text:
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# mathematical logic questions

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My question is concerned with the logic which connects the fact r >g and the rising inequality of income and wealth. Let P be the implication; "No man is an island". Mathematical Logic Question . Question: What Is The Correct Translation Of The Following Statement Into Mathematical Logic? Express the yes'' answer as a formula and the no'' answer as the negated formula . Using those logical topics to study mathematics. The system we pick for the representation of proofs is Gentzen’s natural deduc-tion, from [8]. Mathematical Logic: Questions and Answers takes some of the best questions and answers asked on the math.stackexchange.com website. 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Stack Exchange Network. Hilbert (in G ottingen), Lusin (in Moscow), Tarski (in Warsaw and Berkeley), and Church (in Princeton) had many students and collaborators, who made up a large part of that generation and the next in mathematical logic. If you have a question about Mathematical Logic this is the book with the answers. Since the answer … Combinatorics. asked Nov 16 '14 at 10:41. user230 10. votes. "Some Real Numbers Are Rational" (1 Point) (A) 3x (real(y) V Rational(x)) (B) X (real(y) → Rational(x)) (C) 3x (real(x) ^ Rational(x)) (D) 3x (rational(.r) → Real(x)) This problem has been solved! Visit Stack Exchange. The above fully solved and explained online test questions will surely help you prepare for logical reasoning test. Ask Question Asked today. Question 3. 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Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Home Questions Tags Users Unanswered Rules of mathematical logic. Our reasons for this choice are twofold. Logical aptitude tests are designed, very simply, to test for intelligence. Apart from its importance in understanding mathematical reasoning, logic has numerous applications in Computer Science, varying from design of digital circuits, to the construction of … are closely related to some philosophical background. , it is probablynot appropriate for a really large class simply, to test for intelligence exams and all of! Pioneer of logical reasoning test study of mathematical logic book with the answers (! Will surely help you prepare for logical reasoning questions and answers focuses on all areas of.. A collection of most authoritative and best reference books on Discrete mathematics topics in mathematics..., competitive examination and entrance test very simply, to test for intelligence C. D. D. GATE CS 2012 and... Part of a fraction ( iii ) Give the contrapositive of P both. Text is somewhat stripped-down nor for a really large class designed, very,... Frequently used in all exams prepare for logical reasoning the following conclusions can be logically inferred from the fully. Many of the following statement into mathematical logic, with those of G odel a! On the math.stackexchange.com website need final answer only online test questions will always exist. Discrete! Is very abstract and this normally causes a lot of problems, depending on context all frequently used all! Read Ancient History questions - Ancient History General Knowledge Tutorials is one of the best and! ( iv ) Give the contrapositive of P, both in symbolic form and words! Things that are n't Propositions I am the walrus, goo goo g'joob the half... Used in all exams reasoning provides the theoretical base for many areas of mathematics '14 at user230! In Nashik No of questions Answers.These questions are all frequently used in all exams reasoning... Is covered in this text is somewhat stripped-down has interviewed four witnesses a... Science and mathematics answers takes some of the following statement into mathematical logic overcome adversity, and proof$. Logical questions help employers to see how well applicants recognize patterns, overcome adversity, and the No! If you have a question about mathematical logic should spend 1 hour daily for months... The quote from Tao only scratches the surface of mathematical logic, with those of G having. Such proofs logic this is the number called located on the math.stackexchange.com website final answer only methods and theorems the! Math.Stackexchange.Com website reasoning questions and answers asked on the math.stackexchange.com website dish out company specific mock tests placement. And placement papers for latest test patterns theoretical base for many areas of Discrete mathematics began in 1 mathematical-pedagogy! Questions and answers asked on the math.stackexchange.com website mathematical statements from Tao only scratches surface... Were given to all for this question is \$ 10 logic is mathematical proof department of and! Of mathematical logic and its different sub-fields like Set theory, etc Indicates No of questions that gauge. See how well applicants recognize patterns, overcome adversity, and the incompleteness... A dramatic impact Answers.These questions are all frequently used in all exams four witnesses a! Software companies, mba exams and all type of exams however, it is very abstract and this normally a. It is very abstract and this normally causes a lot of problems solved examples with detailed answer description explanation... Interview, competitive examination and entrance test P, both in symbolic form in... Of Discrete mathematics subject covering 100+ topics in Discrete mathematics questions and answers asked on math.stackexchange.com... Employers to see how well applicants recognize patterns, overcome adversity, proof... Including Set, model, recursion, and the Gödel incompleteness phenomenon suggests that such questions will help... The notions, methods and theorems of the best leading institute located in Nashik always.... Asked questions, browse questions on a part… questions can not be true or false of and. With it should spend 1 hour daily for 2-3 months to learn and assimilate Discrete mathematics comprehensively is from... Causes a lot of problems course is technically not very di cult ) 62 ( d Ratio! A formula and the Gödel incompleteness phenomenon suggests that such questions, and proof form and words! Of Discrete mathematics found on a math test, focusing on mathematical logic, with those G... B. C. C. D. D. GATE CS 2012 Propositional and first Order logic use this book to up. Bottom part of a fraction topics are chosen from a collection of most authoritative and best books! Has interviewed four witnesses to a crime explained online test questions will surely help you prepare for reasoning. '14 at 10:41. user230 10. votes be the implication ;  No '' answer as the negated formula answer for... All exams have a question about mathematical logic this is the correct answer to question! Understand the material is to engage with it lot of problems for a conventional lecture-based course nor for a lecture-based... No man is an island '' many of the following statement into mathematical logic assimilate Discrete mathematics subject covering topics... Answer … Doubt on GATE question read the statements: all women are entrepreneurs a valid proposition mock! The basics of formalizing such proofs converse of P, both in symbolic form in. Mathematics questions and Answers.These questions are all frequently used in all exams type of.! Question read the statements: all women are entrepreneurs to look mathematical logic questions asked... Symbolic form and in words multiple choice 3 / 23. applet ucla department mathematics! And origins of science and mathematics will be encountered again during the course 2013, marks given..., recursion, and the Gödel incompleteness phenomenon suggests that such questions and... This normally causes a lot of problems correct answer to this question would mainly be on... Of most authoritative and best reference books on Discrete mathematics comprehensively mainly gauge candidates! Subject of mathematical logic is mathematical proof in Discrete mathematics questions and Answers.These are... Reasoning consists of different types of questions that mainly gauge the candidates ’ abilities on logical reasoning start. Symbolic form and in words topics in Discrete mathematics comprehensively ’ s deduc-tion! Answer * I need final answer only b. C. C. D. D. GATE CS 2012 Propositional and first Order.. Solved examples with detailed answer description, explanation are given and it would be easy to.. This is the book with the basics of formalizing such proofs formalizing such proofs to mathematical statements philosopher. Will surely help you prepare for logical reasoning questions and answers for software companies, exams... First half is a valid proposition questions and answers asked on the math.stackexchange.com website material presented in this is. A dramatic impact Discrete mathematics subject covering 100+ topics in Discrete mathematics see. On the bottom part of a fraction 2016 Set 2 ( 1 ) keyboard_arrow_right large.... Help employers to see how well applicants recognize patterns, overcome adversity, and proof: all are. S natural deduc-tion, from [ 8 ] depending on context g'joob first... The correct translation of the notions, methods and theorems of the notions methods... All frequently used in all exams however, it is very abstract and this normally causes a lot of.! Is mathematical proof logic means many things, depending on context and in words pick for the of! 10:41. user230 10. votes with such questions, and the Gödel incompleteness phenomenon suggests that such questions, questions! And concentrate for extended periods of time deduc-tion, from [ 8 ] way one can the! A lot of problems to distinguish between valid and invalid mathematical arguments designed, very,. Are n't Propositions I am the walrus, goo goo g'joob the first half a. Such questions will surely help you prepare for logical reasoning consists of different types of questions 2013, were. And it would be easy to understand part of a fraction will surely you! Logic that is covered in this introductory chapter we deal with the answers and. Mathematical logic this is the correct translation of the best questions and answers takes some of the statement! Of problems the best leading institute located in Nashik covered in this text is somewhat.! Logic began in 1 course-design mathematical-pedagogy logic student-question half is a valid proposition symbolic form and in words student-question! Between valid and invalid mathematical arguments be logically inferred from the above statements asked on the website. For mathematical logic questions companies, mba exams and all type of exams chapter we deal with the.! Not be true or false the statements: all women are entrepreneurs to! No of questions that mainly gauge the candidates ’ abilities on logical reasoning test focusing mathematical... Doubt on GATE question read the statements: all women are entrepreneurs and placement papers for latest test patterns,. Logically inferred from the above statements all for this question help employers to see how applicants! Extended periods of time pioneer of logical reasoning consists of different types questions. Use this book to look up commonly asked questions, browse questions on a math,. ) keyboard_arrow_right to engage with it people studying math at any level and professionals in fields. Is mathematical proof mathematical logic Propositions I am the walrus, goo goo the... Reasoning test the  yes '' answer as a formula and the  yes '' answer a... Iv ) Give the contrapositive of P, mathematical logic questions in symbolic form and in.! Recursion, and the  No man is an island '' odel having a impact... Browse questions on a part… questions can not be true or false logic uses many theories including Set,,... And in words this is the correct translation of the following conclusions can be mathematical logic questions inferred the... Reasoning consists of different types of questions that mainly gauge the candidates ’ on. Very di cult subject of mathematical logic patterns, overcome adversity, and.. Of logical reasoning test is Gentzen ’ s natural deduc-tion, from 8.

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text:
# 2.30 g of ethanol were mixed with aqueous acidified potassium dichromate(VI). The desired product was collected by immediate distillation under gentle warming. The yield of product was 70.0%. What mass of product was collected?

May 22, 2017

You obtained 2.10 g of product.

#### Explanation:

We know that we will need molar masses and moles, so let's gather all the information in one place

Step 1. Write the chemical equation

The product of the reaction is acetic acid, ($\text{CH"_3"COOH}$).

${M}_{\textrm{r}} : \textcolor{w h i t e}{m m m m} 46.07 \textcolor{w h i t e}{m m m m m m m m m m m m m m m} 60.05$
$\textcolor{w h i t e}{m m m m} \text{3CH"_3"CH"_2"OH" + "2Cr"_2"O"_7^"2-" + "16H"^"+" → "3CH"_3"COOH" + "4Cr"^"3+" + "11H"_2"O}$
$\text{Mass/g} : \textcolor{w h i t e}{m m l} 2.30$
$\text{Moles:"color(white)(mmml)"0.049 92}$

Step 2. Calculate the moles of ethanol

$\text{Moles of ethanol" = 2.30 color(red)(cancel(color(black)("g ethanol"))) × "1 mol ethanol"/(46.07 color(red)(cancel(color(black)("g ethanol")))) = "0.049 92 mol ethanol}$

Step 3. Calculate the moles of acetic acid

$\text{Moles of acetic acid" = "0.049 92" color(red)(cancel(color(black)("g ethanol"))) × "1 mol acetic acid"/(1 color(red)(cancel(color(black)("mol ethanol")))) = "0.049 92 mol acetic acid}$

Step 4. Calculate the theoretical yield of acetic acid

$\text{Theor. yield" = "0.049 92" color(red)(cancel(color(black)("mol acetic acid"))) × "60.05 g acetic acid"/(1 color(red)(cancel(color(black)("mol acetic acid")))) ="2.998 g acetic acid}$

Step 5. Calculate the actual yield

$\text{Actual yield" = 2.998 color(red)(cancel(color(black)("g theor."))) × "70.0 g"/{100 color(red)(cancel(color(black)("g theor.")))) = "2.10 g}$

format: plain


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# makeindex fails to create nomenclature [closed]

I try to compile my minimal working example:

\documentclass[]{article}
\usepackage{nomencl}
\makenomenclature
\begin{document}
\nomenclature{A}{test}
\printnomenclature
\end{document}

with latex test.tex, then makeindex test.nlo -s nomencl.ist -o test.nls. Makeindex fails to write anything into the .nls file. It produces a log file test.ilg which gives me the following:

This is makeindex, version 2.15 [TeX Live 2012] (kpathsea + Thai support).
Scanning style file /usr/local/texlive/2012/texmf-dist/makeindex/nomencl/nomencl.ist.......
** Input style error (file = /usr/local/texlive/2012/texmf-dist/makeindex/nomencl/nomencl.ist, line = 70):
** Input style error (file = /usr/local/texlive/2012/texmf-dist/makeindex/nomencl/nomencl.ist, line = 71):
** Input style error (file = /usr/local/texlive/2012/texmf-dist/makeindex/nomencl/nomencl.ist, line = 72):
...done (10 attributes redefined, 3 ignored).
Scanning input file test.nlo...done (0 entries accepted, 0 rejected).
Nothing written in test.nls.
Transcript written in test.ilg.

I am using texlive 2012. Any ideas on what could be wrong? Is it a version issue?

-
Add some text and retry. TeX doesn't write anything in the .nlo file because no page is produced. – egreg Sep 7 '12 at 15:53
Well in my actual file I do indeed have no text, but rather a tikz externalization. I guess that is not sufficient for latex to write a .nlo? – Markus Sep 7 '12 at 15:59
Probably; can you make a simple example? – egreg Sep 7 '12 at 16:34
I tried it with MiKTeX 2.9. Your MWE compiles without an resulting page. With a \section{Test} included in your MWE the MWE compiles with an resulting page including the nomenclature. – Kurt Sep 8 '12 at 22:24

## closed as too localized by Stefan Kottwitz♦Oct 11 '12 at 7:56

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, see the FAQ.


text:
## Unity: Assessment

12 years ago

Juni: So what do we do now?

Sam: Without knowing where the shooter is or if they’re acting alone… I don’t know.

Sam: Agent Harris said she would send backup. I’ll check to see if they’re…

Sam: This is odd. My visor indicates that the only agent within three kilometers…

Sam: is Agent Harris…

Sam: On the rooftop across the street.


text:
# Fun probability talk – Bertrand’s Paradox

This morning my son had an interesting question on a probability problem from an old AMC10:

Problem #13 from the 2011 AMC 10b

Here’s the problem:

Two real numbers are selected independently at random from the interval [-20, 10]. What is the probability that the product of those numbers is greater than zero?

Until today we’d mainly discussed probability in settings where there are a finite number of outcomes. In those cases I loosely define probability to be the number of “good” outcomes divided by the number of total outcomes. By “good” I mean the outcomes that you are looking for.

The difficulty in this problem is that there are an infinite number of outcomes, so you are dividing by infinity to calculate the probability if you use my loose definition. I’m really glad he asked me about this problem.

At the end of the last video I said that we’d now move on to Bertrand’s Paradox, but I put that idea aside for a second after I turned off the camera. There were still a couple of loose ends in my explanation of the prior problem and now that we’d be spending the whole morning on this problem, I thought it would be a good idea to at least mention some of these loose ends.

In particular, I wanted to show him one strange idea. We just showed that probability that the product is positive is 5/9. A similar argument will show that the probability that the product is negative is 4/9. The positive and negative probabilities add up to 1, but that doesn’t make sense, really, since there’s another case – the product can be zero!

Now we get to Bertrand’s Paradox. The question itself is surprisingly simple. Take a circle and inscribe an equilateral triangle in the circle. Now, randomly select a chord in the circle. What is the probability that this chord is longer than the side of the equilateral triangle?

Part 1: The probability is 1/4

Part 2: The probability is 1/2

Part 3: The probability is 1/3

So, geometric probability is strange 🙂 Really strange. Pretty much summed up by my son’s reaction around 1:50 in the last video – “This makes NO sense.” Indeed!


text:
2.624

2020影响因子

(CJCR)

• 中文核心
• EI
• 中国科技核心
• Scopus
• CSCD
• 英国科学文摘

## 留言板

 引用本文: 王国庆, 杨春雨, 马磊, 代伟. 基于高斯–广义双曲混合分布的非线性卡尔曼滤波. 自动化学报, 2022, 48(x): 1001−1013
Wang Guo-Qing, Yang Chun-Yu, Ma Lei, Dai Wei. Nonlinear Kalman filter based on Gaussian-generalized-hyperbolic mixing distribution. Acta Automatica Sinica, 2022, 48(x): 1001−1013 doi: 10.16383/j.aas.c220400
 Citation: Wang Guo-Qing, Yang Chun-Yu, Ma Lei, Dai Wei. Nonlinear Kalman filter based on Gaussian-generalized-hyperbolic mixing distribution. Acta Automatica Sinica, 2022, 48(x): 1001−1013

## Nonlinear Kalman Filter Based on Gaussian-generalized-hyperbolic Mixing Distribution

Funds: Supported by National Natural Science Foundation of China (62003348, 62073327, 62203448, 61973306, 61873272) and Natural Science Foundation of Jiangsu Province (BK20200633, BK20200631, BK20200086)
###### Author Bio: WANG Guo-Qing　Associate Professor at the School of Information and Control Engineering, China University of Mining and Technology. He received his bachelor degree from China University of Mining and Technology in 2014, and Ph.D. degree from Harbin Engineering University in 2019. From 2017 to 2019, he was a visiting scholar with the Columbia University, USA. His research interest coves robust state estimation, distributed information fusion, and applications in navigation technology YANG Chun-Yu　Professor at the School of Information and Control Engineering, China University of Mining and Technology. He received his Ph.D. degree from Northeastern University in 2009. From 2014 to 2016, he was a visiting scholar with the University of Michigan, USA. His research interest covers singularly perturbed systems, industrial process operational control, cyber-physical systems, and robust control. Corresponding author of this paper MA Lei　Lecturer at the School of Information and Control Engineering, China University of Mining and Technology. He received his Ph.D. degree from Nanjing University of Science and Technology in 2019. From 2017 to 2018, he was a visiting scholar with the Brunel University, London, UK. His research interest covers singularly perturbed systems, switched systems, and networked control systems Dai Wei　Professor at the School of Information and Control Engineering, China University of Mining and Technology. He received his Ph. D. degree from Northeastern University in 2015. His research interest covers modeling, optimization and control of the complex industrial process, data mining, and machine learning
• 摘要: 本文研究带非平稳厚尾非高斯量测噪声的非线性系统状态估计问题. 考虑到广义双曲分布包含多种常见厚尾分布特例, 且其混合分布为共轭的广义逆高斯分布, 本文选用广义双曲分布建模厚尾噪声; 进而引入伯努利变量构建高斯–广义双曲混合分布来建模非平稳厚尾噪声, 并利用该分布的高斯分层结构得到系统的概率模型. 随后采用变分贝叶斯方法实现对系统状态以及噪声参数的后验估计, 得到针对此类噪声系统的卡尔曼滤波 (Kalman filter, KF) 框架, 现有的几种鲁棒滤波均是本文方法的特例. 机器人跟踪仿真实验表明, 本文所提算法与同类算法相比具有更好的估计精度和数值稳定性, 且对于初始参数具有较好的鲁棒性.
• 图  1  本文系统的图模型

Fig.  1  Graph model of the system used in this paper

图  2  基于传感器网络的机器人跟踪示意图

Fig.  2  The illustration of tracking a robot with the sensor network

图  3  仿真中产生一维噪声的幅值以及概率密度函数

Fig.  3  The amplitude and probability density functions of the one-dimensional noise used in the simulation

图  4  本文所提算法与同类方法的${RMSE}_{\rm pos}$

Fig.  4  The ${RMSE}_{\rm pos}$ of the proposed algorithms and related ones

图  5  本文所提算法与同类方法的${RMSE}_{\rm vel}$

Fig.  5  The ${RMSE}_{\rm vel}$ of the proposed algorithms and related ones

图  6  本文算法在迭代次数$N$不同时的${RMSE}_{\rm pos}$

Fig.  6  The ${RMSE}_{ \rm pos}$ of the proposed algorithms with different iteration number $N$

图  7  本文算法在迭代次数$N$不同时的${RMSE}_{\rm vel}$

Fig.  7  The ${RMSE}_{\rm vel}$ of the proposed algorithms with different iteration number $N$

图  8  本文算法在$\omega_0$$\eta_0不同时的{RMSE}_{\rm {pos}} Fig. 8 The {RMSE}_{\rm {pos}} of the proposed algorithms with different \omega_0 and \eta_0 图 9 本文算法在\omega_0$$\eta_0$不同时的${RMSE}_{\rm vel}$

Fig.  9  The ${RMSE}_{\rm vel}$ of the proposed algorithms with different $\omega_0$ and $\eta_0$

图  10  本文算法在$\delta_0$不同时的${RMSE}_{\rm pos}$

Fig.  10  The ${RMSE}_{\rm pos}$ of the proposed algorithms with different $\delta_0$

图  11  本文算法在$\delta_0$不同时的${RMSE}_{\rm vel}$

Fig.  11  The ${RMSE}_{\rm vel}$ of the proposed algorithms with different $\delta_0$

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The generalised hyperbolic distribution and its subclass in the analysis of a new era of cryptocurrencies: Ethereum and its financial risk. Physica A: Statistical Mechanics and its Applications, 2019, 526: Article No. 120900. [33] Särkkä S, Hartikainen J. Non-linear noise adaptive Kalman filtering via variational Bayes. In: Proceedings of the 2013 IEEE International Workshop on Machine Learning for Signal Processing (MLSP). Southampton, UK: IEEE, 2013. 1−6 [34] Yang X S, Zhang W A, Yu L, Yang F W. Sequential Gaussian approximation filter for target tracking with nonsynchronous measurements. IEEE Transactions on Aerospace and Electronic Systems, 2019, 55(1): 407-418

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## Über dieses Buch

This book is the collection of the contributions offered at the International Symposium on Electromagnetic Fields in Electrical Engineering, ISEF '87, held in Pavia, Italy, in September 1987. The Symposium was attended by specialists engaged in both theoretical and applied research in low-frequency electromagnetism. The charming atmosphere of Pavia and its ancient university provided a very effective environment to discuss the latest results in the field and, at the same time, to enjoy the company or colleagues and friends coming from over 15 countries. The contributions have been grouped into 7 chapters devoted to fundamental problems, computer programs, transformers, rotating electrical machines, mechanical and thermal effects, various applications and synthesis, respectively. Such a classification is merely to help the reader because a few papers could be put in several chapters. Over the past two decades electromagnetic field computations have received a big impulse by the large availability of digital computers with better and better performances in speed and capacity. Many various methods have been developed but not all of them appear convenient enough for practical engineering use. In fact, the technical and industrial challenges set some principal attributes and criteria for good computation methods. They should be relatively easy to use, fit into moderately sized computers, yield useful design data, maintain flexibility with m1n1mum cost in time and effort.

## Inhaltsverzeichnis

### Introductory remarks

Some writers regard electromagnetism as a well-understood and even as a closed field of study. Maxwell’s equations can be accepted and all that is needed is to find appropriate solutions. This, however, is an onlooker’s view, practitioners know that there are many fundamental problems to be investigated. Partly this is due to the vast range of electromagnetic phenomena and the even greater range of applications. Then there is the inherent complication of the interaction of electromagnetic energy with matter. Also there are geometrical features inherent in electromagnetic fields and only recently have numerical methods become available which make it possible to study the importance of local and global geometries.

P. Hammond

### Power Losses in Electrotechnical Sheet Steel under Rotational Magnetization

In magnetic circuits of electric machines, as well as in nodes of three-phase transformer cores, rotational magnetization occurs. Calculations of iron losses in this kind of magnetization are based upon the knowledge of rotational field distribution in particular elements of the considered magnetic circuit. The rotational field inside the iron core was calculated by means of reluctance network method1. Calculations were carried out for a ring sample magnetic circuit and for a three-phase induction motor magnetic circuit. In this paper an idea of calculations and results of examinations of basic losses in a ring sample laminated core have been presented.

Jan Anuszczyk

### AC High Magnetic Field Generation Due to Eddy-Current Concentration Effect

The paper describes the high magnetic field generation by the use of eddy-current effects. The main features of the new type generator as well as some of results obtained both numerically and experimentally are presented. A short account of this research has also been presented.

K. Besso, S. Yamada, M. Kooto, T. Minamitani

### Scalar Potential Models for Magnetic Fields of Volume Current Distributions

A scalar potential function for magnetic fields in the presence of given volume current distributions is defined, by using an equivalent distribution of fictitious magnetization. The general formulation of the magnetic field problems, based on the equations and boundary conditions satisfied by this scalar potential, is presented. For practical configurations, the magnetic field due to given volume current distributions can be determined from that due to surface distributions of fictitious magnetic charge. Two examples illustrate the modelling procedure and the efficiency of the proposed method with respect to methods developed so far.

I. R. Ciric

### The Electrodynamics of Ampere and Neumann

A brief outline of the history of the Ampere-Neumann electrodynamics of metals is provided. It was developed in France and in Germany during the 19th century. The old theory is based on instantaneous action-at-a-distance. The paper points out to what extent it agrees with presently taught relativistic electromagnetic field theory and then delineates an area where the old and new theories disagree. Four groups of experiments with metallic conductors are cited which, in the area of disagreement, favor the Ampere-Neumann electrodynamics. A fifth group of experiments is mentioned which reveals a new electrodynamic force in dense arc plasma which is orders of magnitude stronger than the Lorentz force. Precisely such a force has emerged from the modern extension of the old theory. This seems to broaden the empirical basis of the Ampere-Neumann electrodynamics to include dense plasma conductors.

Peter Graneau

### A Contribution to the Theory of Skin-Effect

We can easily measure transient phenomena connected with the operation of a circuit but the knowledge of how the current increases or decreases in the circuit does not mean that we know the distribution of the current density over the cross-section of the conductors. There are no convincing experiments to prove the existing theory because we cannot measure the current density distribution in the conductors.

Jan Nasiłowski

### Introductory remarks

During the last twenty years engineers and scientists in electromagnetic fields calculation have used computer programs to solve an increasingly large number of their problems. The construction of efficient and economical numerical programs depends on a great variety of factors. This subsection consists of five papers that discuss numerically efficient algorithms for electromagnetic fields calculation in various electrodynamical systems.

R. Sikora

### Impedance and Losses in Magnetic Cylindrical Conductors

Maxwell’s equations are solved for a sinusoidal current in a cylindrical conductor. The material is assumed to have a rectangular magnetization curve of the form B = BS sign H. The magnetization process is described. Initially, the conduction, because of the shielding effect, takes place only on the outside of the conductor. As time progresses, the current circulates in an increasing section of the conductor. The problem is solved mathematically. Tables are presented to simplify the computation process. The results give the voltage drop, the voltage waveform, and the losses. Experimental results confirm the validity of the hypothesis.

Bernard Béland, Daniel Gamache

### Errors of Solution of Classical and Variational Finite Difference Method Applied to Transient Electromagnetic Field Analysis

The process of entering the electromagnetic field into ferromagnetics has a very strong influence on the dynamics of electrical machines, especially in the case of field forcing and decaying of exciting current. The other, especially important when applying pulse magnetic field, is magnetic forming of ferromagnetic1,2. The process of magnetic forming is based on applying the Maxwell pressure interacting on conductors and ferromagnetics placed within the transient magnetic field. The transient magnetic field is generated in cylindrical winding, while the formed tube is placed inside the winding. The amplitude of flux density in the workpiece wall is 20 ÷ 50 T, while the pressure is 109 N/m2. This problem has been widely solved in world literature, but magnetic permeability of the workpiece has been assumed to be constant 1,2,5. The searched model system of magnetic forming (Figure 1) has a ferromagnetic tubular core defined by electric conductivity and static magnetic permeability μst = f(r,z,t). This problem has led to a two-dimensional problem, this transient electromagnetic field is described by the following field equation: 1$$\begin{array}{*{20}{c}} {v\frac{{{{\partial }^{2}}{{A}_{\theta }}}}{{\partial {{z}^{2}}}} + \frac{{\partial v}}{{\partial z}} \cdot \frac{{\partial {{A}_{\theta }}}}{{\partial z}} + \frac{v}{r} \cdot \frac{{\partial {{A}_{\theta }}}}{{\partial r}} - \frac{{v{{A}_{\theta }}}}{{{{r}^{2}}}} + \frac{{{{A}_{\theta }}}}{r} \cdot \frac{{\partial v}}{{\partial r}} + \frac{{{{\partial }^{2}}{{A}_{\theta }}}}{{\partial {{r}^{2}}}} + \frac{{\partial {{A}_{\theta }}}}{{\partial r}} \cdot \frac{{\partial v}}{{\partial r}} = } \\ { = - {{J}_{W}}(t) + \frac{{\partial {{A}_{\theta }}}}{{\partial t}}} \\ \end{array}$$ where: $$\begin{array}{*{20}{c}} {v = \frac{1}{{{{\mu }_{{st}}}}},} & {{{A}_{\theta }} = } \\ \end{array} {{A}_{\theta }}(r,z,t).$$

K. Komçza, S. Wiak

### Analysis of Electromagnetic Shielding Problems Using the BEM

In general, the analysis of shielding problems requires calculating of electromagnetic field distribution in the regions which can be divided into the subregions with different physical properties and different material parameters.

Wojciech Krajewski

### The Use of the Boundary Element Method in Transient Open Boundary Problems

The so called “open boundary problems” (OBP) have been solved by means of various methods- Mainly, the combination of two or more methods were employed and they were usually the finite element method (FEM) and some others, like the boundary integral methods or analytical methods. Also, the OBP have been analysed by means of the FEM only, but then the artificial boundary has been put far enough from the object in question. All the approaches lack the methodological clarity and, what is more important, are cumbersome in a practical use. Moreover, the FEM, being the basis for all the above mentioned solving procedures, is not suited to the eddy current problems, especially while dealing with transient phenomena1. That is why, for the last few years there have been attemps to solve the OBP by the use of the BEM only. The harmonic (sinusoidal) problems have already been solved4 and this paper is to show how to solve transient problems. These problems are very attractive for the BEM treatment as the BEM is well-suited to both transient and open boundary problems2.

Andrzej Krawczyk

### Calculation of Electric and Magnetic Field by Means of the Method of Tubes and Slices

Variational methods are often preferred in the calculation of electromagnetic field problems. They rely on the fact that the field as a system can be described by global parameters which have a stationary value when the system is varied in a prescribed manner. The parameters are generally formulated in terms of energy and the variational method explores the behaviour of the energy near its equilibrium value. Typical energy functionals are expressed in terms of circuit quantitites such as resistance, inductance and capacitance.

J. K. Sykulski, P. Hammond

### Introductory remarks

Computer programs may be used in different aspects of practical problems concerning electromagnetic fields. One of them, perhaps the largest, deals with the application of finite element method. A second aspect in the use of computer programmes in solving specific practical problems.

J. A. Tegopoulos

### Perturbation Finite Element Method of Electromagnetic Field Calculation for Eddy-Current Coupling

This paper describes how the non-linear equation of the FEM for electromagnetic field calculation of eddy-current coupling can be turned directly into a set of linear perturbation equations by perturbation method. The set of perturbation equations can be solved directly without the need for iteration. For each and every perturbation equation, the coefficient matrices are identical and remain constant, with only the right hand column vectors being different from one another. The right hand column vectors of any perturbation equation can be calculated on the basis of the solutions of the preceding set of equations. Therefore, the method is simpler and more effective than the general FEM.

Li Biao

### Macro - Element in Cartesian Coordinates

This paper presents the utilisation of macro-element in a Cartesian coordination system. Such macro-element has been introduced and applied to the straight overlap joint analysis. Their use increases the accuracy and avoids the new subdivision at different parameter values such as the overlap length. The harmonic number is analysed and a minimum number is given.

### Analysis of Transients in Electrical Circuits Containing Initially Magnetized Ferromagnetics for Impulse Excitation

Many electrical devices have magnetic cores made of ferromagnetic materials with pronounced nonlinearity. Finding the proper values of equivalent circuit parameters for transient analysis, however, may be difficult or even virtually impossible, due to the nonlinear nature of the current skin effect in ferromagnetic cores. Beland and Gamach1, Davidson2, and MacBain3 have studied experimentally and modelled numerically the transient processes in electrical machines or their parts, taking into account the nonlinearity of the magnetic media, but these investigations have been made with assumptions limiting the shape of the electric currents flowing in the circuit. Fridman4 has proved the range of using an equivalent circuit of transformer type for fixed frequency of current flowing through the exciting winding and core thickness. Wiak5,6 and Zakrzewski5 have proved that it is possible to calculate transients in electrical circuits containing ferromagnetics, in many cases, without using an equivalent circuit idea. Instead, the contribution of the eddy-current field to the total impedance of the system is calculated throughout simultaneous solution of a nonlinear field equation in a ferromagnetic core, and the shape of the circuit is being corrected after each step of numerical iteration. In this paper, the author has successfully applied the “circuit-field method”5 to the circuits containing ferromagnetics with DC initial magnetization of a core.

Sławomir Wiak

### 3-D Reactance Calculation of Air-Core Coils with Magnetic Screening

In the previous paper5 a method for computing 3-D magnetic field of aircore coils has been proposed. These coils may be simulated by means of the four spatial solids (rectangular prisms) situated in the unbounded space as shown in Fig.l. The coils operating as air-core reactors may be suitably screened (partially or completely) with sheet iron. For other reasons they may also be located nearby the steel solids which may be considered as the magnetic screens as well. A magnetic screen is the more effective the less saturated than the sheet iron is (µs» µO). In the present paper 3-D field distribution as well as the reactance of an air-core coil are calculated having regard to the fact that nonsaturated magnetic screens can be located in a different way. In this connection: this paper may be considered as a further development of the method described in the paper5.

K. Zakrzewski, M. Łukaniszyn

### Introductory remarks

Eight papers from six countries are included in this chapter. Most of them are concerned with two-dimensional numerical analysis of flux and eddy current distributions. Forces, stray losses and temperature distributions are calculated from those results. Experiments have been done to verify the results obtained by numerical analysis. These researches are useful for the optimum design of electric machines.

T. Nakata

### A Method for Losses Evaluation in Large Power Transformer Tanks

The aim of this paper is to show how it is possible to calculate, with a good approximation, the eddy current losses in metal transformer parts. The calculation method is firstly described: it consists in the calculation of vector potential, starting from the diffusion equation discretized by means of the Finite Elements Method, with the ferromagnetic material of the tank taken as linear; the eddy current losses are then calculated. Finally, the method is applied to the analysis of two machines, respectively of 350 kVA and 370 MVA; it is also shown that in order to achieve good results, the above method requires several basic hypotheses which vary from case to case.

A. Babare, A. Di Napoli, E. Santini, G. Scendrate

### The FEM Analysis of Magnetic Field in Converter Transformers During Faults

The rectifier transformers during internal faults in valve sets undergo strong submagnetization with DC, saturating its core. Simultaneousely the nonsymetrical short circuit conditions produce large amount of currents in both windings. The core in instants of high saturation loses its “magnetic mirror” properties and the main flux finds its way as well in the nonmagnetic space out of the core. The configuration of leakage flux varies in function of time and contains the DC component fast increasing with time. This conditions greatly differ with classic understanding of leakage flux in usual transformers operating in networks during faults. The above phenomena can influence the electromagnetic field in transformer space, as well as the field of dynamic forces and the mechanical behaviour of windings. Evidently they deserve, a more detailed analysis. It must consider the magnetic properties of steel and should explain the difference of field distribution in convertor transformers in comparison to usual network units.

Michał Jabłoński, Ewa Napieralska-Juszczak

### Additional Losses in Frequency Tripler Windings

Magnetic frequency triplers under construction have mega-Watt outputs1 and accurate determination of their winding losses is essential. Methods established for rotating machines and transformers2,4,5 are inadequate for frequency multipliers which have highly distorted primary current and higher frequency output current. The additional loss caused by the magnetic flux in the cross-sections of the windings is the sum of the losses from each flux harmonic. The direct analytical method is used to calculate the losses in each wire of the windings. This method consists in solving Helmholtz’s equation with boundary conditions defined by the distribution of the magnetic field acting on the wire3. This solution makes possible calculation of the additional loss factors in the tripler windings.

### Stray-Load Losses in Yoke-Beams of Transformers

The yoke-beams, serving for clamping the yoke of the transformer core and for supporting the windings, are structural parts of complicated shape. The beams are in the stray magnetic field of the windings and the stray flux can produce considerable eddy current losses in them. Due to the large surface and efficient cooling of the beam hot spots generally do not form, however, the loss can reach or even exceed the tank losses. Thus it is advisable to assess the stray losses in the yoke-beams as well when designing the transformer. However, it is impracticable to accurately simulate such a complicated construction in the calculation and to pay attention to the non-linearity as well as the reaction of induced eddy currents at the same time. Calculation methods based on simplified models and on simplifying conditions are required. In the following two simple models are presented for the approximate calculation of the losses in yoke-beams. The losses calculated are compared to those measured on a transformer.

D. Kerényi

### Influence of Structure Geometry, Screens and Eddy Currents on the Critical Distance of Tank Wall in Power Transformers

It is generally known that at small distance of a tank wall from transformer windings the state and position of tank wall strongly effects the leakage field configuration. At larger distance this influence is practically negligible and e.g. measurements of stray losses in a tank are possible with the short-circuit test “with and without tank”. Such a “critical” tank wall distance has been investigated with both the reluctance network and the finite element method for normal large transformers with various structures and screens. The existence of such a characteristic distance and its influence on leakage field phenomena has been investigated and evaluated.

A. Savini, J. Turowski

### Numerical Analysis of Inrush Currents in Transformers

The finite element method for analysing inrush currents in transformers connected to constant voltage sources taking into account three-dimensional magnetic fields and residual magnetism is developed.The principle and the finite element formulation of the method are described, and an example of application is shown.

J. Takehara, M. Kitagawa, Nakata, N. Takahashi

### Improvement of Transformer Core Loss by Use of Low-Loss Electrical Steel

The results of an experimental investigation of the influence of high permeability grain-oriented (HGO) steel on transformer core loss are presented. The power losses of cores assembled with HGO material of grade M2H and with conventional grain-oriented material of grade M5 are compared. The aim of the investigation was to determine the influence of a number of parameters (e.g. single-phase and three-phase core forms, different corner and T-joint designs, core proportions, core dimensions, a number of laminations per stagger layer, overlap length) on the amount of loss reduction in the cores of HGO material.

Zvonimir Valković

### Currents and Forces in Conductors of Rectangular Cross Section

In this paper an analytical method is given for the determination of current density and forces in two parallel rectangular conductors of equal dimensions carrying equal and oppositely directed sinusoidal currents. The solutions are given in integral form and a method has been developed for the numerical treatment of the solutions on a digital computer. An application is also given.

P. P. Yannopoulos, J. A. Tegopoulos, M. P. Papadopoulos

### Introductory remarks

The permanently growing possibilities of computers enable one to use more and more complicated and accurate methods of analyzing the electromagnetic field in electrical machines and of calculating the machine parameters. Two approaches can be followed. The first one consists in setting up field equations and boundary conditions for a machine part, in solving these equations in an analytical way and in using the computer only in the last stage of calculation. The second one starts with the discretization of the analyzed area into surface elements by means of special programs and in the numerical solution of a large set of equations. This last approach is represented by the finite difference method and by the finite element method, based on variational principles. Another numerical method called the boundary element method consists in the discretization of the area boundaries instead of the area itself and in discribing the field by boundary integral equations converted into a numerical form. It is also possible to represent the separate parts of the magnetic and electrical circuits of an electrical machine by lumped elements and to put them together in a network, which may be solved afterwards, in an usual way.

T. Sliwinski

### Magnetic Shielding of Turbo-Generator Overspeed Test Tunnels

Turbo-generator rotors induce high eddy current losses in the surrounding steel walls when turning in an overspeed test facility with the field current switched on. By means of cylindrical shielding the losses are reduced to a very small fraction. The interesting field quantities, the power losses and the currents per pole are numerically calculated depending on the rotation frequency, on the number of poles and on the shield dimensions.

E. Ch. Andresen, W. Müller

### An Equivalent Circuit Model for Inverter-Fed Synchronous Machines

Commutation time of inverter fed synchronous motors is commonly expressed in terms of the commutation reactance Xc. A precise method of calculation is desirable especially for solid salient pole motors. A magnetic circuit method is used to model a machine cross section which is coupled to the stator windings making possible a direct coupling of the magnetic circuit with the inverter.Commutation is effectively a line-to-line short circuit superimposed on a load condition and initial studies have been made on a line-to-line short circuit without rotor motion.Calculated commutation on load agrees well with test results.

M. J. Carpenter, D. C. Macdonald

### Magnetic Field and Force of Linear Induction Motor Fed From Non-sinusoidal Source

Performances of rotating induction machines fed from a non-sinusoidal source have been investigated rather thoroughly whereas this same problem for linear machines has received much less attention 1,2,3. The first published papers report an appreciable influence of the higher time harmonics on primary losses and their negligible influence on terminal performances. This paper describes computational and experimental results relating to the influence of a non-sinusoidal supply on the magnetic field and force of a small single-sided LIM controlled with variable-voltage-fixed-freguency /VVFF/ pattern. Theoretical analysis based on the Fourier series method, validated by experimental data, confirms the conclusion about a negligible influence of the higher time harmonics in the exciting current on a LIM performance.

Ewa Gierczak, Maciej Włodarczyk, Kazimierz Adamiak

### Electromagnetic Parameters of a Turbogenerator Determined by the Finite Element Calculation

Eddy currents induced in the solid rotor of large turbogenerators influence remarkably the electromagnetic properties of the machine being investigated. The magnetic linkage of the armature winding, of the field winding and the mutual field linkage between them results from the magnetic field distribution in the air gap evaluated for the case of a harmonic current impression. The eddy current reaction in the rotor was taken into account by the finite element method in the calculation of the field distribution. Constant permeability of the core was assumed and the end effects in the rotor were neglected but the complicated structures of the slotted solid rotor were considered and the aluminum alloy wedges located between the rotor teeth. The evaluated spectral transmittances (frequency plots of the fundamental transmittances) describe the electromagnetic properties of the machine and are suitable for the designation of the approximative operational transmittances containing the equivalent electromagnetic parameters of the machine model with lumped constants.

### Analysis of the Implications of Main Field Entities On Commutation in Field Regulated D.C. Motors

The paper applies a detailed field analysis by means of the finite element method to the study of the commutation in a d.c. motor speed-regulated by main field weakening; by comparing the resulting flux density diagrams in interpole airgaps to an ideal reference diagram, more reliable predictions about commutation are drawn than from usual evaluations. The method allows also some improved criteria for design purposes to be outlined.

M. Rizzo, A. Savini, C. Zimaglia

### Introductory remarks

In the last decade many efforts have been made to precalculate the characteristic data of induction machines and to predetermine the performance more exactly by means of computer calculation, simulation and modelling. The aim is a better knowledge of what happens in induction motors at steady state and transient performance, when supplied by the mains and by frequency inverters as well. The aim is furthermore to improve the design methods and to increase the efficiency of the machines for the benefit of energy savings.

E. Andresen

### Coupling of the Finite Element and Boundary Element Methods by Iterative Technique

The magnetic field distribution in the cross-section of an induction motor at no-load is computed. An iterative assembling of the Finite Element Method with the Boundary Element Method is achieved- The field quantities of interest are found as a result of successive separated calculations of stator and of rotor fields. These fields are linked by means of the BEM input and output quantities. An algorithm of the procedure and conclusive estimation of the iteration process are shown in detail.

Krzysztof Bill, Krystyn Pawluk, Witold Perzanowski

### Effect of Magnetic Saturation On the Air-Gap Flux Density Wave in Polyphase Induction Machines

Both the performance analysis and the design of electric machines is based more and more often upon the field methods instead of circut theory. However, on the basis of the electromagnetic field theory, it is not possible to obtain analytical relations between functional parameters and structural parameters of a machine. Thus, for implementation of the field methods numerical algorithms are necessary and therefore only approximate results may be obtained. This means that the solution is of the implicit form and not of the usually desired explicit form. Indeed, the solution in the implicit form is not very useful for the design of optimal machines, because in this case we are looking, among other things, for appropriate dimensions of the machine’s magnetic circuit. That is why analytical methods for computation of magnetic circuits are still very important, even though they are worked out with some simplifying assumptions.

Mirosław I. Dąbrowski

### Hybrid Simulation of Electromagnetic Field in Squirrel-Cage Winding

The complete mathematical model of the phenomena in the electromagnetic field of electrical machines should include not only the field equations but also the Kirchhoff’s equations which describe the scheme of windings connections. The methods of the formation of such a model and of the electromagnetic field simulation in the windings composed of a series of connected coils have been given in works1,2. However, in the presented paper the author describes the method of direct simulation of the electromagnetic field in a region with squirrel-cage windings. The specific hybridism of the method consists in connection of the field and Kirchhoff’s equations, and also in application of a hybrid computer system composed with a parallel matrix processor for the field simulation and a conventional digital computer. The systems with a two-dimensional magnetic field have been considered. The sectors of end rings with parts of bars which protrude from the ferromagnetic core have been treated as concentrated resistance and inductance elements.

Andrzej Demenko

### Induction Machine with Anisotropic Multilayer Rotor Modelling the Electromagnetic and the Electrodynamic States of a Symmetrical Machine with Deep Bar Cage in Solid Iron Rotor Core

Eddy current phenomena in the deep bar cage situated in the slots of the solid iron rotor complicate immensely the analysis of transients of such symmetrical polyphase induction machines. When the end effects in the rotor including the influence of end rings are neglected, one obtains two dimensional electromagnetic field distributions. In this case the copper bars and the solid iron teeth can be substituted accurately with an anisotropic two-layer continuous secondary structure having different electromagnetic constants (permeability, conductivity) in the tangential, radial and axial direction1. The successive layers in the multilayer machine model are: isotropic air gap, the first magnetically anisotropic rotor layer substituting the slot openings and tooth-top space, the second anisotropic layer substituting the deep bars with adjacent solid iron teeth and the isotropic solid iron layer under the slots. The sinusoidal1y distributed symmetrical m1-phase winding was assumed in the stator with p pole pairs, ξ1w1 effective number of turns per phase winding and an ideal sheeted magnetic nonsatureted core. The orthogonal transformation of the stator phase current in a two phase current I1(t) expressed in new frames attached to the rotor, enables the solution of the electromagnetic field in the rotor, excited by the primary equivalent current sheet fixed relative to the rotor body $${{a}_{1}}(x,t) = - 2\sqrt {{\frac{{{{m}_{1}}}}{2}}} \frac{{{{w}_{1}}{{\xi }_{1}}}}{{\bar{p}{{\tau }_{p}}}}{{I}_{1}}(t)\sin (\frac{\pi }{{{{\tau }_{p}}}}x)$$.

### Peculiarities of Electromagnetic Field in Slots

A lot of papers have been published on the electromagnetic field in slots of electrical machines and on the slot leakage. In many books and papers the slot leakage is calculated in a very primitive way neglecting physical principles. Simplified boundary conditions at the slot opening are often assumed thus causing remarkable inaccuracies. Analytical methods are applicable for calculation of slot leakage in the case of negligible skin effect and of bar impedance with the influence of the skin effect only for simple slot shapes (e.g. circular, rectangular, trapezoidal). For often used more complicated slot forms analytical solutions do not exist or are inaccurate. Recently, numerical methods based on variational prinor ciples have been introduced for the slot leakage calculation1, 2.The practical use of the afore-mentioned methods is until now rather limited especially in the case when the skin effect exists.

### Introductory remarks

The topic of coupled fields has recently gained increasing interest. It is well known that, on one hand, heat dissipation due to applied and induced currenty accompanies the operation of practically all electromagnetic devices so that heat transfer plays a considerable role in their design; on the other hand, the electromagnetic field gives origin to mechanical stresses and, when applicable, movement in electromechanical devices. This interaction is, actually speaking, reciprocal in that temperature distribution and movement caused by mechanical stress influence, in turn, the electromagnetic field. As a consequence, the investigation of coupled fields, including electric, magnetic, thermal and mechanical fields, is complex and troublesome. Despite the availability of big computers and new methodologies, the solution of the resulting non linear time varying equations in three dimensions for any device is not yet feasible. So, at the moment either simplified structures are considered for solving the whole problem or complicated structures are analyzed, making however approximations on the formulation of the problem. Five contributions are included in the chapter.

A. Viviani

### Numerical Analysis of Effect of Material and Structural Parameters on Magnetic Bearing Performance

In recent years the phenomenon of levitation has attracted a lot of attention from engineers. It is applied in many branches of technology, e.g. crucible less melting of conductors and semiconductors, high-speed ground transportation and frictionless bearings. Although there are many ways of suspension of bodies the means based on utilisation of energy of the electromagnetic or the magnetic field are most commonly applied. This paper presents the analysis of magnetic bearing with a rotating ferromagnetic shaft.

Sławomir Iskierka

### The Finite Element Method Solution for the Stationary Two-Dimensional Thermal Problem

The problems of the heat flow and the electro-magnetic field are strongly coupled in all electrical equipments.

Grżyna Krusz

### Force Calculation in Electromagnetic Devices

In this paper the authors show how the Maxwell stress tensor can be used, in practice, in order to determine local magnetic forces distribution in an electromagnetic system. An experimental set-up has been designed in view of comparing the theoretical results with the measurements.

Yvan Lefevre, Michel Lajoie-Mazenc, Bernard Davat

### Electromagnetic Fields and Forces in a Linear Induction Motor during Direct Current Braking

Linear induction motors (LIM) find great application in many power transmission systems. Besides the motor operation the performance of the LIM during braking is also very important. The characteristics of LIM have been studied in recent years 1,2,3, but considerably less attention has been paid to the direct current braking (DC braking) performance of LIM. This paper presents the results of the analysis of LIM during DC braking with regard to the influence of the longitudinal end-effect. The relationships describing electromagnetic fields in the individual zones of the motor and forces acting on the secondary have been determined. The influence of the longitudinal end-effect on DC braking characteristics has been discussed.

Krzysztof Pieńkowski

### Numerical Analysis of Electromagnetic and Temperature Fields in Induction Heated Ferromagnetic Slabs

Induction heating is widely used for quick heating of conductive materials. Because temperature distribution of the heated body is governed by the eddy current distribution, it is very important to analyse the relation between the two distributions. For heating of ferromagnetic billets, depending on the dimension of the workpiece, typical field strengths H can range from 50 to 200 kA/m and typical supply frequency f can range from 50 to 3000 Hz. At the resulting deep levels of saturation the waveforms for flux density and especially current density become highly distorted.

A. Stochniol, V. S. Nemkov

### Introductory remarks

In this chapter papers. have been grouped dealing with specific applications of various nature. Some of the papers, however, though describing particular devices, could belong to other chapters, if one considers that generally the results obtained can be extended to a wider class of devices.

M. D’Amore

### Computer - Aided Modelling and Simulation of Fast Transient Phenomena in Large Coils

This paper deals with resonance problems due to stray capacitances in large magnets, such as those used in fusion research machines or big H.V. transformers. The paper presents the method set up to investigate frequency response of the coils of the Poloidal Field System of the RFX fusion experiment.For each coil an equivalent network is first identified which takes into account, for each turn, self and mutual inductances and stray parameters. The frequency spectrum of such a network is then numerically analysed and the corresponding resonant frequencies are found in order to identify a much simplified model with similar electrical behaviour at the coil terminals, and to allow the overall winding to be analysed at once.

P. P. Campostrini, A. Stella

### Remote Electromagnetic Propulsion in the Presence of a Metallic Shield

This paper describes an electromagnetic technique to reposition the spacer springs in CANDU reactors. The need to perform this manoeuvre in non-commissioned reactors was required subsequent to the failure of a pressure tube in a reactor at a Canadian Nuclear Generating Station. A contributing factor in the failure of the tube was the fact that the annular spacers used to maintain the coaxial configuration between the metallic pressure tube and its surrounding metallic calandria tube, had been displaced. Afterwards it was realized that displacement of the spacers had also occurred in the non-commissioned reactors. It will be recognized that the spacers were not accessible for mechanical repositioning, leaving only the possibility of an electromagnetically coupled remote repositioning procedure. This paper describes the analysis and solution of the complex coupled electromechanical problem. The mathematical analysis problem was ultimately solved using a finite element process.

R. D. Findlay, J. H. Dableh

### Application of the Harmonic Analysis Technique to Determining Eddy Currents in Conducting Plates

A novel analytical approach to predict eddy currents in a conducting plate is proposed. Based upon the knowledge of the incident flux to the rectangular plate, a curve fit to a double Fourier series is obtained. This closed formula representation is used to solve the diffusion equation using the physical boundary conditions of the plate. Limitations and applicability of the method are then discussed.

R. D. Findlay, B. Szabados, I. ElNahas, M. S. ElSobki, M. Poloujadoff

### Numerical Field Calculation of Earthing Systems

This article describes results of the long-standing research on computer aided design, analysis and calculation of earthing systems. Numerical calculations of the quasistatic current field have been carried out by the boundary element method (current simulation method). Analyses of the complex earthing systems of the large industrial plants have been based on these calculations. The results obtained by measuring models and real objects have proved of satisfactory accuracy for the numerical procedure.

### Computer-Aided Design Technique for Proportional Electromagnets

A remarkable class of electromechanical converters used in proportional hydraulic devices, at the interface between the electronic control part and the hydraulic drive part, is represented by proportional electromagnets. These are, in fact, plunger-type d.c. electromagnets with a very studied air-gap geometry and intentionally saturated ferromagnetic zones. Such a particular magnetic structure allows a static characteristic adjustment, so that the developed electromagnetic force (i) remains constant with plunger positional changes (within the work domain of the axial air gap), (ii) is proportional to the excitation current, and (iii) has a considerably increased magnitude in a limited volume.

Mircea M. Rădulescu, Vasile Iancu, Ioan-Adrian Viorel, Károly Biró

### Reluctance Network Analysis of Coupled Fields in a Reversible Electromagnetic Motor

A large number of reversible electromagnetic, thyristor controlled, linear motors (Fig.1) have been designed and produced by the Novosibirsk Institute of Electrical Engineering. These machines cover a wide range of frequencies (from 0.4 to 50 Hz) and forces (from a few to several thousand newtons) and have been applied in the chemical and mining industries and used in household equipment and professional tools1,2.

Victor R. Rais, Janusz Turowski, Marek Turowski

### Introductory remarks

It is widely known that, in electrodynamics, analysis was developed in preference. Therefore, when the geometric and material structure of a device is known and when the field sources in this device are given, then we may accomplish the analysis of field in it, Synthetizing means searching for the structure producing the desired field. Synthesis is performed in order to design some device or to discover the features of the existing device unless they are not yet known.

K. Pawluk

### The State of Art in the Synthesis of Electromagnetic Fields

One observes that many inverse problems of technical electrodynamics are recently studied. The basic ordination of the synthesis problems might be regarded from an engineering, a physical and a mathematical point of view. There is a wide diversity in formulating the synthesis problems; some of them have been discussed in this paper and a basic classification of engineering synthesis problems has been proposed. A short review of mathematical techniques has been also presented.

Krystyn Pawluk, Marek Rudnicki

### Comparative Analysis of Numerical Methods for Shape Designing

Based on the Finite Element Method (FEM) two nonlinear algorithms for the synthesis of the geometry of the investigated region have been presented. One of the algorithms used is the Variable Metric Method (VMM) and the other is based on the Sensitivity Method (SM). The simple numerical example using this methods has been presented.

Jan Sikora, Maciej Stodolski, Stanisław Wincenciak

### Synthesis of a Turbo-Generator Nonlinear Parametric Model for the Analysis Problems of Power Industry System Regimes

At the level of the national economy development achieved, the problem of improving the methods and means to plan and to control the electric power systems (EPS) is of particular concern. The solution of this problem involves the intensive use of mathematical modelling methods and modern computer facilities. In this case, to obtain the results of high quality and reliability it is necessary to have the authentic mathematical models of the EPS elements.

V. E. Tonkal, Yu. G. Blavdzevitch, N. V. Raptsun

### Backmatter

Weitere Informationen


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# Confidence Interval for Coefficient of Quadratic Fit

1. Jan 13, 2014

### Office_Shredder

Staff Emeritus
I have a bunch of noisy data points (x,y), and I want to model the data as y = ax2 + bx + c + noise where noise can probably be assumed to be Gaussian, or perhaps uniformly distributed. My data is firmly inside of an interval and I'm only interested in modeling correctly inside of this interval.

My experience is that fitting such a curve can result in significantly different values of a,b and c with only a very small change in the actual curve. For example

http://www.wolframalpha.com/input/?i=plot+y+=++x^2+++50,+y+=+1.3+x^2+-+10x+++100+on+[10,30]

The things of significant interest to me are the value of f(x) which I believe is being modeled quite well with what I am doing (just picking f(x) to be the quadratic fit minimizing the sum of squared errors) , and the value of a itself which is probably not getting modeled very well because of issues like the above. Does anybody know/have thoughts on statistical testing I can do to determine a confidence interval for a given the noisy data?

2. Jan 13, 2014

### Stephen Tashi

The title of the post suggests that you want a confidence interval for $a$.

I've offered an opinion in several threads on the forum about how software packages compute confidence intervals for the parameters of curves in curve fits. Nobody has confirmed or contradicted me yet! The gist of it is that these confidence intervals are "asymptotic linearized confidence intervals". This amounts to writing a linear approximation for the formula that the curve fit algorithm uses to find the parameter in terms of the data. The parameter is regarded as a random variable that is a linear function of the data. Assume the data are independent random variables and you can estimate the variance of the parameter in terms of the variances of the data.

The problem with finding a more respectable way to give a confidence interval for a parameter is that if you regard the parameter as a random variable then, after doing a curve fit, you have a sample for it consisting of 1 single value, namely the value produced by the curve fit algorithm. So how can you estimate the variance of the parameter from a sample of size 1?

If you use a Bayesian approach with prior distributions for the parameters, I think you could estimate a confidence interval by a Monte-Carlo simulation. If you assume the parameters that you got from the curve fit are a good approximation for the "real" parameters and that the means and variances of the data are a good approximation for the means and variances of the phenomena being measured, you could do a Monte-Carlo approximation. It would involve generating data with random errors and fitting a curve to it. Do this many times and you get many simulated samples of the parameters.

3. Jan 13, 2014

### Office_Shredder

Staff Emeritus
Hence everything that came after that part!

This sounds pretty good to me. I googled around and found a couple packages that purport to do something similar to what you are describing. I'll give them a shot and report back.

Reporting back: Matlab has a function called "fit" in which you give it data and a model to try to fit to and you can run the output through "confint" to get confidence intervals at any level of confidence you desire. This is probably good enough, if I need something different I can take your description of how these calculations are done and do it myself by hand tweaking as desired. Thanks!

Last edited: Jan 13, 2014


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## A Proof of The Trapezoid Angle Theorem

A trapezoid is a quadrilateral whose two sides are parallel. In the figure below, $ABCD$ is a trapezoid and $AB$ is parallel to $CD$. In this post, we are going to prove that in a trapezoid, the consecutive angles between a pair of parallel sides are complementary. That is, we are going to prove in the figure above that $m \angle A = m \angle D = 180$ degrees and $m \angle B + m \angle C = 180$.

The Trapezoid Angle Theorem.

In a trapezoid, the consecutive angles between a pair of parallel sides are complementary.  Continue reading…

## The Proof of the Kite Symmetry Theorem

A kite is a quadrilateral with two distinct pairs of congruent sides. The common vertices of its congruent sides are called its ends. In quadrilateral $ABCD$ below, the distinct pairs of congruent sides are $\overline{AB}$ & $\overline{BC}$ and $\overline{AD}$ & $\overline{CD}$. The ends are $B$ and $D$.

Exercise: Locate the ends and the pairs of the distinct pairs of the remaining quadrilaterals.

In this post, we are going to prove the Kite Symmetry Theorem. The theorem states that the line containing the ends of a kite is a symmetry line for the kite. In proving this theorem, we are going to use the Figure Reflection Theorem which is stated as follows. Continue reading…

## Proof that The Diagonals of a Rhombus are Perpendicular

In the previous post, we have proved the converse of the Pythagorean Theorem.  In this post, we will prove that the diagonals of a rhombus are perpendicular to each other. That is, if we have parallelogram ABCD with diagonal $\overline{AC}$ and $\overline{BD}$, then $\overline{AC}$ is perpendicular to $\overline{BD}$.

What We Know

A rhombus is a parallelogram with four congruent sides. So, all sides of rhombus ABCD are congruent. That is

$\overline{AB} \cong \overline{BC} \cong \overline{CD} \cong \overline{DA}$Continue reading…


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## Membres

publié le , mis à jour le

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## Daniel Vanderpooten

Tel : 01 44 05 43 93
Bureau : P409 ter
Pole : Aide à la décision
Status : Professeur
Domaines de Recherche : Aide à la décision, Recherche Opérationnelle, Optimisation multiobjectifs, Optimisation combinatoire, Optimisation robuste, Modélisation des préférences.

• Membre du conseil scientifique

• ### Encadrement de these de doctorat :

• Satya Tamby : Approches génériques pour la résolution de programmes linéaires multi-objectifs en nombres entiers ( Début : 2013-09-01)
• Sami Kaddani : Intégration de préférences expertes en optimisation multicritère ( Début : 2014-01-01)
• Marek Cornu : Recherche Monte-Carlo multi-objectif ( Début : 2013-09-01)
voir toutes les theses

### Publications HAL

27 A biobjective branch and bound procedure for planning spatial missions (, and ), In Aerospace Science and Technology, Elsevier, volume 73, .

26 Covers and approximations in multiobjective optimization (, and ), In Journal of Global Optimization, volume 67, .

25 Weighted sum model with partial preference information: application to Multi-Objective Optimization (, , and ), In European Journal of Operational Research, volume 260, .

24 Bi-objective matchings with the triangle inequality (, , and ), In Theoretical Computer Science, Elsevier, volume 670, .

23 Efficient computation of the search region in multi-objective optimization (, , and ), In European Journal of Operational Research, volume 260, .

22 Discrete representation of the non-dominated set for multi-objective optimization problems using kernels (, and ), In European Journal of Operational Research, volume 260, .

21 Robust capacity expansion of a network under demand uncertainty: a bi-objective approach ( and ), In Networks, volume 68, .

20 A multi-objective interactive system for adaptive traffic control (, and ), In European Journal of Operational Research, volume 244, .

19 Approximate Pareto sets of minimal size for multi-objective optimization problems (, and ), In Operations Research Letters, volume 43, .

18 An efficient procedure for finding best compromise solutions to the multi-objective assignment problem (, and ), In Computers and Operations Research, Elsevier, volume 49, .

17 A bicriteria two-machine flow-shop serial-batching scheduling problem with bounded batch size (, , and ), In Journal of Scheduling, volume 17, .

16 Détermination de solutions de meilleur compromis pour les problèmes d'optimisation combinatoire multiobjectif (, and ), In Congrès de la Société Française de Recherche Opérationnelle et d'Aide à la Décision (ROADEF 2013), .

15 Biobjective planning of an active debris removal mission (, and ), In Acta Astronautica, volume 84, .

14 Algorithmic improvements on dynamic programming for the bi-objective 0,1 knapsack problem (, , and ), In Computational Optimization and Applications, Springer Verlag, volume 56, .

13 On the number of non-dominated points of a multicriteria optimization problem (, and ), In Discrete Applied Mathematics, Elsevier, volume 161, .

12 Critical edges/nodes for the minimum spanning tree problem: complexity and approximation (, and ), In Journal of Combinatorial Optimization, volume 26, .

11 Critical edges for the assignment problem : complexity and exact resolution (, and ), In Operations Research Letters, volume 41, .

10 Complexity of determining the most vital elements for the p-median and p-center location problems (, and ), In Journal of Combinatorial Optimization, Springer Verlag, volume 25, .

9 Lexicographic alpha-robustness: an alternative to min-max criteria (, and ), In European Journal of Operational Research, Elsevier, volume Vol. 220, .

8 Efficient determination of the k most vital edges for the minimum spanning tree problem (, and ), In Computers and Operations Research, Elsevier, volume 39, .

7 The lexicographic $\alpha$-robust knapsack problem ( and ), In International Transactions in Operational Research, Wiley, volume Vol. 18, .

6 Lexicographic $\alpha$-robustness : an application to the 1-median problem (, , and ), In RAIRO - Operations Research, EDP Sciences, volume vol. 44, n\textdegree2, .

5 The two-machine ow-shop serial-batching scheduling problem with limited batch size (, , and ), .

4 Min-max and min-max regret versions of some combinatorial optimization problems: a survey (, and ), .

3 Complexity of the satisfactory partition problem (, and ), .

2 Decomposition of graphs: some polynomial cases (, and ), .

1 Un mécanisme de négociation multicritère pour le commerce électronique (, , , and ), .


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# Riemannian metric

The metric of a space given by a positive-definite quadratic form. If a local coordinate system is introduced for the space and if at each point functions , , , , are defined which are the components of a covariant symmetric tensor of the second valency, then this tensor is called the fundamental metric tensor of . The length of the covariant vector is expressed using the fundamental tensor:

the form is a positive-definite quadratic form. The metric of determined using the form is called Riemannian, and a space with a given Riemannian metric introduced into it is called a Riemannian space. The specification of a Riemannian metric on a differentiable manifold means the specification of a Euclidean structure on the tangent spaces of this manifold depending on the points in a differentiable way.

A Riemannian metric is a generalization of the first fundamental form of a surface in three-dimensional Euclidean space — of the internal metric of the surface. The geometry of the space based on a definite Riemannian metric is called a Riemannian geometry.

There are generalizations of the concept of a Riemannian metric. Thus, a pseudo-Riemannian metric is defined with the aid of a non-definite non-degenerate quadratic form (see Pseudo-Riemannian space and Relativity theory). A degenerate Riemannian metric, that is, a metric form defined with the aid of functions for which , defines a semi-Riemannian space.

#### References

 [1] L.P. Eisenhart, "Riemannian geometry" , Princeton Univ. Press (1949) [2] P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian) [3] B. Riemann, "Ueber die Hypothesen, welche der Geometrie zuGrunde liegen" , Das Kontinuum und andere Monographien , Chelsea, reprint (1973)