How to solve recursion $T(n)=T(n/2)+T(n/3)+n?

How to solve recursion T(n)=T(n/2)+T(n/3)+n? I do not really know how to approach this kind of recurrence.

differential equations – Solve boundary value problem with NDSolve. How to print out approximations to a solution?

I solve particular boundary-value-problem for ODE with NDSolve “Shooting” method. The case is that solution is attained very slow, that seems to mean that boundary-value-problem which is supposed to be well-defined enough, in fact, is ill at some place. So i try figure out. First step is to see concrete values of a produced approximations to a solution. What language constructs should i use for that?

Simple example. Suppose we consider particle motion in vertical plane. We throw our particle from initial point with coordinates {x0,y0} := {0,0} and initial trajectory angle 45 degrees. And try to achieve point with coordinates {x1,y1} := {1,0} by varying particle initial velocity. We don’t know two things here: initial velocity and a duration of a motion. Here is how this toy problem can be presented and solved in mathematica:

gravity = 10;
bvpsol = NDSolve(
      (* ODE system (5th order) *)
      {x''(u) / c(u)^2 == 0,
       y''(u) / c(u)^2 == -gravity,
       c'(u) == 0},
      (* boundary conditions (5 items) *)
      {x(0) == y(0) == 0,
       x(1) == 1,
       y(1) == 0,
       x'(0) == y'(0)}
     }, {x(u), y(u), c(u)}, u, 
    Method -> {"Shooting", 
      "StartingInitialConditions" -> {x(0) == y(0) == 0, 
        x'(0) == y'(0) == 1, c(0) == 1}}) // Flatten;

{dxdu, dydu} = D({x(u), y(u)} /. bvpsol, u);
{vx0, vy0} = ({dxdu, dydu} / c(u) /. bvpsol) /. {u -> 0};
duration = c(u) /. bvpsol /. {u -> RandomReal()};

ivpsol = NDSolve({
    (* ODE system *)
    {x''(t) == 0, y''(t) == -gravity},
    (* initial values *)
    {x(0) == y(0) == 0, x'(0) == vx0, y'(0) == vy0}
    }, {x(t), y(t)}, {t, 0, duration});

ParametricPlot({x(t), y(t)} /. ivpsol, {t, 0, duration}, 
 GridLines -> Automatic, AspectRatio -> 1/2)

comprehensive parabolic trajectory

Question: Now what options or language construct should i use to see approximations which are produced NDSolve while solving boundary-value-problem?

linear algebra – Need help to solve simple set of equations

I have below set of equations. All equations are inter-connected to each other and to find out the solution right now I am manually doing the tuning by trial-error method. To make this automated, I would like to know how to solve this for Xt and Xc for Yt=0.1 and Yc=-0.7.

Yt = -0.3759 * Xt + 0.2294

Yt = 0.2744 * Xc + 0.2294

Yc = 0.0325 * Xt – 0.1645

Yc = -0.1006 * Xc – 0.1645


Yt = 0.1

Yc = -0.7

And, I want to find the Xt and Xc which can satisfy above all equations.

Thanks in advance.


How to solve recurrence T(n) <= 2T(n/3) + (c)log3(n) using substitution method

The title wouldn’t let me format correctly, so here’s a better formatting of the question: Show by induction that any solution to a recurrence of the form

T(n) ≤ 2T(n/3) + c log3n

is O(n log3 n).

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How do I solve this integral equation with a DiracDelta?

I have to solve the following integral equation:

$$G_l(E^prime,E)=delta(E-E^prime)+int_E^{E_{max}}dE^{primeprime}G_l(E^{primeprime},E^prime) |E^{primeprime}-E^prime|$$

How can I do it? My problem is the DiracDelta, I do not know how to treat it, if there were no delta I think I could simply derive, but because of the $delta$ this trick won’t work.

Gl(e1, e) == DiracDelta(e - e1) + Integrate(Gl(e2, e1) Abs(e2 - e1), {e2, e, emax})

EDIT This equation come out as an equation to determine the Green’s function for the solution of the Boltzmann equation (BE) for the transport of a low-energy electrons into a metal.

The solution of the BE can be written as:

$$psi_l(E)=int_E^{E_{max}}de^prime G_l(E,E^prime)tilde{S}(E_0,E^prime)$$
where $G_l$ is the Green function and $tilde{S}$ is a source term.

The full problem is treated in this book, in particular see page 45 for these equations.

The function $|E^{primeprime}-E^prime|$ above is not the real function in my equation, but that is too complicated to be added here (see it a MWE).

optimization – Can LASSO algorithm solve for sparse $w$ in an under-determined system?

Consider the model $bf y=Xw+n$ where $bf w$ is a sparse complex number vector of length p, X is a N x p known complex number matrix, and $bf n$ is a complex Gaussian noise.
I’m confused because the literate answers yes to my question (in the title), but when implementing most of the functions provided in:
I get invalid solution.

Is LASSO supposed to solve an under-determined system?

oc.optimization and control – How to solve this non-continuous optimization problem?

I hope you are well. I have a non-continuous optimization problem as follows;

  *The Goal: max ∑i lotsize(i).(35f(i)-30)
  Constraits: lotsize(i) >=0.1,
              lotsize(i) is not continuous.*

where i represents ith trade , f(i) function is as follows;

          *f(i)= 1  ; if a trade is successfull,
                0  ; if a trade is not successfull.*

I try to create a lotsize function. So, I wonder if there are any ways to obtain lotsize function from the aboved optimization problem.

Thanks for your help!

Murat Y.

How to solve this issue

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numerics – How to solve this compilation error message?

I compiled the following Mathematica code

m = 40;
eea = Integrate((x - y)^(-((Alpha) - 1) - 
     1) ((Sum((z^(s))*(Cos(((b)^(s))*((Pi))*(y))), {s, 0, m})* 
        Sum((((Lambda))^((d - 2)*k))*(Sin(((Lambda)^k)*(y))), {k, 1,
           m})) - (Sum((z^(s))*(Cos(((b)^(s))*((Pi))*(0))), {s, 0, 
          m})* Sum((((Lambda))^((d - 2)*
              k))*(Sin(((Lambda)^k)*(0))), {k, 1, m}))), {y, 0, x});
sa = D(eea/ Gamma(-((Alpha) - 1)), {x, 1})

And for the resulting output i gave the following substitution

% /. x -> 1 /. (Lambda) -> 3 /. d -> 15/10 /. 
   z -> 1/10 /. (Alpha) -> 2/10 /. b -> 35

Then for the resulting output i calculated the numerical value using the code


So for the above numerical compilation i got the following error message

General::ovfl: Overflow occurred in computation.
General::stop: Further output of General::ovfl will be suppressed during this calculation.

Could anyone please suggest any answer to solve this issue.

Thank You