Lp Space conjugate function
If $f$ is measurable and $fg$ is in $L^1$ for all $g \in L^q$, must $f \in L^p$?
I remember seeing an upper bound for the binomial $\binom{n}{k}$ with an exponential function, something like $\binom{n}{k}\leq \left(ne/k\right)^k$. What exactly is it, and are there other similar good upper bounds for $\binom{n}{k}$?  Edit: As the link in Macavity's comment shows, the bound is indeed $\binom{n}{k}\leq \left(ne/k\right)^k$. How can we prove this? How do I show that $\frac {a^2}b + \frac {b^2}c + \frac {c^2}d + \frac {d^2}a \ge 4$ for $a^2 + b^2 + c^2 + d^2 = 4$? Let $a, b, c, d$ be positive real numbers such that $a^2 + b^2 + c^2 + d^2 = 4$, show that $$\frac {a^2}b + \frac {b^2}c + \frac {c^2}d + \frac {d^2}a \ge 4.$$  My try:  $$\frac {a^2}b + \frac {b^2}c + \frac {c^2}d + \frac {d^2}a \ge a + b + c + d,$$  yet $$a + b + c + d \le \sqrt{4(a^2 + b^2 + c^2 + d^2)} = 4.$$  Thus, direct application of Cauchy-Schwarz inequality is too weak. I tried other methods but with no significant progress:  $$(\frac {a^2}b + \frac {b^2}c + \frac {c^2}d + \frac {d^2}a)^2 \ge \frac {(a^{4/3} + b^{4/3} + c^{4/3})^3}{a^2 + b^2 + c^2 + d^2} = \frac {(a^{4/3} + b^{4/3} + c^{4/3})^3}4.$$  I also observed that  $$(\frac {a^2}b + \frac {b^2}c + \frac {c^2}d + \frac {d^2}a) + (\frac {a^2}c + \frac {b^2}d + \frac {c^2}a + \frac {d^2}b) + (\frac {a^2}d + \frac {b^2}a + \frac {c^2}b + \frac {d^2}c) + (\frac {a^2}a + \frac {b^2}b + \frac {c^2}c + \frac {d^2}d) = 4 (\frac 1a + \frac 1b + \frac 1c + \frac 1d) \ge 16,$$  since $$\frac 1a + \frac 1b + \frac 1c + \frac 1d \ge \sqrt{\frac {(1 + 1 + 1 + 1)^3}{a^2 + b^2 + c^2 + d^2}} = 4.$$  Now my work might seem stupid or off-topic here, but I provide it here because I wish any of these attempts will lead to a solution. Any hints will be appreciated. A continuous, injective function $f: \mathbb{R} \to \mathbb{R}$ is either strictly increasing or strictly decreasing. The shellscript is very simple:  retroarch $* -c /tmp/retroarch/Data/retroarch/gambatte.cfg   ("retroarch" is another shellscript)  If the filename passed contains spaces it fails:  RetroArch [ERROR] :: Could not read ROM file. biblatex in a nutshell (for beginners) If I need to use Blender within a colour-accurate workflow, what do I have to do to set that up (including hardware calibration, etc)? I wrote a question about blocking certain characters in passwords. It became tremendously unpopular, and it seemed like everyone wanted to rain on my parade every day.  Eventually, I "got it," agreed completely with what people were saying to me, and just wanted to withdraw my question so that my reputation didn't slide even further downhill.  Unfortunately, the system is designed such that I can't withdraw my question or redeem myself. Instead, I just get daily reputation attacks. That's a terrible system, and it needs to be changed. I am looking for a book that goes through the mathematical aspects of neural networks, from simple forward passage of multilayer perceptron in matrix form or differentiation of activation functions, to back propagation in CNN or RNN (to mention some of the topics).  Do you know any book that goes in depth into this theory? I've had a look at a couple (such as Pattern Recognition and Machine Learning by Bishop or Deep Learning by Goodfellow, Bengio and Courville) but still have not found a rigorous one (with exercises would be a plus). Do you have any suggestions? Were I rich, I would live on Long Island.   If I were rich, I would live on Long Island.   Is the first sentence still used, or is used in particular contexts (in example, to give emphasis to the sentence)? Transit Visa in Hong Kong for 15 hour layover between India and USA? We are (2) Indian nation and travelling from India to USA via Hong Kong airport and about 15 hours layover in Hong Kong airport. Do we need a transit visa even if only changing planes? And can we apply in Hong Kong Airport? On what criteria should I evaluate domain registrars? Though I've been a web developer for a fair amount of time, I am going for the first time to buy a few domain names.  I have looked into the domains I'm going to buy and know that they're available, and I've been looking into which sellers to use.  After doing a lot of research, the main ones I'm considering are 1&amp;1, Namecheap, and Gandi.  The problem is, when continuing to research, I'm not really sure what makes one domain seller distinct from another.  I don't need much in the way of services--definitely not hosting, since I plan to use Heroku for that.  I mainly only need the domain itself and DNS management, as well as possibly SSL certificates and WHOIS protection.  Question:  What makes one domain seller different from another?  How can I go about evaluating which one is the best for me?  Note:  This question is not which domain seller is the best, but rather, what criteria can I use to evaluate them and rank one over another.  I'm trying to find out what makes one domain seller different from another, since they all seem to be pretty similar to me right now. What is the difference between .pem, .csr, .key and .crt and other such file extensions? I'm new to SSL / TLS and I want to work with the OpenSSL toolkit.  I don't know what .pem and .csr stands for?  I do know that .key is the private key and .crt is the public key. I am doing some arcpy -functions and afterwards I'm trying to add the calculated results from one table to another table. I'm using the arcpy.AddJoin_management() function.   As input table I'm using a table which is stored in a personal geodatabase. The personal geodatabase is not the workspace environment! As "JoinTable" a Table of a file geodatabase is used. This file geodatabase is the workspace geodatabase environment.  The initialized function looks like this:  (%self.getWorkspaceEnvironment => "W:/a18fb/data/mypersonalgeodatabase.mdb" and tFactsheetHochwasser => table in the personal geodatabase)  arcpy.AddJoin_management("%s/tFactsheetHochwasser"%self.getWorkspaceEnvironment(),"KPG", "GebaeudeKtn_BWVred1Stat","KPG","KEEP_ALL")  If the code is executed the following error-message appears:  Failed to execute.Parameters are not valid. The value cannot be a table     ERROR 000840: The value is not a Raster Layer.      ERROR 000840: The value is not a Raster Catalog Layer.      ERROR 000840: The value is not a Mosaic Layer.      Failed to execute (AddJoin)`. Alternate moves, first one to roll a 6 wins: what's the probability of winning? Question: Players A and B play a sequence of independent games. Player A starts and throws a fair die and wins on a "six". If she fails then player B throws the same die and wins on a "six" or fails otherwise. If she fails then player A gets the die and throws again, and so on. The first player to throw a "six" wins. Find the probability that player A wins.  Here is what I have done:  A wins right away: $\dfrac{1}{6}$  A fails B wins: $\dfrac{5}{6}\times \dfrac{1}{6}$  A fails B fails A wins: $\dfrac{5}{6}\times \dfrac{5}{6}\times \dfrac{1}{6}$  Unlike another similar question which the player B can win on "five" and "six" after player A fails on the first throw, in this case, the games do not seem to have an endpoint. How should I calculate the probability that player A wins then? If I multiclass into 2 or more classes/subclasses that can cast spells, how do I determine what spells I know or can prepare?  For instance, if I am playing a multiclassed character with 1 level in warlock, 2 levels in bard, and 3 levels in paladin, how do I determine which spells I know/have prepared?    Related: Proving any number of the form $2^n$ can't be written as the sum of k consecutive numbers. What does "All things X" mean? In Dark Souls 2 there is a 'Shrine of Winter' near Sinners Rise. When you approach the shrine it will not let you open the door and a message pops up that says      Seek Mightier Souls   What does this mean and how do you open this shrine? Rep graph overview thing™ is white (only in latest Chrome) As you can see, if you go to the reputation graph on your profile on Stack Overflow, the bars in the bottom overview will be white. This makes them invisible unless they are selected.  Eeek!  This does not happen on meta or any other site. Also, I cannot seem to reproduce it on others' profile. Possibly a bug only on your own profile? EDIT: I think I've existed for too long. If I change the CSS on one of the bars to stroke: none, then it becomes green. I think the outline is covering the fill because the bars are too skinny. EDIT2: Confirmed hypothesis by reproducing on Jon Skeet's graph. Proposed solution: enforce minimum bar width and add horizontal scrollbar. EDIT3: Can only repro on Chrome 29.0.1547.57 (Official Build 217859) m (the latest version). Norepro on older versions of Chrome, IE9 (winces), all versions of FF, and all versions of Safari. EDIT4: Can be fixed simply by running the JavaScript $('rect').css('stroke-width', '0') or a userstyle rect { stroke: none }. I may make a user[style|script] to fix this. I have a NetGear WGR614 wireless router.  There is a 'region' setting in the wireless options.  I'm wondering what technical purpose of this setting is?  The admin help states:     It may not be legal to operate the router in a region other than the region shown here.    Which makes me wonder if it could be adjusting it's max power output based on regional radio laws?