tomleslie

Numerically...

Answered: tomleslie 13876

November 20 2021

4 0

as shown in the attached.

Outside the plotted range the solution grows very rapidly and produces "singluarities" at (roughly) -5.5 and 6.8

>	ODE:=diff(u(x),x$4)-sin(x)diff(u(x),x$2)+u(x)diff(u(x),x)-u(x)=1-sin(x); ICS:=u(0)=2, D(u)(0)=0, (D@@2)(u)(0)=-1, (D@@3)(u)(0)=0; sols:=dsolve( [ODE, ICS], numeric); plots:-odeplot( sols, [x, u(x)], x=-2..5);

diff(diff(diff(diff(u(x), x), x), x), x)-sin(x)*(diff(diff(u(x), x), x))+u(x)*(diff(u(x), x))-u(x) = 1-sin(x)

>

Download odeProb2.mw

You state that...

Answered: tomleslie 13876

November 19 2021

2 2

you are using Maple 2021, but whn I check the help for NonIsomorphicGraphs() in my version of Maple 2021, it states (emphasis added)

outputform = graph, adjacency, or bits
The outputform option specifies the form used to represent output graphs. This option is only valid for output = graphs or output = iterator. When outputform=bits (the default) the graph is output as an integer such that the adjacency matrix for the graph can be constructed by the bits set in the output integer. Specifically the integer is the one formed by examining the upper triangular portion of the symmetric adjacency Matrix, excluding the diagonal, and building up, from least to most significant bit order, the entries, moving left to right, then down, by using a 1 digit if the edge is present, and a 0 digit otherwise. There is an example that constructs the adjacency matrix from the bits representation in the examples section below. When outputform=adjacency, the adjacency Matrix of the graph is constructed as a v x v integer[1] Matrix. When outputform=graph, Graph structures are constructed.

From this I would conclude

Identifying a graph by using an integer must to be very efficient in terms of memory usage
The trade-off for having a very memory-efficient storage mechanism, has the side-effect that in order to construct the actual form of the desired object (ie a graph), additional processing is necessary. In this case the 'bits' have to be used to construct an 'adjacency matrix' and the 'adjaceny matrix'. is used to construct the graph
So Maple would seem to be providing a tradeoff between memory storage and speed

You should avoid...

Answered: tomleslie 13876

November 19 2021

3 1

using floating point numbers in mathematical calculations unless it is absolutely necessary (although I would have expected MAple to handle this case).

See the attached for workarounds

>

  restart;
  with(inttrans):
  y := z -> invlaplace(exp(-s)/s, s, z);
#
# Don't use floats unless you *have* to, Use
# rational expressions instead. For example
#
  sol1:=y(t - 1);
  sol2:=y(t - 11/10);
#
# Now plot the above
#
  plot( [sol1,sol2], t=0..3, color=[red, blue]);
#
# alternatively, compute the general result
#
  gen:=y(t-a);
#
# then evaluate for specific values
#
  sol3:=eval(gen, a=1);
  sol4:=eval(gen, a=1.1);
#
# Now p[lot the above
#
  plot( [sol3,sol4], t=0..3, color=[red, blue]);

>

Download lap.mw

Use simplify(value())...

Answered: tomleslie 13876

November 16 2021

0 0

as in the attached (although the answer is still a bit lengthy!

>

restart:
E := 210*10^9:
L := 4:
d1 := L/6:
d2 := (2*L)/6:
d3 := (3*L)/6:
d4 := (4*L)/6:
d5 := (5*L)/6:
b := 0.01:
h := 0.005:
eq := diff(Uy(x), x, x) - piecewise(x < d1, 12*F*x/(E*b*h^3), d1 < x and x < d2, 12*((F + F1)*x - F1*d1)/(E*b*h^3), d2 < x and x < d3, 12*((F + F1 + F2)*x - F1*d1 - F2*d2)/(E*b*h^3), d3 < x and x < d4, 12*((F5 + F4 - F)*x + F*L - F5*d5 - F4*d4)/(E*b*h^3), d4 < x and x < d5, 12*((F5 - F)*x + F*L - F5*d5)/(E*b*h^3), 12*F*(L - x)/(E*b*h^3))

(1)

>	dsolve({eq, Uy(0) = 0, Uy(L) = 0}, Uy(x)): simplify(value(%));

(2)

>

Download odeProb.mw

I'm pretty sure...

Answered: tomleslie 13876

November 15 2021

3 2

that vv has confused "focus" and "vertex".

Anyhow for what it is worth, here is another way to get the equation, using the geometry() package

>

  restart;
  with(geometry):
  with(plots):
  _EnvHorizontalName:= x:
  _EnvVerticalName:= y:
#
# Vertex of parabola
#
  point( vert, [6,-3]):
#
# directrix of parabola
#
  line( dir, x+2*y-1=0, [x,y]):
#
# Compute the focus of the parabola)
#
  PerpendicularLine( axis, vert, dir):
  intersection( P2, dir, axis ):
  point(foc, [ 2*coordinates(vert)[1]-coordinates(P2)[1],
               2*coordinates(vert)[2]-coordinates(P2)[2]
             ]
       ):
#
# Define the parabola in terms of the vertex and focus
#
  parabola( para, [focus=foc, vertex=vert]):
#
# Draw construction and annotation
#
  g:=draw( [ dir(color=red),
             axis(color=blue),
             vert(color=green, symbol=solidcircle, symbolsize=20),
             foc(color=cyan, symbol=solidcircle, symbolsize=20),
             para(color=black)
           ]
         ):
  t:=textplot( [ [coordinates(foc)[], "focus", align=right],
                 [coordinates(vert)[], "vertex", align=right]
               ]
             ):
  display( [g, t]);
#
# Return equation of parabola
#
  Equation(para);

(1)

>

Download eqpara.mw

No!...

Answered: tomleslie 13876

November 14 2021

1 0

from the help (emphasis added)

A return statement causes an immediate return to the point where the current procedure was invoked.

The mere idea of "returning" to somewhere else should scare you as much as it scares me!

To use dsolve/numeric...

Answered: tomleslie 13876

November 14 2021

1 0

you have to supply an ODE rather obviously (as the attached shows) you are supply a "simple" equation, not an ODE!

>

restart:

>

  d := 0.0002:
  kappa := 0.4:
  k__s := 2.5*d:
  Um := 3.266825770:
  Tp := 1.62476126:
  omega := 2*Pi/Tp:
  U0 := Um*sin(omega*t):
  U := Uf*ln(30*y/k__s)/kappa:
  delta := solve(Uf*ln(30*delta/k__s)/kappa = U0, delta):
  t0 := 0.001*Tp:
#
# The following is not an ODE!
#
  ODE := diff(int(U, y = 0 .. delta), t) = delta*diff(U0, t) - Uf^2;
# ICs := Uf(t0) = 0.001*Um;
# dsn1 := dsolve({ICs, ODE}, Uf(t), numeric);

0.2105547463e-3*cos(3.867143725*t)*exp(1.306730308*sin(3.867143725*t)/Uf)*ln(exp(1.306730308*sin(3.867143725*t)/Uf)) = 0.2105547462e-3*exp(1.306730308*sin(3.867143724*t)/Uf)*cos(3.867143724*t)-Uf^2

(1)

>

Download notAnODE.mw

My $0.02...

Answered: tomleslie 13876

November 13 2021

0 0

Using a couple of utility functions which facilitate

Plotting both eigenvalues against a supplied variable name (in the set params) over the supplied range
Plotting both eigenvalues against two supplied valiable names (in the set params) over the supplied ranges

See the attached. NB the plots *look* better in the actual worksheet because no grdlines are used - I have no idea why this site insists on entering gridlines when rendering 2D plots!

>

restart;

>

  with(LinearAlgebra):
  A := Matrix( 5, 5,
               [ [0, 0, 0, 0, 0],
                 [0, 0, 0, 0, 0],
                 [0, 0, tau*PI__u/mu, 0, 0],
                 [0, 0, 0, 0, 0],
                 [0, 0, 0, 0, varpi*PI__g/mu]
               ]
             ):
  B := Matrix( 5, 5,
               [ [mu, 0, 0, gamma, 0],
                 [0, mu + omega, 0, 0, 0],
                 [0, 0, mu + sigma1, theta, 0],
                 [0, 0, 0, alpha2 + gamma + mu, 0],
                 [0, 0, sigma1 + alpha2, 0, theta]
               ]
             ):
  C := A . (1/B):
  Rank(C):
  evs := Eigenvalues(C):
  eig := op({entries(evs, nolist)} minus {0});
  params := { PI__g = 2.2, PI__u = 3.4, alpha2 = 0.33,
              mu = 0.2041, omega = 0.5, sigma1 = 0.72,
              tau = 0.33, theta = 0.9, varpi = 0.096
            }:
#
# Function which will plot the eigenvalues against
# the supplied variable name
#
  plt2:= (z, zrnge) -> plot( [eval(eig, remove(has, params, z))],
                             z=zrnge,
                             color=[red, blue],
                             title=typeset("Eigenvalues versus ", z),
                             titlefont=[times, bold, 18]
                           ):
  plt3:= (z, zrnge, zz, zzrnge)-> plot3d( [ eval(eig, remove(has, params, {z,zz}))],
                                            z=zrnge,
                                            zz=zzrnge,
                                            color=[red, blue],
                                            title=typeset("Eigenvalues versus ", z, " and ", zz),
                                            titlefont=[times, bold, 18]
                                        ):

tau*PI__u/(mu*(mu+sigma1)), varpi*PI__g/(mu*theta)

(1)

>

#
# Examples
#
# Plot both eigenvalues against varpi
# over the range 0..5
#
  plt2(varpi, 0..5);
#
# Plot both eigenvalues agains mu
# over the range -5..5
#
  plt2( mu, -5..5);
#
# Plot both eigenvalues in 3D against varpi and mu
# over the ranges varpi=-5..5 and mu=-3..3
#
  plt3( varpi, -5..5, mu, -3..3);

>

Download plotEigs.mw

Maybe...

Answered: tomleslie 13876

November 11 2021

4 0

the attached will help.

>	restart; with(Student[Calculus1]): ex:=Limit((x^2 - 81)/(sqrt(x) - 3), x = 9); h:=Hint(%); Rule[h[1]](%%); h:=Hint(%); Rule[h](%%); h:=Hint(%); Rule[h](%%); h:=Hint(%); Rule[h](%%); ShowSteps();

ex := Limit((x^2-81)/(sqrt(x)-3), x = 9)

Limit((x^2-81)/(sqrt(x)-3), x = 9) = Limit(4*x^(3/2), x = 9)

Limit((x^2-81)/(sqrt(x)-3), x = 9) = 4*(Limit(x^(3/2), x = 9))

Limit((x^2-81)/(sqrt(x)-3), x = 9) = 4*(Limit(x, x = 9))^(3/2)

Limit((x^2-81)/(sqrt(x)-3), x = 9) = 36*sqrt(9)

"[[(lim)(x^2-81)/(sqrt(x)-3)],[ ,"=",(lim)4 x^(3/2), [lhopital,x^2-81]],[ ,"=",4 ((lim)x^(3/2)), [constantmultiple]],[ ,"=",4 ((lim)x)^(3/2), [power]],[ ,"=",36 sqrt(9), [identity]]]"

(1)

>	# # Use the 'All Steps' button in the pop-up, followed # by the 'Close' buttom to produce the output displayed below # LimitTutor((x^2 - 81)/(sqrt(x) - 3), x = 9);

"[[(lim)(x^2-81)/(sqrt(x)-3)],[ ,"=",(lim)4 x^(3/2), [lhopital,x^2-81]],[ ,"=",4 ((lim)x^(3/2)), [constantmultiple]],[ ,"=",4 ((lim)x)^(3/2), [power]],[ ,"=",36 sqrt(9), [identity]]]"

Limit((x^2-81)/(x^(1/2)-3), x = 9) = 36*9^(1/2)

(2)

>

Download steps.mw

I'm pretty sure this problem...

Answered: tomleslie 13876

November 09 2021

2 7

has been reproted before. Unfoirtuantely I can't find the thread.

I did better by exporting the graph ass encapsulated postscript (EPS), then using GIMP to convert the .ps file to pdf - see the image below

The pdf file produced by GIMP is attached below

test2.pdf

Answers (a)-(f)...

Answered: tomleslie 13876

November 08 2021

2 0

using only the information provided in the question, and some simple Maple manipulations are shown in the attached

And yes, the textbook answer to question 2(c) is incorrect - should be -3 not 3

>	restart: with(IntegrationTools): Given:= [ int(f(x), x=1..4)=7, int(f(x), x=1..6)=4, int(g(x), x=1..6)=5 ];

[int(f(x), x = 1 .. 4) = 7, int(f(x), x = 1 .. 6) = 4, int(g(x), x = 1 .. 6) = 5]

(1)

>

#
# Question a
#
  Ia:= Int( g(x),
            x=4..4
          ):
  Ia = value
       ( int( g(x),
              x=4..4
            )
       );
#
# Question b
#
  Ib:= Int( f(x),
            x=4..1
          ):
  Ib = eval
       ( value(Ib),
         Given
       );
#
# Question c
#
  Ic:= Int(f(x), x=4..6):
  isolate
  ( eval
    ( Split
      ( lhs( Given[2] ),
        4
      )=rhs( Given[2] ),
      Given
    ),
    value(Ic)
  );
#
# Question d
#
  Id:= Int(8*g(x), x=1..6):
  Id = eval
       ( Expand
         ( value(Id)
         ),
         Given
       );
#
# Question e
#
  Ie:= Int( 5+f(x), x=1..6):
  Ie = eval
       ( Expand
         ( value(Ie)
         ),
         Given
       );
#
# Question f
#
  If:= Int(3*f(x)-6*g(x), x=1..6):
  If = eval
       ( Expand
         ( value(If)
         ),
         Given
       );

Int(g(x), x = 4 .. 4) = 0

Int(f(x), x = 4 .. 1) = -7

int(f(x), x = 4 .. 6) = -3

Int(8*g(x), x = 1 .. 6) = 40

Int(5+f(x), x = 1 .. 6) = 29

Int(3*f(x)-6*g(x), x = 1 .. 6) = -18

(2)

>

Download ints.mw

An alternative...

Answered: tomleslie 13876

November 07 2021

1 0

because there are many ways to solve this problem - and I like alternatives!

Download linees.mw

Something like...

Answered: tomleslie 13876

November 06 2021

3 0

the attached perhaps?

Be careful!! As AmirHosein Sadeghimanesh 145 has already noted, the number of such graphs gets very big very quickly as the vertex count goes up . There are more than 11 million ( actually 11716571 ) such graphs with 10 vertices.

>	restart; with(GraphTheory):

>

#
# Return the sequence of the number of non-isomorphic,
# connected graphs with vertices from 1 to 8.
#
# This gives the OEIS sequence st
#
# https://oeis.org/A001349
#
# This number gets very VERY large, very quickly and
# the command becomes very time-consuming for upper
# limits >=10
#
  seq( NonIsomorphicGraphs
       ( j,
         output=count,
         restrictto=connected
       ),
       j=1..9
     );

(1)

>

#
# For given number of vertices, generate the set of edges for
# every non-isomorphic graph, storing this result in an Array()
#
  n:= 7;
  L:= NonIsomorphicGraphs
      ( n,
        output=iterator,
        outputform=graph,
        restrictto=connected
      ):
  Es:= Array
       ( [ seq
           ( Edges( L() ),
              j=1..NonIsomorphicGraphs
                   ( n,
                     output=count,
                     restrictto=connected
                   )
           )
         ]
       ):