Define $\phi={a_1+...+a_n\over n}$ and $(a_n)$ be a sequence of real number. Is it true that $\limsup \phi\le\limsup a_n?$  Intuitively i am wondering why the inequality holds as the sup of a sequence could appear all at the begining and form a decreasing sequence so the average seems larger than the tails of the sequence but this is a question I found in some books, so that's not likely to be false.
This is a question from the book Methods of Real Analysis by R. R. Goldberg.     If $(s_n)$ is a sequence of real numbers and if $$\sigma_n=\frac{s_1+s_2+\cdots+s_n}{n}$$ then prove that: $\operatorname{{lim sup}}\sigma_n \leq \operatorname{lim sup} s_n$.   I don't have any idea how to start working on this problem. Please help. Thanks.
ii) Show that $u$ is not a real part of the function which analytic on $\mathbb{C} \backslash \lbrace 0 \rbrace$ Suppose $u(x,y)=\ln(x^2+y^2)$  i) Show that $u$ is harmonic on $\mathbb{C} \backslash \lbrace 0 \rbrace$  ii) Show that $u$ is not the real part of a function which analytic on  $\mathbb{C} \backslash \lbrace 0 \rbrace$  I manage to show the first part. For the second part, note that $u(x,y)=\ln(x^2+y^2)=\ln(|z|^2)=2 \Re \log(z)$  But this only show that $u$ is not a real part of $\log(z)$. I don know how to show $u$ cannot be the real part of a function which analytic on $\mathbb{C} \backslash \lbrace 0 \rbrace$  Can anyone guide me? Inequality: $x^2+y^2+xy\ge 0$ I want to prove that $x^2+y^2+xy\ge 0$ for all $x,y\in \mathbb{R}$.  My "proof":  Suppose wlog that $x\ge y$, so $x^2\cdot x\ge x^2\cdot y\ge y^2\cdot y=y^3$ (because $x^2\ge 0$ so we can multiply both sides by it without changing the inequality sign) giving $x^3\ge y^3$. Substracting we have $x^3-y^3\ge 0$ or $(x-y)(x^2+xy+y^2)\ge 0$. Since $x\ge y$ we can divide by $x-y$ to get $x^2+xy+y^2\ge 0$.  Is it right?  Thanks for your help! I'm trying to use ARIMA process to predict the behaviour of a time series, the probleme I face is that I can't get the order of each component of ARIMA, the lag is between 0 and 1, same goes for the acf. The time series has no tendency, and I guess no seasonality.     My question is : Does this mean that ARIMA orders are all 0s, and if so, is the HoltWinters a good (or the best) solution for modelling this kind of TS ? UFW Allows 22 for IPv4 and IPv6 but SSH Disconnects When Enabling sudo ufw disable followed by sudo ufw enable kicks me out of SSH  DMESG reports  [UFW BLOCK] IN=eth0 OUT= MAC=30:........ SRC=192.168.1.me DST=192.168.1.server LEN=52 TOS=0x00 PREC=0x00 TTL=128 ID=15776 DF PROTO=TCP SPT=55640 DPT=22 WINDOW=253 RES=0x00 ACK URGP=0  I can log back in without having to change rules via the console (UFW still enabled).  This started after upgrading Xenial (16.04) from kernel 4.4 to 4.15 (HWE). Upgrading to 18.04.1 did not solve the issue.  Versions:   iptables v1.6.1 ufw 0.35 4.15.0-29-generic #31-Ubuntu Ubuntu 18.04.1 LTS   UFW status verbose is (some rules were omitted, but they are all ALLOW)  Logging: on (low) Default: deny (incoming), allow (outgoing), disabled (routed) New profiles: skip  To                         Action      From --                         ------      ---- 22                         ALLOW IN    Anywhere 22 (v6)                    ALLOW IN    Anywhere (v6)   Why is this happening, or at least, how to revert to the expected behavior?  I looked at , and I am not sure it applies, but here's /etc/ufw/before.rules  # # rules.before # # Rules that should be run before the ufw command line added rules. Custom # rules should be added to one of these chains: #   ufw-before-input #   ufw-before-output #   ufw-before-forward #  # Don't delete these required lines, otherwise there will be errors *filter :ufw-before-input - [0:0] :ufw-before-output - [0:0] :ufw-before-forward - [0:0] :ufw-not-local - [0:0] # End required lines   # allow all on loopback -A ufw-before-input -i lo -j ACCEPT -A ufw-before-output -o lo -j ACCEPT  # quickly process packets for which we already have a connection -A ufw-before-input -m conntrack --ctstate RELATED,ESTABLISHED -j ACCEPT -A ufw-before-output -m conntrack --ctstate RELATED,ESTABLISHED -j ACCEPT -A ufw-before-forward -m conntrack --ctstate RELATED,ESTABLISHED -j ACCEPT  # drop INVALID packets (logs these in loglevel medium and higher) -A ufw-before-input -m conntrack --ctstate INVALID -j ufw-logging-deny -A ufw-before-input -m conntrack --ctstate INVALID -j DROP  # ok icmp codes for INPUT -A ufw-before-input -p icmp --icmp-type destination-unreachable -j ACCEPT -A ufw-before-input -p icmp --icmp-type source-quench -j ACCEPT -A ufw-before-input -p icmp --icmp-type time-exceeded -j ACCEPT -A ufw-before-input -p icmp --icmp-type parameter-problem -j ACCEPT -A ufw-before-input -p icmp --icmp-type echo-request -j ACCEPT  # ok icmp code for FORWARD -A ufw-before-forward -p icmp --icmp-type destination-unreachable -j ACCEPT -A ufw-before-forward -p icmp --icmp-type source-quench -j ACCEPT -A ufw-before-forward -p icmp --icmp-type time-exceeded -j ACCEPT -A ufw-before-forward -p icmp --icmp-type parameter-problem -j ACCEPT -A ufw-before-forward -p icmp --icmp-type echo-request -j ACCEPT  # allow dhcp client to work -A ufw-before-input -p udp --sport 67 --dport 68 -j ACCEPT  # # ufw-not-local # -A ufw-before-input -j ufw-not-local  # if LOCAL, RETURN -A ufw-not-local -m addrtype --dst-type LOCAL -j RETURN  # if MULTICAST, RETURN -A ufw-not-local -m addrtype --dst-type MULTICAST -j RETURN  # if BROADCAST, RETURN -A ufw-not-local -m addrtype --dst-type BROADCAST -j RETURN  # all other non-local packets are dropped -A ufw-not-local -m limit --limit 3/min --limit-burst 10 -j ufw-logging-deny -A ufw-not-local -j DROP  # allow MULTICAST mDNS for service discovery (be sure the MULTICAST line above # is uncommented) -A ufw-before-input -p udp -d 224.0.0.251 --dport 5353 -j ACCEPT  # allow MULTICAST UPnP for service discovery (be sure the MULTICAST line above # is uncommented) -A ufw-before-input -p udp -d 239.255.255.250 --dport 1900 -j ACCEPT  # don't delete the 'COMMIT' line or these rules won't be processed COMMIT   PS: I didn't expect this to "fix" the issue but just for reference I changed the port SSHD listens on (and the corresponding rule) and the problem persists. How do I return a longitude and latitude from Google Maps JavaScript geocoder? Can words like "what" be the subject of a sentence? In a question like "Who hears a noise?",  is the subject of the sentence who?  I can think of a few tests for subjects like: "the subject is the phrase that inverts with the auxiliary to form a question". But this is a question and there is no auxiliary. We could say that's because it's a subject, but that would be just to presuppose that it's a subject in the first place.  I can think of tests like: "the form of the verb will agree with the number of the subject", but on the other hand who does not seem to have  fixed grammatical number, and it sometimes seems like who might actually be agreeing with noun phrases later in the sentence:   Who are the most prolific writers of our age? Who is the most prolific writer of our age?   There are also tests like "the subject is usually the first noun phrase in the sentence". But of course if we have a question like:   Who have you bitten?   ... you, not who seems to be the subject. The first noun phrase test is not very good.  I also know that some theories of grammar say that there is a gap in wh- questions that the wh- word is extracted from. So can this gap for instance be the real object of a question? If so, is [See community wiki post in the linked to question]  How can we show whether who is really the subject of my original example question? And how about other wh- words like what? Can they (also) function as subjects? I  to the question "" which was specific to the question asked.  Another user engaged positively with my answer, as there was a small conversation thread.  That person then  and made what I would say are minor modifications (the comments I left in my original code that pertained to the specific case are still in the places where he made edits to be more specific).  The answer does acknowledge that it is a general case of my specific code.  Ignoring the fact that I'm involved in it, what should one do when seeing this situation?  The only thing I can think of, and I'm not sure that it's necessarily the right solution, is to flag the answer for moderator attention.  Other possible actions, which I believe are wrong, are:   Editing the answer; that would change its intent Voting the answer down; it's a technically correct answer   One could leave a comment, but that can lead to confrontation and escalation (depending on the parties involved) which would be counterproductive.  The only thing I've done, which is not really related, is edit the answer to hyperlink the already existing attribution (but I do this as a regular edit for any answer which explicitly references another).  What's the best approach here, if any? Shift + Control + Space doesn't seem to work anymore. Learning Python programming with generic GIS goals in mind? I would like to prove the following equality: $$k^{\phi(l)} + l^{\phi(k)} \equiv1\pmod{lk}$$if $\gcd(l,k)=1$. What methods can I use? Thank you for your help. Is it pythonic for a function to return multiple values? In python, you can have a function return multiple values.  Here's a contrived example:  def divide(x, y):     quotient = x/y     remainder = x % y     return quotient, remainder    (q, r) = divide(22, 7)   This seems very useful, but it looks like it can also be abused ("Well..function X already computes what we need as an intermediate value.  Let's have X return that value also").  When should you draw the line and define a different method? What's the difference between using a 50mm f/1.8G and a 50mm f/1.8D with a Nikon D80? Recently, Nikon announced their , as revamp of the old 50mm f/1.8D. :     The main difference between this new   "nifty fifty" and the almost   decade-old ƒ1.8D is that it has an   autofocus motor inside. This means you   can use it with any current or recent   Nikon SLR. The older lens lacks this   motor and is instead driven by one in   the camera, which cheaper bodies don't   have.   I currently use a Nikon D80 (and I don't plan on changing that any time soon), and I've been considering getting such a lens. Is there any difference between the D and the G lenses when used on a D80? I believe that the D one works just as well, since D80's have internal autofocus motors - is that right?  Besides the autofocus motor aspect, is there any other big difference between them? Since the D is priced at around $120 and the G at $220, should I just get the D? I see an equation like this:  $$y\frac{\textrm{d}y}{\textrm{d}x} = e^x$$  and solve it by "separating variables" like this:  $$y\textrm{d}y = e^x\textrm{d}x$$ $$\int y\textrm{d}y = \int e^x\textrm{d}x$$ $$y^2/2  = e^x + c$$  What am I doing when I solve an equation this way?  Because $\textrm{d}y/\textrm{d}x$ actually means  $$\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$$  they are not really separate entities I can multiply around algebraically.    I can check the solution when I'm done this procedure, and I've never run into problems with it.  Nonetheless, what is the justification behind it?  What I thought of to do in this particular case is write  $$\int y \frac{\textrm{d}y}{\textrm{d}x}\textrm{d}x = \int e^x\textrm{d}x$$ $$\int \frac{\textrm{d}}{\textrm{d}x}(y^2/2)\textrm{d}x = e^x + c$$  then by the fundamental theorem of calculus  $$y^2/2 = e^x + c$$  Is this correct?  Will such a procedure work every time I can find a way to separate variables? I am a graduate student in mathematics who would like to learn some classical mechanics. However, there is one caveat: I am not interested in the standard coordinate approach. I can't help but think of the fields that arise in physics as sections of vector bundles (or maybe principal bundles) and would love an approach to classical mechanics or what have you that took advantage of this.  Now for the questions:   Is there a text book you would recommend that phrases the constructions in classical mechanics via bundles without an appeal to transition functions?   What are the drawbacks to this approach other than the fact that it makes computations less doable? (if it does that)   Are there benefits to thinking about things this way, ie would it be of benefit to someone attempting to learn this material to do it this way? failed to create symbolic link ‘/usr/bin/utserver’: File exists To find minimum distance between two curves How do you keep parents of floated elements from collapsing? Although elements like &lt;div&gt;s normally grow to fit their contents, using the float property can cause a startling problem for CSS newbies: If floated elements have non-floated parent elements, the parent will collapse. For example:   &lt;div&gt;   &lt;div style="float: left;"&gt;Div 1&lt;/div&gt;   &lt;div style="float: left;"&gt;Div 2&lt;/div&gt; &lt;/div&gt;    The parent div in this example will not expand to contain its floated children - it will appear to have height: 0. How do you solve this problem? I would like to create an exhaustive list of solutions here. If you're aware of cross-browser compatibility issues, please point them out. Solution 1 Float the parent.   &lt;div style="float: left;"&gt;   &lt;div style="float: left;"&gt;Div 1&lt;/div&gt;   &lt;div style="float: left;"&gt;Div 2&lt;/div&gt; &lt;/div&gt;    Pros: Semantic code. Cons: You may not always want the parent floated. Even if you do, do you float the parents' parent, and so on? Must you float every ancestor element? Solution 2 Give the parent an explicit height.   &lt;div style="height: 300px;"&gt;   &lt;div style="float: left;"&gt;Div 1&lt;/div&gt;   &lt;div style="float: left;"&gt;Div 2&lt;/div&gt; &lt;/div&gt;    Pros: Semantic code. Cons: Not flexible - if the content changes or the browser is resized, the layout will break. Solution 3 Append a &quot;spacer&quot; element inside the parent element, like this:   &lt;div&gt;   &lt;div style="float: left;"&gt;Div 1&lt;/div&gt;   &lt;div style="float: left;"&gt;Div 2&lt;/div&gt;   &lt;div class="spacer" style="clear: both;"&gt;&lt;/div&gt; &lt;/div&gt;    Pros: Straightforward to code. Cons: Not semantic; the spacer div exists only as a layout hack. Solution 4 Set parent to overflow: auto.   &lt;div style="overflow: auto;"&gt;   &lt;div style="float: left;"&gt;Div 1&lt;/div&gt;   &lt;div style="float: left;"&gt;Div 2&lt;/div&gt; &lt;/div&gt;    Pros: Doesn't require extra div. Cons: Seems like a hack - that's not the overflow property's stated purpose. Comments? Other suggestions? Moving one object causes other to move another (already tried unlinking) Visually stunning math concepts which are easy to explain Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the  or the , which are easy to understand and explain but are mathematically beautiful at the same time.  Do you know of any other concepts like these? Is it possible to extrude a curve in two directions? I want to extrude a curve in two directions.    Or join both curves into one, is it possible? I've tried copying it and then join segments, but I don't see Remove Doubles option while working with curves.