Applications, Examples and Libraries

How to grow and prune a classification tree

by: Markiyan Hirnyk 7819 Product: Maple

December 01 2014

1 0

I'd like to pay attention to the article of David Austin "How to Grow and Prune a Classification Tree"

Here is its introduction:

"

It's easy to collect data these days; making sense of it is more work. This article explains a construction in machine learning and data mining called a classification tree. Let's consider an example.

In the late 1970's, researchers at the University of California, San Diego Medical Center performed a study in which they monitored 215 patients following a heart attack. For each patient, 19 variables, such as age and blood pressure, were recorded. Patients were then grouped into two classes, depending on whether or not they survived more than 30 days following the heart attack.

Assuming the patients studied were representative of the more general population of heart attack patients, the researchers aimed to distill all this data into a simple test to identify new patients at risk of dying within 30 days of a heart attack.

By applying the algorithm described here, Breiman, Freidman, Olshen, and Stone, created a test consisting of only three questions---what is the patient's minimum systolic blood pressure within 24 hours of being admitted to the hospital, what is the patient's age, does the patient exhibit sinus tachycardia---to identify patients at risk. In spite of its simplicity, this test proved to be more accurate than any other known test. In addition, the importance of these three questions indicate that, among all 19 variables, these three factors play an important role in determining a patient's chance of surviving.

Besides medicine, these ideas are applicable to a wide range of problems, such as identifying which loan applicants are likely to default and which voters are likely to vote for a particular political party.

In what follows, we will describe the work of Breiman and his colleagues as set out in their seminal book Classification and Regression Trees. Theirs is a very rich story, and we will concentrate on only the essential ideas"

It would be interesting to compare this approach with discriminant analysis. Hope somebody of the Maple developers will give a concrete example on this theme with Maple.

Games with numbers

by: Kitonum 21675

December 01 2014

5 9

The procedure NumbersGame generalizes the well-known 24 game (implementation in Maple see here), as well as related issues (see here and here).

Required parameters of the procedure:

Result is an integer or a fraction of any sign.

Numbers is a list of positive integers.

Optional parameters:

Operators is a list of permitted arithmetic operations. By default Operators is ["+","-","*","/"]

NumbersOrder is a string. It is equal to "strict order" or "arbitrary order" . By default NumbersOrder is "strict order"

Parentheses is a symbol no or yes . By default Parentheses is no

The procedure puts the signs of operations from the list Operators between the numbers from Numbers so that the result is equal to Result. The procedure finds all possible solutions. The global M saves the list of the all solutions.

Code the procedure:

restart;

NumbersGame:=proc(Result::{integer,fraction}, Numbers::list(posint), Operators::list:=["+","-","*","/"], NumbersOrder::string:="strict order", Parentheses::symbol:=no)

local MyHandler, It, K, i, P, S, n, s, L, c;

global M;

uses StringTools, ListTools, combinat;

MyHandler := proc(operator,operands,default_value)

NumericStatus( division_by_zero = false );

return infinity;

end proc;

NumericEventHandler(division_by_zero=MyHandler);

if Parentheses=yes then

It:=proc(L1,L2)

local i, j, L;

for i in L1 do

for j in L2 do

L[i,j]:=seq(Substitute(Substitute(Substitute("( i Op j )","i",convert(i,string)),"j",convert(j,string)),"Op",Operators[k]), k=1..nops(Operators));

od; od;

L:=convert(L, list);

end proc;

P:=proc(L::list)

local n, K, i, M1, M2, S;

n:=nops(L);

if n=1 then return L else

for i to n-1 do

M1:=P(L[1..i]); M2:=P(L[i+1..n]);

K[i]:=seq(seq(It(M1[j], M2[k]), k=1..nops(M2)), j=1..nops(M1))

od; fi;

K:=convert(K,list);

end proc;

if NumbersOrder="arbitrary order" then S:=permute(Numbers); K:=[seq(op(Flatten([op(P(s))])), s=S)] else K:=[op(Flatten([op(P(Numbers))]))] fi;

else

if NumbersOrder="strict order" then

K:=[convert(Numbers[1],string)];

for i in Numbers[2..-1] do

K:=[seq(seq(cat(k, Substitute(Substitute(" j i","j",convert(j,string)),"i",convert(i,string))), k in K), j in Operators)]

od; else

S:=permute(Numbers);

for s in S do

L:=[convert(s[1],string)];

for i in s[2..-1] do

L:=[seq(seq(cat(k, Substitute(Substitute(" j i","j",convert(j,string)),"i",convert(i,string))), k in L), j in Operators)]

od; K[s]:=op(L) od; K:=convert(K,list) fi;

fi;

M:='M'; c:=0;

for i in K do

if parse(i)=Result then c:=c+1; if Parentheses=yes then M[i]:= convert(SubString(i,2..length(i)-1),symbol)=convert(Result,symbol) else M[i]:=convert(i,symbol)=convert(Result,symbol) fi; fi;

od;

if c=0 then M:=[]; return `No solutions` else M:=convert(M,list); op(M) fi;

end proc:

Examples of use.

Example 1:

NumbersGame(1/20, [$ 1..9]);

1 * 2 - 3 + 4 / 5 / 6 * 7 / 8 * 9 = 1/20

Example 2. Numbers in the list Numbers may be repeated and permitted operations can be reduced:

NumbersGame(15, [3,3,5,5,5], ["+","-"]);

3 - 3 + 5 + 5 + 5 = 15

Example 3.

NumbersGame(10, [1,2,3,4,5]);

1 + 2 + 3 * 4 - 5 = 10

If the order of the number in Numbers is arbitrary, then the number of solutions is greatly increased (10 solutions displayed):

NumbersGame(10, [1,2,3,4,5], "arbitrary order"):

nops(M);

for i to 10 do

M[1+50*(i-1)] od;

If you use the parentheses, the number of solutions will increase significantly more (10 solutions displayed):

NumbersGame(10, [1,2,3,4,5], "arbitrary order", yes):

nops(M);

for i to 10 do

M[1+600*(i-1)] od;

Game.mws

Draghilev’s method. Rolling without slipping....

by: one man 1370

November 21 2014

2 6

On the basis of the Draghilev’s method. Equations: x1^2 + x2^2 – 1 = 0 and x1^4 + x2^4 - 0.12^4 = 0.
Rolling without slipping is nonholonomic constraint.

https://vk.com/doc242471809_343244682

ROLLING.mw

and more:
https://vk.com/doc242471809_343068414
https://vk.com/doc242471809_339747442
https://vk.com/doc242471809_340301415
https://vk.com/doc242471809_341981154
https://vk.com/doc242471809_338645739
https://vk.com/doc242471809_339196723
https://vk.com/doc242471809_339100828
https://vk.com/doc242471809_339058052
https://vk.com/doc242471809_343477990
https://vk.com/doc242471809_389197688
https://vk.com/doc242471809_393738754
https://vk.com/doc242471809_392866881

The comet lander and Maple

by: eithne 2072 Product: Maple

November 17 2014

5 0

We received an interesting and timely submission to the Maple Application Center this morning that I think people might be interested in. It's called:

The Comet 67P/Churyumov-Gerasimenko, Rosetta & Philae, by Dr. Ahmed Baroudy. From the abstract:

Our plan is rather a modest one since all we want is to get , by calculations, specific data concerning the comet and its lander.
We shall take a simplified model and consider the comet as a perfect solid sphere to which we can apply Newton's laws.

We want to find:

I- the acceleration on the comet surface ,
II- its radius,
III- its density,
IV- the velocity of Philae just after the 1st bounce off the comet (it has bounced twice),
V- the time for Philae to reach altitude of 1000 m above the comet .

We shall compare our findings with the already known data to see how close our simplified mathematical model findings are to the duck-shaped comet already known results.
It turned out that our calculations for a sphere shaped comet are very close to the already known data.

Click on the link above if you want to take a look.

eithne

What is Groebner?

by: Markiyan Hirnyk 7819 Product: Maple

November 14 2014

4 18

What is Groebner? That was asked in different forms several times in MaplePrimes and MathStackExchange (for example, see http://math.stackexchange.com/questions/3550/using-gr?bner-bases-for-solving-polynomial-equations ). In view of this I think the presented post on Groebner basis will be useful. This post consists of two parts: its mathematical background and examples of solutions of polynomial systems by hand and with Maple.

Let us start. Up to Wiki http://en.wikipedia.org/wiki/Gr%C3%B6bner_basis ,Groebner basis computation can be seen as a multivariate, non-linear generalization of both Euclid's algorithm for computing polynomial greatest common divisors, and Gaussian elimination for linear systems. This is implemented in Maple trough the Groebner package.
The simplest introduction to the topic I know is a well-written book of Ivan Arzhantsev (https://zbmath.org/?q=an:05864974) which includes the proofs of all the claimed theorems and the solutions of all the exercises. Here is its digest groebner.pdf done by me (The reader is assumed to be familiar with the ideal notion and ring notion (one may refresh her/his knowledge, looking in http://en.wikipedia.org/wiki/Ideal_%28ring_theory%29)). It should be noted that there is no easy reading about this serious matter.
Referring to the digest as appropriate, we solve the system
S:={a*b = c^2+c, a^2 =a+ b*c, a*c = b^2+b} by hand and with the Groebner package.
For the order a > b > c we construct its ideal
J(S):=<f1 = a*b-c^2-c,f2 = a^2-a-b*c, f3 = a*c-b^2-b>.
The link between f1 and f2 gives
f1*a-f2*b = (-c^2-c)*a + (a + b*c)*b = a*b -a*c + b^2*c -
a*c^2 =f4.
The reduction with f1 produces
f4 ->-a*c^2- a*c + b^2*c + c^2 +c =: f4.
Now the reduction with f3 produces
f4 -> -b^2- b - b*c +c^2 + c =:f4.
The link between f2 and f3 gives:
f2*c - f3*a = a*b +a*b^2 -a*c -b*c^2 = f5.
The reduction with f1 produces
f5 -> -a*c + c*b +c^2 +c =:f5.
The reduction with f3 produces
f5 -> -b^2 -b + c*b +c^2 +c =:f5.
The reduction with f4 produces
f5 -> 2b*c =: f5.
The link between f1 and f3
f1*c - f3*b = b^3 + b^2 -c^3 -c^2=:f6.
The reduction with f4 produces
f6 -> 2b*c + 2b*c^2 -2c^3 -2c^2=:f6.
At last, we reduce f6 by f5, obtaining f6:= -2c^3 -2c^2.
We see the minimal reduced Groebner basis of S consists of
a^2 -a -b*c, -b^2 -b- b*c +c^2 +c, -2c^3 - 2c^2.
Now we find the solution set of the system under consideration. The equation -2c^3 - 2c^2 = 0 implies
c=0, c=0, c=-1. The the equation -b^2 - b - b*c +c^2 + c = 0 gives
b = 0 , b = -1, b = 0, b = -1, b = 0, b = 0 respectively.
At last, knowing b and c, we find a from a^2 -a -b*c = 0.
Hence,
[{a = 0, b = 0, c = 0}, {a = 1, b = 0, c = 0}, {a = 0, b = -1, c = 0}], [{a = 0, b = 0, c = 0}, {a = 1, b = 0, c = 0}, {a = 0, b = -1, c = 0}], [{a = 0, b = 0, c = -1}].
The solution of the system under consideration by the Groebner package is somewhat different because Maple does not find the minimal reduced Groebner basis directly.

with(Groebner):

(1)

(2)

(3)

(4)

(5)

(6)

(7)

$S2 := {a*c-b^2-b, a*b-c^2-c, a^2-b*c+a}:$

GB1 := Basis(S2, lexdeg([a, b, c]))

(8)

(9)

${b = -(1/2)*c-1/2+(1/2)*(-3*c^2-2*c+1)^(1/2)}, {b = -(1/2)*c-1/2-(1/2)*(-3*c^2-2*c+1)^(1/2)}$

(10)

${{a = 2*c*(c+1)/(-c-1+(-3*c^2-2*c+1)^(1/2)), b = -(1/2)*c-1/2+(1/2)*(-3*c^2-2*c+1)^(1/2)}, {a = -1/2-(1/2)*(1+2*c*(-3*c^2-2*c+1)^(1/2)-2*c^2-2*c)^(1/2), b = -(1/2)*c-1/2+(1/2)*(-3*c^2-2*c+1)^(1/2)}, {a = -1/2+(1/2)*(1+2*c*(-3*c^2-2*c+1)^(1/2)-2*c^2-2*c)^(1/2), b = -(1/2)*c-1/2+(1/2)*(-3*c^2-2*c+1)^(1/2)}, {a = -(1/2)*c-1/2-(1/2)*(-3*c^2-2*c+1)^(1/2), b = -(1/2)*c-1/2+(1/2)*(-3*c^2-2*c+1)^(1/2)}}$

(11)

Download groebner.mw

Something to discuss about CASes

by: Markiyan Hirnyk 7819 Product: Maple

November 03 2014

5 12

I'd like to pay attention to the recent article "The Misfortunes of a Trio of Mathematicians Using Computer Algebra Systems. Can We Trust in Them?"

In particular, the authors consider the integral

int(abs(exp(2*Pi*Ix)+exp(2*Pi*I*y)),[x=0..1,y=0..1]),

stating "Both Mathematica and Maple return zero as the answer to this calculation. Yet this cannot be correct, because the integrand is clearly positive and nonzero in the indicated region". Unfortunately, they give only the Mathematica command to this end.

Of course, the integral under consideration is complicated so the the simple-minded trials

int(evalc(abs(exp((2*Pi*I)*x)+exp((2*Pi*I)*y))), [x = 0 .. 1, y = 0 .. 1]);

and

VectorCalculus:-int(evalc(abs(exp((2*Pi*I)*x)+exp((2*Pi*I)*y))), [x,y]=Rectangle( 0 .. 1, 0 .. 1));

fail. However,this can be found with Maple (I think with Mathematica too.) in such a way.

Download int.mw

Visualization of meme propagation on the internet

by: ssseahra 35

October 24 2014

7 0

Hi, we recently put together a web video on how memes spread on the internet using several visualizations generated from Maple 18:

http://youtu.be/vEhAkEPwESI

Found the new ability to specify a background image for plots to be very helpful.

MapleSim 7 is now available

by: eithne 2072 Product: MapleSim

October 22 2014

3 0

We have just released a new version of MapleSim.

MapleSim 7 makes it substantially easier to explore and validate designs, create and manage libraries of custom components, and use your MapleSim models with other tools. It includes:

Easy model investigation. A new Results Manager gives you greater flexibility when it comes to investigating your simulation results, including the ability to compare simulation runs on the same axes, instantly plot both probed and unprobed variables, and easily create custom plots.
Convenient library creation. With MapleSim 7, it is significantly easier to create, manage, and share libraries of custom components.
Improved Modelica support. MapleSim 7 expands the support of the Modelica language so that more Modelica definitions can be used directly inside MapleSim.

We have also updated and expanded the MapleSim 7 family of add-on products:

The new MapleSim Battery Library, which is available as a separate add-on, allows you to incorporate physics-based predictive models of battery cells into your system models so you can take battery behavior into account early in the design process.
The MapleSim Connector for FMI, which allows engineers to share very efficient, high-fidelity models created in MapleSim with other modeling tools, has been expanded to support more export formats for co-simulation and model exchange.

See What’s New in MapleSim 7 for more information about these and other improvements in MapleSim.

eithne

Computer Algebra for Theoretical Physics

by: ecterrab 14670

October 15 2014

12 4

Last week the Physics package was presented in a talk at the Perimeter Institute for Theoretical Physics and in a combined Applied Mathematics and Physics Seminar at the University of Waterloo. The presentation at the Perimeter Institute got recorded. It was a nice opportunity to surprise people with the recent advances in the package. It follows the presentation with sections closed, and at the end there is a link to a pdf with the sections open and to the related worksheet, used to run the computations in real time during the presentation.

COMPUTER ALGEBRA FOR THEORETICAL PHYSICS

Generally speaking, physicists still experience that computing with paper and pencil is in most cases simpler than computing on a Computer Algebra worksheet. On the other hand, recent developments in the Maple system implemented most of the mathematical objects and mathematics used in theoretical physics computations, and dramatically approximated the notation used in the computer to the one used in paper and pencil, diminishing the learning gap and computer-syntax distraction to a strict minimum. In connection, in this talk the Physics project at Maplesoft is presented and the resulting Physics package illustrated tackling problems in classical and quantum mechanics, general relativity and field theory. In addition to the 10 a.m lecture, there will be a hands-on workshop at 1pm in the Alice Room.

... Why computers?

We can concentrate more on the ideas instead of on the algebraic manipulations

We can extend results with ease

We can explore the mathematics surrounding a problem

We can share results in a reproducible way

Representation issues that were preventing the use of computer algebra in Physics

Notation and related mathematical methods that were missing:

coordinate free representations for vectors and vectorial differential operators,

covariant tensors distinguished from contravariant tensors,

functional differentiation, relativity differential operators and sum rule for tensor contracted (repeated) indices

Bras, Kets, projectors and all related to Dirac's notation in Quantum Mechanics

Inert representations of operations, mathematical functions, and related typesetting were missing:

inert versus active representations for mathematical operations

ability to move from inert to active representations of computations and viceversa as necessary

hand-like style for entering computations and texbook-like notation for displaying results

Key elements of the computational domain of theoretical physics were missing:

ability to handle products and derivatives involving commutative, anticommutative and noncommutative variables and functions

ability to perform computations taking into account custom-defined algebra rules of different kinds

(problem related commutator, anticommutator, bracket, etc. rules)

	Vector and tensor notation in mechanics, electrodynamics and relativity

	Dirac's notation in quantum mechanics

•

Computer algebra systems were not originally designed to work with this compact notation, having attached so dense mathematical contents, active and inert representations of operations, not commutative and customizable algebraic computational domain, and the related mathematical methods, all this typically present in computations in theoretical physics.

•	This situation has changed. The notation and related mathematical methods are now implemented.

Tackling examples with the Physics package

Classical Mechanics

Inertia tensor for a triatomic molecule

Problem: Determine the Inertia tensor of a triatomic molecule that has the form of an isosceles triangle with two masses in the extremes of the base and mass in the third vertex. The distance between the two masses is equal to a, and the height of the triangle is equal to h.

	Solution

Quantum mechanics

Quantization of the energy of a particle in a magnetic field

Show that the energy of a particle in a constant magnetic field oriented along the z axis can be written as

where and are creation and anihilation operators.

	Solution

	The quantum operator components of satisfy

Unitary Operators in Quantum Mechanics

(with Pascal Szriftgiser, from Laboratoire PhLAM, Université Lille 1, France)

A linear operator U is unitary if , in which case, Unitary operators are used to change the basis inside an Hilbert space, which physically means changing the point of view of the considered problem, but not the underlying physics. Examples: translations, rotations and the parity operator.

	1) Eigenvalues of an unitary operator and exponential of Hermitian operators

	2) Properties of unitary operators

	3) Schrödinger equation and unitary transform

	4) Translation operators

Classical Field Theory

The field equations for a quantum system of identical particles

Problem: derive the field equation describing the ground state of a quantum system of identical particles (bosons), that is, the Gross-Pitaevskii equation (GPE). This equation is particularly useful to describe Bose-Einstein condensates (BEC).

	Solution

	The field equations for the model

	Maxwell equations departing from the 4-dimensional Action for Electrodynamics

General Relativity

Given the spacetime metric,

$g[mu, nu] = (Matrix(4, 4, {(1, 1) = -exp(lambda(r)), (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = -r^2, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -r^2*sin(theta)^2, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = exp(nu(r))}))$

a) Compute the trace of

where is some function of the radial coordinate, is the Ricci tensor, is the covariant derivative operator and is the stress-energy tensor

$T[alpha, beta] = (Matrix(4, 4, {(1, 1) = 8*exp(lambda(r))*Pi, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = 8*r^2*Pi, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 8*r^2*sin(theta)^2*Pi, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 8*exp(nu(r))*Pi*epsilon}))$

b) Compute the components of the traceless part of of item a)

c) Compute an exact solution to the nonlinear system of differential equations conformed by the components of obtained in b)

Background: paper from February/2013, "Withholding Potentials, Absence of Ghosts and Relationship between Minimal Dilatonic Gravity and f(R) Theories", by P. Fiziev.

	a) The trace of

	b) The components of the traceless part of

	c) An exact solution for the nonlinear system of differential equations conformed by the components of

The Physics Project

"Physics" is a software project at Maplesoft that started in 2006. The idea is to develop a computational symbolic/numeric environment specifically for Physics, targeting educational and research needs in equal footing, and resembling as much as possible the flexible style of computations used with paper and pencil. The main reference for the project is the Landau and Lifshitz Course of Theoretical Physics.

A first version of "Physics" with basic functionality appeared in 2007. Since then the package has been growing every year, including now, among other things, a searcheable database of solutions to Einstein equations and a new dedicated programming language for Physics.

Since August/2013, weekly updates of the Physics package are distributed on the web, including the new developments related to our plan as well as related to people's feedback.

Presentation_at_PI_and_UW.pdf Presentation_at_PI_and_UW.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Solutions of Differential Equations with Embedded...

by: Lenin Araujo Castillo 1790 Product: Maple 18

October 15 2014

2 0

The Embedded Components are containers that currently use industries for modeling complex systems to find viable solutions in real time and thus avoid huge wait times and overload our computer; by this paper should show you how to implement a dynamic worksheet through Embedded Components in Maple; it goes from finding solutions to ordinary differential equations partial; which interact with the researcher using different parameters.
Using graphical programming will find immediate solutions to selected problems in science and engineering criteria of variability and boundary conditions evolving development with buttons on multiple actions.

cimac_2014.pdf

(in spanish)

Solutions_of_Differential_Equations_with_Embedded_Components.mw

Lenin Araujo Castillo

Physics Pure

Computer Science

October Live Webinars

by: kkoserski 230 Product: Maple

October 08 2014

1 0

Maplesoft regularly hosts live webinars on a variety of topics. Below you will find details on some upcoming webinars we think may be of interest to the MaplePrimes community. For the complete list of upcoming webinars, visit our website.

Maplesoft Solutions for Math Education

This webinar will demonstrate how Maplesoft’s solutions for mathematics education help teachers bring complex problems to life, allow students to focus on concepts rather than the mechanics of solutions, and offer students the necessary practice to master the concepts being taught.

Key takeaways include:

• How to quickly and painlessly place incoming students in the correct math courses

• How you can use hundreds of intuitive Clickable Math tools to demonstrate and explore up to advanced-level problems and algorithms in the classroom

• How to automate your testing and assessment needs, specifically for math courses

• How to bring your STEM courses to life in an online environment

To join us for the live presentation, please click here to register.

Introduction to Maple T.A. Placement Test Suite 10

This webinar will provide an overview and demonstration of the latest release of the Maple T.A. MAA Placement Test Suite. A result of the ongoing partnership between the Mathematical Association of America (MAA) and Maplesoft, this product gives you the ability to provide the renowned MAA placement tests in an online testing environment. Learn how the Maple T.A. MAA Placement Test Suite can greatly simplify your placement process and explore the latest additions, including a streamlined interface and new tests to determine your students’ readiness for Precalculus and Algebra courses.

To join us for the live presentation, please click here to register.

There is also a recording available from another live webinar we did earlier this month: Introduction to Maple T.A. 10.

Photon Exposure

by: John Dolese 125 Product: Maple 18

September 26 2014

5 0

This application calculates the number of photons reaching a camera sensor for a given exposure. A blackbody model of the sun is generated. The "Sunny 16" rule for exposure is demonstrated. Calculations are done using units.Photon_Exposure_Array.mw

Photon Exposure

Blackbody Model of the Sun

	Plank Constant
	Boltzman Constant
	Light Speed
	Sun Radius
	Earth Orbit
	Sun Color Temperature
	Transmission Factor

Sun: Spectral Radiant Exitance to Earth: Spectral Irradiance

"M(lambda):=(2*Pi*h*c^(2))/((lambda)^(5))*1/((e)^((h*c)/(lambda*kb*Tsun))-1)*(Rsun/(Re_orb))^(2)*tf_atm:"

=

Photopic Relative Response VP vs λ

=

= $Vector(4, {(1) = ` 471 x 2 `*Matrix, (2) = `Data Type: `*float[8], (3) = `Storage: `*rectangular, (4) = `Order: `*Fortran_order})$

(ArrayInterpolation for x,y data VPdata returns y' for new x data lambdaP)

Illuminance in Radiometric and Photometric Units:

E__r := sum(Units:-Standard:-`*`(M(Units:-Standard:-`*`(lambda, Unit('nm'))), Unit('nm')), lambda = 200 .. 4000) =

E__po := Units:-Standard:-`*`(Units:-Standard:-`*`(683.002, Units:-Standard:-`*`(Unit('lm'), Units:-Standard:-`/`(Unit('W')))), sum(Units:-Standard:-`*`(Units:-Standard:-`*`(VP[lambda], M(Units:-Standard:-`*`(lambda, Unit('nm')))), Unit('nm')), lambda = 200 .. 4000)) =

Translation from Illuminance to Luminance for Reflected Light;

Object Reflectance

Object Luminance L__po := proc (R__o) options operator, arrow; R__o*E__po/(Pi*Unit('sr')) end proc: =

Illuminance of a Camera Sensor E_ps applied for time t_expdetermines Luminous Exposure H_p;

Ideal Illuminance is determined by the exposure time t_exp, effective f-number N and to a less extent the angle to the optical axis θ;

•	H Luminous Exposure

•	E_ps Illuminance to the Camera

•	N Effective F-Number

•	t_exp Exposure Time

•	θ Angle to the Optical Axis

E__ps_ideal = Units:-Standard:-`*`(Units:-Standard:-`*`(Units:-Standard:-`*`(Pi, Units:-Standard:-`/`(4)), L__po), Units:-Standard:-`*`(Units:-Standard:-`^`(cos(theta), 4), Units:-Standard:-`/`(Units:-Standard:-`^`(N, 2)))):

The camera meter determines the exposure time t_expto balance the object luminance, reflectance and effective f-number. It does this based on an internal constant k and the camera ISO s.

•	s ISO Gain (Based on saturation at 3 stops above the average scene luminance)

•	k Reflected Light Meter Calibration Constant

for Nikon, Canon and Sekonic

•	c Incident Light Meter Calibration Constant

for Sekonic with flat dome

$N^2/t__exp = `#mrow(mi("\`E__po\`"),mo("⋅"),mi("s"))`/c__m$ (Incident Light Meter)

$Units:-Standard:-`*`(Units:-Standard:-`^`(N, 2), Units:-Standard:-`/`(t__exp)) = Units:-Standard:-`*`(`#mrow(mi("\`L__po\`"),mo("⋅"),mi("s"))`, Units:-Standard:-`/`(k__m)):$ (Reflected Light Meter)

Solve for H in terms of the Camera Meter Constant k and s

Es = Units:-Standard:-`*`(Units:-Standard:-`*`(Units:-Standard:-`*`(Pi, Units:-Standard:-`/`(4)), Lo), Units:-Standard:-`*`(Units:-Standard:-`^`(cos(theta), 4), Units:-Standard:-`/`(Units:-Standard:-`^`(N, 2)))):

H = Units:-Standard:-`*`(Units:-Standard:-`*`(Units:-Standard:-`*`(Units:-Standard:-`*`(Pi, Units:-Standard:-`/`(4)), Lo), Units:-Standard:-`*`(Units:-Standard:-`^`(cos(theta), 4), Units:-Standard:-`/`(Units:-Standard:-`^`(N, 2)))), Units:-Standard:-`*`(Units:-Standard:-`*`(km, Units:-Standard:-`^`(N, 2)), Units:-Standard:-`/`(Units:-Standard:-`*`(Lo, s)))) H = (1/4)*Pi*cos(theta)^4*km/s

H__p := proc (s, theta) options operator, arrow; (1/4)*Pi*k__m*cos(theta)^4/s end proc:

=

Note: Meters are typically set for a scene reflectance 3 stops below 100% or 12.5%.

E__ps := proc (N, R__o, theta) options operator, arrow; (1/4)*Pi*Unit('sr')*R__o*E__po*cos(theta)^4/(Pi*Unit('sr')*N^2) end proc:

=

t__exp_ideal := proc (N, s, R__o) options operator, arrow; H__p(s, theta)/E__ps(N, R__o, theta) end proc:

=

Actual exposure time includes typical lens losses;

	Magnification
	Lens Transmittance
	Lens Flare
	Vignetting

Total Lens Efficiency

=

Replacing Eps with q*Eps we get the "Sunny 16" relation between exposure time and ISO;

t__exp := proc (N, s, R__o) options operator, arrow; H__p(s, theta)/(q*E__ps(N, R__o, theta)) end proc: =

t__exp_alt := proc (N, s, R__o) options operator, arrow; k__m*N^2*Pi/(s*q*R__o*E__po) end proc: =

•	The Number of Photons NP Reaching the Sensor Area A;

•	Circle of confusion for 24x36mm "Full Frame" for 1 arcminute view at twice the diagonal:

A__cc := Units:-Standard:-`*`(Units:-Standard:-`*`(Pi, Units:-Standard:-`^`(Units:-Standard:-`*`(12.6, Unit('`μm`')), 2)), Units:-Standard:-`/`(4)):

•	Sensor Bandwidth Photopic Response VP

•	Exposure Time for Zone 5: Rscene=12.5% , Saturation in Zone 8 Rscene=100%

•	Camera ISO differs from Saturation ISO. Typical Saturation ISO is 2300 when the camera is set to 3200. See DxoMark.

The average number of photons for exposure time based on Reflectance of the scene relative to the metered value:

Zone 5;

NP := proc (s, R__o, theta) options operator, arrow; (1/4)*t__exp(N, s, R__meter)*A__cc*q*R__scene*cos(theta)^4*(sum(VP[lambda]*M(lambda*Unit('nm'))*Unit('nm')*lambda*Unit('nm')/(h*c), lambda = 200 .. 4000))/N^2 end proc:

=

Zone 8; R__meter := Units:-Standard:-`*`(R__scene, Units:-Standard:-`/`(Units:-Standard:-`^`(2, 3))):

NP__sat := proc (s, theta) options operator, arrow; (1/4)*t__exp(N, s, R__meter)*A__cc*q*R__scene*cos(theta)^4*(sum(VP[lambda]*M(lambda*Unit('nm'))*Unit('nm')*lambda*Unit('nm')/(h*c), lambda = 200 .. 4000))/N^2 end proc:

=

Approximate Formula

H__sat := proc (s__sat) options operator, arrow; H__p(s__sat, 0)*E__ps(N, 1, 0)/E__ps(N, 1/8, 0) end proc :

= HFloat(78.53981635)*Units:-Unit(('cd')*('s')/('m')('radius')^2)/s__sat

Average Visible Photon Energy

P__e_ave := Units:-Standard:-`*`(Units:-Standard:-`/`(Units:-Standard:-`+`(850, -350)), sum(Units:-Standard:-`*`(Units:-Standard:-`*`(h, c), Units:-Standard:-`/`(Units:-Standard:-`*`(lambda, Unit('nm')))), lambda = 350 .. 850)): =

NPtyp := proc (s__sat) options operator, arrow; H__sat(s__sat)*A__cc/(683.002*(Unit('lm')/Unit('W'))*P__e_ave) end proc:

=

Download Photon_Exposure_Array.mw

Create DNG Matrices Using Optimization

by: John Dolese 125 Product: Maple 18

September 20 2014

2 0

Obsolete

See my Camera Profiler application instead.

This application creates DNG matrices by optimizing Delta E from a raw photo of x-rites color checker. The color temperature for the photograph is also estimated. Inputs are raw data from RawDigger and generic camera color response from DXO Mark.

	Initialization

XYZoptical to RGB to XYZdata

Sr,g,b is the relative spectral transmittance of the filter array not selectivity for XY or Z of a given color.

Pulling Sr,g,b out of the integral assumes they are scalars. For example Sr attenuates X, Y and Z by the same amount.

Raw Balance is not White Point Adaptation.

The transmission loss of Red and Blue pixels relative to green is compensated by D=inverse(S). The relation to incident chromaticity, xy is unchanged as S.D=1.

(See Bruce Lindbloom; "Spectrum to XYZ" and "RGB/XYZ Matrices" also, Marcel Patek; "Transformation of RGB Primaries")

•	XYZ to RGB

$(Vector(3, {(1) = R_Tbb, (2) = G_Tbb, (3) = B_Tbb})) = (Matrix(3, 3, {(1, 1) = XR*Sr, (1, 2) = YR*Sr, (1, 3) = ZR*Sr, (2, 1) = XG*Sg, (2, 2) = YG*Sg, (2, 3) = ZG*Sg, (3, 1) = XB*Sb, (3, 2) = YB*Sb, (3, 3) = ZB*Sb})).(Vector(3, {(1) = X_Tbb, (2) = Y_Tbb, (3) = Z_Tbb}))$

$(Vector(3, {(1) = R_Tbb, (2) = G_Tbb, (3) = B_Tbb})) = (Matrix(3, 3, {(1, 1) = Sr, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = Sg, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = Sb})).(Matrix(3, 3, {(1, 1) = XR, (1, 2) = YR, (1, 3) = ZR, (2, 1) = XG, (2, 2) = YG, (2, 3) = ZG, (3, 1) = XB, (3, 2) = YB, (3, 3) = ZB})).(Vector(3, {(1) = X_Tbb, (2) = Y_Tbb, (3) = Z_Tbb}))$

$Camera_Neutral = (Matrix(3, 3, {(1, 1) = Sr, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = Sg, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = Sb})).(Matrix(3, 3, {(1, 1) = XR, (1, 2) = YR, (1, 3) = ZR, (2, 1) = XG, (2, 2) = YG, (2, 3) = ZG, (3, 1) = XB, (3, 2) = YB, (3, 3) = ZB})).(Vector(3, {(1) = X_wht, (2) = Y_wht, (3) = Z_wht}))$

•	RGB to XYZ (The extra step of adaptation to D50 is included below)

$(Vector(3, {(1) = X_D50, (2) = Y_D50, (3) = Z_D50})) = (Matrix(3, 3, {(1, 1) = XTbbtoXD50, (1, 2) = YTbbtoXD50, (1, 3) = ZTbbtoXD50, (2, 1) = XTbbtoYD50, (2, 2) = YTbbtoYD50, (2, 3) = ZTbbtoYD50, (3, 1) = XTbbtoZD50, (3, 2) = YTbbtoZD50, (3, 3) = ZTbbtoZD50})).(Matrix(3, 3, {(1, 1) = RX*Dr, (1, 2) = GX*Dg, (1, 3) = BX*Db, (2, 1) = RY*Dr, (2, 2) = GY*Dg, (2, 3) = BY*Db, (3, 1) = RZ*Dr, (3, 2) = GZ*Dg, (3, 3) = BZ*Db})).(Vector(3, {(1) = R_Tbb, (2) = G_Tbb, (3) = B_Tbb}))$

$(Vector(3, {(1) = X_D50, (2) = Y_D50, (3) = Z_D50})) = (Matrix(3, 3, {(1, 1) = XTbbtoXD50, (1, 2) = YTbbtoXD50, (1, 3) = ZTbbtoXD50, (2, 1) = XTbbtoYD50, (2, 2) = YTbbtoYD50, (2, 3) = ZTbbtoYD50, (3, 1) = XTbbtoZD50, (3, 2) = YTbbtoZD50, (3, 3) = ZTbbtoZD50})).(Matrix(3, 3, {(1, 1) = RX, (1, 2) = GX, (1, 3) = BX, (2, 1) = RY, (2, 2) = GY, (2, 3) = BY, (3, 1) = RZ, (3, 2) = GZ, (3, 3) = BZ})).(Matrix(3, 3, {(1, 1) = Dr, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = Dg, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = Db})).(Vector(3, {(1) = R_Tbb, (2) = G_Tbb, (3) = B_Tbb}))$

$(Vector(3, {(1) = X_D50, (2) = Y_D50, (3) = Z_D50})) = (Matrix(3, 3, {(1, 1) = RX_D50, (1, 2) = GX_D50, (1, 3) = BX_D50, (2, 1) = RY_D50, (2, 2) = GY_D50, (2, 3) = BY_D50, (3, 1) = RZ_D50, (3, 2) = GZ_D50, (3, 3) = BZ_D50})).(Matrix(3, 3, {(1, 1) = Dr, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = Dg, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = Db})).(Vector(3, {(1) = R_Tbb, (2) = G_Tbb, (3) = B_Tbb}))$

$(Vector(3, {(1) = X_D50wht, (2) = Y_D50wht, (3) = Z_D50wht})) = (Matrix(3, 3, {(1, 1) = RX_D50, (1, 2) = GX_D50, (1, 3) = BX_D50, (2, 1) = RY_D50, (2, 2) = GY_D50, (2, 3) = BY_D50, (3, 1) = RZ_D50, (3, 2) = GZ_D50, (3, 3) = BZ_D50})).(Matrix(3, 3, {(1, 1) = Dr, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = Dg, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = Db})).Camera_Neutral$

	Functions

	Input Data

	Solve for Camera to XYZ D50 and T

Special bounded partial sums

by: Marvin Ray Burns 550

September 10 2014

0 1

For all real a, the partial sums s_n= sum((-1)^k (k^(1/k) -a), k=1..n) are bounded so that their limit points form an interval [-1.+ the MRB constant +a, MRB constant] of length 1-a, where the MRB constant is limit( $sum((-1)^k*(k^(1/k)), k = 1 ..2*N)$ ,N=infinity).

For all complex z, the upper limit point of s_n= sum((-1)^k (k^(1/k) -z), k=1..n) is the the MRB constant.

We see that maple knows the basics of this because when we enter sum((-1)^k*(k^(1/k)-z), k = 1 .. n)

maple gives

$sum((-1)^k*(k^(1/k)-z), k = 1 .. n)$ .

marvinrayburns.com

Banque de questions MapleTA en français – plus...

by: jzivku 640

September 05 2014

1 0

Aujourd’hui, je suis ravis d’annoncer la disponibilité d’une large banque de questions françaises supportant les enseignements du secondaire et de l’enseignement supérieur. Ce contenu didactique est disponible via le MapleTA Cloud, et également grâce au lien de téléchargement ci-dessous.

Lien de téléchargement de la banque de questions françaises

Ces questions sont librement et gratuitement accessibles, et vous pouvez les utiliser directement sur vos propres évaluations et exercices dans MapleTA, ou les éditer et modifier pour les adapter à vos besoins.

Le contenu de cette banque de questions françaises traite de sujets pour les classes et enseignements pré-bac, post-bac pour en majorité les matières scientifiques.

Les matières traitées par niveaux et domaines sont:

Lycées :

Electricité
Équations Différentielles
Gravitation universelle
Langues
Maths I
Maths II
Physique
Chimie
Mécanique

Enseignement supérieur (Post-Bac) :

Astrobiologie
Introduction au Calcul pour la Biologie
Chimie
Déplacement d'onde
Electricité & Magnétisme
Maths pour l’économie
Maths Post-Bac
Mécanique Angulaire
Mécanique des Fluides
Mécanique linéaire
Physique Post-Bac
Electrocinétique
Matériau
Mécanique des Fluides
Thermodynamique

Jonny Zivku
Maplesoft Product Manager, Maple T.A.

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