Inclusion_exclusion general formula for intersections?

Assume $A_1,, A_2, ldots , A_n$ are subsets of a finite set $S$.
Can we find an expression for the size of $S-{A_1cap A_2 cap ldots
cap A_n}$
in term of the unions of any number of $A_i$‘s
(similar to the one we have for $S-{A_1cup A_2 cup
ldots cup A_n}$
in term of the intersections of the sets
$A_i$‘s)

Rate of convergence of the steepest descent method on a general linear function

Can someone help me with the proof of this theorem (it’s theorem 3.4 from the Nocedal and Wright book – Chapter 3 – Line search methods) regarding the convergence rate of the steepest descent method on a general nonlinear objective function:

T.3.4 Suppose that $f: Bbb R^n to Bbb R$ is twice continuously differentiable function function, and that the iterates generated by the steepest descent method with exact line searches converge to a point x* where the Hessian matrix is $nabla^2f(x^*)$ is positive definite. Then

$$f(x_{k+1})-f(x_k)leleft(frac{lambda_n-lambda_1}{lambda_n+lambda_1}right)^2[f(x_k)-f(x^*)]$$
where $lambda_1leldotslelambda_n$ are the eigenvalues of $nabla^2f(x^*)$.

I know you’re supposed to use the Kantorovich inequality, similar to theorem 3.3, the difference is that here the function is nonlinear, and in T 3.3 the function is strongly convex quadratic (which is what is confusing me).

General solution of ODE of constant 3×3 matrix

Determine the general solution of the system y’=Ay, where A is a constant matrix, defined by

A = begin{bmatrix}-5&-8&4\2&3&-2\6&14&-5end{bmatrix}

After attempting to find the eigenvalues of the system I end up with eigenvalues $lambda$=-1,-3,-3, where -3 has a multiplicity of 2. Then, finding the corresponding vector for $lambda$=-1, u=
begin{bmatrix}3\-1\1end{bmatrix}for $lambda$=-3, v=
begin{bmatrix}-2\1\1end{bmatrix}

and for the $2^{nd} lambda =-3$ the vector w=
begin{bmatrix}-1-2t\frac{1}{2}+t\tend{bmatrix}

I understand how to get the first two eigevectors u and v, but how did they get the third eigenvector to be in terms of t? Is it because it has a multiplicity of 2, so there is a second solution that can be described in terms of a variable? If so, what is the process to finding that third eigenvector? And does this process generalize for an eigenvalue that would have perhaps a multiplicty of 3?

The general solution of the system is

$y(t) = C_{1}e^{-t}u + C_{2}e^{-3t}v + C_{3}e^{-3t}w$

where u,v, and w are defined above.

Are there sola scriptura inerrantists who reject sensus unum as a general hermeneutic?

I have noticed that adherers to sola scriptura that defend biblical inerrancy nearly always argue from a sensus unum understanding scripture as opposed to a sensus plenior understanding of scripture.

Those who argue for a sensus unum will say that a sensus plenior contradicts biblical inerrancy.

Are there any schools of Reformed thought (liberal or otherwise) which have rejected the sensus unum view and yet still affirm biblical inerrancy from a view of sensus plenior? Or for that matter, are there any Reformed persons who argue for such a position?

From the General Thrust Equation towards Tsiolkovsky, how to explain dropping these terms along the way?

The NASA Glen Research Center tutorial page Rocket Thrust Equation links to the General Thrust Equation page which starts with

$$F = (dot{m} V)e – (dot{m} V)0 + (pe – p0) Ae$$

where $e$ and $0$ indicate nozzle exit and free stream, $Ae$ is the nozzle exit area, and $F$ is thrust, the force on the vehicle.

There are three terms on the right side, and as far as I understand it, the derivation of the Tsiolkovsky rocket equation in a vacuum uses only the first term.

If you had to explain the dropping of the 2nd and 3rd terms in a way that could be understood and believed by beginners to rocket science (like me) but without handwaving, “take my word for it”-ing or “go look it up”-ing, or “go google it”-ing, what would you say while holding the chalk and crossing out each of the last two terms?

Are there any general guidelines for proving limits of multivariable functions?

Today I was trying to prove that

$$lim_{(x, y) to (0, 0)}dfrac {x^2y^2}{x^2+y^2} = 0 $$

I got really lucky because the AM-GM inequality directly applies here to give us
$$dfrac {x^2y^2}{x^2+y^2} le dfrac {x^4 + y^4}{x^2+y^2} le dfrac {(x^2+y^2)^2}{x^2+y^2} = x^2+y^2$$

And thus we may choose $delta = sqrt epsilon$.

However, this was really lucky to turn out so cleanly. My question is, when it isn’t so clean, are there any general guidelines?

For example:

  • Do you look at the $|f(x, y)- L| < epsilon$ and try to manipulate it? What exactly is the goal when you try to manipulate this?
  • Do you look the $sqrt {(x-a)^2 + (y-b)^2} < delta$ term and try to manipulate it? What exactly is the goal when you try to manipulate this?

Etc.

Thank you.

integration over general measure space (proposition 9, chapter 18.2 in Royden)

enter image description here

I don’t understand how “$mu(X_infty) le int_X f dmu < infty$” proves that $f$ is finite a.e. on $X$. I think that this just shows $mu(X_infty) < infty$.

I also don’t understand the proof for $sigma$-finite. By Chebychev’s inequality,
$mu(X_n) le n cdot int_X fdmu < infty$. But what if $n = infty$? I think that the inequality does not hold in this case.

I guess $E_n = X_n$ or $X_n setminus [cup_{k=1}^{n-1}X_k]$, right?

I appreciate if you elaborate on this.