what is the barriers of making fast lenses?
What is the limiting factor in lens construction when it comes to f-stop values?
The set of convergent points of a sequence of continuous functions is Borel Suppose that $f_n$ are continuous.  Prove that $E=\{x\in \mathbb{R}:f_n(x)\text{ is convergent}\}$ is Borel.   My first instinct is to take balls around $\lim_{n\to\infty}f_n(x)$, for each $x$ in $E$, and then show that there has to be some ball small enough so that everything inside that ball is a member of $E$. But this would imply not only that $E$ is Borel, but moreover that it is open, so I don't think that can be right.  Can someone show me how to do this? The difference between GPU and CPU I know what a CPU is(I think). It's the thing who's speed is measured in GigaHertz(these days).  However, you hear a lot about a GPU, and letting the GPU take over, not letting the CPU but the GPU do it, GPU-based rendering, etc...  What is this GPU anyway? How can I access it and use it to my advantage? What am I missing out on here? The  from Goblin Punch states the following:     Hungry Coffin   HD 6 AC 14 Slam 2d6 or 2d8 Swallow 1d6 + swallow   Fly 9 Int 5 Morale 12   It says the Hungry Coffin has "HD 6". This doesn't help me at all, because I still don't know what its average HP is, or how to roll its HP. How do I figure this out? Thanks! Can you deliver it [when,after] it [is | will be] assembled? As stated , being on fire means you are doing well, and  we see that you can be commended as an MVP for being on fire for a large portion of the time. This  shows that destroying non-player things gives you fire points, but what all do I get points from, and how many fire points do I need to be on fire/stay on fire? My question is, if you create a new domain user account for each of the SQL Server processes, what permissions should be set for each account? Or does the SQL configuration manager actually take care of this, and I just had an unforeseen issue?  I quite often have to setup Microsoft SQL Server and wondered if anyone can provide advice on configuring the accounts the services should run as. IMO this has been vaguely documented by Microsoft, while they point you in the right direction I have never been able to find any concrete examples.  To summarize what I've seen so far:  For simple deployments\development environments it is OK to use the virtual account defaults the installer uses: e.g. NT SERVICE\MSSQLSERVER  Avoid using the SYSTEM account, this is not secure.  For production and in domain environments it's recommended to use either a Managed Service Account, or create a domain user account (not an admin) for each service. Allegedly if you use a domain account at installation time the installer will set any required permissions for you.  If changing the service account on an existing install from a virtual account to a domain account the recommendation is to use the SQL Server configuration manager to set the new service accounts. Allegedly this will set any required permissions for you.  I just tried changing the service account in an existing install to a domain account and it would give me a logon failure until I granted the account log on as service permission, which contradicts the part where the SQL Server configuration manager will set any required permissions. (Although I'm not sure if a GPO may have interfered with setting this local security policy)  Microsoft does provide a list of the permissions that the SQL Server Setup grants .  But it's not clear to me if that is something I should be doing manually for the user I create to run the service as, or whether using the SQL config manager should automatically set these permissions.  SQL Server 2014, Domain Controller is on Windows Server 2008 R2. Most people spend part of each day standing, and if they have normal anatomy their heads are over their heels in this position.  Even sitting or lying down, the head is higher than the heels (if not over the heels).  Yet the phrase "head over heels" is used in ways implying it is an unusual situation.  How did such usage come to be, and why does it persist?  Why is the more obvious "heels over head" not used? Radius of Convergence of a given Power series Is a current transformer appropriate for a voltage step up flyback converter? I am designing a flyback converter, input 170V output 1200V at 2mA.  I am having trouble finding an appropriate transformer, most do not have a high enough N. Current transformers seem to have ranges in 1:500, so I could work with that.  Is it proper to use a current transformer as a conventional step up voltage transformer? How quickly is the desired state of curl of the Electric Field around a point achieved where a Magnetic Field Vector has just begun changing? My question and context with explanation are given below. Thank you in advance.   Faraday’s Law () states that a time-changing magnetic field vector induces a curl of the Electric field around that point. However it does not specify how quickly the necessary spatial gradient is developed in the Electric Field to satisfy that curl. Of course, the Electric Field vectors around that point cannot just jump to the required values instantaneously in accompaniment as soon as the magnetic field starts changing at a fixed rate, since it is required that the Electric Field has a sensible rate of change to in turn induce a curl of the magnetic Field and continue the propagation of the wave. It would seem the exact rates at which the Electric and Magnetic Fields are changing is critical in the mechanism of an EM wave’s propagation. How would I then calculate the rate of change of E somewhere along the wave given I know the rate of change of the B vector that began it, and given the wave is restricted to one dimension for simplicity? There should after all be some delay in how the EM wave moves from point to point in space, seeing as it moves at the finite speed of causality (3*10^8 m/s), as shown in the first image:    Some of my ideas so far have been that perhaps the rate of change of E is determined such that it exactly cancels the growth of the time-changing E vector that gave rise to a curl of B before, which developed at such a rate to induce a curl of E, which demands this rate of change of E which I am talking about now. In this way the wave might level out the growth of the fields in its wake and spread the change further outwards, in the same way smoothening out a crease spreads it elsewhere. However I am stuck on how to show this rigorously.  I am open to any difficulty of maths, so long as the answer provides the explanation. Below are some diagrams to help visualize what I am talking about:    Faraday's Law requires that a curl of E is developed, which under these restrictions requires a spatial gradient of E to appear across that point. Since the field across that point was previously in some other state, it is required that the Electric Field vectors that are infinitesimally close to that point undergo a rate of change through time to achieve the desired state of curl. The question is, how quickly is this desired state of curl achieved? I'm looking for a good way to perform multi-row inserts into an Oracle 9 database.  The following works in MySQL but doesn't seem to be supported in Oracle.  INSERT INTO TMP_DIM_EXCH_RT  (EXCH_WH_KEY,   EXCH_NAT_KEY,   EXCH_DATE, EXCH_RATE,   FROM_CURCY_CD,   TO_CURCY_CD,   EXCH_EFF_DATE,   EXCH_EFF_END_DATE,   EXCH_LAST_UPDATED_DATE)  VALUES     (1, 1, '28-AUG-2008', 109.49, 'USD', 'JPY', '28-AUG-2008', '28-AUG-2008', '28-AUG-2008'),     (2, 1, '28-AUG-2008', .54, 'USD', 'GBP', '28-AUG-2008', '28-AUG-2008', '28-AUG-2008'),     (3, 1, '28-AUG-2008', 1.05, 'USD', 'CAD', '28-AUG-2008', '28-AUG-2008', '28-AUG-2008'),     (4, 1, '28-AUG-2008', .68, 'USD', 'EUR', '28-AUG-2008', '28-AUG-2008', '28-AUG-2008'),     (5, 1, '28-AUG-2008', 1.16, 'USD', 'AUD', '28-AUG-2008', '28-AUG-2008', '28-AUG-2008'),     (6, 1, '28-AUG-2008', 7.81, 'USD', 'HKD', '28-AUG-2008', '28-AUG-2008', '28-AUG-2008'); Range of $(a_{1}-a_{2})^2+(a_{2}-a_{3})^2+(a_{3}-a_{4})^2+(a_{4}-a_{1})^2$ is If $a_{1},a_{2},a_{3},a_{4}\in \mathbb{R}$ and $a_{1}+a_{2}+a_{3}+a_{4}=0$ and $a^2_{1}+a^2_{2}+a^2_{3}+a^2_{4}=1.$ Then Range of $$E =(a_{1}-a_{2})^2+(a_{2}-a_{3})^2+(a_{3}-a_{4})^2+(a_{4}-a_{1})^2$$ is  Try: From $$(a_{1}-a_{2})^2+(a_{2}-a_{3})^2+(a_{3}-a_{4})^2+(a_{4}-a_{1})^2$$ $$=2(a^2_{1}+a^2_{2}+a^2_{3}+a^2_{4})+2(a_{1}a_{2}+a_{2}a_{3}+a_{3}a_{4}+a_{4}a_{1})$$ $$=2-2(a_{1}+a_{3})(a_{2}+a_{4})=2+2(a_{1}+a_{3})^2\geq 2$$ and equality hold when $\displaystyle a_{1}=-a_{3}$ and $a_{2}=-a_{4}$ Could some help me how to find its upper bound, Thanks So I was reading this:  and while the answers for the first question are good, nobody gave much attention to the second one. In fact, people only said that it can be proved without giving any proof or any.  So, if I have a Lagrangian and ADD an arbitrary function of $\dot{q}$, $q$ and $t$ in such a way that the equations of motion are the same, does this extra function MUST be a total time derivative?   EDIT Ok, I changed my question a little bit:  Question: If I have a function that obeys the Euler-Lagrange equation off-shell, this implies that my function is a time derivative? (This was used in Qmechanic's answer of this other question: , equation 14.)  Also, why people only talk about things that change the Lagrangian only by a total derivative? If this is not always the case that keeps the equation of motion the same, so why is it so important? And why in the two questions I posted about the same statement on Landau&amp;Lifshitz's mechanics book only consider this kind of change in the Lagrangian? Let $G$, $H$, and $K$ be abelian groups. Assume $G \oplus H \cong G \oplus K$. If each are finitely generated, then the structure theorem easily gives us that $H \cong K$. However, there are counterexamples in the general case, namely $H=\mathbb{Z}$, $K=\mathbb{Z} \oplus \mathbb{Z}$, and $G=\bigoplus_{i=1}^\infty\mathbb{Z}$.   I am wondering what goes wrong with the following argument:   Form the split short exact sequence  $$0 \to G \xrightarrow{i} G \oplus H \xrightarrow{\pi} H \to 0$$  where $i$ and $\pi$ are the inclusion and projection, respectively. Then don't we immediately get from the first isomorphism theorem that $H \cong (G \oplus H)/G$, which would contradict the counterexample above?   If we run this for the counterexample given, we get the sequence   $$0 \to \bigoplus_{i=1}^\infty\mathbb{Z} \to \bigoplus_{i=1}^\infty\mathbb{Z} \to \mathbb{Z} \to 0$$  which seems to still be short exact, even though here $G \cong G \oplus H$. But of course here we can see directly that $(G \oplus H)/G \cong G/G$, which is trivial. So what went wrong?   It seems to me to be something along these lines: In this case, the kernel of the projection is not the same as the original group, it is just, say, $0 \oplus \mathbb{Z} \oplus \mathbb{Z} \oplus \dots$, i.e. nontrivial from the second coordinate on, which just happens to be isomorphic to the original group. So if we form the other short exact sequence with $K$ in place of $H$, then the difference seems to be that we have two different projection maps with different (but isomorphic) kernels.   But I'm not really sure how to explain this in the general case, if my reasoning is correct to begin with.   Thanks in advance for any help! I made some scripts on my Raspberry Pi to automate a daily shot at a specific hour.  My DSLR is plugged in through USB. I thought this would keep its battery charged, but no.  Is there really no way automate a DSLR to shoot every 24 hours without having to pick it up, charge/replace its battery, and put in back on the same spot? Leaving the States on Visa Waiver and coming back I'm travelling on the ESTA visa waiver, I'm now in the States but would like to go to Costa Rica for a bit (say three weeks) then come back to the States. All will be within my 90 day limit.  Is that allowed?  (I have my onward ticket home.) Can I have multiple Schengen visas for future travels in my passport? I'm a citizen of Ukraine planning on visiting Italy early in October for tourism. Later that month I'm accepted as a speaker at the conference in Prague, Czech Republic. Both countries are members of Schengen agreement and a visa is required for me to visit them. Therefore, I need to visit Schengen countries 2 times and I need to get my visas ASAP (to confirm my talk to the conference organizers, as well as to avoid overpaying for hotels/flight tickets).   I've heard a number of times that a single person can't apply for a Schengen visa if already in possession of one, but I'd like to clarify this. Say, I've got a visa from Italy for the period of October 1st - October 14th. Can I get another one from Czech Republic for October 25th - October 30th before I visit Italy? Would the answer be different if I were to visit the second country (CZ) on October 13th - October 20th, thus, having a single day overlap between these visits. It would be of a tremendous help if you could link to the official documents that prove your point, but I'd certainly appreciate answers even without such links.  Both times I'm going to apply for a short-term tourist visa covering the dates of my travel. If the answer to both questions above is no - would several visits to multiple Schengen countries over a short period of time be a valid reason to apply for a single multiple-entry visa that covers the whole period from October 1st to October 30th from Italy? Let $G$ be a finite semi-group with identity such that it has only one idempotent.Is $G$ a group?  It only remains to show that for any $a\in G$ $\exists b\in G$ such that $ab=ba=e$ where $e$ is the identity of $G$  Also $e$ is the only idempotent of $G$ .How to proceed next? How to Remove shortcut arrow from desktop icons? How do I turn off the arrow for desktop icons? How to highlight paragraph (linebreak preserving) as well as itemize, tabular etc.? I want to highlight changes which are introduced by myself in the pdf document. So far the soul package is very neat but it does throw errors when I want to highlight not only paragraphs but also some itemize environments, and I don't want to highlight the items separately. Also the \cite{...} command is not supported by \hl. I recently posted a slightly different question:   BTW: What I liked most over other approaches like pdfcomments etc. is that the \hl command from soul has a very exact and clear output, no icons, no clicking etc. involved. Together with todonotes it's very good and almost all I need.