## Can someone give me an example in real numbers where the outer dimension and the inner dimension are the same?

If E is contained in [a, b], then the outer dimension of E is defined as g.l.b the length of G, taking g.l.b over all open sets G that contain E.

The inner measure of E is defined as l.u.b the length of F, where l.u.b is taken over all closed sets F contained in E.

Now my question in real numbers is: Give an example in which outer dimension and inner dimension are the same.
Explain it in detail

## Find the outer measure and the sigma algebra generated by the caratheodal measure

There is the question

Enter image description here

Where $$U _ {*}$$ is defined by outer dimension
$$u _ {*} = inf_ {A_k in mathcal {A} \ E subset cup A_k} sum_ {k} u_ {0} (A_k)$$

With $$Omega$$ denotes the room $$U_0$$ Premass is and $$mathcal {A}$$ Algebra is in $$Omega$$ and
$$sum_ {c} = {E in Omega | E , , {text is caratheodary measure} , , (u _ {*} – measure) }$$

Thank you very much.

## Outer Dimension – Prove that there is a limited set \$ A \$ such that \$ m ^ * (F) le m ^ * (A) -1 \$ for each closed set \$ F subset A \$

Prove that there is a limited amount $$A subset mathbb {R}$$ so that $$m ^ * (F) le m ^ * (A) -1$$ for every closed sentence $$F subset A$$,

Here $$m ^ *$$ is the outer dimension.

I used the limited and immeasurable Vitali set $$V$$and defined $$A = tV$$, from where $$t> 0$$, but I could only prove that
$$m ^ * (A setminus F) ge1$$
Since then, however $$A$$ is also not measurable, I have $$m ^ * (A) le m ^ * (F) + m ^ * (A setminus F)$$So I don't know how to end the evidence.

## unityscript – Unity 3d raycast from local space to outer space

I want a raycast direction that does not relate to the rotation, but what happens is that the raycast points to a point as if it were a transformation

Why does the event help?

``````
if(Physics.Raycast(transform.position, vector3.forward, out hit, SupportedDistance, CanBeSupported))
{
NotSupported = false;

//
}
``````

## Oracle – How do I use values ​​from the outer query in the sub-subquery?

I can not query data in subqueries based on the value of the outer query. How does that work in this query?

``````SELECT
c.txt_description,
COUNT(DISTINCT(j.int_vacancynum) ) AS vacnum,
(
SELECT
COUNT(*)
FROM
(
SELECT
job.txt_appkey,
COUNT(*) repeats
FROM
data_tbl data,
jobapplication_tbl job
WHERE
job.int_vacancynum = data.id
AND data.txt_advcatdtlcode = d.txt_advcatdtlcode //error in this line
AND job.dat_appdate > TO_DATE('01-may-2018','DD-MON-YYYY')
GROUP BY
job.txt_appkey
HAVING
COUNT(*) > 1
)
)
FROM
jobapplication_tbl j,
data_tbl d,
jobcategorydtl_tbl c
WHERE
j.int_vacancynum = d.id
AND d.txt_advcatdtlcode = c.txt_code
AND j.dat_appdate >= TO_DATE('01-MAY-2018','DD-MON-YYYY')
GROUP BY
c.txt_description,
HAVING
COUNT(DISTINCT(j.int_vacancynum) ) > 1
``````

the mistake

"D". "TXT_ADVCATDTLCODE": Invalid identifier

I added a comment to the corresponding line in the error stack.
Many thanks .

## Algorithm – Calculation of the outer polygon

I need to compute an outer polygon from a self-intersecting polygon. I've found that Mathematica has recently implemented the following feature that could solve this problem. Does anyone know how it could be implemented?

https://reference.wolfram.com/language/ref/OuterPolygon.html

Any book or article dealing with this topic would also be helpful.

To illustrate my problem, I have a self-cutting polygon (blue) and want to get the outer polygon (red):

If it helps, this polygon is created by "displacing" * a previous polygon that did not overlap itself:

Could there be a way to increase the polygon that considers self-overlapping?

Thanks a lot!

* not technically balancing, as only some parts of the polygon move in the normal direction.

## magsafe – (2011 macbook pro) outer earthing pins in the plug melt?

(Https://i.stack.imgur.com/3owzo.jpg)

I had to replace my Magsafe charger because the outer pins in the L connector were worn out. A week later, after using my new charger, the same outer pins on the new charger were already worn out. Does anyone know what this could cause?

## Move the outer key inside in an association

I have a list that looks like this:

``````{15000001-><|"Loss" -> 2.85396*10^8, "Exposure" -> 6.61052*10^10|>,
15000002 -> <|"Loss" -> 1.25297*10^8, "Exposure" -> 1.57863*10^11|>,
15000003 -> <|"Loss" -> 2.05979*10^8, "Exposure" -> 6.88024*10^10|>}
``````

I would like to come to the following conclusion:

``````{<|"Event"->15000001,"Loss" -> 2.85396*10^8, "Exposure" -> 6.61052*10^10|>,
<|"Event"->15000002,"Loss" -> 1.25297*10^8, "Exposure" -> 1.57863*10^11|>,
<|"Event"->15000003,"Loss" -> 2.05979*10^8, "Exposure" -> 6.88024*10^10|>}
``````

I tried to make two lists, one with keys and one with values, and put them together, but I could not quite get that to work. I would be happy about suggestions.

## linear algebra – The various products on \$ k \$ -altering forms and their relationship to the outer product

To let $$V$$ be a vector space and $$k in mathbb {N}$$, Describe $$Lambda ^ k V$$ the outer $$k$$-Power of $$V$$,

To let $$f: Lambda ^ k V ^ * to ( Lambda ^ k V) ^ *$$ if the card is such that a $$k$$-Covector $$eta_1 wedge cdots wedge eta_k$$ will be sent to the $$k$$– alternating form (identified as a dual element of $$Lambda ^ k V$$) $$(v_1, cdots, v_k) mapsto sum_ { sigma} epsilon ( sigma) eta_1 (v_ { sigma (1)}) cdots eta_k (v_ { sigma (k) })$$, (It seems that there is another convention with one factor $$frac {1} {k!}$$ in this formula.)

There are two ways (see here) of defining natural products in between $$k$$alternating forms. Let's call $$wedge_1$$ the product is based on the formula with the $$Old$$ Operator (in the previous link) and $$wedge_2$$ the product is based on the formula with a sum via shuffle permutations (in the previous link).

Question: What is true and what is wrong in the following statements? To let $$omega in lambda ^ k V ^ *$$ and $$omega in lambda ^ {k & # 39; v ^ *$$,

1) $$f ( omega wedge omega & # 39;) = f ( omega) wedge_1 f ( omega & # 39;)$$

1 & # 39;) As 1), but with the $$1 / k!$$ Factor in the definition of $$f$$,

2) $$f ( omega wedge omega #) = f ( omega) wedge_2 f ( omega #)$$

2 & # 39;) As 2), but with the $$1 / k!$$ Factor in the definition of $$f$$,

## postgresql – Lateral / Outer Apply Join speed

I have the following Postgres database & query. It takes some `time items` and multiply his hours by a `role tariff` if a matching value for a exists `Employee`& # 39; s `role`otherwise it will multiply by a `personal rate` if there is a match.

Processing 300,000 lines takes about 7 minutes, and I wonder if I can do something to speed it up.

``````Delete table if data cascade exists;
Create the time_entries table (id int, staff_id integer, entry_date date, hours numeric).
insert in time_entries
values
(1, 1, to_date (& # 39; 15 -01-2019 & # 39 ;, & # 39; tt-mm-yyyy & # 39 ;, 1),
(2, 1, to_date (& # 39; 15 -02-2019 & # 39 ;, & # 39; d-mm-yyyy & # 39 ;, 2),
(3, 1, to_date (& # 39; 15 -03-2019 & # 39 ;, & # 39; d-mm-yyyy & # 39 ;, 3)),
(4, 2, to_date (& # 39; 15 -01-2019 & # 39 ;, & # 39; tt-mm-yyyy & # 39;) 4),
(5, 2, to_date (& # 39; 15 -02-2019 & # 39 ;, & # 39; tt-mm-yyyy & # 39;), 5),
(6, 2, to_date (& # 39; 15 -03-2019 & # 39 ;, & # 39; DD-MM-YYYY & # 39 ;, 6);

Create a table staff_rates (id int, staff_id integer, start_date date, rate numeric);
insert in staff_rates
values
(1, 1, to_date (# 01 -01-2019,, # t -t-mm-yyyy &), 1),
(2, 1, to_date (# 01 -02-2019,, # t -t-mm-yyyy &), 2),
(3, 1, to_date (& # 39; 01 -03-2019 & # 39 ;, & # 39; t-mm-yyyy & # 39 ;, 3),
(4, 2, to_date (# 01 -01-2019,, # t -t-mm-yyyy &), 4),
(5, 2, to_date (# 01 -02-2019,, # tt-mm-yyyy #), 5),
(6, 2, to_date (# 01 -03-2019,, TT DD-MM-YYYY)), 6);

Create the role_rates table (id int, role_id integer, start_date date, rate numeric).
insert in role_rates
values
(1, 2, to_date (# 01 -01-2019,, # t -t-mm-yyyy)), 7),
(2, 2, to_date (# 01 -02-2019,, # t -t-mm-yyyy &), 8),
(3, 2, to_date (& # 39; 01 -03-2019 & # 39 ;, & # 39; dd-MM-YYYY & # 39 ;, 9);

Create table staff (id int, role_id integer);
to insert in staff
values
(1, 1),
(2, 2);

CHOOSE
t.staff_id,
staff.role_id,
t.entry_date,
t.hours * (at rr.rate> 0 then rr.rate else sr.rate end) AS total_rate
FROM time_entries t, staff
LEFT JOIN LATERAL (
SELECT sr.rate
FROM staff_rates sr
WHERE sr.staff_id = t.staff_id
AND sr.start_date <= t.entry_date - "at or before this time entry"
ORDER BY sr.start_date DESC NULLS LAST
LIMIT 1
) sr ON true
LEFT JOIN LATERAL (
SELECT rr.rate
FROM role_rates rr, staff
WO rr.role_id = staff.role_id AND t.staff_id = staff.id
AND rr.start_date <= t.entry_date - "at or before this time entry"
ORDER BY rr.start_date DESC NULLS LAST
LIMIT 1
) rr ON true
WO t.staff_id = staff.id
``````