Can the sequence of derivatives $\{f^{(k)}(0)\}_{k\geq 1}$ be any sequence?
Every power series is the Taylor series of some $C^{\infty}$ function
In which ways did scanning Nero's ship enhance/change Starfleet technology? I've heard the explanation that the Enterprise from 2009's Star Trek looked so much more advanced was because data gathered from the Kelvin initially scanning Nero's ship when it first appeared enabled them to upgrade their tech. I'm wondering if they've ever detailed exactly what technologies were advanced or enhanced or discovered. Why do I have so many .Xauthority.* files in my home directory? How secure is Activation Lock on iOS 7 (and later)? Apple showed this new feature in iOS 7 called Activation Lock, that if you enable "Find My iPhone" on a device that is lost or stolen, a person that finds/gets the iPhone will not be able to wipe it or reset it. I'm curious whether the lock is designed to prevent these avenues to defeat the lock:   Can they completely drain the battery and leave the phone in that state for a couple of months and then try it? Or, can they use any third-party software that can access iPhone from outside the iTunes and wipe it that way? Or, can they simply "jailbreak" it and forget about this Activation Lock?   By what methods can this lock be defeated or bypassed? Complicated FOR loop nesting. Having trouble cleaning up legacy code I am trying to tidy up some really bad legacy code from an old developer and I'm trying to get rid of these nested for loops. The code below shows what I'm currently working with. I've fixed this before with nested SOQL queries but nothing like this. How would this need to look to remove these for loops (except Trigger.New)? Would I use Maps? How would I do that?  map&lt;Id,SObject&gt; mapProperyIdBudDetail = util.GetObjFields(setPropertyId, 'Property'); map&lt;Id,SObject&gt; mapSpaceIdBudDetail = util.GetObjFields( PropertyMap.keyset(), 'Space');  for ( Lease__c l: Trigger.New ) {      if(mapProperyIdBudDetail.get(PropertyMap.get(l.space__c))!=null) {          lProperties = new List&lt;SObject&gt;{mapProperyIdBudDetail.get(PropertyMap.get(l.space__c))};              for ( SObject p: lProperties ) {                  for ( String fm: FieldMap.keyset() ) {                      if ( p.get(FieldMap.get(fm)) != Null) {                         l.put(fm,p.get(FieldMap.get(fm)));                     }                  }             }     } } The union of a strictly increasing sequence of $\sigma$-algebras is not a $\sigma$-algebra I am a Chinese citizen holding a HK SAR passport. How long am I able to visit the UK within a 365-day period?  My stamp is for 6 months. Could I leave within 6 months but re-enter again a couple of weeks for another 6 months? Prove Borel Measurable Set A with the following property has measure 0. This question is exercise 4.10 of Richard F. Bass's Real Analysis for Graduate Students, 2nd edition.  Let $\epsilon \in (0,1)$, let $m$ be Lebesgue measure, and suppose $A$ is Borel Measurable subset of $\mathbb R$. Prove that if $$ m(A\cap I)\leq (1-\epsilon)m(I)$$  for every interval $I$, then $m(A)=0$.  Collection of my thoughts  1. Try to prove by contradiction, suppose $m(A)=a&gt;0$ Borel Measurable Set satisfies the condition given. A property of any Borel Measurable Set on $\mathbb R$ is $A$ contains a closed set $F$ and is contained in open set $O$, and the measure of $F$ and $O$ can be arbitrarily close to $a$. 2. Obviously $A$ contains no interval, but it does not lead to contradiction, because we can find an example of $m(A)&gt;0$ that contains no interval(fat Cantor Set(remove middle 1/4 each time) has measure $1/2$ and contains no interval).  2'. For fat Cantor Set $C$ and given $\epsilon&gt;0$ how could we construct an interval I satisfies $m(C \cap I)&gt;(1-\epsilon)m(I) ?$   Maybe try to write Borel measurable set as countable union/intersection of intervals.   I'm stucked here and will appreciate any help, some examples/special case will be great too. Using Fermats prime numbers to prove that there is infinitely many prime numbers Jeffrey Wooldridge in his Econometric Analysis of Cross Section and Panel Data (page 357) says that the empirical Hessian "is not guaranteed to be positive definite, or even positive semidefinite, for the particular sample we are working with.".  This seems wrong to me as (numerical problems apart) the Hessian must be positive semidefinite as a result of the definition of the M-estimator as the value of the parameter which minimizes the objective function for the given sample and the well-known fact that at a (local) minimum the Hessian is positive semidefinite.  Is my argument right?  [EDIT: The statement has been removed in the 2nd ed. of the book. See comment.]  BACKGROUND Suppose that $\widehat \theta_N$ is an estimator obtained by minimizing $${1 \over N}\sum_{i=1}^N q(w_i,\theta),$$ where $w_i$ denotes the $i$-th observation.  Let's denote the Hessian of $q$ by $H$, $$H(q,\theta)_{ij}=\frac{\partial^2 q}{\partial \theta_i \partial \theta_j}$$  The asymptotic covariance of $\widehat \theta_n$ involves $E[H(q,\theta_0)]$ where $\theta_0$ is the true parameter value. One way to estimate it is to use the empirical Hesssian  $$\widehat H=\frac{1}{N}\sum_{i=1}^N H(w_i,\widehat \theta_n)$$  It is the definiteness of $\widehat H$ which is in question. Testing pointers for validity (C/C++) Is there any way to determine (programatically, of course) if a given pointer is "valid"? Checking for NULL is easy, but what about things like 0x00001234? When trying to dereference this kind of pointer an exception/crash occurs.  A cross-platform method is preferred, but platform-specific (for Windows and Linux) is also ok.  Update for clarification: The problem is not with stale/freed/uninitialized pointers; instead, I'm implementing an API that takes pointers from the caller (like a pointer to a string, a file handle, etc.). The caller can send (in purpose or by mistake) an invalid value as the pointer. How do I prevent a crash? Etymology of Tor and Ext Functors The names of the derived functors $\operatorname{Tor}$ and $\operatorname{Ext}$ seem quite cryptic to me. Does anyone know what these abbreviations stand for? I would be glad if someone could tell me where these names come from. When I click the bounty info icon on the iOS app. The added bounty description and reason is not shown.  For example: Preventing applications from stealing focus Are there any solutions to prevent applications stealing focus from the active window?   This is especially annoying when I'm starting an application, switch to do something else and the new application starts receiving half a sentence of text. Can I use a higher rank tensor field as the metric field? What a metric should necessarily do is it should give me a way to associate a frame invariant number with a given pair of spacetime events. Now, if I use a higher rank tensor field (say, for example, a tensor field of rank 3: $g_{\mu \nu \rho}$) then also I can certainly produce a frame invariant scalar out of a given displacement $\vec{A}$ in this trivial way: $ I := g_{\mu \nu \rho} A^{\mu}A^{\nu}A^{\rho}$. Since the components of the higher rank tensor and that of the displacement vector are going to transform in a covariant and a contravariant manner respectively, the cooked up quantity is certainly a scalar.   Another crucial property, which I think has more direct physical content than the previous one, is that between two inertial frames, at least one such transformation should exist that leaves at least one metric invariant. i.e., There should exist at least one combination of transformation matrix $\displaystyle\frac{\partial{x^{\alpha}}}{\partial{x^{\mu '}}}$ and metric $g_{\alpha \beta \gamma}$ that satisfies the following equation:      $\displaystyle\frac{\partial{x^{\alpha}}}{\partial{x^{\mu '}}}\displaystyle\frac{\partial{x^{\beta}}}{\partial{x^{\nu '}}}\displaystyle\frac{\partial{x^{\gamma}}}{\partial{x^{\rho '}}} g_{\alpha \beta \gamma} - \delta_{\mu '}^{\alpha}\delta_{\nu '}^{\beta}\delta_{\rho '}^{\gamma} g_{\alpha \beta \gamma} =0$    The last thing I can think of that can put a restriction on the choice of a tensor as a metric is the existence of a possibility of finding a metric compatible symmetric connection field. Following the usual procedure of finding the expression for a metric compatible symmetric connection field, I reached following condition (unlike the case of the usual two rank metric where we get a full-fledged expression) for the connection in the terms of the metric:      $g_{\nu \rho k_1} \Gamma^{k_1}_{\mu \lambda} - g_{\lambda \mu k_2} \Gamma^{k_2}_{\rho \nu} = \displaystyle\frac{1}{2} (\partial_{\nu}g_{\rho \lambda \mu} + \partial_{\rho}g_{\lambda \mu \nu}  - \partial_{\lambda}g_{\mu \nu \rho}  - \partial_{\mu}g_{\nu \rho \lambda}) $   My question is that if it is possible to satisfy the two highlighted conditions then can we use such a 3 rank (or even higher rank tensors with similarly produced conditions) tensor fields as metric fields?   PS: This is NOT a proposal for a new home-production theory of gravity (or that of anything for that matter) but rather it is just that I am trying to understand why a two rank tensor is used in General Relativity as the metric. Thank you. 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? Multi-core vs Multi-CPU I am requesting a new server from our Server team. Which is better for SQL Server performance: 2 CPUs with 8 cores per CPU or 1 CPU with 16 cores? How do I format a number in Java? Beamer in LyX: no sperator option avaiable I have been using beamer in LyX for a while now. Love it. But I just got a new computer, istalled LyX again and when I create a new beamer file (or open an old one) the "separator" option that is used to end a frame is not avaiable.  I do not know if there is something wrong with my installation or if I should install some package, or something else.  Has anybody ever run into this problem as well?  Thank you for the help.  Edited: Thank you for the answers, I basically need to update lyx on the other computers now. Allow adding bounties from the iOS & Android app (I was surprised I couldn't find this in an existing question, so here goes)    I'd like to add a bounty to a question, but I'm too lazy to go get my computer or a web browser. I like the SE app, and want to use it to do everything. I can't find the button to add a bounty to a question.  Please add this button, I really want to be able to add bounties to questions from the app. Why does DMM show AC and DC current when source is a battery? (How to calculate power in this case?) The circuit is an altered Joule Thief of my design that is driven by an AA or AAA battery. I’m observing the current at the battery anode with my DMM in series. (Extech EX330) This circuit is extremely efficient (In relation to other Joule Thief I've build.) and does some weird stuff - like, it needs a pushbutton sometimes.    Measurements: It is drawing 9.21 mA DC and 6.48 mA AC in series with the battery. It shows 1.364 VDC and 0.010 VAC in parallel with the battery. Output on the LED is 2.572 VDC and 0.011 VAC with no measurable frequency. (with my DMM at least) All measurements are taken with the same configuration and voltage.    But how is it drawing AC and DC and how do I calculate power consumption? --> Could it be because the emitter pin of the NPN is directly connected to ground? --> Or is the circuit just "switching on and off"?     Download schematic:  (I tried to simulate the circuit in LTspice. Thanks for making me aware of this amazing program, Sredni Vashtar!)     What the pushbutton is used for: The pushbutton is used to "kickstart" the oscillation when it wont start by itself. For example: I have to manually "oscillate" using this button to start the circuit if the capacitor is too big and battery voltage is low. Pushing the button releases the energy in the transformer into the capacitor. If it reaches LED forward voltage the circuit keeps oscillating on it's own. It does work without manual intervention when then capacitor in parallel with the LED is not too big, like &lt;5000 µF, and it's a fresh battery.    . Thank you for your answers! I'm learning so much! :D