Question Session = Spørgetime The form of the Exam made precise (en præcissering af eksamensformen) Information about the exam: FINAL Scores!!! (OBS: NEW! - posted on July 03) and Solutions to selected questions
The leading character in the course is Henri Léon Lebesgue (1875-1941). He introduced the Lebesgue integral in his dissertation from 1902, and it was a seminal work. It took some time before his ideas were accepted. A quote from the correspondence between Hermite and Stieltjes from around 1890 indicates a little the squabbles shortly before. Hermite writes: 'Je me detourne avec effroi et horreur de cette plaie lamentable des fonctions qui n'ont pas de derivees'. This was in part a reaction to the discovery by Weierstrasses of a continuous function which is not differentiable in any point. Hermite could hardly have imagined that Lebesgue just 10 years later would introduce a much larger class of functions--the Lebesgue measurable functions--and by doing so would create a completely new foundation for the mathematical analysis in the 20. century. Legesgue himself describes how many of his colleagues would tease him as being ``the man with functions without derivatives''.
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Man tilmelder sig kurset elektronisk
på hjemmesiden https://secure.math.ku.dk/ua. Tirsdag kl. 11-13 i
aud. 4 ved Hans Plesner
Jakobsen, som kan træffes på sit kontor E415, eller
på telefon 35 32 06 89, eller pr. email jakobsen@math.ku.dk. Første forelæsning er tirsdag d. 4. februar.
De to dele af eksamen varer hver præcist 90 minutter. Det er
ikke muligt at overføre tid fra den ene del til den anden.
July 03 2003: Here are the final grades:
DVI-fil - postscript-fil - PDF-fil.
Here are solutions to a few of the questions on the exam:
DVI-fil -postscript-fil - PDF-fil.
The written exam consists of two parts. The first 90 minutes are
without any means of help ("closed book") and during this time an
Essay Problem is do be finished and handed in. Here, one typically is asked
to state and prove one or several specific propositions and theorems in a central topic
from the course. In the remaining 90 minutes more traditional problems
are to be solved, and here all usual means of help are allowed ("open
book").
The essay problem will be one of the listed 14 problems The long
list. Around 1 May 2003 (this will be made more precise) I
choose, by some procedure, 5 of these problems and publish these on
this web page. The idea is that the actual problem will be one of the
5 on this short list. Here it is: The short
list. Notice that it is different from previous exams! In the actual exam, the essay problem will be
accompanied by some remarks aimed at leading the student in the right
direction. Se Old 3MI-exams (in
Danish). There will both be an English and a Danish version of
the exam. The students may use either language in their solutions.
Forelæsninger
Question Session= Spørgetime
Thursday, June 12 at 2:15 p.m. in A103.
(I will be away for a conference in the week before the exam.)
The form of the Exam made precise (en præcissering af eksamensformen)
The two parts of the exam each last exactly 90 minutes. It is not
possible to transfer time from one part to the other.
Information about the exam: FINAL Scores and Solutions to selected questions
OBS: New!
News about the exam!
Syllabus (pensum) for the exam
The course material for the exam consists of the Lecture Notes ``Mål-
og integralteori'' by Berg
and Madsen (2003 edition) with the following EXCEPTIONS:
The following parts of the syllabus are "cursory", i.e. you will be
expected to know the material but not the proofs:
Øvelser
Tid
Lokale
Instruktor
Hold 1
Tirsag kl. 9-11
A107
Morten Overgaard Hansen
Hold 2
Onsdag kl. 15-17
E-37
Klaus Kähler Holst
Bemærk, at Hold 3 er nedlagt.
Øvelsesholdene møder første gang i uge 7.
MatLab er åbent alle hverdage kl. 10-17. Det holder til i lokale
E32 (i
bygningerne på den anden side af Jagtvejen; 155B i gården).
Du kan her sidde og regne opgaver
(lokalet kan altså bruges som læseplads). Endvidere er MatLab
bemandet. Bemandingsplanen er:
Vi benytter bogen "Mål- og integralteori" af Christian Berg og T
age Gutmann
Madsen, som kan købes i Naturfagsbogladen. Den koster 140 kr. Supplerende noter:
Nedenstående supplerende noter har været anvendt i Matematik 2AN
(efterårssemesteret 2001). Noterne om Lebesgueintegralet givet en kort
oversigt
over indholdet af nærværende kursus. Noterne om mængdelære
(herunder om billeder
og urbilleder) og om limes superior og limes inferior forudsættes
kendte i dette
kursus.
Supplerende litteratur:
Hjemmeopgaver
Der vil blive stillet
hjemmeopgaver hver anden uge de første 8 uger af semesteret og hver
uge de
sidste 6 uger (ialt 10 sæt). Opgaverne afleveres til instruktoren, og
du får dem
tilbage i rettet stand. Aflevering af hjemmeopgaver giver dig
mulighed for
løbende at få afprøvet din forståelse af kurset og giver
samtidigt en
fremragende træning i at skrive matematik.
MatLab
Disse vil kunne besvare
spørgsmål om opgaver mv. i 3MI (og andre matematikfag).
Lærebog
| Uge | Dato | Afsnit i bogen | Emne |
| 6 | 4/2 | §§ i, 0, 1 | Introduction, measurable sets, sigma-algebras |
| 7 | 11/2 | § 3 | Measures, examples, almost everywhere |
| 8 | 18/2 | § 2 | Measurabale maps and functions, limits |
| 9 | 25/2 | § 4 | The integral of simple and positive functions; convergence theorems |
| 10 | 4/3 | § 4 | The integral of real and complex functions; convergence theorems |
| 11 | 11/3 | § 4.2, 4.7 | More on integration, "differentiation under the integral sign" |
| 12 | 18/3 | § 4.3--4.6 | More on integration |
| 13 | 25/3 | § 5.1 | Uniqueness of measures |
| 14 | 1/4 | § 5.2, 5.3 | Locally integrable functions, Radon measures |
| 15 | 8/4 | § 6.1, 6.2 | Product measures |
| 16 | 15/4 | § 6.3 | Fubini and Tonelli. (Last day of lectures and problem sessions before vacation.) |
| 17 | 22/4 | No lectures - Easter vacation! (Last day of vacation; so the Wednesday class meets in this week.) | |
| 18 | 29/4 | § 7 | Lp spaces. |
| 19 | 6/5 | § 7 | Lp spaces. Fishers Completeness Theorem. |
| 20 | 13/5 | § 8 | The Fourier transform of integrable functions. |
| 21 | 20/5 | § 8 | The Fourier transform of Schwartz functions. |
| 23/6 | Written Exam! |
Hans Plesner
Jakobsen