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@acer Ok, so, a bug in Integra...

vv 13977
July 31 2020
0 0

@acer Ok, so, a bug in IntegrationTools:-Expand (2015).

builtin...

vv 13977
July 28 2020
0 0

@mthkvv 

modp1 is builtin, so its code is not accessible. You will need to be employed by Maplesoft :-)

search result...

vv 13977
July 28 2020
0 0

@gawati2611 

https://www.maplesoft.com/applications/view.aspx?SID=33406

(it is the first result with any search engine)

Install...

vv 13977
July 28 2020
0 0

@gawati2611

You must download (from Maple Application Center) and install the package. 

LA...

vv 13977
July 28 2020
2 0

I think you are supposed to do the computations by hand (it's very easy) and then check them with Maple.
If you don't, you will never know linear algebra!

Two solutions!...

vv 13977
July 28 2020
0 0

dharr 

 

I solved the problem as a math problem not as a chemical one.
Even if the quantities ar not normalized, working with high precision the system can be managed. Note that even in your solution, if you start with Digits=25, fsolve fails! [So, you had luck in solving the system. The hybrid method (when possible) is much more robust].

The problem is that the system has two very close solutions! The second one has positive components.
Here are both solutions.

 

> 

restart;
> 

dig:=15;
Dig:=500;

15

  

500

 (1)
> 

Digits:=dig;

15

 (2)
> 

eq1:=K_1=(((n_Cl-x)*u_Cl)*(n_H*u_H))/(n_HCl*m);
eq2:=K_2=(((n_Na-x)*u_Na)*(n_OH*u_OH))/(n_NaOH*m);
eq3:=K_w=(n_H*u_H/m)*(n_OH*u_OH/m);
eq4:=(n_NaCl-x)=(n_Na-x)+n_NaOH;
eq5:=(n_NaCl-x)=(n_Cl-x)+n_HCl;
eq6:=(n_Na-x)+n_H=(n_Cl-x)+n_OH;
eq7:=2*ionic=(n_H/m)+((n_Cl-x)/m)+((n_Na-x)/m)+(n_OH/m);
eq8:=u_H=0.4077*ionic^2-0.3152*ionic+0.9213;
eq9:=u_Na=0.0615*ionic^2-0.2196*ionic+0.8627;
eq10:=u_OH=0.1948*ionic^2-0.1803*ionic+0.8887;
eq11:=m=r*V;
eq12:=u_Cl=(1.417625986641341e-01)*exp(-ionic/2.199955601666953e-02)+2.369460669647978e-01*exp(-ionic/3.756472377688394e-01)+5.859738096037875e-01;
 

K_1 = (n_Cl-x)*u_Cl*n_H*u_H/(n_HCl*m)

  

K_2 = (n_Na-x)*u_Na*n_OH*u_OH/(n_NaOH*m)

  

K_w = n_H*u_H*n_OH*u_OH/m^2

  

n_NaCl-x = n_Na-x+n_NaOH

  

n_NaCl-x = n_Cl-x+n_HCl

  

n_Na-x+n_H = n_Cl-x+n_OH

  

2*ionic = n_H/m+(n_Cl-x)/m+(n_Na-x)/m+n_OH/m

  

u_H = .4077*ionic^2-.3152*ionic+.9213

  

u_Na = 0.615e-1*ionic^2-.2196*ionic+.8627

  

u_OH = .1948*ionic^2-.1803*ionic+.8887

  

m = r*V

  

u_Cl = .1417625986641341*exp(-45.4554627939891*ionic)+.2369460669647978*exp(-2.66207201719228*ionic)+.5859738096037875

 (3)
> 

#constants
K_1:=10^(8);K_2:=10^(-0.2);K_w:=10^(-13.995);n_NaCl:=1.2*10^(-4);V:=5*10^(-8);r:=997.0;x:=0;
 

100000000

  

.630957344480193

  

0.101157945425990e-13

  

0.120000000000000e-3

  

1/20000000

  

997.0

  

0

 (4)
> 

eq:={seq(eq||i,i=1..12)}:
> 

Eq:=convert(eq,rational):
> 

nops(Eq);

12

 (5)
> 

X:=indets(Eq,name);nops(%);

{ionic, m, n_Cl, n_H, n_HCl, n_Na, n_NaOH, n_OH, u_Cl, u_H, u_Na, u_OH}

  

12

 (6)
> 

S:=eliminate(Eq, X minus {ionic}):
> 

f:=simplify(S[2])[]:
> 

############
> 

Digits:=Dig;

500

 (7)
> 

############
> 

plot(f, ionic=0..3);

 
> 

ionicS:=fsolve(f):
> 

fsol:=fsolve(f, ionic=2..3, maxsols=4);

2.4072216428762064719709836229735741800889042421577428430537781518115503868004156610600823184790235582817124524344812646563654643777684793591645018003899689015817427173179752634079939003372580778910117251886921767039403180747010437087571428366550700116442476342934147314239344628682714390267374557235869179128731362248614883275894073767620207010250812742674618672565735979954188672644876167737474652628163903812114118227918718074279946285067014424835271403831469999619981894945738392256132856119033804, 2.4072216871137633357650873567132796005943570834918808892029385420170892989399292286850612515531552784097009318682474925120434274561318247604812109570896298690021014741210983094411953617299872175339734761428520595821699622162016473250588750927352871172010279172602164599295625182139979491163163541079616912134910472650620522616220553867460689623033644828672852349620815922422127489013599363589101476837483454485781158832844608657444742327219956690842787006059290738120338229107614498999976795218845853

 (8)
> 

numsol:=nops([fsol]);
fsol[numsol]-fsol[1];

2

  

0.442375568637941037337397054205054528413341380461491603902055389121395135676249789330741317201279884794337662278556779630783633454013167091566996609674203587568031230460332014613927291396429617509541598828782296441415006036163017322560802171055567802829668017285056280553457265100895788983843747733006179110402005639340326480099840482612782832085998233677055079942467938816368723195851626824209319550673667040604925890583164796042152942266007515602227820738500356334161876106743843939099812049e-7

 (9)
> 

for i to numsol do
SOL[i]:= [ionic=fsol[i], eval(S[1], ionic=fsol[i])[]]
od:
> 

for i to numsol do evalf[dig](SOL[i]) od;

[ionic = 2.40722164287621, m = 0.498500000000000e-4, n_Cl = 0.120000000000000e-3, n_H = -0.570189355755727e-11, n_HCl = -0.203222877739516e-18, n_Na = 0.120000004599272e-3, n_NaOH = -0.459927245018502e-11, n_OH = -0.110262131059512e-11, u_Cl = .586364294954344, u_H = 2.52504946683014, u_Na = .690449163557180, u_OH = 1.58348862197850]

  

[ionic = 2.40722168711376, m = 0.498500000000000e-4, n_Cl = 0.120000000000000e-3, n_H = 0.570189329858645e-11, n_HCl = 0.203222874359743e-18, n_Na = 0.119999995400728e-3, n_NaOH = 0.459927219629856e-11, n_OH = 0.110262130551076e-11, u_Cl = .586364294908359, u_H = 2.52504953971809, u_Na = .690449166940834, u_OH = 1.58348865549082]

 (10)
> 

 

> 

 

> 

seq(  evalf[dig](evalf(eval((lhs-rhs)~(eq),SOL[i]))),  i=1..numsol);

{-0.9e-14, -0.68940488e-18, 0., 0.203222877739516e-18, 0.45018502e-18, 0.1e-14, 0.2e-14, 0.5e-14, 0.225595811e-13}, {-0.124575880e-13, -0.1e-14, -0.30551076e-18, -0.203222874359743e-18, -0.19629856e-18, -0.1e-27, 0., 0.1e-14, 0.2e-14, 0.7e-14}

 (11)
> 

# So, the second solution has all components >0. !!!
# It is very strange that the two solutions are so close!!!
> 

 

 

=0...

vv 13977
July 27 2020
0 0

@ReactionUra 

Some variables are mathematically negative; they are small in absolute value, so probably they can be considered 0.

Note that dharr's solution is similar; actually if you start his worksheet with Digits:=25, fsolve fails just because of those variables.

tested for 2015...

vv 13977
July 26 2020 
Explore(plot([[[0,0],[1,0],[1/3,2],[0,0]],[(12*s*(t-1)*(s-1)*xi^2+t)/(3+12*(t-1)*(s-1)*xi^2+12*(t-1)*(s-1)*xi),2*t/(1+4*(t-1)*(s-1)*xi^2+4*(t-1)*(s-1)*xi), xi=-infinity..infinity]]), parameters=[s=0. .. 1, t=0. .. 1]);

 

Sudoku...

vv 13977
July 25 2020
0 0

@mmcdara 

1. Not for permutations. E.g. it is known the number of invertible nxn matrices over Zp (i.e. the order of the group GL(n, Zp)).

2. Yes, I have mysef a post here with a Sudoku solver using a Groebner basis: 
https://www.mapleprimes.com/posts/207910-A-Compact-Sudoku-Solver-Via-Groebner

inellipse...

vv 13977
July 25 2020

@one man 

The parametric equation of an inellipse is here. Probably in Maple 17 this will work:

restart;
Explore(plot([[[0,0],[1,0],[1/3,2],[0,0]],[(12*s*(t-1)*(s-1)*xi^2+t)/(3+12*(t-1)*(s-1)*xi^2+12*(t-1)*(s-1)*xi),2*t/(1+4*(t-1)*(s-1)*xi^2+4*(t-1)*(s-1)*xi), xi=-infinity..infinity]]), s=0. .. 1, t=0. .. 1);


 

Explore...

vv 13977
July 25 2020 
restart;
A:=[1,0]: B:=[1/3, 2]:
K:= 1/(s-1)/(t-1): U:=s*~A: V:=t*~B:
ell := (4*xi^2*~U+K*~V ) /~ (4*xi^2+4*xi+K):
Explore(plot([ [[0,0],A,B,[0,0]], [ell[], xi=-infinity..infinity]]),s=0.0 .. 1, t=0.0..1);


 

N^2 to N...

vv 13977
July 24 2020
0 0

This reminds me of the simplest bijection f : N^2 --> N.
f(n,k) = 2^(k-1) * (2*n-1).

 

Unfortunately ......

vv 13977
July 23 2020
0 0

@dharr 

Nice, but unfortunately for n=4 the  reduction is almost the same and n>4 is out of the question.
Also, rhe diagonal dominance could just eliminate a "few" nonsingular matrices.

n!*n...

vv 13977
July 23 2020
0 0

@dharr 

It is possible a reduction by a factor of n!*n.
I was curious about a compiled version. I have used a custom "nextperm" and "det" and the runtime was 0.015 sec for n=3.

But it is not very useful. Trying n=4, the estimated runtime is about a week!!! Without compilation it would be years!
It would be interestiong a theoretical approach, but I have no idea about that.
 

 

A. Heck...

vv 13977
July 20 2020
0 0

@janhardo 

If you cannot find the book, you can at least download (from the author's site) and work the exercises and their solutions. They are for Maple 8 but are valid for Maple 2020 too.

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