Units in Euler Math Toolbox

Units have long been an integral part of EMT. I want to show you a few examples. First, let us talk about the non-metric system the US is still using. Quote: „Nobody understands the metric system! We use teaspoons!“.

>1inch
   0.0254
>1in -> " cm"
   2.54 cm
>inch$, in$
   0.0254
   0.0254
>1ft -> inch
   12
>1yard -> ft
   3
>1mile -> yard
   1760
>1mile -> km
   1.609344

As you see, you simply append the unit to a number and it will be converted into the metric system (in case you do not know that’s meters and kilograms). The units are stored in global variables which end with $. These variables are visible even within functions. So units work in functions too.

There is also a conversion operator -> which can either convert to text or to numbers, depending on the type of the parameter after the ->. See the examples.

But there is more to it. Often we use fractions or powers of units. EMT can handle that too. Let us start with fractions. In the following example, we introduce a new unit „nautical mile“ which is missing in EMT but will be present in future versions.

>sm$ = 1852.216;
>160sm/h -> " m/sec"
   82.3207111111 m/sec

As you see, we can now easily convert nautical miles per hours (knots) to meter per second. Assume an airplane departing with 90 knots is forced to use a climb of 200 feet per nautical mile. We want to compute the feet per minute climb which we can read from the VSI.

>200ft/sm * 90sm/h -> " ft/min"
   300 ft/min

Of course, you can do this (at least approximately) in the head, if you multiply the numbers and divide by 60. E.g., at 120 knots, you need to double the ft/sm requirement. For just another example, let us compute the sink rate which is necessary at 190 knots to obtain a 3° glide path.

>200ft/sm * 90sm/h -> " ft/min"
   300 ft/min
>190sm/h * tan(3°) -> ft/min; print(%,unit=" ft/min")
      1008.50 ft/min

The example shows still another way to print with units. By the way, the degree symbol behaves like a unit in many ways.

For further examples, here are some american kitchen units. You will also notice that EMT can handle powers in units on both sides.

>10cm^3
   1e-005
>cup$ = 236.5882365liter/1000
   0.0002365882365
>tablespoon$ = cup$/16;
>teaspoon$ = tablespoon$/2;
>1teaspoon -> " cm^3"
   7.39338239062 cm^3
>1liter -> teaspoon
   135.256090807

08. April 2016 von mga010
Kategorien: English, Euler | Schreibe einen Kommentar

Solving Partition Problems with Linear Programming

Yesterday I promised to show how to solve partition problems with integer linear programming. Let us take the problem of yesterday: We want to split the numbers from 1 to 30 in 10 triplets (x,y,z) such that x+y+z=0 modulo 31. We will use linear problems of the following type.

\(Ax=b, \quad x_i \in \{0,1\}, \quad c^T x \to \text{max.}\)

There will be one variable in x for each possible triplet. The value of this variable determines if the triplet is in the selection (1) or not (0). So the j-th column of A refers to the triplet with number j. The i-th row of A is a linear constraint which makes sure that the number i is used only in one triplet. A contaoins only 0-1-values, and

\(a_{i,j} = 1 \quad\Leftrightarrow\quad i \in T_j = \{n_{1,j},n_{2,j},n_{3,j}\}.\)

The constraints will be

\(\sum_j a_{i,j} x_j = 1.\)

We just need any feasible point and could use any target function. Alternatively, we could use „less equal“ in the previous line and maximize the number of triplets used.

How to do that in EMT? We first store all possible triplets into one matrix.

>function makeT (n:index=31) ...
$T=[];
$for i=1 to n-2;
$   for j=i+1 to n-1;
$       k=n-mod(i+j,n);
$       if k!=i and k!=j then T=T_[i,j,k]; endif;
$   end;
$end;
$return T
$endfunction
>makeT(7)
             1             2             4 
             1             4             2 
             1             6             7 
             2             4             1 
             2             5             7 
             3             4             7 
             3             5             6 
             3             6             5 
             5             6             3 
>T=makeT(31);

Then we define the matrix A.

>function makeA (T) ...
$v=sort(unique(flatten(T)));
$A=zeros(cols(v),rows(T));
$for i=1 to cols(v);
$   for j=1 to rows(T);
$       if any(v[i]==T[j]) then A[i,j]=1; endif;
$   end;
$end;
$return A;
$endfunction
>makeT(5)
             1             4             5 
             2             3             5 
>fraction makeA(makeT(5))
         1         0 
         0         1 
         0         1 
         1         0 
         1         1 
>A=makeA(T);
>size(A)
 [31,  405]

The function makeA() can take any matrix with partitions in its rows. The first command simply selects the numbers that appear in any partition. In our case, A is simply the vector [1,…,30].

Now we need to solve the problem with integer linear programming. We use the LPSOLVE library for this (given to EMT by the developers).

>function solveP (A,T) ...
$  x=ilpsolve(A,ones(rows(A))',ones(cols(A)),
$         vlb=zeros(cols(A)),vub=ones(cols(A)),>max);
$  return T[nonzeros(x')];
$  endfunction
>solveP(A,T)
               1             5            25 
               2             7            22 
               3            11            17 
               4             6            21 
               8             9            14 
              10            23            29 
              12            24            26 
              13            19            30 
              15            20            27 
              16            18            28 

The return value of the function ilpsolve() is a 0-1-vector. We want to print the triplets which are marked by 1 in this vector. The variables vlb and vub are lower and upper bounds for the variables. Interestingly, the solution works without these restrictions. Nevertheless, more restrictions usually mean shorter calculations.

30. März 2016 von mga010
Kategorien: English, Euler | Schreibe einen Kommentar

Modulo Arithmetic with Euler Math Toolbox

Recently, I came across the problem to compute expressions in the modulo arithmetic with integer numbers. Here is an example: We like to compute expressions modulo 31. Since 31 is prime the numbers 0 to 30 with addition and multiplication modulo 31 will be a field. Each number a between 1 and 31 will have a unique multiplicative inverse b, such that ab=1 modulo 31. E.g.

>mod(27*23,31)
     1

The inverse of 27 is 23 modulo 31.

How can one find the inverse element? The fastest way uses Euclid’s algorithm. If you compute the greatest common divider of 27 and 31 using Euclid’s algorithm you can derive x and y such that 27x+31y=1. Then x (taken modulo 31) will be the inverse of 27 modulo 31. Here is the computation for these numbers:

\(31=27+4, \quad 27=6 \cdot 4 + 3, \quad 4 = 3 + 1,\)

and by computing backwards

\(1 = 4 – 3 = 7 \cdot 4 – 27 = 7 \cdot 31 – 8 \cdot 27.\)

Note -8=23 modulo 31. We get our result 23.

Of course, we could program that in EMT or Maxima. Maxima contains already a function for the inverse modulo a number.

>&inv_mod(27,31)
                                   23

We also know the small Lemma of Fermat.

\(a^{p-1} = 1 \text{ mod } p.\)

Thus

\(a \cdot a^{p-2} = 1 \text{ mod } p.\)

This can also be used to compute the inverse. To get the result in EMT we need a trick which I am going to explain below. But in Maxima we can compute very large powers, albeit very slowly.

>&mod(27^29,31)
                                   23

For single computations or small numbers, symbolic computation is okay. But assume we need more speed. The computation above does not work in EMT, because the power is too large. Python, by the way, has infinite integers too. So the computation works there too.

>>> print 27**29
 323257909929174534292273980721360271853387
>>> print 27**29 % 31
 23

One of the reasons of this posting is to show how this can be done in EMT and other languages without an infinite integer arithmetic, and moreover much faster. The trick is the fact that we can as well take the modulo in each step of the multiplication. Here is the first simple code.

>mod(27^29,31) // WRONG RESULT!
   12
>function map powmod (a,n,p) ...
$  b=1;
$  loop 1 to n; b=mod(b*a,p); end;
$  return b
$  endfunction
>powmod(27,29,31) // CORRECT RESULT!
   23

However, this is still not the fastest way. For powers, we can use this trick.

\(x^{2y} = \left(x^y\right)^2, \quad x^{2y+1} = \left(x^y\right)^2 \cdot x.\)

This leads to the following code. By the way, the function matrixpower() of EMT uses the same technique. To understand the code, note that it is not efficient to use a mapping function recursively. Thus we call it from another function.

>function powmodrek (a,n,p) ...
$  if n==0 then return 1
$  elseif n==1 then return a
$  elseif mod(n,2)==0 then
$     b=powmodrek(a,n/2,p); return mod(b*b,p);
$  else
$     b=powmodrek(a,(n-1)/2,p); return mod(mod(b*b,p)*a,p);
$  endfunction
>function map powmod (a:number,n:nonnegative integer,p:index) ...
$  return powmodrek(a,n,p);
$  endfunction
>powmod(27,29,31)
   23
>powmod(27,1:30,31)
   [27,  16,  29,  8,  30,  4,  15,  2,  23,  1,  27,  16,  29,  8,  30,
   4,  15,  2,  23,  1,  27,  16,  29,  8,  30,  4,  15,  2,  23,  1]

The last computation is not effective, of course. If we need to compute all powers we should use a simple loop instead. But I included that example to show that powmod() can map to vector input.

As you see, the powers of 27 do not cycle through all the number 1 to 30. For other base numbers, they do.

>powmod(3,1:30,31)
 [3,  9,  27,  19,  26,  16,  17,  20,  29,  25,  13,  8,  24,  10,  30,
 28,  22,  4,  12,  5,  15,  14,  11,  2,  6,  18,  23,  7,  21,  1]

The second reason for this posting is to demonstrate the solution to another problem. We want to partition the numbers 1 to 30 into triplets with a+b+c=0 modulo 31. This is surprisingly easy. We have for all q different from 1

\(1+q+q^2 = \dfrac{q^3-1}{q-1}\)

Thus by the Lemma of Fermat

\(3^0 + 3^{10} + 3^{20} = \dfrac{3^{30}-1}{3^{10}-1} = 0 \text{ mod }31.\)

Our triples can be found easily from this. Just multiply the equation with powers of 3.

>for k=0 to 9;  ...
>  powmod(3,k,31)+"+"+powmod(3,10+k,31)+"+"+powmod(3,20+k,31)+"=0",  ...
>end;
 1+25+5=0
 3+13+15=0
 9+8+14=0
 27+24+11=0
 19+10+2=0
 26+30+6=0
 16+28+18=0
 17+22+23=0
 20+4+7=0
 29+12+21=0

It is also possible to find such a partition with any desired property using linear programming and LPSOLVE in EMT. This should be the topic of another posting, however.

29. März 2016 von mga010
Kategorien: English, Euler | Schreibe einen Kommentar

Mathematik oder Pädagogik?

Ich lese in ZEIT online das Interview mit einer ehemaligen Schulleiterin in einer Hamburger „Stadtteilschule“. In einem Bildblock herausgestellt wird folgender Satz:

Die Ausbildung unserer Lehrer ist praxisfern: Schüler brauchen keine Fachgenies, sondern Pädagogen, die ihre Probleme verstehen.

Wie so oft ist das so zugespitzt formuliert, dass jeder zustimmen kann. „Fachgenies“ sind ganz offensichtlich in der Schule fehl am Platz. Ebenso wenig bestreitet jemand, dass Lehrer und Lehrerinnen die Probleme der Schüler verstehen sollten. Der Satz entlarvt sich damit als Meinungsmache, deren wesentliches Merkmal ist Sachverhalte zu unterstellen, die so nicht der Realität entsprechen. Stimmt man dem Satz zu, dann stimmt man auch der tendenziösen Botschaft zu, die ganz einfach lautet: Weniger Fachausbildung für Lehrer!

Nun versuchen wir es damit in Deutschland seit Jahren. Die Lehramtsausbildung für das Gymnasium entspricht inzwischen nur mit zusätzlichen Modulen einem Bachelor-Abschluss im Fach. Die Zulassungsarbeit kann durch eine Bachelorarbeit ersetzt werden. Im Vergleich dazu war es noch vor wenigen Jahren ohne Problem möglich, mit dem ersten Staatsexamen in die Promotion einzusteigen. In den anderen Schularten sieht es nicht besser aus. Fachinhalte wurden und werden dort bis auf das Schulniveau reduziert. Wurde die Schule dadurch besser? Nach Aussage derselben Kritiker unserer Lehramtsausbildung ist gerade das nicht der Fall. Sie empfehlen allerdings nur mehr von einer Medizin, die bisher auch nicht gewirkt hat.

Fragen Sie doch einmal einen beliebigen Schulleiter, was ihn an der real existierenden Schule am meisten bedrückt. Über zu große Fachkenntnisse des Lehrpersonals werden Sie da nichts hören. Im Gegenteil gibt es Schulleiter, die sich über mangelnde fachliche Professionalität von Referendaren beschweren. Das führt zu Ärgernissen mit Eltern und Probleme im Unterricht bis hin zur Notwendigkeit, die Klausuren dieser angehenden Kollegen nachkorrigieren lassen zu müssen. Die wirklichen Klagen der Schulleiter betreffen eher Schüler, Eltern und die Schulbehörde.

Nun ist gerade die veränderte Zusammensetzung der Klassen ein Argument der Pädagogen für mehr Pädagogik und weniger fachliche Ausbildung. Nutzt das etwas? Ich glaube nicht. Auch ich kann provokante Sätze formulieren und textuell herausstellen:

Es ist nicht hilfreich, nützliche Fachvorlesungen in Mathematik durch verkopfte Fachvorlesung in Pädagogik zu ersetzen.

Wir sollten allerdings von einer übermäßigen Konfrontation wieder ein wenig herunterkommen. Wir sind uns schließlich einig darüber, dass der Erfolg der Schule, und zwar nicht nur bei den unteren Schichten, zu wünschen übrig lässt. Die Kompetenzen in den MINT-Fächern ebenso wie die Lesekompetenzen, die ja ständig gemessen werden, sind im Schnitt unzureichend. Wir sollten uns auch einig darüber sein, dass ein Lehrer sowohl pädagogisch-didaktische, also auch fachliche Qualitäten haben muss. Ein guter Lehrer benötigt beides, und dazu noch menschliche Fähigkeiten, die man nicht antrainieren kann.

Ein guter Lehrer besitzt Sachkenntnis sowie pädagogische und didaktische Kompetenzen in gleichem Maße. Im optimalen Fall hat er auch besondere menschliche Qualitäten. 

Ich schlage erneut eine zweiphasige Ausbildung als Kompromiss vor, bestehend aus einer Fachausbildung in der ersten Phase und Ausbildung zum Lehrer in der zweiten Phase. Die Fachausbildung besteht aus einer Ausbildung zum Bachelor in den zu unterrichtenden Fächern, mit einem Fach als Hauptfach. Das gilt zumindest für die Lehrämter am Gymnasium. In der zweiten Phase werden dann Schulpraktika, pädagogische Inhalte und didaktische Module miteinander zu einem Master of Education verzahnt. Das erste Staatsexamen wird abgeschafft. Dieser Vorschlag wurde schon öfter gemacht, auch von prominenter Seite. Er scheiterte aber immer an universitären Lehramtsexperten, die glauben, einen Lehrer von der Schule direkt wieder in die Schule abholen zu müssen. Im Unterschied dazu glaube ich als Fachdozent nicht, bei der mich nicht betreffenden anderen Phase, der eigentlichen Lehramtsausbildung, hineinreden zu müssen.

Was die Lehrämter an Grund- und Hauptschulen angeht, so bin auch ich der Meinung, dass hier die Fachinhalte auf das Notwendigste zu reduzieren sind. Bei der Realschule allerdings ist die Lage kritischer zu sehen. Dies ist eine Schulart, für der sich zu viele Studentinnen und Studenten entscheiden, ohne allerdings die fachlichen Voraussetzungen für einen erfolgreichen Unterricht mitzubringen oder zu erwerben. Der Überhang an Bewerbern ist hier so groß, dass eine fachliche Hürde nicht schaden kann. Derzeit machen etwas zehnmal mehr Kandidatinnen und Kandidaten einen Abschluss als Stellen zur Verfügung stehen. Ob man hier einen Bachelor vorsehen sollte, mag diskutierbar sein. Aber es muss klar sein, dass selbst an Realschulen ein Mathematiklehrer ein Mathematiker sein muss. Amateure richten zu viel Schaden an.

Ein Mathematiklehrer muss Mathematiker sein, kein Amateur. Er muss sich im Studium die Grundsätze seinen Faches angeeignet haben. Die lassen sich eben nur in Fachvorlesungen üben.

Schließlich möchte ich noch präzisieren, was ich unter „Fachinhalten“ verstehe. Keineswegs müssen das fortgeschrittene Forschungsinhalte des jeweiligen Fachs sein. Auch sind es nicht unbedingt die klassischen Inhalte ohne Rücksicht darauf, wie sich Fächer in der modernen Zeit wissenschaftlich wandeln. Und die Curricula der Bachelor in Mathematik sind in der Tat moderner geworden. Wo etwa Funktionentheorie und Algebra noch immer eine Hauptrolle spielen, ist das dem Lehramtsstudium und dem schriftlichen Staatsexamen geschuldet. Diese scharfe, aber unsachgemäße Akzentuierung würde mit dem obigen zweistufigen Modell sofort aufhören. Das ist übrigens ein weiterer Grund, warum sich selbst Fachvertreter gegen eine Reform des Lehramtsstudiums wenden.

Eine bessere Schule tut Not. Sie ist eine gesellschaftliche Herausforderung. Die Lehramtsausbildung ist nur eine Facette des Problems. Vermutlich wird man ohne eine Aufwertung des Lehrerberufs nicht auskommen. Die erfolgreichen Länder vereinen strikte Zugangsprüfungen mit guter Bezahlung und guten Arbeitsbedingungen. Vielleicht sollten wir ein wenig von ihnen lernen.

25. Februar 2016 von mga010
Kategorien: Deutsch, PISA | Schreibe einen Kommentar

Euler Math Toolbox and Matlab

Recently, this site got more traffic than usual. So I thought I welcome all of you in the new year (which is already three weeks old now). And I provide some new content for you to read. For me, the selected topic is important. I welcome very much any discussion about it. I am currently writing a requested paper for an on-line journal about Euler Math Toolbox, so I could benefit from your input.

What is the difference between Euler Math Toolbox and Matlab? Or rather, what is the intended difference? And are there alternatives to both?

First of all the following points are obvious to me.

  • Matlab is profiling itself in the professional market. It is called the „industry standard“, and it is advertised as a must-know to students. Matlab tries very much to serve the needs of the industry. The best example is its simulation toolbox which connects Matlab to hardware. Matlab is backed up by a group of commercial programmers.
  • Euler Math Toolbox is for education and research. It is not intended to be used as a professional tool in industry. EMT is backed up by open software.
  • Both systems fall short for specialized applications. Most professional or research software is compiled in a general programming language and does things that a one-for-all software cannot do.
  • We need to educate our students in general programming, not in software systems, especially not in commercial systems. With a good background in programming any software can be adapted easily.
  • That said, a tool like Euler Math Toolbox or Matlab can be useful for quick computations or to mathematical demos. And for this, both systems are equally well equipped with an advantage for EMT.
  • We should have free and open, reliable numerical libraries for our main computer languages.

If you are interested in the primary differences between EMT and Matlab, here is a list.

  • EMT is open and free, Matlab is commercial with a considerable pricing.
  • The user interface of EMT is notebook oriented similar to Maple or Mathematica, Matlab is command oriented. Both have scripts, of course, and are similar in this respect.
  • Both systems can do symbolic computations, EMT with the free Maxima, Matlab with an additional package you need to buy.
  • EMT can use many open external programs like Povray, Python, C, Scilab, Latex. For Matlab, you can buy powerful external packages to do industry strength computations.
  • Matlab has an optional compiler. In EMT, you need to write C or Python for more speed.
  • Matlab is used all around the world, and you will easily find someone with the same problem as you have. EMT has a community, but it is way smaller.

There might be more differences that I forgot to mention. But all in all it is clear to me that EMT deserves to be used in education much more than it is right now. Matlab, on the other hand, should be removed from general education. It should be reserved for specialized courses done by engineers who have used the program for their work.

Alternatives? There are many.

  • R is a statistical package featuring a matrix language as well. It is free and has a big community. The programming in R is not as easy to learn as they claim, but to get statistics done it is not necessary. The interface is ugly and cumbersome to use.
  • Maple, Mathematica or other algebra software are commercial packages in the higher price range. They offer more advanced features than Maxima and often yield better results. But they are not designed to be used as numerical software.
  • Geogebra tries to be the knife for all purposes. It is a beautiful software for schools and has a large community. Clearly, it falls short in the area of numerical and other programming. Actually, it is a whole different category of software.
  • Python is probably the best alternative to big software packages. There are very nice packages for plotting and for numerical computations. The symbolic package falls short in comparison, however. The Sage project relies completely on Python and has a web interface which clearly is the proper way in the future.
  • General programming langes are the way to do software education in universities for engineers and math students. Most courses have an obligatory course in Java, and rightly so.

Open for discussion.

25. Januar 2016 von mga010
Kategorien: English, Euler | 2 Kommentare

Pure Symbolic Functions in Maxima and EMT

I like to demonstrate a few tricks with Maxima and EMT. Assume you want to generate the Vandermonde matrix, and check the identity

\(\det \begin{pmatrix} 1 & x_0 & \ldots & x_0^n \\ \vdots & \vdots &  & \vdots \\ 1 & x_n & \ldots & x_n^n \end{pmatrix} = \prod\limits_{0 \le i<j \le n} (x_j-x_i)\)

You can generate this matrix easily in EMT for specific values.

>n=3; v=0:3; V=v'^(0:3)
             1             0             0             0 
             1             1             1             1 
             1             2             4             8 
             1             3             9            27 
>det(V)
 12

It is a bit more difficult to check the right hand side of the equation. Of course, you can simply write a function

>function vprod (v:vector) ...
$p=1; n=length(v);
$for i=1 to n;
$   for j=i+1 to n;
$       p=p*(v[j]-v[i]);
$   end;
$end;
$return p;
$endfunction
>vprod(v)
 12

It is also possible with Matrix tricks. We generate all differences in a matrix, square the matrix, set the diagonal to 1, take the total product of all matrix elements and take the 4-th root.

>(prod(prod(setdiag((v-v')^2,0,1))'))^(1/4)
 12

But that is not the general case with variables. To generate this we use purely symbolic functions in EMT. These functions evaluate in Maxima at the time when they are used, not when they are defined.

>function mrow (k,v,n) &&= create_list(v[k]^m,m,0,n)

                                     m
                        create_list(v , m, 0, n)
                                     k

>function Vand (v,n) &&= apply(matrix,create_list(mrow(k,v,n),k,0,n))

           apply(matrix, create_list(mrow(k, v, n), k, 0, n))

>&Vand(v,2)

                             [         2 ]
                             [ 1  v   v  ]
                             [     0   0 ]
                             [           ]
                             [         2 ]
                             [ 1  v   v  ]
                             [     1   1 ]
                             [           ]
                             [         2 ]
                             [ 1  v   v  ]
                             [     2   2 ]

This is almost self explaining, once you know that „matrix(a,b,c)“ generates a matrix in Maxima with rows „a,b,c“. The creation takes place for the specific symbolic v when „Vand(v,2)“ is called.

Maxima prints the result with indices. But it is a string of the following form.

>V &= Vand(v,2);
>""+V
 matrix([1,v[0],v[0]^2],[1,v[1],v[1]^2],[1,v[2],v[2]^2])

Since EMT vectors start with index 1 by default, we can only evaluate this expression for zero based vectors (unless we modify the functions above a little bit).

>v=0:2; zerobase v;
>V()
             1             0             0 
             1             1             1 
             1             2             4

Maxima can factor the determinant of any Vandermonde determinant of specific size.

>&factor(determinant(Vand(v,3)))

         (v  - v ) (v  - v ) (v  - v ) (v  - v ) (v  - v ) (v  - v )
           1    0    2    0    2    1    3    0    3    1    3    2

This is in accordance with the theory. Let me try the following matrix.

\(\begin{pmatrix} (x_0-x_0)^n & \ldots & (x_0-x_n)^n \\ \vdots & & \vdots \\ (x_n-x_0)^n & \ldots & (x_n-x_n)^n \end{pmatrix}\)

>function mrow (k,v,n) &&= create_list((v[k]-v[m])^n,m,0,n)
 
                                         n
                    create_list((v  - v ) , m, 0, n)
                                  k    m
 
>function Vand (v,n) &&= apply(matrix,create_list(mrow(k,v,n),k,0,n))
 
           apply(matrix, create_list(mrow(k, v, n), k, 0, n))
 
>&Vand(v,2)
 
                 [                      2           2 ]
                 [     0       (v  - v )   (v  - v )  ]
                 [               0    1      0    2   ]
                 [                                    ]
                 [          2                       2 ]
                 [ (v  - v )       0       (v  - v )  ]
                 [   1    0                  1    2   ]
                 [                                    ]
                 [          2           2             ]
                 [ (v  - v )   (v  - v )       0      ]
                 [   2    0      2    1               ]

I am not sure if you are aware of the following formula.

>&factor(determinant(Vand(v,3)))
 
                    2          2          2          2          2
         9 (v  - v )  (v  - v )  (v  - v )  (v  - v )  (v  - v )
             1    0     2    0     2    1     3    0     3    1
                                                                      2
                                                             (v  - v )
                                                               3    2

The factor 9 is interesting. For n=4, it is 96, and for n=5, it is 2500. Do you guesswork. The factorization of the case n=4 takes some seconds. I did not try higher degrees.

Of course, it is easy to compute this numerically, even for very large matrices.

>function d1(v) := det((v'-v)^(cols(v)-1))
>d1(0:10)
 1.43679021499e+072

 

12. Dezember 2015 von mga010
Kategorien: English, Euler | Schreibe einen Kommentar

December Problem

I found the following problem on the page of the wonderful Science Blogs. It is in German. Let me translate it. Bernd writes two different numbers between 1 and 1000 on two cards of paper. Anna selects one, and after seeing it has to decide if she wants to keep it or change to the other card. The player with the higher card wins.

This is a case of a two person game with the following tactics for each player.

  • Bernd: He can write any combination (i,j) with 1 <= i < j <= N.
  • Anna: She can decide to change up to a number A with 1 <= A <= N.

Note that we simplified the situation for Anna. It does not make sense to change with 3, but not with a smaller number. Weather or not this simplification makes the outcome worse for Anna remains to be investigated.

Now, Bernd can select any of his tactics (i,j) with probability p(i,j), and Anna can select any of her tactics with probability q(A). The problem to select the optimal strategy (i.e. the distribution of probabilities) for each player can be solved with optimization. I have done similar problems with EMT on this page. I solved the problem above for (N=10) with a bit of code.

However, I want to spoil the solution here. If you want to try on your own, do not read further.

  • Strategy for Bernd: Bernd should select the pairs (1,2),…,(N-1,N) at random with equal probability.
  • Strategy for Anna: Before each game, Anna should select a number A from 1 to N-1 at random, and switch if her number is less or equal A.

Let us prove that this is optimal.

Assume, Anna selects tactics A. So she will change cards at any number smaller or equal A. Then Bernd will lose 1/(N-1) on average with his strategy. This is so, since the only case that makes a difference for him is the pair (A,A+1), where Anna wins all the time. So, no matter what tactics A Anna selects, the worst case is an average loss of 1/(N-1). Only if Anna would select A=N, he would do better.

Now assume, Bernd selects tactics (i,j). Then Anna, with her strategy, will win in any case if i<=A<j. In all other cases, the average result is 0. We need to find the worst case for Anna based on her strategy. That is indeed the case, if Bernd selects a pair (i,i+1) and it is equal to a win of 1/(N-1).

It is now clear that there cannot be a better strategy for either player. Anna cannot win more than Bernd loses on average.

With a bit more care in the proof, it is also clear that it would not help Anna to add more strategies. Assume, she rejects some numbers, and accepts others, instead of rejecting the numbers 1 to A altogether for some A. Then Bernds tactics will lose less or equal 1/(N-1) on average. For an example, assume Anna rejects 1 and 3. Then (1,2) will be a loss for Bernd, (2,3) will be a win, and (3,4) will be a loss, all other cases are 0 on average.

This is a very nice problem. I could not find a reference in the net, however. But it should be known.

Let me add the code for EMT.

>v1=flatten(dup(1:10,10));
>v2=flatten(dup((1:10)',10));
>i=nonzeros(v1<v2); v1=v1[i]; v2=v2[i]; 
>A=(1:10)';
>M=-((v1<=A)&&(A<v2)); 
>function game (R) ...
$## Maximizes G such that R'.q>=G, sum q_i = 1, q_i>=0.
$## Returns {q,G}.
$    m=rows(R); n=cols(R);
$    M=R|-1_ones(1,n); b=zeros(m,1)_1; eq=ones(m,1)_0;
$    c=zeros(1,n)|1; restr=ones(1,n)|0;
$    res=simplex(M,b,c,eq,restr,>max);
$    return {res[1:n]',res[-1]};
$endfunction
>{pB,res}=game(M); fraction res
 -1/9
>k=nonzeros(pB); fraction (pB[k]_v1[k]_v2[k])'
       1/9         1         2 
       1/9         2         3 
       1/9         3         4 
       1/9         4         5 
       1/9         5         6 
       1/9         6         7 
       1/9         7         8 
       1/9         8         9 
       1/9         9        10 
>{pA,res}=game(-M'); fraction res
 1/9
>fraction pA'|(1:10)'
       1/9         1 
       1/9         2 
       1/9         3 
       1/9         4 
       1/9         5 
       1/9         6 
       1/9         7 
       1/9         8 
       1/9         9 
         0        10

This looks a bit difficult. Here are some explanations.

The vectors v1 and v2 simply contain the pairs (i,j) with 1 <= i < j <= N = 10. The trick to get all pairs is to create matrices with 1 to 10 in each row resp. column and to „flatten“ them to vectors. Then we select only the pairs with i < j using the nonzeros() function of EMT.

The matrix M is then constructed in the following way: The element in row A and column (i,j)  is -1 (a loss for Bernd) if i <= A < j. This is the matrix of average outcomes if Anna selects A and Bernd selects (i.j).

The game() function calls the simplex algorithm for the following problem.

\(Mp \ge G, \quad \sum_k p_k = 1, \quad G \to \text{Max.}\)

The probabilities p(k) are then maximizing the worst case G, for all rows of M (tactics of Anna). The result is p and G. We print the nonzero elements of p along with the corresponding i and j.

The same is finally done with the problem seen from Anna’s side.

By the way, the fact that both results must be equal (besides the sign) is an application of duality in optimization.

Really a nice problem!

 

07. Dezember 2015 von mga010
Kategorien: English, Euler | Schreibe einen Kommentar

Today in the Classroom

Me: Instead of the sum of squares we can also try to minimize the maximal distance

\(f(c) = \max_\limits{1 \le k \le n} |x_k-c|\)

What would be the result, assuming we have the points in order

\(x_1 < \ldots < x_n\)

Silence …

After two painful minutes, Me: Okay, let’s make a sketch. If c is here

Unbenannt-1

What is the value of f(c)?

Silence …

I can see from some faces that they know the answers, or at least have an idea about it. But still, silence.

Only after a long while with me impatiently waiting for any response, I can hear a whisper: In the middle.

So what is this? Are they afraid to show silliness? No posing? Is this problem so hard? I cannot say.

Currently, I assume that these Bachelor students are trained to passively absorb math. It starts in school, and continues, much worse, in University. How we can break through this I do not know. There are some legal problems involved. You can no longer enforce active participation. That was an illusion anyway. But we should generate a climate where students don’t feel bad about making mistakes. After all, learning math is not flight training. How to do that against the habits of all other classes is something I would love to learn.

09. November 2015 von mga010
Kategorien: English | 2 Kommentare

Numerical versus Symbolic Solutions

I did a short video about symbolic solutions versus numerical methods. The point is that a solution such as

\(x=\left(\frac{\sqrt{5339}}{8\,3^{\frac{3}{2}}}+\frac{379}{216}\right)^{\frac{1}{3}}-\dfrac{2}{9\,\left(\frac{\sqrt{5339}}{8\,3^{\frac{3}{2}}}+\frac{379}{216}\right)^{\frac{1}{3}}}-\dfrac{1}{3}\)

are not helpful, unless you are a number theorist. The solutions of equations of degree 5 are usually not available at all.

This has to be continued, because often symbolic computations can be useful if done in a clever way. Stay tuned.

03. November 2015 von mga010
Kategorien: Uncategorized | Schreibe einen Kommentar

Floating Point Arithmetic – Youtube Video

I demonstrate floating point arithmetic and big floats in this video on Youtube. I will continue this in a series with videos about EMT.

02. November 2015 von mga010
Kategorien: Uncategorized | Schreibe einen Kommentar

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