binomial coefficients by taking out factors and equating coefficients.
Suppose we state with the expression
$(1+x)^n$
.
$(1+x)^n=(1+x)^2(1+x)^{n-2} =(1+2x+x^2)(1+x)^{n-2}$

$\sum_{r=0}^n n \begin{pmatrix}n\\r\end{pmatrix}x^r=(1+2x+x^2) \sum_{r=0}^{n-2} \begin{pmatrix}n-2\\r\end{pmatrix} x^r$

$\sum_{r=0}^n \begin{pmatrix}n\\r\end{pmatrix}x^r= \sum_{r=0}^{n-2} \begin{pmatrix}n-2\\r\end{pmatrix}x^r + 2 \sum_{r=0}^{n-2} \begin{pmatrix}n-2\\r\end{pmatrix}x^{r+1} + \sum_{r=0}^{n-2} \begin{pmatrix}n-2\\r\end{pmatrix}x^{r+2}$

$\sum_{r=0}^n \begin{pmatrix}n\\r\end{pmatrix}x^r= \sum_{r=0}^{n-2} \begin{pmatrix}n-2\\r\end{pmatrix}x^r + 2 \sum_{r=1}^{n-1} \begin{pmatrix}n-2\\r-1\end{pmatrix}x^r + \sum_{r=2}^n \begin{pmatrix}n-2\\r-2\end{pmatrix}x^r$

Now equating coefficients.
$r=0$
: Both sides equal 1 (last two terms are zero).
$r=1$
:
$n=(n-2)+2(1)$
(last term is zero).
$r \ge 2$
:
$\begin{pmatrix}n\\r\end{pmatrix}= \begin{pmatrix}n-2\\r\end{pmatrix}+ 2 \begin{pmatrix}n-2\\r-1\end{pmatrix} + \begin{pmatrix}n-2\\r-2\end{pmatrix}$

Refresh


text:
# MTI Improvement Factor for Land-Based Radar

This example discusses the moving target indication (MTI) improvement factor and investigates the effects of the following on MTI performance:

• Frequency

• Pulse repetition frequency (PRF)

• Number of pulses

• Coherent versus noncoherent processing

This example also introduces sources of error that limit MTI cancellation. Lastly, the example shows the improvement in clutter-to-noise ratio (CNR) for a land-based, MTI radar system.

### MTI Improvement Factor

At a high-level, there are two types of MTI processing, coherent and noncoherent. Coherent MTI refers to the case in which the transmitter is coherent over the number of pulses used in the MTI canceler or when the coherent oscillator of the system receiver is locked to the transmitter pulse, which is also known as a coherent-on-receive system. A noncoherent MTI system uses samples of the clutter to establish a reference phase against which the target and clutter are detected.

The MTI improvement factor ${\mathit{I}}_{\mathit{m}}$ is defined as

${\mathit{I}}_{\mathit{m}}=\frac{{\mathit{C}}_{\mathit{i}}}{{\mathit{C}}_{\mathit{o}}}$,

where ${\mathit{C}}_{\mathit{i}}$is the clutter power into the receiver and ${\mathit{C}}_{\mathit{o}}$is the clutter power after MTI processing.

#### Effect of Frequency

Investigate the effect of frequency on MTI performance using the mtifactor function. Use a pulse repetition frequency (PRF) of 500 Hz and analyze for 1- through 3-delay MTI cancelers for both the coherent and noncoherent cases.

% Setup parameters m = 2:4; % Number of pulses in (m-1) delay canceler freq = linspace(1e9,10e9,1000); % Frequency (Hz) prf = 500; % Pulse repetition frequency (Hz) % Initialize outputs numM = numel(m); numFreq = numel(freq); ImCoherent = zeros(numFreq,numM); ImNoncoherent = zeros(numFreq,numM); % Calculate MTI improvement factor versus frequency for im = 1:numM % Coherent MTI ImCoherent(:,im) = mtifactor(m(im),freq,prf,'IsCoherent',true); % Noncoherent MTI ImNoncoherent(:,im) = mtifactor(m(im),freq,prf,'IsCoherent',false); end % Plot results helperPlotLogMTI(m,freq,ImCoherent,ImNoncoherent);

There are several takeaways from the results. First, the difference between the coherent and noncoherent results decreases with increasing frequency for the same m for the m = 3 and 4 cases. The results for the m = 2 case show that the improvement factor is very similar at lower frequencies but performance diverges at higher frequencies. Second, increasing m improves the clutter cancellation for both coherent and noncoherent MTI. Third, when the PRF is held constant, the MTI improvement factor decreases with increasing frequency. Lastly, for m = 3 and 4, the coherent performance is better than the noncoherent performance.

#### Effect of PRF

Next, consider the effect of PRF on the performance of the MTI filter. Calculate results for an L-band frequency at 1.5 GHz.

% Set parameters m = 2:4; % Number of pulses in (m-1) delay canceler freq = 1.5e9; % Frequency (Hz) prf = linspace(100,1000,1000); % Pulse repetition frequency (Hz) % Calculate MTI improvement factor versus PRF for im = 1:numM ImCoherent(:,im) = mtifactor(m(im),freq,prf,'IsCoherent',true); ImNoncoherent(:,im) = mtifactor(m(im),freq,prf,'IsCoherent',false); end % Plot results helperPlotMTI(m,prf,ImCoherent,ImNoncoherent,'Pulse Repetition Frequency (Hz)','PRF versus MTI Improvement');

When frequency is held constant, there are several results to note. First, the difference between the coherent and noncoherent improvement factors increase for increasing frequency for the same m for the m = 3 and 4 cases. The results for the m = 2 case show that the improvement factor is very similar over the majority of PRFs investigated. Second, the MTI performance improves with increasing PRF. Lastly, for m = 3 and 4, the coherent performance is better than the noncoherent performance.

#### Combined Effects of Frequency and PRF

Next, consider the combined effects of frequency and PRF on the MTI improvement factor. This will allow a system analyst to get a better sense of the entire analysis space. Perform the calculation for a coherent MTI system using a 3-delay canceler.

% Set parameters m = 4; % Number of pulses in (m-1) delay canceler freq = linspace(1e9,10e9,100); % Frequency (Hz) prf = linspace(100,1000,100); % Pulse repetition frequency (Hz) % Initialize numFreq = numel(freq); numPRF = numel(prf); ImCoherentMatrix = zeros(numPRF,numFreq); % Calculate coherent MTI improvement factor over a range of PRFs and % frequencies for ip = 1:numPRF ImCoherentMatrix(ip,:) = mtifactor(m,freq,prf(ip),'IsCoherent',true); end % Plot results helpPlotMTImatrix(m,freq,prf,ImCoherentMatrix);

Note that the same behaviors as previously noted are shown here:

• MTI performance improves with increasing PRF

• MTI performance decreases with increasing frequency

### MTI Performance Limitations

MTI processing is based on a requirement of target and clutter stationarity within a receive window. When successive returns are received and subtracted from one another, clutter is canceled. Any effect, whether internal or external to the radar, that inhibits stationarity within the receive window will result in imperfect cancellation.

A wide variety of effects can decrease the performance of the MTI cancellation. Examples include but are not limited to:

• Transmitter frequency instabilities

• Pulse repetition interval (PRI) jitter

• Pulse width jitter

• Quantization noise

• Uncompensated motion either in the radar platform or the clutter

The next two sections will discuss the effects of null velocity errors and clutter spectrum spread.

#### Null Velocity Errors

MTI performance declines when the clutter velocity is not centered on the null velocity. The effect of these null velocity errors results in a decreased MTI improvement factor since more clutter energy exists outside of the MTI filter null.

Consider the case of a radar operating in an environment with rain. Rain clutter has a non-zero average Doppler as the clutter approaches or recedes from the radar system. Unless the motion of the rain clutter is detected and compensated, the cancellation of the MTI filtering will be worse.

In this example, assume a null velocity centered at 0 Doppler. Investigate the effects on the improvement factor given clutter velocities over the range of -20 to 20 m/s for the coherent MTI processing case.

% Setup parameters m = 2:4; % Number of pulses in (m-1) delay canceler clutterVels = linspace(-20,20,100); % MTI null velocity (m/s) nullVel = 0; % True clutter velocity (m/s) freq = 1.5e9; % Frequency (Hz) prf = 500; % Pulse repetition frequency (Hz) % Initialize numM = numel(m); numVels = numel(clutterVels); ImCoherent = zeros(numVels,numM); ImNoncoherent = nan(numVels,numM); % Compute MTI improvement factor for im = 1:numM for iv = 1:numVels ImCoherent(iv,im) = mtifactor(m(im),freq,prf,'IsCoherent',true,'ClutterVelocity',clutterVels(iv),'NullVelocity',nullVel); end end % Plot results nullError = (clutterVels - nullVel).'; helperPlotMTI(m,nullError,ImCoherent,ImNoncoherent,'Null Error (m/s)','MTI Improvement with Null Velocity Errors');

The coherent MTI experiences a rapid decrease in the improvement as the null error increases. The rate at which the improvement suffers increases with increasing number of pulses in the (m-1) delay canceler. In the case where m = 4, only a slight offset of 1.1 m/s results in a loss in the improvement factor of 3 dB.

A wider clutter spread results in more clutter energy outside of an MTI filter null and thus results in less clutter cancellation. While clutter spread is due in part to the inherent motion of the clutter scatterers, other sources of clutter spread can be due to:

• Phase jitter due to sampling

• Phase drift, which can be due to instability in coherent local oscillators

Consider the effect of the clutter spread on the MTI improvement factor.

% Setup parameters m = 2:4; % Number of pulses in (m-1) delay canceler sigmav = linspace(0.1,10,100); % Standard deviation of clutter spread (m/s) freq = 1.5e9; % Frequency (Hz) prf = 500; % Pulse repetition frequency (Hz) % Calculate MTI improvement numSigma = numel(sigmav); for im = 1:numM for is = 1:numSigma ImCoherent(is,im) = mtifactor(m(im),freq,prf,'IsCoherent',true,'ClutterStandardDeviation',sigmav(is)); ImNoncoherent(is,im) = mtifactor(m(im),freq,prf,'IsCoherent',false,'ClutterStandardDeviation',sigmav(is)); end end % Plot results helperPlotMTI(m,sigmav,ImCoherent,ImNoncoherent,'Standard Deviation of Clutter Spread (m/s)','MTI Improvement versus Clutter Spread');

As can be seen from the figure, the standard deviation of the clutter spread is a great limiting factor of the MTI improvement regardless of whether the MTI is coherent or noncoherent. As the standard deviation of the clutter spread increases, the MTI improvement factor decreases significantly until the improvement falls below 5 dB for all values of m in both coherent and non-coherent cases.

### Clutter Analysis for a Land-Based, MTI Radar

Consider a land-based MTI radar system. Calculate the clutter-to-noise ratio with and without MTI processing.

First, setup the radar and MTI processing parameters.

% Radar properties freq = 1.5e9; % L-band frequency (Hz) anht = 15; % Height (m) ppow = 100e3; % Peak power (W) tau = 1.5e-6; % Pulse width (sec) prf = 500; % PRF (Hz) Gtxrx = 45; % Tx/Rx gain (dB) Ts = 450; % Noise figure (dB) Ntx = 20; % Number of transmitted pulses % MTI settings nullVel = 0; % Null velocity (m/s) m = 4; % Number of pulses in the (m-1) delay canceler N = Ntx - m; % Number of coherently integrated pulses

Consider a wooded hills operating environment with a clutter spread of 1 m/s and a mean clutter velocity of 0 m/s. Calculate and plot the land surface reflectivity for the grazing angles of the defined geometry. Use the landroughness and landreflectivity functions for the physical surface properties and reflectivity calculation, respectively.

% Clutter properties sigmav = 1; % Standard deviation of clutter spread (m/s) clutterVel = 0; % Clutter velocity (m/s) landType = 'Wooded Hills'; % Get the surface standard deviation of height (m), surface slope (deg), % and vegetation type [surfht,beta0,vegType] = landroughness(landType); % Calculate maximum range for simulation Rua = time2range(1/prf); % Maximum unambiguous range (m) Rhoriz = horizonrange(anht,'SurfaceHeight',surfht); % Horizon range (m) RhorizKm = Rhoriz.*1e-3; % Horizon range (km) Rmax = min(Rua,Rhoriz); % Maximum range (m) % Generate vector of ranges for simulation Rm = linspace(100,Rmax,1000); % Range (m) Rkm = Rm*1e-3; % Range (km) % Calculate land clutter reflectivity grazAng = grazingang(anht,Rm,'TargetHeight',surfht); nrcs = landreflectivity(landType,grazAng,freq); nrcsdB = pow2db(nrcs); helperPlot(grazAng,nrcsdB,'NRCS','NRCS (dB)','Reflectivity \sigma_0 (dB)','Land Reflectivity \sigma_0');

Next, calculate the radar cross section (RCS) of the clutter using the clutterSurfaceRCS function. Note the drop in the clutter RCS as the radar horizon range is reached.

% Calculate azimuth and elevation beamwidth azbw = sqrt(32400 / db2pow(Gtxrx)); elbw = azbw; % Calculate clutter RCS rcs = clutterSurfaceRCS(nrcs,Rm,azbw,elbw,grazAng(:),tau); % Plot clutter RCS including horizon line rcsdB = pow2db(rcs); % Convert to decibels for plotting hAxes = helperPlot(Rkm,rcsdB,'RCS','Range (km)','Clutter RCS (dBsm)','Clutter Radar Cross Section (RCS)'); helperAddHorizLine(hAxes,RhorizKm);

Since the propagation path deviates from free space, include the clutter propagation factor and atmospheric losses in the calculation.

The default permittivity calculation underlying the radarpropfactor function is a sea water model. In order to more accurately simulate the propagation path over land, calculate the permittivity for the vegetation using the earthSurfacePermittivity function.

% Calculate land surface permittivity for vegetation temp = 20; % Ambient temperature (C) wc = 0.5; % Gravimetric water contnt epsc = earthSurfacePermittivity('vegetation',freq,temp,wc);

Calculate the clutter propagation factor using the radarpropfactor function. Include the vegetation type in the calculation. At higher frequencies, the presence of vegetation can cause additional losses.

% Calculate clutter propagation factor Fc = radarpropfactor(Rm,freq,anht,surfht, ... 'SurfaceHeightStandardDeviation',surfht, ... 'SurfaceSlope',beta0, ... 'VegetationType',vegType, ... 'SurfaceRelativePermittivity',epsc, ... 'ElevationBeamwidth',elbw); helperPlot(Rkm,Fc,'Clutter Propagation Factor','Range (km)', ... 'Propagation Factor (dB)', ... 'One-Way Clutter Propagation Factor F_C');

Next, calculate the atmospheric losses in this simulation. Assume the default standard atmosphere. Perform the calculation using the tropopl function.

% Calculate the atmospheric loss due to water and oxygen attenuation elAng = height2el(surfht,anht,Rm); % Elevation angle (deg) numEl = numel(elAng); Latmos = zeros(numEl,1); Llens = zeros(numEl,1); for ie = 1:numEl Latmos(ie,:) = tropopl(Rm(ie),freq,anht,elAng(ie)); end hAxes = helperPlot(Rkm,Latmos,'Atmospheric Loss','Range (km)','Loss (dB)','One-Way Atmospheric Losses'); ylim(hAxes,[0 0.1]);

Calculate the CNR using the radareqsnr function and plot results with and without MTI. Again, note the drop in CNR as the simulation range approaches the radar horizon.

% Calculate CNR lambda = freq2wavelen(freq); cnr = radareqsnr(lambda,Rm(:),ppow,tau, ... 'gain',Gtxrx,'rcs',rcs,'Ts',Ts, ... 'PropagationFactor',Fc, ... 'AtmosphericLoss',Latmos); coherentGain = pow2db(N); cnr = cnr + coherentGain; hAxes = helperPlot(Rkm,cnr,'CNR','Range (km)','CNR (dB)','CNR Clutter-to-Noise Ratio'); helperAddHorizLine(hAxes,RhorizKm); % Calculate CNR with MTI Im = mtifactor(m,freq,prf,'IsCoherent',true,... 'ClutterVelocity',clutterVel, ... 'ClutterStandardDeviation',sigmav, ... 'NullVelocity',nullVel)
Im = 55.3986
cnrMTI = cnr - Im; helperAddPlot(Rkm,cnrMTI,'CNR + MTI',hAxes);

Lastly, calculate the MTI improvement factor assuming a null error exists due to the true clutter velocity being 3 m/s while the null velocity remains centered at 0 m/s.

% Calculate CNR with null velocity error trueClutterVel = 3; % Clutter velocity (m/s) nullError = trueClutterVel - nullVel; % Null error (m/s) ImNullError = mtifactor(m,freq,prf,'IsCoherent',true,... 'ClutterVelocity',trueClutterVel, ... 'ClutterStandardDeviation',sigmav, ... 'NullVelocity',nullVel)
ImNullError = 33.6499
cnrMTINullError = cnr - ImNullError; helperAddPlot(Rkm,cnrMTINullError, ... sprintf('CNR + MTI with %.1f (m/s) Null Error',nullError), ... hAxes);

ImLoss = Im - ImNullError
ImLoss = 21.7488

Note the dramatic decrease in the CNR due to the MTI processing. When the null velocity is set to the clutter velocity, the improvement is 55 dB. When there is an uncompensated motion, the cancellation decreases to 34 dB. This is a loss of 21 dB of cancellation. This shows the need to properly compensate for motion or to steer the null to the appropriate velocity.

### Summary

This example discusses the MTI improvement factor and investigates a multitude of effects on MTI performance. Using the mtifactor function, we see that MTI performance:

• Improves with increasing PRF

• Decreases with increasing frequency

• Improves with increasing the number of pulses in the (m - 1)-delay canceler

Additionally, we see that the performance of coherent MTI is generally better than noncoherent MTI.

Lastly, we investigate limitations of MTI performance within the context of a land-based MTI radar system, showing the need to properly compensate for unexpected clutter velocities.

### References

1. Barton, David Knox. Radar Equations for Modern Radar. Artech House Radar Series. Boston, Mass: Artech House, 2013.

2. Richards, M. A., Jim Scheer, William A. Holm, and William L. Melvin, eds. Principles of Modern Radar. Raleigh, NC: SciTech Pub, 2010.

function helperPlotLogMTI(m,freq,ImCoherent,ImNonCoherent) % Used for the MTI plots that have a logarithmic x axis hFig = figure; hAxes = axes(hFig); lineStyles = {'-','--','-.'}; numM = numel(m); for im = 1:numM semilogx(hAxes,freq.*1e-9,ImCoherent(:,im),'LineWidth',1.5, ... 'LineStyle',lineStyles{im}, 'Color',[0 0.4470 0.7410], ... 'DisplayName',sprintf('Coherent, m = %d',m(im))) hold(hAxes,'on') semilogx(hAxes,freq.*1e-9,ImNonCoherent(:,im),'LineWidth',1.5, ... 'LineStyle',lineStyles{im}, 'Color',[0.8500 0.3250 0.0980], ... 'DisplayName',sprintf('Noncoherent, m = %d',m(im))) end grid(hAxes,'on'); xlabel(hAxes,'Frequency (GHz)') ylabel(hAxes,'MTI Improvement Factor (dB)') title('Frequency versus MTI Improvement') legend(hAxes,'Location','Best') end function helperPlotMTI(m,x,ImCoherent,ImNonCoherent,xLabelStr,titleName) % Used for the MTI plots that have an x-axis in linear units hFig = figure; hAxes = axes(hFig); lineStyles = {'-','--','-.'}; numM = numel(m); for im = 1:numM plot(hAxes,x,ImCoherent(:,im),'LineWidth',1.5, ... 'LineStyle',lineStyles{im}, 'Color',[0 0.4470 0.7410], ... 'DisplayName',sprintf('Coherent, m = %d',m(im))) hold(hAxes,'on') if any(~isnan(ImNonCoherent)) % Don't plot if NaN plot(hAxes,x,ImNonCoherent(:,im),'LineWidth',1.5, ... 'LineStyle',lineStyles{im}, 'Color',[0.8500 0.3250 0.0980], ... 'DisplayName',sprintf('Noncoherent, m = %d',m(im))) end end grid(hAxes,'on'); xlabel(hAxes,xLabelStr) ylabel(hAxes,'MTI Improvement Factor (dB)') title(titleName) legend(hAxes,'Location','Best') end function helpPlotMTImatrix(m,freq,prf,ImMat) % Creates image of MTI improvement factor with Frequency on the x-axis and % PRF on the y-axis hFig = figure; hAxes = axes(hFig); hP = pcolor(hAxes,freq.*1e-9,prf,ImMat); hP.EdgeColor = 'none'; xlabel(hAxes,'Frequency (GHz)') ylabel(hAxes,'PRF (Hz)') title(sprintf('Coherent MTI Improvement, m = %d',m)) hC = colorbar; hC.Label.String = '(dB)'; end function varargout = helperPlot(x,y,displayName,xlabelStr,ylabelStr,titleName) % Used for CNR analysis % Create new figure hFig = figure; hAxes = axes(hFig); % Plot plot(hAxes,x,y,'LineWidth',1.5,'DisplayName',displayName); grid(hAxes,'on'); hold(hAxes,'on'); xlabel(hAxes,xlabelStr) ylabel(hAxes,ylabelStr); title(hAxes,titleName); ylims = get(hAxes,'Ylim'); set(hAxes,'Ylim',[-100 ylims(2)]); % Add legend legend(hAxes,'Location','Best') % Output axes if nargout ~= 0 varargout{1} = hAxes; end end function helperAddPlot(x,y,displayName,hAxes) % Add additional CNR plots plot(hAxes,x,y,'LineWidth',1.5,'DisplayName',displayName); end function helperAddHorizLine(hAxes,val) % Add vertical line indicating horizon range xline(hAxes,val,'--','DisplayName','Horizon Range','LineWidth',1.5); end


text:
# 5.1: Energy

Learning Objectives

• Define energy and work
• Define basic terms of thermodynamics like system and surroundings
• Describe the nature of energy transfers
• Compare and contrast exothermic and endothermic processes sign conventions associated with these process.

## Introduction

In this Chapter we will introduce thermodynamics, the science of energy and its transformations. Energy can be considered to be the capacity to do work or transfer heat.

Definition: Energy

Energy is the capacity to do work (w) or transfer heat (q)

In Chapter 1 we introduced the concepts of kinetic and potential energy, and we will now take a deeper look at these. From a macroscopic perspective we sense objects as hot or cold as represented by the temperature, and recognize that heat will flow from hot to cold, but often mistake this to mean temperature is energy, which it is not. Thermal energy can be related to the random motion of molecules as they vibrate, rotate and translate in space.

## Molecular Kinetic Energy

### Translational Kinetic Energy

$$E_{k} = \frac{1}{2}mv^{2}$$, is the energy associated with the movement of a chemical entity's center of mass, where m is the mass of the chemical entity (molecule, atom or ion) and v is the velocity of its center of mass. In the fluid phases (gas and liquids) there will be a distribution of translational energies as different molecules randomly move around each other.

### Vibrational Kinetic Energy

$$E_{k} = \frac{1}{2}kx^{2}$$, is the energy associated with atom's vibrational motion, which can be modeled as a oscillating spring with k being Hooke's law constant and x is the displacement from the equilibrium location. That is, even the atoms in a solid, be it a covalent bond or ionic bond (crystal lattice) are not static, and they vibrate back and forth around equilibrium positions like in a spring, with the the average distance being the "bond distance". Note, atoms oscillate because there are attractive and repulsive forces acting on them which change as their location changes. For example, in the diatomic hydrogen bond, the nuclei are accelerated towards each other if the attractive forces are greater than the repulsive, but as they get closer and closer, the nuclear/nuclear repulsion becomes stronger, and they start to be repulsed. As they move farther and farther apart, these forces become weaker, and the attractive forces once again begin to dominate, causing them to approach each other, with the resulting motion like the oscillation of a spring. Hooke's law equation actually describes the potential energy of the oscillator, when the distance is a maximum from the equilibrium, but as energy is conserved, this is equal to the kinetic energy that atoms feel in the midpoint of the oscillation. The hotter a substance the faster the vibrational frequency.

### Rotational Kinetic Energy

$$E_{\omega} = \frac{1}{2}I\omega^{2}$$ , is the energy of a chemical entity associated with rotation, where $$\omega$$ is the rotational frequency (how fast the molecule spins) and I is the moment of inertia (in a sense, its resistance to spinning, the farther apart the hydrogen the greater I, the closer together, the smaller I. There can be multiple axis of rotation.

### Molecular Potential Energy

Coulombic Energy is the energy between charged particles. There are both repulsive forces (between like charged particles) and attractive forces (between opposite charged particles).

### Atomic Coulombic Energy

The coulombic energy of attraction of the electron to the nucleus of an atom. These are weakest for outer shell electrons and strongest for core electrons.

### Bond Coulombic Energies

The energies of covalent and ionic bonds. In the case of ionic compounds they are the attractive and repulsive interactions of the various cations and anions. For covalent, they are the +/- attractions of electrons to nuclei of different atoms, but there are also repulsive interactions of the -/- electrons and +/+ nuclei. The equilibrium distance is considered as the potential energy of the bond. As bonds are broken and formed during a chemical reaction energy the potential energy may be released or absorbed from the surroundings which either gain or lose kinetic energy.

### Nuclear Energy

There are "strong" and "weak" forces holding the nucleus together and these will not be covered in this class.

Exercise $$\PageIndex{1}$$

Identify the type of energy described by each of the following statements as potential or kinetic

1. Vibrational motion of atoms in a bond that are stimulated by infrared radiation
2. Energy of an electron in an atomic orbit
3. Energy of a covalent bond
4. Rotation of a molecule induced by microwave radiation

Kinetic

Potential

Potential

Kinetic

## Units of Energy

Work based definition of energy

This is the SI unit of energy and one Joule is a Newton meter (J = N.m), where a Newton is the SI unit of force. The Newton (named after Isaac Newton) comes from classical mechanics and Newton's second Law of motion, F = ma (mass times acceleration), and acceleration is the change in velocity per unit time, $$a=\frac{\Delta v}{\Delta t}$$ and velocity is the vectorial change in displacement per unit time, $$v=\frac{\Delta x}{\Delta t}$$ , where x is the displacement in a specific direction. So like any derived SI unit, the Joule can be described in terms of the SI base units, $$1J=1kg\frac{m^{2}}{sec^{2}}$$

### Calorie

Heat based definition of energy

Historically, the calorie is the energy required to raise one gram of water one degree celsius (from 14.5oC to 15.5oC), but today the calorie is considered to be a depreciated unit, and according to NIST Special Publication 1038, "The International System of Units (SI)-Conversion Factors of General Use", the definition of the calorie is based on the Joule:

1cal ≡ 4.184 Joules

Note, the above is a defined number and should not be treated as if it has significant digits. Also, the calorie is actually a very small unit of measurement and it should be noted that the nutritional Calorie (written with a capital C) is really a kilocalorie.

Exercise $$\PageIndex{2}$$

The NIH provides a table of Estimated Calorie Requirements (kilocalories) for gender and age groups that indicates an active 19-30 year old male should needs around 3,000. calories. How many joules is this? Use appropriate SI Prefix in your answer.

12.55MJ

## What is Work?

Work has many forms, of which mechanical(translational) work can be described as the energy used to displace an object a distance d when opposed against a force, like that of gravity.

$w=Fd$

where

• $$w$$ is the work
• $$F$$ is the opposing force
• $$d$$ is the distance the object was displaced.

Note, there are other types of work, like the expansion of a gas and electric work (the work done to move a charged particle in an electric field). In this class we will mostly be interested in PV work, which can be visualized as the work done through the translational motion of molecules moving in three dimensional space.

### Work of Expansion: Pressure-Volume (PV) Work

To fully understand this we need to have covered gasses, but we can imagine molecules in a gas as having translational motion, which could do work against the surroundings. If we could imagine a frictionless piston in a cylinder there would be three possibilities where the pressure inside is either equal to, greater than or less than the pressure outside. If the pressure inside is greater, the gas inside will expand moving the piston and doing work on the surroundings. If the pressure outside is greater, the piston will move inward compressing the gas inside as work is done on the system.

$$P_{ext} = P_{int}$$ The system is at equilibrium, no work is done
$$P_{ext} > P_{int}$$ The system contracts and work is done on the system
$$P_{ext} < P_{int}$$ The system expands and work is done on the environment

Pressure is defined as Force per unit area $$P = \dfrac{F}{A}$$, so F=PA, and if the external pressure in Figure 5.1.5 is greater than the internal pressure, work will be done on the system, causing it to contract by the volume AΔh, so ΔV = AΔh

$w = F d = P_{ext}AΔh$

The change in the volume of the cylinder (ΔV) as the piston moves a distance d is ΔV = AΔh, as shown in Figure 5.1.5. The work performed is thus

$w = P_{ext}ΔV \label{7.4.4}$

The units of work obtained using this definition are correct for energy: pressure is force per unit area (newton/m2) and volume has units of cubic meters, so

$w=\left(\dfrac{F}{A}\right)_{\textrm{ext}}(\Delta V)=\dfrac{\textrm{newton}}{\textrm m^2}\times \textrm m^3=\mathrm{newton\cdot m}=\textrm{joule}$

Whether work is defined as having a positive sign or a negative sign is a matter of convention. In section 5.4 we will learn that positive work or heat is defined as when work or heat is added to the system, which in this case is the gas in the cylinder. So a contracting gas correlates to positive work (the system gains energy) and an expanding gas correlates to negative work, where the system does work on the surroundings. In the above case, the ΔV < 0, but it relates to a positive work flow with energy being added to the system. So based on this convention,

$w = −PΔV$

Exercise $$\PageIndex{3}$$

A system containing oxygen gas is heated at a constant pressure of 40.0 atm so that its volume increases 177 L to 458 L. Express the amount of work that the system did in kilo-joules.

The formula for work from the expansion of a gas at constant pressure

$w=-P_{ext} \Delta V\nonumber$

$w=-40\; atm \times (458\; L - 177\; L) = -11240\; L\; atm \nonumber$

Convert from L atm to joules

$-11240 \; L\; atm \times 101.325\; \dfrac{J}{L\ atm} = -1138893\; J \nonumber$

Convert from J to kJ and round to get the final answer

$-1138893\; J \times \dfrac{1\; kJ}{1000\; J}=-1140\; kJ \nonumber$

$w=-1140\; kJ \nonumber$