Applications, Examples and Libraries

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We've just launched Maple Flow 2024!

You're in the driving seat with Maple Flow - each new feature has a straight-line connection to a user-driven demand to work faster and more efficiently.

Head on over here for a rundown of everything that's new, but I thought I'd share my personal highlights here.

If your result contains a large vector or matrix, you can now scroll to see more data. You can also change the size of the matrix to view more or fewer rows and columns.

You can resize rows and columns if they're too large or small, and selectively enable row and column headers.

If the vector or matrix in your result contains a unit, you can now rescale units with the Context Panel (for the entire matrix) or inline (for individual entries).

A few releases ago, we introduced the Variables palette to help you keep track of all the user-defined parameters at point of the grid cursor.

You can now insert variables into the worksheet from the Variables palette. Just double-click on the appropriate name.

Maple Flow already features command completion - just type the first few letters of a command, and a list of potential completions appears. Just pick the completion you need with a quick tap of the Tab key.

We've supercharged this feature to give potential arguments for many popular functions. Type a function name followed by an opening bracket, and a list appears.

In case you've missed it, the argument completion list also features (when they make sense) user-defined variables.

You can now link to different parts of the same worksheet. This can be used to create a table of contents that lets you jump to different parts of larger worksheets.

This page lists everything that's new in the current release, and all the prior releases. You might notice that we have three releases a year, each featuring many user-requested items. Let me know what you want to see next - you might not have to wait that long!

In our recent project, we're diving deep into understanding the SIR model—a fundamental framework in epidemiology that helps us analyze how diseases spread through populations. The SIR model categorizes individuals into three groups: Susceptible (S), Infected (I), and Recovered (R). By tracking how people move through these categories, we can predict disease dynamics and evaluate interventions.

Key Points of the SIR Model:

  • Susceptible (S): Individuals who can catch the disease.
  • Infected (I): Those currently infected and capable of spreading the disease.
  • Recovered (R): Individuals who have recovered and developed immunity.

Vaccination Impact: One of the critical interventions in disease control is vaccination, which moves individuals directly from the susceptible to the recovered group. This simple action reduces the number of people at risk, thereby lowering the overall spread of the disease.

We're experimenting with a simple model to understand how different vaccination rates can significantly alter the dynamics of an outbreak. By simulating scenarios with varying vaccination coverage, students can observe how herd immunity plays a crucial role in controlling diseases. Our goal is to make these abstract concepts clear and relatable through practical modeling exercises.


 

In this exercise, we are going back to the simple SIR model, without births or deaths, to look at the effect of vaccination. The aim of this activity is to represent vaccination in a very simple way - we are assuming it already happened before we run our model! By changing the initial conditions, we can prepare the population so that it has received a certain coverage of vaccination.

We are starting with the transmission and recovery parameters  b = .4/daysand c = .1/days . To incorporate immunity from vaccination in the model, we assume that a proportion p of the total population starts in the recovered compartment, representing the vaccine coverage and assuming the vaccine is perfectly effective. Again, we assume the epidemic starts with a single infected case introduced into the population.​
We are going to model this scenario for a duration of 2 years, assuming that the vaccine coverage is 50%, and plot the prevalence in each compartment over time.

 

restart
with(plots)

b := .4; c := .1; n := 10^6; p := .5

deS := diff(S(t), t) = -b*S(t)*I0(t); deI := diff(I0(t), t) = b*S(t)*I0(t)-c*I0(t); deR := diff(R(t), t) = c*I0(t)

diff(R(t), t) = .1*I0(t)

(1)

F := dsolve([deS, deI, deR, S(0) = 1-p, I0(0) = 1/n, R(0) = p], [S(t), I0(t), R(t)], numeric, method = rkf45, maxfun = 100000)

odeplot(F, [[t, S(t)], [t, I0(t)], [t, R(t)]], t = 0 .. 730, colour = [blue, red, green], legend = ["S(t)", "I0(t)", "R(t)"], labels = ["Time (days)", "  Proportion\nof Population "], title = "SIR Model with vaccine coverage 50 %", size = [500, 300])

 

F(100)

[t = 100., S(t) = HFloat(0.46146837378273076), I0(t) = HFloat(0.018483974421123688), R(t) = HFloat(0.5200486517961457)]

(2)

eval(S(:-t), F(100))

HFloat(0.46146837378273076)

(3)

Reff := proc (s) options operator, arrow; b*(eval(S(:-t), F(s)))/(c*n) end proc; Reff(100)

HFloat(1.845873495130923e-6)

(4)

plot(Reff, 0 .. 730, size = [500, 300])

 

Increasing the vaccine coverage to 75%

NULL

restart
with(plots)

b := .4; c := .1; n := 10^6; p := .75

deS := diff(S(t), t) = -b*S(t)*I0(t); deI := diff(I0(t), t) = b*S(t)*I0(t)-c*I0(t); deR := diff(R(t), t) = c*I0(t)

diff(R(t), t) = .1*I0(t)

(5)

NULL

F1 := dsolve([deS, deI, deR, S(0) = 1-p, I0(0) = 1/n, R(0) = p], [S(t), I0(t), R(t)], numeric, method = rkf45, maxfun = 100000)

odeplot(F1, [[t, S(t)], [t, I0(t)], [t, R(t)]], t = 0 .. 730, colour = [blue, red, green], legend = ["S(t)", "I0(t)", "R(t)"], labels = ["Time (days)", "  Proportion\nof Population "], title = "SIR Model with vaccine coverage 75%", size = [500, 300])

 

F(1100)

eval(S(:-t), F1(100))

HFloat(0.249990000844159)

(6)

Reff := proc (s) options operator, arrow; b*(eval(S(:-t), F1(s)))/(c*n) end proc; Reff(100)

HFloat(9.99960003376636e-7)

(7)

plot(Reff, 0 .. 730, size = [500, 300])

 

Does everyone in the population need to be vaccinated in order to prevent an epidemic?What do you observe if you model the infection dynamics with different values for p?

No, not everyone in the population needs to be vaccinated in order to prevent an epidemic . In this scenario, if p equals 0.75 or higher, no epidemic occurs - 75 % is the critical vaccination/herd immunity threshold . Remember,, herd immunity describes the phenomenon in which there is sufficient immunity in a population to interrupt transmission . Because of this, not everyone needs to be vaccinated to prevent an outbreak .

What proportion of the population needs to be vaccinated in order to prevent an epidemic if b = .4and c = .2/days? What if b = .6 and "c=0.1 days^(-1)?"

In the context of the SIR model, the critical proportion of the population that needs to be vaccinated in order to prevent an epidemic is often referred to as the "herd immunity threshold" or "critical vaccination coverage."

• 

Scenario 1: b = .4and c = .2/days

``

restart
with(plots)

b := .4; c := .2; n := 10^6; p := .5``

deS := diff(S(t), t) = -b*S(t)*I0(t); deI := diff(I0(t), t) = b*S(t)*I0(t)-c*I0(t); deR := diff(R(t), t) = c*I0(t)

diff(R(t), t) = .2*I0(t)

(8)

F1 := dsolve([deS, deI, deR, S(0) = 1-p, I0(0) = 1/n, R(0) = p], [S(t), I0(t), R(t)], numeric, method = rkf45, maxfun = 100000)

odeplot(F1, [[t, S(t)], [t, I0(t)], [t, R(t)]], t = 0 .. 730, colour = [blue, red, green], legend = ["S(t)", "I0(t)", "R(t)"], labels = ["Time (days)", "  Proportion\nof Population "], title = "SIR Model with vaccine coverage 50 %", size = [500, 300])

 


The required vaccination coverage is around 50% .

• 

Scenario 1: b = .6and c = .1/days

restart
with(plots)

b := .6; c := .1; n := 10^6; p := .83NULL

deS := diff(S(t), t) = -b*S(t)*I0(t); deI := diff(I0(t), t) = b*S(t)*I0(t)-c*I0(t); deR := diff(R(t), t) = c*I0(t)

diff(R(t), t) = .1*I0(t)

(9)

NULL

F1 := dsolve([deS, deI, deR, S(0) = 1-p, I0(0) = 1/n, R(0) = p], [S(t), I0(t), R(t)], numeric, method = rkf45, maxfun = 100000)

odeplot(F1, [[t, S(t)], [t, I0(t)], [t, R(t)]], t = 0 .. 730, colour = [blue, red, green], legend = ["S(t)", "I0(t)", "R(t)"], labels = ["Time (days)", "  Proportion\nof Population "], title = "SIR Model with vaccine coverage 83% ", size = [500, 300])

 

"The required vaccination coverage is around 83 `%` ."


Download SIR_simple_vaccination_example.mw


 

When a derivative can be written as a function of the independent variable only for example

y'=f(x)

y''=f(x)

y'''(x)=f(x)

etc.

 

We call that a directly integrable equation.

 

Example 1:

 

Find the general solution for the following directly integrable equation

diff(y, x) = 6*x^2+4*y(1) and 6*x^2+4*y(1) = 0

That means

int(6*x^2+4, x)

y = 2*x^3+c+4*x", where" c is an arbitary solution

``

 

 
equation1 := diff(y(x), x) = 6*x^2+4

diff(y(x), x) = 6*x^2+4

(1)

NULL

NULL

sol1 := dsolve(equation1, y(x))

y(x) = 2*x^3+c__1+4*x

(2)

And  if we have the initial condition
y(1) = 0
particular_sol1 := dsolve({equation1, y(1) = 0}, y(x))

y(x) = 2*x^3+4*x-6

(3)

"(->)"

 

 

 

 

 

"Example 2:"NULL

NULL

"  Find the particular solution for the following equation with condition"

 

x^2*(diff(y(x), x)) = -1

y(1)=3

So we will need to get the y' by itself

int(-1/x^2, x)

so,

y = 1/x+c , where c is an arbitary constant

And this is our general solution. Now we plug in the initial condition when x = 1, y = 3.

 

That means c = 2.

 

Thus, the particular solution is

 

y = 1/x+2``

eq := x^2*(diff(y(x), x)) = -1

x^2*(diff(y(x), x)) = -1

(4)

NULL

NULL

sol := dsolve(eq, y(x))

y(x) = 1/x+c__1

(5)

NULL

particular_sol := dsolve({eq, y(1) = 3}, y(x))

y(x) = 1/x+2

(6)

NULL

NULL

plot(1/x+2, x = -20 .. 20, color = "Red", axes = normal, legend = [typeset(1/x+2)])

 

NULL

NULL

NULL

NULL

" Example 3:"

 

" Find the particular solution for the following equation with condition"

 

diff(y, t, t) = cost, (D(y))(0) = 0, y(0) = 1

eq1 := diff(y(t), t, t) = cos(t)

diff(diff(y(t), t), t) = cos(t)

(7)

particular_sol := dsolve({eq1, y(0) = 1, (D(y))(0) = 0}, y(t))

y(t) = -cos(t)+2

(8)

"(->)"

 

 

 

 

NULL


 

Download integral.mw

 

Hello,

Attached I am sending several procedures for curves in 3D space. They were written without using Maple's built-in DiffGeo procedures and functions. As an example of their use, I made several animations - Maple worksheets are attached. I hope that maybe they will be useful to someone.

Regards.

ClsDGproc-Curves.zip

restart;

torus:= (x^2+y^2+z^2 + R^2-r^2)^2 - 4*R^2*(x^2+y^2):

torus1:=eval(torus,[r=1,R=4]);

(x^2+y^2+z^2+15)^2-64*x^2-64*y^2

(1)

plots:-implicitplot3d(torus1=0, x=-6..6,y=-6..6,z=-6..6, numpoints=5000, scaling=constrained, view=-2..2);

 

sol1:=solve(torus1<0, [x,y,z]); # It should be easy for Maple

[[x = 0, y < -3, -5 < y, z < (-y^2-8*y-15)^(1/2), -(-y^2-8*y-15)^(1/2) < z], [x = 0, y < 5, 3 < y, z < (-y^2+8*y-15)^(1/2), -(-y^2+8*y-15)^(1/2) < z]]

(2)

eval(torus1, [x=3,y=2,z=0]); # in the interior

-48

(3)

eval(sol1, [x=3,y=2,z=0]);   # ???

[[3 = 0, 2 < -3, -5 < 2, 0 < (-35)^(1/2), -(-35)^(1/2) < 0], [3 = 0, 2 < 5, 3 < 2, 0 < (-3)^(1/2), -(-3)^(1/2) < 0]]

(4)

 

 

Download Bug-solve_ineqs.mw

This Maplesoft guest blog post is from Prof. Dr. Johannes Blümlein from Deutsches Elektronen-Synchrotron (DESY), one of the world’s leading particle accelerator centres used by thousands of researchers from around the world to advance our knowledge of the microcosm. Prof. Dr. Blümlein is a senior researcher in the Theory Group at DESY, where he and his team make significant use of Maple in their investigations of high energy physics, as do other groups working in Quantum Field Theory. In addition, he has been involved in EU programs that give PhD students opportunities to develop their Maple programming skills to support their own research and even expand Maple’s support for theoretical physics.


 

The use of Maple in solving frontier problems in theoretical high energy physics

For several decades, progress in theoretical high energy physics relies on the use of efficient computer-algebra calculations. This applies both to so-called perturbative calculations, but also to non-perturbative computations in lattice field theory. In the former case, large classes of Feynman diagrams are calculated analytically and are represented in terms of classes of special functions. In early approaches started during the 1960s, packages like Reduce [1] and Schoonship [2] were used. In the late 1980s FORM [3] followed and later on more general packages like Maple and Mathematica became more and more important in the solution of these problems. Various of these problems are related to data amounts in computer-algebra of O(Tbyte) and computation times of several CPU years currently, cf. [4].

Initially one has to deal with huge amounts of integrals. An overwhelming part of them is related by Gauss’ divergence theorem up to a much smaller set of the so-called master integrals (MIs). One performs first the reduction to the MIs which are special multiple integrals. No general analytic integration procedures for these integrals exist. There are, however, several specific function spaces, which span these integrals. These are harmonic polylogarithms, generalized harmonic polylogarithms, root-valued iterated integrals and others. For physics problems having solutions in these function spaces codes were designed to compute the corresponding integrals. For generalized harmonic polylogarithms there is a Maple code HyperInt [5] and other codes [6], which have been applied in the solution of several large problems requiring storage of up to 30 Gbyte and running times of several days. In the systematic calculation of special numbers occurring in quantum field theory such as the so-called β-functions and anomalous dimensions to higher loop order, e.g. 7–loop order in Φ4 theory, the Maple package HyperLogProcedures [7] has been designed. Here the largest problems solved require storage of O(1 Tbyte) and run times of up to 8 months. Both these packages are available in Maple only.

A very central method to evaluate master integrals is the method of ordinary differential equations. In the case of first-order differential operators leading up to root-valued iterative integrals their solution is implemented in Maple in [8] taking advantage of the very efficient differential equation solvers provided by Maple. Furthermore, the Maple methods to deal with generating functions as e.g. gfun, has been most useful here. For non-first order factorizing differential equation systems one first would like to factorize the corresponding differential operators [9]. Here the most efficient algorithms are implemented in Maple only. A rather wide class of solutions is related to 2nd order differential equations with more than three singularities. Also here Maple is the only software package which provides to compute the so-called 2F1 solutions, cf. [10], which play a central role in many massive 3-loop calculations

The Maple-package is intensely used also in other branches of particle physics, such as in the computation of next-to-next-to leading order jet cross sections at the Large Hadron Collider (LHC) with the package NNLOJET and double-parton distribution functions. NNLOJET uses Maple extensively to build the numerical code. There are several routines to first build the driver with automatic links to the matrix elements and subtraction terms, generating all of the partonic subprocesses with the correct factors. To build the antenna subtraction terms, a meta-language has been developed that is read by Maple and converted into calls to numerical routines for the momentum mappings, calls to antenna and to routines with experimental cuts and plotting routines, cf. [11].

In lattice gauge calculations there is a wide use of Maple too. An important example concerns the perturbative predictions in the renormalization of different quantities. Within different European training networks, PhD students out of theoretical high energy physics and mathematics took the opportunity to take internships at Maplesoft for several months to work on parts of the Maple package and to improve their programming skills. In some cases also new software solutions could be obtained. Here Maplesoft acted as industrial partner in these academic networks.

References

[1] A.C. Hearn, Applications of Symbol Manipulation in Theoretical Physics, Commun. ACM 14 No. 8, 1971.

[2] M. Veltman, Schoonship (1963), a program for symbolic handling, documentation, 1991, edited by D.N. Williams.

[3] J.A.M. Vermaseren, New features of FORM, math-ph/0010025.

[4] J. Blümlein and C. Schneider, Analytic computing methods for precision calculations in quantum field theory, Int. J. Mod. Phys. A 33 (2018) no.17, 1830015 [arXiv:1809.02889 [hep-ph]].

[5] E. Panzer, Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals, Comput. Phys. Commun. 188 (2015) 148–166 [arXiv:1403.3385 [hep-th]].

[6] J. Ablinger, J. Blümlein, C .Raab, C. Schneider and F. Wissbrock, Calculating Massive 3-loop Graphs for Operator Matrix Elements by the Method of Hyperlogarithms, Nucl. Phys. 885 (2014) 409-447 [arXiv:1403.1137 [hep-ph]].

[7] O. Schnetz, φ4 theory at seven loops, Phys. Rev. D 107 (2023) no.3, 036002 [arXiv: 2212.03663 [hep-th]].

[8] J. Ablinger, J. Blümlein, C. G. Raab and C. Schneider, Iterated Binomial Sums and their Associated Iterated Integrals, J. Math. Phys. 55 (2014) 112301 [arXiv:1407.1822 [hep-th]].

[9] M. van Hoeij, Factorization of Differential Operators with Rational Functions Coefficients, Journal of Symbolic Computation, 24 (1997) 537–561.

[10] J. Ablinger, J. Blümlein, A. De Freitas, M. van Hoeij, E. Imamoglu, C. G. Raab, C. S. Radu and C. Schneider, Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams, J. Math. Phys. 59 (2018) no.6, 062305 [arXiv:1706.01299 [hep-th]].

[11] A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover, A. Huss and T.A. Morgan, Precise QCD predictions for the production of a Z boson in association with a hadronic jet, Phys. Rev. Lett. 117 (2016) no.2, 022001 [arXiv:1507.02850 [hep-ph]].

     Happy Easter to all those who celebrate! One common tradition this time of year is decorating Easter eggs. So, we’ve decided to take this opportunity to create some egg-related math content in Maple Learn. This year, a blog post by Tony Finch inspired us to create a walkthrough exploring the four-point egg. The four-point egg is a method to construct an egg-shaped graph using just a compass and a ruler, or in this case, Maple Learn. Here's the final product: 

     The Maple Learn document, found here, walks through the steps. In general, each part of the egg is an arc corresponding to part of a circle centred around one of the points generated in this construction. 

     For instance, starting with the unit circle and the three red points in the image below, the blue circle is centred at the bottom point such that it intersects with the top of the unit circle, at (0,1). The perpendicular lines were constructed using the three red points, such that they intersect at the bottom point and pass through opposite side points, either (-1,0) or (1,0). Then, the base of the egg is constructed by tracing an arc along the bottom of the blue circle, between the perpendicular lines, shown in red below.

 

     Check out the rest of the steps in the Maple Learn Document. Also, be sure to check out other egg-related Maple Learn documents including John May’s Egg Formulas, illustrating other ways to represent egg-shaped curves with mathematics, and Paige Stone’s Easter Egg Art, to design your own Easter egg in Maple Learn. So, if you’ve had your fill of chocolate eggs, consider exploring some egg-related geometry - Happy Easter!  

Let N=pq be an odd semi-prime; What is the distribution of  integers that has a common divisor with N. We have shown that the distribution in [1,N-1] is a symmetric one, and there exsits a multiple of p lying to a multiple of q. We post the Maple source here.

 

gap := proc(a, b) return abs(a - b) - 1; end proc

HostsNdivisors := proc(N)

local i, j, g, d, L, s, t, m, p, q, P, Q, np, nq;

m := floor(1/2*N - 1/2);

L := evalf(sqrt(N));

P := Array();

Q := Array();

s := 1; t := 1;

for i from 3 to m do

   d := gcd(i, N);

    if 1 < d and d < L then P(s) := i; s++;

    elif L < d then Q(t) := i; t++; end if;

end do;

  np := s - 1;

  nq := t - 1;

 for i to np do printf("%3d,", P(i)); end do;

  printf("\n");

  for i to nq do printf("%3d,", Q(i)); end do;

  printf("\n gaps: \n");

  for i to np do

     for j to nq do

      p := P(i); q := Q(j);

      g := gap(p, q);

      printf("%4d,", g);

  end do;

    printf("\n");

end do;

end proc

 

HostOfpq := proc(p, q)

local alpha, s, t, g, r, S, T, i, j;

   S := 1/2*q - 1/2;

   T := 1/2*p - 1/2;

   alpha := floor(q/p);

    r := q - alpha*p;

   for s to S do

     for t to T do

       g := abs((t*alpha - s)*p + t*r) - 1;

        printf("%4d,", g);

      end do;

     printf("\n");

 end do;

end proc

 

MapleSource.pdf

MapleSource.mw

 



(EDITED 2024/03/11  GMT 17H)

In a recent Question@cq mentionned in its last reply "In fact, I wanted to do parameter sensitivity analysis and get the functional relationship between the [...] function and [parameters]. Later, i will study how the uncertainty of [the parameters] affects the [...] function".
I did not keep exchanging further on with @cq, simply replying that I could provide it more help if needed.

In a few words the initial problem was this one:

  • Let X_1 and X_2 two random variables and G the random variable defined by  G = 1 - (X_1 - 1)^2/9 - (X_2 - 1)^3/16.
     
  •  X_1 and X_2 are assumed to be gaussian random variables with respective mean and standard deviation equal to (theta_1, theta_3) and (theta_2, theta_4).
     
  • The four theta parameters are themselves assumed to be realizations