## co.combinatorics – Optimization problem on sums of differences between real numbers with combinatorial constraints

We are given a sequence $$S_n$$ of $$n$$ points on a straight line $$L$$, whose coordinates are denoted by $$x_1, x_2, ldots, x_n$$ in non-decreasing order (i.e., the corresponding non-decreasing ordered sequence of Euclidean distances between each point of $$S_n$$ and an arbitrarily chosen point of $$L$$). Let $$D$$ be equal to $$max_{i,jin (n)} |x_i-x_j|$$.

We denote by $$tau$$ a threshold point on $$L$$ maximizing the following sum $$R(tau)$$:

$$R(tau):=left(sum_{substack{1le itau}} D-(x_j-x_i)right)+ left(sum_{substack{1le itau}} D-(x_k-x_j)right)~.$$

Given three distinct points with indices $$i, j$$ and $$k$$, we define

$$R(i,j,k):={sum_{1le i

and

$$R'(tau):=left(sum_{substack{1le i

Question: What is the minimum value for the ratio $$frac{R(tau)}{R'(tau)}$$ over all possible sequences $$S_n$$, asymptotically for $$ntoinfty$$?

## mysql – Why some DATETIME columns return numbers instead of UTC strings?

Given the following table,

``````CREATE TABLE test (
id int unsigned NOT NULL AUTO_INCREMENT,
created_at datetime NOT NULL DEFAULT CURRENT_TIMESTAMP,
updated_at datetime DEFAULT CURRENT_TIMESTAMP ON UPDATE CURRENT_TIMESTAMP,
completed_at datetime NOT NULL DEFAULT CURRENT_TIMESTAMP,
PRIMARY KEY(id)
);
``````

when I `SELECT * FROM test` from the MySQL terminal, the date columns all show string dates as expected.

When I use the same `select` query in a Node backend using the MySQL xDevAPI, the `created_at` and `completed_at` columns return numbers while the `updated_at` column returns a UTC string.

Why does this happen?

I want to only use UTC strings, and only want to change the `updated_at` date on `update` statements but MySQL docs seem to not address what’s happening here so I don’t know if it’s MySQL or the xDevAPI.

## Does the étale topos determine the Hodge numbers?

Does the étale topos (considered as an abstract Grothendieck topos) of a smooth proper variety determine its Hodge numbers?

## Wallace tree – multiplying two 3 bit numbers

I’m trying to draw a Wallace Tree to multiply two 3 bit numbers, I wonder if it’s correct or not.

## finite element method – Problem with NDSolveValue : “The function value {\$Failed} is not a list of numbers with dimensions…”

I was having fun modifying a code given to me as an answer to a previous problem here, courtesy of user Alex Trounev (Thank you again), when I encountered a certain error which I had never seen before.

Here is the aforesaid code :

``````(*parameters*)
r0 = 0.5;
h = 1;
(Alpha) = 0.8;

(*region definition*)
reg = Cuboid({.5, 0., 0.}, {1., 2 Pi, 1.});

reg3D = ImplicitRegion(
r0^2 <= x^2 + y^2 <= 1 && 0 <= z <= 1, {x, y, z});

(*equation + conditions*)
eq1 = D(u(t, r, (Theta), z),
t) - (D(u(t, r, (Theta), z), r, r) +
1/r*D(u(t, r, (Theta), z), r) -
1/((Alpha)^2 r^2) D(u(t, r, (Theta), z), (Theta), (Theta)) +
D(u(t, r, (Theta), z), z, z));

ic = u(0, r, (Theta), z) == 1;

bc = DirichletCondition(u(t, r, (Theta), z) == Exp(-5 t), r == r0);
nV = NeumannValue(1, r == 1);
pbc = PeriodicBoundaryCondition(u(t, r, (Theta), z), (Theta) == 0,
TranslationTransform({0, 2 (Pi)*(Alpha), 0}));

(*solution computation*)
sol = NDSolveValue({eq1 == nV, ic, bc, pbc},
u, {t, 0, 2}, {r, (Theta), z} (Element) reg);

(*frames=Table(DensityPlot3D(sol(t,Sqrt(x^2+y^2),ArcTan(x,y),z),{x,y,
z}(Element)reg3D,ColorFunction(Rule)"Rainbow",OpacityFunction(Rule)
None,Boxed(Rule)False,Axes(Rule)False,PlotRange(Rule){0,1.5},
PlotPoints(Rule)50,PlotLabel(Rule)Row({"t =
",t}),ColorFunctionScaling(Rule)False),{t,.05,1,.05})
ListAnimate(frames)*)
``````

When I run the code, after some time, I get greeted with the following error :

`NDSolveValue::nlnum: The function value {\$Failed} is not a list of numbers with dimensions {39639} at {t,u(t,r,(Theta),z),(u^(1,0,0,0))(t,r,(Theta),z)} = {0.0138161,{<<1>>},{-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,<<15>>,-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,-4.66626,<<39589>>}}.`

When I click on the three dots next to the error, I don’t find any information on the error like it’s usually the case. I then decide to google some answers.
I found some answers here while also trying to comprehend the error by looking at this and finally that answer here.

So if I did understand it correctly, such error arises when you use NDSolve (or NDSolveValue) to get a symbolical solution to your equation, but problems come up when you try to numerically evaluate it for plotting purpose, or when trying to get a symbolical result with a function that requires numerical values ?

In any case, I do not really understand why I get such error as my plot part is currently between (* … *) so it shouldn’t matter. As for the rest of the code, I do not really see an error but I am just a beginner so…

Anyway, can a kind fellow enlighten me please ?

## dg.differential geometry – Instanton numbers for diverse gauge bundles on diverse manifolds — their relations to characteristic classes

It is standard (?) that the $$SU(N)$$ gauge theory has the instanton number $$n$$ quantized as $$n in mathbb{Z}$$
$$n = { 1 over 8pi^2} int_{mathcal{M}_{4}} text{tr} left(F wedge Fright) = {1 over 64 pi^2} int_{mathcal{M}_{4}} d^4 x , epsilon^{munurholambda} F_{munu}^alpha F_{rholambda}^alpha in mathbb{Z}$$

Here $$F$$ is the curvature 2-form $$F=d A + A wedge A$$ and $$A$$ is the gauge 1-connection of the gauge bundle of gauge group $$G$$. Also $$F=F^alpha T^alpha$$ where $$alpha$$ is the Lie algebra indices, with repeated indices summed over.

We know that:

• If the gauge group is $$G=SU(2)$$, the instanton number $$n in mathbb{Z}.$$ I think this can be understood as
$$n = c_2(V_{SU(2)}) in mathbb{Z}?$$
• If the gauge group is $$G=SO(3)$$ on the spin 4-manifold $$mathcal{M}_{4}$$, then $$n in frac{1}{2}mathbb{Z}$$. (I use the notation to mean that $$n in frac{1}{2}mathbb{Z}$$ as $$n$$ takes the half integer values.)
$$n = p_1(V_{SO(3)})/4 in frac{1}{2}mathbb{Z}?$$
• If the gauge group is $$G=SO(3)$$ on the non-spin 4-manifold $$mathcal{M}_{4}$$, then $$n in mathbb{Z}/4$$.
$$n = p_1(V_{SO(3)})/4 in frac{1}{4}mathbb{Z}?$$

What are the general statements we can make for other general $$G$$ and other manifolds?

Questions:

1. If the gauge group is $$G=U(1)$$, the instanton number $$n in 2 mathbb{Z}.$$ True or False? We can express this $$n$$ as the first Chern class $$c_1$$ square of associated vector bundle $$V_{U(1)}$$ as
$$n = c_1(V_{U(1)})^2 in 2 mathbb{Z}?$$
Is this correct?

2. If the gauge group is $$G=SU(N)$$, the instanton number $$n in mathbb{Z}.$$ True or False? We can express this $$n$$ as the second Chern class $$c_2$$ of associated vector bundle $$V_{SU(N)}$$ as
$$n = c_2(V_{SU(N)}) in mathbb{Z}?$$

3. If the gauge group is $$G=PSU(N)$$ on the spin 4-manifold $$mathcal{M}_{4}$$, the instanton number can be $$1/N$$ fractional of $$mathbb{Z}$$ values:
$$n in frac{1}{N} mathbb{Z},$$
True or False? What is the precise mathematical invariant to characterize this $$n in frac{1}{N} mathbb{Z}$$? Is that Pontryagin class $$p_1$$ when $$G=PSU(N)$$ is real $$mathbb{R}$$? Or some $$c_2(V_{PSU(N)})$$ when $$PSU(N)$$ is complex $$mathbb{C}$$?

4. If the gauge group is $$G=PSU(N)$$ on the non-spin 4-manifold $$mathcal{M}_{4}$$, the instanton number can be $$1/N^2$$ fractional of $$mathbb{Z}$$ values: $$n in frac{1}{N^2} mathbb{Z},$$
True or False? What is the precise mathematical invariant to characterize this $$n in frac{1}{N^2} mathbb{Z}$$? Is that Pontryagin class $$p_1$$ when $$G=PSU(N)$$ is real $$mathbb{R}$$? Or some $$c_2(V_{PSU(N)})$$ when $$PSU(N)$$ is complex $$mathbb{C}$$?

5. If the gauge group is $$G=U(N)$$ on the spin or non-spin 4-manifold $$mathcal{M}_{4}$$, the instanton numbers carry both the $$U(1)$$ and $$PSU(N)$$ part with constraints. So there are two instanton numbers
$$n_{U(1)}$$ and $$n_{PSU(N)}$$.
What can be their constraints?
$$n_{U(1)} in ?$$
$$n_{PSU(N)} in ?$$

## postgresql – Best field type for cryptocurrency big numbers?

I’m using PostgreSQL for my cryptocurrency exchange database, the question is: for saving currency amount (numbers) with their precision (like 323232323232323.45454545 ~ 23 digit + 1 dot: 15 digit before dot and 8 digit after that), Should I use `varchar(24)` type for them or `double precision` or `numeric(15,8)` ?

Note: It seems that `double precision` type cant properly save big numbers like example above and it will be rounded to 323232323232323!

Witch one has better performance (speed) and needs less resources?

## How can I calculate variance under certain condition in APPLE numbers

I want to do something like STDEVA(IF()) or STDEVA(FILETER()) in APPLE’s numbers, is it possible?

thank you very much

## linear algebra – Given a chain of commuting matrices over the complex numbers, can we build one over the real numbers?

Suppose we have two $$ntimes n$$ matrices $$A$$ and $$B$$ with entries in $$mathbb{R}$$, and two non-scalar matrices $$X$$ and $$Y$$ with entries in $$mathbb{C}$$, such that $$AX=XA$$, $$XY=YX$$, and $$BY=YB$$.

Is it necessarily the case that there exist non-scalar matrices $$X’$$ and $$Y’$$ with entries in $$mathbb{R}$$ such that $$AX’=X’A$$, $$X’Y’=Y’X’$$, and $$BY’=Y’B$$?

(Here “non-scalar” just means that the matrices aren’t scalar multiples of the identity matrix.)

## sass – A styled ordered list whose outer list should have numbers and inner list should have letters using CSS counter property

I am trying to styles an ordered list where i want all the outer list elements to have numeric list style and inner list item to have letters.

The below is my structure.

`````` <div class="subitems">
<ol><li>This is a list with numeric list style
<ol><li>This is a list with numeric list style
<ol class="rulebook-alpha"><li>This is a list with numeric list style</li>
<li>This is a list with numeric list style</li>
</ol></li>
<li>This is a list with numeric list style</li>
</ol></li>
</ol>
</div>
``````

This is my css

``````div.subitems {
ol {
list-style-type: none;
counter-reset: item;
li {
counter-increment: item;
position: relative;
z-index: 0;

&::before {
content: counters(item, ".") ". ";
display: table-cell;
position: absolute;
left: -22px;
}
ol.rulebook-alpha {
counter-reset: alpha;

li {

&::before {
counter-increment: alpha;
content: "("counter(alpha, lower-alpha) ") ";
}
}
}

}
}

}
``````

This is my output