Why does the Trinitarian Formula start with "In the NAME…" and not "In the NAMES…"?

Question: Today the Universal Church celebrates the Solemnity of the Holy Trinity. In the Trinitarian Formula, we invoke the NAME (singular) of the Three Persons of the Trinity. In English we say: "In the NAME OF the Father, and OF the Son and OF the Holy Spirit." Note the repetition of the preposition 'OF' prefixed to each of the Three Persons, to reinforce individuality. But at the same time, we do not say: "In the NAMES of the Father, etc." Does it imply that there is but one name for all the Three Persons, and that Father, Son and Holy Spirit are not individual names? My question therefore, is: What according to Trinitarians is the reason for starting of the Trinitarian Formula with "In the name…" (singular) and not "In the names…" (plural)?
Comment: The question tags do not indicate which Christian Group is under analysis. What is meant by 'The Universal Church' ?
Comment: Nigel, by Universal Church , I mean the entire Church including, but not limited to, the Catholic Church. 4thJune, 2023 is celebrated as Trinity Sunday by both the Western  and the Eastern Liturgical Calendars.  ( Courtesy; Wikipedia ).
Comment: **Synecdoche** The word, "Name" is used, but it is a synecdoche representing the full "Authority and Divine Power" of the Trinity!
Answer:
Does it imply that there is but one name for all the Three Persons, and that Father, Son and Holy Spirit are not individual names?

Sort of; it's not that the individual persons don't have names, but that God has one Name, and the formula is as it is to emphasize that there is one God.
However...

What is the reason for starting with 'in the Name' and not 'in the Names'?

...the best answer is because that's what Jesus told us to do.

Matthew 28:19

[Jesus said,] "Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit..."

"Name" here ("ὄνομα") is singular.
Answer: "In the name…" (singular) shows the unity of the divine essence.
Pohle-Preuss, The Divine Trinity: A Dogmatic Treatise, § "The threefold personality of God as taught in the New Testament — Texts treating of the three Divine Persons together":

The essential identity of the three Divine Persons follows further from the singular form "in nomine" ["in the name"], because throughout the Bible "nomen Domini" ["name of the Lord"] signifies God's power, majesty, and essence.13 As the Three have but one name, so They have but one essence, one nature, one substance. St. Augustine beautifully observes:

Iste unus Deus, quia non in nominibus Patris et Filii et Spiritus Sancti, sed in nomine Patris et Filii et Spiritus Sancti. Ubi unum nomen audis, unus est DeusThis is one God, for it is not in the names of the Father, and of the Son, and of the Holy Ghost, but in the name of the Father, and of the Son, and of the Holy Ghost. Where thou hearest one name, there is one God.14

13. "Nomen est numen." ["The name is the divinity."]14. August., Tract. in Ioa., 6, n. 9.

Baptism is invalidated by saying "In the names…" (plural).
Pohle-Preuss, The Sacraments: A Dogmatic Treatise (vol. 1): The Sacraments in General, Baptism, Confirmation, § "Matter and Form" of baptism:

Baptism would still be invalid if the minister would introduce a phrase embodying an anti-Trinitarian heresy,54 e.g., "I baptize thee in the names of the Father, and of the Son, and of the Holy Ghost."5554. Tritheism, Arianism, etc.55. "Baptizo te in nominibus Patris et Filii et Spiritus Sancti."

Father, Son, and Holy Ghost are the names of the Divine Persons, not of the Trinity.
God is one

not in the oneness of a single Person, but in the Trinity of one substance. […] distinction in persons, unity in essencenon in uníus singularitáte persónæ, sed in uníus Trinitáte substántiæ. […] in persónis propríetas, et in esséntia únitas—Preface of the Holy Trinity
Answer: The three Persons in the one, blessed Godhead all have a variety of names and titles. There is no one single name for the Father, or for the Son or for the Holy Spirit. I won't develop that here as it is self-evident to all who are familiar with the Scriptures.
The origin of the phrase in question, "In the name of..." is Jesus Christ himself. He was the first to command it, and for that to be written down by the apostle Matthew, under the direction  of the Holy Spirit.

"Go ye therefore, and teach all nations, baptizing them in  the name
of the Father, and of the Son, and of the Holy Ghost, teaching
them..." Matthew 28:19-20 A.V.

Had Jesus meant to tell Christians to baptize in three names, to indicated three distinctives with regard to baptism, he would have said, "baptizing them in the name of the Father, and in the name of the Son, and in  the name of the Holy Ghost". But the fact that all three persons in the blessed Trinity are of the one divine substance, or being; all equally God, means that there is this one, over-arching 'Name' that covers all of that, in one go. It is the Name of supreme authority of the one, true God. The entirety of the Godhead is encapsulated by this power of the authority of the Divine Name.
Of course, non-trinitrians will not have that, but this question is asked from, and looking to, the perspective of mainstream Christianity which is based on this foundational doctrine of the Trinity.
That is why all mainstream Christian groups baptise using the formula that starts, "In the name of the Father, and of the Son, and of the Holy Spirit".
Comment: Up-voted +1. _onoma_ means not only name, but 'authority' and 'cause' [Strong 3686](https://biblehub.com/greek/3686.htm). The baptism is in the name of, under the authority of and unto the cause of  . . . . . . . . .
Answer: To be honest, I suspect this is a question of English grammar rather than theology.
The expression could be taken as an abbreviation of "In the name of the Father and {in the name] of the Son and [in the name] of the Holy Spirit". In other words, the word "name" at the beginning would be singular in form even if they were different names.
I can't speak for Latin grammar, but the point is worth checking.
Comment: Foxe's Book of Martyrs (published in 1563, not too long before the King James Version of the Bible), uses a similar phrase – "name" meaning "authority": "in the name of the king and queen" (https://www.ccel.org/ccel/foxe/martyrs/files/fox116.htm).
Comment: Latin and Greek have parallel grammar in that place. It would be unusual and therefore require justification for that type of elision
Comment: Agreed. I admittedly come from a non-Trinitarian viewpoint, but this seems like a question of grammar. If it said "in the names of the Father, and of the Son, and of the Holy Spirit," it would imply that each of the three persons has multiple names. Also, Father, Son, and Holy Ghost aren't names anyway, they're titles – "in the name of" essentially means "by the authority of." There's no need to say "by the authorities of" because it's the same authority from all three.
Comment: You have a point, Samuel Bradshaw. Hope you develop it into an answer.
Comment: @SamuelBradshaw Not necessarily. While a singular noun related to a plural is typically "shared", whether a plural noun related to a plural is specifically one each or multiple depends at least in part on particular grammar rules in languages. If your suggestion were generally the case, how would languages express the idea of a plurality with one per each (e.g. names for 3 persons, 1 name per person).
Answer:
Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost - Matthew 28:19

We bring a lot of assumptions to this verse which, while not necessarily wrong, may serve to hamper understanding at depth.
Baptizing

The above mentioned verb βαπτιζω (baptizo), which is really the same verb as the parent but with more dynamic, deliberate or willful action: to do an immersion, to plunge in. This verb is used in the classics to describe the deliberate sinking of ships, the "inundating" of a city by throngs of people, or a being up to the ears in debt. It's used 80 times in the New Testament, see full concordance, but translators should avoid using the verb "to baptize", since in English that verb doesn't do anything other than refer to a relatively modern religious ritual, namely Christianity's ritual of water baptism. As we describe above, there are quite a few mediums into which one may be immersed, and it's the willful and total immersing that this verb speaks about, not the medium.

In

The ubiquitous preposition εις (eis) describes a motion into any place or thing, and can often be translated with "in", "into" or "at". It is the opposite of εκ (ek), which describes a motion out of anything, and differs from προς (pros) in that the latter describes the approach while εις (eis) describes the arrival.

The Name

The noun ονομα (onoma) means name, and it and our English word "name" stem from the same ancient Proto-Indo-European root "nomn-" that left its mark in languages from Sanskrit (nama) and Avestan (nama) to Russian (imya) and all European languages. That means that our word describes something that was among the very first concepts to be named; a necessarily ancient idea that sits at the very core of what defines humanity.
Our word "noun" comes from the Latin word for "name," namely nomen, and sure enough, technically speaking, one's personal name is a "proper noun" and nouns like "lamp" or "mouse" are really the "names" of things. That means that a human name creates a generic category that contains one member: someone named Barney is as unique as the lamp and the mouse.

And

The ubiquitous copulative particle και (kai) simply means "and" or "also".  And και (kai) is altogether not very exciting either. It's mostly used to tie words or statements together in much the same way as does our English word "and"

Baptize is most often assumed to mean water and "in the name of" is assumed to be a single phrase meaning something like "by the authority of".  However, Jesus came to baptize with the Holy Spirit and with fire, so water need not always be in view.
"Name" can indicate authority but it most often is simply a definition of who or what something is.  Since 'and' is copulative and 'name' is singular, the 'name' (or defining nature) of Father, Son, and Spirit is the same.  It is a decidedly Trinitarian verse directly from the mouth of the Lord Jesus.
Therefore, this verse can be understood as follows:
"having gone, then, disciple all the nations, totally and willingly immersing them within the defining nature of the Father, and of the Son, and of the Holy Spirit".
This, then, is not a call to go about performing a water baptism ritual but a directive that, through the teaching of the gospel, members of all tribes and tongues and nations may willingly and completely be immersed in the Triune God.  You must be born again.
If water immersion is the clear, inextricable intention of the wording of the "great commission" then Paul has been exempted from Jesus' command by Jesus Himself:

I thank God that I baptized none of you except Crispus and Gaius,so that no one can say you were baptized in my name.
I did, in fact, baptize the household of Stephanas; beyond that, I don't recall if I baptized anyone else.  **For Christ did not send me to baptize, but to preach the gospel — **not with eloquent wisdom, so that the cross of Christ will not be emptied of its effect. - 1 Corinthians 1:14-17

I've not yet seen anyone insisting that the Apostle Paul was not fulfilling the "great commission".

Incidentally, this understanding brings great depth to the command against 'taking the Lord's name in vain'.  It is not a prohibition against verbal misuse of the words God or Jesus (although as a matter of love and respect one should not) but it is a prohibition against making an empty (vain) confession, as those who honor Him with their lips but their hearts are far from Him.
Comment: Up-voted +1. Very astute. Very individual. Whether all is technically correct, I am not qualified to say. But it has made me to ponder and I shall ponder it more. Much appreciated, Brother Michael.
Comment: Down-voted:  While the beginning part has some validity, the last part, concluding that baptism = immersion (instead of the application of water more broadly) is sheer conjecture with no supporting details. So also, "not a call to go about performing a water baptism ritual" is a preaching with no proof. Up until relatively recently Baptism = putting water on people of all ages in the name of the Triune God. This was the universal practice for well over 1500 years.
Comment: @Epimanes I appreciate the reason for the down-vote.  Would you argue that being baptized with the Holy Spirit still requires water?
Comment: @MikeBorden My issue is not that you have your own opinion. My issue is that there is no support given from scripture (in that second half).
Comment: @Epimanes. I've edited in response.
Answer: The fact that it is a singular name thrice, means that it is not one baptism but 3 baptism you take in the name of each: Father, Son, Holy Spirit. Ellipsis was very common in ancient writings, and it must have been used the same way in verbal teaching or tradition in the early church, as the construct would be unnecessary to be spoken in minute detail: (baptise in the name of the father, in the name of the Son etc). Thus, you baptise thrice in the water for each name.
It could be said (either sarcastically or seriously) that the one baptism expresses a unification of the three into one name and one person, hence has been committing a Unitarian heresy. In the Catholic Church, even the use of wrong pronoun "we" instead of "I", nullifies the baptism ritual.
To some it may seem that the English translation is imitating the grammatical discord or irregular Greek pattern of collective singular nouns or verbs in this for singular subject. However, some have suggested that the ellipsis is apparent, and the sentence is legal according to English rules. The repetition of in the name of has been elided.
I found a great answer on BHSE:

Grammatically it is natural to view the prepositional phrase "in the name" as being "applied" to all three terms independently.
This could be an example of the adjunctive και[and] or of ellipsis, or both. The adjunctive και [and] lends itself to ellipsis and also serves to reduce repetition.
This would result in:
In the name of [εἰς τὸ ὄνομα] the Father and [in the name of] the Son and [in the name of] the Holy Spirit. (NA28) εἰς τὸ ὄνομα τοῦ πατρὸς καὶ τοῦ υἱοῦ καὶ τοῦ ἁγίου πνεύματος,

An even better source confirming the ellipsis is from the 19th century scholar Heinrich Meyer's NT commentary, in the footnote we read:

Had Jesus used the words τὰ ὀνόματα [names] instead of τὸ ὄνομα [name], then, however much He may have intended the names of three distinct persons to be understood, He would still have been liable to be misapprehended, for it might have been supposed that the plural was meant to refer to the various names of each separate person. The singular points to the specific name assigned in the text to each of the three respectively, so that εἰς τὸ ὄνομα is, of course, to be understood both before τοῦ υἱοῦ and τοῦ ἁυίου πνεύματος; comp. Rev 14:1 : τὸ ὄνομα αὐτοῦ καὶ τὸ ὄνομα τοῦ πατρὸς αὐτοῦ. We must beware of making any such dogmatic use of the singular as to employ it as an argument either for (Basilides, Jerome, Theophylact) or against (the Sabellians) the orthodox doctrine of the Trinity.

The example can be confused with the grammatical discord found in Greek, Beginning with NT Greek where we often see singular noun or verbs for plural subjects.
To use the example of 1 John 4:1 "Beloved, believe not every spirit, but prove the spirits, whether they are of God: because many false prophets are gone out into the world." The Greek has a singular verb "is" (estin) for plural spirits, implying implicitly to test each spirit individually. We also see the discord in Matt 10:2 (ὀνόματά ἐστιν) names is; Luke 10:20 names written(sg). However, in the case of the baptismal formula verse, it is not an example of grammatical discord, but of ellipsis.
Although one answer says that the English of grammar requires plural names for plural nouns, there are better experts like Meyer who have settled any doubt among those with poor English. We should conclude that the ellipsis is natural in English as it is in Greek. The verse should be understood as baptising in all the three names.
Comment: "The fact that it is a singular name thrice, means that it is not one baptism but 3 baptism" and yet Scripture is clear there is "one faith, one baptism"
Comment: That doesn't clarify *at all*. You need to support the claim that an expression in the form "in NOUN of NOUN1, and of NOUN2 and of NOUN3" can be understood due to elision as "in NOUN of NOUN1 and in NOUN of NOUN2..."; that's not a ordinarily elision pattern. You also need to clarify what you mean by discord and collective nouns.
Comment: " that the one baptism expresses a unification of the three into one name and one person, hence has been committing a Unitarian heresy." say what? "a baptism" is the entire act, not a single pouring of water.
Comment: Michel16, I beg to agree with the idea of correctness of English style of expression, than of grammar. But in legal terms , we use plural, for instance: This property is registered in the names of Mr Smith and his wife, with equal shares.
Comment: Yes, the translations are therefore must be wrong for following the Greek pattern.
Comment: "following the irregular Greek pattern of collective singular nouns and verbs in this instance" This point needs some serious elaboration as does some of the discussion around ellipsis, namely what is elided.
Comment: I clarified a bit in the answer, the grammatical discord is the irregular form of Greek; and the repetition is elided. There are 3 baptism in each name. Jews had already known and baptise in Father's name. New believers first repented in John's repentance baptism, then Jesus, and then holy spirit. The nations without god needs the baptism of all 3.
Answer: Synecdoche The single, unassuming word, Name is used, but it is being used as a synecdoche representing the full "Authority and Divine Power " of the Trinity. It is a singular word meant to represent the full Almighty power of God.
To understand this concept (figure of speech) simply recall Sgt. Preston of the Yukon (a Mounted Policeman in Canada) yelling to a fleeing criminal, "Stop in the Name of the Queen!" The lone policeman had a very limited amount of power himself.  But the minute he mentioned the "Name" he had the full weight and authority of the British Empire behind him!!!
In the same vein, a policeman not in a Kingdom, may yell, "Halt in the Name of the Law!" Again, the minute he mentions the "Name" he has total control over the fleeing suspect granted by the Law of the land.
When a pastor/priest/bishop baptizes a believer in the "Name of the Father, and of the Son, and of the Holy Spirit" he is adjuring the full power of God to witness the public conversion, and he is baptizing the born-again person into nothing less than the spiritual, honorable Family of God.
As a synecdoche, the "Name" represents all that the Almighty Jehovah God of Eternity is in essence---no matter how mysterious and ineffable that glorious trinitarian Being is. The single "Name" can represent the "Trinity" in its entirety because it is a synecdoche.


What are the valuable benefits of having a node?

Question: Supposedly I'd like to help Ethereum to expand (in terms of the World Computer), so I'd like to setup a Linux machine which can run smart contracts.

What kind of benefits I can get from doing that?
Do I earn anything particular?
If so, what I can get and would it increase over time?
Answer: [EDIT] this answer was edited to add more information about non monetary valuable benefits provided by a full node. This edit was triggered by one of @fraxture comment on this answer.

TL;DR
No, you won't earn "old world" money or Ether (ETH).
However you would have a lot of other benefits from helping the Ethereum project grow and increasing your investment's value to technical benefits like fast transaction sending and validation or gaining even better security and more privacy. These reasons alone are worth having a full node for, even if you don't earn direct money.

1 - What kind of benefits I can get from doing that?
There is a nice explanation by Anthony on Ethereum Classic forum (but it's also true for Ethereum for the moment):

You have a few reasons why you'd run a node, including:

You're solo mining or running a mining pool, if this is the case then you need a node in order to talk to the network.
You want to make sure your transactions to the network yourself. Remote nodes are generally reliable but are controlled by 3rd parties and typically throttle heavy usage.
You want to help secure the network; the more independent nodes running the more copies there are of the blockchain and the more resilient it is.
You want to make the network faster and secure; the more nodes the lower the latency in sharing blocks and the more copies of the blockchain that exist.

A first and not the least important incentive is then to help Ethereum to become more secure (you verify transactions) and faster. This doesn't give you any reward at first but if that means that the Ethereum project can live and grow, what you invested as knowledge, time and money will become more and more valuable. This is a big incentive at least for me.
Also, having a full node is mandatory if you are solo mining (not mining in a pool) but even without mining, running a full node makes you able to have a direct access to the Ethereum blockchain and being connected to other pears directly. If you don't have one, you will have to connect to another node, mainly using RPC connection.
It means that one direct benefit for you is that you won't have problem with overloaded tier nodes that would prevent you from including your transactions on the network. You'll still have problem including your transaction to a block as mining nodes may include transactions with higher fees first, but at least you will skip the first obstacle.
Also if you have a look at this nice article on Ethereum transaction lifecycle in chapter "2.3 Transaction is validated locally", you'll notice that if you have a local node, it will validate your transaction prior to broadcast it. This can make you save some time.
Indeed, if you don't use your own full node, you'll have to trust someone's node for this validation. Running your own removes your need to trust anyone except you and your hardware.
You will also be ensured that the node you send your transaction to is on the chain you expect it to be. This is not very important in normal days, but think about it when a fork occurs.
Also, having your own node, and to me that's probably the more valuable benefit, ensures more privacy.
(note that no absolute privacy is possible. You can mitigate the risk of any curious person trying to find you but not completely stop a well organized group or person).
When you connect a third party node, trustful or not, you have to send your signed transaction to it to be broadcasted to the whole Ethereum network.
Doing so will expose you, even if you use secured HTTP communication, because not only the data can give information about your stake or your actions on the network but also meta data like your IP, time of the connection, size of data, target server, ...
It won't be very hard for anyone having access to your internet log (your internet provider, government, pirates, spies, ...) on the time range of a block to know that you did an action in this block, filter by content size and then find a few sender addresses or even just one that could link you to your Ethereum account. Once you're linked to an account it's an opened door to multiple side effects from social engineering to funds extortion or for special cases like in censorship contexts to link someone to some messages on the internet and then put him in jail... bad...
When you have a local node, you send your transaction to it and then your transaction is broadcasted to the whole network with all other transactions. It's more difficult for one to say which transaction you added to your local node pool as they would have to check the difference between all incoming and outgoing transactions on your node by decrypting the P2P datas exchanged with all your peers, not impossible but hard to reach I think, see for this how P2P security on Ethereum works for now.
There's probably some more benefits but I will edit this post when I find them. Also switch to Ethereum 2.0 with Sharding and Casper FFG will change many things so I'll edit it when it will be in production anyway.
2 - Do I earn anything particular?
If you talk about money (ETH) and you setup a full node alone today, you won't get any ether reward.
But you can run it AND mine using GPUs. You will earn ether as block reward. Note that it's not very valuable today due to the difficulty to mine in front of the cost of purchasing mining hardware and and the cost of electricity. But you can get pleasure to do it and learn a lot of things. Just be sure to buy hardware that perhaps you can reuse for gaming when you'll notice that it's a bit pointless to expect rewards with a small setup.
In a near future, Ethereum will move to PoS (Proof of Stake) system instead of PoW (Proof of Work). So mining will stop working. Having a node will only make you able to access the network except if you are one of the lucky validators that will stake their Ether and be "the miners of PoS", but without GPUs need. However, hardware requirements to be a validator will be a bit hard to reach for regular people (needs of security, availability, many RAM and CPU, ... and of course some Ether to stake).
3 - If so, what I can get and would it increase over time?
No, as you won't earn anything by simply running a full node without mining, this won't increase over time.
Comment: @fraxture well, this is the whole purpose of all the crypto thing: not having to trust anyone. If one single part of the trust less chain is not trusted, all the trust is broken all along the chain. It then begins with trusting the node you use to communicate with the blockchain.
Comment: @NicholasMassart thanks for the explanation. A follow-on question: since these nodes exist in a network, if one node isn't trustworthy, couldn't that corrupt the whole network of nodes? Or is it that your point of entry onto that network is a particularly vulnerable spot?
Comment: @NicolasMassart I read your answer and it helped me a lot. I wanted to ask you if you can help me with this other doubt. In the ethereum documentation I read that infura is classified as Ethereum API as a service, while BlockCypher as Ethereum Web APIs. What is the difference
Comment: @MarioRoma I'm very happy of that!
However I can't really tell about your other question... the best would be to ask Infura directly to their community https://community.infura.io/
Comment: Can you describe some scenarios in which trusting/not trusting someone can make a crucial difference?
Comment: @fraxture I'm sorry for not answering you. I didn't forget you but I'm working on a real answer (the question is a bit tricky and very interesting) but please allow me some more time as I have a lot of work aside...
Comment: Looking forward :)
Comment: @fraxture I edited my answer, I hope it will meet your expectations ;)
Comment: Adding more nodes slows down the network, it doesn't make it faster. It is easy to see why. When a miner finds a block, it has to propagate to other nodes. The more nodes there are, the longer time it is going to take (even if the time is O(log(n)).

The same goes for zero-confirmation transactions.
Comment: @LarsPensjö speed depends a lot of the number of peers each node have. More peers the faster it is to spread a tx to a majority of nodes. I did not do the math but if you have any to support your sayings I would be glad to try to read them.
Comment: @NicolasMassart no math needed. The important thing is the time it takes for a new block to propagate to other nodes, or the time it takes for a new transaction request to propagate to the other nodes. More nodes takes more time, isn't that obvious?
Comment: @LarsPensjö not obvious to me. More nodes means more peers and so more  ways to share the tx. Also, the validation of a tx doesn't need all nodes to validate the mined block. You can consider that, given the randomness and decentralisation of peers, once you reached a certain number of validation your tx can safely be considered as finalized. This number doesn't change when the number of nodes of the network grows. So having the math is required imo.
Comment: @NicolasMassart If you have one single node in the whole network, block propagation is immediate as there is no propagation. Same goes for transactions. Add one extra node to this, and you need to make exactly one hop. Adding another 15 nodes or so, will possibly need the same time (depending on fan-out). Adding 15*15 nodes would require 2 hops or so. Adding 15*15*15 nodes would require 3 hops. 1 hop is more than 0 hops. 2 hops is more than 1 hop.  As you see, more nodes delay the propagation. More delays mean a slower network.
Comment: @LarsPensjö but we don't currently have only one node, at least on public networks that are what we talk about. So you can't compare a 1 node "network" speed with an already large network. Adding nodes in a network where the required number of hops to propagate the tx enough is already reached just helps having the tx propagated. But this makes me think that perhaps we should specify that this answer applies to networks with already a significant size and probably only for mainnet as having a node on testnet has some slightly different incentives.
Comment: @NicolasMassart When you think about "faster network", you probably think about adding a new node. It will then have to download the whole chain. That will go faster with more nodes to load from. But, although it is convenient, it isn't an important use case. The important use case is the block propagation times.  You won't find any supporting documentation that shows this goes faster with more nodes.

There is one exception that I know of. The number of nodes serving light clients were not enough.


Big O and Asymptotic Notation

Question: So I am currently working in the Apostol Book "Introduction to Analytic Number Theory", and am attempting to prove one of the major results regarding the prime number theorem. Specifically I am trying to show that given:
$$\lim_{x \to \infty} \frac{\pi(x)\log x}{x} = 1$$
show that:
$$\lim_{x\to \infty} \frac{\vartheta (x)}{x} = 1$$
What I have done so far is this: We know for $x > 2$ that:
$$\vartheta (x) = \pi (x) \log x + \int_{2}^{x}\frac{\pi(t)}{ t}dt$$
This a result derived previously in the book. So we must show:
$$\lim_{x \to \infty} \frac{\pi(x)\log x +\int_{2}^{x}\frac{\pi(t)}{t}dt}{x} = 1$$
Clearly, it suffices to show that:
$$
\lim_{x \to \infty} \frac{1}{x}\int_{2}^{x}\frac{\pi(t)}{ t}dt = 0
$$
Apostol then discusses a bound on the integrand. Specifically, states that:
$$\lim_{x \to \infty} \frac{\pi(x)\log x}{x} = 1
\implies \frac{\pi(t)}{t} = \mathcal{O} \left(\frac{1}{\log t} \right)
$$
for $t>2$. At first, I was unsure as to where this result comes from, but then I realized one could get it by dividing the $\log x$ from both sides of the limit definition. However, it still does not make sense as to what the result means/wether it is rigorous or not. I guess the question I am asking is, what is the connection between big $\mathcal{O}$ notation and asymptotic notation (limit of a quotient)?
Comment: @Did Looking back at the definition I feel quite stupid for asking this. It becomes fairly obvious. Thanks!
Comment: Did you simply look for a definition of [Big-O asymptotics](https://en.wikipedia.org/wiki/Big_O_notation)? It should allay all your doubts...
Comment: He does not need something to raise doubts.  He needs something to allay doubts.
Comment: @GEdgar Indeed. Thanks.
Answer: This is quite rigorous:
All you need is that the stated limit implies that $\pi(x)\log x/x<M$ for $x$ large enough. ($M$ can be any number greater than $1$.)
From this you get $\pi(x)/x<M/\log x$ for $x$ large enough, and that is precisely the meaning of the big $\mathcal{O}$ notation.
Answer: You have that
$$
\lim_{x\to\infty}\frac{\pi(x)\log x}{x} = 1,\tag{1}
$$
and the claim is that
$$
\frac{\pi(t)}{t} = \mathcal O\left(\frac{1}{\log t}\right)\qquad\text{as $t\to\infty$,}
$$
which is to say that there exists $B>0$ and $M>0$ such that $\pi(t)/t \le B/\log t$ for all $t>M$. Clearly the claim is true if and only if
$$
\frac{\pi(t)\log t}{t} \le B \tag{2}
$$
for all $t$ sufficiently large. Since you are given $(1)$, you have $(2)$ since eventually the quotient $\frac{\pi(t)\log t}{t}$ is $\le 2$, say.

Big O and Asymptotic Notation


Is it possible to connect NRF24L01 to localhost in laptop without wifi router?

Question: I want to send data from arduino to my localhost port:8080(with ipython notebook) via modul NRF24L01 without WiFi router. Is it possible? If it possible, how to do that?
Thank  you...
Answer: It is possible to have 2 NRF24L01 radios to talk to each other without a WIFI router.
However, is it possible to get your notebook to talk to a NRF24L01?  Probably not unless you can write a device driver for it.
You could use two Arduinos each with a NRF24L01, and get one to receive the data and then send it via Serial to you laptop, but you would be better looking at an ESP8266, set that up as an access point, connect your laptop to that network and job done.
(By the way you won't be able to call you laptop localhost)


How does the GARCH part affect the ACF/PACF of an ARMA-GARCH process?

Question: I need some help with fitting an ARMA-GARCH model.
I'm analyzing a daily time series. I don't understand how the order of the GARCH impacts ACF or PACF plots. I mean: what is the difference I should notice in the ACF plot when I'm using GARCH(1,1)+ARMA(1,1) instead of GARCH(2,1)+ARMA(1,1)? What 'part' of the ACF plot should change when I modify the GARCH or/and ARMA orders?
Answer: An ARMA(p,q)-GARCH(r,s) model looks like this:
\begin{aligned}
x_t &= \mu_t + u_t, \\
\mu_t &= c + \varphi_1 x_{t-1} + \dots + \varphi_p x_{t-p} + \theta_1 u_{t-1} + \dots + \theta_q u_{t-q} \quad \text{("ARMA")}, \\
u_t &= \sigma_t \varepsilon_t, \\
\sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 + \dots + \alpha_s u_{t-s}^2 + \beta_1 \sigma_{t-1}^2 + \dots + \beta_r \sigma_{t-r}^2 \quad \text{("GARCH")}, \\
\varepsilon_t &\sim i.i.D(0,1),
\end{aligned}
where $D$ is some probability distribution with zero mean and unit variance.
The ACF/PACF of the original series $x_t$ would mostly be determined by the ARMA part, not the GARCH part. The GARCH part implies nonconstant variance of $u_t$, so you have an ARMA model with the noise part being a bit ill-behaved (not i.i.d.). This might blur the ACF/PACF a little bit, but the general patterns would be the same.
If you are interested in selecting and fitting an ARMA-GARCH model, then you would probably also look at the ACF/PACF plots for the model's residuals with the aim of model diagnostics. You may look at the raw standardized residuals $\hat\varepsilon_t$. If you change something in the equation for $\sigma_t^2$ (the GARCH part of ARMA-GARCH), you may not notice much difference in their ACF/PACF plots. However, you should notice some changes in the ACF/PACF plots of squared standardized residuals $\hat\varepsilon_t^2$.
Answer: As explained by @Richard Hardy above. The time series, $x_t$ is first decomposed into the conditional mean process $\mu{_t}$ and residual/error part $u_t$. The conditional mean process can be modelled by zero, constant or ARMA. The residual $u_t$ by GARCH. Actually, you can start by assuming a constant or a zero conditional mean process (instead of ARMA) if you are using financial returns time series. Then subtract the fitted conditional mean process, $\mu_t$ from the original series, $x_t$. Then square this residual term and plot it to get a sense of the variance (squared volatility) in the series. Let's call this series $u_t{^2}$. The GARCH model is trying to model this new series. The ACF/PACF of $u_t{^2}$ will help to choose the order of $\alpha$ and $\beta$
Note the stochastic nature of the conditional variance term, $\sigma_t$ due to the random variable, $\epsilon{_t}$ term.
Comment: *you can start by assuming ARMA to be a constant or a zero mean process* This sounds contradictory, as ARMA is nonconstant. Did you perhaps mean $\mu_t$ instead of ARMA? Regarding your question, simulataneous estimation is more efficient and gives correct standard errors and such. Stepwise estimation ignores the uncertainty about the first-stage conditional mean when fitting the second-stage zero-mean GARCH model. See e.g. [this](https://stats.stackexchange.com/questions/41509/what-is-the-difference-between-garch-and-arma/231512#231512) for some more detail.
Comment: Now that I have answered your question, you might wish to remove it from your answer (and perhaps replace it with what I responded).
Comment: @RichardHardy I have corrected above.


English (GB) Language pack for Magento 1.9.1

Question: I have been unable to find the GB English language pack for Magento 1.9.1 and have searched high and low, but it seems it has been removed since August 2014 (ref: http://www.magentocommerce.com/translations). I checked magento connect but nothing there either. I am hoping someone can confirm if I am doing the right thing to translate my store from US english to GB, I am relatively new to Magento but happy to work with system files, and I have done this:
I have set the default local to English GB in my back end (system-configuration-general-general-locale). I copied the contents of the folder app/locale/en_US to a new folder, app/locale/en_GB (which did not exist in my fresh install of Magento 1.9.1). Having found some of the necessary changes (cart=basket etc) I've amended it in there and it is working fine on the front end. The instructions on adding a language pack manually to Magento are far more complex, but does what I have done work for en_GB at least for such a common language? I assume since en_GB is listed in magento then all I need to do is create the necessary translation files in the app/loale/en_GB folder. Any help or advise on the correct way or an alternative resource for the language pack will help!
Answer: Had the same issue whilst trying to locate the GB language pack. I am aware that just posting links as an answer is NOT the stack overflow way but this is where the language pack can be found --
ALL credit to Lightnest - www.lightnest.co.uk
http://www.lightnest.co.uk/magento-en_gb-united-kingdom-language-pack/ I also uploaded to GitHub https://github.com/ecommerceheroes/magento_GB_LP .
Answer: You can also create in your theme folder a locale folder where you create a translate.csv. At the backend you can go to system->configuratation->general->design and set the theme folder after translation (on storeview). It is one file with all your translation lines.
Answer: Yep, en_GB has been removed. Clearly the American developers are determined to force us Brits to accept the terms "Cart", "Zip" and "Shipping" instead of "Basket", "Postcode" and "Delivery". Unfortunately those terms don't apply here or just sound plain wrong, so now every British developer has to spend hours making their own translation files, when some of the most obscure countries in the world have language packs readily available on Magento Connect.
Fortunately I have kept the old British translation file from an older installation, it's somewhat incomplete but it's a start. When I have the time I will create a complete pack and make it available via my website.
Comment: u wot m8???????


Why Muslims are not allowed to read Torah?

Question: I found many references in the holy Qur'an about reading Torat\Torah and also about believing in Torah. So should Muslims read Torah or just deny it on the basis of assumed corruption?
Comment: Thanks for your cooperation in [islam.stackexchange](http://islam.stackexchange.com/review/first-posts/7374). Please [be specific](http://islam.stackexchange.com/help/how-to-ask) while asking your question and provide references in order for your intentions to be clarified to the readers.
Comment: Would u plz term where have u heard that Muslims are not allowed to read Torah? As you asked this interesting question?
Comment: Please share one of the many references which as you say have found in Quran, as not to read torah. There are many Muslim scholars who have byhearted Hindu vedas,bible and so on like Dr.zakir naik, M.M Akber and so on.By saying this I don't wish to say whether it's allowed or not, as I have not come across this question before and haven't researched on this matter...I would like to get any one reference from Quran which you have found.
Comment: I believe as a muslim we can read whatever text/books. Allah has provide `fuurqan` in this Ramadan to guide one who always seek the truth
Answer: Well There is no harm in reading but you should remember that Shaytan (Satan) is a psychological manipulator and you should keep this in mind that Bible came to correct the induced corruption in torah, and now Qur'an is the replenishment of the true message and has a
privilege by Allah himself that it is being guarded from corruption by Allah Azawajjal himself
as this message is intended for all the coming generations till day of judgement and is not for specific civilization of some specific time like torah and bible.
And if you read you will find some statements similar to Quran as they MIGHT(stress on this) have not been corrupted to that great extent.
Be aware Satan is to you an avowed enemy, and he would go to any length to misguide you.
So I would say unless it isn't damn necessary for you to read the torah or are in such a situation then seek refuge with Allah first by saying
"A - uzu billah hi minash Shaytan ir rajeem".
Comment: Are you concern more about syaitan rather than the truth? Allah will always guide one who seek the truth
Answer: I searched to some extent, but I did not find any specific related issue like what you said. But there could be a reason (if e.g. that is a right word), so logically it could be related to the risk of Satan who mislead people by paying attention to them and even changing their beliefs. So for instance if a Moslem has no strong faith in Islam, afterwards (perhaps would be trended to it (Torah) and actually leaves his righteous religion. Thus it could be counted as a reason if somebodies recommend not to recite Torah (although as I said I didn't find such a matter).
To see more related questions/answers, and likewise asking your question:

http://www.islamquest.net/
Comment: Hmm, the reason u gave is exactly right. But, I too heard from someone, that Prophet(pbuh) prohibited someone when that someone was reading Torah, saying that you should read Qur'an. Albeit, the answer you gave is right, but half the reason only. :)
Comment: Perhaps it is true which u uttered. Maybe I mentioned half of the reasons as you said. And God knows best.
Answer:
Jewish people should be encouraged to read the Quran and and Islamic people should be encouraged to read the Torah/Tanakh how wicked and naive to rubbish the name of Allahs other great works. There Torah and Quran are in alignment with each other, ye of little faith what is it you are afraid of? Read the books.. read the books.. the New Testament is based around the destruction and completion of the word of God and they use the good amens set out in the Torah to give their writing legitimacy.. without the Torah.. there is no Abrahamic religion. You should read the Torah but also understand one thing.. both Arabic and Hebrew have the same roots in fact Hebrew is an ancestor of Arabic. Many of the letters and words are the exact same. If possible.. download the text from Sefaria.com and translate each letter from Hebrew to Arabic and notice how similar your languages are. Either translate each letter or each word to Arabic.. Arabic people are lucky in this way.. they can read the Torah in a unique way. One final thing.. it is a codex.. Allah made every single letter important. The Gematria in the Torah is a big part of it. It's the book that willl not change it reveals its infinite wisdom over and over and over .
Answer: There is no prohibition of muslims to read Torah or Injeel.  However, if you don't posses a solid foundation  in Islamic knowledge, this is not recommended according to hadith. If you are an "aleem", then you can use the Quran as a "Furkhan" to sort the authentic from the questionable or blasphmous.
There are gems of knowledge in the Torah that are in agreement to and complement what is in the Quran.  On the other hand, there is alot of slander against the prophets, and blasphemous statements about Allah that totally contradict the Quran.
Following the Torah and other parts of Old Testament can and did lead mankind astray.
examples:
Deuteronomy chapter 20 orders genocide of the worst kind; to "leave nothing left breathing" of people or animals.
Numerous verses in the Torah encourage slavery and human bondage and endorse cruel treatment of slaves.  A google search on "Torah and slavery" will return numerous such verses.
These scriptures were used by Western colonialists to justify their crimes against humanity throughout history.
One of the essential tenents of Islamic faith is beleif in the Day of Resurrection, "Yawmi Deen" where all of humanity will be resurrected and judged according to their deeds.  Even an atom's weight (zerre) of good deed (hayer) or bad deed (sher) will be accounted for.
The Yawmi Deen is strongly emphasized and repeated in the Quran.  The message is clear and unambigous that we will be rewarded or punished accordingly for our actions in this present life.
By contrast, there is not one verse in the entire 5 books of Torah mentioning Yawmi Deen or heaven or hell. The hereafter (ahira) is not mentioned in the 5 books of Torah.
It is beyond the scope of this answer to expound on the details of how essential beleif in Ahira, Yawmideen, Jannah (heaven), and Jahanem (hell) are to Islam.  The conduct of persons or organizations are greatly influenced by the accountability for their actions on yawmideen.
The fact that such an essential element of faith is not even mentioned anywhere in any of the 5 books of Torah calls to question the authenticity of the Torah as exists today.
Comment: Your "examples" sound more like fallacious arguments from incredulity than anything else, and [fallacious arguments are bad](https://islam.meta.stackexchange.com/q/1438/22).
Comment: @goldPseudo I tried to see how you thought my points were fallacious.  I think what happened is you are using deductive reasoning, whereas with Islamic matters you should be using "inductive" reasoning. Deductive reasoning may lead you to assume my points are fallacious whereas they would be perfectly logical using inductive reasoning.


reference or references for resumé

Question: I'm preparing my resumé but I'm stuck on a point. I'll write a reference and as far as I see usually there is a references title in most resumés then details. But, I have just one reference. What should I write references or reference for the title?
Comment: In what country is this? In most cases, you don't list the references in your resume. You only give your reference(s) later in the hiring process.
Comment: why do you only have one reference?
Comment: @StephanBranczyk It's about Turkey. I'm an undergraduate student and think to apply summer internship
Answer: For internships, you'll probably be asked for your references during the application process (either on a form online, or on a form on paper).
I can only speak about the US, not Turkey.
But in my opinion, a candidate should never plaster the contact information of his references on his resume for everyone to see. I've seen candidates do that, and usually I think to myself that such candidates are disrespectful to their references for having done that.
If you want, you can include that "references are available upon request", but in your case, I would just omit that part. That "references are available upon request" is a given for any resume.
And the fact that you only have one reference to give is a red flag. So do not call attention to that fact early in the process if you can help it. That being said, consider doing some volunteer work, or getting some personal references from older friends/mentors.
At your age, it's expected that you may not have any work experience yet, but you should still try to have a minimum of two references, even if they're not professional references. Also, know that references are only usually checked once they've decided to hire you, or once you've made it to the final round. Checking references too early in the process just takes too much time otherwise. That is why, it would be premature to list them in your resume.
That being said, as a sanity check, verify that what I am saying applies to Turkey as well. If out of three Turkish people, two of them tell you to include such a header, then do as they say, and forget what I said. Follow the most common conventions in your country.
Comment: I'm taking a course that is english for bussiness life. Its lecturer said don't write on the cv references are available upon request. Because, it evokes asperity on interviewers. Besides, the reference is my another lecturer. However, as far as I know, references are generally listed on cvs in Turkey. So, when we consider the information, can it really be a problem if I write just one reference instead of two?
Comment: Asperity? Well, if that's the case, then write down the one you have. Put down "Reference" as its header. It may not matter, but you should still work on getting a second reference as soon as you can.
Comment: See previous comment (that I edited while you replied).

reference or references for resumé


How can I see which contacts have been assigned which roles and permissions?

Question: How can i find out which contacts have been assigned which roles? I am looking to ensure that only a few key staff can access and edit our database...
I can see we have two roles set up but I don't know how to find out who has been assigned which role...
Comment: What CMS are you on? In Drupal you can see this via the Drupal menu --> People
I dont know how to do this in Wordpress or Joomla.
Answer: Welcome to SE! The place to start is in the documentation about roles and permissions at https://docs.civicrm.org/user/en/latest/initial-set-up/permissions-and-access-control/
It does depend on the CMS as Jaap says, but they are all covered here. You will see who is assigned to which role in the CMS (Its under Users in WordPress) but then you need to allocate permisssions to these rolls to control which CiviCRM features each role has. There are some more complex option too, with roles and permissions specific to CiviCRM, but you probably don't need these.


Center of mass on an unbounded interval

Question: I am having a hard time locating the center of mass of $y=\frac{1}{x}$ on $(1,\infty)$. I used the formula but I get a value of infinity for the mass of the lamina alone. I was able to get an improper integral after using the formula but then it just gave me an infinity as value at the end. I tried to check it with the graph and it seems that it isn't at infinity. How do we solve center of mass when the interval is unbounded? Any idea is greatly appreciated. Thank you so much.
Comment: Where did this question come from?
Answer: If your lamina is assumed to have finite mass $M$, then it is not possible that its  linear density $\lambda$ is constant, as you have verified. However, it can have e.g. density
$$\lambda(x) = \frac{Mxe^{1-x}}{2\sqrt{1+\frac1{x^4}}}.$$
Then parametrizing your lamina with $\gamma : [1,+\infty\rangle \to \Bbb{R}^2$, $\gamma(t)=\left(t,\frac1t\right)$ gives that the mass is indeed
$$M = \int_\gamma dm = \int_1^\infty \lambda(t)\gamma'(t)\,dt = \frac{M}2\int_1^\infty te^{1-t}\,dt= M$$
and the center of mass is at
\begin{align}
(x,y) &= \left(\int_\gamma x\,dm, \int_\gamma y\,dm\right) \\
&= \left(\int_\gamma t\lambda(t)\gamma'(t)\,dt, \int_\gamma \frac1t\lambda(t)\gamma'(t)\,dt\right) \\
&= \frac{M}2\left(\int_1^\infty t^2e^{1-t}\,dt, \int_1^\infty e^{1-t}\,dt\right)\\
&= \frac{M}2(5,1).
\end{align}

Center of mass on an unbounded interval


Monospace font sometime loses pixels

Question: I'm trying to implement a html/css/js editor, to make things easier I decided to use a monospace font, in this case the JetBrains font. But as I tried setting the caret position to a multiple of the size of one letter, I noticed that the caret was, when the text got big enough, way further then what I expected.
Playing a bit with this, I figured a really weird thing, here is an example:

Do you know why this is happening and if there is a way of implementing it like that, or should I always recalculate the width of the whole line?
By the way, here is the css code of the sizer:
.__editor-sizer {
    opacity: 0;
    position: absolute;
    top: -200%;
    padding: 0;
}

Answer: Ok so after a lot of trials, I finally came up to the conclusion that the issue could only come from the .clientWidth property, which appears to convert the result to an int. That is why I tried using .getBoundingClientRect().width instead. This is the result, not perfect but enough close for me:

Monospace font sometime loses pixels


What is the optimal angle when throwing a stone on a slope?

Question: The question what-is-the-optimum-angle-of-projection-when-throwing-a-stone-off-a-cliff was asked and answered a while back.  This one has a much cleaner answer.  Now you are on a uniform slope, a line through the origin that is not horizontal and you want to throw a stone as far as possible.  As a function of the angle of the slope, what angle should you throw at?  Usual assumptions:  uniform gravity, no friction.
Comment: Just setting a challenge.  I remember the answer.
Comment: Are you asking this question or just setting it as a challenge?
Comment: Should we take into account anatomic concerns? Because for example to throw a stone at a 45° angle, one have to release it quite early. Which would cause a lower speed compared to a later release on a lower angle. So we might have a speed/angle trade-off here. Also, the arm and wrist configuration should cause a non linear acceleration curve. (peaking at lower angles?)
Answer: I am surprised no one seems to have answered this.
I get $$\pi/4 + \alpha/2$$.
If the line is given by $y = x \tan \alpha$, with $\alpha$ acute and we throw from the origin at an angle $\theta$ from the x-axis, at velocity $1$, then we have that the projectile satisfies, assuming gravity $g=2$ (in appropriate units)
$\displaystyle y = t\sin\theta - t^2$, $\displaystyle x = t\cos \theta$
The time at which it intersects the line again is given by
$\displaystyle t\sin\theta - t^2 = t \tan\alpha \cos \theta$ and so
$\displaystyle t = \sin \theta - \tan \alpha \cos \theta$
It is enough to maximize the horizontal distance travelled by the projectile, which is given by
$\displaystyle \cos \theta (\sin \theta - \tan \alpha \cos \theta)$
With little manipulation, we need to maximize
$\displaystyle  \sin(2\theta - \alpha)$
which gives the result.
Comment: That's what I got.
Answer: When the solution is so neat, I feel like it should have a high-level explanation "from the book" that makes it immediately obvious. Sadly, I was not able to find one, but I'm posting what I did get in hopes that it might inspire someone else.
Imagine firing projectiles at all angles from the origin. Assume unit speed and gravity $g = 2$, as in Moron's answer. Consider a frame of reference which starts at the origin and falls freely. There is no gravity in this frame of reference, so all projectiles expand radially outward forming a cone,
$$x^2 + y^2 = t^{2}.$$
However, the ground is no longer given by $y = x \tan\alpha$ but rather by
$$y = x\tan\alpha + t^{2},$$
which is a parabolic cylinder. The projectiles hit the ground at the points where the cone and the parabolic cylinder intersect. Subtracting the two equations, we find that this intersection satisfies
$$\left(x^2 + x\tan\alpha\right) + \left(y^2 - y\right) = 0,$$
i.e. on the $xy$ plane it projects to a circle. On physical grounds, one can see that the circle passes through the origin, and is tangent to the line $y = x\tan\alpha$ there. The projectile that reaches the farthest corresponds to the point with the largest $x$-coordinate, and the angle of the projectile is simply the angle of the line joining that point to the origin (because projectiles just move radially outward in this frame of reference). I encourage you to draw a little diagram and see that this line bisects the angle between the vertical, at $\pi/2$, and the ground, at $\alpha$.
Comment: You sound like a junior Einstein thinking about this the falling frame.  Thanks.


Magento: Add two products for a category on same product page

Question: I would like to show 2 products on a product page in magento for a category, both having complete details and separate "Add to Cart" buttons. Can anyone give me some directions how can I get this done? I am a newbie and i have a project with this problem.
Answer: Assign the Second product as related product from backend or any other relationship  , Replicate the Code for that Product as well
Comment: Both the products have the same Part Number with different prefix. Will this work? And also it would be of great help if you could provide some code to implement.


How to find all occurrences of a variable within a source code?

Question: The title is pretty self-explanatory. I have a minified JavaScript code (100k+ lines) and would like to find all occurrences of a particular variable so I could rename it to better understand code. Is there a way to do that?
Comment: Does your text editor have a search function?
Comment: Try to work on the non-minified form of the source code.
Comment: grep -i "variable_name"
Comment: Rename the variable and find where your program fails?
Answer: Unless this variable is global, there is no way of doing that because with javascript's closure, you can have a tons of "a" variable that doesn't match to the same things.
Comment: Yes, but in that case I could "divide" file into constituent parts to apply changes and "glue" it back together. You actually gave me an idea.
Comment: well you can cut to everywhere the debogguer pass from a localized entry point but that's a long way to go.
Answer: There isn't really a foolproof way to do this, as mentioned in Walfrats answer, closures and the like make this too difficult.
However, it is possible if the library created a min.map file to map the minification process when it was minified. This is called source mapping and allows the minification process to be reversed. Many big JS libraries such as jQuery have this.
You can read more about this here: Introduction to JavaScript Source Maps
However, if the library has a min.map file, then it is probably open source anyway. Searching for the name of the minified JS file will probably lead you to its Github (or similar) page where you would be able to see the unminified source.
Unfortunately, if there is no min.map file, and the code isn't open source, then you are probably out of luck.
Comment: jQuery offers the unminified source for download anyway.
Comment: I would say it serves the same purpose as 'Remote Debugging' and .pdb symbol maps do for compiled C# code (in release mode)
Comment: @RobertHarvey hence my third paragraph 'However, if the library has a min.map file, then it is probably open source anyway.' - I was pointing out that although there exists a way to un-minify code it is mostly pointless if you just want to study the code as the unminified version is usually available. Not sure how to edit my answer to make that point clearer?
Comment: I'm not sure how important the min.map file is.  Why does that even exist, if the unminified source is probably available in any case?
Comment: @RobertHarvey the only use I can think of is the ability to debug on production. I sometimes come across bugs on our live website that I can't initially reproduce on dev or test, and often curse the fact that all my functions and variables are called a, b and c as it can take me a while to work out what is going on, especially if it isn't code I've seen in a while... we don't have a source mapping files though so I havn't used it myself! But I believe that is the intention.


No neutral in metal switch box for three-way switches in an older 1930's home

Question: This is a 1936 home in Virginia with BX wiring and metal switch and outlet boxes.  We went to replace original three-way light switches located at the top and the bottom of the stairs controlling a hall light and found that the neutral wire from switch at the bottom of the stairs is connected to the metal box at the top of the stairs.  There is no other neutral in the box at the top of the stairs to connect it to otherwise.  With the new switch, connecting it to the metal box is only way to make it work.  Is that o.k.?  It seems to have worked for 80 years.
Answer: No, you cannot bootleg neutral from the very rickety ground BX provides.
It worked for 80 years because normal switches don't need neutral.   You are introducing a new wrinkle: a smart switch that does.  Here is how classical 3-way circuits work.  Remember current flows in loops, and the route may differ, certainly as will the color codes.

Note that neither 3-way switch uses neutral, and the "remote" 3-way doesn't even have neutral anywhere near. Remember: installers are stuck with the colors that cables are manufactured with, so white wires may not be be neutral.  Travelers are the ones that land on brass screws, they can be any color. I prefer to mark wires with colored tape to indicate their actual function, but apparently, I'm weird.
The problem is you want to put the smart switch somewhere it can't go.  You need to rethink.
Generally, you handle this by carefully selecting a smart-switch master that goes at the switch location that does have a neutral. Then you either accept a plain switch at the remote location, or use a matching smart switch "remote".  In the latter case, you re-task the 3 wires to be whatever you need for the remote to work.


USB drive emulator

Question: My company uses a software that needs an "original" USB drive inserted, but that's an old software, and currently we only have 1 working pen left. Is there any way to create an image of the pen, or something like that so we can have the software on 2 of our PC's?
Comment: There are services that will "hack a dongle" for obsolete unsupported dongle secured software, search for Dongle dupe, duplication, or emulation. The costs of a custom hack can be more than the original software costs, so if it is not obsolete, unsupported, very mission important sofware, dont bother.  Just purchace another dongle, even if that means buying the package one more time, or using a replacement softwares.
Comment: There is either a hidden file on the pen drive or the key is sotred in the firmware etc and is enumerated by PnP (etc). It sounds like a hardware security dongle and as such is probably off topic for the site(?)
Comment: Make sure they have the same filesystem and volume label, and then simply copy all the files from the working USB to the other one... Let us know if that works.
Comment: doesn't work, the software works in any folder i place it, but the original usb drive needs to be inserted, even if i put another usb inside ,as FAT and no label, it doesn't work
Answer: Are you sure this is a USB mass storage device and not a security key?
Security keys can be duplicated but it requires intimate knowledge of the controller chip and likely is in violation of any license you're using the software with.
If this is truly a USB flash drive, with no other additional devices (USB devices can present themselves as multiple devices to Windows, so open Device Manager and watch for new devices) - try to find a device of the same exact make and model and use an imaging utility such as dd for windows or similar to copy it on the sector level.  At the very least try to make sure it's the same capacity.

USB drive emulator


Taylor series expansion limitation

Question:
Appplying taylor to any functions its not necessary that that polynomial which gets formed from taylor will work for all x , taylor doesnt gives a condition for when the series is always equal to the function right ?

We would need some other methods to check whether the series always matches with the function right ? Like in case of ln(1+x) where expanding using taylor around x=0 we will get a series which is only valid for |x|<=1 , or is it that we can have a series around some number like x= 5 for ln(1+x) then the series will always converge ? If yes when and when not to check for series convergence ?

Is there a deep reason as to why taylor fails to give the whole function in terms of polynomial when expanded around x= 0 but ut might work when expanded around some other point ?
Comment: @QC_QAOA Poles are not the only type of complex singularity.
Comment: @ProblemDestroyer Not really. The fact that $i$ has anything to do with the failure of the Maclaurin series of $\frac{1}{x^2+1}$ to converge at $x=2$ is not obvious.
Comment: Taylor series fail for some functions when there is a pole of that function in the complex plane. That is, Taylor series work everywhere if and only if the function is entire (no poles). See https://en.wikipedia.org/wiki/Laurent_series for more information
Comment: @QC_QAOA can this working and not working can be showed from the derivation of taylor series itself? I mean from the assumption taken while deriving taylor series we may get why pole might cause problem etc.?
Comment: I see understood @Ian thanks
Answer: Surprisingly, the answers to your questions require some complex analysis, even though your questions are ostensibly about real-valued functions of a real variable.
That said:
1:
Taylor's theorem with the Lagrange remainder says that if $\lim_{n \to \infty} \frac{|f^{(n)}(x_1)|}{n!} (x-x_0)^n = 0$ whenever $x \in (a,b)$ and $x_1$ is between $x$ and $x_0$ then the Taylor series centered at $x_0 \in (a,b)$ converges to $f$ at all points of $(a,b)$. However this result can be extremely hard to check.
2:
Complex analysis tells us that the radius of convergence of the Taylor series at a given $x_0$ is the distance from $x_0$ to the nearest complex singularity. For $\operatorname{Log}(z)$ (the principal complex logarithm) for example this is the distance from $x_0$ to $0$, which is a singularity of $\operatorname{Log}$ because it is a branch point. Thus for instance $\ln(1+x)$ centered at $x_0=5$ has radius of convergence $6$.
Complex analysis also tells us that if $f$ is complex differentiable on a disk of radius $r$ centered at $z_0$, then the Taylor series of $f$ at $z_0$ converges to $f$ on a disk of radius $R=\min \left \{ r,\left ( \limsup_{n \to \infty} \left ( \frac{|f^{(n)}(z_0)|}{n!} \right )^{1/n}  \right )^{-1} \right \}$, where $0^{-1}$ is understood as $+\infty$. We can't freely drop this "if $f$ is complex differentiable..." assumption. If we do drop it, we can still say the series converges on a disk of radius $\left ( \limsup_{n \to \infty} \left ( \frac{|f^{(n)}(z_0)|}{n!} \right )^{1/n} \right )^{-1}$ but we cannot be sure that the limit is actually $f$.
3:
I think this is answered by my answer to #2.
This gets much more counterintuitive with functions that look completely innocuous on the real line, such as $\frac{1}{x^2+1}$. Here the series at $x_0=0$ has radius of convergence $1$ even though the restriction to the real line has no singularities anywhere. In some sense the restriction "knows" about the singularity at $\pm i$.
Comment: @ProblemDestroyer What I said in #1 will guarantee convergence when it works. I don't think it guarantees divergence when it doesn't work (except in the trivial setting where this limit isn't zero at $x=x_0$, in which case the Taylor series doesn't converge to anything, much less the original function).
Comment: @ProblemDestroyer I looked back and spotted *another* error, fixed now.
Comment: Sir actually i dont know much about poles and basically the complex analysis , i was into real numbers only and also till high school level , but yeah a bit real analysis i know . Can this be a bit easily explained using the knowledge without complex analysis ?
Comment: @ProblemDestroyer Only at the level of my point #1. The "deep explanation" needs complex analysis, period.
Comment: I think i got a bit of understanding from #1 thanks Sir .  Although its possible right that the series might converge at some other values of a where its being evaluated (expanded) at? For all values of x ?
Comment: @ProblemDestroyer Check my revised #1, I had an error.
Comment: Thank you , i understood :)

Taylor series expansion limitation


New Job Pay Schedule

Question: I recently got a new job at a small company and earn a salary rate. They pay on the 15th and 30th of each month however the HR employee explained to me that although the pay week ends on the 15th and 30th of each month, I won't get the check until almost 5 days later for those pay weeks. I am sure they will pay me but according to HR employee it allows them to compensate for any "outstanding balances". Although I'm sure there is not much I can do and honestly I don't like their pay schedule at all however my question is this normal?
Comment: In your Q you said you will get the check after no more than 5 days later. I would consider that "a few days later". Just think of it this way - you get paid on (say) the 19th and the 4th of each of month.
Comment: "I don't like their pay schedule at all". <-- Fair enough to have this opinion. But ask yourself if this fact is enough reason for you to turn down the job offer.
Comment: @Brandin, I didn't turn down the offer. I am just use to getting paid the same day for the pay week or a few days later not almost a week later. This is all new to me.
Answer: It seems perfectly normal to me.  I've always worked at places that paid either weekly or bi-weekly; and they always paid 1 week after.  This gives time to sort out any problems like your forgetting to complete a timesheet, filling it out incorrectly (working on project X instead of Y, got sick and left early, etc).  It's much easier for bean counters to fix things before the money's gone out than after the fact.
If anything, I'm mildly surprised your payday is only offset 5 days instead of 7; which would give the bean counters a consistent 5 days to do their work, instead of 5 days some weeks and 3 days others.  Are you sure they meant 5 calendar days, not 5 working days?
Comment: My company's pay periods are 1st - 15th and 16th - end of month. We get paid on the 25th (or earlier if 25th is on a weekend) and the 10th (or earlier), so I get usually 7 working days after the end of the pay period.
Answer:
is this normal?

Yes, its normal.
I've also worked for companies that paid twice per month - there's nothing odd or nefarious about that. I've also worked for companies that paid once per month, every other week, and a few other arrangements.
And every company I've ever worked for held the paycheck for some number of days after the end of the pay period. Usually, it was around 5 days to 1 week.
In my last company, we were paid on Fridays for the previous week and the week before that (so 7 days after the end of the pay period for many, and 5 days after the end of the pay period for those who worked on Sunday). Timesheets needed to be completed and approved by the end of the day on Tuesdays, so they actually turned them around fairly quickly.
Nothing unusual here at all.

New Job Pay Schedule


VB6 Error - Cannot load .ocx File

Question: I have been trying to load a OCX file into one of my VB6 projects for most of the day today.  I've tried checking the COM registration in the registry the best that I know how, and have attempted several times using REGSVR32 on the file to no avail.  All I keep getting from VB6 when I try to load the component (Under Project --> Components) is 'FilePath\FileName.ocx' could not be loaded.  Has anyone else seen this before, and if so, any ideas how to fix it?
Answer: Just a couple of tips to help isolate the problem....
Look at the control with OleView
Examine the name: if it is one of the reserved words such as 'Menu', vb6 will fail to load it.  (Although vb6 will happily let you create (and use) one with a bad name until it is saved and reloaded)
Examine the interface: does it describe the functionality?  You may have a damaged or unlicensed ocx.
Attempt to load it with the ActiveX Control Test Container...
If you can, the problem is with VB6 and not the ocx.
Comment: In OleView use File->View Typelib on the OCX and notice the importlib lines. For instance importlib("oleguids3.tlb"); means that this tlb is needed to be registered on the dev machine before referencing the OCX. The reason is that if a library is referenced by the [external] interfaces of the OCX then it should be registered on the dev machine or the OCX is unusable in the IDE.
Answer: This it may be a dependency issue -- some other component needed by the OCX is not present. Dependency Walker will find any static dependencies.
Comment: Sure enough, this turned out to be a dependency issue.  It sure has been a while since I fought with this one, completely forgot I had asked about it.
Answer: We fixed similar error recently. In our case the error was in the Visual Basic project (.vbp) file. Project file had been edited outside standardized build machine and contained references to OCX components - including workstation specific paths & versions of the components.
You can either edit the the project file in text editor or get a working version from version control.
Answer: Un-registering and registering again worked for me too.
Please note that you need to register this component using this syntax:
<code>regsvr32 /i:design olch2x8.ocx
</code>
Answer: When you say you have used "REGSVR32 on the file to no avail" do you mean that you get a file name succeeded message, or an error message? If registering the file succeeds CMB is likely right about the ocx being unlicensed.
Answer: Licensing aside, there are a couple of other trip points with vb6.  Especially if the OCX was developed in VB6.
If so, the 'officially' system registered version may conflict with the specific ocx you're using.    In other words, both of them might have the same proper name (or classid) but the one you're attempting to use may not implement all the functionality described in the system registered one (i.e. you have a less evolved version).  This arises when the author desires to maintain binary compatibility while enhancing the functionality of a control.  As long as the public interfaces remain compatible, vb will not recalculate the classid.
You can fix this by forcibly unregistering the specific control (actually unregister all instances of the control). (regsvr32  /u control.ocx )   Then re-register the one you intend to use.  Be certain that no running instance of VB6 exists when you do this or the results might not work.  (Check your task list)
Hope that helps...
Answer: The control may be reliant on another DLL or OCX that may be missing or not registered and this could manifest itself as VB reporting that the control you're trying to use is missing. However, I'm not sure how you would identify the dependencies. As far as I know, dependency walker identifies static dependencies; I'm not sure if it can identify COM dependencies.
Answer: Un register the Ocx
re register the ocx
then loaded the project again in VB6, it worked for me.
Thanks everyone

VB6 Error - Cannot load .ocx File


How can we use digital signatures for mutual authentication between two computers?

Question: I get the fact that we can use digital signatures for authenticating the signer (since they are using their private key to create the signature, and this is known only to the signer ). But how can the verifier also be authenticated using digital signature? I have a feeling that we could use digital certificates for this but I am confused about how such a protocol would work.
Comment: If A is the signer and B verifies A's signature then just let B be a signer too and A verify B's signature. It works the same in both directions and is commonly used in TLS (mutual authentication with client certificates), IPSec VPN, GPG and S/MIME email signatures ...
Comment: If what I wrote answers your question, please click the check mark beside it to select it as the accepted answer. This will make it easier for others to find it when searching similar questions
Answer: Preface
Digital signatures and signing is not a one way street. A public private keypair is actually just a mathematically linked keypair, with one key designated as the private key and the other designated as the public key.
When you sign a message with the private key, the public key can be used to check the message and thus validate the sender. Additionally, when you encrypt a message with the public key only the private key can decrypt it. This can be used to ensure only one receiver can read the message. What some people don't realize though, is that this process is actually mathematically reversible. You could sign a message with your public key (making it only verifiable to yourself), or encrypt a message with your private key (making it readable to the world). Mathematically, keypairs are interchangeable and can both perform the same operations. However, it is worth noting that when it comes to practical applications, these keys should not be interchanged. You can read more about why a private key is chosen to be private in this StackOverflow answer. In short, both keys are derived by an exponent. In real world cryptographic applications like RSA, the exponent for the private key is made extremely large, thus causing the public key exponent to be extremely small. This makes guessing the private key WAY harder, while making the task of guessing the public key WAY easier (which is fine because everyone already knows the public key). So, if you gave out your private key and kept your public key secret, it would be easy for an attacker to derive your privately held public key when compared to the proper method of doing things.
Also, something to touch on: Mathematically, signing and encrypting is the same process. Signing a message is simply taking a hash of the message and then encrypting the hash using your private key. This encrypted hash is sent along with the message. Other users will verify this signature by decrypting it using your public key. They will then hash the message and compare the hash they computed with the hash from your signature. If they match you are verified because they only you could have used your private key to encrypt the correct hash of the message.
Note: You mention digital certificates as being used to solve your problem as if it is a separate thing from public-private key cryptography. It is not. These are both examples of asymmetric cryptography. Digital certificates contain a public key along with other metadata, all of which is usually signed by another authority (using asymmetric cryptography)
To answer your question
There are multiple ways you can do this with asymmetric cryptography. In my following examples we will have Alice and Bob who both want to authenticate the other user.
Example 1
Oddly enough, in the same SO thread I linked to explain digital certificates there is one answer to your question. Similarly, to achieve mutual authentication between two entities we can sign an encrypted message. Suppose Alice initiates a conversation with Bob. Alice encrypts the message "respond with the codeword potato" using Bob's public key. Alice then signs this encrypted message with her private key. When Bob receives the message, he can verify that Alice sent it by checking the signature against Alice's public key. Bob can then decrypt the message with his private key and respond to Alice with the codeword "potato" (encrypting his response with Alice's public key). When Alice receives the message from Bob and reads the codeword, she can validate it is Bob on the other end.
**Alternatively, Bob could just sign his response message and Alice could verify him that way (a more secure method) but this scenario has the downfall of not involving the word "potato".
Example 2
Because a public-private keypair is simply a mathematically linked keypair, we could conceivably have two private keys. In this implementation, the size of the exponent would not be chosen to be overly small or large for either key. This would prevent one key from being "weaker"/"stronger" than the other. If we give Alice the first private key, and Bob the second private key, they would be able to sign their messages with their own private key, thus making it verifiable to the other user's private key. AKA, Alice sends a "hello" message to bob, and signs it with her private key. When Bob receives the message, he can validate it using his own private key (which is the partner key to Alice's) and then respond with a "hello", signing this message with his private key. Both parties would know that the other is who they say they are. Additionally, because both of these keys are private, Alice and Bob could encrypt their messages using their own private keys, ensuring that they are only readable to the other person.
**Disclaimer: This second example came straight out of my head. There are probably implications to signing/encrypting messages back and forth like this which I am unaware of, but this is a simple to digest example.


Security work as a freelancer

Question: I am confined to a wheelchair (paralyzed) so I have very extensive care needs. I'm very interested in computer security, but I am not able to work a traditional 9-to-5 job onsite. Is it possible (how?) to become a security consultant that works freelance from home? What specialism (If any) would offer me the best chance?
Answer: Are you able to see people at your place? This might help lower the personal issues a customer could have about sharing security details with you. With a certain expertise and reputation it shouldn't be hard to even skip that part. I worked with many people I never met because they got recommended by someone I trust.
What do you mean with interested? No experience yet? If not, you can start look into securing servers, those are usually remote from the office too, so it doesn't really matter where you work. Then you need to decide which kind of servers you wanna specialize on, Linux, Windows, all. Get to know the basic tools like OpenVPN, SSH login with certificates, Firewall-Rules, SSL/Https in Webservers like Nginx and Apache, port blocking tools like fail2ban, maybe things like XSS and SQL injections, depending what kind of security you prefer. From there your interests will take you in the right direction.
Start training on an own server, either as a VM or a real one at a provider, help friends look into their setups, and so on. To really consult people you should know what you are talking about in general. You don't need to know all specific commands and it's okay to learn during the job. Security is more about planning ahead and preventing problems than actually dealing with live hacks, so you are not really on a hot schedule.
You should have a broad knowledge of all possible security threats and either know how to handle those or who to use for subtasks you are not familiar with.
For "on-site" working there are tools like remote desktop in the windows world or on linux you open a terminal anyway. That's not really an issue if the client trusts you.
Answer: I'm biased.
If I were hiring a security consultant, the first thing I'd have to recognize for myself is the fact that for my business, there are resources available:

In-the-office for inside use only (no remote access)
In-the-office with remote access as an option
External facing stuff (i.e. websites) where this is the original intention

It's going to be hard to make the first two items above available for someone who doesn't work in the office.  Additionally it will be super difficult to share the "tribal knowledge" about such systems.  Personally, I wouldn't want to share even login and password information with someone remote, as it's very, very vulnerable.
The skill you're proposing to utilize touches very, very private information within a business.  I'm sure there's training for it somewhere but you must consider the practical social limitations of a client's willingness to hand you the keys to the entire kingdom, sight unseen.  Usually a consultant in this situation would be paired with someone from the inside, for balance's sake.
Comment: I think one can do security work remotely. That said, it's getting enough experience under your belt to do so remotely. Catch-22 situation.

Security work as a freelancer


Can I make a colour gradient or color change base on the number of visit to some specific country?

Question: i have been trying with geopandas and folium and other common libraries of python. I create myself a little challenge, I make a map and highlight the countries I have been to with a single colour. However, I would like to add more features into it, I want to present a different color of the country if I have been there more than once. For example, taiwan, i have been there once, so it is a orange color, japan, twice, so is a red colur, korea, 3 times, so is a purple color etc...
I took reference from matplotlib recently but the code isnt working, is it possible to give me some advice?
here is my code
the excel file has 2 columns Column A called 'Country', consists of the name of country like china, japan, germany etc, Column B called 'Visits', consists of how many times I have travelled to a corresponding country.
from google.colab import files
import pandas as pd
import geopandas as gpd
import folium
import matplotlib.cm as cm
import matplotlib.colors as colors

# Uploads file
uploaded = files.upload()

filename = next(iter(uploaded))

# Read  Excel file to a pandas DataFrame
df = pd.read_excel(filename)

# world geometry
world = gpd.read_file(gpd.datasets.get_path('naturalearth_lowres'))

# Merge data with the world data
visited = world.merge(df, left_on='name', right_on='Country')

# map createw
m = folium.Map(location=[0, 0], zoom_start=2)

# boundaries
for _, r in world.iterrows():
    sim_geo = gpd.GeoSeries(r['geometry']).simplify(tolerance=0.001)
    geo_j = folium.GeoJson(data=sim_geo.to_json(), style_function=lambda x: {'color': 'black'})
    folium.Popup(r['name']).add_to(geo_j)
    geo_j.add_to(m)

# Function to set the color based on the number of visits
def color(visits):
    # Create a colormap for 4+ visits
    colormap = cm.get_cmap('YlOrRd')

    # Normalize the visits to 0-1
    norm = colors.Normalize(vmin=1, vmax=4)

    # Get the color from the colormap and convert to hex
    rgba = colormap(norm(visits), bytes=True)
    return '#{:02x}{:02x}{:02x}'.format(rgba[0], rgba[1], rgba[2])

for _, r in visited.iterrows():
    sim_geo = gpd.GeoSeries(r['geometry']).simplify(tolerance=0.001)
    geo_j = folium.GeoJson(data=sim_geo.to_json(), style_function=lambda x: {'fillColor':                     color(r['Visits'])})
    folium.Popup(r['name']).add_to(geo_j)
    geo_j.add_to(m)

#  map
m

Answer: It looks like you could make such a map in a simpler way, just with GeoPandas plotting tools, i.e. without using folium. Say you have these data:

Country
Visits

China
1

Iran
3

Japan
2

North Korea
3

South Africa
2

Venezuela
4

If you store it in a CSV-file, you can extract and show it like this:
import pandas as pd
import geopandas
import matplotlib.pyplot as plt
import matplotlib.cm as cm
import matplotlib.colors as colors

# read a CSV file to a pandas DataFrame
df = pd.read_csv('visited_countries.csv', delimiter='\t')

visited = list(df['Country'])  # countries visited
num = list(df['Visits'])  # number of visits

# get the countries dataset from Natural Earth
file_with_all_countries = geopandas.datasets.get_path('naturalearth_lowres')
all_gdf = geopandas.read_file(file_with_all_countries)
# set the names of the countries as the index
all_gdf = all_gdf.set_index("name")
# get the data for the visited countries
visited_gdf = all_gdf.loc[visited]
# create a column for the number of visits
visited_gdf["num"] = num
cmap = cm.YlOrRd
# show all the countries with the minimum color from the cmap, indicating 0 visits by default
ax = all_gdf.plot(facecolor=cmap(0), edgecolor='grey')
# paint the visited countries
norm = colors.BoundaryNorm(sorted(set(num)), ncolors=cmap.N, extend='both')
visited_gdf.plot(ax=ax, column='num', legend=True, vmin=min(num), vmax=max(num), cmap=cmap, norm=norm)
plt.title('Country Visits')
plt.show()

Here is the result:


Should I retire followers after I've leveled up significantly since their joining?

Question: Some brief research has lead me to the conclusion that, while the strengths of a follower match your current level at the time of their joining you on your quest, they do not level up through the rigors of questing as does your character.  If this is true, it would appear to make sense to retire a follower after you've gone up X number of levels.  For example, it wouldn't be practical to take Faendal from Riverwood with you from the start of your quest all the way up to your character's 50th level since your follower would be uselessly outmatched probably at around level 10 or 15.
Answer: Yes, that's mostly correct.  However, my understanding is that it's your level at the time they're first loaded (i.e. when you visit the area they're in) that controls their level, not your level when you hire/enlist them.
Answer: While this is generally true, there is one item that will forcefully reset your followers (to reflect your new, higher level) - the humble Wabbajack!
To use, you want to wabbajack your follower (might want to save first, so you don't accidentally slay them) until they get hit by one of the transformation effects - when they turn back into their normal self, they'll have been updated in level, the same as if you had clicked on them and typed (in sequence) enable disable into the console.
Comment: Marked as answer since other people have confirmed that this indeed works.  I've just been using the disable/enable console cheat and haven't felt the least bit bad about it.
Comment: @oscilatingcretin Gotcha, yup, that works too. Mentioned wabbajack specifically for those folks on the 360 or PS3.


Which church traditions regard Constantine the Great as a Saint?

Question: Constantine  was the emperor of the Roman Empire who legalized Christianity and called the Counsel of Nicaea, among other thing. Which church traditions regard Constantine the Great as a Saint?
Comment: Do you have a reason to think any do? I've never heard of a "Saint Constantine".
Comment: I'm pretty sure some Orthodox venerate him
Comment: I'll have to take your word for it. Or you can link a source. Wikipedia lists "Saint Constantine" as an alternative name. I believe you; just wasn't sure at first, so others may not be as well.
Comment: I think this is a great question that may have the cart before the horse.  It would be better phrased, "why does X regard Constantine the Great as a Saint?"  You would have to do the research to find the denomination that does.  Asking for denominations is like a polling question.  Let's say there are two denominations, who would be the accepted answer?
Answer: It appears that Emperor Constantine the Great is venerated as a saint in the Catholic Church, according to catholic.org:

Constantine was the most dominating figure of his lifetime, towering over his contemporaries, including Pope Sylvester I. He presided over the Council of Nicaea, gave extensive grants of land and property to the Church, founded the Christian city of Constantinople to serve as his new capital, and undertook a long-sighted program of Christianization for the whole of the Roman Empire. While he was baptized a Christian only on his deathbed, Constantine nevertheless was a genuinely important figure in Christian history and was revered as a saint, especially in the Eastern Church.

Catholic.org gives his feast day as May 21.
A page on the official website of The Antiochian Orthodox Church describes Constantine as a saint:

The Orthodox Church sees Constantine as the emperor who assisted the early Church in evangelizing the Roman Empire. For this reason it honors him as Saint Constantine Equal-to-the-Apostles [Constantinos Isapostolos].

Although I was unable to find an official statement by the Oriental Orthodox Church to the effect that they revere Constantine as a saint, the website of the Western Prelacy of the Armenian Apostolic Church of America, one of the churches in the Oriental Orthodox Communion, lists June 16 as the feast day of "Sts. King Constantine and his mother Helena". Thus, at least one of the Oriental Orthodox Churches considers him a saint.
The Wikipedia article on Constantine states that he is venerated as a saint by "Eastern Orthodox Christians, Byzantine Catholics, and Anglicans" (and in a footnote that he is recognized by the Oriental Orthodox Church), but I can find no specific backing for the "Anglican" claim from any reliable Anglican source. It also puts Constantine on a list of people celebrated in the Lutheran calendar, but notes that the Lutheran church "does not officially recognize saints".
My best supported answer, then, is that Constantine is recognized as a saint by the Eastern Orthodox Church, by the Catholic Church (particularly the Oriental sui iuris churches), by at least one of the (several) Oriental Orthodox churches, and perhaps by the Anglican Communion churches.
Answer: Constantine has never been added to the Roman Martyrology and thus has never been recognized officially as a saint. Nevertheless, Eastern Rite Catholics follow the Orthodox traditions and are permitted to do from Rome. His feast day is May 21. On a personal and private level Catholics are permitted to honor Constantine as a saint.
Comment: Welcome to Christianity.SE, and thanks for offering another answer. Can you provide quotes or links to church materials that support your answer and provide more information? If so, it would make this answer a better fit for this site. See: [How we are different than other sites](http://meta.christianity.stackexchange.com/questions/1808/how-we-are-different-than-other-sites). Thanks!


After reinstalling Linux on dual Boot, I have multiple boot options

Question: I have a Lenovo Yoga c940 with eOS dual boot. I cleared the eOS partition using Windows disk management tool and reinstalled eOS on the empty space. I want to do it one more time (for reasons that are not important). However, during boot now, I have multiple instances of Linux boot hard drive (with the same ID). Is there a way to clear the Linux partition, delete all options except windows, so that I can reinstall eOS and have only 2 options, Win and eOS?
Comment: You may not need to remove and reinstall eOS. This may be something relatively simple. Can you update your question to show a list of all storage devices and their partitions? This will make it easier to outline the next steps
Comment: Thank you for the comment Matigo. Actually I want to reinstall eOS anyway. I made a mistake during partitioning and the root folder is now full.
But now I am afraid that if I clear the Linux partitions with Windows Disk utilty and resize again, I will have 3 Ubuntu UEFI options.
Answer: Did you try $ update-grub2 or $ update-initramfs -u?
Comment: I dont think this will work unless you delete the windows partitions first ... the -u (update) looks for partitions to include for the boot
Answer: U can use efibootmgr command to clear the unnecessary entries in your bios :
https://doc.ubuntu-fr.org/efibootmgr
for usage


add run capacitor to capacitor start motor to increase efficiency

Question: Can I add a run capacitor to a capacitor start motor to increase its efficiency? I remember doing this 20 years ago and the run current dropped 20% or so, but I didn't measure the power consumed. The motor was quieter and cooler. The run cap value was 0.1 x start cap value, connected across the start cap and start switch. This is an offgrid water pump application.
Comment: +1 good question .You are right ,I dont have definative test results to back this up so I wont answer your question .
Comment: The question needs to be rearanged, since I don't understand what you mean by start cap and run cap. The terminology used is usualy related to: run cap - always connected, start cap - disconnected trough centrifugal switch when the motor gains rpms. Now where do you intend to add capacitor, and to which one, how many caps do you have now?
Comment: You won't always have access to both terminals of the centrifugal switch, but if you do, this should work.
Answer: That would be worth investigating. You could try it again and make measurements this time. If you do it, you would be risking overheating the start winding. If you have information that the motor can be reversed by switching the capacitor from one winding to the other, you can be sure that the two windings are identical. If to windings are not designed to be interchangeable, the start winding has smaller wire and will likely overheat fairly quickly if connected continuously with the start value of capacitor. It may overheat eventually even with a X 0.1 capacitor.
If the motor is reversible in the manner described, there will probably be a diagram for that on the nameplate or on a label on the inside of the wiring box cover. You could also check the instruction sheet if you have it or can find it on the manufacturer's web site.