for any $a,b\in R^+$, how to prove that $$a b \leq \frac{a^{3}}{3}+\frac{b^{3 / 2}}{3 / 2}$$ my try: if I can prove that $\frac{a^2+b^2}{2}\leq \frac{a^{3}}{3}+\frac{b^{3 / 2}}{3 / 2}$then this inequality can be proved but it seems that $\frac{a^2+b^2}{2}&gt; \frac{a^{3}}{3}+\frac{b^{3 / 2}}{3 / 2}$  I had no clue then.
states that if $a, b \geq 0$, $p, q &gt; 0$, and $\frac{1}{p} + \frac{1}{q} = 1$, then $$ab\leq \frac{a^p}{p} + \frac{b^q}{q}$$ (with equality only when $a^p = b^q$).  Back when I was in my first course in real analysis, I was assigned this as homework, but I couldn't figure it out.  I kept trying to manipulate the expressions algebraically, and I couldn't get anywhere.  But every proof that I've seen since uses calculus in some way to prove this.  For example, a common proof is based on this  and integration.  The proof on Wikipedia uses the fact that $\log$ is concave, which I believe requires the analytic definition of the logarithm to prove (correct me if I'm wrong).  Can this be proven using just algebraic manipulations?  I know that that is a somewhat vague question, because "algebraic" is not well-defined, but I'm not sure how to make it more rigorous. But for example, the proof when $p = q = 2$ is something I would consider to be "purely algebraic":  $$0 \leq (a - b)^2 = a^2 + b^2 - 2ab,$$ so $$ab \leq \frac{a^2}{2} + \frac{b^2}{2}.$$
In theory, is it possible that the loss on the train set will increase together with the train set accuracy? (in the normal case the train set loss should go down) And vice versa, can the train loss decrease along with the accuracy?   Note: This is obviously different than the classic underfit / overfit cases which are easier to grasp. How to write (La)TeX (with parentheses) [or any other TeX-related logo] I'm looking after a macro to write the (La)TeX logo (with parentheses), i.e. something similar to \TeX, \LaTeX, but where the output includes ()'s. I've noticed that, if I add \n to a pattern for substituting using sed, it does not match. Example:  $ cat &gt; alpha.txt This is a test Please do not be alarmed  $ sed -i'.original' 's/a test\nPlease do not/not a test\nBe/' alpha.txt  $ diff alpha.txt{,.original}  $ # No differences printed out   How can I get this to work? When should I use  or >()? I know that &lt;, &lt;(), &lt;&lt; all are used to give input and  &gt;, &gt;() which are used to redirect output.  But I don't know what is clear difference between them and when to use them.  Please explain in detail and along with references if possible. Thanks How would I be able to create a grill of sorts found on the front of some computer cases such as this one Why use the Conjure Minor Elemental spell when its results are so unpredictable? The  spell states:  You summon elementals that appear in unoccupied spaces that you can see within range. You choose one the following options for what appears:  One elemental of challenge rating 2 or lower Two elementals of challenge rating 1 or lower Four elementals of challenge rating 1/2 or lower Eight elementals of challenge rating 1/4 or lower.   It seems like the only two things a player can select are the maximum CR of the elemental conjured, and, arguably, the location its summoned in. I am confused as to why you would want to use this spell given that its results are so unpredictable. Why would a wizard cast a spell that's essentially an open invitation to whatever elemental from a list decides to show up? You can't strategize around the spell very much because, as written, you don't choose what gets summoned, and you can easily end up with something you weren't counting on. It won't be useless, but why would you cast a spell that has such an unpredictable result? It's like the difference between the chromatic orb and chaotic bolt - if the chaotic bolt didnt have the chance to bounce. You'd never pick a random damage type over a damage type of choice. Why would you choose a spell that might do something you want over something that does exactly what you intend it to do? What are the situations in which you would use this spell that wouldn't be better solved by a spell of equal or lower level that has more predictable results? Analyzing a sentence Do my analyze is correct? If don't then correct it.  Sentence: The tsunami and the undersea earthquake which triggered it, the largest ever recorded in the region, killed nearly 20,000 people, destroyed over 100,000 homes and threw the lives of tens of millions into turmoil.  Subject: The tsunami and the undersea earthquake Principal verb: triggered Adjective clause of subject:the largest ever recorded in the region, killed nearly 20,000 people, destroyed over 100,000 homes Use of parallelism:threw the lives of tens of millions into turmoil. Why the rows returns by "explain" is not equal to count()? Light emission from the vicinity of a black hole The photograph of the black hole, at the centre of galaxy M87, is now a part of human history. We are told that the red-colour is caused by hot gas, matter &amp; radiation around the "edge", the Event Horizon. Given the power of the gravitational forces emanating from black hole, why aren't these three things drawn into the abyss? If light is bent/ drawn into the BH, how does this red image arise? The logic, here, would be that a body, on the Event Horizon, is immune from the gravity--which sounds pretty fatuous? Basis of cotangent space Prove $ \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B $ How would I verify the following double angle identity. $$ \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B $$ So far I have done this. $$ (\sin A\cos B+\cos A\sin B)(\sin A\cos B-\cos A\sin B)  $$But I am not sure how to proceed. How do I set up a "secure" open resolver? This is a  about securing public DNS resolvers   Open DNS servers seem pretty neat and convenient, as they provide IP addresses that we can use consistently across our company regardless of where they are located. Google and OpenDNS provide this functionality, but I'm not sure that I want these companies to have access to our DNS queries.  I want to set up something like this for use by our company, but I hear a lot about this being dangerous practice (particularly in regards to ) and I want to make sure that we do this right. What things do I need to keep in mind when building this type of environment? Word for "moving up one conceptual level" Suppose there are $2$ permutations given by $p$ and $q$. I need to check whether $p$ belongs to the group generated by $q$, and if so, it's representation in a power of $q$. In other words, I've been given $p$ and $q$, I need to check whether $\exists i \in \{ 0,...,|q|-1 \} \text{ such that } p = q^i$ I know that if $m = \gcd(i, |q|)$, then $|q^i| = \frac{|q|}{m}$. Now, say that such an $i$ exists. Then, $$ |p| = \frac{|q|}{\gcd(i, |q|)} \Rightarrow \gcd(i, |q|) = \frac{|q|}{|p|} $$ Thus, a necessary condition for such an $i$ to exist is: $|p|$ divides $|q|$ Say that this holds, and $\frac{|q|}{|p|} = r$, where r is an integer. Then we just have to solve $\gcd(i, |q|) = r$ for $ i \in \{ 0,...,|q|-1 \}$. Can this be done? If not manually, then by some efficient algorithm? (Note: Just looping over all values of $i$ is not efficient as $i$ may be very large) Edit: Another way I can think of solving this is by writing both $p$ and $q$ as product of disjoint cycles: $$ p = c_1c_2\ldots c_k \text{ and } q = d_1d_2\ldots d_l $$ And thus, as disjoint cycles commute: $$ p^i = c_1^ic_2^i\ldots c_k^i $$ How to proceed after this? Why has Steve Nash moved to the LA Lakers in sign-and-trade deal? Congratulations to Steve Nash on his move to Lakers !  His contract with the Suns has expired so I don't understand why he moved in a sign-and-trade deal.    Why resign and move in trade? Why didn't the Lakers sign Nash without any trade? Differentiability of this picewise function $$f(x,y) = \left\{\begin{array}{cc} \frac{xy}{x^2+y^2} &amp; (x,y)\neq(0,0) \\ f(x,y) = 0 &amp; (x,y)=(0,0) \end{array}\right.$$  In order to verify if this function is differentiable, I tried to prove it by the theorem that says that if $\frac{∂f}{∂x}$ and $\frac{∂f}{∂y}$ exist and are continuous at the point $(x_0,y_0)$, then the function is differentiable at this point. So I did:  $$\frac{\partial f}{\partial x}(0,0) = \lim_{h\to 0}\frac{f(0+h,0)-f(0,0)}{h} = 0$$ $$\frac{\partial f}{\partial y}(0,0) = \lim_{h\to 0}\frac{f(0,0+h)-f(0,0)}{h} = 0$$  so we have that the partial derivatives at point $(0,0)$ is $0$. Now, if we take the derivative at $(x,y)\neq (0,0)$ and then take the limit of it as $(x,y)\to(0,0)$, we can see if the derivatives are continuous or not. So here it is:  $$\frac{\partial f}{\partial x}(x,y) = \frac{y(y^2-x^2)}{(x^2+y^2)}$$  but   $$\lim_{(x,y)\to(0,0)} \frac{y(y^2-x^2)}{(x^2+y^2)} $$ does not exist (by wolfram alpha... but can anybody tell me an easy way to prove this limit does not exist? easier than taking the limit in different directions?), therefore the derivative is not continuous at $(0,0)$, so we can't say $f$ is differentiable at $(0,0)$, but for $(x,y)\neq (0,0)$ the function is continuous, as it is a quotient of continuous functions. So $f$ is at least differentiable at $(x,y)\neq (0,0)$.   Now, to verify differentiability at $(0,0)$ I think we must use the limit definition of differentiablity: A function is differentiable at $(0,0)$ iff: $$\lim_{(h,k)\to (0,0)} \frac{f(0+h,0+k)-f(0,0)-\frac{\partial f}{\partial x}(0,0)-\frac{\partial f}{\partial y}(0,0)}{\|(h,k)\|} = 0$$  Let's calculate this limit:  $$\lim_{(h,k)\to (0,0)} \frac{f(0+h,0+k)-f(0,0)-\frac{\partial f}{\partial x}(0,0)-\frac{\partial f}{\partial y}(0,0)}{\|(h,k)\|} = \\ \lim_{(h,k)\to (0,0)} \frac{\frac{hk}{h^2+k^2}}{\sqrt{h^2+k^2}} = \\ \lim_{(h,k)\to (0,0)} \frac{hk}{(h^2+k^2)\sqrt{h^2+k^2}}$$  which I think, it's a limit that does not exist, therefore the function isn't differentiable at $(0,0)$ Changing my Apple ID's email: address is not available Is "double hashing" a password less secure than just hashing it once? Is hashing a password twice before storage any more or less secure than just hashing it once?  What I'm talking about is doing this:  $hashed_password = hash(hash($plaintext_password));   instead of just this:  $hashed_password = hash($plaintext_password);   If it is less secure, can you provide a good explanation (or a link to one)?  Also, does the hash function used make a difference?  Does it make any difference if you mix md5 and sha1 (for example) instead of repeating the same hash function?  Note 1:  When I say "double hashing" I'm talking about hashing a password twice in an attempt to make it more obscured.  I'm not talking about the .  Note 2:  I know I need to add a random salt to really make it secure.  The question is whether hashing twice with the same algorithm helps or hurts the hash. How to find a computer name in a LAN from the IP address? My network includes machines running Linux and others running Windows. And my machine is running Linux. Remove empty directory from CVS? I'm not quite sure how this happened, but somehow a completely empty hierarchy of directories has ended up in my repository:  com/ com/companyname/ com/companyname/blah/ com/sun/ com/sun/java/ com/sun/java/jax_rpc_ri/   I think what happened was that these directories did have files in them, but then a developer realized he/she shouldn't have checked them in in the first place since these are by-products of the build process, so he/she removed the files but somehow the empty directories are left in the repository as ancient relics.  How can I remove this from CVS? The only results I seem to be able to find on google say that there shouldn't be a need to remove empty directories as CVS won't keep them around in the first place, and that the -P (prune) options to cvs update should remove them from the working directory  - which is zero help if you actually have empty directories in your repository.  A cvs remove and cvs commit doesn't seem to take care of this situation:  $ cvs remove -Rf com cvs remove: Removing com cvs remove: Removing com/companyname cvs remove: Removing com/companyname/blah cvs remove: Removing com/sun cvs remove: Removing com/sun/java cvs remove: Removing com/sun/java/jax_rpc_ri $ cvs commit com cvs commit: Examining com cvs commit: Examining com/companyname cvs commit: Examining com/companyname/blah cvs commit: Examining com/sun cvs commit: Examining com/sun/java cvs commit: Examining com/sun/java/jax_rpc_ri $ ls -l com total 24 drwxrwxr-x  2 matt matt 4096 Oct 15 14:38 CVS drwxrwxr-x  9 matt matt 4096 Oct 15 14:38 companyname drwxrwxr-x  4 matt matt 4096 Oct 15 14:38 sun   It's still there!  Does SVN have this weird behavior too?