In Boyd and Vandenberghe's textbook on Convex Optimization, is claimed that:      We always have $$ \underset{x,y}{\text{inf}\; f(x, y)} = \underset{x}{\text{inf}}\;\tilde{f}(x) $$   where $$ \tilde{f}(x) = \text{inf}_y\;f(x, y).$$ In other words, we can always minimize a function by   first minimizing over some of the variables, and then minimizing over the remaining   ones.    My question is the following (since it is unclear to me from the context): does this applies only to convex functions or to every function?
I am wondering if I can minimize a multi-variable function one variable at a time. In other words, is it true that:  $min_{x_1,x_2} f(x_1,x_2)=min_{x_1} min_{x_2} f(x_1,x_2)$
Understanding Clip Warning 000117 for Polygons? This error makes little sense because 95% of the time this works without any problem, but here is the situation:  I am clipping polygon A by polygons B-H.  While most of the time this works perfectly, occasionally this happens:   All layers are from the same coordinate system, and the processing extent is the same as polygon A when I setup the clip. Polygons C-H clip just fine without any errors, so it is usually polygon B that has the issue; however, most of the time there are no issues whatsoever. The polygons do overlap each other. I have repaired the geometry, but the results are the same.  Given that it isn't a coordinate problem, a processing extent problem, or even a geometry problem, what else could it be causing this error, and what would I have to do to resolve this problem?  UPDATE:  The error happened again with my largest set of Polygon A; however, I then deleted 75% of the Polygon A set and it worked just fine. So my question now is: how limited is the Clip feature when it comes to large areas, and is there a way around this without having to divide the Polygon A shapefile into several segments? If F is a finite field, then $F^F=P_F$. Factoring $x^n + y^n$ over the integers Remove preinstalled apps on Big Sur MacBook Pro M1 Sorry for my English, I'm French and  I have  tried a lot (even with SIP disabled) but nothing works. I'm kind very desperate since I’m not sure how I would set up screen time to limit my children’s usage of the Pre-installed apps. The reason why I need to do so, is that my kid waste time in stead of studying. Is there a guide how to set this up? Checking via ArcPy if ArcMap is in edit session? C++ pointer address issue I can get the element like this $("#txtEmail") but I'm not sure how to get the actual value. What's the difference between "speak" and "talk", grammatically speaking? There are a number of questions e.g.  and  that deal with the slightly different connotations of the words "speak" and "talk". However, there also seem to be some grammatical differences between the two words. This question is about whether there's a way to formally pin down these differences, or whether they're just contingent features of the two words having different histories.  Some examples of grammatical differences are as follows:   be talkative ... be speakative speak French ... talk French give a talk ... give a speech (as opposed to a speak) speak up ... talk up (the meanings of the two phrases being completely different, with "up" being a preposition only in the latter case) speaking of which ... talking of which grammatically speaking ... grammatically talking   On the other hand, many other constructions work just as well with either (though they might have subtly different meanings), for example   speak to ... talk to speak with ... talk with speak about ... talk about   I'm interested in whether there's a way to pin down these differences (e.g. are the two words classified as different types of verb in some way?), or whether they're essentially just arbitrary. I'm also interested in why we have these two different words with subtly different meanings. Is the distinction between “speak” and “talk” a feature of many languages, or is it just a peculiarity of English?  Etymologically, both words are from Germanic origins. “Talk” seems to have been formed from the Middle English tale, even though “speak” already existed in the English language by then (as far as I can tell). This makes it even more mysterious: Why did we form a new word as a synonym of one that was already established? Or were the meanings different at that time? Is there any example of a non-measurable set whose proof of existence doesn't appeal to the Axiom of choice? Is there any example of a non-measurable set whose proof of existence doesn't appeal to the Axiom of choice?  What would it imply if there was such an example?  EDIT: For instance, maybe this will help understand the kind of example I had in mind, it is known that an important feature to determine that the AC is needed in the Banach-Tarski case is the non-transitivity of rotations on Euclidean space, one might find an example of a transitive group for some given space keeping the rest equal and make the AC unnecessary?  I guess then it might be said that this transitivity will be in this particular case equivalent to the AC or some amount of it, but I guess that it would be important to show this if it hadn't been realized before. What happens with the force of gravity when the distance between two objects is 0? so I had my first approximation to the gravity equation  \begin{equation} F=GmM/r^2 \end{equation} and some questions arose that my teacher couldnt respond: if r approximates to 0 with mM being constant, then the force should tend to infinity, right? so that being true, then if we would travel to the center of the earth, we would be crushed by gravity? if so, how can there be anything right in the center of the earth? wouldnt anything there be compressed and destroyed by the infinite force?  having that in mind, my next question is as gravity force is measured from the center of an object, any part of the same object which is not at the center would be affected by the same object's gravity? if so, how can we differentiate between two objects that are not in a vacuum? my question originated thinking in us standing on the earth, in contact with it, are we, in a gravitational sense, one object? this lead to the last question, if gravity is a property of matter, then how applying the same logic as before, how does it work between two neutrons?(chose neutrons because i assumed it would be more difficult obtaining gravitational force if there was electrostatic forces from charges inbetween) do two newtrons in contact count as 1 gravitational object? and how thats its own gravity affect the newtron from center and out? Is my C1/D visa valid for transit in the US, if I want to go on a vacation to Mexico? I have been reading Geoff Cumming's 2008 paper  [~200 citations in Google Scholar] -- and  am confused by one of its central claims. This is one in the series of papers where Cumming argues against $p$-values and in favour of confidence intervals; my question, however, is not about this debate and only concerns one specific claim about $p$-values.   Let me quote from the abstract:     This article shows that, if an initial   experiment results in two-tailed $p= .05$, there is an $80\%$   chance the one-tailed $p$-value from a replication will fall in   the interval $(.00008, .44)$, a $10\%$ chance that $p &lt; .00008$,   and fully a $10\%$ chance that $p &gt; .44$. Remarkably, the   interval—termed a $p$ interval—is this wide however large   the sample size.   Cumming claims that this "$p$ interval", and in fact the whole distribution of $p$-values that one would obtain when replicating the original experiment (with the same fixed sample size), depend only on the original $p$-value $p_\mathrm{obt}$ and do not depend on the true effect size, power, sample size, or anything else:     [...] the probability distribution of $p$ can be derived without   knowing or assuming a value for $\delta$ (or power). [...] We do not assume any prior knowledge   about $\delta$, and we use only the information $M_\mathrm{diff}$ [observed between-group difference] gives about $\delta$ as   the basis for the calculation for a given $p_\mathrm{obt}$ of the distribution of   $p$ and of $p$ intervals.      $\quad\quad\quad$    I am confused by this because to me it seems that the distribution of $p$-values strongly depends on power, whereas the original $p_\mathrm{obt}$ on its own does not give any information about it. It might be that the true effect size is $\delta=0$ and then the distribution is uniform; or maybe the true effect size is huge and then we should expect mostly very small $p$-values. Of course one can start with assuming some prior over possible effect sizes and integrate over it, but Cumming seems to claim that this is not what he is doing.  Question: What exactly is going on here?    Note that this topic is related to this question:  with an excellent answer by @whuber. Cumming has a paper on this topic to:  -- but that one is clear and unproblematic.  I also note that Cumming's claim is repeated several times in the 2015 Nature Methods paper  that some of you might have come across (it already has ~100 citations in Google Scholar):     [...] there will be substantial variation in the $P$ value of repeated experiments. In reality, experiments are rarely repeated; we do not know how different the next $P$ might be. But it is likely that it could be very different. For example, regardless of the statistical power of an experiment, if a single replicate returns a $P$ value of $0.05$, there is an $80\%$ chance that a repeat experiment would return a $P$ value between $0$ and $0.44$ (and a $20\%$ change [sic] that $P$ would be even larger).   (Note, by the way, how, irrespective of whether Cumming's statement is correct or not, Nature Methods paper quotes it inaccurately: according to Cumming, it's only $10\%$ probability above $0.44$. And yes, the paper does say "20% change". Pfff.) I bought a Magic the Gathering display box today with cards from the new edition Innistrad. Unfortunately one of the cards in a booster pack was damaged. It is not just a simple scratch - one corner is bent and a small part of the card is missing. According to different trading platforms this card would be worth about 6 Dollars minimum in mint condition. So it is not really expensive but also not totally worthless.  What can I do about this? Can I obtain a replacement card from the producer (WotC) or from the store I bought it from? How can I prove that the card was damaged when I opened the booster pack? I am sure I am not the first person who got a damaged card from a booster pack. How are such issues usually handled? How to SEO a Single-Page website How do you find all $n$ such that $\phi(n)|n$ What is ambient occlusion? I've heard about ambient occlusion and it looks nice, but what exactly is it? I installed Oneiric as a guest OS in VirtualBox using the Mini CD. I installed the ubuntu-desktop package and everything seems to work fine - well, except for a font issue with the terminal:    As you can see, some of the letters overlap and it's difficult to read. I checked the settings and it the checkbox "Use the system fixed width font" is checked with "Monospace | 12" selected underneath it.    Edit: running the commands mentioned in the comments:  $ fc-match "Monospace" DejaVuSansMono.ttf: "DejaVu Sans Mono" "Book" $ gconftool --get /desktop/gnome/interface/monospace_font_name Monospace 10 $ debsums ttf-dejavu-core /usr/share/doc/ttf-dejavu-core/AUTHORS                                        OK /usr/share/doc/ttf-dejavu-core/BUGS                                           OK /usr/share/doc/ttf-dejavu-core/NEWS.gz                                        OK /usr/share/doc/ttf-dejavu-core/README                                         OK /usr/share/doc/ttf-dejavu-core/changelog.Debian.gz                            OK /usr/share/doc/ttf-dejavu-core/copyright                                      OK /usr/share/doc/ttf-dejavu-core/langcover.txt.gz                               OK /usr/share/doc/ttf-dejavu-core/status.txt.gz                                  OK /usr/share/doc/ttf-dejavu-core/unicover.txt.gz                                OK /usr/share/fonts/truetype/ttf-dejavu/DejaVuSans-Bold.ttf                      OK /usr/share/fonts/truetype/ttf-dejavu/DejaVuSans.ttf                           OK /usr/share/fonts/truetype/ttf-dejavu/DejaVuSansMono-Bold.ttf                  OK /usr/share/fonts/truetype/ttf-dejavu/DejaVuSansMono.ttf                       OK /usr/share/fonts/truetype/ttf-dejavu/DejaVuSerif-Bold.ttf                     OK /usr/share/fonts/truetype/ttf-dejavu/DejaVuSerif.ttf                          OK Why does SQL Server use a better execution plan when I inline the variable? Reputation and inbox notification shows up again after refresh Modal vs Non-Modal vs Auxiliary modal vs Conjugated Verb Modal vs Non-Modal vs Auxiliary modal vs Conjugated Verb   How are these kinds of verb related to each other? Specially Non-Modals and Conjugated Verb. Is the "Auxiliary modal" just another name for "Auxiliary verb"?