My understanding of Hawking radiation is probably extremely lackluster but from my understanding, two virtual particles pop into existence just like anywhere in the universe. Except due to the extreme gravitational influence of the black hole one particle escapes at c and the other is pulled in overtime this causes the black hole to lose mass.  So this event can only take place at the event horizon but from my understanding supermassive black holes radiate away much slower than smaller black holes (the extreme mass difference aside.)   How come supermassive black holes radiate away at a slower rate than small black holes despite having more surface areas?  And if the particle escapes and doesn't destroy each other how come this doesn't create infinite energy?

Pedestrian question from a non-physicist:  I read on Wikipedia that larger black holes emit less net Hawking radiation than smaller black holes.  This seems counterintuitive to me. If black holes are essentially sucking in mass and converting it to Hawking radiation, why wouldn't a more massive black hole emit more Hawking radiation?  Moreover, how is it that Hawking radiation can escape black holes in the first place when visible light (which I know to be another form of radiation) cannot?


What's the best stack up for a four layer board with two power planes (plus and minus a dc voltage), a ground plane, and a signal plane? Can anyone share a resource that talks about the best practices for stack up and why they are the best practices? I can't find many things that address having two power planes. Is the following an acceptable stack up, or should the signal and ground planes be adjacent?  Signal +VDC -VDC Ground

I'm designing a 4 layer PCB and I know that the standard stack-up is  Signals GND VCC Singals  (GND and VCC can be switched depending on the layer with more signals) The problem is, I don't really want to connect all ground pins through vias, there are just too many of them ! maybe because I'm not used to 4 layer PCBs, anyway, I've read a  by Henry W. Ott about a different stack-up  GND Signals Signals GND  (Where the power is being routed with wide traces on the signal planes) According to him, this is the best stack-up possible with a four-layer PCB, for the following reasons: 1.Signal layers are adjacent to ground planes. 2.Signal layers are tightly coupled (close) to their adjacent planes. 3.The ground planes can act as shields for the inner signal layers. (I think this requires stitching ??) 4.Multiple ground planes lower the ground (reference plane) impedance of the board and reduce the common-mode radiation. (don't really understand this one) One problem is cross-talk, but I really don't have any signals in the third layer, so I don't think that corss-talk will be an issue with this stack-up,am I correct in my assumption ? Note: The highest frequency is 48MHz, there's a wifi module on the board too.


I am Bangladeshi, have Portuguese D visa(PhD purpose). Booked a ticket by Qatar Airways Dhaka to Lisbon.The travel itinary is Dhaka - Doha - Barcelona - Lisbon. But in Barcelona I have 6 hours layover and it is mentioned that Qatar airways operated by Vueling. Do I need a transit visa for Spain?  I have tried to contact as many places as possible but still get no response. If it is possible then response please. Thank you!

I found many related questions on this site but I am still not sure about the rules. How can I decide if I need a visa to transit? Schengen members as of May 2021 are as follows:  Austria Belgium Czech Republic Denmark (excluding Greenland and the Faroe Islands - but an open border with the Schengen Area is maintained) Estonia Finland France (excluding overseas departments and collectivities) Germany Greece Hungary Iceland Italy Latvia Liechtenstein Lithuania Luxembourg Malta Netherlands (excluding Aruba, Curaçao, Sint Maarten and the Caribbean Netherlands) Norway (excluding Svalbard) Poland Portugal Slovakia Slovenia Spain (except Ceuta and Melilla) Sweden Switzerland


First off I am not sure whether I have the correct terminology. What I mean by certificate authority is a verified certificate distributor such that if I goto a website in my Browser that is HTTPS (with an SSL certificate) it doesn't give a warning that it might be unsafe.  There is two things I need to do here.  I have a Windows XP machine and a Linux Mint (Ubuntu 14 base) machine. The Windows XP machine has a certificate authority installed such that it authorizes use of certificates it distributes so that I don't get any warnings in my browsers. I need to extract this from the Windows XP machine  Second, I need to "install" this onto my Ubuntu machine such that it when I visit a website via any browser on my Ubuntu machine that has a certificate issued by the CA, that I don't get  a warning/have to add an exception.  I hope I have asked the question properly - please say if you don't understand.

My work has decided to issue their own  (CA) to handle different aspects of our work securely without paying for certificates.   Cryptographically sign emails Encrypt email contents Make access to things like the company  client-certificate based. Revoke the keys of former employees automatically   They sent me a .pem file, and I'm not sure how to add it to my Ubuntu install. The instructions sent were: "Double-clicking on it on a Mac should install it."   How do I proceed? Do I need to do something with  to create a .key, .csr, or .crt file?


How do I prove that for any $a, b, c \in \mathbb{R}$ the inequality $a^2+b^2+c^2 \geq ab+bc+ac$ is true?

How can the following inequation be proven?  $$a^2 + b^2 + c^2 \ge ab + bc + ca$$


I been trying to find the title of the movie involving these 2 scenes and the best description from what I can remember. I can’t  think who the 2 kids I think both are male and have a watch alarm set to watch a specific tv show. But I am thinking pretty sure came out in the 90s and had to do with time travel somehow to the past. Only one of the friends goes back not sure how.  So I only remember parts involving a wrist watch. First in the beginning two friends both have a wrist watch alarm set to remind them both to watch the same TV show. Then when one of the friends that goes into the past, his alarm goes off on his watch to remind him to watch the TV show but somehow helps by getting an idea using the watch battery.  The watch didn’t cause the time traveling only that the battery of the watch was used in the past after traveling there.  Both friends that had a alarm that set for both friends to watch a specific tv show  But can’t find those scenes when I think I am close to finding the movie the scenes aren’t there and I kept thinking it’s A Kid in King Arthur’s Court but something like that kind of movie feel.  Yea kind of but it has more a 90s feel. It’s not clockstoppers is slowing down time.

I have been trying to find the title of this kid's movie involving these 2 scenes and the best description from what I can remember. Each of the scenes are connected by the use of the watch but in two different time periods in 1980s/1990s and then in the past.  First it’s about two best friends (I think both are male) who both set their wrist watches at a certain time to remember to watch a TV show (I think wrestling?) each week and after confirming the exact times on their wrist watches they say yeah air alright and possibly high five.  The show somehow had to do with time travel to the past. Only one of the friends goes back but I'm not sure how. While being stuck in the past and trying to solve a problem to help the people in the past, his alarm for the TV show goes off and he remembers the title of the TV show by hearing the alarm uses the battery to solve the problem. The watch didn’t cause the time traveling only that the battery of the watch was used in the past after traveling there.  But other then those scenes I don’t know the whole main plot and around the time it came out in the early 90s or late 80s.  It's not A Kid In King Arthur’s Court.


When a UK visitors visa is rejected with comments no right to appeal or to administrative review can one apply afresh and, after how long?

I am from Nigeria and I'm doing a one year Internship program with The Marriott hotel in Wisconsin, I applied for the UK visitor visa two weeks ago to visit my fiance who lives and work in the UK, but I was refused the visa because the letter of my workplace that I provided was back dated of 2014 which they say they are not satisfy it accurately reflects my current employment circumstance. I also provided my pay slips, but they said they were not satisfied with it demonstrating the terms and conditions of my employment, and they also said I did not present any evidence to satisfy me having sufficiently strong ties and commitments in my country of habitual residence or elsewhere to motivate my timely departure from the UK. They said the documents my fiance presented were OK.  How soon can I re-apply for a UK visa after getting a refusal?


Can you explain which one is correct or if both are correct, under which scenario one is more suitable than the other? I have this doubt in general, when to use nouns as adjectives or possession like in this example. Can you give me some pointers?

How would you explain to a person who is learning English, and whose native language does not have attributive nouns, when the possessive should be used instead of an attributive noun?  In particular, how would you explain it referring to the following list of sentences?   Today's news is bad This week's schedule has been changed Tomorrow's lunch will be at the French restaurant This year's report will be communicated in the reunion room Last summer's vacations were memorable Last year's meetings were a complete disaster


Problem 4. use the principle of induction to verify:   $$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}6$$     base case is obviously easy, but I don't know how to prove the inductive case

I encountered the following induction proof on a practice exam for calculus:  $$\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$$  I have to prove this statement with induction.  Can anyone please help me with this proof?


I have a lengthy two-column document where some of the figures end up taking up the entire column (centered in the column), but there ought to be room for text either above or below the figure.  Help?  Here's an example:   The preamble is reasonably straightforward:  \documentclass[letterpaper,twocolumn]{article} \usepackage{amsmath} \usepackage{luatextra} \usepackage{hyperref} \usepackage{fontspec} \usepackage{listings} \usepackage[margin=1in]{geometry} \usepackage{unicode-math} \usepackage{booktabs} \usepackage{threeparttable} \usepackage{siunitx} \usepackage{topcapt} \usepackage[version=4]{mhchem} \defaultfontfeatures{Ligatures=TeX} \setmainfont{TeX Gyre Termes} \setmonofont{TeX Gyre Cursor} \setmathfont{TeX Gyre Termes Math} \renewcommand\thesection       {\Roman{section}} \newcommand{\matr}[1]{\mathbf{#1}} \renewcommand{\vec}[1]{\bf{#1}}  %float placement parameters \renewcommand{\topfraction}{.85} \renewcommand{\bottomfraction}{.7} \renewcommand{\textfraction}{.15} \renewcommand{\floatpagefraction}{.66} \renewcommand{\dbltopfraction}{.66} \renewcommand{\dblfloatpagefraction}{.66} \setcounter{topnumber}{9} \setcounter{bottomnumber}{9} \setcounter{totalnumber}{20} \setcounter{dbltopnumber}{9}   and the offending figure is defined as:  \begin{figure}[tbp]   \centering   \includegraphics[width=\linewidth]{spectra.pdf}   \caption{K, Th, and U basis spectra. Although drawn as smooth curves, the       x-axis (energy deposited) is divided into 0.012 MeV wide channels,       and the corresponding y-axis value is the average number of counts       recorded during a one-second measurement having an energy that falls       within that channel.  Solid curves correspond to \ce{NaI}, while       dotted curves correspond to \ce{CeBr3}. The dashed K curve in the top       panel corresponds to a Radian-sized \ce{NaI} spectrum that has had       the count rate in each channel multiplied by a factor of 1.3 (so that       the low-energy peak roughly overlaps those of its larger breathren).       The bottom panel is just a blow-up of the high-energy portion of the   spectra presented in the top panel. Vertical dashed lines indicate window   boundaries (see Sec.\ \ref{sec:fitting}) with the lower limit of each   window located at 0.072, 0.372, 0.696, 1.032, 1.344, 1.548, 1.944, and   2.208 MeV.}   \label{fig:basisspectra} \end{figure}   Thanks!

How to influence the position of float environments like figure and table in LaTeX?   This is a general question and should collect useful answers for all users. I hope we can use this as a reference


I'm not really understanding the difference between $F[\alpha$] and $F(\alpha$), where $\alpha$ is an element of $E$, a field extension of $F$. From what I read, $F[\alpha$] = $F(\alpha$) when $\alpha$ is an algebraic over $F$. But why is that? And if $\alpha$ is transcendental, then it seems to be a completely different case.   Really, any clarification on what the difference between $F[\alpha$] and $F(\alpha$) is would be tremendously helpful. Thank you!

I've seen similar questions asked here, but I've not been able to find a comprehensive answer.  I know that for a ring $R$, $R[X]$ denotes the ring of polynomials over $R$ and $R(X)$ denotes the field of fractions of $R[X]$.  But if $\alpha \in S$, where $S \supseteq R$ are rings, what is the distinction between $R[\alpha]$ and $R(\alpha)$?  It is my understanding that if $R$ is a field then $R[\alpha] \cong R(\alpha)$, but that this is not generally true for any ring.  Is this correct?  Adding to my confusion is the fact that $\mathbb Z[i]$ and $\mathbb Z(i)$ are used interchangeably, despite $\mathbb Z$ not being a field.  However, it's clear to me that they are equivalent ( i.e. $\mathbb Z[i] = \mathbb Z(i) = \{ a + bi | a,b \in \mathbb Z \}$); is this because $i$ is algebraic over $\mathbb Z$?  Or for some other reason?  Thanks for your help!


this is the test code I am using to understand how java handles object memory.  public class TestCode {    public static void main(String[] args) {     TestCode obj = new TestCode();      CustomClass cs1 = new CustomClass(5);     obj.updateExistingObj(cs1);     System.out.println(cs1.val);      CustomClass cs2 = new CustomClass(5);     obj.instantiateExistingObj(cs2);     System.out.println(cs2.val);      CustomClass cs3 = null;     obj.updateNullObj(cs3);     System.out.println(cs3.val);   }    void updateExistingObj(CustomClass cs1) {     cs1.val = 9;   }    void instantiateExistingObj(CustomClass cs2) {     cs2 = new CustomClass(9);   }    void updateNullObj(CustomClass cs3) {     cs3 = new CustomClass(9);   } }  class CustomClass {   int val;   CustomClass next;   CustomClass(int x) { val = x; } }   The output of the first syso where I am printing cs1.val I am getting expected value which is 9.  The output of the second syso where I am printing cs2.val I am getting 5 as output instead of 9.  The output of the third syso where I am printing cs3.val I am getting a null pointer exception.   Can anybody help me understand what is happening here under the hood? How exactly java handles the memory location when we pass an object as a function parameter? Thanks for helping!!

I always thought Java uses pass-by-reference. However, I've seen a couple of blog posts (for example, ) that claim that it isn't (the blog post says that Java uses pass-by-value). I don't think I understand the distinction they're making. What is the explanation?


What leads us to the conclusion that spin is angular momentum? Could it not be some other quantity? Sorry if this is a rookie level question.

If I understand correctly, spin is an intrinsic property of particles, which follows the algebra of angular momentum, but has nothing to do with an "orbital angular momentum" in that the particle is not like a small sphere that rotates on itself, to which we could attribute an angular momentum as usual in classical mechanics, cf.  and  Phys.SE posts.   My question is: why does spin combine with orbital angular momentum to give total angular momentum? To me this is surprising, as spin, in fact, has nothing to do with a classical angular momentum. Any insight on that?


The assertion: $n^3 + 5n$ is divisible by $6$  I have completed the basis step $(n=1)$ and the first part of the induction step $(n=k)$, but I am stuck on the second part $(n=k+1)$.  This is what I have so far:  For $n=k$: $k^3 + 5k = 6t$  For $n= k+1$:  $(k+1)^3 + 5(k+1)$  $= k^3 + 3k^2 + 8k + 6$  $= (k^3 + 5k) + 3k + 3k^2 + 6$  $= 3k^2 + 3k + 6t + 6$ $= ???$  I cannot pull out a $6$ because that would leave halves that cannot be counted as integers. How should I proceed?

Prove that $ n^3 + 5n  $ is divisible by 6 for all $ n \in \textbf{N} $.  I provide my proof below.


This might be a bit of an odd question, but I figured that at least one or more persons here have been in the situation that I am now.   My question is as follows:     What is the (if any) 'proper etiquette' for telling a   loved one that you wish to record them as they tell their life story,   before they die and all familial information is gone?   I am struggling with how to word it without it coming out so bluntly or being seen as a rude and/or crude question to ask. Obviously, from the standpoint of a genealogist and someone in general that loves and wants to remember their family members, recording someone would seem to be common sense. However, bringing up the subject of death or phrasing your question so that it doesn't come out as, "I know you're going to die soon, so can I record you?", can be quite the task (with respect to each personality type).

I have a couple of elderly relatives who have not kept a journal and refuse to write down their "life story." I really would like to know their stories and who they were and who they are. They have their favorite stories that they rehash every time I see them, but I'd like to get more. I have no experience with "interviewing" and am looking for information on how to get their stories. Any websites, books, and or tips\techniques would be greatly appreciated.


If I attempt to find the displacement $x$, from equilibrium when a mass, $M$ is attached to an upside down vertical spring with spring constant $k$ I get two different answers when I use energy considerations and when I use forces. There is probably some simple conceptual mistake I am making but I can't identify it. Here are the equations I am referring to  $0=\frac{1}{2}kx^2-mgx$, (making our equilibrium point, $y = 0$) $$x=\frac{2mg}{k}$$ Now using forces, $$mg=kx$$ $$mg/k=x$$ Obviously $2mg/k$ is not equal to $mg/k$ so what is my mistake?

If a spring is stretched by a weight of mass m, (so the extension is $\Delta x$) then $ k = \frac W{\Delta x} = \frac {mg}{\Delta x}$. So $ k\Delta x= mg $.  When the Spring is stretched by distance $\Delta x$ (by the weight) then it looses gravitational potential enegry. ($ \Delta GPE= mg\Delta x$)  But, when we calculate the change in elastic potential energy, We get  $ U = \frac 12 k(\Delta x)^2 $. Since $ k\Delta x = mg$, $U=\frac 12 mg\Delta x$  At, equilibrium we have no kinetic energy $E_k = 0$  Doesn't that violate Energy Conservation?  Where does the rest of lost GPE goes?  Here's a illustration


In Notepad++, I have many lines in the format:  abc.xyz   where "abc" might have space or - or numbers.  How do I delete lines without "."? So if a line does not have any ".", I want it removed completely, not even leave an empty line.

I have a text file and want to keep lines started with <Path>, and delete all the other lines.  How can I do it?


I have a remote Ubuntu 16.04 server and try to ssh into it over VPN connection.  From Windows Putty it works fine but from Ubuntu 16.04 Workstation terminal not! Debug show me this:  ssh -vvv -p 33333 karl@177.177.177.177 OpenSSH_7.6p1 Ubuntu-4ubuntu0.3, OpenSSL 1.0.2n  7 Dec 2017 debug1: Reading configuration data /etc/ssh/ssh_config debug1: /etc/ssh/ssh_config line 19: Applying options for * debug2: resolving "177.177.177.177" port 33333 debug2: ssh_connect_direct: needpriv 0 debug1: Connecting to 177.177.177.177 [177.177.177.177] port 33333. debug1: connect to address 192.168.1.100 port 33333: Connection refused ssh: connect to host 192.168.1.100 port 33333: Connection refused   some Ideas what I can do?

I have an Ubuntu Server 10.10 32-bit in my home. I'm making SSH connections to it from my PC via Putty.  The problem is, sometimes I'm able to login seamlessly. However, sometimes it gives me an error like this: Network error: Connection refused.  Then, I dont't change anything, try to login a few times more, wait a while and try again. Sometimes I can log in, sometimes I cannot. It seems pretty random to me.  What can I do to solve this?  Edit:  And Sometimes, Putty gives Network error: Software caused connection abort error after displaying login as: text.  Here is the ping -t output:  Pinging 192.168.2.254 with 32 bytes of data: Reply from 192.168.2.254: bytes=32 time=6ms TTL=64 Reply from 192.168.2.254: bytes=32 time=65ms TTL=6 Reply from 192.168.2.254: bytes=32 time=88ms TTL=6 Reply from 192.168.2.254: bytes=32 time=1ms TTL=64 Reply from 192.168.2.254: bytes=32 time=3ms TTL=64 Reply from 192.168.2.254: bytes=32 time=1ms TTL=64 Reply from 192.168.2.254: bytes=32 time=1ms TTL=64 Reply from 192.168.2.254: bytes=32 time=1ms TTL=64 Reply from 192.168.2.254: bytes=32 time=1ms TTL=64     I turned off firewall of router, and everything seems to work now. Except for that, I still can't enter my web server by typing external IP from my PC.


Is there a way to see if a field exists in an IDataReader-based object w/o just checking for an IndexOutOfRangeException?  In essence, I have a method that takes an IDataReader-based object and creates a strongly-typed list of the records.  In 1 instance, one data reader has a field that others do not.  I don't really want to rewrite all of the queries that feed this method to include some form of this field if I don't have to.  The only way I have been able to figure out how to do it so far is to throw the 1 unique field into a try/catch block as shown below.  try {     tmp.OptionalField = reader["optionalfield"].ToString(); } catch (IndexOutOfRangeException ex) {     //do nothing }   Is there a cleaner way short of adding the "optional field" to the other queries or copying the loading method so 1 version uses the optional field and the other doesn't?  I'm in the 2.0 framework also.

How do I check to see if a column exists in a SqlDataReader object?  In my data access layer, I have create a method that builds the same object for multiple stored procedures calls.  One of the stored procedures has an additional column that is not used by the other stored procedures.  I want to modified the method to accommodate for every scenario.  My application is written in C#.


I have an EEE PC 701 and want to record video using the built-in webcam. I have installed Lubuntu 14.10. I already have installed avconv and I want to use avconv.

What software options are there for taking videos and or pictures within Ubuntu (Much like Apple's Photobooth)?  Please limit one software solution per answer and include key features/description of the software


When I visit YouTube, this is what I see in the upper left corner:    At first I thought that YouTube was thinking I'm in France, but the URL is still https://www.youtube.com and the rest of the site appears in English with no French anywhere.

I see a new logo with CA at youtube.com. Does anyone know what it means?


Find $$\lim_{n\to \infty}\int_1^{\infty}\frac{n}{1+x^n}dx$$  Lebesgue dominated convergence does not work. My next thought was to use Fatou's lemma and reverse Fatou's lemma to bound the integral, but I haven't the faintest idea how to calculate the liminf or limsup of the integrand. Wolfram Alpha tells me that the value of the integral decreases (starting with n=2) from about 1.2 to about 0.7, but I can't even show that the sequence is decreasing right now.  Any hints would be appreciated.

I am particularly curious about exchanging the sum and integral in the third equation, if this is unjustified then please let me know.  We have that   $$\lim_{n \to \infty} n \int_{1}^{\infty}\frac{dx}{1+x^n} =\lim_{a \to \infty}\lim_{n \to \infty} n \int_1^{a}\frac{dx}{1+x^n} $$  Also   $$\frac{1}{1+x^n} = \frac{1}{x^n}\frac{1}{1+\frac{1}{x^n}} = \sum_{i=0}^{\infty}\bigg(\frac{-1}{x^n}\bigg)^i\frac{1}{x^n} =\sum_{i=0}^{\infty}(-1)^i \frac{1}{x^{(i+1)n}}$$  Since we are integrating for $x \geq 1$, the above series converges for every $x$ so we may integrate term-by-term to obtain   $$n \int_1^{a}\frac{1}{1+x^n}dx = n\sum_{i=0}^{\infty}\int_1^{a}\frac{1}{x^{(i+1)n}} \, dx$$  $$= n\left[\frac{1}{n-1}-\frac{1}{2n-1}+\cdots\right]-n\left[\frac{a^{1-n}}{n-1}-\frac{a^{1-2n}}{2n-1}+\cdots\right] $$  where the second term goes to zero as $n$ goes to infinity. Further, we have that   $$\lim_{n \to \infty}\frac{n}{nk-1} = \frac{1}{k}$$  so the first term becomes $1 -\frac{1}{2}+\frac{1}{3}-... = \ln(2)$. In conclusion we get  $$\lim_{a \to \infty}\lim_{n \to \infty} n \int_1^{a}\frac{dx}{1+x^n} =\lim_{a \to\infty} \ln(2) = \ln(2) $$  independently of $a$.


Consider the  determinant of the matrix $A = [x_{ij}]_{n\times n}$ as a polynomial in the ring $\Bbb{R}[x_{11},\dots,x_{nn}]$. How can I show that it is irreducible ?

A polynomial $p$ over a field $k$ is called irreducible if $p=fg$ for polynomials $f,g$ implies $f$ or $g$ are constant. One can consider the determinant of an $n\times n$ matrix to be a polynomial in $n^2$ variables. Does anyone know of a slick way to prove this polynomial is irreducible?  It feels like this should follow quite easily from basic properties of the determinant or an induction argument, but I cannot think of a nice proof. One consequence of this fact is that $GL_n$ is the complement of a hypersurface in $M_{n}$. Thanks.


So I have a bunch of files in a folder, where a leading number represent the order of the file, but they are not the usual 1-n, like this:  2-file.txt 4-file.txt 7-file.txt 12-file.txt   and I want to have them in a sequential order from 1 to "number of files", preserving the numeric order, so:  1-file.txt 2-file.txt 3-file.txt 4-file.txt   where 1-file.txt is the previous 2-file.txt, 2-file.txt is the previous 4-file.txt and so on

I have the following files   SSt12.precip.374.sub.climatology.ctl SSt12.precip.2874.sub.climatology.ctl SSt12.precip.3764.sub.climatology.ctl SSt12.precip.6774.sub.climatology.ctl   I want to rename the files as   SSt12.precip.1.sub.climatology.ctl SSt12.precip.2.sub.climatology.ctl SSt12.precip.3.sub.climatology.ctl SSt12.precip.4.sub.climatology.ctl


I have a question about uncountable nouns in a sentence. For example, which of the following sentences is correct?  This is an important future work.  This is important future work.  It seems that according to the rules the second one is correct, but it looks somewhat absurd to me. Which one is correct?  p.s. This is not a duplicate question. It's not about the general rule. It's about the absurdity of using a rule.

I can’t for the life of me figure out where to use a and where to use the — and where there is no article at all. Is there a simple rule of thumb to memorize?   The standard rule you always hear:     “If a person knows which item you are talking about then use "the"   . . . doesn’t clear things up for me, as I have no idea whether or not they know.


If gravitons (or some other quantum particle) are indeed responsible for the effect of gravity, can gravity be blocked, defected or trapped somehow?  Is there any evidence that a gravitational field can be augmented at all, other than by another massive object?

If I remember well, they said that it can't, but I do not know why.  Yes, I meant if gravity can be shielded using something like a Faraday cage (or something else?).  Thank you.


I recently started using Ubuntu 20.04, and I want to customise date & time format (in Windows I could write it down using a "formatted string" like %YYYY-%MM-%day). I am referring to the clock at the top on the panel.  I've found a gnome shell extension (Format Clock), but it's not compatible with gnome-shell versions >= 3.14, and I have 3.36.2. I would want to customise the format differently from the options available on the list in Region & Language. How could I do that?

On previous versions I had the date displayed. Now it just shows the current day but not the day's number or the month.   Is there a way to customize the clock or tray/taskbar more in GNOME shell on Ubuntu 17.10 and later?


Related to  and  - why do so many entries in the newsletters I get include questions for tags I explicitly hide?  For example, in the Board & Card Games Weekly, there's all kinds of  questions hilit - a tag I hide on BoardGames.SE.  Including questions related to hidden tags only serves to clutter-up the newsletter and my inbox.

Stack Overflow is huge and varied. Wouldn't it be great if I could just subscribe to a newsletter with a list of my favorite tags? If I love dealing with , I think it would make sense for my version of the newsletter to focus more on Android questions.  Or, if I sign up for the  newsletter, but don't own a Mac, it would be nice if I my newsletter would focus on the part of the site I do follow (i.e., iOS).  So I would like to suggest that the newsletters for each site (optionally) take into account the user's favorited and ignored tags for that site.  I know that you can receive emails about specific filters, but I don't think that's a good solution for users who already receive the newsletter. Most of the questions it highlights cover topics that I have no interest in. It would be much better if I could filter out all the noise and get down to my likely favorites.


I've got  Ubuntu 18.04 Server in Virtual Box, and once I've ran out of 10 GB of allocated space on virtual disk, I've increased the size of the virtual disk to 17 GB and resized the root partition with parted's resizepart  command (and done so while partition is in use, and yes I know it's bad practice, but it's a virtual machine for development, all code committed to github and I've a clone of the machine just in case).   Everything seems to have worked: lsblk and udisksctl properly report the new size, but df is still stuck at reporting the old 10 GB and 100% usage (even after reboot).  Why is it so and what can I do about this odd df's behavior ?

How can I resize partitions from the command line? I've heard of , but I don't want to use a GUI program.


I have an text file named answer.txt. And i'm trying to read the contents from a file and trying to insert in the below yaml file (above 'kind:')  For that, I'm reading the file and assigning to a variable. file= 'cat answer.txt'   If, I print this using echo, echo $file I can able to see the contents.  But, when i try to insert this in SED command, the actual variable itself is getting inserted but not the actual contents. I'm not sure, what i'm doing wrong here.   Command - sed  '/kind/i $file' input  Yaml file name - input  apiVersion: v1 data:   Corefile: |     .:53 {         errors         health         kubernetes cluster.in-addr.arpa ip6.arpa {            pods insecure            upstream            fallthrough in-addr.arpa ip6.arpa            ttl 30         }         prometheus :1020         forward . /etc/resolv.conf         cache 30         loop         reload         loadbalance     } kind: ConfigMap metadata:   creationTimestamp: "2020-01-24T15:19:51Z"   name: core   namespace: system   resourceVersion: "166"   selfLink: /api/v1/system/configmaps/core   uid: b80b35ad-8718-4e88-8asbef-bceb289d50ee     After applying SED command,   apiVersion: v1     data:       Corefile: |         .:53 {             errors             health             kubernetes cluster.in-addr.arpa ip6.arpa {                pods insecure                upstream                fallthrough in-addr.arpa ip6.arpa                ttl 30             }             prometheus :1020             forward . /etc/resolv.conf             cache 30             loop             reload             loadbalance         }     $file     kind: ConfigMap     metadata:       creationTimestamp: "2020-01-24T15:19:51Z"       name: core       namespace: system       resourceVersion: "166"       selfLink: /api/v1/system/configmaps/core       uid: b80b35ad-8718-4e88-8asbef-bceb289d50ee   Could some one please correct me as i'm a beginner?  Thanks in advance  Operating System - MAC

If I run these commands from a script:  #my.sh PWD=bla sed 's/xxx/'$PWD'/' ... $ ./my.sh xxx bla   it is fine.  But, if I run:  #my.sh sed 's/xxx/'$PWD'/' ... $ ./my.sh $ sed: -e expression #1, char 8: Unknown option to `s'    I read in tutorials that to substitute environment variables from shell you need to stop, and 'out quote' the $varname part so that it is not substituted directly, which is what I did, and which works only if the variable is defined immediately before.  How can I get sed to recognize a $var as an environment variable as it is defined in the shell?


Prove that f0 – f1 + f2 - … - f2n-1 + f2n = f2n-1 – 1.  For n is all positive numbers. I have an idea to what I must do, but I can't figure what the base case is. I think it is f(0) = 0 and f(1) = 1.  But then idk how to prove these?

Possible Duplicate:          This is a homework question so I'm looking to just be nudged in the right direction, I'm not asking for my work to be done for me.      The Fibonacci numbers are defined as follows: $f_0 = 0$, $f_1 = 1$, and $f_{n+2} = f_n + f_{n+1}$ whenever $n \geq 0$. Prove that when $n$ is a positive integer:  \begin{equation*} f_0 - f_1 + f_2 + \ldots - f_{2n-1} + f_{2n} = f_{2n-1} - 1 \end{equation*}    So as I understand it, this is an induction problem. I've done the basis step using $n = 1$:    \begin{align*} - f_{2(1)-1} + f_{2(1)} &= f_{2(1)-1} - 1\newline - f_1 + f_2 &= f_1 - 1\newline - 1 + 1 &= 1 - 1\newline 0 &= 0 \end{align*}  I've concluded that the inductive hypothesis is that $- f_{2n-1} + f_{2n} = f_{2n-1} - 1$ is true for some $n \geq 1$. From what I can gather, the inductive step is:    \begin{equation*} f_0 - f_1 + f_2 + \ldots - f_{2n-1} + f_{2n} - f_{2n+1} = f_{2n} - 1 \end{equation*}  However, what I find when I try to prove it using that equation is that it is incorrect. For example, when I take $n = 1$    \begin{align*} - f_{2(1)-1} + f_{2(1)} + f_{2(1)+1} &\neq f_{2(1)} - 1\newline - f_1 + f_2 - f_3 &\neq f_2 - 1\newline - 1 + 1 - 2 &\neq 1 - 1\newline -2 &\neq 0 \end{align*}  I suppose that my inductive step is wrong but I'm not sure where I went wrong. Maybe I went wrong elsewhere. Any hints?


I am using the following query (in the top-right search field): [worklight] is:question answers:0.  Some of the results are marked with [duplicate].  Is it possible to exclude these? I could not see a proper option for this in the search help section.

I often search for closed questions and sift through the ones that are not duplicates to see if any of them need to be killed. Technically, closure is the first step towards deletion and the only closed posts that are actually useful are duplicates (unless it's a crappy one).  For the rest, if the community hasn't bothered to reopen it after a while or if the OP has made no effort to edit their question, I delete it if it has no redeeming content. For example, the only closed posts seen on  are exact dupes.  But this does become tedious when there are several results for various close reasons with duplicates interspersed between them. I'd like to be able to search as:  closed:1 duplicate:0   so that it gives a list of questions closed as OT/NaRQ/NC/TL. This will be similar in behaviour to migrated:0|1.   This will also be useful for 10k+ users on all sites (2k+ in beta) in the network, especially SO, so that they can search for questions to spend their delete votes on— especially closed, non-dupe questions in the tag they specialize in. More often than not, I hesitate to vote on questions that are outside my area of interest (unless it is crappy).


I saw a very tiny black bug crawling over my computer screen the other day. Having recently acquired a "" for smartphone, I started investigating. On a white paper it seemed like a small black dot. I think its in the 0.3 - 0.8 millimeter range. It would have been fairly unnoticeable if it weren't for it to crawl over my screen. What could it be? I was thinking about dust mites, but they say it should appear as white. So I'm not entirely sure. I live in Western Europe. Note: don't look at the scale. It is wrong.

I found these bugs walking on the floor/ wall and on my laptop.  And unfortunately I think they could be in my bed, too, since I bring my laptop to bed sometimes.   I live in a big city in the Netherlands without pets. My house has humidity issues.  I have noticed that the bugs are very tough to exterminate; meaning that when trying to squeeze them it really needs a lot of effort for their size. Their size is about 1 - 2 mm and their color is whitish.    Please note that my lightbulb is having a yellow color.so it makes it look darker than normal. It's more white than looks in the picture          This is a pic of it walking on my screen...


Performed time series clustering where I used DTW to generate a distance matrix.  The distance matrix was then given as an input to t-SNE where the two-dimensional results from t-SNE were used for clustering with DBSCAN.  Does it make sense?

I've got an application where it'd be handy to cluster a noisy dataset before looking for subgroup effects within the clusters.  I first looked at PCA, but it takes ~30 components to get to 90% of the variability, so clustering on just a couple of PC's will throw away a lot of information.    I then tried t-SNE (for the first time), which gives me an odd shape in two dimensions that is very amenable to clustering via k-means.  What's more, running a random forest on the data with the cluster assignment as the outcome shows that the clusters have a fairly sensible interpretation given the context of the problem, in terms of the variables that make up the raw data.  But if I'm going to report on these clusters, how do I describe them?  K-means clusters on principal components reveal individuals who are nearby to one another in terms of the derived variables that comprise X% of the variance in the dataset.  What equivalent statement can be made about t-SNE clusters?  Perhaps something to the effect of:     t-SNE reveals approximate contiguity in an underlying high-dimensional manifold, so clusters on the low-dimensional representation of the high-dimensional space maximize the "likelihood" that contiguous individuals will not be in the same cluster   Can anyone propose a better blurb than that?


I run the two logistic regressions on different time windows (let say 1/2015 and 3/2015) and get two sets of parameters estimates. They differ a little.   Is it possible to use any statistical test to find out whether the difference between every respective two parameters (from the first model and the second model) is significant?

So I run logistic regressions of same dependent variables and independent variables on two sets of data $S_1$ and $S_2$. I obtained two sets of coefficients $\beta_1$ and $\beta_2$. Now I suspect that $\beta_1$ and $\beta_2$ should really be equal, since I don't think there are distinctive difference between $S_1$ and $S_2$.   My question is, how do I test the significant differences between the coefficients?   Thanks!


I have a table that has columns like following. Batch number is a database design thing like row number  dm_id        type_in_id    type_student_id     amount  36              1               1               1  36              1               2               2  36              1               3               3  36              2               1               4  36              2               2               5  36              2               3               6  36              3               1               7  36              3               2               8  36              3               3               9   I want to create a query that would return this:  dm_id  amount1 amount2 amount3 amount4 amount5 amount6 amount7 amount8 amount9  36       1       2       3       4       5       6       7       8       9   How can I do that?  P.S. Version 5.5.56-MariaDB - MariaDB Server

Been looking to pivot rows (progress_check) into columns check 1, check 2 etc... No sum or totals required, just to display the results hopefully... Can anyone help, Thanks Ad


Find $\lim_{x\to 1}(\frac{p}{1-x^p}-\frac{q}{1-x^q})$    My attempt: I took LCM and applied lhospital but not getting the answer.Please help

How does one evaluate the following limit? $$\lim_{x \to 1} \left(\frac{23}{1-x^{23}}-\frac{11}{1-x^{11}} \right)$$ The answer is $6$.  How does one justify this answer?  Edit: So it really was just combine the fraction and use L'hopital's rule twice (because function and its first derivative are of indeterminate form at $x=1$). This problem is more straightforward than it seems at first.


Does the temperature of a body depend on the frame of reference?

If an observer starts moving at relativistic speeds will he observe the temperature of objects to change as compared to their rest temperatures? Suppose the rest temperature measured is $T$ and the observer starts moving with speed $v$. What will be the new temperature observed by him?


I just figured that, we can redirect both System.out and System.err messages to  a file, am not able to understand this command which does redirecting while running this below file using : java StandarProgram> output.log 2>error.log  What is 2? and if am not using 2, error messages are displayed on console and not redirected to file? am not able to get this at all.  public class StandardProgram{      public static void main(String[] args){         System.out.println("Hello");        System.err.println("World");      } }

In bash, calling foo would display any output from that command on the stdout.  Calling foo > output would redirect any output from that command to the file specified (in this case 'output').  Is there a way to redirect output to a file and have it display on stdout?


Can we create a system of recurrences that exhibits chaotic behavior, with the restriction that we are only allowed to use the integers?  For example, we could create a system:  $$ \begin{align} a(t) &= a(t-1)^2 +b(t-2) + 3c(t-1)  \\ b(t) &= 3 + a(t-1)  \\ c(t) &= b(t-1)b(t-2)^2 \\ & \vdots \end{align} $$  I'm of course looking for a system that stays within some boundaries for certain intial conditions.  In other words, maybe:  $$ \begin{align} -20 \le a(t) \le 100  \\ 15 \le b(t) \le 50  \\ -100 \le c(t) \le 100 \\ & \vdots \end{align} $$  Of course, all of the functions I mentioned are just examples.  The main problem is that the functions must exhibit what I will call "totally integer functions".  That is to say that they must be functions from integers to integers.  That disallows division, since we could get a non-integer result.  It also disallows floors, ceilings, and modulo functions, for example.  I also cannot use function composition, unless the total depth of recursion is less than $\log{(\log{(n)})}$ for a function with all values between $-n$ and $n$.

I'll summarize first and then add details.  Please forgive me as I'm delving into the field of chaos without much knowledge of it, seeking an expert to help point me in the right direction.  Are there dynamic maps with strange attractors whose coordinates (or defining functions) are almost integers, in the sense that they could be described in the decimal system with very few digits to the right of the decimal point?  To make this idea of "almost integers" more precise, I'm looking for systems that essentially require coordinates with $n$ digits to the right of the decimal point, and $\exp(n)$ digits to the left of the decimal point.  For example, in the Mathematica 9 documentation of "RecurrenceTable", they delve into a system of the form:  $$ \begin{align} x_{n+1} &= \alpha x_n + y_n \\ y_{n+1} &= \beta + \left(x_n\right)^2 \\ y_0     &= \delta \\ x_0     &= \gamma \end{align} $$  I'm looking to find dynamic maps like this example that possess at least one strange attractor.  The important part is that this strange attractor can be described by large real or complex numbers that are "almost integers".


In indirect speech present perfect tense is back shifted to past perfect tense.  For example -  John said, "I have written a letter." (Direct speech) John said that he had written a letter. (Indirect speech)  But why I see in news articles that the present perfect tense is often not back shifted to past perfect tense ?  For example -  The President said that investigators have conducted hearings on different issues. (Direct speech - President said "The investigators have conducted hearings on different issues"  Why often I see in news articles that present perfect tense is not back shifted to past perfect tense though in almost every book its written present perfect tense is to be converted to past perfect tense in reported speech ? Please advise me on it. Thank You

While changing direct to indirect speech, we generally back shift the tense.  Example -  "I have finished my work."             (Direct speech) He said that he had finished his work.  (Indirect speech)  In this example present perfect tense is back shifted to past perfect tense.     NOTE : There is an exception to this rule : At the time of reporting, if the reported words are still true or still relevant then back shifting is optional. For example: "He said that he has lost his keys." Here present perfect tense is not back shifted because at the time of reporting situation is still relevant. By relevant I mean that the consequence of losing his key can still be seen or heard.    Now my main question(s) is:   Context: Recently an earthquake came in India and Nepal. Thousands of people have lost their lives. Rescue operation is on its way.   Now consider the example which is not an indirect speech: "The President learned that this earthquake has caused destruction all across Nepal and India."   First: tell me, here is it correct to use "has caused" under the context that situation is still relevant, i.e. the aftermath of the earthquake can still be seen?  Second: Here is back shifting optional or obligatory?    Another question:   You see, my intention of asking question is -  In sentence structure of indirect speech like - (He said +........... , He informed +....... , He claimed + .......... ), I am aware that if the situation is still relevant then back shifting of present perfect tense is optional.   But what if the sentence structure is something like - ( He learned + ........... , He noticed +.......... , He found +....... ) here also is back shifting of present perfect tense optional if the situation is still relevant ?  P.S. : My question concerns only present perfect tense.


There are two words on either side of the word I'm looking for: synonym and antonym. Of course, a synonym is a word that means the same thing (automobile vs. vehicle), and an antonym is a word that means the opposite (male vs. female). However, let's use the example of automobiles. There are different types: car, truck, van, SUV, etc. They're neither synonyms or antonyms of each other. But is there a word that describes their similarities, given they're all automobiles?  The word should fit this sentence:     A car is a _______________ of a truck.

If north is the antonym of south, then what is the relationship between north and all other non-north directions such as east, west, south, south-east, south-west, etc.?  Similarly, if male is the antonym of female, then what is the relationship between male and the other non-male genders such as transgender variants, female, etc.?  Perhaps I can illustrate my question as:          antonym  north<--------->south            ? north<--------->everything non-north   They are perhaps mutually exclusive. But I expect that there's a semantic term that better describes this relationship. If there's no straightforward technical term available, I'd appreciate non-technical alternatives as well.


Regarding the product notation... if we have non-commutative things (for example matrices): $$\prod_{i=1}^N {\bf A}_i$$ Is it by default assumed to mean $${\bf A}_1{\bf A}_2\cdots {\bf A}_N$$ or $${\bf A}_N{\bf A}_{N-1} \cdots {\bf A}_1$$ or maybe even an arbitrary order?

Does a sum or product symbol, $\Sigma$ or $\Pi$, imply an ordering?  Clearly if $\mathbf{x}_i$ is a matrix then:  $$\prod_{i=0}^{n} \mathbf{x}_i$$  depends on the order of the multiplication. But, even if one accepts that it has a sequence, it is not clear if it should mean $\mathbf{x}_0\mathbf{x}_1 \cdots \mathbf{x}_{n-1}\mathbf{x}_n$ or $\mathbf{x}_n\mathbf{x}_{n-1} \cdots \mathbf{x}_{1}\mathbf{x}_0$.  A similar question, is there a "big" wedge product convention?  $$\overset{n}{\underset{i=0}{\Huge\wedge}} \;{}^{\Large{\mathbf{x}_i} \;=\; \mathbf{x}_0 \wedge \mathbf{x}_1 \;\cdots \mathbf{x}_{n-1}\; \wedge \mathbf{x}_{n}} $$


Is a statistically correct to find the parameters of the model $$ Y_t = \beta_1 Y_{t-1} + \beta_2 X + \epsilon $$ by just using OLS? For example, there won't be any problems with bias of the coefficients?

I am doing auto-regress by usual linear regression package. e.g.  $y_t=φx+ε_t$ with $x =y_{t-1}$  My reason is that, Auto-regression does assumes iid errors, same for linear regression. Linear Regression doesn't have assumption on independent variables. What's different is merely that the independent variables is replaced with lagged dependent variable y.  But I see AR always is fitted by specific methods, I am afraid I'm doing wrong.


I'm trying to understand 'classes' in JavaScript.  Usually emulating classes in JavaScript looks something like this:  function MyClass(someParameter) {     this.someValue = someParameter;     this.someOtherValue = "green"; }  MyClass.prototype.someMethod = function() {     console.log("Hello!"); };   While the non-function members are set inside the constructor, the method is added outside of the constructor, to the .prototype property of the constructor.  I'd like to understand why. Why not create the methods inside the constructor, just like the rest of the members? Meaning:  function MyClass(someParameter) {     this.someValue = someParameter;     this.someOtherValue = "green";      this.someMethod = function() {         console.log("Hello!");     } }

What's the difference between  var A = function () {     this.x = function () {         //do something     }; };   and  var A = function () { }; A.prototype.x = function () {     //do something };


Let $(X, \|\cdot\|_X), (Y, \|\cdot\|_Y)$ normed spaces, operator $A: X \longrightarrow Y$ weakly continuous. Is $A$ continuous with respect to the strong topology?  Thanks in advance for your ideas.

Suppose I have a weakly sequentially continuous linear operator T between two normed linear spaces X and Y (i.e. $x_n \stackrel {w}{\rightharpoonup} x$ in $X$ $\Rightarrow$ $T(x_n) \stackrel {w}{\rightharpoonup} T(x)$ in $Y$). Does this imply that my operator T must be bounded?


I simply want to create a buffer of a specific size, around a single point. When I create a buffer it uses degrees/ or radians, or whatever, and the buffer is created that covers the whole screen.   I am working on a South African map, so I have tried setting the project CRS to WGS84/UTM30S.  I then load my shapefile, and set the shapefile CRS to WGS84/UTM30S. Still the buffering tool creates this massive buffer.   I just want to be able to create a buffer, of known radius in meters.    I am using QGIS 1.8.0 Lisboa  Anyone with ideas?

I  have had no luck getting the buffer tool to accept anything but degrees as units of measure.  I have found lots of stuff saying the layer needs to be reprojected and saved but it hasn't worked at all for me.   Is there a way I could create a buffer without using ftools or at least force the units to meters somehow?  As a workaround I converted meters to degrees (lat) and used that but the final product needs to be as close to reality as possible.  Things I've tried:   setting every unit option I could find to meters (where possible). setting everything to NAD83/Maryland (data is for Washington, DC) and saving it as such (as layers in ESRI shape files). reimporting the reprojected layers setting relevant layers to Google Mercator   The was tried followed by creating a buffer. Many were tried in combination. QGIS 1.7.3 Slackware64 current (qgis from SBo-13.37 repo, tried on multilib and plain 64it with same results)


I want to add javascript to page footer if the shortcode is used into the page content.  add_shortcode('show-popup', 'show_aip_popup');  function show_aip_popup($args){     // HTML Comes here for popup }

I've got shortcode that needs to include JS library only once and only there where it's used.  function shortcode_function($atts, $content = null) {     $class = shortcode_atts( array('something' => '0'), $atts );      return "<script src=\"http://maps.googleapis.com/maps/api/js?sensor=false\"></script>              And here goes HTML code for the map."; } add_shortcode("sh", "shortcode_function");   The problem is that above example is going to include library several times if maps shortcode is used several times. How can I prevent that? Or more precisely: how can I do it "the correct way"? Can I use wp_enqueue_script() in that place?  Just one requirement: I really need this to include that library only-when-shortcode-is-used.


I have read somewhere that the radical $\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}=1$ and I don't understand it. How do you solve this(when the RHS is unknown)?

We have $$ \sqrt[3]{2 +\sqrt{5}} + \sqrt[3]{2-\sqrt{5}} = 1 $$  Is there any way we can get this results through algebraic manipulations rather than just plugging it into a calculator?  Of course, $(2 +\sqrt{5}) + (2-\sqrt{5}) = 4 $, maybe this can in some way help?


Me and my friends are looking into starting to play D&D. We try to get the minimal necessary books to start whit. I know that the easyest way is to get a red box but its currently out of our price range.   I was looking into the starter kit book ->   But i fell like i would need the player guid and monster guide to because they are offen bought in a bundle. Is there a way to get D&D book in digital, except the novels of course.   What is the minimal books we nead to start playing ? Of course we would get more stuff later on. And for the field we would use some home made tokens and re-drawable mat.   We would like to start whit 4.0. Thats the lattest right ?

I've asked several questions about 4th Edition D&D because I may be able to start a game. However, I'd need to purchase (some) materials. I am aware of the 4 book set (core) and the expected Essentials Red Box.  So now I need to know: presumably buying either the 4 book set or essentials will be enough to get my game on. Would there be something 'lacking' though? Would I be missing character types, for instance, as I've seen that the splatbook strategy is still in effect?  I'm an experienced player and GM and plenty creative, so many holes I'd be able to fill in myself. I just don't want to get sucked into too much financial impact.


Can anyone explain the difference between   $ gedit ~/.bashrc   and  $sudo -H gedit /etc/environment   in setting environment variables?

Where is the preferred place to set the PATH envvar?  ~/.profile or /etc/environment?  What is the case when PATH is set in both places? Is the end result a concatenation of both values set in those two places?


I'm studying the final topology on a set $X$ induced by a family $\{f_\alpha:X_\alpha\rightarrow X\mid \alpha\in A\}$ and know of its universal property, namely that given a function $g:X\rightarrow Z$, $g$ is continuous iff $g\circ f_\alpha$ is continuous for all $\alpha\in A$.  Showing this is not difficult, but it is not constructive.    What bothers me about this is that my usual understanding of universal properties is that they refer to initial or terminal properties and morphisms; typically, they involve the existence of a unique morphism, as the definitions  suggest.  As this  points out, we may use the category of cones along with the forgetful functor from $\mathbf{Top}$ to $\mathbf{Set}$ to show that the above universal property is indeed as the above definitions give.  To be honest, I don't know enough category theory to completely understand the process, which may be where my issue lies.  The question I have is then:  How (or if) the universal property of the disjoint union topology can be realized as a special case of the final topology's universal property, since the disjoint union topology is the final topology induced by the canonical injections of $X_\alpha$ into $X=\amalg_{\alpha\in A}{X_\alpha}$, which is constructive in nature, i.e. that it guarantees the existence of a unique morphism such that a certain diagram commutes.

I'm learning some category theory and I thought I had understood universal objects but maybe I have not because I cannot write down the definition of initial topology in terms of categories.  My understanding of universal objects so far: I thought universal objects are either initial or terminal objects in a comma category $(O \downarrow F)$ where $O$ is a selection functor $\textbf{1} \to O \in C$ and $F$ is a forgetful functor into $C$. In the example of a free group, $F: \textbf{Group} \to \textbf{Set}$ and $O$ is the set that generates the free group. The objects in this category are pairs $(f, G)$ where $f$ is a (set-theoretic) map from $S \to G$.   Now I would like to do the same for the initial topology but I'm struggling to see what the objects are and also what $F$ should be. $O$ is probably going to be $X$, the space we would endow with the topology. So we probably want $F$ to be a functor $\textbf{Top} \to \textbf{Set}$. The objects are probably $(f,Y)$ where $f$ is a map $X \to Y$ (or $Y \to X$?) such that there exists a continuous map $f^\prime$ with $F(f^\prime) = f$.  Is this right? If not: what are the objects? And is the initial topology an initial or a terminal object?  Edit  I think what I wrote above is wrong: I get one commutative diagram for each $Y_i$ in the family with respect to which the initial topology is defined. This means that the objects are $(g_i, Z)$ where $g_i : Z \to Y_i$ and the comma category would be $(Y_i, F)$. Is this right?  The diagram for $(Y_i \downarrow F)$ would look like this:  X --(f_i)--> Y_i               ^               |                |             (g_i)                |               |               Z   Then the question is whether the unique continuous map $\varphi$ is $Z \to X$ or $X \to Z$.


Given: $a = qb + r$. Then it holds that $\gcd(a,b)=\gcd(b,r)$. That doesn't sound logical to me. Why is this so?    Addendum by LePressentiment on 11/29/2013: (in the interest of  and averting a duplicate)  What's the intuition behind this result? I only recognise the proof and examples solely due to algebraic properties and formal definitions; I'd like to apprehend the result naturally.

How can I prove that $\gcd(a, b) = \gcd (a, b + ma)$?  I have tried this:  let $g = \gcd(a, b)$, then $g \mid a$ and $g \mid b$. This means that $g \mid ax+by$.  I don't know what to do next.  Thanks.


If I was a god that could see the state of a particle in superposition without touching it, wouldn't I know what the particle is at that moment? Wouldn't that make the superposition concept just switching between states very fast and being fragile to being observed? Do we call that a particle is in superposition, just because if we measure it, the measured value is not relevant anymore to calculate the next state? So basically we cannot measure it, but in the background the particle is still changing it's state right? So, the particle is not actually in superposition. All the sites just talk about this concept as, "the particle being in the two states at the same time, because we cannot predict it" but that's like very vague for me.

I was wondering if quantum particles do actually  exists in two different states simultaneously and if it has been proven they do indeed exists in a superposition of states.  How has it been figured out since observing it would collapse the wave-function into one single state (two superposition of states into one ) as it has been mentioned in the Schrodinger's cat experiment?


Using pdftex i can  cross ref to a label say \ref{label1} for example. Clickin on it in the .pdf document you 'll be linked to the label. I'd like there to be a "back reference link" to the point that called \ref{label1}. Is this possible without adding another label to that point? Thank you.

I'm including \usepackage{hyperref} so that each instance of \ref (as well as each page number in the index and the table of contents) automatically links to that page.  When I'm viewing the document (in ) and I click a link, I jump to the linked page.  Is there any easy way to go "back" to where I was before I clicked the link?  (And is the answer any different when using Adobe Acrobat reader?)


What kind or ordering is used for questions in month/week tabs? Tooltip text says:        But I can't understand sort criteria. All of questions presented in the following picture were asked in December:

It has come to my attention that on the homepage on any given SE you can sort by "active", "hot", "week", and "month".  The tooltip says the latter three sort by "most views, answers, and votes" over different periods of time.  So it seems that this is uses a different formula than the "arbitrary hotness points" that create hot network questions.  What is the formula and what is the rationale for using views here, but not in the network hot questions?    Also, what is the rationale for offering such different sorting options between the homepage and the questions tab?  I have been active for ~4 months and never realized there was a sort by hotness option on a different screen (as I have always looked at the questions tab to see what was new) until today.


Let $A$ be a square matrix such that $AA^T = I$. If $\det{A} < 0$, find $\det{A}$ and $\det{(I + A)}$.  I managed to get $\det{A} = -1$ but had difficulties when trying to calculate $\det{(I + A)}$.

Given a real valued matrix $A$ that satisfies $AA^{\top} = I$ and $\det(A)<0$, calculate $\det(A + I)$.  My start : Since $A$ satisfies  $AA^{\top} = I$, $A$ is a unitary matrix. The determinant of a unitary matrix with real entries is either $+1$ or $-1$. Since we know that $\det(A)<0$, it follows that $\det(A)=-1$.


How do you vertically center align a :before element. I have tried using vertical-align: bottom, vertical-align:middle, vertical-align: -50% and vertical-align:super. Nothing I do is bringing the plus symbol down to be vertically center aligned with the text next to it.  Any ideas?    .infoTitles {  color: #2E393F;  font-family: 'Muli', sans-serif;  font-size: 1.5rem;  line-height: 1.5em; } .infoTitles:before {  content: '';  vertical-align: super;  float: left;  margin-right: 8px;  background-image: url('https://cdn4.iconfinder.com/data/icons/ios7-essence/22/add_plus-512.png');  background-size: 15px 15px;  width: 15px;  height: 15px;  display: block; }

How long will I expect to wait for a quote?

I have a div with two images and an h1. All of them need to be vertically aligned within the div, next to each other. One of the images needs to be absolute positioned within the div. What is the CSS needed for this to work on all common browsers?

testing...


My input is a set of objects having different values with respect to a set of features (F1, F2, F3, F4, F5). The output should be a set of minimal features allowing to distinguish all these objects or if not possible, the features that distinguishes most of these objects.    Fo example, F1 distinguishes objects 4 (value of 2) and 5 (value of 11) from the rest of objects. We need to distinguish obj1, obj2 and obj3. F2 distinguishes Obj2 (value of 4) from the obj1 (value of 6) and Obj3 (value of 6). After combining both features F1 and F2, we can distinguish obj4, obj 5, obj4. I need an additional feature that allows to distinguish obj1 from obj3. F4 would do the job, since obj1 (value of 9) and obj 3 (value of 10). Our set allowing to distinguish all objects is then F1, F2, F4. Note that F6 is useless since it does not help distinguish our objects.  Is there a way to obtain this set of features that allows to distinguish all objects, knowing that I have more than 500 features and more than 500 objects.   Thanks in advance for your help.

My input is a set of objects having different values with respect to a set of features (F1, F2, F3, F4, F5). The output should be a set of minimal features allowing to distinguish all these objects or if not possible, the features that distinguishes most of these objects.   For example F1 distinguishes objects 4 (value of 2) and 5 (value of 11) from the rest of objects. We need to distinguish obj1, obj2 and obj3. F2 distinguishes Obj2 (value of 4) from the obj1 (value of 6) and Obj3 (value of 6). After combining both features F1 and F2, we can distinguish obj4, obj 5, obj4. I need an additional feature that allows to distinguish obj1 from obj3. F4 would do the job, since obj1 (value of 9) and obj 3 (value of 10). Our set allowing to distinguish all objects is then F1, F2, F4. Note that F6 is useless since it does not help distinguish our objects.  Is there a way to obtain this set of features that allows to distinguish all objects, knowing that I have more than 500 features and more than 500 objects.   Thanks.


Very new to drupal, I am playing around with the mail function to test how I can send mails using drupals API.   I used this tutorial from .   Whenever I try to call the PHPScript which should send the test mail, i get this error:      Fatal error: Call to undefined function drupal_mail() in   /drupal_dev/sites/all/modules/configurator/Trader/PHPScripts/testMail.php   on line 14   Fyi: my configurator.module is located in /drupal_dev/sites/all/modules/configurator/. I saw a couple of examples where the drupal_mail() function is implemented in the .module file, but the tutorial mentioned above said I may place this function anywhere.   So what am I doing wrong?   Here goes my code:   $to = "recipient@host.com"; $email_from = "donotreply@host.com";  $params['var'] = 'test_var1'; $params['more'] = 'test_var2';  $sent = drupal_mail('testMail', 'key', $to, language_default(), $params, $email_from, TRUE );  if (!$sent) {     echo "Failure"; } else {     echo "Success"; }  //drupal mail looks for a function that matches the first parameter _ mail to build the message function testMail_mail($key, &$message, $params) {     $language = $message['language'];     switch ($key)      {         case 'key':              $message['subject'] = t('Subject', $vars, $language->language);             $message['body'][] = "Hallo Jannik. This is the body of the email, and might contain vars from params.";             break;      } }

This is what I have right now:  <?php     define('DRUPAL_ROOT', '/var/www/clients/client1/web15/web');     include_once(DRUPAL_ROOT . '/includes/bootstrap.inc');     drupal_bootstrap(DRUPAL_BOOTSTRAP_FULL); //  drupal_flush_all_caches();        $users = views_get_view_result('userlist', 'uid_list');     $i = 0;     foreach ($users as $user) {         $filename = 'sites/default/files/generated_pdfs/individual_pdf_report_' . $user->uid . '.pdf';         generate_employee_pdf($user->uid, $filename);         error_log('Generated ' . $filepath);         $i++;     }     // create object     $zip = new ZipArchive();     // open archive     if ($zip->open("sites/default/files/generated_zips/individual-reports.zip", ZIPARCHIVE::CREATE) !== TRUE) {         die ("Could not open archive");     }     // initialize an iterator     // pass it the directory to be processed     $iterator = new RecursiveIteratorIterator(new RecursiveDirectoryIterator("sites/default/files/generated_pdfs/"));     // iterate over the directory     // add each file found to the archive     foreach ($iterator as $key=>$value) {         $zip->addFile(realpath($key), $value->getFilename()) or die ("ERROR: Could not add file: $key");     }     // close and save archive     $zip->close(); ?>   I'm getting a bunch of errors from the above when run standalone. Works fine in my module. Some of the errors are:  Warning: Invalid argument supplied for foreach() in drupal_depth_first_search() (line 56 of /var/www/clients/client1/web15/web/includes/graph.inc). Warning: Invalid argument supplied for foreach() in _module_build_dependencies() (line 229 of /var/www/clients/client1/web15/web/includes/module.inc). Warning: include_once(sites/all/modules/views_bulk_operations/actions/archive.action.inc): failed to open stream: No such file or directory in views_bulk_operations_load_action_includes() (line 56 of /var/www/clients/client1/web15/web/sites/all/modules/views_bulk_operations/views_bulk_operations.module).


All I want to do is install Linux Ubuntu 12.04 LTS as another OS on my laptop (which already has Windows 8). I realize this thread might be deemed as a duplicate of  but I don't care about back-up or Lubuntu, nor can I disable UEFI because I can't get into BIOS no matter how hard I try.  I found out about this option in Windows 8, which for some reason has something to do with recovery: "Change advanced start-up settings"/"Advanced startup". Seems like at this point I do not have to deal with BIOS, UEFI, and Secure Boot, nor do I want to; this installation process is already complicated enough. (I'm writing this because I don't know if this is relevant to the problem I'm about to address.) I boot from the DVD with Ubuntu .iso image properly burnt on it. At the start-up I saw a suspicious line:  Could not open "\EFI\BOOT\fallback.efi": 14   Then it moves on to the boot partitions menu without Windows 8 on it. Now I'm booting into the trial mode of Ubuntu. Even during the installation process it does not recognize Windows 8:    On  the author says that she "had to manually partition the hard drive, because the Ubuntu installer didn't recognize that Windows 8 was already on the computer." I don't know if this is safe for an amateur like myself to carry it out, considering that partitioning cannot be undone and I don't know how to do it right:    Problem/question: How do I get it to recognize Windows 8 so that I can partition the memory correctly?  Hope I didn't put up too many findings. Any help would be appreciated.

I'm absolutely new to Linux. I would like to know how to install Ubuntu alongside the pre-installed Windows 8+ OS.  Should I do it with Wubi, or through the Live USB/DVD?  What steps do I need to take to correctly install Ubuntu?


If someone was incredibly well trained with a regular sword, but had no connection to the Force, would that person be able to wield a lightsaber?  Are there any examples of this or something similar to this in any canon?  I don't know if a Jedi's / Sith's connection to the Force enables them to use a lightsaber, or just increases their abilities with a lightsaber.  I guess  is the obvious example of a non-Force user being pretty badass with a lightsaber, but I'm looking for a regular human here.

This is a two part question, with the two parts being mirror images to each other.  Is the ability to use the Force a requirement, for some reason, to using a lightsaber?  And if so, why?  And on the other side, on Hoth, when Luke is hanging in ice in the wampa's cave, he uses the Force to bring his lightsaber into his hand.  But, as best I can remember and find, the Jedi always use a lightsaber like a "real" sword and keep it in their hands (although I think there's one case in a movie where a lightsaber is thrown to cut down support for a walkway).  Can a lightsaber be wielded with the Force only?  For instance, when Anakin's hand was cut off, could he have controlled the movement of his lightsaber the way Vader did with the objects he Force-tossed objects at Luke in the Cloud City?  While that's two questions (that I think are closely related), another way to put it might be to ask, "How closely is the use of the Force related to wielding a lightsaber?"  Addendum: No, this is NOT a duplicate of .  Note I am the one who asked both questions.  I asked both because they cover different groups.  See the Venn diagram.  I allow room for possible members of Sith and Jedi who do not use the Force.  (Hey, in the Star Wars universe, almost anything can happen.)  If that is not possible, then just imagine the black box not intercepting either or both of the Jedi and Sith boxes.  One can be non-aligned (with Jedi or Sith) and still use the Force and it's possible, at some point, we may find some odd story about a Jedi or Sith who cannot use the Force.  (There may be some story about how such a person is not felt by Force users, so is valuable to one side or the other.)  This question overs the black box.  The other question covers the green box.  Yes, there is overlap, but there are also distinct separate groups covered by each question that are not covered in the other question.


five month old newbie who's having some fun, here. I've been experimenting a bit, and got a problem. I was following this tutorial from Zacharias Reinhardt on how to create massive background forests via alpha channels. Seems to me to be the best method there is, if you're aiming for not too much processing.   And then I got stuck in the node editor when removing the black alpha channel part of the png. I'm 99% certain I did everything exactly like Zacharias showed in the tutorial, but somehow, I'm getting a somewhat different result? And yes, RGBA was selected during export of the original file.  I've added a screenshot of the section in the tutorial and a screenshot of what I've got on my screen. Any help would be greatly appreciated! :)

I must be missing something here.  I'm doing an extremely simple Import Image as Plane using the AddOn and the imported image has an alpha background but it renders (and even previews in the material preview) as black where the alpha background should be.  When I import, I select Emission & Transparent so the material setup is automatic.  I've read similar issues but people are typically missing the transparent node and connecting the Alpha channel from the image into the Mix Node factor.  But this is all here.  I must be missing something really stupid but I've wasted so much time trying to fix this thing.  Any help would be greatly appreciated.  Not sure if it matters but I'm using 2.74 RC2.  Many thanks!  I created this dummy blend file to test outside my project but problem still exists.  Here is the simple .blend file:


I want to place a limit under a word, as done  except with an arbitrary text rather than an existing math operator like lim or max. \mathop{avg}\limits_{y\in U(x)} gives me the following, but I would like "avg" to be in non-italicized.  I have also tried to define \def\avg{\text{avg}} and \newcommand{\avg}{\text{avg}} but then the subscript is no longer underneath the text, even with \avg\limits_{y\in U(x)}:

Mathematical operators, such as function names, should be set in roman type, not italics. Latex already has commands for some operators, including \max, \min, and \log.  How can I define additional such commands?


Ever wondered where those pinned shortcuts are stored in Win7/8/8.1/10? This way you can edit, delete or create new shortcuts. Sometimes they get duplicated but they don't appear as pinned so this is the only way to delete them.

Where, in the Windows 7 file system, are the taskbar shortcuts stored (per user)?


I getting a wifi signal at a specific distance from router in windows (computer connects and works) But I am unable to connect to the same wifi at that range in Ubuntu 16.04.   $ sudo lshw -class network   description: Wireless interface        product: RTL8723BE PCIe Wireless Network Adapter        vendor: Realtek Semiconductor Co., Ltd.        physical id: 0        bus info: pci@0000:08:00.0        logical name: wlo1        version: 00         width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress bus_master cap_list ethernet physical wireless        configuration: broadcast=yes driver=rtl8723be driverversion=4.4.0-66-generic firmware=N/A ip=192.168.0.11 latency=0 link=yes multicast=yes wireless=IEEE 802.11bgn        resources: irq:18 ioport:5000(size=256) memory:c6100000-c6103fff   *-network        description: Ethernet interface        product: RTL8101/2/6E PCI Express Fast/Gigabit Ethernet controller        vendor: Realtek Semiconductor Co., Ltd.        physical id: 0        bus info: pci@0000:09:00.0        logical name: eno1        version: 0a        size: 10Mbit/s        capacity: 100Mbit/s        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress msix bus_master cap_list ethernet physical tp mii 10bt 10bt-fd 100bt 100bt-fd autonegotiation        configuration: autonegotiation=on broadcast=yes driver=r8169 driverversion=2.3LK-NAPI duplex=half firmware=rtl8107e-2_0.0.2 02/26/15 latency=0 link=no multicast=yes port=MII speed=10Mbit/s        resources: irq:49 ioport:4000(size=256) memory:c6004000-c6004fff memory:c6000000-c6003fff

I have installed Ubuntu 15.04 on my laptop with RTL8723BE Wi-fi card. But it is always disconnecting from network. I have tried echo "options rtl8723be fwlps=N ips=N" | sudo tee /etc/modprobe.d/rtl8723be.conf but that didn't helped. What can I do to prevent the wifi from disconnecting? If I upgrade to kernel 4.X, will it help?  ~$ lspci -knn | grep Net -A2  09:00.0 Network controller [0280]: Realtek Semiconductor Co., Ltd. RTL8723BE PCIe Wireless Network Adapter [10ec:b723]     Subsystem: Hewlett-Packard Company Device [103c:2231]     Kernel driver in use: rtl8723be   ~$ rfkill list 0: hci0: Bluetooth     Soft blocked: yes     Hard blocked: no 1: phy0: Wireless LAN     Soft blocked: no     Hard blocked: no   ~$ ifconfig && iwconfig && route -n && ping -c 1 google.com eth0      Link encap:Ethernet  HWaddr 38:63:bb:cd:4a:7e             UP BROADCAST MULTICAST  MTU:1500  Metric:1           RX packets:3 errors:0 dropped:0 overruns:0 frame:0           TX packets:53 errors:0 dropped:0 overruns:0 carrier:0           collisions:0 txqueuelen:1000            RX bytes:752 (752.0 B)  TX bytes:8445 (8.4 KB)  lo        Link encap:Local Loopback             inet addr:127.0.0.1  Mask:255.0.0.0           inet6 addr: ::1/128 Scope:Host           UP LOOPBACK RUNNING  MTU:65536  Metric:1           RX packets:3870 errors:0 dropped:0 overruns:0 frame:0           TX packets:3870 errors:0 dropped:0 overruns:0 carrier:0           collisions:0 txqueuelen:0            RX bytes:314613 (314.6 KB)  TX bytes:314613 (314.6 KB)  wlan0     Link encap:Ethernet  HWaddr c0:38:96:6d:c4:83             inet addr:192.168.1.205  Bcast:192.168.1.255  Mask:255.255.255.0           inet6 addr: fe80::c238:96ff:fe6d:c483/64 Scope:Link           UP BROADCAST RUNNING MULTICAST  MTU:1500  Metric:1           RX packets:15240 errors:0 dropped:0 overruns:0 frame:0           TX packets:14627 errors:0 dropped:0 overruns:0 carrier:0           collisions:0 txqueuelen:1000            RX bytes:14410285 (14.4 MB)  TX bytes:2192744 (2.1 MB)  eth0      no wireless extensions.  wlan0     IEEE 802.11bgn  ESSID:"DIR-615"             Mode:Managed  Frequency:2.437 GHz  Access Point: 00:90:4C:08:00:0D              Bit Rate=150 Mb/s   Tx-Power=20 dBm              Retry short limit:7   RTS thr=2347 B   Fragment thr:off           Power Management:off           Link Quality=70/70  Signal level=-22 dBm             Rx invalid nwid:0  Rx invalid crypt:0  Rx invalid frag:0           Tx excessive retries:0  Invalid misc:186   Missed beacon:0  lo        no wireless extensions.  Kernel IP routing table Destination     Gateway         Genmask         Flags Metric Ref    Use Iface 0.0.0.0         192.168.1.1     0.0.0.0         UG    400    0        0 wlan0 169.254.0.0     0.0.0.0         255.255.0.0     U     1000   0        0 wlan0 192.168.1.0     0.0.0.0         255.255.255.0   U     0      0        0 wlan0 PING google.com.Dlink (172.26.136.19) 56(84) bytes of data. 64 bytes from 19.136.26.172.in-addr.arpa (172.26.136.19): icmp_seq=1 ttl=249 time=102 ms  --- google.com.Dlink ping statistics --- 1 packets transmitted, 1 received, 0% packet loss, time 0ms rtt min/avg/max/mdev = 102.547/102.547/102.547/0.000 ms


I just installed Blender with brew cask install blender. I tried to import an SVG with File > Import > Scalable Vector Graphics and it did not show. I made a simple SVG file with a circle in both Illustrator and Inkscape and I don't see it in the window. I zoomed out and still could not find it. Here is my screen:    I read  and . I confirm that I don't have raster graphics and that I exported the Illustrator files with those directions.  Here are dropbox links to the SVG files created: Illustrator circle, Inkscape circle (604B and 2kB respectively).  After importing, the Outliner adds a Curve item:    Clicking the eye of this item to show or hide makes no difference in the viewport.  I am new to Blender. Am I doing something wrong or is this a bug?  Update: I removed the Drobpox links because the question already has a solution, and one can easily redo them with Illustrator or Inkscape.

I'm definitely a newbie with blender still, as I've dipped my toe in a bit being a video editor.  My issue is importing .svg saved from illustrator.  Just simple vectors with not too many paths.  Just a thick text company logo.  I want to bring in and extrude, light, etc.  I've done it before in older versions of blender with no problem.  Am I not doing something right? Can someone tell me how to get an eps saved as an svg from illustrator to size correctly?


"Rogue One: A Star Wars Story" film has a companion canon prequel novel, "Catalyst" by James Luceno.  Should "Catalyst" be read before or after watching the film and why?  (the question assumes that the story will be read, one way or another)

I have acquired a copy of  (in a game of chance) and I'm due to see  on opening weekend.   Has there been any specific advice offered (by Disney) as to whether or not I should read Catalyst before going to see the Rogue One movie?


I currently have a valid UK and US Business visit Visa (Both are multiple entry visa's). I will be travelling to Portugal next month and my flights are from Mumbai through London to finally Porto. My travel agent was booking a ticket for a flight which will be landing in London heathrow airport and then get on another plane from Gatwick airport (with 5 hrs transit  to get to gatwick)... my question is ...do I need get a transit visa or can I manage with the UK business visa which I already have and have been granted a business visit visa to enter UK.... It would be few hours in London just to get to another airport within London. I will be getting a  Schengen visa to enter Portugal. .

Is there a website or some other way to find out if I need a transit visa for a short stop in the UK on my way somewhere else? Ideally this should give me the answer for all nationalities, and take into account any other visas I may have.


Let $C^1([a,b])$ denote the space of all continously differentiable functions on $[a,b], a,b\in\mathbb{R}$.  On this space, define the following metric: $$ d(f,g)=d_{\infty}(f,g)+d_{\infty}(f',g'), $$ where $$ d_{\infty}(f,g):=\lVert f-g\rVert_{\infty}:=\sup_{x\in [a,b]}\lvert f(x)-g(x)\rvert $$ and, similarly, $d(f',g')_{\infty}$ is defined.    So I have to show that each Cauchy sequence $(f_n)$ in $C^{1}([a,b])$ converges in $C^1([a,b])$. Don't really have an idea, but at least I can write down what Cauchy sequence means.  Let $(f_n)$ be some Cauchy-sequence in $C^1([a,b])$, i.e. for all $\varepsilon >0$ there is some $N\in\mathbb{N}$ such that $$ \lVert f_n-f_m\rVert_{\infty}+\lVert f_n'-f_m'\rVert_{\infty}\leq\varepsilon~~\forall n,m\geq N. $$  Now, I have to find some $f\in C^1([a,b])$ such that $f_n\to f$, i.e. for all $\varepsilon >0$ there exists some $N\in\mathbb{N}$ such that $$ \lVert f_n-f\rVert_{\infty}+\lVert f_n'-f'\rVert_{\infty}\leq\varepsilon~~\forall n\geq N. $$

Consider the space of continuously differentiable functions, $$C^1([a,b]) = \{f:[a,b]\rightarrow \mathbb{R}\mid f,f' \text{ are continuous}\}$$ with the $C^1$-norm $$\lVert f\rVert := \sup_{a\leq x\leq b}|f(x)|+\sup_{a\leq x\leq b}|f'(x)|.$$ Prove that $C^1([a,b])$ is a Banach Space.   This was the proof we were given: Assuming $C^1([a,b])$ is a normed linear space all we need to show is completeness. Let $(f_n)$ be a Cauchy Sequence in $C^1([a,b])$ with respect to the $C^1$-norm. Then each $f_n,f'_n\in (C([a,b]),\|\cdot\|_{\sup})$. We know that $C([a,b])$ is complete and thus there exists $f,g\in C([a,b])$ such that $f_n\rightarrow f$, and $f'_n\rightarrow g$ (uniformly) with respect to $\|\cdot\|_{\sup}$. If we let  $$ F_n(x) = \int_a^x f_n(t)dt, \hspace{2mm}  F(x) = \int_a^x f(t)dt $$ then $F_n\rightarrow F$ uniformly because  $$\lVert F_n-F\rVert_{\sup}\leq \sup_{a\leq x\leq b}\int_a^x|f_n(t)-f(t)|dt\leq \lVert f_n-f\rVert_{\sup}<\epsilon.$$ From the fundamental theorem of calculus: $$f_n(x)-f_n(a) = \int_a^x f'_n(t)dt $$ Since $f'_n\rightarrow g$ uniformly then  $$ \int_a^xf'_n(t)dt\rightarrow \int_a^x g(t)dt $$ Since we know that $f_n\rightarrow f$ uniformly, $$f(x)-f(a) = \int_a^x g(t) dt $$ which by the fundamental theorem of calculues implies $f'=g$. So we know have $f_n\rightarrow f$ and $f'_n\rightarrow g=f'$ which mean $f_n\rightarrow f\in C^1([a,b])$ with respect to $C^1$-norm. So every cauchy sequence converges. Hence $C^1([a,b])$ is a Banach Space.   So I understand most of the proof. Where I get confused is that how did we actually show this satisfies the $C^1$-norm? Maybe I don't understand what this norm actually does.   Thank you for any help, comments and advice!


There's food debris in the butter. My knowledge says that the oxygen deprived liquid gets bacteria in it and it turns into botulism same as vacuum sealed individual bags of fish.  My bosses don't care and leave it like this every day. The bacon sits there indefinitely as well. No date. There's no timestamp on either butter or bacon.    Is this safe?

If I left food out of the refrigerator for some period of time, is it still safe? If I left it out too long, can I salvage it by cooking it more?


UPDATE     I found an answer to this question for myself and I posted it, but i can only accept it in two days..     I create this multidimensional array:  array (   0 =>    array (     'OrderDate' => '02.11.2018',     'ClientNumber' => 5500   ) )  array (   0 =>    array (     'OrderDate' => '02.11.2018',     'ClientNumber' => 5500   ) )  array (   0 =>    array (     'OrderDate' => '05.03.2018',     'ClientNumber' => 5500   ) )  array (   0 =>    array (     'OrderDate' => '10.12.2018',     'ClientNumber' => 2200   ) )  array (   0 =>    array (     'OrderDate' => '10.12.2018',     'ClientNumber' => 2200   ) )   The array shouldn't contain duplicate OrderDate's from each ClientNumber.  I want the output to look like this:  array (   0 =>    array (     'OrderDate' => '02.11.2018',     'ClientNumber' => 5500   ) )  array (   0 =>    array (     'OrderDate' => '05.03.2018',     'ClientNumber' => 5500   ) )  array (   0 =>    array (     'OrderDate' => '10.12.2018',     'ClientNumber' => 2200   ) )   What if-condition should i use to make my output look like this? Any help will be appreciated, thank you!  This is my function that is called in a foreach. I tried my best to check with in_array and array_column but the output didn't look like the code above.   $array = array(); function createArray($clientNr, $orderDate){  if(empty($array)){ // if array is empty, just add   $array[] = array(               "OrderDate"=>$orderDate,               "ClientNumber"=>$clientNr                  );  } else {     if(??????)<--------------------------- // check here if date from clientnumber exists       $array[] = array(                    "OrderDate"=>$orderDate,                    "ClientNumber"=>$clientNr                   );     }  } }

How can I remove duplicate values from a multi-dimensional array in PHP?  Example array:  Array (     [0] => Array     (         [0] => abc         [1] => def     )      [1] => Array     (         [0] => ghi         [1] => jkl     )      [2] => Array     (         [0] => mno         [1] => pql     )      [3] => Array     (         [0] => abc         [1] => def     )      [4] => Array     (         [0] => ghi         [1] => jkl     )      [5] => Array     (         [0] => mno         [1] => pql     )  )


My working example is:  \documentclass[a4paper, 12pt]{scrreprt}  \usepackage{amsmath} \usepackage{braket} \usepackage{mleftright}  \begin{document}  \begin{subequations} \label{ComputationInnProd}     \begin{align}         \label{ComputationInnProdDiscr}         \langle \underbrace{ \mleft( \sum_{\alpha'} \psi_{\alpha'}^* \bra{\alpha'} \mright) }_{\bra{\psi}} \vert \underbrace{ \mleft( \sum_{\alpha''} \varphi_{\alpha''} \ket{\alpha''} \mright) }_{ \ket{\phi} } \rangle = \sum_{\alpha', \alpha''} \psi_{\alpha'}^* \varphi_{\alpha''} \braket{\alpha' \vert \alpha''} = \\         \sum_{\alpha',\alpha''} \psi_{\alpha'}^* \varphi_{\alpha''} \delta_{\alpha' \alpha''} = \sum_{\alpha'} \psi_{\alpha'}^* \varphi_{\alpha'},  \\         \label{ComputationInnProdCont}         \langle \underbrace{ \mleft( \int {d\alpha'} \psi^* \mleft( \alpha' \mright) \bra{\alpha'} \mright) }_{\bra{\psi}} \vert \underbrace{ \mleft( \int {d\alpha''} \varphi \mleft( \alpha'' \mright) \ket{\alpha''} \mright) }_{ \ket{\phi} } \rangle = \int {d\alpha'} \int {d\alpha''} \psi^* \mleft( \alpha' \mright) \varphi \mleft( \alpha'' \mright) \braket{\alpha' \vert \alpha''} = \\         \int {d\alpha'} \int {d\alpha''} \psi^* \mleft( \alpha' \mright) \varphi \mleft( \alpha'' \mright) \delta \mleft( \alpha' - \alpha'' \mright) = \int {d\alpha'} \psi^* \mleft( \alpha' \mright) \varphi \mleft( \alpha' \mright),     \end{align} \end{subequations}  \end{document}   I want the first two equations to belong to subequation 0.1a and have the ComputationInnProdDiscr label and the latter two to be grouped as subequation 0.1b and have the label ComputationInnProdCont for future reference.  This code produces the following output:   I've marked each of the equality signs that I want to be aligned with each other in red and green, respectively. I'm also open to suggestions for what would make this more readable because it does not look particularly pretty.

I want the equation number to be shown only at the last line, or better, somewhere in the middle of all. So instead of (1), (2), ... only (1). I use \begin{align} ... \end{align}


I was upgrading from 13.10 t0 14.04 and the system said:  "Could not install the upgrades The upgrade has aborted. Your system could be in an unusable state. A recovery will run now (dpkg --configure -a)."  After running this recovery it said the upgrade was complete but that there may be problems.  Any advice? (I am an ignorant man so PLEASE say it in very simple, step by step terms! :-) Thank you!

I decided to update to Oneiric using the update manager. At some point, I got this message:     The upgrade has aborted. Your system could be in an unusable state. A recovery will run now (dpkg --configure -a).   Then, later on:     The upgrade has completed but there were errors during the upgrade process.   What can I do?  EDIT: Update manager offers to perform a 'Partial Upgrade'. "865 updates have been selected. The updates have already been downloaded, but not installed". It still gives the option to upgrade to 11.10


$A$ is a set of integers and $\lvert A \rvert = 501$.   For any $a \in A$, $1 \le a \le 1000$.  Prove that there are integers $a,b \in A$, such that $a \vert b$. (a divides b)  The question doesn't specify it, but I assume that $a \neq  b$.  (Hint: Note that every integer $x$ can be uniquely presented as $x = y · 2 ^k$ where $k > 0$ and $y$ is odd. Consider the function $x \to y$)  Can I get some help?

This is exercise 1.6.1 of   Suppose that 501 distinct integers are selected from 1 . . . 1000. Show that there are distinct selected integers $a$ and $b$ such that $a | b$. Show that this is not always true if 500 integers are selected.  This seems like it should be simple but it's has been stumping me for a few days.  Here's my idea for a proof:  Let $S$ be a set of 500 distinct integers between 1 and 1000 that don't divide each other. Let $S_{1-500}$ and $S_{501-1000}$ partition $S$ into sets containing the elements ≤ and > than 500, respectively. Then consider $2 S_{1-500}$ and $\frac{1}{2}S_{501-1000}$, the result of multiplying the elements of these sets by 2 and 1/2, respectively. Because elements of S are coprime, $S_{1-500}$, $S_{501-1000}$, $2 S_{1-500}$, and $\frac{1}{2}S_{501-1000}$, are disjoint. Also note that each of these sets is a subset of the integers from 1 to 1000. One can deduce that the cardinality of the union of these sets is 1000, and therefore they form a partition of the integers from 1 to 1000.  Now, I know this demonstrates that you can't add another number to $S$ that is coprime with all the elements in $S$, which means that a coprime subset of the integers from 1 to 1000 can have at most 500 elements. Can someone articulate why this is true for me? Also, this was in a section on the Pigeonhole principle, can someone rework the proof to utilize that, and possibly make it simpler? Thanks.


I'm seeking a word that best describes the point of land created by the confluence of two rivers. The parcel of land in question would be situated in between the two rivers as their flow combined to form one river. Is there such a name?  This is not a duplicate of the question asking about the word confluence. This question refers to the point of land created by the confluence.

I've googled for a while and on some sites I've found the word "watershed" as the proposed word. Is it the word that best suits it?


I have a script in a folder:  /home/ubuntu/jasperreports-server-cp-6.3.0/ctlscript.sh   I need this script to run every time the server starts. Usually the server shuts down on Friday evenings and starts on Monday morning. What do I need to do in order to make this happen?.  How I manually start the script:  I will ssh to the server, and then go to this location /home/ubuntu/jasperreports-server-cp-6.3.0  and then run ./ctlscript.sh* start command manually.  But I am unable to start this script at the server start on Monday automatically.   I tried using rc.local , and also creating a script in /etc/init.d, I did follow this but it didn't work either .  Does anyone have any other suggestions? Or is my procedure wrong?

How can I run scripts automatically when Ubuntu starts up so I don't have to run them manually after startup?


In the Anonymous Window of the developer console, I am attempting to execute the following:  System.debug(     JSON.serialize([         SELECT OpportunityLineItemSchedule.Revenue, OpportunityLineItemSchedule.ScheduleDate, OpportunityLineItemSchedule.OpportunityLineItem.TotalPrice, OpportunityLineItemSchedule.OpportunityLineItem.Opportunity.Id, OpportunityLineItemSchedule.OpportunityLineItem.Opportunity.CloseDate, OpportunityLineItemSchedule.OpportunityLineItem.Opportunity.Amount         FROM OpportunityLineItemSchedule         WHERE OpportunityLineItemSchedule.OpportunityLineItem.Opportunity.Amount != OpportunityLineItemSchedule.OpportunityLineItem.TotalPrice     ]) );   I receive this error:  Line: 5, Column: 78 expecting a colon, found 'OpportunityLineItemSchedule.OpportunityLineItem.TotalPrice'   I understand this error typically pops up when one is attempting to use a variable in a query. However, I am not: I am using standard fields. Why am I receiving this error?

When I run the following query in query editor I get this error.  Error     Unknown error parsing query   Query  SELECT Id, OwnerId,Account.OwnerId FROM Contact where OwnerId = Account.OwnerId limit 100   Does anybody know what the problem is?


I have a valid Schengen visa which was used for a business trip to Sweden in Oct 2019. It's valid upto 1 year and is a type 'C' visa with multiple entries allowed.  Now, I want to visit Italy(maximum stay) and Switzerland in February 2020 for tourism purpose. Do I need to get fresh visa for this visit or will the old visa suffice.

My wife was just issued a multiple entry visa by France, which is good for one year.  Our first trip will be to France, the issuing country.  My question is: after we leave the EU and return, must future trips also start in France? Or may we go directly to (say) Greece from the US on a later visit?