Energy Conservation in Holding Weights This is a two part question, and it might have to do with biology a bit.  Part 1: I grab a heavy dumbbell and curl it. Halfway up I stop the curl and just hold it there with my elbow at 90 degrees. There is no net acceleration, so there is no net force on the object, so no work is being done. But clearly I am using energy to do this, as my muscles are tensed. Since my chemical energy is not going into any sort of mechanical energy, is it simply going into heat?  Part 2: If we assume that all of the chemical energy of my muscles turns to heat, what if I now lift the weight? If I accelerate this weight up, now I'm converting some of this chemical energy into mechanical energy, so less can go to heat. Doesn't that mean your arms get hotter if you don’t do any work onto the object than if you do? This just seems counterintuitive to me.
Why does holding something up cost energy while no work is being done? I read the definition of  as  $$W ~=~ \vec{F} \cdot \vec{d}$$  $$\text{ Work  =  (Force) $\cdot$ (Distance)}.$$ If a book is there on the table, no work is done as no distance is covered. If I hold up a book in my hand and my arm is stretched, if no work is being done, where is my energy going?
Making the bibliography appear in the table of contents Confidence interval interpretation in linear regression when errors are not normally distributed Dates and times: "on", "in", "at"? I’m often confused when I speak about times and dates. What is the rule for using on, in, and at in the following sentences?   I will do it ___ Tuesday. We married ___ March. He returned ___ the same day. Every day ___ the same time, I walk the dog. Set shortcuts to change keyboard layout in Windows 10? Is there any way to set keyboard shortcuts to quickly change to a different language/keyboard layout? The old shortcuts still seem to work (when set back in Windows 7), but I can't seem to find anything to change it in Windows 10. I have to solve: $\lim\limits_{x\to \infty} \frac{\ln(x+1)}{\ln(x)}$. Can you give me any hints to go? Thanks a lot! PBKDF vs HKDF for pretty long key I'm developing a messenger application with encrypted chats.   In the first version of the app I've used PBKDF2 (10000 iterations, SHA1, random salt) to extend a short user password and generate keys to encrypt (AES256) and sign message (HMAC).  In the current version of the app I generate a random 130-bit key and use it instead of a user password. I think that using PBKDF2 with such a key has some overhead.   I see 2 possible ways to improve the current approach:   Keep using PBKDF2 and decrease iterations to 1 or 10. Don't use PBKDF2 and use HKDF instead.   What are the advantages and disadvantages of each approach? Are there any other options? Can the relative pronoun "whose" be used for animals, things and countries? When I was a student of English as a foreign language, more years ago than I care to count, I was taught that the relative pronoun “whose” could only be used for human beings, i.e., when someone possesses something.   My teachers were native speakers from England, then.   Since that time, I’ve seen “whose”, as a relative pronoun, used in several contexts where no human beings are mentioned, e.g. with animals, objects, countries, abstract nouns, etc, in American newspapers and magazines. Unfortunately, I can’t remember any of the sentences and cannot give examples.  What is acceptable in written and spoken English, and what isn’t, concerning the use of “whose” ? What is `^M` and how do I get rid of it? I am a little bit confused about the Area Under Curve (AUC) of ROC and the overall accuracy.  Will the AUC be proportional to the overall accuracy? In other words, when we have a larger overall accuracy will we definitely a get larger AUC? Or are they by definition positively correlated?  If they are positively correlated, why do we bother reporting both of them in some publications?  In real case, I performed some classification task and got the results as follows: classifier A got an accuracy 85% and AUC of 0.98 and classifier B got an accuracy of 93% and AUC of 0.92. Question is, which classifier is better? Is it possible to get results similar to these (or do my results indicate a bug in my implementation)? Newton's third law does not apply to the kinetic energy formula? Here is a classic problem:     A 60kg boy and a 40kg girl are sitting in front of each other on skateboards on a frictionless horizontal surface. The boy pushes the girl and the girl gains a velocity of 0.3m/s during the 0.5s the boys hand was in contact with her. What is the boy's resulting velocity?   Considering Newton's Third Law, the best way is to find the total force applied and rearrange the formula to find the boy's speed.  Newton's Law Method:  $F = m*a$  $a=\frac{0.3m/s}{0.5s}$  $F = 40kg * 0.6m/s^2$  $F = 24N$  Therefore the boy's speed will be:   $\frac{24N}{60kg} = a$  $a = 0.4m/s^2$  $v = 0.4m/s^2 * 0.5s$  $v = 0.2m/s$  Kinetic Energy Method:  $E_{k}=\frac{1}{2}*m*v^2$ the girl's kinetic energy  $E_{k}=\frac{1}{2}*40kg*(0.3m/s)^2$  $E_{k}=1.8J$  Since every action has an equal and opposite reaction the boy should have the same amount of kinetic energy.  $E_{k}=\frac{1}{2}*60kg*(0.2m/s)^2$  $E_{k}=1.2J$  As you can see, using Newton's Third Law, the kinetic energy was not the same. So what's going on here? I thought that in this case they should both have the same amount of kinetic energy due to Newton's Law. I must be missing something since there is more energy going in one direction than the other after meeting. My book used the first method as the answer and, well, it feels wrong. Understanding divison by monic polynomial in $R[x]$ where $R$ is an arbitrary ring I read "Algebra: Chapter 0" by P.Aluffi. I encountered a topic where it says you can divide any polynomial in $R[x]$($R$ is any ring) by a monic polynomial(that is, a polynomial of the form $x^d + \sum\limits_{i=0}^{d-1} a_i x^{i}$).   It says if $g(x)$ is any polynomial and $f(x)$ is a monic polynomial. Then $\exists \ \ \ q(x), r(x) \in R[x],\: \deg r(x) &lt; \deg f(x): g(x) = f(x)q(x) + r(x).$  I have two questions regarding this:  1) Why $f(x)$ needs to be monic?   2) How can we prove it? The book talk about induction and that if $\deg  g(x) = n &gt; d = deg \ f(x)$, then $\forall a \in R$ we have $ax^n = ax^{n-d}f(x) + h(x)$ for some $h(x): \deg \ h(x) &lt; n$. It's true, but how do I take it from here? Executing cmd.exe commands from Java A positive harmonic function on the punctured plane is constant How do I get my Mac to boot from an Ubuntu USB key? I am planning an hour and a half ride with my 1 month old baby girl. I heard that being in the car seat for a long time is not good for newborns.  Is an hour and a half too long? Should we do a stop in the middle of the trip? Taking screenshot of a specific area from the command line I know how to detect when resistors are arranged in parallel or series arrangement and I can also find their equivalent resistance in simple circuits or when resistances are connected in form of triangle but what happens when the arrangement is complex like this :   Which resistors are parallel and which are in series ? How can I find the equivalent resistance in such cases ? Is there rule or method for figuring this out ? ravel: opposite meanings? Let $U$ be connected. A function $u \in H^1(U)$ is a weak solution   of Neumann's problem   \begin{equation} (*)\qquad\left\{ \begin{array}{rl} -\Delta = f &amp; \text{in } U \\ \frac{\partial u}{\partial \nu} = 0 &amp; \text{on } \partial U \end{array} \right. \end{equation}   if   $$ \int_U Du \cdot Dv \; dx = \int_Ufv \; dx $$   for all $v \in H^1(U)$. Let $f\in L^2(U)$. Prove $(*)$ has a weak solution   if and only if   $$ \int_U f \; dx =0. $$   For the only if part I set $v=1$. However I do not see where to start the if part. I was thinking of the Lax-Milgram theorem. That's where I'm now. Near-universally applicable alternative to "sanity check" The phrase "sanity check" is  as it implies that there is something wrong with people who have mental illnesses and the word "sanity" has been used to discriminate against such people.  Therefore, it should should be avoided.  What is a better alternative in the context of software?  Example usage:     _: if this is not what it should be, we have memory corruption (a fatal   error) and should terminate the program.