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    Transitioning to Jordan Form

    by: rlopez 3025 Maple 2015
    November 02 2015
    5  10

    I have two linear algebra texts [1, 2]  with examples of the process of constructing the transition matrix Q that brings a matrix A to its Jordan form J. In each, the authors make what seems to be arbitrary selections of basis vectors via processes that do not seem algorithmic. So recently, while looking at some other calculations in linear algebra, I decided to revisit these calculations in as orderly a way as possible.

     

    First, I needed a matrix A with a prescribed Jordan form. Actually, I started with a Jordan form, and then constructed A via a similarity transform on J. To avoid introducing fractions, I sought transition matrices P with determinant 1.

     

    Let's begin with J, obtained with Maple's JordanBlockMatrix command.

     

    • 

    Tools_Load Package: Linear Algebra

    Loading LinearAlgebra

    J := JordanBlockMatrix([[2, 3], [2, 2], [2, 1]])

    Matrix([[2, 1, 0, 0, 0, 0], [0, 2, 1, 0, 0, 0], [0, 0, 2, 0, 0, 0], [0, 0, 0, 2, 1, 0], [0, 0, 0, 0, 2, 0], [0, 0, 0, 0, 0, 2]])
     

    ``

    The eigenvalue lambda = 2 has algebraic multiplicity 6. There are sub-blocks of size 3×3, 2×2, and 1×1. Consequently, there will be three eigenvectors, supporting chains of generalized eigenvectors having total lengths 3, 2, and 1. Before delving further into structural theory, we next find a transition matrix P with which to fabricate A = P*J*(1/P).

     

    The following code generates random 6×6 matrices of determinant 1, and with integer entries in the interval [-2, 2]. For each, the matrix A = P*J*(1/P) is computed. From these candidates, one A is then chosen.

     

    L := NULL:
     

     

    After several such trials, the matrix A was chosen as

     

    A := Matrix(6, 6, {(1, 1) = -8, (1, 2) = -8, (1, 3) = 4, (1, 4) = -8, (1, 5) = -1, (1, 6) = 5, (2, 1) = -1, (2, 2) = 3, (2, 3) = 1, (2, 4) = -2, (2, 5) = 2, (2, 6) = -1, (3, 1) = -13, (3, 2) = -9, (3, 3) = 8, (3, 4) = -11, (3, 5) = 1, (3, 6) = 5, (4, 1) = 3, (4, 2) = 3, (4, 3) = -1, (4, 4) = 4, (4, 5) = 1, (4, 6) = -2, (5, 1) = 7, (5, 2) = 5, (5, 3) = -3, (5, 4) = 6, (5, 5) = 2, (5, 6) = -3, (6, 1) = -6, (6, 2) = -2, (6, 3) = 3, (6, 4) = -7, (6, 5) = 2, (6, 6) = 3})
     

     

    for which the characteristic and minimal polynomials are

     

    factor(CharacteristicPolynomial(A, lambda))

    (lambda-2)^6

    factor(MinimalPolynomial(A, lambda))

    (lambda-2)^3
     

     

    So, if we had started with just A, we'd now know that the algebraic multiplicity of its one eigenvalue lambda = 2 is 6, and there is at least one 3×3 sub-block in the Jordan form. We would not know if the other sub-blocks were all 1×1, or a 1×1 and a 2×2, or another 3×3. Here is where some additional theory must be invoked.

    ``

    The null spaces M[k] of the matrices (A-2*I)^k are nested: `⊂`(`⊂`(M[1], M[2]), M[3]) .. (), as depicted in Figure 1, where the vectors a[k], k = 1, () .. (), 6, are basis vectors.

     

    Figure 1   The nesting of the null spaces M[k] 
     

     

    The vectors a[1], a[2], a[3] are eigenvectors, and form a basis for the eigenspace M[1]. The vectors a[k], k = 1, () .. (), 5, form a basis for the subspace M[2], and the vectors a[k], k = 1, () .. (), 6, for a basis for the space M[3], but the vectors a[4], a[5], a[6] are not yet the generalized eigenvectors. The vector a[6] must be replaced with a vector b[6] that lies in M[3] but is not in M[2]. Once such a vector is found, then a[4] can be replaced with the generalized eigenvector `≡`(b[4], (A-2*I)^2)*b[6], and a[1] can be replaced with `≡`(b[1], A-2*I)*b[4]. The vectors b[1], b[4], b[6] are then said to form a chain, with b[1] being the eigenvector, and b[4] and b[6] being the generalized eigenvectors.

     

    If we could carry out these steps, we'd be in the state depicted in Figure 2.

     

    Figure 2   The null spaces M[k] with the longest chain determined
     

     

    Next, basis vector a[5] is to be replaced with b[5], a vector in M[2] but not in M[1], and linearly independent of b[4]. If such a b[5] is found, then a[2] is replaced with the generalized eigenvector `≡`(b[2], A-2*I)*b[5]. The vectors b[2] and b[5] would form a second chain, with b[2] as the eigenvector, and b[5] as the generalized eigenvector.

    ``

    Define the matrix C = A-2*I by the Maple calculation

     

    C := A-2

    Matrix([[-10, -8, 4, -8, -1, 5], [-1, 1, 1, -2, 2, -1], [-13, -9, 6, -11, 1, 5], [3, 3, -1, 2, 1, -2], [7, 5, -3, 6, 0, -3], [-6, -2, 3, -7, 2, 1]])
     

    ``

    and note

     

    N := convert(NullSpace(C), list)

    [Vector(6, {(1) = 1/2, (2) = 1/2, (3) = 1, (4) = 0, (5) = 0, (6) = 1}), Vector(6, {(1) = -1/2, (2) = -1/2, (3) = -2, (4) = 0, (5) = 1, (6) = 0}), Vector(6, {(1) = -2, (2) = 1, (3) = -1, (4) = 1, (5) = 0, (6) = 0})]

    NN := convert(LinearAlgebra:-NullSpace(C^2), list)

    [Vector(6, {(1) = 2/5, (2) = 0, (3) = 0, (4) = 0, (5) = 0, (6) = 1}), Vector(6, {(1) = 0, (2) = 0, (3) = 0, (4) = 0, (5) = 1, (6) = 0}), Vector(6, {(1) = -1, (2) = 0, (3) = 0, (4) = 1, (5) = 0, (6) = 0}), Vector(6, {(1) = 2/5, (2) = 0, (3) = 1, (4) = 0, (5) = 0, (6) = 0}), Vector(6, {(1) = -3/5, (2) = 1, (3) = 0, (4) = 0, (5) = 0, (6) = 0})]
     

    ``

    The dimension of M[1] is 3, and of M[2], 5. However, the basis vectors Maple has chosen for M[2] do not include the exact basis vectors chosen for M[1].

     

    We now come to the crucial step, finding b[6], a vector in M[3] that is not in M[2] (and consequently, not in M[1] either). The examples in [1, 2] are simple enough that the authors can "guess" at the vector to be taken as b[6]. What we will do is take an arbitrary vector in M[3] and project it onto the 5-dimensional subspace M[2], and take the orthogonal complement as b[6].

    ``

    A general vector in M[3] is

     

    Z := `<,>`(u || (1 .. 6))

    Vector[column]([[u1], [u2], [u3], [u4], [u5], [u6]])
     

    ``

    A matrix that projects onto M[2] is

     

    P := ProjectionMatrix(NN)

    Matrix([[42/67, -15/67, 10/67, -25/67, 0, 10/67], [-15/67, 58/67, 6/67, -15/67, 0, 6/67], [10/67, 6/67, 63/67, 10/67, 0, -4/67], [-25/67, -15/67, 10/67, 42/67, 0, 10/67], [0, 0, 0, 0, 1, 0], [10/67, 6/67, -4/67, 10/67, 0, 63/67]])
     

    ``

    The orthogonal complement of the projection of Z onto M[2] is then -P*Z+Z. This vector can be simplified by choosing the parameters in Z appropriately. The result is taken as b[6].

     

    b[6] := 67*(eval(Z-Typesetting:-delayDotProduct(P, Z), Equate(Z, UnitVector(1, 6))))*(1/5)

    Vector[column]([[5], [3], [-2], [5], [0], [-2]])

    NULL
     

    ``

    The other two members of this chain are then

     

    b[4] := Typesetting:-delayDotProduct(C, b[6])

    Vector[column]([[-132], [-12], [-169], [40], [92], [-79]])

    b[1] := Typesetting:-delayDotProduct(C, b[4])

    Vector[column]([[-67], [134], [67], [67], [0], [134]])
     

    ``

    A general vector in M[2] is a linear combination of the five vectors that span the null space of C^2, namely, the vectors in the list NN. We obtain this vector as

     

    ZZ := add(u || k*NN[k], k = 1 .. 5)

    Vector[column]([[(2/5)*u1-u3+(2/5)*u4-(3/5)*u5], [u5], [u4], [u3], [u2], [u1]])
     

    ``

    A vector in M[2] that is not in M[1] is the orthogonal complement of the projection of ZZ onto the space spanned by the eigenvectors spanning M[1] and the vector b[4]. This projection matrix is

     

    PP := LinearAlgebra:-ProjectionMatrix(convert(`union`(LinearAlgebra:-NullSpace(C), {b[4]}), list))

    Matrix([[69/112, -33/112, 19/112, -17/56, 0, 19/112], [-33/112, 45/112, 25/112, 13/56, 0, 25/112], [19/112, 25/112, 101/112, 1/56, 0, -11/112], [-17/56, 13/56, 1/56, 5/28, 0, 1/56], [0, 0, 0, 0, 1, 0], [19/112, 25/112, -11/112, 1/56, 0, 101/112]])
     

    ``

    The orthogonal complement of ZZ, taken as b[5], is then

     

    b[5] := 560*(eval(ZZ-Typesetting:-delayDotProduct(PP, ZZ), Equate(`<,>`(u || (1 .. 5)), LinearAlgebra:-UnitVector(4, 5))))

    Vector[column]([[-9], [-59], [17], [58], [0], [17]])
     

    ``

    Replace the vector a[2] with b[2], obtained as

     

    b[2] := Typesetting:-delayDotProduct(C, b[5])

    Vector[column]([[251], [-166], [197], [-139], [-112], [-166]])
     

     

    The columns of the transition matrix Q can be taken as the vectors b[1], b[4], b[6], b[2], b[5], and the eigenvector a[3]. Hence, Q is the matrix

     

    Q := `<|>`(b[1], b[4], b[6], b[2], b[5], N[3])

    Matrix([[-67, -132, 5, 251, -9, -2], [134, -12, 3, -166, -59, 1], [67, -169, -2, 197, 17, -1], [67, 40, 5, -139, 58, 1], [0, 92, 0, -112, 0, 0], [134, -79, -2, -166, 17, 0]])
     

    ``

    Proof that this matrix Q indeed sends A to its Jordan form consists in the calculation

     

    1/Q.A.Q = Matrix([[2, 1, 0, 0, 0, 0], [0, 2, 1, 0, 0, 0], [0, 0, 2, 0, 0, 0], [0, 0, 0, 2, 1, 0], [0, 0, 0, 0, 2, 0], [0, 0, 0, 0, 0, 2]])``
     

    NULL

    The bases for M[k], k = 1, 2, 3, are not unique. The columns of the matrix Q provide one set of basis vectors, but the columns of the transition matrix generated by Maple, shown below, provide another.

     

    JordanForm(A, output = 'Q')

    Matrix([[-5, -43/5, -9/5, 7/5, -14/5, -3/5], [10, -4/5, -6/25, 1/5, -6/25, -3/25], [5, -52/5, -78/25, 13/5, -78/25, -39/25], [5, 13/5, 38/25, -2/5, 38/25, 4/25], [0, 6, 42/25, -1, 42/25, 21/25], [10, -29/5, -11/25, 1/5, -11/25, 7/25]])
     

    ``

    I've therefore added to my to-do list the investigation into Maple's algorithm for determining an appropriate set of basis vectors that will support the Jordan form of a matrix.

     

    References
     

    NULL

    [1] Linear Algebra and Matrix Theory, Evar Nering, John Wiley and Sons, Inc., 1963

    [2] Matrix Methods: An Introduction, Richard Bronson, Academic Press, 1969
     

    NULL

    ``

    Download JordanForm_blog.mw

    Day of the year - bug

    by: Christopher2222 6035
    November 01 2015
    2  1

    There are few ways to determine the day of the year.  The first way I show here is a bug.

    with(Finance):
    FormatDate("Aug-4-2015","%j")
                                                 "%j"

    it was described on the help page that the options can be of the form of any described in StringTools[ParseTime] of which %j did not work.  Fortunately there's other ways.

    with(StringTools):
    ParseTime("%m-%d-%Y", "8-4-2015"):-yearDay
                                                                           216

    with(Finance):
    DayCount("jan-1-2015", "aug-4-2015")+1
                                                                      216

     

     

                                                                  

     

     

    Sending Emails from the Maple Command Line

    by: Samir Khan 2151 Maple
    url
    October 30 2015
    2  6

    Some time ago, @marc005 asked how he could send an email from the Maple command line.

    Why would you want to do this? Using Maple's functionality, you could programatically construct an email - perhaps with the results of a computation - and email it yourself or someone else.

    I originally posted a solution that involved communicating with a locally-installed SMTP server using the Sockets package. But of course, you need to set up an SMTP server and ensure the appropriate ports are open.

    I recently found a better solution. Mailgun (http://mailgun.com) is a free email delivery service with an web-based API. You can communicate with this API via the URL package; simply send Mailgun a URL:-Post() message that contains account-specific information, and the text of your email.

    The general steps and Maple commands are given below, and you can download the worksheet here.

    Note: Maplesoft have no affiliation with Mailgun.

    Step 1:
    Sign up for a free Mailgun account.

    Step 2:
    In your Mailgun account, go to the Domains section - it should look like the screengrab below (account-specific information has been blanked).

    Note down the API Base URL and the API key.

    • the API Base URL looks like https://api.mailgun.net/v3/sandboxXXXXXXXXXXXXXXXXXXXXXXXXXXX.mailgun.org.  
    • •the API Key looks like key-XXXXXXXXXXXXXXXXXXXXXXXXXX

    Step 3:

    In Maple, define strings containing your own API Base URL and API Key. Also, define the recipient's email address, the email you want the recipient to reply to, the email subject and email body.

    >restart:
    >APIBaseURL := "https://api.mailgun.net/v3/sandboxXXXXXXXXXXXXXXXXXXXXXXXX.mailgun.org":
    >APIKey:="key-XXXXXXXXXXXXXXXXXXXXXXXX":
    >toEmail := "xxxxx@xxxxxx.com":
    >fromEmail:="First Last <FirstLast@Domain.com>":
    >subject:= "Email Subject Goes Here":
    >emailBody := "I'd rather have a bottle in front of me than a frontal lobotomy":

    Step 4:
    Run the following code

    > URL:-Post(cat(APIBaseURL,"/messages"),[
    "from"=fromEmail,"to"=toEmail,
    "subject"=subject,
    "text"= emailBody],
    user="api",password=APIKey);

    If you've successfully sent the email, you should see something like this (account-specific information is blanked out)

    You can also send HTML emails by replacing the "text" line with "html" = str, where str is a string with your HTML code.

    Mac OS X 10.11

    by: eithne 2072 Maple , MapleSim
    osx
    October 23 2015
    0  1

    For those who are running Maple and/or MapleSim on the Mac, and who may have missed an earlier question on this site, we wanted to let you know that there are some problems with running Maple and MapleSim on the new Mac OS, Mac OS X 10.11 (El Capitan). We’re working on a solution, which we expect will be ready in a few weeks. We’ll keep you posted, but in the meantime, please delay updating your Mac OS for now to avoid problems.

     

    eithne

    MaplePrimes Update - New Tools for Spam, Saving...

    by: Bryon 1728 MaplePrimes
    October 21 2015
    4  2

    Earlier today, we published some changes to MaplePrimes in order to add some new features that have been requested by our users.

    Delete as Spam

    Users with a reputation score above 500 have always had the ability to remove inappropriate or spam posts using the Delete function. As of this update, a Delete as Spam feature has been added that provides some additional capabilities targeted directly at the spam messages that are posted on MaplePrimes. When this feature is used, the following occurs:

    • The message is deleted
    • The author's account is marked as a spam account and blocked
    • All other messages from the author are removed

    Since this is a powerful feature, there are also some safeguards in place to prevent accidents from occurring:

    • Only users with a reputation less than 10 can be removed in this way
    • In the event that content is ever inadvertently removed, MaplePrimes administrators have the ability to restore content removed in this way.

    Hopefully, these features make it much more difficult for spam accounts to get traction on MaplePrimes, and also reduce the time involved in policing the spam that is posted.

    Draft Messages

    You now have the ability to save a message as a draft so that you can come back to it later. All draft messages are saved in your profile and can be managed by clicking the account option at the top of every page. 

    Advanced Search

    Some powerful searching capabilities have been added to MaplePrimes with the addition of a new Advanced Search feature. Using this search allows you to quickly find questions or posts within a particular date range, by author and/or by tag.

     

    Save page as PDF

    We know that some of our members maintain personal archives of MaplePrimes content, and to make this easier, we have added a feature that generates a PDF copy of a page. This option is available in the More area that appears under every post and question.

     

    View your Replies

    Your MaplePrimes user profile page has always included archives of your posts, questions and answers. With today's update, replies are also available.

     

    In addition to these, some other minor changes & fixes were made.

    As always, thank you very much for your dedication to MaplePrimes and for your suggestions. We hope that you find these features useful and look forward to other ideas that you have!

    They're watching...

    by: Kat 232 MaplePrimes
    October 19 2015
    6  5

    Like most companies today, Maplesoft monitors its website traffic, including the traffic coming to MaplePrimes. This allows us to view statistical data such as how many total visits MaplePrimes gets, how many unique visitors it gets, what countries these visitors come from, how many questions are asked and answered, how many people read but never respond to posts, etc. 

    Recently one of our regular MaplePrimes users made the comment that MaplePrimes does not reach new Maple users. We found this comment interesting because our data and traffic numbers show a different trend. MaplePrimes gets unique visitors in the hundreds of thousands each year, and since its inception, it has welcomed unique visitors in the many millions. Based on these unique visitor numbers and the thousands of common searches specifically about Maple that people are doing, we can see that many of these unique visitors are in fact new Maple users looking for resources and support as they begin using Maple. Other visitors to MaplePrimes include people who use Google (or other search engines) to find an answer to a particular mathematics or engineering question, regardless of what mathematics software they choose to use, and Google points them to MaplePrimes. There are some popular posts that were written months, even years ago, that are still getting high visitor views today, showing the longevity of the information on MaplePrimes. 

    MaplePrimes gets the majority of its visible activity from a small number of extremely active members. In public user forums around the world, these types of members are given many names – power users, friendlies, evangelists. Every active public user forum has them. On MaplePrimes, it’s this small number of active members that are highly visible. But, what our traffic data reveals is the silent majority. These people, many of them repeat visitors, are quietly reviewing the questions and answers that our evangelists are posting. The silent majority of MaplePrimes visitors are the readers; they are the quiet consumers of information. For every person that writes, comments on, or likes a post, there are thousands more that read it. 

    Here are a few more MaplePrimes traffic data points for your reference:

    • MaplePrimes is very international and draws people from all around the world. Here are the top 10 countries where the most MaplePrimes visitors come from:
      1. USA
      2. India
      3. Canada
      4. Germany
      5. China
      6. United Kingdom
      7. Brazil
      8. Australia
      9. France
      10. Denmark
    • Here are the top 5 keywords people are using in their searches on MaplePrimes:
      1. Data from plot
      2. Physics
      3. Sprintf
      4. Size of plot
      5. Fractal
    • MaplePrimes is growing at a very fast rate: Traffic (visitors to the site) and membership size is growing at nearly double the pace it was last year. The total number of posts and questions this year is also much higher compared to the same timeframe last year. 
    • Our top 5 MaplePrimes members have each visited MaplePrimes more than 1200 times and viewed a combined total of more than 10,000 pages (that is total page views, not unique page views). Our top 25 MaplePrimes members have visited at least 250 times each (many of them nearly 1000 times each) and our top 50 MaplePrimes members have visited a combined total of over 23,000 times, visiting nearly 200,000 pages. Thank you! We’re glad you like it. :-)

     

    Mapleprimes updates and spam

    by: Christopher2222 6035
    October 19 2015
    2  1

    Why hasn't mapleprimes yet been able to solve the spam problem.  There was never this long a problem with Mapleprimes 1, every morning I find spam for the last few weeks. 

    Also the timing still has yet to be fixed.  After 1 hour the times for when a post was made goes back to 0 minutes ago.

     

     

    New features in Mathematica 10.3.0

    by: Markiyan Hirnyk 7819 Maple
    mathematica residue
    October 16 2015
    2  16

    Mathematica 10.3.0 was announced yesterday. This is the 6th release of Mathematica 10 during 16 months. I wonder its  MathematicaFunctionData and   FindFormula . At first sight, the former is an analog of FunctionAdvisor of Maple, but the latter isn't any analog. Also compare the outputs of

    Residue[Binomial[n,k],{n,-j}]

    (-1)^j/(j!*k!*(-j-k)!)

     and

    >`assuming`([residue(binomial(n, k), n = -j)], [integer, j > 0]);

                    residue(binomial(n, k), n = -j)
    Let us wait for Maple 2016.

     

    Exact solutions to Einstein's equations

    by: ecterrab 14630
    differential-geometry physics
    October 14 2015
    9  19

    “Exact solutions to Einstein’s equations” is one of those books that are difficult even to imagine: the authors reviewed more than 4,000 papers containing solutions to Einstein’s equations in the general relativity literature, collecting, classifying, discarding repetitions in disguise, and organizing the whole material into chapters according to the physical properties of these solutions. The book is already in its second edition and it is a monumental piece of work.

     

    As good as it is, however, the project resulted only in printed material, a textbook constituted of paper and ink. In 2006, when the DifferentialGeometry package was rewritten to enter the Maple library, one of the first things that passed through our minds was to bring the whole of “Exact solutions to Einstein’s equations” into Maple.

     

    It took some time to start but in 2010, for Maple 14, we featured the first 26 solutions from this book. In Maple 15 this number jumped to 61. For Maple 17 we decided to emphasize the general relativity functionality of the DifferentialGeometry package, and Maple 18 added 50 more, featuring in total 225 of these solutions - great! but still far from the whole thing …

     

    And this is when we decided to “step on the gas” - go for it, the whole book. One year later, working in collaboration with Denitsa Staicova from Bulgarian Academy of Sciences, Maple 2015 appeared with 330 solutions to Einstein’s equations. Today we have already implemented 492 solutions, and for the first time we can see the end of the tunnel: we are targeting finishing the whole book by the end of this year.

     

    Wow2! This is a terrific result. First, because these solutions are key in the area of general relativity, and at this point what we have in Maple is already the most thorough digitized database of solutions to Einstein’s equations in the world. Second, and not any less important, because within Maple this knowledge comes alive. The solutions are fully searcheable and are set by a simple call to the Physics:-g_  spacetime metric command, and that automatically sets the related coordinates, Christoffel symbols , Ricci  and Riemann  tensors, orthonormal and null tetrads , etc. All of this happens on the fly, and all the mathematics within the Maple library are ready to work with these solutions. Having everything come alive completely changes the game. The ability to search the database according to the physical properties of the solutions, their classification, or just by parts of keywords also makes the whole book concretely more useful.

     

    And, not only are these solutions to Einstein’s equations brought to life in a full-featured way through the Physics  package: they can also be reached through the DifferentialGeometry:-Library:-MetricSearch  applet. Almost all of the mathematical operations one can perform on them are also implemented as commands in DifferentialGeometry .

     

    Finally, in the Maple PDEtools package , we already have all the mathematical tools to start resolving the equivalence problem around these solutions. That is: to answer whether a new solution is or not new, or whether it can be obtained from an existing solution by transformations of coordinates of different kinds. And we are going for it.

     

    What follows is a basic illustration of what has already been implemented. As usual, in order to reproduce these results, you need to update your Physics library from the Maplesoft R&D Physics webpage.

     

    Load Physics , set the metric to Schwarzschild (and everything else automatically) in one go

    > 

    with(Physics)
    > 

    g_[sc]

    `Systems of spacetime Coordinates are: `*{X = (r, theta, phi, t)}

    		  

    `Default differentiation variables for d_, D_ and dAlembertian are: `*{X = (r, theta, phi, t)}

    		  

    `The Schwarzschild metric in coordinates `[r, theta, phi, t]

    		  

    `Parameters: `[m]

    		  

    g[mu, nu] = (Matrix(4, 4, {(1, 1) = r/(-r+2*m), (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) = (r-2*m)/r}))

    		 (1)

    And that is all we do :) Although the strength in Physics  is to compute with tensors using indicial notation, all of the tensor components and related properties of this metric are also derived on the fly (and no, they are not in any database). For instance these are the definition in terms of Christoffel symbols , and the covariant components of the Ricci tensor

    > 

    Ricci[definition]

    Physics:-Ricci[mu, nu] = Physics:-d_[alpha](Physics:-Christoffel[`~alpha`, mu, nu], [X])-Physics:-d_[nu](Physics:-Christoffel[`~alpha`, mu, alpha], [X])+Physics:-Christoffel[`~beta`, mu, nu]*Physics:-Christoffel[`~alpha`, beta, alpha]-Physics:-Christoffel[`~beta`, mu, alpha]*Physics:-Christoffel[`~alpha`, nu, beta]

    		 (2)
    > 

    Ricci[]

    Physics:-Ricci[mu, nu] = Matrix(%id = 18446744078179871670)

    		 (3)

    These are the 16 Riemann invariants  for Schwarzschild solution, using the formulas by Carminati and McLenaghan

    > 

    Riemann[invariants]

    r[0] = 0, r[1] = 0, r[2] = 0, r[3] = 0, w[1] = 6*m^2/r^6, w[2] = 6*m^3/r^9, m[1] = 0, m[2] = 0, m[3] = 0, m[4] = 0, m[5] = 0

    		 (4)

    The related Weyl scalars  in the context of the Newman-Penrose formalism

    > 

    Weyl[scalars]

    psi__0 = 0, psi__1 = 0, psi__2 = -m/r^3, psi__3 = 0, psi__4 = 0

    		 (5)

     

    These are the 2x2 matrix components of the Christoffel symbols of the second kind (that describe, in coordinates, the effects of parallel transport in curved surfaces), when the first of its three indices is equal to 1

    > 

    "Christoffel[~1,alpha,beta,matrix]"

    Physics:-Christoffel[`~1`, alpha, beta] = Matrix(%id = 18446744078160684686)

    		 (6)

    In Physics, the Christoffel symbols of the first kind are represented by the same object (not two commands) just by taking the first index covariant, as we do when computing with paper and pencil

    > 

    Christoffel[1, alpha, beta, matrix]

    Physics:-Christoffel[1, alpha, beta] = Matrix(%id = 18446744078160680590)

    		 (7)

    One could query the database, directly from the spacetime metrics, about the solutions (metrics) to Einstein's equations related to Levi-Civita, the Italian mathematician

    > 

    g_[civi]

    ____________________________________________________________

    		  

    [12, 16, 1] = ["Authors" = ["Bertotti (1959)", "Kramer (1978)", "Levi-Civita (1917)", "Robinson (1959)"], "PrimaryDescription" = "EinsteinMaxwell", "SecondaryDescription" = ["Homogeneous"]]

    		  

    ____________________________________________________________

    		  

    [12, 18, 1] = ["Authors" = ["Bertotti (1959)", "Kramer (1978)", "Levi-Civita (1917)", "Robinson (1959)"], "PrimaryDescription" = "EinsteinMaxwell", "SecondaryDescription" = ["Homogeneous"]]

    		  

    ____________________________________________________________

    		  

    [12, 19, 1] = ["Authors" = ["Bertotti (1959)", "Kramer (1978)", "Levi-Civita (1917)", "Robinson (1959)"], "PrimaryDescription" = "EinsteinMaxwell", "SecondaryDescription" = ["Homogeneous"]]

    		 (8)

    These solutions can be set in one go from the metrics command, just by indicating the number with which it appears in "Exact Solutions to Einstein's Equations"

    > 

    g_[[12, 16, 1]]

    `Systems of spacetime Coordinates are: `*{X = (t, x, theta, phi)}

    		  

    `Default differentiation variables for d_, D_ and dAlembertian are: `*{X = (t, x, theta, phi)}

    		  

    `The Bertotti (1959), Kramer (1978), Levi-Civita (1917), Robinson (1959) metric in coordinates `[t, x, theta, phi]

    		  

    `Parameters: `[k, kappa0, beta]

    		  

    g[mu, nu] = (Matrix(4, 4, {(1, 1) = -k^2*sinh(x)^2, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = k^2, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = k^2, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = k^2*sin(theta)^2}))

    		 (9)

    Automatically, everything gets set accordingly; these are the contravariant components of the related Ricci tensor

    > 

    "Ricci[~]"

    Physics:-Ricci[`~mu`, `~nu`] = Matrix(%id = 18446744078179869750)

    		 (10)

    One works with the Newman-Penrose formalism frequently using tetrads (local system of references); the Physics subpackage for this is Tetrads

    > 

    with(Tetrads)

    `Setting lowercaselatin letters to represent tetrad indices `

    		  

    0, "%1 is not a command in the %2 package", Tetrads, Physics

    		  

    0, "%1 is not a command in the %2 package", Tetrads, Physics

    		  

    [IsTetrad, NullTetrad, OrthonormalTetrad, SimplifyTetrad, TransformTetrad, e_, eta_, gamma_, l_, lambda_, m_, mb_, n_]

    		 (11)

    This is the tetrad related to the book's metric with number 12.16.1

    > 

    e_[]

    Physics:-Tetrads:-e_[a, mu] = Matrix(%id = 18446744078160685286)

    		 (12)

    One can check these directly; for instance this is the definition of the tetrad, where the right-hand side is the tetrad metric

    > 

    e_[definition]

    Physics:-Tetrads:-e_[a, mu]*Physics:-Tetrads:-e_[b, `~mu`] = Physics:-Tetrads:-eta_[a, b]

    		 (13)

    This shows that, for the components given by (12), the definition holds

    > 

    TensorArray(Physics:-Tetrads:-e_[a, mu]*Physics:-Tetrads:-e_[b, `~mu`] = Physics:-Tetrads:-eta_[a, b])

    Matrix(%id = 18446744078195401422)

    		 (14)

    One frequently works with a different signature and null tetrads; set that, and everything gets automatically recomputed for the metric 12.16.1 accordingly

    > 

    Setup(signature = "+---", tetradmetric = null)

    [signature = `+ - - -`, tetradmetric = {(1, 2) = 1, (3, 4) = -1}]

    		 (15)
    > 

    eta_[]

    eta[a, b] = (Matrix(4, 4, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (1, 4) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0, (3, 4) = -1, (4, 1) = 0, (4, 2) = 0, (4, 3) = -1, (4, 4) = 0}))

    		 (16)
    > 

    e_[]

    Physics:-Tetrads:-e_[a, mu] = Matrix(%id = 18446744078191417574)

    		 (17)
    > 

    TensorArray(Physics:-Tetrads:-e_[a, mu]*Physics:-Tetrads:-e_[b, `~mu`] = Physics:-Tetrads:-eta_[a, b])

    Matrix(%id = 18446744078191319390)

    		 (18)

    The related 16 Riemann invariant

    > 

    Riemann[invariants]

    r[0] = 0, r[1] = 1/k^4, r[2] = 0, r[3] = (1/4)/k^8, w[1] = 0, w[2] = 0, m[1] = 0, m[2] = 0, m[3] = 0, m[4] = 0, m[5] = 0

    		 (19)

    The ability to query rapidly, set things in one go, change everything again etc. are at this point fantastic. For instance, these are the metrics by Kaigorodov; next are those published in 1962

    > 

    g_[Kaigorodov]

    ____________________________________________________________

    		  

    [12, 34, 1] = ["Authors" = ["Kaigorodov (1962)", "Cahen (1964)", "Siklos (1981)", "Ozsvath (1987)"], "PrimaryDescription" = "Einstein", "SecondaryDescription" = ["Homogeneous"], "Comments" = ["All metrics with _epsilon <> 0 are equivalent to the cases _epsilon = +1, -1, _epsilon = 0 is anti-deSitter space"]]

    		  

    ____________________________________________________________

    		  

    [12, 35, 1] = ["Authors" = ["Kaigorodov (1962)", "Cahen (1964)", "Siklos (1981)", "Ozsvath (1987)"], "PrimaryDescription" = "Einstein", "SecondaryDescription" = ["Homogeneous", "SimpleTransitive"]]

    		 (20)
    > 

    g_[`1962`]

    ____________________________________________________________

    		  

    [12, 13, 1] = ["Authors" = ["Ozsvath, Schucking (1962)"], "PrimaryDescription" = "Vacuum", "SecondaryDescription" = ["Homogeneous", "PlaneWave"], "Comments" = ["geodesically complete, no curvature singularities"]]

    		  

    ____________________________________________________________

    		  

    [12, 14, 1] = ["Authors" = ["Petrov (1962)"], "PrimaryDescription" = "Vacuum", "SecondaryDescription" = ["Homogeneous", "SimpleTransitive"]]

    		  

    ____________________________________________________________

    		  

    [12, 34, 1] = ["Authors" = ["Kaigorodov (1962)", "Cahen (1964)", "Siklos (1981)", "Ozsvath (1987)"], "PrimaryDescription" = "Einstein", "SecondaryDescription" = ["Homogeneous"], "Comments" = ["All metrics with _epsilon <> 0 are equivalent to the cases _epsilon = +1, -1, _epsilon = 0 is anti-deSitter space"]]

    		  

    ____________________________________________________________

    		  

    [12, 35, 1] = ["Authors" = ["Kaigorodov (1962)", "Cahen (1964)", "Siklos (1981)", "Ozsvath (1987)"], "PrimaryDescription" = "Einstein", "SecondaryDescription" = ["Homogeneous", "SimpleTransitive"]]

    		  

    ____________________________________________________________

    		  

    [28, 16, 1] = ["Authors" = ["Robinson-Trautman (1962)"], "PrimaryDescription" = "Vacuum", "SecondaryDescription" = ["RobinsonTrautman"], "Comments" = ["The coordinate zeta is changed to xi", "AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)"]]

    		  

    ____________________________________________________________

    		  

    [28, 26, 1] = ["Authors" = ["Robinson, Trautman (1962)"], "PrimaryDescription" = "Vacuum", "SecondaryDescription" = ["RobinsonTrautman"], "Comments" = ["One can use the diffeo r -> -r and u -> -u to make the assumption r > 0", "The case _m = 0 is Stephani, [28, 16,1]", "The metric is type D at points where r = 3*_m/(xi1+xi2) and type II on either side of this hypersurface. For convenience, it is assumed that 3*_m - r*(xi1 + xi2) > 0", "AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)"]]

    		  

    ____________________________________________________________

    		  

    [28, 26, 2] = ["Authors" = ["Robinson, Trautman (1962)"], "PrimaryDescription" = "Vacuum", "SecondaryDescription" = ["RobinsonTrautman"], "Comments" = ["One can use the diffeo r -> -r and u -> -u to make the assumption r > 0", "The case _m = 0 is Stephani, [28, 16,1].", "AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)"]]

    		  

    ____________________________________________________________

    		  

    [28, 26, 3] = ["Authors" = ["Robinson, Trautman (1962)"], "PrimaryDescription" = "Vacuum", "SecondaryDescription" = ["RobinsonTrautman"], "Comments" = ["One can use the diffeo r -> -r and u -> -u to make the assumption r > 0", "The case _m = 0 is Stephani, [28, 16,1].", "AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)"]]

    		  

    ____________________________________________________________

    		  

    [28, 43, 1] = ["Authors" = ["Robinson, Trautman (1962)"], "PrimaryDescription" = "EinsteinMaxwell", "SecondaryDescription" = ["PureRadiation", "RobinsonTrautman"], "Comments" = ["h1(u) is the conjugate of h(u)"]]

    		 (21)

     

    The search can be done visually, by properties; this is the only solution in the database that is a Pure Ratiation solution, of Petrov Type "D", Plebanski-Petrov Type "O" and that has Isometry Dimension equal to 1:

    > 

    DifferentialGeometry:-Library:-MetricSearch()

     

    Set the solution, and everything related to work with it, in one go

    > 

    g_[[28, 74, 1]]

    `Systems of spacetime Coordinates are: `*{X = (u, eta, r, y)}

    		  

    `Default differentiation variables for d_, D_ and dAlembertian are: `*{X = (u, eta, r, y)}

    		  

    `The Frolov and Khlebnikov (1975) metric in coordinates `[u, eta, r, y]

    		  

    `Parameters: `[kappa0, m(u), b, d]

    		  

    "`Comments: `With _m(u) = constant, the metric is Ricci flat and becomes 28.24 in Stephani."

    		  

    g[mu, nu] = (Matrix(4, 4, {(1, 1) = (2*m(u)^3-6*m(u)^2*eta*r-r^2*(-6*eta^2+b)*m(u)+r^3*(-2*eta^3+b*eta+d))/(r*m(u)^2), (1, 2) = -r^2/m(u), (1, 3) = -1, (1, 4) = 0, (2, 1) = -r^2/m(u), (2, 2) = r^2/(-2*eta^3+b*eta+d), (2, 3) = 0, (2, 4) = 0, (3, 1) = -1, (3, 2) = 0, (3, 3) = 0, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = r^2*(-2*eta^3+b*eta+d)}))

    		 (22)

     

    The related Riemann invariants:

    > 

    Riemann[invariants]

    r[0] = 0, r[1] = 0, r[2] = 0, r[3] = 0, w[1] = 6*m(u)^2/r^6, w[2] = -6*m(u)^3/r^9, m[1] = 0, m[2] = 0, m[3] = 0, m[4] = 0, m[5] = 0

    		 (23)

    To conclude, how many solutions from the book have we already implemented?

    > 

    DifferentialGeometry:-Library:-Retrieve("Stephani", 1)

    [[8, 33, 1], [8, 34, 1], [12, 6, 1], [12, 7, 1], [12, 8, 1], [12, 8, 2], [12, 8, 3], [12, 8, 4], [12, 8, 5], [12, 8, 6], [12, 8, 7], [12, 8, 8], [12, 9, 1], [12, 9, 2], [12, 9, 3], [12, 12, 1], [12, 12, 2], [12, 12, 3], [12, 12, 4], [12, 13, 1], [12, 14, 1], [12, 16, 1], [12, 18, 1], [12, 19, 1], [12, 21, 1], [12, 23, 1], [12, 23, 2], [12, 23, 3], [12, 24.1, 1], [12, 24.2, 1], [12, 24.3, 1], [12, 26, 1], [12, 27, 1], [12, 28, 1], [12, 29, 1], [12, 30, 1], [12, 31, 1], [12, 32, 1], [12, 34, 1], [12, 35, 1], [12, 36, 1], [12, 37, 1], [12, 37, 2], [12, 37, 3], [12, 37, 4], [12, 37, 5], [12, 37, 6], [12, 37, 7], [12, 37, 8], [12, 37, 9], [12, 38, 1], [12, 38, 2], [12, 38, 3], [12, 38, 4], [12, 38, 5], [13, 2, 1], [13, 2, 2], [13, 2, 3], [13, 7, 1], [13, 7, 2], [13, 7, 3], [13, 7, 4], [13, 7, 5], [13, 7, 6], [13, 7, 7], [13, 7, 8], [13, 14, 1], [13, 14, 2], [13, 14, 3], [13, 19, 1], [13, 31, 1], [13, 32, 1], [13, 46, 1], [13, 48, 1], [13, 49, 1], [13, 49, 2], [13, 51, 1], [13, 53, 1], [13, 59, 1], [13, 59, 2], [13, 60, 1], [13, 60, 2], [13, 60, 3], [13, 60, 4], [13, 60, 5], [13, 60, 6], [13, 60, 7], [13, 60, 8], [13, 61, 1], [13, 61, 2], [13, 62, 1], [13, 62, 2], [13, 62, 4], [13, 62, 6], [13, 63, 1], [13, 63, 2], [13, 63, 3], [13, 63, 4], [13, 64, 1], [13, 64, 2], [13, 64, 3], [13, 64, 4], [13, 65, 1], [13, 69, 1], [13, 71, 1], [13, 72, 1], [13, 73, 1], [13, 74, 1], [13, 74, 2], [13, 74, 3], [13, 76, 1], [13, 77, 1], [13, 77, 2], [13, 79, 1], [13, 79, 2], [13, 80, 1], [13, 81, 1], [13, 83, 1], [13, 84, 1], [13, 84, 2], [13, 84, 3], [13, 85, 1], [13, 85, 2], [13, 86, 1], [13, 87, 1], [14, 6.1, 1], [14, 6.2, 1], [14, 6.3, 1], [14, 7, 1], [14, 8.1, 1], [14, 8.2, 1], [14, 8.3, 1], [14, 9.1, 1], [14, 9.2, 1], [14, 10, 1], [14, 10, 2], [14, 12, 1], [14, 12, 2], [14, 12, 3], [14, 14, 1], [14, 14, 2], [14, 15, 1], [14, 15.1, 2], [14, 15.2, 2], [14, 15.3, 2], [14, 16, 1], [14, 16, 2], [14, 17, 1], [14, 18, 1], [14, 18, 2], [14, 19, 1], [14, 20, 1], [14, 21, 1], [14, 21, 2], [14, 21, 3], [14, 22, 1], [14, 23, 1], [14, 24, 1], [14, 25, 1], [14, 26, 1], [14, 26, 2], [14, 26, 3], [14, 26, 4], [14, 27, 1], [14, 28, 1], [14, 28, 2], [14, 28, 3], [14, 29, 1], [14, 30, 1], [14, 31, 1], [14, 32, 1], [14, 33, 1], [14, 35, 1], [14, 37, 1], [14, 38, 1], [14, 38, 2], [14, 38, 3], [14, 39, 1], [14, 39, 2], [14, 39, 3], [14, 39, 4], [14, 39, 5], [14, 39, 6], [14, 40, 1], [14, 41, 1], [14, 42, 1], [14, 46, 1], [15, 3, 1], [15, 3, 2], [15, 4, 1], [15, 4, 2], [15, 4, 3], [15, 9, 1], [15, 10, 1], [15, 12, 1], [15, 12, 2], [15, 12, 3], [15, 12, 4], [15, 12, 5], [15, 12, 6], [15, 17, 1], [15, 17, 2], [15, 17, 3], [15, 17, 4], [15, 18, 1], [15, 19, 1], [15, 19, 2], [15, 20, 1], [15, 21, 1], [15, 21, 2], [15, 22, 1], [15, 23, 1], [15, 23, 2], [15, 24, 1], [15, 24, 2], [15, 25, 1], [15, 25, 2], [15, 26, 1], [15, 26, 2], [15, 27, 1], [15, 27, 2], [15, 27, 3], [15, 27, 4], [15, 27, 5], [15, 27, 6], [15, 27, 7], [15, 27, 8], [15, 28, 1], [15, 29, 1], [15, 30, 1], [15, 31, 1], [15, 32, 1], [15, 34, 1], [15, 34, 2], [15, 34, 3], [15, 43, 1], [15, 43, 2], [15, 43, 3], [15, 50, 1], [15, 50, 2], [15, 50, 3], [15, 50, 4], [15, 50, 5], [15, 50, 6], [15, 62, 1], [15, 62, 2], [15, 62, 3], [15, 63, 1], [15, 63, 2], [15, 63, 3], [15, 65, 1], [15, 65, 2], [15, 66, 1], [15, 66, 2], [15, 66, 3], [15, 75, 1], [15, 75, 2], [15, 75, 3], [15, 77, 1], [15, 77, 2], [15, 77, 3], [15, 78, 1], [15, 79, 1], [15, 81, 1], [15, 81, 2], [15, 81, 3], [15, 82, 1], [15, 82, 2], [15, 82, 3], [15, 83, 1.1], [15, 83, 1.2], [15, 83, 2], [15, 83, 3.1], [15, 83, 3.2], [15, 83, 4], [15, 84, 1], [15, 85, 1], [15, 85, 2], [15, 85, 3], [15, 86, 1], [15, 86, 2], [15, 86, 3], [15, 87, 1], [15, 87, 2], [15, 87, 3], [15, 87, 4], [15, 87, 5], [15, 88, 1], [15, 89, 1], [15, 90, 1], [16, 1, 1], [16, 1, 2], [16, 1, 3], [16, 1, 4], [16, 1, 5], [16, 1, 6], [16, 1, 7], [16, 1, 8], [16, 1, 9], [16, 1, 10], [16, 1, 11], [16, 1, 12], [16, 1, 13], [16, 1, 14], [16, 1, 15], [16, 1, 16], [16, 1, 17], [16, 1, 18], [16, 1, 19], [16, 1, 20], [16, 1, 21], [16, 1, 22], [16, 1, 23], [16, 1, 24], [16, 1, 25], [16, 1, 26], [16, 1, 27], [16, 14, 1], [16, 14, 2], [16, 14, 3], [16, 14, 4], [16, 14, 5], [16, 14, 6], [16, 14, 7], [16, 14, 8], [16, 14, 9], [16, 14, 10], [16, 14, 11], [16, 14, 12], [16, 14, 13], [16, 14, 14], [16, 14, 15], [16, 14, 16], [16, 14, 17], [16, 14, 18], [16, 14, 19], [16, 14, 20], [16, 18, 1], [16, 19, 1], [16, 20, 1], [16, 22, 1], [16, 24, 1], [16, 24, 2], [16, 43, 1], [16, 45, 1], [16, 45, 2], [16, 46, 1], [16, 47, 1], [16, 50, 1], [16, 51, 1], [16, 54, 1], [16, 61, 1], [16, 63, 1], [16, 66, 1], [16, 66, 2], [16, 66, 3], [16, 67, 1], [16, 71, 1], [16, 72, 1], [16, 73, 1], [16, 74, 1], [16, 75, 1], [16, 76, 1], [16, 77, 1], [16, 77, 2], [16, 77, 3], [16, 78, 1], [17, 4, 1], [17, 4, 2], [17, 5, 1], [17, 9, 1], [17, 14, 1], [17, 15, 1], [17, 15, 2], [17, 16, 1], [17, 20, 1], [17, 22, 1], [17, 23, 1], [17, 24, 1], [17, 24, 2], [17, 26, 1], [17, 27, 1], [17, 27, 2], [17, 28, 1], [17, 28, 2], [17, 29, 1], [17, 29, 2], [17, 30, 1], [17, 31, 1], [18, 2, 1], [18, 2, 2], [18, 2, 3], [18, 2, 4], [18, 2, 5], [18, 2, 6], [18, 2, 7], [18, 2, 8], [18, 48, 1], [18, 48, 2], [18, 49, 1], [18, 50, 1], [18, 64, 1], [18, 64, 2], [18, 64, 3], [18, 65, 1], [18, 66, 1], [18, 67, 1], [18, 71, 1], [18, 75, 1], [19, 17, 1], [19, 17, 2], [19, 21, 1], [20, 3, 1], [20, 4, 1], [20, 5, 1], [20, 7, 1], [20, 8, 1], [20, 9, 1], [20, 10, 1], [20, 11, 1], [20, 12, 1], [20, 13, 1], [20, 15, 1], [20, 16, 1], [20, 17, 1], [20, 20, 1], [20, 21, 1], [20, 23, 1], [20, 27, 1], [20, 28, 1], [20, 29, 1], [20, 32, 1], [20, 34, 1], [20, 36, 1], [20, 38, 1], [20, 38, 2], [20, 38, 3], [20, 44, 1], [20, 46, 1], [20, 54, 1], [20, 57, 1], [20, 57, 2], [28, 16, 1], [28, 17, 1], [28, 21, 1], [28, 21, 2], [28, 21, 3], [28, 21, 4], [28, 21, 5], [28, 21, 6], [28, 21, 7], [28, 24, 1], [28, 25, 1], [28, 26, 1], [28, 26, 2], [28, 26, 3], [28, 41, 1], [28, 43, 1], [28, 44, 1], [28, 44, 2], [28, 44, 3], [28, 44, 4], [28, 44, 5], [28, 44, 6], [28, 45, 1], [28, 45, 2], [28, 46, 1], [28, 46, 2], [28, 53, 1], [28, 53, 2], [28, 55, 1], [28, 55, 2], [28, 56.1, 1], [28, 56.2, 2], [28, 56.2, 3], [28, 56.3, 1], [28, 56.4, 1], [28, 56.5, 1], [28, 56.6, 1], [28, 58.2, 1], [28, 58.3, 1], [28, 58.3, 2], [28, 58.4, 1], [28, 60, 1], [28, 61, 1], [28, 64, 1], [28, 66, 1], [28, 67, 1], [28, 68, 1], [28, 72, 1], [28, 73, 1], [28, 74, 1]]

    		 (24)
    > 

    nops([[8, 33, 1], [8, 34, 1], [12, 6, 1], [12, 7, 1], [12, 8, 1], [12, 8, 2], [12, 8, 3], [12, 8, 4], [12, 8, 5], [12, 8, 6], [12, 8, 7], [12, 8, 8], [12, 9, 1], [12, 9, 2], [12, 9, 3], [12, 12, 1], [12, 12, 2], [12, 12, 3], [12, 12, 4], [12, 13, 1], [12, 14, 1], [12, 16, 1], [12, 18, 1], [12, 19, 1], [12, 21, 1], [12, 23, 1], [12, 23, 2], [12, 23, 3], [12, 24.1, 1], [12, 24.2, 1], [12, 24.3, 1], [12, 26, 1], [12, 27, 1], [12, 28, 1], [12, 29, 1], [12, 30, 1], [12, 31, 1], [12, 32, 1], [12, 34, 1], [12, 35, 1], [12, 36, 1], [12, 37, 1], [12, 37, 2], [12, 37, 3], [12, 37, 4], [12, 37, 5], [12, 37, 6], [12, 37, 7], [12, 37, 8], [12, 37, 9], [12, 38, 1], [12, 38, 2], [12, 38, 3], [12, 38, 4], [12, 38, 5], [13, 2, 1], [13, 2, 2], [13, 2, 3], [13, 7, 1], [13, 7, 2], [13, 7, 3], [13, 7, 4], [13, 7, 5], [13, 7, 6], [13, 7, 7], [13, 7, 8], [13, 14, 1], [13, 14, 2], [13, 14, 3], [13, 19, 1], [13, 31, 1], [13, 32, 1], [13, 46, 1], [13, 48, 1], [13, 49, 1], [13, 49, 2], [13, 51, 1], [13, 53, 1], [13, 59, 1], [13, 59, 2], [13, 60, 1], [13, 60, 2], [13, 60, 3], [13, 60, 4], [13, 60, 5], [13, 60, 6], [13, 60, 7], [13, 60, 8], [13, 61, 1], [13, 61, 2], [13, 62, 1], [13, 62, 2], [13, 62, 4], [13, 62, 6], [13, 63, 1], [13, 63, 2], [13, 63, 3], [13, 63, 4], [13, 64, 1], [13, 64, 2], [13, 64, 3], [13, 64, 4], [13, 65, 1], [13, 69, 1], [13, 71, 1], [13, 72, 1], [13, 73, 1], [13, 74, 1], [13, 74, 2], [13, 74, 3], [13, 76, 1], [13, 77, 1], [13, 77, 2], [13, 79, 1], [13, 79, 2], [13, 80, 1], [13, 81, 1], [13, 83, 1], [13, 84, 1], [13, 84, 2], [13, 84, 3], [13, 85, 1], [13, 85, 2], [13, 86, 1], [13, 87, 1], [14, 6.1, 1], [14, 6.2, 1], [14, 6.3, 1], [14, 7, 1], [14, 8.1, 1], [14, 8.2, 1], [14, 8.3, 1], [14, 9.1, 1], [14, 9.2, 1], [14, 10, 1], [14, 10, 2], [14, 12, 1], [14, 12, 2], [14, 12, 3], [14, 14, 1], [14, 14, 2], [14, 15, 1], [14, 15.1, 2], [14, 15.2, 2], [14, 15.3, 2], [14, 16, 1], [14, 16, 2], [14, 17, 1], [14, 18, 1], [14, 18, 2], [14, 19, 1], [14, 20, 1], [14, 21, 1], [14, 21, 2], [14, 21, 3], [14, 22, 1], [14, 23, 1], [14, 24, 1], [14, 25, 1], [14, 26, 1], [14, 26, 2], [14, 26, 3], [14, 26, 4], [14, 27, 1], [14, 28, 1], [14, 28, 2], [14, 28, 3], [14, 29, 1], [14, 30, 1], [14, 31, 1], [14, 32, 1], [14, 33, 1], [14, 35, 1], [14, 37, 1], [14, 38, 1], [14, 38, 2], [14, 38, 3], [14, 39, 1], [14, 39, 2], [14, 39, 3], [14, 39, 4], [14, 39, 5], [14, 39, 6], [14, 40, 1], [14, 41, 1], [14, 42, 1], [14, 46, 1], [15, 3, 1], [15, 3, 2], [15, 4, 1], [15, 4, 2], [15, 4, 3], [15, 9, 1], [15, 10, 1], [15, 12, 1], [15, 12, 2], [15, 12, 3], [15, 12, 4], [15, 12, 5], [15, 12, 6], [15, 17, 1], [15, 17, 2], [15, 17, 3], [15, 17, 4], [15, 18, 1], [15, 19, 1], [15, 19, 2], [15, 20, 1], [15, 21, 1], [15, 21, 2], [15, 22, 1], [15, 23, 1], [15, 23, 2], [15, 24, 1], [15, 24, 2], [15, 25, 1], [15, 25, 2], [15, 26, 1], [15, 26, 2], [15, 27, 1], [15, 27, 2], [15, 27, 3], [15, 27, 4], [15, 27, 5], [15, 27, 6], [15, 27, 7], [15, 27, 8], [15, 28, 1], [15, 29, 1], [15, 30, 1], [15, 31, 1], [15, 32, 1], [15, 34, 1], [15, 34, 2], [15, 34, 3], [15, 43, 1], [15, 43, 2], [15, 43, 3], [15, 50, 1], [15, 50, 2], [15, 50, 3], [15, 50, 4], [15, 50, 5], [15, 50, 6], [15, 62, 1], [15, 62, 2], [15, 62, 3], [15, 63, 1], [15, 63, 2], [15, 63, 3], [15, 65, 1], [15, 65, 2], [15, 66, 1], [15, 66, 2], [15, 66, 3], [15, 75, 1], [15, 75, 2], [15, 75, 3], [15, 77, 1], [15, 77, 2], [15, 77, 3], [15, 78, 1], [15, 79, 1], [15, 81, 1], [15, 81, 2], [15, 81, 3], [15, 82, 1], [15, 82, 2], [15, 82, 3], [15, 83, 1.1], [15, 83, 1.2], [15, 83, 2], [15, 83, 3.1], [15, 83, 3.2], [15, 83, 4], [15, 84, 1], [15, 85, 1], [15, 85, 2], [15, 85, 3], [15, 86, 1], [15, 86, 2], [15, 86, 3], [15, 87, 1], [15, 87, 2], [15, 87, 3], [15, 87, 4], [15, 87, 5], [15, 88, 1], [15, 89, 1], [15, 90, 1], [16, 1, 1], [16, 1, 2], [16, 1, 3], [16, 1, 4], [16, 1, 5], [16, 1, 6], [16, 1, 7], [16, 1, 8], [16, 1, 9], [16, 1, 10], [16, 1, 11], [16, 1, 12], [16, 1, 13], [16, 1, 14], [16, 1, 15], [16, 1, 16], [16, 1, 17], [16, 1, 18], [16, 1, 19], [16, 1, 20], [16, 1, 21], [16, 1, 22], [16, 1, 23], [16, 1, 24], [16, 1, 25], [16, 1, 26], [16, 1, 27], [16, 14, 1], [16, 14, 2], [16, 14, 3], [16, 14, 4], [16, 14, 5], [16, 14, 6], [16, 14, 7], [16, 14, 8], [16, 14, 9], [16, 14, 10], [16, 14, 11], [16, 14, 12], [16, 14, 13], [16, 14, 14], [16, 14, 15], [16, 14, 16], [16, 14, 17], [16, 14, 18], [16, 14, 19], [16, 14, 20], [16, 18, 1], [16, 19, 1], [16, 20, 1], [16, 22, 1], [16, 24, 1], [16, 24, 2], [16, 43, 1], [16, 45, 1], [16, 45, 2], [16, 46, 1], [16, 47, 1], [16, 50, 1], [16, 51, 1], [16, 54, 1], [16, 61, 1], [16, 63, 1], [16, 66, 1], [16, 66, 2], [16, 66, 3], [16, 67, 1], [16, 71, 1], [16, 72, 1], [16, 73, 1], [16, 74, 1], [16, 75, 1], [16, 76, 1], [16, 77, 1], [16, 77, 2], [16, 77, 3], [16, 78, 1], [17, 4, 1], [17, 4, 2], [17, 5, 1], [17, 9, 1], [17, 14, 1], [17, 15, 1], [17, 15, 2], [17, 16, 1], [17, 20, 1], [17, 22, 1], [17, 23, 1], [17, 24, 1], [17, 24, 2], [17, 26, 1], [17, 27, 1], [17, 27, 2], [17, 28, 1], [17, 28, 2], [17, 29, 1], [17, 29, 2], [17, 30, 1], [17, 31, 1], [18, 2, 1], [18, 2, 2], [18, 2, 3], [18, 2, 4], [18, 2, 5], [18, 2, 6], [18, 2, 7], [18, 2, 8], [18, 48, 1], [18, 48, 2], [18, 49, 1], [18, 50, 1], [18, 64, 1], [18, 64, 2], [18, 64, 3], [18, 65, 1], [18, 66, 1], [18, 67, 1], [18, 71, 1], [18, 75, 1], [19, 17, 1], [19, 17, 2], [19, 21, 1], [20, 3, 1], [20, 4, 1], [20, 5, 1], [20, 7, 1], [20, 8, 1], [20, 9, 1], [20, 10, 1], [20, 11, 1], [20, 12, 1], [20, 13, 1], [20, 15, 1], [20, 16, 1], [20, 17, 1], [20, 20, 1], [20, 21, 1], [20, 23, 1], [20, 27, 1], [20, 28, 1], [20, 29, 1], [20, 32, 1], [20, 34, 1], [20, 36, 1], [20, 38, 1], [20, 38, 2], [20, 38, 3], [20, 44, 1], [20, 46, 1], [20, 54, 1], [20, 57, 1], [20, 57, 2], [28, 16, 1], [28, 17, 1], [28, 21, 1], [28, 21, 2], [28, 21, 3], [28, 21, 4], [28, 21, 5], [28, 21, 6], [28, 21, 7], [28, 24, 1], [28, 25, 1], [28, 26, 1], [28, 26, 2], [28, 26, 3], [28, 41, 1], [28, 43, 1], [28, 44, 1], [28, 44, 2], [28, 44, 3], [28, 44, 4], [28, 44, 5], [28, 44, 6], [28, 45, 1], [28, 45, 2], [28, 46, 1], [28, 46, 2], [28, 53, 1], [28, 53, 2], [28, 55, 1], [28, 55, 2], [28, 56.1, 1], [28, 56.2, 2], [28, 56.2, 3], [28, 56.3, 1], [28, 56.4, 1], [28, 56.5, 1], [28, 56.6, 1], [28, 58.2, 1], [28, 58.3, 1], [28, 58.3, 2], [28, 58.4, 1], [28, 60, 1], [28, 61, 1], [28, 64, 1], [28, 66, 1], [28, 67, 1], [28, 68, 1], [28, 72, 1], [28, 73, 1], [28, 74, 1]])

    492

    		 (25)
    > 

    NULL

    :)


    Download Exact_Solutions_to_Einstein_Equations.mw

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

    In-depth Design Using Multidimensional Visualizati...

    by: Alan Elbanhawy 65 Maple
    October 07 2015
    5  0

    The engineering design process involves numerous steps that allow the engineer to reach his/her final design objectives to the best of his/her ability. This process is akin to creating a fine sculpture or a great painting where different approaches are explored and tested, then either adopted or abandoned in favor of better or more developed and fine-tuned ones. Consider the x-ray of an oil painting. X-rays of the works of master artists reveal the thought and creative processes of their minds as they complete the work. I am sure that some colleagues may disagree with the comparison of our modern engineering designs to art masterpieces, but let me ask you to explore the innovations and their brilliant forms, and maybe you will agree with me even a little bit.

    Design Process

    Successful design engineers must have the very best craft, knowledge and experience to generate work that is truly worthy of being incorporated in products that sell in the tens, or even hundreds, of millions. This is presently achieved by having cross-functional teams of engineers work on a design, allowing cross checking and several rounds of reviews, followed by multiple prototypes and exhaustive preproduction testing until the team reaches a collective conclusion that “we have a design.” This is then followed by the final design review and release of the product. This necessary and vital approach is clearly a time consuming and costly process. Over the years I have asked myself several times, “Did I explore every single detail of the design fully”? “Am I sure that this is the very best I can do?” And more importantly, “Does every component have the most fine-tuned value to render the best performance possible?” And invariably I am left with a bit of doubt. That brings me to a tool that has helped me in this regard.

    A Great New Tool

    I have used Maple for over 25 years to dig deeply into my designs and understand the interplay between a given set of parameters and the performance of the particular circuit I am working on. This has always given me a complete view of the problem at hand and solidly pointed me in the direction of the best possible solutions.

    In recent years, a new feature called “Explore” has been added to Maple. This amazing feature allows the engineer/researcher to peer very deeply into any formula and explore the interaction of EVERY variable in the formula. 

    Take for example the losses in the control MOSFET in a synchronous buck converter. In order to minimize these losses and maximize the power conversion efficiency, the most suitable MOSFET must be selected. With thousands of these devices being available in the market, a dozen of them are considered very close to the best at any given time. The real question then is, which one is really the very best amongst all of them? 

    There are two possible approaches - one, build an application prototype, test a random sample of each and choose the one that gives you the best efficiency.  Or, use an accurate mathematical model to calculate the losses of each and chose the best. The first approach lacks the variability of each parameter due to the six sigma statistical distribution where it is next to impossible to get a device laying on the outer limits of the distribution. That leaves the mathematical model approach. If you take this route, you can have built-in tolerances in the equations to accommodate all the variabilities and use a simplified equation for the control MOSFET losses (clearly you can use a very detailed model should you chose to) to explore these losses. Luckily you can explore the losses using the Explore function in Maple.

    The figure below shows a three dimensional plot, plus five other variables in the formula that the user can change using sliders that cover the range of values of interest including Minima and Maxima, while observing in real time the effects of the change on the power loss.

    This means that by changing the values of any set of variables, you can observe their effect on the function. To put it simply, this single feature helps you replace dozens of plots with just one, saving you precious time and cost in fine-tuning your design. In my opinion, this is equivalent to an eight-dimensional/axes plot.

    I used this amazing feature in the last few weeks and I was delighted at how simple it is to use and how much it simplifies the study of my approach and my components selection, in record times!

    Resequencing Concepts and Skills via Maple's Clickable...

    by: kkoserski 230
    October 06 2015
    0  0

    This October 21st, Maplesoft will be hosting a full-production, live streaming webinar featuring Dr. Robert Lopez, Emeritus Professor and Maple Fellow. You might have caught Dr. Lopez's Clickable Calculus webinar series before, but this webinar is your chance to meet the man behind the voice and watch him use Clickable Math techniques live!

    In this webinar, Dr. Lopez will present examples of what "resequencing concepts and skills" looks like when implemented with Maple's point-and-click syntax-free paradigm. He will demonstrate how Maple can not only be used to elucidate the concept, but also, how it can be used to illustrate and implement the manipulations that ultimately the student must master.

    Click here for more information and registration.

    Rearranging the “expression of equations”: Further...

    by: rlopez 3025
    October 05 2015
    4  5

    A wealth of knowledge is on display in MaplePrimes as our contributors share their expertise and step up to answer others’ queries. This post picks out one such response and further elucidates the answers to the posted question. I hope these explanations appeal to those of our readers who might not be familiar with the techniques embedded in the original responses.

    Before I begin, a quick note that the content below was primarily created by one of our summer interns, Pia, with guidance and advice from me.

    The Question: Rearranging the expression of equations

    SY G wanted to be able to re-write an equation in terms of different variables.  SY G presented this example: 

    I have the following two equations:

    x1 = a-y1-d*y2;
    x2 = a-y2-d*y1;

    I wish to express the first equation in terms of y1 and x2, so that

    x1 = c - b*y1+d*x2;

    where c=a-a*d and b=1-d^2. How can I get Maple to rearrange the original equation x1 in term of y1, x2, c and b?

    This question was answered by nm who provided code with a systematic approach:

    restart;
    eq1:=x1=a-y1-d*y2:
    eq2:=x2=a-y2-d*y1:
    z:=expand(subs(y2=solve(eq2,y2),eq1)):
    z:=algsubs((a-a*d)=c,z):
    algsubs((1-d^2)=b,z);

    On the other hand, Carl Love answered this enquiry using a more direct and simple code:

    simplify(x1=a-y1-d*y2, {a-y2-d*y1= x2, 1-d^2= b, a-a*d= c});

    Let’s talk more about the expand, algsubs, subs, and simplify commands

    First let’s take a look at the method nm used to solve the problem using the commands expand, subs, solve and algsubs.

    The expand command, expand(expr, expr1, expr2, ..., exprn), distributes products over sums. This is done for all polynomials. For quotients of polynomials, only sums in the numerator are expanded; products and powers are left alone.

    The solve command, solve(equations, variables), solves one or more equations or inequalities for their unknowns.

    The subs command, subs(x=a,expr), substitutes a for x in the expression expr.

    The function algsubs, algsubs(a = b, f),performs an algebraic substitution, replacing occurrences of a with b in the expression f.  It is a generalization of the subs command, which only handles syntactic substitution.

    Let’s tackle the Maple code written by nm step by step:

    1) restart;
    The restart command is used to clear Maple’s internal memory

    2)  eq1:=x1=a-y1-d*y2:
          eq2:=x2=a-y2-d*y1:
    The names eq1 and eq2 were assigned to the equations SY G provided.

    3) z:=expand(subs(y2=solve(eq2,y2),eq1)):
    A new variable, z, was created, which will end up being x1 written in the terms SY G wanted.

    • solve(eq2,y2)
      • the solve command was used to solve the expression eq2 for the variable y2.

    • subs(y2=solve(eq2,y2),eq1)
      • The subs command was used to replace in expression eq1, y2 as determined by the solve step. 

    • expand(subs(y2=solve(eq2,y2),eq1))
      • The expand command was used to distribute products over sums. Note: this step served to ensure that the final output looked exactly how SY G wanted.

    4) z:=algsubs((a-a*d)=c,z):
    First, nm equated a-a*d to c, so later the algsubs command could be applied to substitute the new variable c into the expression z.

    5) algsubs((1-d^2)=b,z);
    Again, nm equated 1-d^2 to b, so later the algsubs command could be applied to substitute the new variable b into the expression z.

    An alternate approach

    Now let us check out Carl Love’s approach. Carl Love uses the simplify command in conjunction with side relations.

    The simplify command has many calling sequences and one of them is the simplify(expr,eqns), that is known as simplify/siderels. A simplification of expr with respect to the side relations eqns is performed. The result is an expression which is mathematically equivalent toexpr but which is in normal form with respect to the specified side relations. Basically you are telling Maple to simplify the expression (expr) using the parameters (eqns) you gave to it.

     

    I hope that you find this useful. If there is a particular question on MaplePrimes that you would like further explained, please let me know. 

    Visual Gamut and RGB Color Spaces

    by: John Dolese 125 Maple 18
    pointplot3d colortools
    October 04 2015
    2  6

     Obtain the tri-stimulus XYZ values from the CIE Color matching functions.

     Show the gamut of maximum chroma for the standard observer model with a D65 Illuminant.

     Approximate the white point of a Planckian source and compare to D65.

     Translate the maximum chroma gamut in xy to Lab (CIE L*a*b*) for perceived gamut (Violet and Magenta come together)

     Map the RGB color cube of fully saturated color into Lab and compare to perceivable colors.

    10/6/15  Initial Document

    •12/28/15 Improve RGB gamut with more data points: Procedures added for RGB to Lab: Wavlength Colors now based on CIEDE2000 model for Lab.                   

     

     Here is the latest version of this document, the MSL_data must be in a directory set in the mw file;

    MSL_data.xlsx    Vision_RGB_Gamut.mw

    Modeling and analysis of mechanical systems with...

    by: Lenin Araujo Castillo 1790 Maple 2015
    engineering embedded-components
    October 04 2015
    2  0

    In this paper we will demonstrate the many differences of implementation in the modeling of mechanical systems using embedded components through Maplesoft. The mechanical systems are used for different tasks and therefore have different structure in its design; as to the nature of the used functional elements placed on them, they vary greatly. This diversity is reflected in approaches and practices in modeling.

    The following cases focus on mechanical components of the units manufacturing and processing machines. We can generate graphs for analysis using different dynamic pair ametros; all in real-time considerations in its manufacturing costs from the equations of conservation of energy.
    Therefore modeling with Maplesoft ensures the smooth optimum performance in mechanical systems, highlighting the sustainability criteria for other areas of engineering.

     

    XXXIII_Coloquio_SMP_2015.pdf

    XXXIII_Coloquio_UNASAM_2015.mw

    (in spanish)

    L.AraujoC.

     

     

    BesselJ evaluation problem

    by: mas1jd 20 Maple 2015
    infinity bessel
    October 02 2015
    1  4

    Apparently inconsistent behaviour of the BesselJ() function.

    Examples: BesselJ(-3, 0)  ... gives 0 (correct)

    but BesselJ(-3.0, 0), BesselJ(-3, 0.0)  and BesselJ(-3, 0.0) all give Float(infinity) (wrong! - should be 0.0)

    The problem seems to occur for all negative integer values of the first argument (the order) when the second argument is 0 or 0.0.

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