logic – A necessary condition for a relation to be in 2NF but not in 3NF is that some non-prime attribute must be determined by a non-prime attribute

I will state the complete question now, since it did not fit in the title.

Is the statement given below correct?

A necessary condition for a relation to be in 2NF but not in 3NF is that some non-prime attribute must be determined by a non-prime attribute or a set containing a non-prime attribute.


This is how I view the above statement:

Breaking the statement into parts:

  1. A necessary condition for a relation to be in 2NF is that some non-prime attribute must be determined by a non-prime attribute or a set containing a non-prime attribute.
  2. A necessary condition for a relation to not be in 3NF is that some non-prime attribute must be determined by a non-prime attribute or a set containing a non-prime attribute.

I will conclude that the statement is correct if and only if both the parts of the statement are individually correct.

Is my view of the statement correct?

If it is correct, then

Part two of the statement fails 3NF test, therefore making part two true.

Part one of the statement does not necessarily mean that the relation is in 2NF, therefore making part one false, because every non-prime attribute $A$ in $R$ may not be fully functionally dependent on every key of R.

So the statement should be false.


What I understand about necessary and what I understand about sufficient condition is this

A necessary condition is a condition that must be present for an event to occur. A sufficient condition is a condition or set of conditions that will produce the event. A necessary condition must be there, but it alone does not provide sufficient cause for the occurrence of the event. Only the sufficient grounds can do this. In other words, all of the necessary elements must be there.

differential topology – Transitivity of Equivalence Relation of Tangent Vectors on Manifold

I’m just starting to learn differential topology, and came across a notion when defining tangent vectors on a smooth manifold that confused me a bit. Let $M$ be a smooth ($C^infty$) manifold, with $p in M$, and define a smooth curve at $p$ as a smooth map $gamma colon (-1, 1) to M$ such that $gamma(0) = p$. Let $(U, phi)$ be any coordinate chart on $M$ containing $p$. Two smooth maps at $p:,$ $gamma_1, gamma_2$ are called equivalent if $(phi circ gamma_1)'(0) = (phi circ gamma_2)'(0)$; this is denoted $gamma_1 sim_p gamma_2$. A tangent vector is defined as an equivalence class of all smooth curves at $p$ under this relation.

The problem I’m having with this is proving that it is an equivalence relation. Obviously, reflexivity and symmetry follow immediately from the definition, but I’m trying to prove the transitivity of this relation. I wrote out a proof but wasn’t sure if it was valid because of the use of the chain rule outside Euclidean space. Here it is:

Let $gamma_1, gamma_2, gamma_3 colon (- 1, 1) to M$ be smooth curves at $p$. Assume that $gamma_1 sim_p gamma_2$ and $gamma_2 sim_p gamma_3$. Then there exists coordinate charts $(U, phi), (V, psi)$ containing $p$ such that:

$$(phi circ gamma_1)'(0) = (phi circ gamma_2)'(0) ,,text{ and } ,, (psi circ gamma_2)'(0) = (psi circ gamma_3)'(0)$$

Using the chain rule, the first equality implies that

$$ phi'(gamma_1(0)) cdot gamma_1′(0) = phi'(gamma_2(0)) cdot gamma_2′(0)$$
As $gamma_1(0) = gamma_2(0) = p$, this means that $gamma_1′(0) = gamma_2′(0)$. Likewise, we have that

$$(psi'(gamma_2(0)) cdot gamma_2′(0) = (psi'(gamma_3(0)) cdot gamma_3′(0) Rightarrow gamma_2′(0) = gamma_3′(0)$$

This means that $gamma_1′(0) = gamma_3′(0)$, meaning that

$$phi'(gamma_1(0))cdot gamma_1′(0) = phi'(gamma_3(0)) cdot gamma_3′(0),$$

so that $$(phi circ gamma_1)'(0) = (phi circ gamma_3)'(0) Rightarrow gamma_1 sim_p gamma_3$$

I feel like it isn’t legal to use the chain rule there because the codomain of our smooth maps isn’t Euclidean space, but I can’t think of any other way to prove the transitivity of this relation. Is my proof correct? If not, could someone give me a hint as to how to prove transitivity? Thanks!

co.combinatorics – how to use concept of equivalence relation to answer question

i’ve been trying to solve this question the last 3 days -_-

so the question is to use concept of Equivalence Relation to get the amount of callback routines a program need to perform if buffers can be processed simultaneously but a thread is only executed in the assigned buffers one at a time. the given buffer memory allocation:

B1 [ T1 ][ T4 ][ T5 ]
B2 [ T4 ][ T2 ][ T3 ]
B3 [ T5 ][ T6 ]
B4 [ T5 ][ T3 ][ T2 ]
B5 [ T1 ][ T5 ][ T3 ][ T6 ]
B6 [ T1 ][ T5 ]
B7 [ T6 ]
B8 [ T4 ][ T1 ][ T3 ][ T6 ]

extra info:

  • each process consist of threads that are allocated shared memory
    space call buffer memory.
  • an empty buffer is allocated a certain number of thread and become
    full when multiple threads are executed to store the data. then
    process executes callback to retrieve data from any available full
    buffer.

I have tried directed graph but i’m not sure about the relation whether i should use Bn as vertices or Tn. can someone please help me get an idea how to solve this TT_TT

Recurrence relation with alternating inequality

Let’s define the recurrence relation:

$$
begin{align}
a_0 &= 1 \
c(n) cdot a_n &= sumlimits_{k=1}^n frac{1}{k} a_{n-k}
end{align}
$$

where $c(n)$ is a function which depends on $n$.

Lets define a constant $d$, where $d < c(n)$ if $n$ is even and $d > c(n)$ otherwise.

Question: Can I say then that:

$$
begin{align}
d cdot a_n &< sumlimits_{k=1}^n frac{1}{k} a_{n-k}, quad text{if n is even} \
d cdot a_n &> sumlimits_{k=1}^n frac{1}{k} a_{n-k}, quad text{if n is odd} \
end{align}
$$

One-to-many relationship in django models (Relation of one Model with another two Models)

I want to save all the likes in a Like Table.

Likes of both Comments and Post, in one table.

Is their something like this to achieve this thing:

Class Post:
   ...

Class Comment:
   ...

Class Like(models.Model):
   '''Either of Post or Comment will have to be selected in this field'''
   post_or_comment = models.Something((Post, Comment), on_delete=models.CASCADE)

the database system to store any value associated with such external writes temporarily in a special relation in the database

The text below is from the book Database System Concepts by Silberschatz.
Why the values associated with external writes need to be stored in a special relation in the database? Can’t we behave with these values like other values?

We must be cautious when dealing with observable external writes, such as writes
to a user’s screen, or sending email. Once such a write has occurred, it cannot be
erased, since it may have been seen external to the database system. Most systems
allow such writes to take place only after the transaction has entered the committed
state. One way to implement such a scheme is for the database system to store any value
associated with such external writes temporarily in a special relation in the database
,
and to performthe actual writes only after the transaction enters the committed state. If
the system should fail after the transaction has entered the committed state, but before
it could complete the external writes, the database system will carry out the external
writes (using the data in non-volatile storage) when the system is restarted.

mysql – Bug Tracking System entities: How to design the relation between Projects, Users and Ticket?

Im working in a Bug Tracking System. But Im not sure if the Project has Tickets and Users or the Users have Projects and Tickets. Not sure how to show it in the Front-end. If you show a list of Projects and then you click on each Ticket or show both Users and Tickets within the Project list.

My EER diagram of my database is the following:

enter image description here

exchange rate – Where is some kind of reliable resource which displays the actual Bitcoin price curve in relation to predictions?

I’m looking for a webpage which shows the actual price data for Bitcoin with one or more curves overlaid over this which shows the most accurate curve(s) predicted by others before they knew for a fact. I want these curves to extend into the future as well.

If there is at least one such prediction which has been roughly correct so far, I would be at the very least very interested in seeing what they think will happen to the price in the next few weeks, months and years.

I’ve stared at the Bitcoin price curve for the entire time it’s existed, and for the last 3 years, last 1 year, last 6 months, last week, last day, etc. I truly see no pattern whatsoever. It looks like it’s steadily but slowly going up and up forever, based on the current data.

It really frustrates me that I have no idea if Bitcoin will stay like the current price for ages or go to 1 million USD in the next few months. I genuinely have no idea at all. I frankly started to believe it would have reached over 1 million USD by the end of 2020… but that didn’t happen.