ordinary differential equations – ODE eigenvalue problem with unusual boundary conditions

I am given:

y”+λy=0, y(0)=0, (1−λ)y(1)+y′(1)=0

As usual we are looking for not trivial solutions.
Looks like a standard eigenvalue problem and yet I am totally stuck.
The case when lambda = 0 is rather obvious. A=B=0. Not much fun.
But when I start trying for lambda greater or smaller than zero, I get to this:

  1. λ < 0, the solution is of the form:
    B((1+ω^2)sinh(ω)+ωcosh(ω))=0 where λ=-ω^2

  2. λ > 0, the solution is of the form:

    B((1-ω^2)sin(ω)+ωcos(ω))=0 where λ=ω^2

The question states:Find the nontrivial stationary paths,
stating clearly the eigenfunctions y. In the case 1) I cant see any non trivial solutions but… well in the second case I cant see either. I know there are solutions.
Any help would be highly appreciated

mathematical philosophy – Is there an analogue of the Lost Melody Theorem in ordinary recursion theory and if not, why not?

In their arXiv preprint, “Infinite Time Turing Machines” (arXiv:math/9808093v1 (math.LO) 21 Aug 1998) Hamkins and Lewis state the Lost Melody Theorem for ITTM’s as follows:

Lost Melody Theorem 4.9 (pg. 28 in the preprint above–my comment) There is a real, $c$, such that {$c$} is decidable, but $c$ is not writable. Consequently, there is a constant, total function which is not computable, but whose graph is nevertheless decidable: $f$($x$)= $c$.

Consider also the following quote from the Carl, Schlicht, and Welch paper, “Recognizable Sets and Wooden Cardinals: Computations Beyond the Constructible Universe” (pg. 5 in my copy)

A typical phenomenon for infinitary computations is the existence of sets of ordinals which are recognizable, but not computable. Following (HL00)(that is, the Hamkins /Lewis paper just referred to–my comment), we call such sets lost melodies .

Finally, to return to the Hamkins/Lewis paper, consider further their motivation for the Lost Melody Theorem found in the little paragraph directly above the statement of the theorem:

Like the previous theorem, the next identifies the a surprising divergence (as in ‘not in’?–my comment) from the classical theory (the “classical theory” presumably being ordinary recursion theory–my comment). The real $c$ in the theorem is like a forgotten melody that you cannot produce on your own but which you can recognize when someone sings it to you.

My motivation for the question hangs both on this metaphor and on the fact two groups of authors seem to claim that such ‘lost melodies’ are an infinitary phenomenon, yet I would wager that many (including myself) have had times where we could not remember some melody but recognized it when hummed (or sung) back to us (a decidably finite phenomenon). Indeed, if one assumes that the human brain can be instatiated by a turing machine (otherwise we would have to accept J.R. Lucas-like arguments as valid), there would have to be an analogue of the Lost Melody Theorem in ordinary recursion theory, wouldn’t there (perhaps it would a type of computational-complexity theorem)?

ordinary differential equations – How to avoid the problem of a non-numerical value problem in mathematica?

I have been trying to solve this system of Differential equations for a lot of time now but the Mathematica software is always showing the message that it has encountered a non-numerical data point for the derivative at x=0.

System of Differential equations

The system of equations exists for the value at x=0 but the system keeps showing the same message. How can I avoid this? Any help would be deeply appreciated.

python – Guidance to implement Ordinary least squares more efficiently

I have implemented a program that carries out ordinary least squares in raw python (but using numpy arrays rather than lists). I would like for my code to be critiqued by the community.

What is a faster/more efficient way to implement ordinary least squares than what I have done in my code snippet below?

The assumption is that there is one feature dimension. (A.T * A is 2 X 2)

My goal is to implement this in C, and then the parallel-izable parts in CUDA C


mat = np.array((
    (1.,1.),
    (1.,2.),
    (1.,3.),
))

arr = np.array((
                (1., 2., 4.)
)).T


def least_squares(mat, arr):
    #matrix for A.T * A
    new_arr = np.array((
                (0.,0.),
                (0.,0.)
    ))
    #Right hand side vector
    new_right = np.array((
            (0., 0.)
    )).T

    #Solution Vector
    x_hat = np.array((
            (0., 0.)
    )).T
    #Calculate A.T * A
    for i in range(3):
        new_right(0)(0) += mat(i)(0) * arr(i)
        new_right(1)(0) += mat(i)(1) * arr(i)
        for j in range(2):
            new_arr(0)(j) += mat(i)(0) * mat(i)(j)
            new_arr(1)(j) += mat(i)(1) * mat(i)(j)

    #Perform Elimination on A.T * A & right side b        
    row_multiple = new_arr(1)(0)/new_arr(0)(0)
    new_arr(1) -= (row_multiple)*new_arr(0)
    new_right(1,0) -= row_multiple*new_right(0,0)

    
    #Solve for x-hat
    x_hat(1,0) = new_right(1,0) / new_arr(1,1)
    x_hat(0,0) = (new_right(0,0) - new_arr(0,1)*(new_right(1,0) / new_arr(1,1))) / new_arr(0,0)
    return x_hat

ct.category theory – Is the notion of a 2-category introduced to fix/forget the size issues in the definition of (an ordinary) category?

A category $mathcal{C}$ consists of pair of classes $(mathcal{C}_0, mathcal{C}_1)$, along with maps $$mathcal{C}_1times_{mathcal{C}_0}mathcal{C}_1rightarrow
mathcal{C}_1rightrightarrows mathcal{C}_0rightarrowmathcal{C}_1.$$

A category is called small, locally small, large, depending on the size of the class $mathcal{C}_1$.

  • If $mathcal{C}_1$ is a set (which would imply $mathcal{C}_0$ is a set), then, $mathcal{C}$ is called a small category.

  • If $mathcal{C}_1$ is a proper class, but $mathcal{C}(a,b)$ is a set for each $a,bin mathcal{C}_0$, then, $mathcal{C}$ is called a locally small category.

  • Anything that does not fall in above two cases is called a large category.

A $2$-category is described to be something, that comes with a class $mathcal{C}_0$, a category $mathcal{C}(a,b)$ for each $a,bin mathcal{C}_0$, some more data satisfying some conditions.

If I start with a category that is not locally small, I know that, $mathcal{C}(a,b)$ is not a set. Is it then a “better idea” to consider it as a category, along with some compatibility conditions, instead of “feeling bad” that it is not a set?

Is this how the notion of a $2$-category came into picture?

Is the size issue, in any way, related to the necessity of introducing the notion of a $2$-category?

Does the same justification hold in introducing the notion of an $n$-category?

If none of the above has a fairly positive answer, is there a way to introduce the notion of a $2$-category as an outcome of trying to fix the size issue in ordinary category theory? Is there a way to introduce the notion of a $n+1$-category as an outcome of trying to fix the size issue in an $n$-category?

ordinary differential equations – Can you solve a linearly independent ODE’s homogeneous part without the particular solution.

For example $y”-6y’+9y=xe^{3x},hspace{5mm} y(0)=0,y'(0)=0 $

I was wondering if it is possible to compute the homogenous solution even before working out the particular solution to this equation.

As an aside, I was wondering if the initial conditions are (0,0) for y’ and y then is it not the case that I don’t need the particular solution. As, in effect I am only lookin at the homogenous side of the equation.

I think my misunderstanding comes form the fact that in $y_{c}=y_{p}+y_{h}$ i see $y_{p}$ as a translation or rotation of the solution space away from the origin, and therefore if the inital condition are centered at (0,0) then I know the solution must exist in $y_{h}$.

ordinary differential equations – Invariant Lines

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ordinary differential equations – How to solve $dy/dx = f(g(x,y))$

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ordinary differential equations – Formula for the derivative of finite power series in reversed order of terms.

I wanted to solve the polar part in Schrödinger’s wave equation for the H-atom by direct substitution of functions of form:-
$$
Theta_{lm}(theta) = a_{lm} sin^{|m|}theta sum_{r≥0}^{r≤(l-|m|)/2}(-1)^rb_r cos^{l-|m|-2r}theta
$$

The $a$‘s are normalisation constants, no problem there. However, the problem of determining the $b$‘s ultimately drops down to finding the first and second derivatives of the polynomial in $z=cos theta$:
$$
P(z)=sum_{r≥0}^{r≤(l-|m|)/2} (-1)^rb_r z^{l-|m|-2r}
$$

Which is a finite power series written in decreasing order of powers. I couldn’t find a formula for so (well sometimes I get that dumb), but I think it does exist, maybe some reference book or website. I emphasize that what I’m doing is right the reverse of Frobenius-method. Thanks.

ag.algebraic geometry – Canonical lift of the deformation of an ordinary abelian variety

If $A/k$ is a principally polarised ordinary abelian variety ($k$ a perfect field of characteristic $p$, we may assume it is finite for simplicity), we have a canonical lift $hat{A}/W(k)$.
Now if I take a deformation $A_{epsilon}/k(epsilon)$, does there still exists a canonical lift of this deformation to a deformation of (the generic fiber of) $hat{A}$?

By the Kodaira-Spencer mapping deformations are essentially encoded by differentials of $A$, so the question boils down to whether differentials on $A$ lifts canonically to differentials on $hat{A}$. By Katz, Serre-Tate local moduli, Section 3, differentials on any lift $tilde{A}$ correspond to points in $T_p(A^vee)(k)$, and his main theorem 3.7.1 describe the compatibility of this identification with the Kodaira-Spencer map. Is there a way to use this to lift differentials canonically on the canonical lift?