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dharr

Dr. David Harrington

8917 Reputation

22 Badges

21 years, 140 days
University of Victoria
Professor or university staff
Victoria, British Columbia, Canada

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Contact Dr. David Harrington
I am a retired professor of chemistry at the University of Victoria, BC, Canada. My research areas are electrochemistry and surface science. I have been a user of Maple since about 1990.

MaplePrimes Activity


These are replies submitted by dharr

Remark...

dharr 8917
7 hours ago
0 0

I understand the 100 possibilities to be 10 x 10, with 10 being the ways to choose 2 things out of 5. Is it 2 because it is d2 - d1? or is it 2 regardless of the size of the matrices.

I don't know of such a formula, but seems bear to  some resemblance to Cauchy-Binet.

Simplified code:

Adjoint353_3.mw

Remark...

dharr 8917
12 hours ago
0 0

@salim-barzani I don't understand what you mean. I don't understand why the paper doesn't use Eq 2.5 to change Eq 2.8 to an algebraic equation in R and just solve it (for a chosen n). The paper is carelessly prepared and there are some sign issues especially with Eqs 2.2 and 2.5 and the phi equivalent to 2.5 (that @janhardo used, and is not what the paper says). I don't understand about eq 2.2 and the chirp, which just seems to come out of nowhere. Maybe you understand that and can fix the sign issue. Maybe you know how to go from 2.8 to 2.9. This seems like a poor paper to work from to learn this method; I certainly am not spending any more time on it without an explanation of these unresolved issues. Probably just get a different paper that uses the same method and work from that.

Don't know...

dharr 8917
22 hours ago
0 0

@salim-barzani I don't know how they got from 2.8 to 2.9 so I stopped there.

nice substitution...

dharr 8917
January 11 2026
0 0

@janhardo I was stuck until I saw your answer, substituting R' instead of phi'. Vote up. Like many similar papers there are lots of typos and poor descriptions.

s1.mw

Weierstrass function...

dharr 8917
January 07 2026
0 0

@salim-barzani The wikipedia page has something about the analogy with trig functions. Its doubly-periodic nature can be shown in a plot:

Doubly periodic nature of the Weierstrass function. For example a hexagonal lattice with unit cell vectors in the complex plane at 60degrees apart. The poles are at the lattice points

See DLMF - 23.5.8 for lattice with omega__1  real, periodicity along the real axis = 2*omega__1. The g-invariants are

> 

restart
> 

g__2 := 0; g__3 := GAMMA(1/3)^18/(4*Pi*`ω__1`)^6

0

(64/19683)*Pi^12/(GAMMA(2/3)^18*omega__1^6)

omega_1 = 1.

> 

f := WeierstrassP(z, g__2, eval(g__3, `ω__1` = 1))

WeierstrassP(z, 0, (64/19683)*Pi^12/GAMMA(2/3)^18)

> 

p := plots:-complexplot3d(f, z = -5-5*I .. 5+5*I, view = [default, default, 0 .. 4])

Looking down

> 

plots:-display(p, orientation = [-90, 0], style = surfacecontour)

> 

NULL

Download Weierstrass.mw

WeierstrassP...

dharr 8917
January 07 2026
0 0

@salim-barzani I did Eq 47 at the bottom, as in table 2. odetest gives zero.

ode-17.mw

not a solution...

dharr 8917
January 04 2026
1 0

If I put in some random values for the parameters, I don't get zero, so it doesn't seem to be a solution.

T-pde.mw

much smaller example...

dharr 8917
December 29 2025

Interesting bug. I reduced it down to a smaller example within a worksheet.

This version works correctly - the bitmask bm has 1000 1's so Anding with num1 (which has fewer bits) just returns num1

restart

n := 1000; num := 124345994492475404798327875362902901992795666778727657534420884655177904531686659133645511903099791240872149553399687857146297830364878763673608694434636759773303896159506085472672670558713210501921333058462227444188822916868591893071268581049687884472831284477016171744356197692209391893001975814590875407081409079843847570753624652978353229534727345944100084962186575445080168254068349693213343402194051212518373483099158508416844494586587575956364301894537801311066848168445514196432228108255100694654760323061911666708425773177386359200866161779630276830551354956041869854482996129392080604969770318232847290083683448702192322340; bm := integermul2exp(1, n)-1

10715086071862673209484250490600018105614048117055336074437503883703510511249361224931983788156958581275946729175531468251871452856923140435984577574698574803934567774824230985421074605062371141877954182153046474983581941267398767559165543946077062914571196477686542167660429831652624386837205668069375

NULL

NULL

operation := proc (num, n) local q, k; q := num; for k to 2 do q := integerdivq2exp(q, n) end do; q end proc

NULL

num1 := 1083029963437854242395921050992

1083029963437854242395921050992

num1_And_result := Bits:-And(num1, bm)

1083029963437854242395921050992

NULL

Now we do exactly the same but running "operation"  (line uncommented) before defining num1. The value of num1_And_result is now much longer - the low order bits match those of num1 but there are extra high order bits

restart

n := 1000; num := 124345994492475404798327875362902901992795666778727657534420884655177904531686659133645511903099791240872149553399687857146297830364878763673608694434636759773303896159506085472672670558713210501921333058462227444188822916868591893071268581049687884472831284477016171744356197692209391893001975814590875407081409079843847570753624652978353229534727345944100084962186575445080168254068349693213343402194051212518373483099158508416844494586587575956364301894537801311066848168445514196432228108255100694654760323061911666708425773177386359200866161779630276830551354956041869854482996129392080604969770318232847290083683448702192322340; bm := integermul2exp(1, n)-1

10715086071862673209484250490600018105614048117055336074437503883703510511249361224931983788156958581275946729175531468251871452856923140435984577574698574803934567774824230985421074605062371141877954182153046474983581941267398767559165543946077062914571196477686542167660429831652624386837205668069375

NULL

NULL

operation := proc (num, n) local q, k; q := num; for k to 2 do q := integerdivq2exp(q, n) end do; q end proc

NULL

operation(num, n)

1083029963437854242395921050992

num1 := 1083029963437854242395921050992

1083029963437854242395921050992

num1_And_result := Bits:-And(num1, bm)

1830994333800291384327304938937691735750004293893503924104067280873348896706437751150658315935255321166852220199930976133533210823991007641683152920140445237629081271967190811426086258505503763155530649996669558866130785263471490070113476827138676322045464116977583554908224267729573167448333358722416

NULL

But if we define num1 before running operation, it is OK.

restart

n := 1000; num := 124345994492475404798327875362902901992795666778727657534420884655177904531686659133645511903099791240872149553399687857146297830364878763673608694434636759773303896159506085472672670558713210501921333058462227444188822916868591893071268581049687884472831284477016171744356197692209391893001975814590875407081409079843847570753624652978353229534727345944100084962186575445080168254068349693213343402194051212518373483099158508416844494586587575956364301894537801311066848168445514196432228108255100694654760323061911666708425773177386359200866161779630276830551354956041869854482996129392080604969770318232847290083683448702192322340; bm := integermul2exp(1, n)-1

10715086071862673209484250490600018105614048117055336074437503883703510511249361224931983788156958581275946729175531468251871452856923140435984577574698574803934567774824230985421074605062371141877954182153046474983581941267398767559165543946077062914571196477686542167660429831652624386837205668069375

NULL

NULL

operation := proc (num, n) local q, k; q := num; for k to 2 do q := integerdivq2exp(q, n) end do; q end proc

NULL

num1 := 1083029963437854242395921050992

1083029963437854242395921050992

operation(num, n)

1083029963437854242395921050992

num1_And_result := Bits:-And(num1, bm)

1083029963437854242395921050992

It is also OK if we alter the call to operation so the output is different - argument 1001 instead of n = 1000

restart

n := 1000; num := 124345994492475404798327875362902901992795666778727657534420884655177904531686659133645511903099791240872149553399687857146297830364878763673608694434636759773303896159506085472672670558713210501921333058462227444188822916868591893071268581049687884472831284477016171744356197692209391893001975814590875407081409079843847570753624652978353229534727345944100084962186575445080168254068349693213343402194051212518373483099158508416844494586587575956364301894537801311066848168445514196432228108255100694654760323061911666708425773177386359200866161779630276830551354956041869854482996129392080604969770318232847290083683448702192322340; bm := integermul2exp(1, n)-1

10715086071862673209484250490600018105614048117055336074437503883703510511249361224931983788156958581275946729175531468251871452856923140435984577574698574803934567774824230985421074605062371141877954182153046474983581941267398767559165543946077062914571196477686542167660429831652624386837205668069375

NULL

NULL

operation := proc (num, n) local q, k; q := num; for k to 2 do q := integerdivq2exp(q, n) end do; q end proc

NULL

operation(num, 1001)

270757490859463560598980262748

num1 := 1083029963437854242395921050992

1083029963437854242395921050992

num1_And_result := Bits:-And(num1, bm)

1083029963437854242395921050992

NULL

Download Bits_bug2.mw

Maple Flow...

dharr 8917
December 29 2025
0 0

There are instructions for migrating to Maple Flow (not Maple) and downloading the Migration Assistant at https://www.maplesoft.com/products/mapleflow/migrating-to-maple-flow/

(I don't use Maple Flow and don't have any experience with this.)

polynomial systems...

dharr 8917
December 29 2025
0 0

@Alfred_F I like to distinguish polynomial systems from more complicated nonlinear systems, since Maple is very good at dealing with polynomials systems (including with inequalities). 

multiplication...

dharr 8917
December 28 2025
0 0

Your expand(dio1) didn't work because you entered the multiplcations in dio1 as "." (matrix multiplication) rather than "*". If you change those then expand(dio1) works as expected. Now for the harder bit...

multiplicative group...

dharr 8917
December 27 2025
0 0

@Blanc In the GroupTheory package you can set up the finite field with GeneralLinearGroup(1,53), but the elements are permutations, so not so easy to work with. But outside of the GroupTheory package you can find the primitive elements with the GF commands.

> 

restart;

Set up the finite field.

> 

F := GF(53, 1);

_m2143189235968

If we want to represent the elements by integers we need to use the ConvertIn, ConvertOut mechanism. For example

> 

2*31 mod 53;

9

> 

two := F:-ConvertIn(2);
thirty_one := F:-ConvertIn(31);

modp1(ConvertIn(2, T), 53)

modp1(ConvertIn(31, T), 53)

Shorthand way

> 

use F in
two*thirty_one
end use;

modp1(ConvertIn(9, T), 53)

Longer way

> 

F:-`*`(two, thirty_one);
F:-ConvertOut(%);

modp1(ConvertIn(9, T), 53)

9

We can find a primitive element at random and check its multiplicative order is 52

> 

F:-PrimitiveElement();
F:-order(%);

modp1(ConvertIn(21, T), 53)

52

To find all the primitive elements - there are 24 of them

> 

[seq(F:-ConvertIn(i), i = 1..52)]:
map(F:-ConvertOut, select(F:-isPrimitiveElement, %));
nops(%);

[2, 3, 5, 8, 12, 14, 18, 19, 20, 21, 22, 26, 27, 31, 32, 33, 34, 35, 39, 41, 45, 48, 50, 51]

24

 

Download Field.mw

%0...

dharr 8917
December 26 2025
0 0

%0 in the message string in userinfo would give the comma

> 

restart;
> 

showargs:=proc()
userinfo(2,procname,"1. arguments: ",_passed);
userinfo(2,procname,"2. arguments: %0",_passed);
NULL
end proc;

proc () userinfo(2, procname, "1. arguments: ", args); userinfo(2, procname, "2. arguments: %0", args); NULL end proc

> 

infolevel[showargs]:=2;

2

> 

showargs(y(x),sin(y));

showargs: 1. arguments:  y(x) sin(y)

showargs: 2. arguments: y(x), sin(y)

> 

NULL

Download userinfo.mw

Like this?...

dharr 8917
December 26 2025
0 0

@Blanc Is this what you want?

> 

restart
> 

with(GroupTheory)

Define group operation and inverse for additive group mod 53.

> 

n := 53

53

> 

Z := module() export `.`, `/`;
`.`:=(a,b)->a+b mod n; # multiplication rule
`/`:=a->-a mod n;      # inverse of a
end module;

_m2104944203328

Use 1 to generate the group

> 

G := CustomGroup([1], Z); GroupOrder(G)

_m2104873192640

53

> 

els := [Elements(G)[]]

[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52]

As expected the orders of all elements except the identity are equal to the group order. Therefore they can all generate the group

> 

map(ElementOrder, els, G)

[1, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53]

We can check by seeing if we can generate the same group from one of these elements

> 

G2 := CustomGroup([23], Z); Elements(G2)

_m2104979438784

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52}

Some group tests etc are easier if we have a permutation group. Convert it to a permutation group

> 

PG := PermutationGroup(G); g := GroupOrder(PG); IsCyclic(PG)

_m2104894529568

53

true

Find the elements that have ElementOrder equal to the group order

> 

gens, notgens := selectremove(proc (i) options operator, arrow; ElementOrder(i, PG) = g end proc, [Elements(PG)[]])

The identity is not a generator; the other 52 are.

> 

notgens; nops(gens)

[_m2104893251488]

52

We could check by generating the group with an element and see if we get the same group

> 

el27 := Elements(PG)[28]

_m2104972448896

> 

PG2 := Group({el27})

_m2104815721152

> 

evalb(Elements(PG) = Elements(PG2))

true

> 

NULL

Download CyclicGp.mw

same output...

dharr 8917
December 25 2025
0 0

@nm Since you were originally interested in the integral, why not just

seq(evalf(Int(sqrt(1+sin(x)^2),x=0..phi)),phi=-3..3,0.5);

which gives the same as MMA. But if you really want to go from the EllipticE, then

> 

restart;
> 

[seq(evalf(Int(sqrt(1+sin(x)^2),x=0..phi)),phi=-3..3,0.5)];

[-3.678135309, -3.140058363, -2.507982114, -1.810019561, -1.123887723, -.5191743566, 0., .5191743566, 1.123887723, 1.810019561, 2.507982114, 3.140058363, 3.678135309]

> 

q:=[seq(int(sqrt(1+sin(x)^2),x=0..phi), phi=-3..3, 1/2)];

[-2^(1/2)*EllipticE((-sin(3)^2+1)^(1/2), (1/2)*2^(1/2))-2^(1/2)*EllipticE((1/2)*2^(1/2)), -2^(1/2)*EllipticE((-sin(5/2)^2+1)^(1/2), (1/2)*2^(1/2))-2^(1/2)*EllipticE((1/2)*2^(1/2)), -2^(1/2)*EllipticE((-sin(2)^2+1)^(1/2), (1/2)*2^(1/2))-2^(1/2)*EllipticE((1/2)*2^(1/2)), -2^(1/2)*EllipticE((1/2)*2^(1/2))+2^(1/2)*EllipticE((1-sin(3/2)^2)^(1/2), (1/2)*2^(1/2)), -2^(1/2)*EllipticE((1/2)*2^(1/2))+2^(1/2)*EllipticE((1-sin(1)^2)^(1/2), (1/2)*2^(1/2)), -2^(1/2)*EllipticE((1/2)*2^(1/2))+2^(1/2)*EllipticE((1-sin(1/2)^2)^(1/2), (1/2)*2^(1/2)), 0, 2^(1/2)*EllipticE((1/2)*2^(1/2))-2^(1/2)*EllipticE((1-sin(1/2)^2)^(1/2), (1/2)*2^(1/2)), 2^(1/2)*EllipticE((1/2)*2^(1/2))-2^(1/2)*EllipticE((1-sin(1)^2)^(1/2), (1/2)*2^(1/2)), 2^(1/2)*EllipticE((1/2)*2^(1/2))-2^(1/2)*EllipticE((1-sin(3/2)^2)^(1/2), (1/2)*2^(1/2)), 2^(1/2)*EllipticE((-sin(2)^2+1)^(1/2), (1/2)*2^(1/2))+2^(1/2)*EllipticE((1/2)*2^(1/2)), 2^(1/2)*EllipticE((1/2)*2^(1/2))+2^(1/2)*EllipticE((-sin(5/2)^2+1)^(1/2), (1/2)*2^(1/2)), 2^(1/2)*EllipticE((1/2)*2^(1/2))+2^(1/2)*EllipticE((-sin(3)^2+1)^(1/2), (1/2)*2^(1/2))]

> 

evalf(q);

[-3.678135308, -3.140058362, -2.507982114, -1.810019561, -1.123887723, -.519174356, 0., .519174356, 1.123887723, 1.810019561, 2.507982114, 3.140058362, 3.678135308]

> 

q:=signum(phi)*(sqrt(2)*EllipticE(sqrt(2)/2) + sqrt(2)*EllipticE(sqrt(-sin(3)^2 + 1), sqrt(2)/2));

signum(phi)*(2^(1/2)*EllipticE((1/2)*2^(1/2))+2^(1/2)*EllipticE((-sin(3)^2+1)^(1/2), (1/2)*2^(1/2)))

> 

[seq(evalf(q),phi=-3..3, 0.5)];

[-3.678135308, -3.678135308, -3.678135308, -3.678135308, -3.678135308, -3.678135308, 0., 3.678135308, 3.678135308, 3.678135308, 3.678135308, 3.678135308, 3.678135308]

> 

NULL

Download EllipticE.mw

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