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MaplePrimes Posts are for sharing your experiences, techniques and opinions about Maple, MapleSim and related products, as well as general interests in math and computing.

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  • We assume that the radius of the outer stationary circle is  1. If we set the radius  x  of the inner stationary circle, all the other circles are uniquely determined by solving the system Sys.  Should be  x<=1/3 . If  x=1/3  then all the inner circles have a radius  1/3 . The following picture explains the meaning of symbols in the procedure Circles:

                                       

     

     

    Circles:=proc(x)

    local OO, O1, O2, O3, O4, O2x, O2y, O3x, O3y, OT, T1, T2, T3, s, t, dist, Sys, Sol, sol, y, u, v, z, C0, R0, P;

    uses plottools, plots;

    OO:=[0,0]: O1:=[x+y,0]: O2:=[O2x,O2y]: O3:=[O3x,O3y]: O4:=[-x-z,0]: OT:=[x+2*y-1,0]:

    T1:=(O2*~y+O1*~u)/~(y+u): T2:=(O3*~u+O2*~v)/~(u+v): T3:=(O4*~v+O3*~z)/~(v+z):

    solve({(T2-T1)[1]*(s-((T1+T2)/2)[1])+(T2-T1)[2]*(t-((T1+T2)/2)[2])=0, (T3-T2)[1]*(s-((T2+T3)/2)[1])+(T3-T2)[2]*(t-((T3+T2)/2)[2])=0}, {s,t}):

    assign(%);

    dist:=(A,B)->sqrt((B[1]-A[1])^2+(B[2]-A[2])^2):

    Sys:={dist(O1,O2)^2=(y+u)^2, dist(OO,O2)^2=(x+u)^2, dist(O2,O3)^2=(u+v)^2, dist(OO,O3)^2=(x+v)^2, dist(O3,O4)^2=(z+v)^2, x+y+z=1, dist(O2,OT)^2=(1-u)^2, dist(O3,OT)^2=(1-v)^2};

    Sol:=op~([allvalues([solve(Sys)])]);

    sol:=select(i->is(eval(convert([y>0,u>0,v>0,z>0,O2y>0,x<=y,u<=y,v<=u,z<=v],`and`),i)), Sol)[];

    assign(sol);

    O1:=[x+y,0]: O2:=[O2x,O2y]: O3:=[O3x,O3y]: O4:=[-x-z,0]: OT:=[x+2*y-1,0]:

    C0:=eval([s,t],sol);

    R0:=eval(dist(T1,C0),sol):

    P:=proc(phi)

    local eq, r1, r, R, Ot, El, i, S, s, t, P1, P2;

    uses plots,plottools;

    eq:=1-dist([r*cos(s),r*sin(s)],OT)=r-x;

    r1:=solve(eq,r);

    r:=eval(r1,s=phi);

    R[1]:=evalf(r-x);

    Ot[1]:=evalf([r*cos(phi),r*sin(phi)]);

    El:=plot([r1*cos(s),r1*sin(s),s=0..2*Pi],color="Green",thickness=3);

    for i from 2 to 6 do

    S:=[solve({1-dist(OT,[s,t])=dist(Ot[i-1],[s,t])-R[i-1], 1-dist(OT,[s,t])=dist(OO,[s,t])-x})];

    P1:=eval([s,t],S[1]); P2:=eval([s,t],S[2]);

    Ot[i]:=`if`(evalf(Ot[i-1][1]*P1[2]-Ot[i-1][2]*P1[1])>0,P1,P2);

    R[i]:=dist(Ot[i],OO)-x;

    od;

    display(El,seq(disk(Ot[k],0.012),k=1..6),circle(C0,R0,color=gold,thickness=3),circle([x+2*y-1,0],1, color=blue,thickness=4), circle(OO,x, color=red,thickness=4), seq(circle(Ot[k],R[k], thickness=3),k=1..6), scaling=constrained, axes=none);

    end proc:

    animate(P,[phi], phi=0..Pi, frames=120);

    end proc:  

     

    Example of use (I got  x=0.22  just by measuring the ruler displayed original animation):

    Circles(0.22);

                                   

     

     

    The curve on the following animation is an astroid (a special case of hypocycloid). See wiki for details. Hypocycloid procedure creates animation for any hypocycloid.  Parameters of the procedure: R is the radius of the outer circle, r is the radius of the inner circle.

    Hypocycloid:=proc(R,r)

    local A, B, f, g, F;

    uses plots,plottools;

    A:=circle(R,color=green,thickness=4):

    B:=display(circle([R-r,0],r,color=red,thickness=4),line([R-r,0],[R,0],color=red,thickness=4)):

    f:=t->plot([(R-r)*cos(s)+r*cos((R-r)/r*s),(R-r)*sin(s)-r*sin((R-r)/r*s),s=0..t],color=blue,thickness=4):

    g:=t->rotate(rotate(B,-R/r*t,[R-r,0]),t):

    F:=t->display(A,f(t),g(t),scaling=constrained):

    animate(F,[t], t=0..2*Pi*denom(R/r), frames=90);

    end proc:

     

    Examples of use:

    Hypocycloid(4,1); 

                                          

     

     

    Hypocycloid(5,3);

                                          

     

     

     Круги.mw

    The Saturday edition of our local newspaper (Waterloo Region Record) carries, as part of The PUZZLE Corner column, a weekly puzzle "STICKELERS" by Terry Stickels. Back on December 13, 2014, the puzzle was:

    What number comes next?

    1   4   18   96   600   4320   ?

    The solution given was the number 35280, obtained by setting k = 1 in the general term k⋅k!.

    On September 5, 2015, the column contained the puzzle:

    What number comes next?

    2  3  3  5  10  13  39  43  172  177  ?

    The proposed solution was the number 885, obtained as a10 from the recursion

    where a0 =2.

    As a youngster, one of my uncles delighted in teasing me with a similar question for the sequence 36, 9, 50, 55, 62, 71, 79, 18, 20. Ignoring the fact that there is a missing entry between 9 and 50, the next member of the sequence is "Bay Parkway," which is what 22nd Avenue is actually called in the Brooklyn neighborhood of my youth. These are subway stops on what was then called the West End line of the subway that went out to Stillwell Avenue in Coney Island.

    Armed with the skepticism inspired by this provincial chestnut, I looked at both of these puzzles with the attitude that the "next number" could be anything I chose it to be. After all, a sequence is a mapping from the (nonnegative) integers to the reals, and unless the mapping is completely specified, the function values are not well defined.

    Indeed, for the first puzzle, the polynomial f(x) interpolating the points


    (0, 1), (1, 4), (2, 18), (3, 93), (4, 600), (5, 4320)

    reproduces the first six members of the given sequence, and gives 18593 (not 35280) for f(7). In other words, the pattern k⋅k! is not a unique representation of the sequence, given just the first six members. The worksheet NextNumber derives the interpolating polynomial f and establishes that f(n) is an integer for every nonzero integer n.

    Likewise, for the second puzzle, the polynomial g(x) interpolating the points

    (1, 2) ,(2, 3) ,(3, 3) ,(4, 5) ,(5, 10) ,(6, 13) ,(7, 39) ,(8, 43), (9, 172) ,(10, 177)

    reproduces the first ten members of the given sequence, and gives -7331(not 885) for g(11). Once again, the pattern proposed as the "solution" is not unique, given that the worksheet NextNumber contains both g(x) and a proof that for integer n, all values of g(n) are integers.

    The upshot of these observations is that without some guarantee of uniqueness, questions like "what is the next number" are meaningless. It would be far better to pose such challenges with the words "Find a pattern for the given members of the following sequence" and warn that the function capturing that pattern might not be unique.

    I leave it to the interested reader to prove or disprove the following conjecture: Interpolate the first n terms of either sequence. The interpolating polynomial p will reproduce these n terms, but for k>n, p(k) will differ from the corresponding member of the sequence determined by the stated patterns. (Results of limited numerical experiments are consistent with the truth of this conjecture.)

    Attached: NextNumber.mw

    Bruce Jenkins is President of Ora Research, an engineering research and advisory service. Maplesoft commissioned him to examine how systems-driven engineering practices are being integrated into the early stages of product development, the results of which are available in a free whitepaper entitled System-Level Physical Modeling and Simulation. In this series of blog posts, Mr. Jenkins discusses the results of his research.

    This is the third entry in the series.

    My last post, System-level physical modeling and simulation: Adoption drivers vs. adoption constraints, described my firm’s research project to investigate the contemporary state of adoption and application of systems modeling software technologies, and their attendant methods and work processes, in the engineering design of off-highway equipment and mining machinery.

    In this project, I interviewed some half-dozen expert practitioners at leading manufacturers, including both engineering management and senior discipline leads, to identify key technological factors as well as business and competitive issues driving adoption and use of systems modeling at current levels.

    After identifying present-day adoption drivers as well as current constraints on adoption, finally I sought to learn practitioners’ visions, strategies and best practices for accelerating and institutionalizing the implementation and usage of systems modeling tools and practices in their organizations.

    I was strongly encouraged to find a wealth of avenues and opportunities for exploiting enterprise business drivers, current industry disruptions, and related internal realignments and change-management initiatives to help drive introduction—or proliferation—of these technologies and their associated new ways of working into engineering organizations:

    • Systems modeling essential to compete by creating differentiated products
    • Mechatronics revolution in off-highway equipment
    • Industry downturns and disruptions create opportunities for disruptive innovation
      • Opportunities to leverage change in underlying industry competitive dynamics
      • Mining industry down-cycle creates opportunity to innovate, find new ways of working
      • Some manufacturers are using current down-cycle in mining industry to change their product innovation strategy
    • Strategies of manufacturers pursuing disruptive innovation
      • Best odds are in companies with deep culture of continually inculcating new skills into their people, and rethinking methods and work processes
      • Some managements willing to take radical corporate measures to replace old-thinking engineering staff with “systems thinkers”
      • Downsizing in off-highway equipment manufacturers may push them to seek more systems-level value-add from their component suppliers
    • New technology opportunities inside manufacturers ready to move more deeply into systems modeling
      • Opportunities in new/emerging industries/companies without legacy investments in systems modeling tools and libraries
      • Best practice for introducing systems modeling: start with work process, then bring in software
      • Capitalizing on engineering’s leeway and autonomy in specifying systems modeling software compared with enterprise-standard CAD/PLM tools
    • Systems modeling technology advances anticipated by practitioner advocates
      • Improving software integration, interoperability, data interchange
      • Improving co-simulation across domain tools
      • Better, more complete FMI (Functional Mock-up Interface) implementation/compliance
      • Higher-fidelity versions of FMI or similar

    The white paper detailing the findings of this research is intended to offer guidance and advice for implementing change, as well as documentation to help convince colleagues, management and partners that new ways of working exist, and that the software technologies to support and enable them are available, accessible, and delivering payback and business advantage to forward-thinking engineering organizations today.

    My hope is that this research finds utility as a practical, actionable aid for engineers and engineering management in helping their organizations to adopt and implement—or to strengthen and deepen—a simulation-led, systems-driven approach to product development.

    You can download the full white paper reporting our findings here.

    Bruce Jenkins, Ora Research
    oraresearch.com

    most effective built in operator code award goes to ppl that wrote the code for the union and intercect set operations for maple. Very important simple example below of  one of its applications.

     

    When i work with algorithms, probably one of my most primary ports of enquiry (figuratively jeez skynet)  is to set up and if statement triggered to terminate the loop once the operations performed for any further cycles is INDEMPOTENT. this doesnt always mean your output is convergent in every case but it allows you to minimize the amount of time the cpu needs to collect data( ie the point at which it would produce that same set as it did in the last most loop)

     

     

    Y := proc (X) local N, S1, `&Sopf;`; if X <> `union`(X, S1[N]) then N := (rand(1 .. NrANGE))(); S1[N] := {K[1](4+N), K[1](5+N), K[1](6+N), K[1](7+N), K[1](8+N), K[1](9+N), K[1](10+N), K[1](11+N), K[1](12+N), K[1](13+N), K[1](14+N), K[1](15+N)}; `&Sopf;` := `union`(X, S1[N]) else  end if end proc

    proc (X) local N, S1, `&Sopf;`; if X <> `union`(X, S1[N]) then N := (rand(1 .. NrANGE))(); S1[N] := {K[1](4+N), K[1](5+N), K[1](6+N), K[1](7+N), K[1](8+N), K[1](9+N), K[1](10+N), K[1](11+N), K[1](12+N), K[1](13+N), K[1](14+N), K[1](15+N)}; `&Sopf;` := `union`(X, S1[N]) else  end if end proc

    (1)

    ``

     

    Download idempotency.mwidempotency.mw

     

     

    A prime producing polynomial.

     

    Observations on the trinomial n2 + n + 41.

     

    by Matt C. Anderson

     

    September 3, 2016

     

    The story so far

     

    We assume that n is an integer.  We focus our attention on the polynomial n^2 + n + 41.

     

    Furthur, we analyze the behavior of the factorization of integers of the form

     

    h(n) = n2 + n + 41                                          (expression 1)

     

    where n is a non-negative integer.  It was shown by Legendre, in 1798 that if 0 ≤ n < 40 then h(n) is a prime number.

     

    Certain patterns become evident when considering points (a,n) where

     

    h(n) ≡ 0 mod a.                                             (expression 2)

     

    The collection of all such point produces what we are calling a "graph of discrete divisors" due to certain self-similar features.  From experimental data we find that the integer points in this bifurcation graph lie on a collection of parabolic curves indexed by pairs of relatively prime integers.  The expression for the middle parabolas is –

     

    p(r,c) = (c*x – r*y)2 – r*(c*x – r*y) – x + 41*r2.           (expression 3)

     

    The restrictions are that 0<r<c and gcd(r,c) = 1 and all four of r,c,x, and y are integers.

     

    Each such pair (r,c) yields (again determined experimentally and by observation of calculations) an integer polynomial a*z2 + b*z + c, and the quartic h(a*z2 + b*z + c) then factors non-trivially over the integers into two quadratic expressions.  We call this our "parabola conjecture".  Certain symmetries in the bifurcation graph are due to elementary relationships between pairs of co-prime integers.  For instance if m<n are co-prime integers, then there is an observable relationship between the parabola it determines that that formed from (n-m, n).

     

    We conjecture that all composite values of h(n) arise by substituting integer values of z into h(a*z2 + b*z + c), where this quartic factors algebraically over Z for a*z2 + b*z + c a quadratic polynomial determined by a pair of relatively prime integers.  We name this our "no stray points conjecture" because all the points in the bifurcation graph appear to lie on a parabola.

     

    We further conjecture that the minimum x-values for parabolas corresponding to (r, c) with gcd(r, c) = 1 are equal for fixed n.  Further, these minimum x-values line up at 163*c^2/4 where c = 2, 3, 4, ...  The numerical evidence seems to support this.  This is called our "parabolas line up" conjecture.

     

    The notation gcd(r, c) used above is defined here.  The greatest common devisor of two integers is the smallest whole number that divides both of those integers.

     

    Theorem 1 - Consider h(n) with n a non negative integer. 

    h(n) never has a factor less than 41.

     

    We prove Theorem 1 with a modular construction.  We make a residue table with all the prime factors less than 41.  The fundamental theorem of arithmetic states that any integer greater than one is either a prime number, or can be written as a unique product of prime numbers (ignoring the order).  So if h(n) never has a prime factor less than 41, then by extension it never has an integer factor less than 41.

     

    For example, to determine that h(n) is never divisible by 2, note the first column of the residue table.  If n is even, then h(n) is odd.  Similarly, if n is odd then h(n) is also odd.  In either case, h(n) does not have factorization by 2.

     

    Also, for divisibility by 3, there are 3 cases to check.  They are n = 0, 1, and 2 mod 3. h(0) mod 3 is 2.  h(1) mod 3 is 1. and h(2) mod 3 is 2.  Due to these three cases, h(n) is never divisible by 3.  This is the second column of the residue table.

     

    The number 0 is first found in the residue table for the cases h(0) mod 41 and h(40) mod 41.  This means that if n is congruent to 0 mod 41 then h(n) will be divisible by 41.  Similarly, if n is congruent to 40 mod 41 then h(n) is also divisible by 41.

    After the residue table, we observe a bifurcation graph which has points when h(y) mod x is divisible by x.  The points (x,y) can be seen on the bifurcation graph.

     

    < insert residue table here >

     

    Thus we have shown that h(n) never has a factor less than 41.

     

    Theorem 2

     

    Since h(a) = a^2 + a + 41, we want to show that h(a) = h( -a -1).

     

    Proof of Theorem 2

    Because h(a) = a*(a+1) + 41,

    Now h(-a -1) = (-a -1)*(-a -1 +1) + 41.

    So h(-a -1) = (-a -1)*(-a) +41,

    And h(-a -1) = h(a).

    Which was what we wanted.

    End of proof of theorem 2.

     

    Corrolary 1

    Further, if h(b) mod c ≡ = then h(c –b -1) mod c ≡ 0.

     

    We can observe interesting patterns in the “graph of discrete divisors” on a following page.

     

    Walking into the big blue Maplesoft office on August 3rd was a bit nerve wracking. I had no idea who anyone was, what to expect, or even what I would be doing. As I sat in the front hall waiting for someone to receive me, I remember thinking, “What have I gotten myself into?”. Despite my worries on that first day, interning at Maplesoft has been a great experience! I never knew that I would be able to learn so much about programming and working in a company in such a short amount of time. Although Maple was a programming language that was foreign to me a couple weeks ago, I feel like I’m relatively well versed in it now. Trying to learn a new language in this short timespan hasn’t been easy, but I think that I picked it up quickly, even if I’ve had my fair share of frustrations.

    Chaos Game example on Rosetta Code

    At Maplesoft, I’ve been contributing to the Rosetta Code project by writing short programs using Maple. The Rosetta Code project is dedicated to creating programming examples for many different tasks in different programming languages. My summer project has been to create solutions using Maple for as many tasks as possible and to post these to Rosetta Code; the goal being to have the list of tasks without Maple implementation shrink with each passing day. It’s nice to feel like I’m leaving a mark in this world, even if it is in such a small corner of the internet.

    Flipping Bits example on Rosetta Code/MapleCloud

    This internship, of course, came with its share of challenges. During my work on the Rosetta Code project, I posted solutions for a total of 38 tasks. Some of them were easy, but some of them took days to complete. On some days, I felt like I was on top of the world. Everything I made turned out great and I knew exactly how to tackle each problem. Other days were slower. I’ve spent ages just staring at a computer monitor trying to figure out just how on earth I was going to make this machine do what I wanted it to do! The 24 Game task was particularly hard, but also very educational. Through this task, I learned about modules, a concept previously unknown to me. I’m fairly sure that the 24 Game also took me the longest, whereas the Increment a numerical string task took me no time at all. Despite it being easy, the Increment a numerical string task wasn’t particularly fun; a bit of a challenge is required for something to be entertaining, after all. My personal favourite was the Fibonacci n-step number sequences task. It was the first really challenging task I encountered, and for after which the feeling of finally completing a task that I spent so long on, of finally overcoming that mountain, was extremely satisfying. Not all challenges end in satisfaction, however. I often found myself accidentally doing something that made the window freeze. I would close the program, then cry a bit on the inside when I realized I just lost the past half an hour’s worth of unsaved work. Nevertheless, I’m glad I got to face all these obstacles because they have made me more resilient and a better programmer.

    The following is the code for the Fibonacci n-step number sequences task

    numSequence := proc(initValues :: Array)
    	local n, i, values;
    n := numelems(initValues);
    values := copy(initValues);
    for i from (n+1) to 15 do
    values(i) := add(values[i-n..i-1]);
    end do;
    return values;
    end proc:
     
    initValues := Array([1]):
    for i from 2 to 10 do
    initValues(i) := add(initValues):
    printf ("nacci(%d): %a\n", i, convert(numSequence(initValues), list));
    end do:
    printf ("lucas: %a\n", convert(numSequence(Array([2, 1])), list));

    Maple was a great software to program with and a fairly straightforward language to learn. Having previously programmed in Java, I found Maple similar enough that transitioning wasn’t too difficult. In fact, every once in a while when I didn`t know what to do for a task, I would take a look at the Java example in Rosetta Code and it would point me in a direction or give me some hints. While the two languages are similar, there are still many differences. For example, I liked the fact that in Maple, lists started at an index of 1 rather than 0 and arrays could an arbitrary starting index. Although it was different from what I was used to, I found that it made many things much less confusing. Another thing I liked was that the for loop syntax was very simple. I never once had to run through in my head how many times something would loop for. There were such a wide variety of commands in Maple. There was a command for practically anything, and if you knew that it existed and how to use it, then so much power could be at your fingertips. This is where the help system came in extremely handy. With a single search you might find that the solution to the exact problem you were trying to solve already existed as a Maple command. I always had a help window open when I was using Maple.

    Multiplication Tables example on Rosetta Code

    Spending my summer coding at Maplesoft has been fun, sometimes challenging, but an overall rewarding experience. Through contributing to the Rosetta Code project, I’ve learned so much about computer programming, and it certainly made the 45 minute drive out to Waterloo worth it!

    Yili Xu,
    Maplesoft SHAD Intern

    The presentation below is on undergrad Quantum Mechanics. Tackling this topic within a computer algebra worksheet in the way it's done below, however, is an exciting novelty and illustrates well the level of abstraction that is now possible using the Physics package.

     

    Quantum Mechanics: Schrödinger vs Heisenberg picture

    Pascal Szriftgiser1 and Edgardo S. Cheb-Terrab2 

    (1) Laboratoire PhLAM, UMR CNRS 8523, Université Lille 1, F-59655, France

    (2) Maplesoft

     

    Within the Schrödinger picture of Quantum Mechanics, the time evolution of the state of a system, represented by a Ket "| psi(t) >", is determined by Schrödinger's equation:

    I*`&hbar;`*(diff(Ket(psi, t), t)) = H*Ket(psi, t)

    where H, the Hamiltonian, as well as the quantum operators O__S representing observable quantities, are all time-independent.

     

    Within the Heisenberg picture, a Ket Ket(psi, 0) representing the state of the system does not evolve with time, but the operators O__H(t)representing observable quantities, and through them the Hamiltonian H, do.

     

    Problem: Departing from Schrödinger's equation,

      

    a) Show that the expected value of a physical observable in Schrödinger's and Heisenberg's representations is the same, i.e. that

    Bra(psi, t)*O__S*Ket(psi, t) = Bra(psi, 0)*O__H(t)*Ket(psi, 0)

      

    b) Show that the evolution equation of an observable O__H in Heisenberg's picture, equivalent to Schrödinger's equation,  is given by:

    diff(O__H(t), t) = (-I*Physics:-Commutator(O__H(t), H))*(1/`&hbar;`)

    where in the right-hand-side we see the commutator of O__H with the Hamiltonian of the system.

    Solution: Let O__S and O__H respectively be operators representing one and the same observable quantity in Schrödinger's and Heisenberg's pictures, and H be the operator representing the Hamiltonian of a physical system. All of these operators are Hermitian. So we start by setting up the framework for this problem accordingly, including that the time t and Planck's constant are real. To automatically combine powers of the same base (happening frequently in what follows) we also set combinepowersofsamebase = true. The following input/output was obtained using the latest Physics update (Aug/31/2016) distributed on the Maplesoft R&D Physics webpage.

    with(Physics):

    Physics:-Setup(hermitianoperators = {H, O__H, O__S}, realobjects = {`&hbar;`, t}, combinepowersofsamebase = true, mathematicalnotation = true)

    [combinepowersofsamebase = true, hermitianoperators = {H, O__H, O__S}, mathematicalnotation = true, realobjects = {`&hbar;`, t}]

    (1)

    Let's consider Schrödinger's equation

    I*`&hbar;`*(diff(Ket(psi, t), t)) = H*Ket(psi, t)

    I*`&hbar;`*(diff(Physics:-Ket(psi, t), t)) = Physics:-`*`(H, Physics:-Ket(psi, t))

    (2)

    Now, H is time-independent, so (2) can be formally solved: psi(t) is obtained from the solution psi(0) at time t = 0, as follows:

    T := exp(-I*H*t/`&hbar;`)

    exp(-I*t*H/`&hbar;`)

    (3)

    Ket(psi, t) = T*Ket(psi, 0)

    Physics:-Ket(psi, t) = Physics:-`*`(exp(-I*t*H/`&hbar;`), Physics:-Ket(psi, 0))

    (4)

    To check that (4) is a solution of (2), substitute it in (2):

    eval(I*`&hbar;`*(diff(Physics[Ket](psi, t), t)) = Physics[`*`](H, Physics[Ket](psi, t)), Physics[Ket](psi, t) = Physics[`*`](exp(-I*H*t/`&hbar;`), Physics[Ket](psi, 0)))

    Physics:-`*`(H, exp(-I*t*H/`&hbar;`), Physics:-Ket(psi, 0)) = Physics:-`*`(H, exp(-I*t*H/`&hbar;`), Physics:-Ket(psi, 0))

    (5)

    Next, to relate the Schrödinger and Heisenberg representations of an Hermitian operator O representing an observable physical quantity, recall that the value expected for this quantity at time t during a measurement is given by the mean value of the corresponding operator (i.e., bracketing it with the state of the system Ket(psi, t)).

    So let O__S be an observable in the Schrödinger picture: its mean value is obtained by bracketing the operator with equation (4):

    Dagger(Ket(psi, t) = Physics[`*`](exp(-I*H*t/`&hbar;`), Ket(psi, 0)))*O__S*(Ket(psi, t) = Physics[`*`](exp(-I*H*t/`&hbar;`), Ket(psi, 0)))

    Physics:-`*`(Physics:-Bra(psi, t), O__S, Physics:-Ket(psi, t)) = Physics:-`*`(Physics:-Bra(psi, 0), exp(I*t*H/`&hbar;`), O__S, exp(-I*t*H/`&hbar;`), Physics:-Ket(psi, 0))

    (6)

    The composed operator within the bracket on the right-hand-side is the operator O in Heisenberg's picture, O__H(t)

    Dagger(T)*O__S*T = O__H(t)

    Physics:-`*`(exp(I*t*H/`&hbar;`), O__S, exp(-I*t*H/`&hbar;`)) = O__H(t)

    (7)

    Analogously, inverting this equation,

    (T*(Physics[`*`](exp(I*H*t/`&hbar;`), O__S, exp(-I*H*t/`&hbar;`)) = O__H(t)))*Dagger(T)

    O__S = Physics:-`*`(exp(-I*t*H/`&hbar;`), O__H(t), exp(I*t*H/`&hbar;`))

    (8)

    As an aside to the problem, we note from these two equations, and since the operator T = exp((-I*H*t)*(1/`&hbar;`)) is unitary (because H is Hermitian), that the switch between Schrödinger's and Heisenberg's pictures is accomplished through a unitary transformation.

     

    Inserting now this value of O__S from (8) in the right-hand-side of (6), we get the answer to item a)

    lhs(Physics[`*`](Bra(psi, t), O__S, Ket(psi, t)) = Physics[`*`](Bra(psi, 0), exp(I*H*t/`&hbar;`), O__S, exp(-I*H*t/`&hbar;`), Ket(psi, 0))) = eval(rhs(Physics[`*`](Bra(psi, t), O__S, Ket(psi, t)) = Physics[`*`](Bra(psi, 0), exp(I*H*t/`&hbar;`), O__S, exp(-I*H*t/`&hbar;`), Ket(psi, 0))), O__S = Physics[`*`](exp(-I*H*t/`&hbar;`), O__H(t), exp(I*H*t/`&hbar;`)))

    Physics:-`*`(Physics:-Bra(psi, t), O__S, Physics:-Ket(psi, t)) = Physics:-`*`(Physics:-Bra(psi, 0), O__H(t), Physics:-Ket(psi, 0))

    (9)

    where, on the left-hand-side, the Ket representing the state of the system is evolving with time (Schrödinger's picture), while on the the right-hand-side the Ket `&psi;__0`is constant and it is O__H(t), the operator representing an observable physical quantity, that evolves with time (Heisenberg picture). As expected, both pictures result in the same expected value for the physical quantity represented by O.

     

    To complete item b), the derivation of the evolution equation for O__H(t), we take the time derivative of the equation (7):

    diff((rhs = lhs)(Physics[`*`](exp(I*H*t/`&hbar;`), O__S, exp(-I*H*t/`&hbar;`)) = O__H(t)), t)

    diff(O__H(t), t) = I*Physics:-`*`(H, exp(I*t*H/`&hbar;`), O__S, exp(-I*t*H/`&hbar;`))/`&hbar;`-I*Physics:-`*`(exp(I*t*H/`&hbar;`), O__S, H, exp(-I*t*H/`&hbar;`))/`&hbar;`

    (10)

    To rewrite this equation in terms of the commutator  Physics:-Commutator(O__S, H), it suffices to re-order the product  H  exp(I*H*t/`&hbar;`) placing the exponential first:

    Library:-SortProducts(diff(O__H(t), t) = I*Physics[`*`](H, exp(I*H*t/`&hbar;`), O__S, exp(-I*H*t/`&hbar;`))/`&hbar;`-I*Physics[`*`](exp(I*H*t/`&hbar;`), O__S, H, exp(-I*H*t/`&hbar;`))/`&hbar;`, [exp(I*H*t/`&hbar;`), H], usecommutator)

    diff(O__H(t), t) = I*Physics:-`*`(exp(I*t*H/`&hbar;`), H, O__S, exp(-I*t*H/`&hbar;`))/`&hbar;`-I*Physics:-`*`(exp(I*t*H/`&hbar;`), Physics:-`*`(H, O__S)+Physics:-Commutator(O__S, H), exp(-I*t*H/`&hbar;`))/`&hbar;`

    (11)

    Normal(diff(O__H(t), t) = I*Physics[`*`](exp(I*H*t/`&hbar;`), H, O__S, exp(-I*H*t/`&hbar;`))/`&hbar;`-I*Physics[`*`](exp(I*H*t/`&hbar;`), Physics[`*`](H, O__S)+Physics[Commutator](O__S, H), exp(-I*H*t/`&hbar;`))/`&hbar;`)

    diff(O__H(t), t) = -I*Physics:-`*`(exp(I*t*H/`&hbar;`), Physics:-Commutator(O__S, H), exp(-I*t*H/`&hbar;`))/`&hbar;`

    (12)

    Finally, to express the right-hand-side in terms of  Physics:-Commutator(O__H(t), H) instead of Physics:-Commutator(O__S, H), we take the commutator of the equation (8) with the Hamiltonian

    Commutator(O__S = Physics[`*`](exp(-I*H*t/`&hbar;`), O__H(t), exp(I*H*t/`&hbar;`)), H)

    Physics:-Commutator(O__S, H) = Physics:-`*`(exp(-I*t*H/`&hbar;`), Physics:-Commutator(O__H(t), H), exp(I*t*H/`&hbar;`))

    (13)

    Combining these two expressions, we arrive at the expected result for b), the evolution equation of a given observable O__H in Heisenberg's picture

    eval(diff(O__H(t), t) = -I*Physics[`*`](exp(I*H*t/`&hbar;`), Physics[Commutator](O__S, H), exp(-I*H*t/`&hbar;`))/`&hbar;`, Physics[Commutator](O__S, H) = Physics[`*`](exp(-I*H*t/`&hbar;`), Physics[Commutator](O__H(t), H), exp(I*H*t/`&hbar;`)))

    diff(O__H(t), t) = -I*Physics:-Commutator(O__H(t), H)/`&hbar;`

    (14)


    Download:    Schrodinger_vs_Heisenberg_picture.mw     Schrodinger_vs_Heisenberg_picture.pdf

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

    Hanze University of Applied Sciences Groningen has created 105 questions related to engineering mechanics for structures (statics/construction). These 105 randomised questions with graphics are used for first year students in civil engineering, structural engineering, architectural engineering and building engineering.

    The topics of the course modules are as follows:
    - Force Vectors (10)
    - Support Reactions (26)
    - Internal Forces (31)
    - Stress (21)
    - Trusses (17)

    All questions have a translation button which makes it easy to switch from English to any other language. The questions are first shown in Dutch [NL] but by clicking [UK] in the Preview, the English version is shown. The text can easily be edited and changed into the language of choice in the Maple T.A. question editor. Only the button needs adjustment in the question source.

    60 questions are “exercises“ which means that these questions have extended feedback. The remaining questions (45) are “tests” meaning that the questions include no feedback.

    Cone.zip - construction exercises (60 questions)

    Cont.zip - construction tests (45 questions)

    Jonny
    Maplesoft Product Manager, Online Education Products

    Bruce Jenkins is President of Ora Research, an engineering research and advisory service. Maplesoft commissioned him to examine how systems-driven engineering practices are being integrated into the early stages of product development, the results of which are available in a free whitepaper entitled System-Level Physical Modeling and Simulation. In this series of blog posts, Mr. Jenkins discusses the results of his research.

    This is the second entry in the series.

    My last post, Strategies for accelerating the move to simulation-led, systems-driven engineering, described my firm’s research project to investigate the contemporary state of adoption and application of systems modeling software technologies, and their attendant methods and work processes, in the engineering design of off-highway equipment and mining machinery.

    Adoption drivers

    In this project, I conducted in-depth, structured but open-ended interviews with some half-dozen expert practitioners at leading manufacturers, including both engineering management and senior discipline leads. These interviews identified the following key technological factors as well as business and competitive issues driving adoption and use of systems modeling tools and methods at current levels:

    • Fuel economy and emissions mandates, powertrain electrification and autonomous operation requirements
    • Software’s ability to drive down product cost of ownership and delivery times
    • Traditional development processes often fail to surface system-level issues until fabrication or assembly, or even until operational deployment
    • Detailed analysis tools such as FEA and CFD focus on behaviors at the component level, and are not optimal for studies of the complete system
    • Engineering departments/groups enjoy greater freedom in systems modeling software selection and purchase decisions than in enterprise-controlled CAD/PDM/PLM decisions
    • Good C/VP-level visibility of systems modeling tools, especially in off-highway equipment

    Adoption constraints

    At the same time, there was widespread agreement among all the experts interviewed that these tools and methods are not being brought to bear with anywhere near the breadth or depth that practitioner advocates would like, and that they believe would be greatly beneficial to their organizations and industries.

    In probing why this is, the interviews revealed an array of factors constraining broader adoption at present. These range from legacy engineering culture issues, through human resource limitations and constraints imposed by business models and corporate cultures, to entrenched shortcomings in how long-established systems modeling software toolsets have been deployed and applied to the product development process:

    • Legacy engineering culture constraints
      • Conservatism of mining machinery product development culture
      • Engineering practices in long-standardized industries grounded in handbook formulas and rules of thumb
      • Perceived lack of time in schedule to do systems modeling
    • Human resource constraints
      • Low availability of engineers with systems modeling skills
      • Shortage of engineers trained in systems thinking
      • Legacy engineering processes compound shortage of systems-thinking engineers
      • Industry downturns put constraints on professional staff development
    • Business-model and corporate-culture constraints
      • Culture of seeking to mitigate cost and risk by staying with legacy designs instead of advancing and innovating the product
      • Corporate awareness of need to innovate in mining machinery gets stifled at engineering level
      • Low C/VP-level visibility of systems modeling tools in mining machinery
    • Engineering organization constraints on innovating/modernizing their systems modeling technology infrastructure
      • Power users wedded to legacy systems modeling tools
      • Weak integration at many/most points of the engineering digital toolset chain
      • Implementing systems modeling software as a sales configuration/costing aid seen as taking too much time

    My next post will detail practitioners’ visions, strategies and best practices for accelerating and institutionalizing the implementation and usage of systems modeling tools and practices in their organizations.

    You can download the full white paper reporting our findings here.

    Bruce Jenkins, Ora Research
    oraresearch.com

    In order to change Maple for the better, I use to submit SCRs. However, as i was kindly
    informed by Bryon (a copy of his e-letter on demand), MaplePrimes are under reconstruction and do not
    work properly. At least my three messages sent through the Contact button were lost.
    I have  unsuccessfully tried to reach beta.maplesoft.com (see the result of ping in the screen screen.29.08.16.docx).
    Please, help me!

    Hi,

    In the following example I introduce some commutation rules that are standard in Quantum Mechanics. A major feature of the Maple Physics Package, is that it is possible to define tensors as Quantum Operators. This is of great interest because powerful tensor simplification rules can then be used in Quantum Mechanics. For an example, see the commutation rules of the components of the angular momentum operator in ?Physics,Examples. Here, I focus on a possible issue: when destroying all quantum operators, the pre-defined commutation rules still apply, which should not be the case. As shown in the post, this is link to the fact that these operators are also tensors.
     

    NULL

     

    Physics:-Version()[2]

    `2016, August 16, 18:56 hours`

    (1)

    NULL

    NULL

    restart; with(Physics); interface(imaginaryunit = I)

    First, set a 3D Euclidian space

    Setup(mathematicalnotation = true, dimension = 3, signature = `+`, spacetimeindices = lowercaselatin, quiet)

    [dimension = 3, mathematicalnotation = true, signature = `+ + +`, spacetimeindices = lowercaselatin]

    (2)

    Define two rank 1 tensors

    Define(x[k], p[k])

    `Defined objects with tensor properties`

     

    {Physics:-Dgamma[a], Physics:-Psigma[a], Physics:-d_[a], Physics:-g_[a, b], p[k], x[k], Physics:-KroneckerDelta[a, b], Physics:-LeviCivita[a, b, c]}

    (3)

    Now, further define these tensors as quantum operators and gives the usual commutation rule between position and momentum operators (Quantum Mechanics).

    Setup(hermitianoperators = {p, x}, algebrarules = {%Commutator(p[k], p[n]) = 0, %Commutator(x[k], p[l]) = I*`&hbar;`*KroneckerDelta[k, l], %Commutator(x[k], x[l]) = 0}, realobjects = {`&hbar;`})

    [algebrarules = {%Commutator(p[k], p[n]) = 0, %Commutator(x[k], p[l]) = I*`&hbar;`*Physics:-KroneckerDelta[k, l], %Commutator(x[k], x[l]) = 0}, hermitianoperators = {p, x}, realobjects = {`&hbar;`}]

    (4)

    As expected:

    (%Commutator = Commutator)(p[a], x[b])

    %Commutator(p[a], x[b]) = -I*`&hbar;`*Physics:-KroneckerDelta[a, b]

    (5)

    Now, reset all the Hermitian operators, so that all quantum operators are destroyed. This is useful if, for instance, one needs to compare some the result with the commutative case.

    Setup(redo, hermitianoperators = {})

    [hermitianoperators = none]

    (6)

    As expected, there are no quantum operators anymore...

    Setup(quantumoperators)

    [quantumoperators = {}]

    (7)

    ...so that the following expressions should commute (result should be true)

    Library:-Commute(p[a], x[b])

    false

    (8)

    Result should be 0NULL

    Commutator(p[a], x[b])

    -I*`&hbar;`*Physics:-KroneckerDelta[a, b]

    (9)

    p[a], x[b]

    p[a], x[b]

    (10)

    NULL

    NULL

    ``

    NULLNULL

    Below is just a copy & paste of the above section. The only difference, is that "Define(x[k], p[k])" has been commented, so that x[k]and p[k] are not a tensor. In that case, everything behaves as expected (but of course, the interesting feature of tensors is not available).

    ````

    NULL

    restart; with(Physics); interface(imaginaryunit = I)

    First, set a 3D Euclidian space

    Physics:-Setup(mathematicalnotation = true, dimension = 3, signature = `+`, spacetimeindices = lowercaselatin, quiet)

    [dimension = 3, mathematicalnotation = true, signature = `+ + +`, spacetimeindices = lowercaselatin]

    (11)

    #Define two rank 1 tensors

    Now, further define these tensors as quantum operators and gives the usual commutation rule between position and momentum operators (Quantum Mechanics)

    Physics:-Setup(hermitianoperators = {p, x}, algebrarules = {%Commutator(p[k], p[n]) = 0, %Commutator(x[k], p[l]) = Physics:-`*`(Physics:-`*`(I, `&hbar;`), Physics:-KroneckerDelta[k, l]), %Commutator(x[k], x[l]) = 0}, realobjects = {`&hbar;`})

    [algebrarules = {%Commutator(p[k], p[n]) = 0, %Commutator(x[k], p[l]) = I*`&hbar;`*Physics:-KroneckerDelta[k, l], %Commutator(x[k], x[l]) = 0}, hermitianoperators = {p, x}, realobjects = {`&hbar;`}]

    (12)

    As expected:

    (%Commutator = Physics:-Commutator)(p[a], x[b])

    %Commutator(p[a], x[b]) = -I*`&hbar;`*Physics:-KroneckerDelta[a, b]

    (13)

    Now, reset all the Hermitian operators, so that all quantum operators are destroyed.

    Physics:-Setup(redo, hermitianoperators = {})

    [hermitianoperators = none]

    (14)

    As expected, there are no quantum operators anymore...

    Physics:-Setup(quantumoperators)

    [quantumoperators = {}]

    (15)

    ...so that the following expressions should commute (result should be true)

    Physics:-Library:-Commute(p[a], x[b])

    true

    (16)

    Result should be 0``

    Physics:-Commutator(p[a], x[b])

    0

    (17)

    p[a], x[b]

    p[a], x[b]

    (18)

    NULL

    ``

    NULL``

    NULL


    Download Quantum_operator_as_Tensors_August_23_2016.mw

    Bruce Jenkins is President of Ora Research, an engineering research and advisory service. Maplesoft commissioned him to examine how systems-driven engineering practices are being integrated into the early stages of product development, the results of which are available in a free whitepaper entitled System-Level Physical Modeling and Simulation. In the coming weeks, Mr. Jenkins will discuss the results of his research in a series of blog posts.

    This is the first entry in the series.

    Discussions of how to bring simulation to bear starting in the early stages of product development have become commonplace today. Driving these discussions, I believe, is growing recognition that engineering design in general, and conceptual and preliminary engineering in particular, face unprecedented pressures to move beyond the intuition-based, guess-and-correct methods that have long dominated product development practices in discrete manufacturing. To continue meeting their enterprises’ strategic business imperatives, engineering organizations must move more deeply into applying all the capabilities for systematic, rational, rapid design development, exploration and optimization available from today’s simulation software technologies.

    Unfortunately, discussions of how to simulate early still fixate all too often on 3D CAE methods such as finite element analysis and computational fluid dynamics. This reveals a widespread dearth of awareness and understanding—compounded by some fear, intimidation and avoidance—of system-level physical modeling and simulation software. This technology empowers engineers and engineering teams to begin studying, exploring and optimizing designs in the beginning stages of projects—when product geometry is seldom available for 3D CAE, but when informed engineering decision-making can have its strongest impact and leverage on product development outcomes. Then, properly applied, systems modeling tools can help engineering teams maintain visibility and control at the subsystems, systems and whole-product levels as the design evolves through development, integration, optimization and validation.

    As part of my ongoing research and reporting intended to help remedy the low awareness and substantial under-utilization of system-level physical modeling software in too many manufacturing industries today, earlier this year I produced a white paper, “System-Level Physical Modeling and Simulation: Strategies for Accelerating the Move to Simulation-Led, Systems-Driven Engineering in Off-Highway Equipment and Mining Machinery.” The project that resulted in this white paper originated during a technology briefing I received in late 2015 from Maplesoft. The company had noticed my commentary in industry and trade publications expressing the views set out above, and approached me to explore what they saw as shared perspectives.

    From these discussions, I proposed that Maplesoft commission me to further investigate these issues through primary research among expert practitioners and engineering management, with emphasis on the off-highway equipment and mining machinery industries. In this research, focused not on software-brand-specific factors but instead on industry-wide issues, I interviewed users of a broad range of systems modeling software products including Dassault Systèmes’ Dymola, Maplesoft’s MapleSim, The MathWorks’ Simulink, Siemens PLM’s LMS Imagine.Lab Amesim, and the Modelica tools and libraries from various providers. Interviewees were drawn from manufacturers of off-highway equipment and mining machinery as well as some makers of materials handling machinery.

    At the outset, I worked with Maplesoft to define the project methodology. My firm, Ora Research, then executed the interviews, analyzed the findings and developed the white paper independently of input from Maplesoft. That said, I believe the findings of this project strongly support and validate Maplesoft’s vision and strategy for what it calls model-driven innovation. You can download the white paper here.

    Bruce Jenkins, Ora Research
    oraresearch.com

    just reading this story... a large quantity of information does not add up for me, like he denies it is his work and was done by hamilton? also its blasted across the net that he his proof is valid, so he cant say he doesnt want fame because its been all over the internet, and the sheer lack of logic of refusing money based on a moral code of conduct? then give it to charity, pay for a dozen scholarships but what the money is dirty?

     

    pretty sure this whole thing is a pile of crap made up by someone.

    A more honest and specific version of lemma 3.

    CONGRUENT_FUNCTIONS_OF_THE_FRACTIONAL_PART_OVER_Q_LEMMA_4.mw

    Maple Worksheet - Error

    Failed to load the worksheet /maplenet/convert/CONGRUENT_FUNCTIONS_OF_THE_FRACTIONAL_PART_OVER_Q_LEMMA_4.mw .

    Download CONGRUENT_FUNCTIONS_OF_THE_FRACTIONAL_PART_OVER_Q_LEMMA_4.mw

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