Alternative Function for EREGI in PHP
How can I convert ereg expressions to preg in PHP?
How to interpret regression coefficients in logistic regression? If $\sum a_n b_n <\infty$ for all $(b_n)\in \ell^2$ then $(a_n) \in \ell^2$ I'm trying to prove the following:     If $(a_n)$ is a sequence of positive numbers such that $\sum_{n=1}^\infty a_n b_n&lt;\infty$ for all sequences of positive numbers $(b_n)$ such that $\sum_{n=1}^\infty b_n^2&lt;\infty$, then $\sum_{n=1}^\infty a_n^2 &lt;\infty$.   The context here is functional analysis homework, in the subject of Hilbert spaces.  Here's what I've thought:  Let $f=(a_n)&gt;0$. Then the problem reads: if $\int f\overline{g}&lt;\infty$ for all $g&gt;0,g\in \ell^2$, then $f\in \ell^2$. This brings the problem into the realm of $\ell^p$ spaces.  I know the inner product is defined only in $\ell^2$, but it's sort of like saying: if $\langle f,g\rangle &lt;\infty$ for all $g&gt;0,g\in \ell^2$ then $f\in \ell^2$.  I read this as: "to check a positive sequence is in $\ell^2$, just check its inner product with any positive sequence in $\ell^2$ is finite, then you're done", which I find nice, but I can't prove it :P  From there, I don't know what else to do. I thought of Hölder's inequality which in this context states: $$\sum_{n=1}^\infty a_nb_n \leq \left( \sum_{n=1}^\infty a_n^2 \right)^{1/2} \left( \sum_{n=1}^\infty b_n^2 \right)^{1/2}$$  but it's not useful here. Associativity of symmetric difference of sets By $A\oplus B$ we denote the symmectric difference of two sets. The definition is $A\oplus B =(A\setminus B) \cup (B\setminus A)$. Now I hope to show that $A\oplus (B\oplus C) = (A\oplus B)\oplus C$. I remember that there's an elegant proof, but I forget its detail. (The first step is to show $A\oplus (A\oplus B) = B$, and maybe applying this formula, we can obtain associativity.) What is the difference between “do” and “make”? I am taking a course in Mathematical Methods for physicsPhysics (Junior Level).   We are working from   > Mathematical Methods for physicists, George B.Arfken .    I just need online resources covering ( Matrices, Determinants, Vector Analysis, Tensors and differential forms and Vector spaces...etc) to study and understand from. How does the telepathy supernatural ability work? Sum of the alternating harmonic series $\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k} = \frac{1}{1} - \frac{1}{2} + \cdots $ Is there a way to calculate a sorted field with sequential numbers?  I have seen  that outlines how to calculate sequential numbers, but this is always calculated on FID order, not on sorted order.  #Pre-logic Script Code: rec=0 def autoIncrement():      global rec      pStart = 1       pInterval = 1      if (rec == 0):           rec = pStart       else:           rec += pInterval       return rec  #Expression: autoIncrement()   An example of what I'm trying to do.  I've used an advanced sort to sort by year, month, day, and now want to have sequential numbers in the Seq field.  You'll see that my OBJECTID field is not in order, so the above code won't work.    Can this be done either in the Field Calculator or using an Update Cursor in arcpy? What is the intuition behind the unit normal vector being the derivative of the unit tangent vector? I've seen the math, but... It just doesn't make sense to me. How is the slope going to point perpendicular to the vector that is clearly a straight line not going in that direction? In the story I'm trying to recall, everybody on their thirteenth birthday (?) is assigned by some kind of brain analysis method a profession and given some kind of pill (?) that instantly (?) grants them the required knowledge and skills to start working immediately. Some professions are more sought after than others, especially in view of the olympic (?) games regularly held on a profession-by-profession basis.  The protagonist, however, is turned down from this system and dismissed with some general reason and has to work his way to adulthood the boring old way: studying from books. Eventually he rejects from his status as a "special" student and goes on the run, hiding from the teachers and envying the normal people.      Eventually, it turns out that the protagonist was actually smart enough that being spoonfed premade information was a waste. What he actually was assigned to was advancing the bundles of knowledge themselves, as well as being front line in the technological progress.  "Somebody has to write those textbooks! You can infuse knowledge about how to use today's technology, but you can't do that for tomorrow's - people that get assigned a profession become tied with their assigned technology and are only as relevant as that. If you want more, you must learn how things actually work, and there's no shortcut for that." (not an actual quote, but a specific passage as I remember it.)  The troubles the protagonists underwent were actually planned and necessary for him to acknowledge and accept his role in society.   This story was certainly longer than a couple pages, but it wasn't very long either. If it's a short story, it was probably written by Asimov, but I can't be sure. It rings me of Brave New World given the theme of government-issued population planning, but it certainly doesn't fit with anything else in the actual plot. Any idea? How to properly render smoke on a transparent background? When rendering volumes with Film > Transparent enabled, the alpha appears to be the wrong type (premultiplied/straight). In the 3D view it works as expected:    This is a known .  Is there any way to work around this? I am currently in a clan with three members. When we tried to initiate a clan war it said we had insufficient clan requirements. Does anyone know what the full requirements are? Is there a minimum clan member requirement? Why does HTML require that multiple spaces show up as a single space in the browser? How to run find -exec? I'd like to find the files in the current directory that contain the text "chrome".  $ find . -exec grep chrome find: missing argument to `-exec'   What am I doing wrong? Texture Channel animated, doesn't appear in viewport I'm creating an educational video, and for efficiency, I'm using a Cyclic repeater to repeat an on-page/off-page effect.  Every loop, I have the texture channel switch to the next image, rather than duplicating all the objects/animations. This works properly and appears when rendering the animation, but does not appear in the viewport. It sticks to the original UV image that was mapped.  Is there a way to have this change appear in the viewport? How do I identify the C terminal on my HVAC? How do I identify the C terminal on an HVAC, furnace, or boiler? If the terminals don't have a screw labeled C what should I look for on the wiring diagram? What steps should I take if a wiring diagram cannot be found or if it indicates that a C terminal doesn't exist?  Note: This is an extension to  which asks about running the wires through the wall. This question is about where to connect the wire at the HVAC. Would it be possible to break an RSA key, in for example 1 week of time, if the cracker have already spent X number of years building an index of primes by performing every permutation of existing prime keys up to $2^{2048}$?  I understand this would take an immense amount of time to do, but it would only be done once. Obviously, computing such an index would take a considerable amount of time and would have to be done with someone having the right amount of resource. Given the public key, you would be able to look up what the two primes are, and hence retrieve the private key instantly.  Breaking an 256 AES would then be easier, as AES keys are often generated using the RSA private key.  Would it be possible to build such an index? How long would that take in permutations and perhaps with the fastest computer in the world? I need to reset my password. I have followed these steps:    However, then I go to "root" or "netroot" recovery options, it tells me:  Give root password for maintenance (or type Control-D to continue)   Clearly, I do not know the root password. If I type CTRL+D, I return to the list of options. From  I read:  Under chapter 'The Other Way':  4. Highlight the line that begins kernel and press 'e' to edit`   But in the grub configuration file I have no line that starts with kernel. Only:  setparams 'Ubuntu...' recordfail set gxfpayload... insmod part_msdos insmod ext2 set root=... search --no-floppy... linux /boot/vmlinuz-2.6.38... initrd /boot/initrd.img-2.6....   Those are all lines in my GRUB. Which line should I edit? Or is there another way to reset my password? YA book about a young powerful female caster attending a school that trains in magical arts and duels I read a young adult novel around the turn of the century. This story features a young female protagonist that is very naturally gifted in magic. She is attending a  to learn mastery and control over her gifts.  A lot of her school revolves around magic dueling. The participants are taught to extend their inner aura like a second magical skin, possibly referred to as creating a shield, or . The user can increase the strength of the shield by focusing harder on it, but doing so means you have less ability to be offensive. The amount of energy needed to create a shield was half the energy a foe would need to focus at you to break it.  Ex: Mage A and B are equally matched. Mage A focuses half his power into a shield. Mage B would then need to focus 100% of his power to break A’s shield.  Another aspect I remember is that when dueling, a school master would extend a personal shield on to one of the combatants as a sort of secondary shield that goes underneath the combatants. When one mage broke the others mages personal shield, the school masters shield protects the losers life from harm, and the breaker of the shield is declared the winner.  I distinctly remember one match in which the protagonist is having a duel with another strong student. She creates a wind storm around herself that picks up the sand from the arena floor and makes her impossible to see and thus makes it hard for her opponent to focus on her shield to break it. She then lashes out at her opponent with incredibly powerful strikes with the final strike being so hard it shatters the opponents shield and then breaks the school masters shield as well and stops just shy of killing her dueling partner.  Any assistance on this one? I think my daughter would love to read it. Particular solution of $y''-4y'+5y = 4e^{2x} (\sin x)$ How do I find the particular solution of this second order inhomogenous differential equation? (Using undetermined coefficients).  $y''-4y'+5y = 4e^{2x} (\sin x)$  I can find the generel homogenous solutions but I need help for the particular.