Maple 2015 Questions and Posts

These are Posts and Questions associated with the product, Maple 2015

Dear Users!
Hope you all are fine with everything. How we can identify the same equations from a number of equations using maple command, like
Eq1:=5.09295817894067*`τu`[1, 1]-30.5577490736439*`τu`[2, 1]+178.253536262923*`τu`[3, 1]-30.5577490736439*`τu`[1, 2]+183.346494441862*`τu`[2, 2]-1069.52121757753*`τu`[3, 2]+178.253536262923*`τu`[1, 3]-1069.52121757753*`τu`[2, 3]+6238.87376920228*`τu`[3, 3];
Eq2:=5.09295817894067*`τu`[1, 1]+10.1859163578814*`τu`[2, 1]+15.2788745368241*`τu`[3, 1]-30.5577490736439*`τu`[1, 2]-61.1154981472883*`τu`[2, 2]-91.6732472209439*`τu`[3, 2]+178.253536262923*`τu`[1, 3]+356.507072525849*`τu`[2, 3]+534.760608788841*`τu`[3, 3]-3/7;
Eq3:=5.09295817894067*`τu`[1, 1]-30.5577490736439*`τu`[2, 1]+178.253536262923*`τu`[3, 1]+10.1859163578814*`τu`[1, 2]-61.1154981472883*`τu`[2, 2]+356.507072525849*`τu`[3, 2]+15.2788745368241*`τu`[1, 3]-91.6732472209439*`τu`[2, 3]+534.760608788841*`τu`[3, 3]-9/7;
Eq4:=5.09295817894067*`τu`[1, 1]+10.1859163578814*`τu`[2, 1]+15.2788745368241*`τu`[3, 1]+10.1859163578814*`τu`[1, 2]+20.3718327157631*`τu`[2, 2]+30.5577490736484*`τu`[3, 2]+15.2788745368241*`τu`[1, 3]+30.5577490736484*`τu`[2, 3]+45.8366236104784*`τu`[3, 3]-12/7;
Eq5:=5.09295817894067*`τu`[1, 1]-30.5577490736439*`τu`[2, 1]+178.253536262923*`τu`[3, 1]+50.9295817894067*`τu`[1, 2]-305.577490736439*`τu`[2, 2]+1782.53536262923*`τu`[3, 2]+504.202859715131*`τu`[1, 3]-3025.21715829077*`τu`[2, 3]+17647.1000900295*`τu`[3, 3]-18/7;
Eq6:=5.09295817894067*`τu`[1, 1]+10.1859163578814*`τu`[2, 1]+15.2788745368241*`τu`[3, 1]+50.9295817894067*`τu`[1, 2]+101.859163578814*`τu`[2, 2]+152.788745368241*`τu`[3, 2]+504.202859715131*`τu`[1, 3]+1008.40571943027*`τu`[2, 3]+1512.60857914560*`τu`[3, 3]-3;
Eq7:=5.09295817894067*`τu`[1, 1]+10.1859163578814*`τu`[2, 1]+15.2788745368241*`τu`[3, 1]-30.5577490736439*`τu`[1, 2]-61.1154981472883*`τu`[2, 2]-91.6732472209439*`τu`[3, 2]+178.253536262923*`τu`[1, 3]+356.507072525849*`τu`[2, 3]+534.760608788841*`τu`[3, 3]-3/7;
Eq8:=5.09295817894067*`τu`[1, 1]+10.1859163578814*`τu`[2, 1]+15.2788745368241*`τu`[3, 1]+10.1859163578814*`τu`[1, 2]+20.3718327157631*`τu`[2, 2]+30.5577490736484*`τu`[3, 2]+15.2788745368241*`τu`[1, 3]+30.5577490736484*`τu`[2, 3]+45.8366236104784*`τu`[3, 3]-12/7;
Eq9:=41.7622570673196*`τu`[3, 1]+41.7622570673196*`τu`[1, 3]+15.2788745368220*`τu`[1, 1]+83.5245141346398*`τu`[2, 3]+30.5577490736443*`τu`[2, 1]+113.063671572516*`τu`[3, 3]+83.5245141346398*`τu`[3, 2]+30.5577490736443*`τu`[1, 2]+61.1154981472892*`τu`[2, 2];
In above equations Eq2 and Eq7; Eq4 and Eq8 are same. If I have set of 100 equation how I can identify similar equations?
@acer @Kitonum @Preben Alsholm


While trying to convert into integers a sample S drawn from a binomial distribution,  I've observed that  round~(S) didn't do the job while map(round, S) did it.

First question: why the first syntax and the second one are not equivalent on this case?

I investigated a little bit further by applying round~ on a row vector T of Hfloats (thus T and S are "identical")

Second question: While  round~(S) doesn't work but round~(T) does?



S := Statistics:-Sample(Binomial(10, 0.5), 2);
round~(S);      # Why round~(S) doesn't return integers
map(round, S);  # but map(round, S) does?


S := Vector[row](2, {(1) = 6.0, (2) = 7.0}, datatype = float[8])


Vector[row]([6., 7.])


Vector[row]([6, 7])


Vector[row](2, {1 = HFloat(6.), 2 = HFloat(7.)}, datatype = float[8], storage = rectangular, order = Fortran_order, shape = [])


# Evaluation of round~(T) on a vector of Hfloats

T := Vector[row](2, [HFloat(6.), HFloat(3.)]);
round~(T);   # round~(T) returns integers,

T := Vector[row](2, {(1) = HFloat(6.0), (2) = HFloat(3.0)})


Vector[row](2, {1 = HFloat(6.), 2 = HFloat(7.)}, datatype = float[8], storage = rectangular, order = Fortran_order, shape = [])


Vector[row]([6, 3])






Maple doesn't completely check the condition on the number of trials "n" for Binomial and NegativeBinomial distributions (package Statistics).
The attribute "Conditions" explicitely says that n must be a strictly positive integer but no strictly positive real valuereturna an error (ok, it would be stupid to set n to a non integer value !!!).

I think it is a default that ought to be corrected in future releases (this default still exists in Maple 2018)




`Maple 2015.2, APPLE UNIVERSAL OSX, Dec 20 2015, Build ID 1097895`




X := RandomVariable(Binomial(n, p)):
L := [attributes(X)][3]:
A := exports(L)

Conditions, ParentName, Parameters, CDF, CharacteristicFunction, Kurtosis, Mean, Median, Mode, MGF, ProbabilityFunction, Skewness, Support, Variance, VariationCoefficient, CDFNumeric, QuantileNumeric, RandomSample, RandomSampleSetup, RandomVariate



[0 <= p, p <= 1, n::posint]


# Maple should return an error for N is not of type posint
# It seems that Sample uses floor(N)

N := 10.49; type(N::posint);
P := 1/2:
X := RandomVariable(Binomial(N, P)):
Mean(X), N*P;
ProbabilityFunction(X, k);
S := Sample(X, 10^6):

# A non consistent result (only non negative values of k should be accepted)

eval(ProbabilityFunction(X, k), k=evalf(Pi));

N := 10.49




5.245000000, 5.245000000


piecewise(k < 0, 0, binomial(10.49, k)*(1/2)^k*(1/2)^(10.49-k))







X := RandomVariable(NegativeBinomial(n, p)):
L := [attributes(X)][3]:
A := exports(L):

[0 < n, 0 < p, p <= 1]


N := 10.49:
P := 1/2:
X := RandomVariable(NegativeBinomial(N, P)):






When creating a user random variable, I would like to instanciate some of its attributes, for instance ParentName.
But it seems that it's not always possible.

​​​​​​​Is it a Maple's limitation or am I not doing the things correctly ?



U := RandomVariable(Uniform(0, 1)):


A := attributes(U)[3]



AllAttributes := with(A);

[CDF, Conditions, HodgesLehmann, InverseSurvivalFunction, MGF, MaximumLikelihoodEstimate, Mean, Median, Mode, PDF, Parameters, ParentName, Quantile, RandomSample, RandomSampleSetup, RandomVariate, RousseeuwCrouxSn, Support, Variance]





# Define a user random variable

v := Distribution(PDF = (z -> piecewise(0 <= t and t < 1, 1, 0))):
V := RandomVariable(v):
A := attributes(V)[3];
AllAttributes := with(A);



[Conditions, PDF]




# its definition can be augmented by adding some recognized attributes...
# even if the result returned by Mean is strange

v := Distribution(PDF = (z -> piecewise(0 <= t and t < 1, 1, 0)), 'Mean'=1/Pi, 'Median'=exp(-1)):
V := RandomVariable(v):
A := attributes(V)[3];
AllAttributes := with(A);
[Median, Mean](V)



[Conditions, Mean, Median, PDF]


[exp(-1), 1/Pi(_R1)]


# but not all the recognized attributes seem to be able to be instanciated:

v := Distribution(PDF = (z -> piecewise(a <= t and t < b, 1/(b-a), 0)), 'Parameters'=[a, b]);
v := Distribution(PDF = (z -> piecewise(a <= t and t < b, 1/(b-a), 0)), 'ParentNames'=MyDistribution);

Error, (in Statistics:-Distribution) invalid input: too many and/or wrong type of arguments passed to NewDistribution; first unused argument is Parameters = [a, b]


Error, (in Statistics:-Distribution) invalid input: too many and/or wrong type of arguments passed to NewDistribution; first unused argument is ParentNames = MyDistribution







WE20 Oceanography Historical Perspective


Maple is a trademark of Waterloo Maple Inc.


Adapted from Introductory Oceanography 10th ed. by Harold V. Thurman and Alan P. Trujillo


1.  How did the view of the ocean by early Mediterranean cultures influence the naming of planet Earth?


Early cultures envisioned the world as composed of large landmasses surrounded by marginal bodies of water.


2.  What is the difference between an ocean and a sea?  Which ones are the seven seas?


A sea is

   •  Smaller and shallower than an ocean (this is why the Arctic Ocean might be more appropriately considered a sea)

   •  Composed of salt water (many "seas" such as the Caspian Sea in Asia, are actually large fresh water lakes)

   •  Somewhat enclosed by land (but some seas, such as the Sargasso Sea in the Atlantic Ocean, are defined by strong ocean currents rather than land.


The seven seas:  the North Pacifiic, the South Pacific, the North Atlantic, the South Atlantic, the Indian, the Arctic, and the Southern or Antarctic.


3.  Describe the development of navigation techniques that have enabled sailors to navigate the open ocean far from land.


Initially, navigators in the Northern Hemisphere measured the angle between the horizon and the North Star (Polaris), which is directly above the North Pole.  Latitude could be determined by noting the angular difference between the horizon and the North Star.  In the Southern Hemisphere, the angle between the horizon and the Southern Cross was used because the Southern Cross is directly overhead at the South Pole.


There was no method of determining longitude until one based on time was developed at the end of the 18th century.  Pendulum-driven clocks in use in the early 1700s would not work for long on a rocking ship at sea.  In 1728 a cabinetmaker in Lincolnshire, England, named John Harrison, began working on a new type of time peace.  The "chronometer" was driven by a helical balance spring that remained horizontal and independent of ship motion.


As Earth turns on its rotational axis it moves through 2Pi radian every 24 hours (one complete rotation).  Earth, therefore, rotates through (1/12)*Pi radian of longitude per hour.  As a result, a navigator need only know the time at the prime or Greenwich at the exact same time the sun is at its highest point (local noon) at the ship's location.


Suppose a ship sets sail west across the Atlantic Ocean from Europe, checking its longitude each day when the sun is at its highest point (solar noon) at the ship's location.  On one day the chronometer reads 16:18 hours.  What is the location of the ship?


The ship is 4 hours and 18 minutes behind (west of) Greenwich time.  To determine the longitude we must convert time into longitude.  


(1/12)*Pi radian of longitude per hour

(1/12)*Pi*(1/60) = (1/720)*Piradian of longitude per minute

(1/720)*Pi*(1/60) = (1/43200)*Piradian of longitude per second


4*((1/12)*Pi)+18*((1/720)*Pi) = 43*Pi*(1/120)


180*(43*Pi*(1/120))/Pi = 7740*(1/120) and 7740*(1/120) = 129/2 and 129/2 = 64.5degree west of the Greenwich meridian.


In Maple:

dg := proc (hr, mn, sc) local r, d, d_dec; r := (1/12)*hr*Pi+(1/720)*mn*Pi+(1/4320)*sc*Pi; d := 180*r/Pi; d_dec := evalf[5](d); print(d, d_dec) end proc

dg(4, 18, 0)

129/2, 64.500


4.  Construct a time line that includes the major events of human history that have resulted in greater understanding of our planet in general and the oceans in particular.







5.  Using a diagram, illustrate the method used by Eratosthenes to calculate the circumference of Earth.


Eratoshenes (276–192 BCE) had heard that at noon on the longest day of the year the Sun shone directly into the waters of a deep,vertical well in Syene (Aswan), which was 800 kilometers (500 miles) to the south.  At the same time in Alexandria, he noticed a vertical pole cast a slight shadow when the Sun was at its apex in the sky over Alexandria.  He accurately measured the shadow the pole cast, which was 7.2 degree.


Convert 7.2 ° to radian  alpha^R = ((1/180)*Pi*`#msup(mi("&alpha;",fontstyle = "normal"),mo("&compfn;"))`*Pi)*`#msup(mn("7.2"),mo("&compfn;"))` and ((1/180)*Pi*`#msup(mi("&alpha;",fontstyle = "normal"),mo("&compfn;"))`*Pi)*`#msup(mn("7.2"),mo("&compfn;"))` = (1/25)*Pi




convert(0.4000000000e-1, 'rational', 'exact')



(2*Pi*800)*km/((1/25)*Pi) = 40000*km




 The equatorial circumference of Earth is about 24,901 miles (40,075 km). However, from pole-to-pole — the meridional circumference — Earth is only 24,860 miles (40,008 km) around. This shape, caused by the flattening at the poles, is called an oblate spheroid.  Since Eratoshenes was calculating the meridional circumference, his calculation was only 8 km different from the accepted modern measurement.


The following calculations are for the diagram.


NULLNULL`implies`(40000/(2*Pi) = 20000*alpha/Pi and 20000*alpha/Pi = (1/25)*Pi*sin(alpha) and (1/25)*Pi*sin(alpha) = y/c, c*sin(alpha) = y)

`&approx;`(20000*sin((1/25)*Pi)/Pi, 798*kg)NULLNULL







`implies`(tan(alpha) = y/x implies tan(alpha)/y = 1/x, y/tan(alpha) = 20000*x*sin((1/25)*Pi)/(Pi*tan((1/25)*Pi)) and `&approx;`(20000*x*sin((1/25)*Pi)/(Pi*tan((1/25)*Pi)), 6316*km))









with(plottools); with(plots)

p1 := circle([0, 0], 20000/Pi)

p2 := line([0, 0], [6316, 798])

p3 := line([0, 0], [6316, 0])

p4 := pointplot([[6316, 0], [6316, 798]], color = yellow, symbol = solidcircle, symbolsize = 25, transparency = .3)

p5 := textplot({[4400, -600, "Syene (Aswan)"], [4400, 1300, "Alexandria"]}, font = [Perpetua, bold, 18])

display(p1, p2, p3, p4, p5, scaling = constrained, size = [350, 350], axes = none)



6.  While the Arabs dominated the Mediterranean region during the Middle Ages, what were the most significant ocean-related events taking place in Europe?


Between 831 and 1071, the Emirate of Sicily was one of the major centers of Islamic culture in the Mediterranean. After its conquest by the Christian Normans, the island developed its own distinct culture with the fusion of Latin and Byzantine influences. Palermo remained a leading artistic and commercial center of the Mediterranean well into the Middle Ages.


Europe was reviving, however, as more organized and centralized states began to form in the later Middle Ages after the Renaissance of the 12th century. Motivated by religion and dreams of conquest, the kings of Europe launched a number of Crusades to try to roll back Muslim power and retake the holy land. The Crusades were unsuccessful in this goal, but they were far more effective in weakening the already tottering Byzantine Empire that began to lose increasing amounts of territory to the Seljuk Turks and later to the Ottoman Turks. They also rearranged the balance of power in the Muslim world as Egypt once again emerged as a major power in the eastern Mediterranean.


7.  Describe the important events in oceanography that occurred during the Age of Discovery in Europe.


One of the most famous voyages during the Age of Discovery began in 1768 when the HMS Endeavour left Portsmouth, England, under the command of Captain James Cook. Over 10 years Cook led three world-encircling expeditions and mapped many countries, including Australia, New Zealand and the Hawaiian Islands. He was an expert seaman, navigator and scientist who made keen observations wherever he went. He was also one of the first ship captains to recognize that a lack of Vitamin C in sailors’ diets (due mostly to a lack of fresh fruit) caused scurvy, a serious disease that killed many sailors in those times. Cook always sailed with lots of pickled cabbage, which he insisted that the sailors eat. Scurvy was never a problem on his ships because the cabbage contained lots of Vitamin C.


8.  List some of the major achievements of Captain James Cook.


Three important voyages commanded by Captain Cook:  

1771–1768  Endeavor observes the transit of Venus in Tahiti; charts New Zealand; charts the eastern coastline of Australia and continues on to New Guinea and Java.


1772–1776 Resolution crosses the Anartic circle for the first time in history.  On returning to England Cook writes about preventing scurvy on long voyages.


1776–1779 Resolution and Discovery visit New Zealand, Tonga, Tahiti, and Christmas Island; Hawaiian islands discovered; expedition crosses the Pacific eastwards to make landfall off the coast of Oregon.  Cook sails into the Bering Straits but foiled by ice he returns to Hawaii where he is killed by natives on 14 Feb 1779.


9.  Describe, in general, the voyages of  HMS Challenger and HMS Beagle.


The Beagle sailed from Plymouth Sound on 27 December 1831 under the command of Captain Robert FitzRoy. While the expedition was originally planned to last two years, it lasted almost five—the Beagle did not return until 2 October 1836. Darwin spent most of this time exploring on land (three years and three months on land; 18 months at sea). The book he wrote is a vivid travel memoir as well as a detailed scientific field journal covering biology, geology, and anthropology that demonstrates Darwin's keen powers of observation, written at a time when Western Europeans were exploring and charting the whole world.


Darwin's notes made during the voyage include comments hinting at his changing views on the inflexibility of species. On his return to England, he wrote the book based on these notes, at a time when he was first developing his theories of evolution through common descent and natural selection. The book includes some suggestions of his ideas, particularly in the second edition of 1845 (Thurman and Trujillo, p. 23).


On December 21, 1872 the HMS. Challenger sailed from Portsmouth, England, for an epic voyage which would last almost three and a half years. It  was the first expedition organized and funded for a specific scientific purpose: to examine the deep-sea floor and answer questions about the ocean environment.  Although the mission was scientific, another purpose was to collect information that could be used in the laying of communication cables along the sea floor.


The expedition covered 69,000 miles (about 130,000 km) and gathered data on currents, water chemistry, temperature, bottom deposits and marine life at 362 oceanographic stations. More than 4700 new species of marine animals were discovered during the course of the voyage, many of which were found on the seafloor – an environment that scientists originally believed to be too inhospitable to support life (


10.  Describe the voyages of the Fram and how it helped prove there was no continent beneath the Arctic ice pack.


The Fram was the first ship specially built (in Norway) for polar research. She was used on three important expeditions: with Fridtjof Nansen on a drift over the Arctic Ocean 1893-96, with Otto Sverdrup to the arctic archipelago west of Greenland - now the Nunavut region of Canada - 1898-1902, and with Roald Amundsen to Antarctica for his South Pole expedition 1910-12. The Fram is now housed and exhibited in the Fram Museum at Bygdøynes, Oslo



11.  Why did Benjamin Franklin want to know about the surface current pattern in the North Atlantic Ocean?


As Deputy Postmaster General of the American colonies, Franklin promoted using the Gulf Stream to speed up delivery of mail from America to Europe, as well as to improve other commercial shipping (


12.  What was Alexander Agassiz's major contribution to an increased knowledge of the oceans?


Agassiz is widely acknowledged as the driving force that brought oceanography recognition as a science.  His training as a mining engineer led him to develop ingenious oceanographic sampling devices—the prototypes for many devices in use on research vessels today—that improved the quantitative value of biological sampling (Thurman and Trujillo, p. 27).


13.  What important oceanographic inventions and data came out of Word Wars I and II?

The need to detect submarines led the navies of the world to greatly expand their studies of the sea. This led to the founding of oceanography departments at state universities, including Oregon State, Texas A&M University, University of Miami, and University of Rhode Island,

and the founding of national ocean laboratories such as the various Institutes of Oceanographic Science ( stewart_ocean_book.pdf).


14.  List features of the oceans that can be studied remotely by use of satellites.


• phytoplankton concentrations

• heat storage and aerosol formation and other ocean influences on climate processes

• cycles of carbon, sulfur, and nitrogen concentrations


• Bathymetry/Seafloor (Bathymetry is the measurement of depth of water in oceans seas, or lakes)

• Topography

• Costal Process

• Marine Geophysics

• Ocean Acoustics

• Ocean Chemistry

• Ocean Circulation

• Ocean Heat Budget

• Ocean Optics

• Ocean Pressure

• Ocean Temperature

• Ocean Waves

• Ocean Winds

• Salinity/Density

• Sea ice

• Sea Surface Topography



15.  Discuss what problems the human body can experience as a result of diving underwater.


Nitrogen narcosis (also referred to as inert gas narcosis, raptures of the deep, and the Martini effect) :  While scuba diving does sometimes involve breathing air that’s mixed with nitrogen, this condition is caused when the diver goes deeper into the water, where the partial pressure of nitrogen increases and more nitrogen ends up getting absorbed into the bloodstream. The higher the nitrogen concentration in the bloodstream, the slower the nervous system will be, and the more likely the diver will experience intoxicating effects that can seriously impair their judgment underwater—possibly enough to lead them into dangerous situations (


Barotrauma: Trauma caused by rapid or extreme changes in air pressure, especially affecting enclosed cavities within the body such as the middle ear (otic barotrauma), the sinuses (sinus barotrauma), and the lungs (pulmonary barotrauma) (


Decompression illness is caused by intravascular or extravascular bubbles that are formed as a result of reduction in environmental pressure (decompression). The term covers both arterial gas embolism, in which alveolar gas or venous gas emboli (via cardiac shunts or via pulmonary vessels) are introduced into the arterial circulation, and decompression sickness, which is caused by in-situ bubble formation from dissolved inert gas. Both syndromes can occur in divers, compressed air workers, aviators, and astronauts, but arterial gas embolism also arises from iatrogenic causes unrelated to decompression. Risk of decompression illness is affected by immersion, exercise, and heat or cold. Manifestations range from itching and minor pain to neurological symptoms, cardiac collapse, and death. First-aid treatment is 100% oxygen and definitive treatment is recompression to increased pressure, breathing 100% oxygen. Adjunctive treatment, including fluid administration and prophylaxis against venous thromboembolism in paralysed patients, is also recommended. Treatment is, in most cases, effective although residual deficits can remain in serious cases, even after several recompressions (





How to show the y-axis numbers in the second plot ?   y is in [0,3]. 

I try the following but it shows a negative axis. 

display(P, L, axis = [tickmarks = [5, subticks = 3], thickness = 2]);  this is not correct

The file is attached.





Dear Users!

I have made a code using loops. But when I exceute it I go unwanted expression please see the files and try to fix it. I shall be very thankful.

Special request to:

@acer @Kitonum @Preben Alsholm @Carl Love

Dear Users!

Hope everyone should fine here. I want to know how to multiple the entries of a matrix with a matrix like If A is matrix as:

A:= Matrix(2, 2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = 0, (2, 2) = 2})
I want nonzero entries should multiply with m*m identity matrix and zero entries multiply with null matrix of order m*m. for m=2 the desired results i calculated manually as:

Matrix(4, 4, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 2, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 2});

I want to know the general way. Thanks in advance


Dear Users!

Hoped you will be fine. I want to define a block matrix A for any value of M like this way

A = [A[0], A[1], A[2], ..., A[M]]

A[i] = [A[i,0], A[i,1], A[i,2], ..., A[i,M]]


A[i,j] = Transpose([a[i,j,0], a[i,j,1], a[i,j,2], ..., a[i,j,M]]);

Kindly help me in this matter. Thanks in advance.

I have a system of ODEs with parameters, p[i], and variables, x[i].

f := [

associated with the innitial conditions:
[x[1](0) = p[2], x[2](0) = p[3]].

I am interested in sets of parameters where the solution x[1](t) is the same; if [p[1],p[2],p[3]] is associated with a solution x(p,t), and [ph[1],ph[2],ph[3]] is asociated with the solution x(ph,t); then x[1](p,t)=x[1](ph,t) for all t if and only if

[ph[1] = ph[1],
ph[2] = p[2],
ph[3] = -p[1]*p[2]^2+p[2]^2*ph[1]+p[3]]

i.e. ph[1] takes any real value, ph[2] takes the same values as p[2] and ph[3] takes a value determined by the original parameter vector and ph[1].

In a previous question it was demonstrated that x[2](ph,t)/x[2](p,t) rapidly converge on p[1] as t increases for a specific parameter vector that was given in the question (see graph below)


This raises the question does this limit generally hold?

I have struggled to do this in maple and I am suspicious of the answer i have got
limit (x[2](ph,t)/x[2](p,t),t=infinity)=+/- infinity

My question is
+ when does a finite limit exist?
+ what is the finite limit?


Dear Users!

Hoped everything going fine with you. I constrcut the following code to construct the polynomials for any M1 and M2

printlevel := 2; M1 := 3; M2 := 3; nu := 1;
for k1 from 0 while k1 <= M1-1 do
for k2 from 0 while k2 <= M2-1 do
GP[k1+1, k2+1] := simplify(sum((-1)^(k1-i1)*GAMMA(k1+i1+2*nu)*GAMMA(nu+1/2)*x^i1*(sum((-1)^(k2-i2)*GAMMA(k2+i2+2*nu)*GAMMA(nu+1/2)*y^i2/(GAMMA(i2+nu+1/2)*factorial(k2-i2)*factorial(i2)*GAMMA(2*nu)), i2 = 0 .. k2))/(GAMMA(i1+nu+1/2)*factorial(k1-i1)*factorial(i1)*GAMMA(2*nu)), i1 = 0 .. k1))
end do end do;
I want to put this polynomials in a vector like
[GP[1,1]   GP[1,2]   GP[1,3]   GP[2,1]   GP[2,2]   GP[2,3]   GP[3,1]   GP[3,2]   GP[3,3]]
Similarlty I want to construct a vector having constants for any values of M1 and M2 like this
[a[1,1]   a[1,2]   a[1,3]   a[2,1]   a[2,2]   a[2,3]   a[3,1]   a[3,2]   a[3,3]]

I have a system of ODEs with parameters, p[i], and variables, x[i].

f := [

associated with the innitial conditions:
[x[1](0) = p[2], x[2](0) = p[3]].

I have sets of parameters that i am interested in

[p[1] = 1, p[2] = 2, p[3] = 3]
[p[1] = 2, p[2] = 2, p[3] = 7]
[p[1] = 3, p[2] = 2, p[3] = 11]
[p[1] = 4, p[2] = 2, p[3] = 15]
[p[1] = 5, p[2] = 2, p[3] = 19]
[p[1] = 6, p[2] = 2, p[3] = 23]
and so on.

The solutions for these equations appear to be anisotropic scalings of each other.


/xh[1]\ =/1 0\ /x[1]\      
\xh[2]/   \0 q/ \x[2]/

where x is the variable for one parameter vector and xh is the variable for another; and q is a nuimber I am interesting in working out (and think may be p[1]).

how do you work out q?

[this is in fact a complicated way of asking when I numerically integrate an ODE on maple how do i get access to the sequence of numbers in the result]

Dear Users!

Hope you are doing well. Below is the code to solve system of equations

u[0, 0] := tau[1]-2*tau[2]+3*tau[3] = 1;
u[0, 1] := tau[1]-tau[3] = 1/2;
u[0, 2] := tau[1]+2*tau[2]+3*tau[3] = 2;
SOL[0] := fsolve({seq(`$`(u[0, l1], l1 = 0 .. M-1))});

the solutions SOL[0] := {tau[1] = .7500000000, tau[2] = .2500000000, tau[3] = .2500000000}
I want to define a vector of order M*1 having the solution of of tau's like that

[.7500000000  .2500000000  .2500000000]

I'm waiting for you positive response. Thanks in advance.
Special request to:
@acer @Carl Love @Kitonum @Preben Alsholm



I need help to approximate Fn(x,y) and Sn(y) for n>=3 to see the convergence. 

Here is the problem:

I have an integral equation Fn(x, y) defined by

F0(x,y) = 0,

F1(x,y) = (9/8)*(y+1) - (21/8)*( y*exp(-2*x) + exp(-x) )

Fn+1(x,y) = (9/8)*(y+1) - (21/8)*( y*exp(-2*x) + exp(-x) ) + (3/4)*int ( Fn( x-z, 2*y*exp(-z) ), z=0..Un(x) ), where the function Un(x) is defined below.

where I suppose that Fn(x,y) >=0 and y>=0 and x>=0. I am iterating the integral equation and find the zeros Sn(y) solutions of Fn(S(y) , y) = 0. The goal is to show that the sequence Fn(x,y) converges to some F(x,y) and Sn(y) converges to some S(y). We can also plot the curves Sn(y) to show the convergence. Notice that Sn(y) >= 0 is increasing in n and decreasing in y.

What I did :

If n = 0, I solve F1(x,y) = 0 and find analytically x = S1(y) = -ln((1/14)*(-7+sqrt(84*y^2+84*y+49))/y). Notice that F1(x,y) = 0 is quadratic in exp(-x) and then F1(x,y) = 0 is easy to solve. S1(y) is the positive function for y>=0.

If n = 1, we need to solve F2(x,y) = (9/8)*(y+1) -(21/8)*( y*exp(-2*x) + exp(-x) ) + (3/4)*int ( F1( x-z, 2*y*exp(-z) ), z, 0, U1(x) ). The function U1(x) is obtained as follows :

i) We define g( U ) = U + S1( 2*y*exp(-U) )

ii) We find the function U1(x,y) solution of g( U1(x,y) ) = x


A generalization of Un(x) is given by

i) Define g( U ) = U + Sn( 2*y*exp(-U) )

ii) Find the function Un(x,y) solution of g( Un(x,y) ) = x

The curve S2(y) :

I find the analytic function F2(x, y), but it is impossible to solve F2(x,y) = 0 and find x = S2(y) analytically. So I choose an interval [0, 10] for y and discretize that interval. Now at each point yi in [0, 10], the function F2(x, yi) depends only on x. I apply the bisection method (dichotomic method) and find the value xi such that F2(xi, yi) = 0. The data (yi, xi) gives the curve S2(y). I plot the data (yi, xi) and (y, S1(y)) in the same curve.


It is impossible to iterate the process below and find F3(x, y), F4(x,y), etc... because we cannot find U2(x), U3(x), etc... analytically. Any suggestions or helps to plot the curves (y, Sn(y)) for n = 3, 4, etc. to see the convergence ?

Thanks !


I have an ODE with 3 parameters
diff(B[1](t), t) = piecewise(t < 1000, kaC*(R-B[1](t))-k[d1]*B[1](t), 1000 < t, -k[d1]*B[1](t))

I'd like to make a graph that shows how its solution vary as Kac and Kd vary. This could have an axis of the form:

            | /k[d]
  B(t)  |/_ __

which could contain a surface composed of the solutions as k[d] varies. Then a series of surfaces could be put together on the same axis to show what happens as kaC varies.

Some typical values are:

kaC = 6*10^(-2),
k[d1] = 7*10^(-3),
R = 1

I'd like to graph everything  in two orders of magnitude of these values for KaC and k[d1].

Currently I think the key obstacle is making a surface of solutions to the ODE; as once I can do that I think making a sequence of them on the same axis should be quite simple with Display




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