calculation – How to find the minimum equation with an unknown matrix

Suppose I want to minimize the equation

$ A = 1 ^ T_ {n1} B * C1_ {n_1} $

where B is a matrix of unknowns of dimension (n1 x n2) and C is a constant matrix of dimension (n2 x n1), and $ 1_ {n1} $ is a vector of ones (length $ n_1 $), so A is a scalar

subject to certain constraints, here all the columns of B are equal to 1 and all the lines are equal to a certain value (for example, w).

Are there any standard methods I could use to find the elements of B that minimize A?

to solve – Solve does not work with this equation, case of trigonometry

I was trying to solve this equation, but Mathematica 12 continues to try but does not give an answer.

Is there a problem in my system or something with Mathematica 12?

Solve[E^((-Log[5]/ 2) * t) * 3 * Cos[330*Pi*t]== 6, t]

I entered this equation as part of a problem and wanted to draw a point, but it turned out that Mathematica did not respond. It is curious to see that such a complex software is connected to it.

Thank you all for your help.

plot – sine-Gordon equation using the pseudospectral method

I want to try this code what is the problem?
I've tried that
https://reference.wolfram.com/language/tutorial/NDSolveMethodOfLines.html
When I tried, I had enter the description of the image here

sol = NDSolve[{D[u[t, x, y], t, t]==
re[u[t, x, y], x, x]+ D[u[t, x, y], y, y]- Peach[u[t, x, y]],
you[0, x, y] == Exp[-(x^2 + y^2)],
Derivative[1, 0, 0][u][0, x, y]    == 0, u[t, -10, y] == u[t, 10, y],
you[t, x, -10] == u[t, x, 10]}
u, {t, 0, 6}, {x, -10, 10}, {y, -10, 10},
Method -> {"MethodOfLines",
"SpatialDiscretization" -> {"TensorProductGrid",
"DifferenceOrder" -> "Pseudospectral"}}]Plot3D[First[u[6, x, y] /. ground], {x, 20, 40}, {y, -15, 15},
PlotRange -> All, PlotPoints -> 40]

enter the description of the image here

plot – Resolution of the Laplace equation using the collocation method in Mathematica

I want to try this code using mathematica Solve the Laplace equation using the collocation method

data = Flatten[Table[{x, y}, {x, 0, 1, 0.1}, {y, 0, 1, 0.1}], 1];
b1 = table[{0, y}, {y, 0, 1, 0.1}];
b2 = table[{1, y}, {y, 0, 1, 0.1}];
b3 = table[{x, 0}, {x, 0, 1, 0.1}];
boundarydata = Flatten[{b1, b2, b3}, 1];
b4 = table[{x, 1}, {x, 0, 1, 0.1}];
Rate @ Thread @ g[b4] = RandomReal[{0, 1}, 11];
region = implicit region[0<=x<=1&&0<=y[0<=x<=1&&0<=y[0<=x<=1&&0<=y[0<=x<=1&&0<=y<= 1, {x, y}];
MeshCoordinates@
DiscretizeRegion[region , MaxCellMeasure -> .001], 165];
solution =
FunctionBasis[{{Laplacian[u[x, y], {x, y}]== 0, data}, {u[x, y] == x * y, bornedata},
{u[x, y] == g[{x, y}], b4}}, u, {x, y}, "Base" -> {{"Gaussian", nodes, 0.7623}, {"Polynomial", 2}}];
Plot3D[u[u[u[u[x, y] /. solution, {x, 0, 1}, {y, 0, 1}, ColorFunction -> Hue, AxesLabel -> Automatic]lsol = evaluate[Laplacian[u[Laplacian[u[Laplacien[u[Laplacian[u[x, y] /. solution, {x, y}

];
Plot3D[{lsol, 0}, {x, 0, 1}, {y, 0, 1}, PlotRange -> All]
Plot3D[Abs[lsol - 0], {x, 0, 1}, {y, 0, 1}, PlotRange -> All]

enter the description of the image here

And the exit should be like that
enter the description of the image here

What's the problem ?

operators – Implicit definition of a coordinate in a differential equation

I would like to solve the following differential equation

$ frac {d ^ 2 psi (r _ *)} {dr ^ 2_ *} + ( omega 2 – V (r)) psi (r _ *) = 0 $

and the boundary conditions are $ psi ( inf) = psi (-inf) = 0 $. The problem is that the function V (r) is defined in $ r $, do not $ r _ * $. However I have the following relationship

$ r_ * = r + 2M ln (r-2M) $

which, clearly, can not be solved for $ r $.

My real problem is finding the eigenvalues ​​for $ omega $, but any help to deal with this implicit definition of $ r $ can help me.

PS: I post this here and not in physical / mathematical exchange because the function $ V (r) $ It's complicated and I need to solve it computerically. And no, I do not want to turn the whole equation $ r $ coordinate and solve here.

How to solve this differential equation analytically?

How can I Resolution the following differential equation?
I need the answer to this equation or only the first integration with respect to r.

Resolution[1/(G L (n - r) (n + r))
    Sin[[Theta]1]Peach[[Theta]2](54 (n ^ 2 - r ^ 2) ^ 6 +
2 L ^ 2 (9 r ^ 4 (9 q0 ^ 2 + 2 r ^ 6) + 36 n ^ 6 r ^ 4 (-5 + 4 V0 ^ 2) -
18 n + 10 (1 + 4 V0 ^ 2) + 18 n + 8 r ^ 2 (5 + 16 V0 ^ 2) -
2 n ^ 2 (9 q0 ^ 2 r ^ 2 + 96 q0 r ^ 5 V0 + r ^ 8 (45 + 4 V0 ^ 2)) +
## EQU1 ##
9 L ^ 2 (12 L ^ 4 m (14 n ^ 6 - 51 n ^ 4 r ^ 2 + 5 r ^ 6) V[
          r]Derivative ^ 3 + (n - r) ^ 3 (n + r) ^ 3[1][V][
          r]    (4 r (n ^ 2 - r ^ 2) ^ 2 +
3 L ^ 4 m derivative[1][V][
             r]    (n ^ 2 + r ^ 2 + (3 n ^ 2 r + r ^ 3) derivative[1][V][r])) +
6 L ^ 4 m (n - r) (n + r) V[
          r]^ 2 (12 n ^ 2 r (-5 n ^ 2 + r ^ 2) derivative[1][V][
             r]    + (n - r) (n + r) (10 n ^ 2 - 6 r ^ 2 -
6 (n ^ 4 - n ^ 2 r ^ 2 + 2 r ^ 4) (V ^ [Prime][Prime])[r] +
3 (n - r) r (n + r) (2 n 2 + r 2)
!  ( * SuperscriptBox[(V), 
TagBox[
RowBox[{"(", "3", ")"}],
Derivative],
MultilineFunction-> None])[r])) - (n - r) ^ 2 (n + r) ^ 2 V[
          r] (4 (n + 6 + (-3 L + 4 m + n + 4) r ^ 2 - 5 n + 2 r ^ 4 + 3 r ^ 6) +
3 L ^ 4 m (-36 n ^ 2 r ^ 2 derivatives[1][V][

                r]^ 2 + (n - r) (n +
r) (2 (n ^ 2 + 4 r ^ 2) (V ^ [Prime][Prime])[r] +
3 r ^ 2 (-n + r) (n + r) (V ^)[Prime][Prime])[r]^ 2 +
2 r (-n + r) (n + r)
!  ( * SuperscriptBox[(V), 
TagBox[
RowBox[{"(", "3", ")"}],
Derivative],
MultilineFunction-> None])[r]+
3 r derivatives[1][V][
                r]    (8 n ^ 2 -
2 (4 n ^ 4 - 5 n ^ 2 r ^ 2 + r ^ 4) (V ^ [Prime][Prime])[r] -
r (n ^ 2 - r ^ 2) ^ 2
!  ( * SuperscriptBox[(V), 
TagBox[
RowBox[{"(", "3", ")"}],
Derivative],
MultilineFunction-> None])[r]))))) == 0, V[r], r]
`` `

reference request – Neumann equation on manifold with edge or angle

Let $ (M, g) $ to be a compact Riemannian variety with boudnary and corner, that is, locally inserted into $[0infty)^1timesmathbbR^{n-1}$[0infty)^1timesmathbbR^{n-1}$[0infty)^1timesmathbbR^{n-1}$[0infty)^1timesmathbbR^{n-1}$ or $[0infty)^ktimesmathbbR^{n}$k[0infty)^ktimesmathbbR^{n}$k[0infty)^ktimesmathbbR^{n-k}$[0infty)^ktimesmathbbR^{n-k}$, or $ n = dim (M) $.

Consider the Neumann equation, that is to say
$$ begin {cases} Delta u = f & mbox {in} M, \
frac { partial u} { partial nu} = g & mbox {on} partial M end {comments}, $$

where the second equation is defined smoothly-wisely on $ partial M $, that is, defined on the smooth part of $ partial M $.

Q assume $ int_M f + int _ { partial M} g = 0 $ hold, can we find a solution $ u $, or can we have
$$ | u | _ {L ^ p_ {k + 2}} leq C ( | f | _ {L ^ p_k} + | g | _ {L ^ p_ {k + 1, delta}} + | u | _ {L ^ p_ {k + 1}}), $$
or $ | g | _ {L ^ p_ {k + 1, delta}} = inf { | G | _ {L ^ p_ {k + 1} (M)} big | G | _ { partial M} = g } $.

Any reference is welcome.

Differential equation without solution that satisfies a certain condition

This is the first week that I deal with differential equations and I am stuck at the next question. I do not know how to approach that, and any help would be greatly appreciated.

Let $ L $ to be a positive number.

Please show that the equation $ ( sqrt {x} + sqrt {y}) sqrt {y} dx = xdy $ There is no solution that satisfies $ lim_ {x to infty} frac {y (x)} {x} = L $.

Number Theory – Is it possible to obtain an interesting statement about even perfect numbers from the equation $ 1 / operatorname {rad} (n) = 1 / 2-2 varphi (n) / sigma (n) $?

It is well known that the problem concerning even perfect numbers is to prove or disprove whether they are infinitely numerous. A few weeks ago, I wrote the following conjecture: $ varphi (n) $ denotes the total function of Euler, $ sigma (n) = sum_ {1 leq d mid n} d $ the sum of the divisors and $$ operatorname {rad} (n) = prod _ { substack {p mid n \ p text {prime}}} p $$
is the product of the division of distinct prime numbers $ n> $ 1 with the definition $ operatorname {rad} (1) = 1 $, see the Radical Wikipedia of an integer.

The total function of Euler and the sum of the divisors can be found in formulas of equivalence to the Riemann hypothesis, and the radical of an integer is the famous arithmetic function that appears in the formulation of the conjecture abc.

Conjecture. An integer $ n geq 1 $ is an even number if and only if
$$ operatorname {rad} (n) = frac {1} { frac {1} {2} -2 frac { varphi (n)} { sigma (n)}}. tag {1} $$

I quoted this conjecture a few days ago in MSE. My intention is to know if it is possible to obtain a statement on the problem of even perfect nubmers, if there is an infinity of them, using the equation or if you can argue that it seems that the equation $ (1) $ is not useful for this purpose.

Question. Is it possible to get an interesting statement about the infinitude of even perfect numbers, or a fact about their distribution, using this equation $ (1) $ or invoke previous Conjecture (It's easy to prove that even perfect numbers $ n $ satisfy, but my proof attempt for the other part of the conjecture failed)? If you think this is not possible for an obstruction, please explain it. Thank you so much.

You can invoke propositions about perfect numbers, tools or conjectures from the analytic number theory (we can search for and read these statements in the literature). I hope this is a nice exercise for this site, in any case I hope comments.

Apple Pages inserts gray equation after 8.1 update

I'm using the Page Equation insert feature to insert LaTeX equations for a long time now. However, recently after the new update, I am unable to do so because the option is grayed out.
enter the description of the image here

I tried to copy an equation from one of my old files to see if I could change it. And I can! enter the description of the image here

I do not know why I can not insert new equations. Any help is appreciated.

Thank you