CATAM News 1997-1998
Old News Headlines
1997 - 1998
22 Apr 1998 Problem with Part II project 5.3 22 Apr 1998 Confusion in Part II project 23.1 (Astrophysics) 22 Apr 1998 Last-minute CATAM queries 01 Apr 1998 Part II project 2.8 (Reaction-Diffusion Equations) 01 Apr 1998 Corrections to Part II project 5.2 (S-wave scattering) 30 Mar 1998 New Museums Site, Mond Room and Old Music School PWFs 30 Mar 1998 Minor Errors in Part II projects 11.1, 11.2 30 Mar 1998 Minor error in IB project 2.5 (Schrodinger's equation) 16 Mar 1998 Major Error in Part II project 4.5 (Euler's Equations) 11 Mar 1998 Easter closing arrangements 11 Mar 1998 Correction to Part II project 3.9 05 Mar 1998 Correction to Part II project 17.1 (Graph Colouring) 05 Mar 1998 Correction to Part IB project 2.7 (Numerical Quadrature) 13 Feb 1998 Correction to IB Project 2.1 (Random Samples) 30 Jan 1998 Part II projects: unit allocations 23 Jan 1998 CATAM Room PCs reserved on Wednesday mornings 23 Jan 1998 Part IB: Rewording of compulsory project 1.2 23 Jan 1998 Part II: correction to project 5.3 02 Jan 1998 Large arrays in CATSL for Windows 19 Nov 1997 Errors in Part II projects 2.8 and 14.3 14 Nov 1997 Maple Computer Algebra System 20 Oct 1997 Demonstrator availability 20 Oct 1997 Part II Projects Manual corrections 14 Oct 1997 Welcome message 14 Oct 1997 Out of date manuals 03 Oct 1997 Mill Lane Lecture Rooms security
22 Apr 1998 Problem with Part II project 5.3
Quantum Mechanics: Bound State Energies
The rubric to this project states that it can be done with only a knowledge of the course Foundations of Quantum Mechanics. However, a small part of the project (near the end) requires you to use the variational method, which is no longer taught in that course. If you have already done the project using the variational method now taught in Applications of Quantum Mechanics, you can ignore the rest of this notice.
The following changes allow the project to be done without a knowledge of the variational method. You can obtain full credit by answering the questions in their new form given below. The version of the project on the web pages has been corrected.
On page 4, the paragraph starting 'Derive explicitly an upper bound...' should be replaced by the following (here _ means a subscript and pm refers to the plus/minus symbol):
Consider the possible trial wavefunctions
phi_pm = C_pm (psi_plus pm psi_minus),
where C_pm are normalization constants to be determined. Compute the expectation values of the Hamiltonian,
E_pm = .
You should find these values to be
(followed by the equation for E_pm as at present).
Question (b) on page 4 should be replaced by:
By expanding phi_pm on the complete set of energy eigenstates of H and being careful to take into account the symmetry of the state in each case, show that E_pm are respectively upper bounds for the ground state and first excited state energies.
Dr R. E. Hunt
CATAM Director
22 Apr 1998 Confusion in Part II project 23.1 (Astrophysics)
Some notational confusion has occured in this project. In question 1, students should note that epsilon is the dimensionless BINDING energy and has the opposite sign to total energy. Throughout the question, epsilon is used to represent ANY energy which has been rescaled to make a dimensionless quantity. This is equivalent to choosing a=G=M=1.
In question 3, the notation "epsilon_min" is misleading; the question should say just that the minimum energy is -1/2 and the square of the maximum speed of a particle is -2 times the minimum energy.
In question 5, "actual mass" should read "actual (unrenormalized) mass".
Dr R. E. Hunt
CATAM Director
22 Apr 1998 Last-minute CATAM queries
The e-mail address for CATAM queries, catam@maths.cam.ac.uk, will continue to operate up until (and, in fact, after) the submission deadlines. However, students should note that after Thursday 23rd April it is unlikely to be worth sending ACADEMIC queries; these usually take a few days to answer satisfactorily, excluding weekends. It is, however, still worth addressing queries about the submission procedure to this address.
On the day of submission, problems are best sorted out by trying to find me in person in DAMTP (my office is G23); I shall certainly be there from 2-4pm.
Dr R. E. Hunt
CATAM Director
01 Apr 1998 Part II project 2.8 (Reaction-Diffusion Equations)
A further minor correction.
Following on from the correction published in CATAM News on 19th November 1997, the mesh-size suggested in the last paragraph of 2.8(b) should be N=100 (rather than N=101). Solutions will be allowed which use the old size, however, and no credit will be lost.
Dr R. E. Hunt
CATAM Director
01 Apr 1998 Corrections to Part II project 5.2 (S-wave scattering)
In equation (4), U(x) should read U(r). In the sentence immediately following, chi_0'(0) (rather than chi_0(0)) should be set to an arbitrary value if using the Runge-Kutta method.
Dr R. E. Hunt
CATAM Director
30 Mar 1998 New Museums Site, Mond Room and Old Music School PWFs
The PWF-MATHS computers are in great demand in the fortnight between Easter and the Project submission dates on 27th/28th April, but you can also use the PCs in the New Museums Site, Mond Room and Old Music School PWFs (about 90 PCs and 30 Macs in total).
The Old Music School is booked for 16th/17th April AM and 23rd all day (as of 30 March). Check the bookings notice in the OMS for latest news.
The CATAM software library and Borland Pascal are available at these PWFs, but note that no demonstrators are available there, and a charge is made for printing.
Minor Errors in Part II projects 11.1, 11.2
Very minor error in Part II project 11.1 (Statistical Mechanics)
The reference on page 2 to a "library unit CUBUNIT" should read "CATSL procedure CUBIC".
Minor error in Part II project 11.2 (The Ising Model)
On page 4 of this project, just below the equation for kappa_i, the equation sigma_{-1} = sigma_0 should read sigma_{-1} = sigma_{N-1}.
Dr R. E. Hunt
CATAM Director
Minor error in IB project 2.5 (Schrodinger's equation)
In part 1 of this project, you are asked to run the program for E=4.6 with a step-size of 0.6. This should read 0.4. Students who have already done the project and who used the old step size will not be penalized.
The software note (erroneously referred to as S1 in the project) should explain that RKf is an adaptive step method, but that RK4 can be used if a fixed step method is preferred.
The web version of this project has been corrected.
Dr R. E. Hunt
CATAM Director
16 Mar 1998 Major Error in Part II project 4.5 (Euler's Equations)
A major error has come to light in the above project: equation (6) is incorrect, and this affects some subsequent questions. The following corrections have been made to the project; the web version reflects these changes and can be used for reference in case the explanation below is unclear.
The CATAM Examiners accept that this rather late major change may cause students problems, particularly if they have already written up the project. Consequently, considerable leeway will be allowed in write-ups; in particular, answers to the original (rather than corrected) questions will be allowed. Please e-mail catam@maths.cam.ac.uk if you wish your project to be marked in this way. It would also be in your own best interests to attach a short explanatory note to the start of your project if this affects you.
In the following, _ refers to a subscript and ^ to a superscript.
Equation (6) should read (Aw_1^2+Bw_2^2+Cw_3^2)/2=E - energy is conserved (rather than the magnitude of the angular velocity, as incorrectly stated).
Question 2 has been reworded as follows: For given values of A/B and C/B, explain why we may take B=1 without loss of generality. Show further that choosing E=1 is equivalent to re-scaling the time variable t, and give the scaling factor.
In the first paragraph of section 2.2, replace "Omega=1" by "E=1", twice.
In question 3, replace "(1,0,0)" by "(sqrt(2/A),0,0) (i.e., (1,0,0) before scaling)".
In question 4, replace "w_1 approximately equal to 1" by "w_1 approximately equal to sqrt(2/A)" and "w_3 approximately equal to 1" by "w_3 approximately equal to sqrt(2/C)".
The last sentence of question 6 has been reworded as follows: What value must H take (refer to equation 7) for such a solution?
Dr R. E. Hunt
CATAM Director
11 Mar 1998 Easter closing arrangements
The Mill Lane Lecture Rooms building will be closed over the Easter period as follows:
- Thursday 9 April to Monday 13th April inclusive
The Mill Lane Lecture Rooms building is closed on Saturdays outside Full Term.
When the building is closed, the PWF will be available for use by MLLR keyholders, but a continuous service cannot be guaranteed during the Easter weekend and faults are unlikely to be fixed with the usual efficiency! The Mill Lane PWF will operate normally at all other times.
Please note that the demonstrators will be available during the week after Lent Full Term and the week before Easter Full Term.
11 Mar 1998 Correction to Part II project 3.9
Minor error in Part II project 3.9
In section 2, second line, the equation
H''(1) = P_d
should read
H''(1) = -P_d.
Dr R. E. Hunt
CATAM Director
05 Mar 1998 Correction to Part II project 17.1 (Graph Colouring)
Note that in the first paragraph of section 2 of this project, the word "size" should read "order".
Dr R. E. Hunt
CATAM Director
05 Mar 1998 Correction to Part IB project 2.7 (Numerical Quadrature)
Just before question 8 of this project, it is stated that the file glquad.dat contains data for N=1,...,30; and that you should calculate the two integrals for values of N from 1 to 30. In each case, "1" should read "2".
It has been pointed out that for some larger values of N, the data in glquad.dat is not as accurate as it should be: for N=30, some of the x_i are accurate only to 4 significant figures. This should not worry you, and WILL NOT affect the accuracy of your results to any real extent.
Dr R. E. Hunt
CATAM Director
13 Feb 1998 Correction to IB Project 2.1 (Random Samples)
The last sentence of question 2(i) should read:
Find theta-hat_n for which this l_n(theta) is maximised and comment on the location of theta-hat_n with reference to your answer to question 1.
Dr R. E. Hunt
CATAM Director
30 Jan 1998 Part II projects: unit allocations
At the beginning of the year, I indicated that a few particularly long projects might have their unit allocations altered. In the cold light of day, I have decided not to make any alterations this year because of the possible confusion and subsequent complaints that could result. Apologies for any inconvenience.
Dr R. E. Hunt
CATAM Director
23 Jan 1998 CATAM Room PCs reserved on Wednesday mornings
Please note that the CATAM PCs are reserved for a Statistics course on Wednesday mornings 9-11am for the rest of this term.
David Harris
23 Jan 1998 Part IB: Rewording of compulsory project 1.2
Part IB: Rewording of compulsory project 1.2
To clear up some misunderstandings over project 1.2 (Linear algebra and matrices), the first paragraph of section 4 (Kernels and annihilators) has been reworded as follows:
The kernel of A is the set ker A of all column vectors x such that Ax=0. If A is p x q then ker A is a subspace of F^q. Row operations correspond to pre-multiplying by invertible matrices, so do not change the kernel. To find a basis for ker A, first reduce A to row echelon form, and call x_i (the i-th component of x) pivotal if the i-th column of A contains a pivot. The kernel is then defined by expressing the pivotal x_i in terms of the other (non-pivotal) x_j, which can be chosen freely.
Here _ means a subscript and ^ a superscript.
Dr R. E. Hunt
CATAM Director
23 Jan 1998 Part II: correction to project 5.3
Part II: correction to project 5.3
The last line of caution (b) on page 2 of project 5.3 should read "before computing e_m" rather than "before computing e_{m+1}".
Here _ means a subscript and ^ a superscript.
Dr R. E. Hunt
CATAM Director
02 Jan 1998 Large arrays in CATSL for Windows
In Borland Pascal (BP) and Turbo Pascal for Windows (TPW) the size of any single data structure is limited to 64 kilobytes of memory, even though your computer may have many Megabytes of virtual memory available.
A new example program shows how to access data arrays of any size, using Turbo Pascal for Windows or Borland Pascal on Windows 3.1, 95 or NT, limited only by the amount of virtual memory available on the computer.
The file is I:\CATAM\EXAMPLES\A_DEMOS\BIGARRAY.PAS on the PWF.
Click here to download a copy.
Click here for more indformation.
Note that most Computational Projects do not need huge amounts of storage when programmed correctly. You may sometimes need to use the technique shown in procedure testpointer, but it's very unlikely that you will need to use the techniques shown in testhuge and testdispose.
The CATSL installation file I:\CATAM\INSTALL\CATAM.ZIP has been updated to include this file.
David Harris - DAMTP - 2 Jan 98
19 Nov 1997 Errors in Part II projects 2.8 and 14.3
Some errors have been found in project 2.8 (Reaction-diffusion equations in population genetics). The right hand side of equation (12c) should read -s*sdot (where sdot means ds/dt) rather than -sdot/s; correspondingly, in the right hand side of (14d), s_j (i.e. s subscript j) should be replaced by (1/s_j). The definition of Delta y which immediately follows should be 1/N rather than 1/(N-1).
The web version of the project has been corrected, in case the above is not clear.
Any students intending to attempt project 14.3 (Petrov classification of a gravitational field) may be interested in some tutorial notes explaining how to use Maple to perform calculations with the tensors needed for the project, and including some Maple routines to work with such tensors. These notes may be obtained by e-mailing catam@maths.cam.ac.uk ; they will be sent to you via the University mail.
Dr R. E. Hunt
CATAM Director
14 Nov 1997 Maple Computer Algebra System
In several Part II projects use of a Computer Algebra System (CAS) is required or recommended. Maple V.4 (the latest version) is available on the PWF PCs (an older version is also available on the Macintosh computers in the Mill Lane CATAM room).
To start up Maple click on the icon in the CATAM group. You will see a short introductory worksheet which shows how to run the comprehensive online tutorials which are supplied with Maple.
Note that you are free to use Maple for any project, whether or not a CAS is actually recommended, but you should be aware that:
- for intensive numerical calculations Maple should be told to use the hardware floating-point unit (see help on evalhf)
- the CATAM demonstrators may be less familiar with Maple than they are with the CATSL software.
- few (if any) colleges provide Maple on college computers
- if you choose to use Maple to do a project for which a CAS is not suggested in the project booklet, you should bear in mind that you may not be allowed to use some of the Maple built-in functions. As a rule of thumb, do not use a built-in function if there is no equivalent CATSL routine and if use of the built-in function would make the programming considerably easier. For example, use of the Maple "isprime" operator to test whether an integer is prime would not be allowed in a project which required you to write a program to find prime numbers. Please e-mail catam@maths.cam.ac.uk if you need any clarification in specific cases.
The previous version (Maple V.3) is still available for those who have old .m files which version V.4 cannot read (but it's unlikely that any such files actually exist).
Several Maple books are available for reference in the Mill Lane Computer room. Details are given in the on-line introduction.
20 Oct 1997 Demonstrator availability
Due to time constraints on our demonstrators, we have had to change the days on which they are available in the Mill Lane CATAM room: Friday has been replaced by Wednesday.
The times are now therefore Monday to Thursday, 2-4pm.
Dr R. E. Hunt
CATAM Director
20 Oct 1997 Part II Projects Manual corrections
Error in Part II manual, contents list:
Project 3.6, listed as "Fluid particle drift in a periodic flow field" in the contents, does not exist. Please ignore it!
------
Errors have come to light in two projects. The corrected versions have been put on the web. Here are the details:
Project 19.3 (The weight enumerator polynomial), question 1: exchange W(C;1,0) and W(C;0,1).
Project 15.7 (Continued fractions): the convergents are incorrectly defined; the right-hand sides of the given expressions for P_{n+1} and Q_{n+1} should have plus instead of minus signs.
Dr R. E. Hunt
CATAM Director
14 Oct 1997 Welcome message
Welcome back to Cambridge, and good luck with the CATAM projects! You can now access CATAM News via the web, at http://www.maths.cam.ac.uk/catam - please read it regularly for important messages.
Demonstrators are available throughout Michaelmas Term in the CATAM Room from 2 - 4 p.m. on Mondays, Tuesdays, Thursdays and Fridays. Please use them as your first port of call for any queries or problems with the software or the projects.
If you find any misprints or other errors in the CATAM booklets, please e-mail catam@maths.cam.ac.uk so that we can issue corrections. The same e-mail address should be used for any problems with hardware, software or the mathematical content of the projects with which the demonstrators are unable to help; we will aim to respond within two days (though for complicated questions this may only be an acknowledgement).
Dr R. E. Hunt
CATAM Director
14 Oct 1997 Out of date manuals
Important warning:
Please check that you have the CURRENT (i.e., 1997/98) version of your projects booklet. This applies to both Parts IB and II. Some old copies of the booklets were erroneously stored near the DAMTP reception, and may have been sold to or taken by students. If you were sold an out of date copy, please e-mail catam@maths.cam.ac.uk and we will authorize a free exchange.
Dr R. E. Hunt
CATAM Director
03 Oct 1997 Mill Lane Lecture Rooms security
The Maths Public Workstation Facility (PWF) is closed at 5pm, Monday-Friday and at 1pm on Saturday. The ground floor doors are deadlocked at closing time so please leave promptly otherwise you will be TRAPPED inside the building!
The hours for which Mill Lane PWF is open are restricted by the University insurers. If theft or damage should occur in a building which is not properly secured the University would not receive full compensation from the insurers.
Only authorised keyholders are allowed to work in the Catam Room at night when the MLLR building is locked. Students MUST NOT UNDER ANY CIRCUMSTANCES be in the Mill Lane building outside the official hours - see the Part IA/IB and Part IIA/IIB Project Booklets for details. If you do find yourself locked in the building at night telephone the University Security Centre on 31818. The PWF logon records are monitored and anyone found working without permission when the building is locked will be reported to University authorities.
Users of the Maths PWF may also use the New Museums Site PWF (off Pembroke Street) which is available 24 hours/day. Keys are available from the Computing Service Reception, Pembroke Street.