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Course
Profile for MATH4401 Advanced Analysis, Semester 1, 2005
(2 unit, 3L 1T)
Course Objective
- To introduce students to some of the basic
techniques of Nonlinear Analysis and their
applications to existence of solutions for
differential equations.
- Students should acquire a good knowledge and
appreciation of Degree Theory and associated
fixed point principles and gain some initial
knowledge and appreciation of Bifurcation Theory.
They should gain an appreciation of some of the
central ideas such as modification arguments and
the use of cones (for positive solutions). They
should gain an appreciation of the role of
lipshitz conditions and compactness in existence
theory. Students will be able to apply the theory
to the analysis of concrete model problems such
as, for example, proving existence of solutions
of operator equations and ordinary differential
equations (including existence of periodic
solutions).
Contact and Advice
Assumed Background
- The formal prerequisites are MP212 or MP282 or
MP202 or MP272 or MATH2400. Specifically, based
on these courses students should have a basic
appreciation of multivariable calculus. A
knowledge of basic facts about Banach spaces,
Hilbert spaces, compactness, openness, closedness
will be assumed, although these topics will
be reviewed briefly in class. Results developed during the
course will be applied to ODEs PDEs and other
problems in the setting of functional analysis.
It is a student's own responsibility to fill in
any gaps in their assumed knowledge. You may need
to undertake background reading to understand the
lecture material.
Teaching Mode
- Three hours of lectures and one hour of
tutorial per week.
- Lectures: Monday 10am-12noon 67-641,
Tuesday 10am-11am 67-342
- Tutorials: Thursday 11am-12noon, 67-341
- There are no tutorials in week 1.
- Public holidays: March 25, April 25, May
2, 2005.
- Midsemester Break: March 26-April 3, 2005
- Examination period: Revision period is
June 5-12, 2005. Examination period is June
13-25, 2005
Syllabus
MATH4401 will cover the following topics. Specific
details are covered in lectures.
There will be a total of 35 lectures and 13
tutorials. The course aims to cover Brouwer
degree, Leray-Schauder degree, fixed point
theorems, iterative methods, and
applications such as Krasnoselskii
bifurcation and Rabinowitz bifurcation.
- Norm; Closed and Open Sets; Boundary;
Distance from a set to a point; Compactness in
n-dimensional Euclidean space;
Sard, Weierstrass and Tietze's Theorems.
- Axioms satisfied by Brouwer Degree; Properties of
Brouwer Degree including Poincare-Bohl; Weighted
Sum Formula; Borsuk Theorem; Reduction Theorem;
Applications.
- Cones in n-dimensional Euclidean space; Existence of nonnegative solutions.
- Banach Spaces; Norms; Compactness; Space of
Continuous functions, Arzela --Ascoli Compactness; Bounded
linear mappings; Differentiability of nonlinear
mappings; Banach Contraction Mapping Principle;
Examples.
- Schauder Degree theory; Applications.
- Statement of the initial value problem (IVP) for
first order systems and equivalence with a system
of integral equations, Existence for IVP,
Boundary Value problems. Periodic
solutions via Brouwer's Fixed Point Theorem and
Degree Theory.
- Basic bifurcation problem, Bifurcation from a
simple eigenvalue, Krasnoselskii's bifurcation
results. Rabinowitz' bifurcation theorem.
Examples.
Information Changes
- Any changes to course information will be
announced in lectures and the information will be
reproduced on the web page (
http://www.maths.uq.edu.au/courses/MATH4401).
It is your responsibility to keep up to date with
all information presented.
Resources
Assessment
Students should be familiar with the assessment rules
in their degrees as well as general
university policy such as found in the General Award
Rules. These are all set out on the
Program and Course Information page on the UQ website
http://www.uq.edu.au/student/courses/
- Assessment Scheme:
Assignments will count for 50% of
assessment, and a final examination
will count for the remaining 50%
of the assessment.
- Assignments: Assignments
will be due approximately fortnightly. Provisional due dates:
- Assignment 1: 17th March
- Assignment 2: 7th April
- Assignment 3: 21st April
- Assignment 4: 5th May
- Assignment 5: 19th May
- Assignment 6: 2nd June
Late assignments attract a 10% mark penalty if
submitted within 24 hours of the deadline. Assignments
submitted later than that will receive a mark of
0.
In determining your overall assignment grade, the
worst mark will be dropped, and the remaining marks averaged.
Marks will be awarded for correctness, clarity and
efficiency of presentation, use of appropriate
mathematical language, and the level of
innovation. Part marks will be awarded for the
successful completion of subtasks.
- If you miss an assignment: In case of
illness (or bereavement) you may be exempted from
an assignment if a medical certificate (or other
documentation) is received by the course
co-ordinator within one week of the due
date of the assignment. If you are exempted, then
your assignment marks are weighted on a pro-rata
basis. Note that ad hoc excuses (car trouble and
similar) will not be accepted; only documentation
in connection with illness or bereavement
. If you enrolled late then exemption will
automatically be granted for anything missed
before the date of enrolment.
- Discussion of the assignment questions prior to writing up
is permitted, even encouraged. However, writing up should be done alone,
and students should be aware of the university's policy on
plagiarism as outlined below.
- Final Examination: The modality of the
final exam, after discussion with the class, has been set as follows:
closed book, 3 hours long. (The duration is pending approval from
the Faculty).
It will be held in
the examinations period at a time and place to be
advised by the examinations section. Calculators
will not be permitted.
Anything discussed in lectures is examinable. In
addition problems from assignments and tutorials
as well as similar problems may form part of the
written exam.
- If you miss the final exam: See the
Official Examination Policies at the Resources
Page http://www.maths.uq.edu.au/courses/MATH4091/Resources.html.
This page contains a lot of useful information
about examinations as well as about available
resources. See also the
Supplementary and Special examination information
below.
- Failure to complete assessment items:
Failure to complete any item of assessment will
result in a weighting of zero for that item
except as provided for under the heading "If
you miss the final exam".
Supplementary
examinations
A
supplementary examination may be awarded in one course to
students who obtain a grade of 2 or 3 in the final
semester of their program and require this course to
finish their degree. You should check the rules for your
degree program for information on the possible award of
supplementary examinations. Applications for
supplementary examinations must be made to the Director
of Studies in the Faculty.
EPSA Faculty policy on the award of supplementary exams
may be found via the Faculty Guidelines on Examinations
from the EPSA student page
http://www.epsa.uq.edu.au/index.html?page=7640&pid=7563
Special examinations
If a student is unable to sit a
scheduled examination for medical or other adverse
reasons, she/he can and should apply for a special
examination. Applications made on medical grounds should
be accompanied by a medical certificate; those on other
grounds must be supported by a personal declaration
stating the facts on which the application relies.
Applications for special examinations for central and
end-of-semester exams must be made through the Student
Centre. Applications for special examinations in school
exams are made to the course coordinator.
More information on the University’s assessment
policy may be found
http://www.uq.edu.au/hupp/index.html?page=25113&pid=25075
EPSA Faculty policy on the award of
special exams may be found via the Faculty Guidelines on
Examinations from the EPSA student page
http://www.epsa.uq.edu.au/index.html?page=7640&pid=7563
Plagiarism:
The University has adopted the following definition:
Plagiarism is the action or practice of taking and using
as one’s own the thoughts or writings of another,
without acknowledgment. The following practices
constitute acts of plagiarism and are a major
infringement of the University's academic values:
- Where
paragraphs, sentences, a single sentence or
significant parts of a sentence are copied
directly, and are not enclosed in quotation marks
and appropriately footnoted;
- Where
direct quotations are not used, but are
paraphrased or summarised, and the source of the
material is not acknowledged either by footnoting
or other simple reference within the text of the
paper; and
- Where
an idea which appears elsewhere in printed,
electronic or audio-visual material is used or
developed without reference being made to the
author or the source of that material.
When a student knowingly plagiarises
someone’s work, there is intent to gain an advantage
and this may constitute misconduct.
Students are encouraged to study together and to discuss
ideas, but this should not result in students handing in
the same or similar assessment work. Do not allow another
student to copy your work.While students may discuss
approaches to tackling a tutorial problem, care must be
taken to submit individual and different answers to the
problem. Submitting the same or largely similar answers
to an assignment or tutorial problem may constitute
misconduct.
If a deliberate act of plagiarism
resulting from particular, submitted assessment item is
proven the results of that assessment item may be
annulled and other action may be taken as is considered
appropriate in the circumstances of the case.
Any student with a disability who
may require alternative academic arrangements in the
course is encouraged to seek advice at the commencement
of the semester from a Disability Adviser at Student
Support Services.
Assistance for Students:
Students with English language difficulties should
contact the course coordinator or tutors for the course.
Students with English language difficulties who require
development of their English skills should contact the
Institute for Continuing and TESOL Education on extension
56565.
The Learning Assistance Unit located in the Relaxation
Block in Student Support Services.
You may consult learning advisers in the unit to provide
assistance with study skills, writing assignments and the
like. Individual sessions are available. Student Support
Services also offers workshops to assist students. For
more information, phone 51704 or on the web http://www.sss.uq.edu.au/index.html.
Feedback on
assessment:
You may
request feedback on assessment in this course
progressively throughout the semester from the course
coordinator. Feedback on assessment may include
discussion, written comments on work, model answers,
lists of common mistakes and the like.
Students may peruse examinations scripts and obtain
feedback on performance in a final examination provided
that the request is made within six months of the release
of final course results. After a period of six months
following the release of results, examination scripts may
be destroyed.
Information on the University’s policy on access to
feedback on assessment may be found at
http://www.uq.edu.au/hupp/index.html?page=25114&pid=25075
EPSA Faculty policy on assessment feedback and re-marking
may be found at
http://www.epsa.uq.edu.au/index.html?page=7674&pid=7564
Library
contact:
The liaison librarian for the physical sciences
disciplines is located in the Physical Sciences and
Engineering Library in the Hawken Building and may be
consulted for assistance in the course:
Leith Woodall
Email: l.woodall@library.uq.edu.au
Extension: 52367
For library queries also visit their
Frequently Asked Questions page,
http://www.library.uq.edu.au/skills/question.html which is also accessible from the library
Homepage,
http://www.library.uq.edu.au/index.html.
Student
Liaison Officer:
The School of Physical Sciences has
a Student Liaison Officer as an independent source of
advice to assist students with resolving academic
difficulties.
The Student Liaison officer during 2005 will be Dr Peter
Adams, Room 547 Priestley building, (email
pa@maths.uq.edu.au)
Assessment Criteria:
- To earn a Grade of 7, a student
must demonstrate an excellent understanding of
all of the theory of the stochastic calculus and
its applications to finance as discussed in the
course and its associated materials. This
includes clear expression of nearly all their
deductions and explanations, the use of
appropriate and efficient mathematical techniques
and accurate answers to nearly all questions and
tasks with appropriate justification.
- To earn a Grade of 6, a student
must demonstrate a comprehensive understanding of
the theory of the stochastic calculus and its
applications to financeas discussed in the course
and its associated materials. This includes clear
expression of most of their deductions and
explanations, the general use of appropriate and
efficient mathematical techniques and accurate
answers to most questions and tasks with
appropriate justification.
- To earn a Grade of 5, a student
must demonstrate an adequate understanding of the
course theory and its applications to finance as
discussed in the course and its associated
materials. This includes clear expression of some
of their deductions and explanations, the use of
appropriate and efficient mathematical techniques
in some situations and accurate answers to some
questions and tasks with appropriate
justification.
- To earn a Grade of 4, a student
must demonstrate an understanding of the basic
concepts of the stochastic calculus and its
applications to finance as discussed in the
course and its associated materials. This
includes occasionally expressing their deductions
and explanations clearly, the occasional use of
appropriate and efficient mathematical techniques
and accurate answers to a few questions and tasks
with appropriate just ification. They will have
demonstrated knowledge of techniques used to
solve problems and successfully applied this
knowledge in some cases.
- To earn a Grade of 3, a student
must demonstrate some knowledge of the basic
concepts of the stochastic calculus and its
applications to finance as discussed in the
course and its associated materials. This
includes occasionally expressing their deductions
and explanations clearly, the use of a few
appropriate and efficient mathematical techniques
and attempts to answer a few questions and tasks
accurately and with appropriate justification.
They will have demonstrated knowledge of
techniques used to solve problems.
- To earn a Grade of 2, a student
must demonstrate some knowledge of the basic
concepts of the stochastic calculus and its
applications to finance as discussed in the
course and its associated materials. This
includes attempts at expressing their deductions
and explanations as well as attempts to answer a
few questions accurately.
- A student will receive a Grade of
1 if they demonstrate extremely poor knowledge of
the basic concepts in the course material. This
includes attempts at answering some questions but
showing an extremely poor understanding of the
key concepts.
Graduate Attributes:
The following graduate attributes will be developed in
the course –
In-Depth, Comprehensive and Well founded
Knowledge of the Field of Study including the
ability to collect, analyse, and organise information and
mathematical ideas
• through the subject matter covered in lectures
including the way it is organised and presented
• through understanding and synthesising pieces of
information as a result of problem solving and group
discussions in tutorials underpinned by cooperative
learning strategies
• through reading texts and references to collect,
analyse and synthesise information and mathematical ideas
to problem solve for assignments and to prepare for
tutorials and exams
Effective Communication including
the ability to convey those ideas clearly and
fluently, in both written and spoken forms as well as
including the ability to interact effectively with others
in order to work towards a common outcome
• through questioning, listening, and presenting in
tutorials
• through examples of the use the appropriate level
and style of language in lectures and tutorials
• by effective interaction in tutorials to
acclomplish set tasks
• through writing solutions to assignments in the
professionally accepted form
An understanding of how other disciplines relate
to the field of study
• through applying the mathematical techniques of
the course to simple problems from other disciplines.
An international perspective on the field of
study
• through the lecture material and
completing tutorial and assignment work using
internationally accepted standards of mathematical rigour
and notation
Independence and Creativity
• The ability to work and learn
independently through tutorial tasks, assignment work and
exam preparation.
• The ability to generate ideas and to identify
problems, and create solutions based on examples given in
lectures and tutorials including tutorial solutions.
Critical Judgement
• Including the ability to define problems
and to apply critical reasoning to analyse these problems
through examples given in lectures and tutorials and
through independent thought underpinned by informed
judgement
• Including the ability to evaluate mathematical
arguments and to reflect critically on the correctness of
the arguments through examples given in lectures and
tutorials as well as in tutorial solutions
Ethical And Social Awareness
• through interacting with a class involving
students from diverse social and cultural backgrounds
For more information on the University policy on
development of graduate attributes in courses, refer to
the web
http://www.uq.edu.au/hupp/contents/view.asp?s1=3&s2=20&s3=5.
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