## co.combinatorics – Trade-off between radius and covering number using \$d\$-dimensional balls

Given any $$d$$-dimensional solid $$X$$, let the length of the longest line segment connecting two points of $$X$$ be equal to $$1$$. How can we prove the following conjecture?

For any integer $$n ge 1$$, there exists a radius $$r$$ and a positive constant $$c$$ (i.e., independent of $$X$$, $$n$$, $$d$$ and $$r$$) such that

$$rlefrac{c}{n^{1/d}}$$

where $$n$$ is the number of $$d$$-dimensional balls having radius $$r$$ that completely cover $$X$$, with possible overlaps.

If the conjecture is false in general, it is possible to add a condition bounding $$d$$ (as a function of $$n$$ or $$r$$) such that the inequality holds?

## mysql – covering index mystery?

So has anyone else applied a covering index in MySQL and gotten improved performance but when you run explain, it shows using where, using temporary under the extra column? Any thoughts why this is the case? I’m happy about the performance improvement but the engineer in me wants to know why the extra column is not saying “using index”. If you need me to provide details, I will but this question is directed to the experts who have already experienced this with MySQL and know why. I would add that in my particular case, I applied a covering index to an update statement. All the columns in the query were included in the composite (covering) index. Here is an example of exactly what I did:

Table A has the following properties:
25,000 rows.
Has the following columns: one, two, three, four, five, six, seven, eight, and nine.

My covering index query target:

update table A set one = ‘somevalue’, two = ‘somevalue2’, four = ‘somevalue3’, five = ‘somevalue4’
where seven = ‘somevalue5’ and eight = ‘somevalue6’

covering index on table A:
idx_covering(seven,eight,
one,two,four,five)

## real analysis – A Covering Lemma for Arbitrary Measures

In the book “Harmonic Measure” by Garnett and Marshall, we have the following result:

Lemma I.2.3 Let $$mu$$ be a positive Borel measure on $$partial{mathbb{D}}$$ and let $${I_{j}}$$ be a finite sequence of open intervals in $$partial{mathbb{D}}$$. Then $${I_{j}}$$ contains a pairwise disjoint subfamily $${J_{k}}$$ such that

$$sumlimits_{k}mu(J_{k})geq{frac{1}{3}mubig(cup_{j}I_{j}big)}$$

By repeating the argument in Garnett and Marshall we get the same result for positive measures $$mu$$ on $$mathbb{R}$$, only with a better constant- $$1/2$$ instead of $$1/3$$. Does an analogous result hold in $$mathbb{R}^{n}$$ for $$n>2$$? Namely, does there exist an absolute constant $$c=c(n)>0$$ such that if $$mu$$ is a positive Borel measure on $$mathbb{R}^{n}$$ and $${Q_{i}}$$ is a finite sequence of balls or cubes, then there exists a pairwise disjoint subfamily of balls or cubes $${J_{k}}$$ such that

$$sumlimits_{k}mu(J_{k})geq{ccdot{}mubig(cup_{j}I_{j}big)}$$

If this fails in higher dimensions, does there exist a characterization of the measures for which such a condition holds?

## ag.algebraic geometry – Bounded non-symmetric domains covering a compact manifold

This question is somewhat related to this other question of mine.

I was wondering which are the known examples of bounded domains $$Omega$$ in $$mathbb C^n$$ admitting a compact free quotient.

By a theorem of Siegel, such a domain must be holomorphically convex. Moreover, if the boundary is sufficiently regular, say $$C^2$$ (even if, by a recent theorem of A. Zimmer, $$C^{1,1}$$ suffices), by the classical theorem of Wong-Rosay, then $$Omega$$ must be biholomorphic to the unit ball.

Of course all bounded symmetric domains give such examples, by a classical theorem of E. Borel. But I am interested in more “exotic” examples, specifically non symmetric examples.

The only I am aware of live in $$mathbb C^2$$ and are the universal covers of Kodaira fibrations (see this question for more details).

1. Is it possibile for instance to construct higher dimensional analogous of the universal cover of a Kodaira fibration?
2. Which are the non symmetric but still homogeneous bounded domains in $$mathbb C^n$$? Do they admit in some case a compact free quotient?

Remark that I am really looking for compact, free quotients.

Thank you very much in advance.

## postgresql – Postgres 11+: Are covering indexes (INCLUDE) useful for join / where conditions?

I want to better understand when index covering can be useful to enable index-only scans in Postgres 11+. As the documentation says, given the cover index

``````CREATE INDEX tab_x_y ON tab(x) INCLUDE (y);
``````

Queries like this can only use it for index scans:

``````SELECT y FROM tab WHERE x = 'key';
``````

Now I'm wondering if such a cover index could also allow index scans if the cover columns are shown as conditions. For example, assume a cover index:

``````CREATE INDEX tab_x_y_z ON tab(x) INCLUDE (y, z);
``````

Would this only allow index scans for the following queries?

``````SELECT z FROM tab WHERE x = 'key' AND y = 1;

SELECT x, y, z FROM (VALUES ('key1'),('key2'),('key3')) sub(id)
JOIN tab ON tab.x = sub.id WHERE y = 1;
``````

## How does Rank I of the Heavy Gunner talent Covering Fire react with melee attacks?

Rank I of the Heavy Gunner Talent Covering Fire is on page 97 of the Lancer Core Book First Edition PDF as follows (focus on mine):

As a quick action, choose a character within the line of sight and range of one of your HEAVY ranged weapons and within 10 spaces: you will be affected until the start of your next turn. If your target moves more than 1 field for the duration, Impaired will be deleted You can attack them in response to a HEAVY-RANGE weapon to do half damage, (heat), or (burning).and then this effect ends. You can carry out this attack at any time during your movement (e.g. wait until you leave the cover).

COVERING FIRE can only affect one character at a time – subsequent uses replace previous ones – and ends immediately if your target damages you.

How p. With 104 of the core book notes under the "Patterns" heading in the "Weapons Tags" section, some weapons affect an area rather than a single target:

Some weapons and systems have special attack patterns: LINE, CONE, BLAST and BURST. These attacks affect all targets in a defined area and require a separate attack roll for each target. The damage is rolled once and the bonus damage is halved if multiple targets are affected.

The section then describes how these 4 types of AoEs work in particular.

How does Covering Fire's response to melee attacks work?

• First of all, AoE attacks (with heavy ranged weapons) can even be
made with the Covering Fire reaction?
• If so, they must be centered (or otherwise focused) on the
Character selected for covering fire? Or they can be placed in any one
Configuration as long as the selected character is in the AoE? (Tut
The answer depends on which AoE pattern the attack must use.)
• Does the reaction attack affect only the selected character? Or do it
Impact on other goals in AoE how else? (But if
affect other AoE targets, take full damage / heat / burn,
or is it also halved?)

## Euclidean geometry – covering the disk with a family of infinite dimensions – the convex continuation

To let $$(U_n) _n$$ be an arbitrary sequence of open convex subsets of the unit disk $$D (0,1) subseteq mathbb {R} ^ 2$$ s.t. $$sum_ {n = 0} ^ infty lambda (U_n) = infty$$ (Where $$lambda$$ is the Lebesgue measure). Is there a sequence? $$(q_n) _n$$ in the $$mathbb {R} ^ 2$$ s.t. $$D (0,1) subseteq bigcup_ {n = 0} ^ infty (q_n + U_n)$$?

With the notation $$q_n + U_n$$, I mean
$$q_n + U_n: = {x in mathbb {R} ^ 2 | x-q_n in U_n }$$

This question is very similar to this one, but I was asked in the comments to ask anyway.

## mysql – Is the order of the columns in a covering index important?

I have the following coverage indexes:

``````INDEX (col1, col3); -- index 1
INDEX (col1, col2, col3); -- index 2
``````

because I want to support the following types of queries:

1)

``````  SELECT ...
FROM my_table
WHERE col1 = ... AND
col2 = ...
ORDER BY col3
``````

2)

``````  SELECT ...
FROM my_table
WHERE col1 = ...
ORDER BY col3
``````

I am not familiar with how a cover index works. Is index 1 redundant? Or does a coverage index require the columns to be next to each other?

## Euclidean geometry – covering the disk with a family of infinite dimensions

To let $$(U_n) _n$$ an arbitrary sequence of open subsets of the unit disk $$D (0,1) subseteq mathbb {R} ^ 2$$ s.t. $$sum_ {n = 0} ^ infty lambda (U_n) = infty$$ (Where $$lambda$$ is the Lebesgue measure). Is there a sequence? $$(q_n) _n$$ in the $$mathbb {R} ^ 2$$ s.t. $$D (0,1) subseteq bigcup_ {n = 0} ^ infty (q_n + U_n)$$?

With the notation $$q_n + U_n$$, I mean
$$q_n + U_n: = {x in mathbb {R} ^ 2 | x-q_n in U_n }$$

## Ag.algebraic geometry – Galois covering, which corresponds to the finite quotient of the étale basic group

To let $$X$$ be a connected scheme,$$pi_1 (X, bar {x})$$ his étale fundamental group for a geometric point $$bar {x}: Spec (K) rightarrow X$$
and $$E = pi_1 (X, bar {x}) / N$$ a finite quotient of $$pi_1 (X, bar {x})$$

I'm looking for a book or a work that describes the explicit structure of the Galois cover $$Y rightarrow X$$ corresponding to $$E$$ unlike the book by Grothendieck's SGA or Tamas Szamuely