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 $R(tau)$:

$$R(tau):=left(sum_{substack{1le i<j<kle n: \ x_i,x_j<tau,~~x_k>tau}}
D-(x_j-x_i)right)+
left(sum_{substack{1le i<j<kle n: \ x_i<tau,~~x_j,x_k>tau}}
D-(x_k-x_j)right)~.
$$

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

$$R(i,j,k):={sum_{1le i<j<kle n}
max{D-(x_j-x_i),D-(x_k-x_j)}}~,$$

and

$$R'(tau):=left(sum_{substack{1le i<j<kle n: \ x_i,x_j<tau,~~x_kgetau}}
R(i,j,k)right)+
left(sum_{substack{1le i<j<kle n: \ x_i<tau,~~x_j,x_kgetau}}
R(i,j,k)right)~.
$$


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.enter image description here

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;
        padding-left: 20px;
        li {
            counter-increment: item;
            padding: 0 0 20px 5px;
            position: relative;
            z-index: 0;

            &::before {
                content: counters(item, ".") ". ";
                display: table-cell;
                position: absolute;
                left: -22px;
            }
            ol.rulebook-alpha {
                counter-reset: alpha;
                padding-left: 30px;

                li {
                    padding: 20px 0 0 10px;

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

This is my output
enter image description here

My aim is to have 1.2 instead of 1.4 could you anyone please help me