amazon web services – Will critical security updates get applied even with “auto minor version upgrade” disabled?

RDS offers an “auto minor version upgrade” setting, described in the docs, which causes AWS to automatically upgrade your database engine from time to time:

If you want Amazon RDS to upgrade the DB engine version of a database automatically, you can enable auto minor version upgrades for the database.

When Amazon RDS designates a minor engine version as the preferred minor engine version, each database (with this setting turned on) is upgraded to the minor engine version automatically

However, some other AWS docs also describe something called “required software patching”, framed as distinct from other automatic engine upgrades:

Maintenance events that require Amazon RDS to take your DB instance offline are (bla bla bla), database engine version upgrades, and required software patching. Required software patching is automatically scheduled only for patches that are security and durability related.

(bolding mine)

I’m not sure how to interpret these two passages, taken together. One possibility is that if I turn off the “auto minor version upgrade” setting, no engine updates whatsoever will be applied automatically. Another possibility is that some engine updates – such as important security fixes – are “required” and thus will be applied even if I have the setting turned off, and that turning it on simply means that I’ll get some other engine updates applied too, even though they don’t involve important security fixes.

Understanding which interpretation is the correct one seems important when evaluating the security implications of turning “auto minor version upgrade” off. Which interpretation is true?

reference request – Question about an exact expression for the root-mean-square of the distances of the critical points to a given zero of a polynomial

Let $p(z) = prod_{j=1}^{l+1} (z – z_j)^{M_j}$ be a complex polynomial of degree $n$, where the $z_j$ are distinct for $1, ldots, l+1$. The first $l$ entries in the list ${z’_1, ldots, z’_{n-1} }$ of the not necessarily distinct critical points, $(p'(z’_k) = 0)$, shall be of the second kind, hence $p(z’_k) neq 0$, while the rest will be of the first kind, $p(z’_k) = 0$.

Consciously deviating slightly from
“Borcea’s variance conjectures on the critical points of polynomials” by D. Khavinson, R.Pereira, M. Putinar, E. Saff and S. Shimorin in Notions of Positivity and the Geometry of Polynomials, Series: Trends in Mathematics, Birkhäuser, Basel, 2011, pp. 283 – 309,
we define the augmented Gauss-Lucas-matrix $D_{ps}$ by

mathbf{D_{ps}} :=
1 – C_1 & ldots & 1 – C_{l+1} \
d_{11} & ldots &
d_{1(l+1)} \
vdots & & vdots \
d_{l1} & ldots &
d_{l(l+1)} \
end{pmatrix} , ,

where $$ d_{kj} := frac{M_j}{n} frac{H(z’_k)}{|z_j – z’_k|^2} , , qquad H(z’_k) := frac{n}{sum^{l+1}_{t=1} frac{M_t}{|z_t – z’_k|^2}} , , qquad C_j := sum_{k=1}^l d_{kj} .

This implies all row and column sums of $D_{ps}$ to equal 1, so it is a positive doubly stochastic matrix, if the answer to the following question is affirmative:

Is $C_j < 1$ true for each $j in { 1 , ldots, l+1 }$ of every complex polynomial $p(z)$ which has at least two distinct zeros?

This is an addition at the end of my old question The Poisson-kernel in the plane and polynomials, but it is of a different nature, I have a lot of additional info and even though there was recent progress on related questions, Sendov’s conjecture, the issue is not quite resolved, so I want to draw some attention to this topic again; the cited article can be retrieved here:

More background/related results:

  1. Connection to Sendov’s conjecture, Borcea’s variance conjecture:
    From $sum_{j=1}^{l+1} frac{z_j}{z_j – z’_k} = n$ one derives by an elementary, simple calculation $$D_{ps}(|z_1|^2, ldots, |z_{l+1}|^2)^t = (star, H(z’_1) + |z’_1|^2, ldots, H(z’_l) + |z’_l|^2)^t , (ast) , .$$
    As the entries of $D_{ps}$ are invariant with respect to translations in the complex plane of the zeros of $p(z)$, this implies
    $$D_{ps}(|z_1 – c|^2, ldots, |z_{l+1} – c|^2)^t = (star, H(z’_1) + |z’_1 – c|^2, ldots, H(z’_l) + |z’_l – c|^2)^t $$
    for an arbitrary complex $c$. Putting $c = z_i$ and summing over $k$ excluding the first row, we obtain
    $$ sum_{j=1}^{l+1} frac{C_j}{l} |z_j – z_i|^2 = frac{1}{l} sum_{k=1}^l (H(z’_k) + |z_i – z’_k|^2) , . $$
    We note the convex combination of the $|z_j – z_i|^2$ on the left and that equation $(ast)$ implies $ frac{1}{l}sum_{k=1}^l H(z’_k) leq 1$, if $|z_j| leq 1$ for all $j$, where equality occurs if and only if $p(z) = z^n – d$ and $|d| = 1$.
    Proceeding as in part 2 of the cited article, we get
    $$ sum_{j=1}^{l+1} |z_j – z_i|^2 = sum_{j=1}^{n} |z_j – S|^2 + n|z_i – S|^2 – sum_{k=l+1}^{n-1} |z’_k – z_i|^2 , , $$
    where $S$ is the centroid of the zeros $z_j$, using the relation between zeros and critical points of the first kind. Now comes the catch of our definition of $D_{ps}$: As all row and column sums of $D_{ps}$ equal 1, it preserves the component sum of each vector it is multiplied to, so without further effort, we obtain the equation
    $$ sum_{k = 1}^{n-1} ( H(z’_k) + |z_i – z’_k|^2) = r_1 + n|z_i – S|^2 + sum_{j=1}^{n}|z_j – S|^2 , , $$
    in general, where $$r_1 = sum_{j=1}^{l+1} (1-C_j)|z_j – z_i|^2 $$
    and $H(z’_k) = 0$ for the critical points $z’_k$ of the first kind, corresponding neatly to the usual convention for harmonic mean values. This gives a general exact expression for the root mean square of the distances of the critical points to a given zero of an arbitrary complex polynomial with at least two distinct zeros and an affirmative answer to our question above results in a sufficient though not necessary condition for $r_1$ to be strictly positive, implying
    $$frac{1}{n-1} sum_{k = 1}^{n-1} |z_i – z’_k|^2 < frac{n}{n-1} |z_i – S|^2 + frac{1}{n-1}sum_{j=1}^{n}|z_j – S|^2 – frac{1}{n-1} sum_{k = 1}^{n-1} H(z’_k) $$
    for every $z_i$, no matter the multiplicities of the zeros.
  2. If $D_{ps}$ is doubly stochastic, it gives a direct approach to the majorization of the critical points and the centroid by the zeros, which was used by Pereira and independently Malamud to settle a number of open problems in the early 2000s.
  3. $C_j = frac{n – M_j}{n}$ if all zeros are on a straight line, see the cited article, or on a circle, see my old question.
  4. Using $p'(z’_k)/p(z’_k) = 0$ and the inequality for the arithmetic mean and the root mean square, I have proven
    $$d_{kj} leq frac{n – M_j}{n} , ,$$
    where equality occurs if and only if there are exactly two distinct zeros.
  5. There is no other reasonable inequality for a single $d_{kj}$, but the $C_j$ seem to be bounded from below about as tightly as they are from above.
  6. Let all zeros be of the same multiplicity, then the perpendicular bisector theorem in polynomial geometry implies the existence of a row $k$, where $d_{ki} geq d_{kj}$, if a fixed column $i$ is compared to any single column $j$, but the row $k$ cannot always be chosen to be the same for all columns $j$. Apart from this result, I have only empirical knowledge of the entries of $D_{ps}$ in general.
  7. Examples of the type $p(z) = (z^n – 1)(z – gamma)$, where $gamma$ is a complex number of small absolute value illustrate the points 4, 5 and 6.
  8. The question about the $C_j$ has been checked for a number of parametric families of polynomials of low degree and some random configurations of low degree by my pupils and me, the answer has always been affirmative up to now, the usual symmetric extremal configurations $p(z) = z^n – 1$ and $z^n – z$ also fit in nicely.
  9. For large degree $n$, the bounds get arbitrarily tight and counterexamples may well exist.

fa.functional analysis – Morse approximation with bounded number of critical points

Let $(M^3,g)$ be a compact Riemannian 3-manifold and let $fin C^{infty}(M)$ be a smooth function. Does there exist a constant $k>0$ (possibly depending on $M$ and $g$) such that $f$ can be $C^2$-approximated by Morse functions whose number of critical points is at most $k$?

In other words, if we define $mathrm{Morse}_k(M,g)subset C^{infty}(M)$ to be the subset of Morse functions which have at most $k$ critical points, then can $k$ be chosen large enough so that $mathrm{Morse}_k(M,g)$ is dense in $C^2(M)$?

If such a $k$ exists, then the Morse inequalities give a necessary lower bound for $k$ in terms of the total Betti number, namely
$$k geq b_0(M)+b_1(M)+b_2(M)+b_3(M).$$
I am, however, asking about the existence of an upper bound for $k$.

Any references would be appreciated.

design – How to avoid points of critical failure?

Use Sqlite as your persisted outbound-queue.

You already have a conceptual outbound-queue from each device, which is implemented in memory (not persisted).

Sqlite is very very reliable. It has been pounded by billions of devices, and they have reached safety-critical certifications for the software. It also can practically run on almost every device. If you have limited disk space, you can quite easily run an SQL query to implement whatever data-trimming policy you design.

The internet itself is quite unreliable, particularly from the perspective of IoT devices. So the only way to avoid the biggest point of failure (prolonged internet unavailability) is to persist the outbound queue.

Even better if you can connect directly to the devices and query the sqlite database. Because the queue is already persisted, there’s no need for an intermediate queue (RabbitMQ). (My company has plans to do this for our software, and to opensource the tooling).

algebraic topology – Critical subgraph of the Kneser graph

Let the Kneser graph $X_{n,k}$ be the graph of subsets of $(n)$ with $k$ elements, such that two subsets are considered neighbors if they are disjoint.
One can prove, using some formulations of the Borsuk Ulam theorem, that the Kneser graph has chromatic number $n-2k+2$
A vertex critical subgraph is the smallest subgraph (smallest number of vertices) that has the same chromatic number as the full graph.

Question: what is the vertex critical subgraph of Kneser graph?
I read that it is the subgraph generated by the subsets which doesn’t contain consecutive numbers, But I don’t know how to prove it.

Connecting paths of all residues modulo k in color critical graphs

I recently stumbled over the following question, and could not really find helpful references.

For an integer $k$ a graph $G$ is called k-color critical if $chi(G)=k$ but $chi(G’)<k$ for every proper subgraph $G’ subseteq G$.

Is there an absolute constant C>0 such that the following holds:

If G is a k-color critical graph, then for every two distinct vertices $u$ and $v$ in $G$, for every number $q<k/C$, and every integer $r le q$, there exists a path $P$ in $G$ connecting $u$ and $v$ such that $P$ has length congruent to $r$ modulo $q$?

integration – Duhamel’s formula for a critical operator

I have an operator that has a non-trivial solution to the homogeneous equation.

Let $hat{e}_0$ be the critical eigenvector and C(t) be the corresponding amplitude of an operator $mathcal{L}$ of an operator $frac{d}{dt} – mathcal{L}$ such that $(frac{d}{dt} – mathcal{L}) C(t) hat{e}_0 = 0$.

Otherwise, $mathcal{L}$ has a discrete spectrum and complete orthonormalised basis of eigenvectors.

I know by the Fredholm alternative, that I cannot now solve this equation uniquely
(frac{d }{d t} – mathcal{L}) u = f
where $f$ is a forcing term. However, can I still write the solution in the following form?

u = e^{tmathcal{L}} u_0 + int^t_0 e^{mathcal{L}(t-s)}f(s) ds

How do I specify the homogeneous solution in the above format?

Best wishes,


calculus – Finding the critical numbers for a function of several variables

I have a function $f(x,y) = x^3+y^3 – 3x^2 – 3y^2 -9x$. I want to the find the critical values of the function so I got $f_x = 3x^2 – 6x – 9=0$ and $f_y = 3y^2 – 6y = 0$.

I have determined that $x=3,x=-1$ and $y=0,y=2$.

Are the critical numbers to be paired up: $(3,0), (3,2), (-1,2),(-1,0)$? I am quite confused about why I need to pair these up.