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Jul 15, 2008, 6:15:36 AM7/15/08

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Hi, I'm working on the Sage website.

I am searching for interesting content and this time I want to ask

everyone who has used Sage for his or her research or in education in

class to write a short success story. It should talk about how it was

used and the general and personal experience. Just some sentences to

give new users a first impression of Sage.

thanks and greetings, Harald

Jul 15, 2008, 7:10:46 AM7/15/08

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In teaching:

I used SAGE for some sections of a differential equations course

I was teaching last fall. Wasn't sure if the students would just

"take the hit" and refuse to learn SAGE so I made the assignments

worth very little and very easy. Also, I made extra assignments for

extra credit.

It turned out not only did most do the assignment but a lot more

than I expected did the extra credit ones too and say that they liked SAGE and

its philosophy. I would call it a success in teaching.

I used SAGE for some sections of a differential equations course

I was teaching last fall. Wasn't sure if the students would just

"take the hit" and refuse to learn SAGE so I made the assignments

worth very little and very easy. Also, I made extra assignments for

extra credit.

It turned out not only did most do the assignment but a lot more

than I expected did the extra credit ones too and say that they liked SAGE and

its philosophy. I would call it a success in teaching.

In research:

I wrote some time ago a procedure for computing Duursma zeta functions of

a linear error-correcting code. AFAIK, no other program does this (not

GUAVA nor

MAGMA, ...). These zeta functions are very similar to the zeta

function of a curve

(they have a functional equation, a "Riemann hypothesis", etc), except

that no one knows

why the "Riemann hypothesis" doesn't always hold. There is a

conjecture that it holds

for a certain "extremal" class of codes, so maybe the RH holds for "good" or

"optimal" codes (since optimal codes are sometimes extremal)? A year or 2

ago, I found an example of a formally self-dual with optimal

parameters which violates

the RH. I would call that a success in research.

Aug 14, 2008, 1:25:13 AM8/14/08

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to infinity really” to motivate six year olds. Let’s now move on by

motivating all ages using SAGE. I just give examples and not lines of

code, mainly because to promote the usage of SAGE and to encourage

others to check the results found. {Note: I encountered several bugs

in SAGE, but release 3.1 should be fine}

1. After counting we learn our youngsters to use "*","+","-". Not

always easy to find motivating exercises. The following question

proved to be fun for either an individual or a group. Especially in

cases where people had to wait for one hour or longer, e.g. bus trip

et cetera.

a. Easy start. Take the for instance the numbers 3,5,7,8. The

challenge is: make as many numbers as possible (i) starting from 0,

(ii) you may use each number only once, and (iii) using the three

operands "*","+","-".

The amazing fact is that you can produce consecutive numbers in the

range 0..65. The following output comes from a SAGE notebook:

1 = 3+5-7 , 2 = 3+7-8 , 3 = 3 , 4 = 3+8-7 , 5 = 5 , 6 = 3+8-5 , 7 =

7 , 8 = 8 , 9 = 3+5-7+8 , 10 = 3+7 , 11 = 3+8 , 12 = 3*8-(5+7) , 13 =

3+7-5+8 , 14 = 3*5+7-8 , 15 = 3*5 , 16 = 3*5-7+8 , 17 = 3*8-7 , 18 =

3*7+5-8 , 19 = 3*8-5 , 20 = (3+8-7)*5 , 21 = 3*7 , 22 = 3*5+7 , 23 =

3+5+7+8 , 24 = 3*7-5+8 , 25 = (8-3)*5 , 26 = 3*7+5 , 27 = 5*7-8 , 28 =

(5+7)*3-8 , 29 = 3*7+8 , 30 = 3*5+7+8 , 31 = 3*8+7 , 32 = 5*7-3 , 33 =

5*8-7 , 34 = 3*7+5+8 , 35 = 5*7 , 36 = 3*8+5+7 , 37 = 5*8-3 , 38 =

5*7+3 , 39 = (5+8)*3 , 40 = (3+7-5)*8 , 41 = 7*8-3*5 , 42 =

(3+7)*5-8 , 43 = 5*7+8 , 44 = (5+7)*3+8 , 45 = (7+8)*3 , 46 =

3+8+5*7 , 47 = 5*8+7 , 48 = (3+5)*7-8 , 49 = (3*5-8)*7 , 50 =

(3+7)*5 , 51 = 7*8-5 , 52 = (5+8)*(7-3) , 53 = 7*8-3 , 54 = 7*8+3-5 ,

55 = (3+8)*5 , 56 = (3+5)*7 , 57 = (3+5)*8-7 , 58 = (3+7)*5+8 , 59 =

3*8+5*7 , 60 = (5+7)*(8-3) , 61 = 3*7+5*8 , 62 = (3+8)*5+7 , 63 =

3*7*(8-5) , 64 = (3*5-7)*8 , 65 = (3*7-8)*5 , 66 has no expression

b. Less easy. Is 3,5,7,8 the best 4-tuple? Two ways to answer this

using SAGE:

i. All (reasonable) combinations of 4 different numbers

ii. Hard: Prove it by (a) produce symbolic expressions – there are

many! – and (b) just try to solve them.

c. You can play around by also allowing other operands like “/”, “^”,

“mod or % in SAGE”, or you can allow also negative numbers and look

for the longest consecutive row. In the case of 3,5,7,8 and the three

operands "*","+","-" I found 94 consecutive numbers.

d. If you add “^” (so 2^3=8), you find that the 4-tuple 2,3,7,8

delivers a nice row of numbers 0..75 as the following SAGE output

shows:

1 = 2**3-7 , 2 = 2 , 3 = 3 , 4 = 2+3+7-8 , 5 = 2*3+7-8 , 6 = 2*3 , 7 =

7 , 8 = 8 , 9 = 2**3-7+8 , 10 = 2+8 , 11 = 2*7-3 , 12 = 2+3+7 , 13 =

2*3+7 , 14 = 2*3+8 , 15 = 2**3+7 , 16 = 2**3+8 , 17 = 2*7+3 , 18 =

(2*7-8)*3 , 19 = 2*7-3+8 , 20 = 2+3+7+8 , 21 = 2*3+7+8 , 22 = 2*7+8 ,

23 = 2**3+7+8 , 24 = 3**2+7+8 , 25 = 2*7+3+8 , 26 = 2*8+3+7 , 27 =

(2+3)*7-8 , 28 = (3+7)*2+8 , 29 = 3*7+8 , 30 = (2+8)*3 , 31 =

2+8+3*7 , 32 = (7-3)*8 , 33 = (2+3)*8-7 , 34 = 2*3*7-8 , 35 =

(2+3)*7 , 36 = (3+7+8)*2 , 37 = 2*8+3*7 , 38 = 2*7+3*8 , 39 =

(7-2+8)*3 , 40 = (2+3)*8 , 41 = 2*3*8-7 , 42 = 2*3*7 , 43 =

(2+3)*7+8 , 44 = 7**2+3-8 , 45 = (2+7)*(8-3) , 46 = 7**2-3 , 47 =

(2+3)*8+7 , 48 = (2**3)*7-8 , 49 = (2+8-3)*7 , 50 = 2*3*7+8 , 51 =

(2+7+8)*3 , 52 = 7**2+3 , 53 = 7*8-3 , 54 = 7**2-3+8 , 55 = 2*3*8+7 ,

56 = (2**3)*7 , 57 = (2**3)*8-7 , 58 = (3*7+8)*2 , 59 = 7*8+3 , 60 =

(3+7)*(8-2) , 61 = 2+3+7*8 , 62 = 2*3+7*8 , 63 = (3**2)*7 , 64 =

(2**3)*7+8 , 65 = 3**2+7*8 , 66 = (2*7+8)*3 , 67 = (2+8)*7-3 , 68 =

7-3+8**2 , 69 = (2+7)*8-3 , 70 = 2*7*(8-3) , 71 = (2**3)*8+7 , 72 =

(2+7)*8 , 73 = (2+8)*7+3 , 74 = 3+7+8**2 , 75 = (2+3)*(7+8) , 76 has

no expression

e. Allowing negative numbers, I found that 103 consecutive numbers can

be produced. Such an exercise proved to be fun with motivated adults

and some good drinks.

2. Playing around with four number is relatively easy. More difficult

is the 5-tuple question, even in the simple case by only allowing the

operands "*","+","-". The reason is mainly because the number of

possibilities ‘explodes’.

To conclude: SAGE helped me to find relative simple exercises and

questions to motivate people of all ages (i) to do exercises “using

pen and paper”, and (ii) to become aware of the depth of even simple

mathematics. I learned from SAGE that “mathematics on a computer is

even better then gaming”. Roland

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