Does that a continuous function $f$ of $R$ into $R$ has a local maximum at $(p,f(p))$ implies that there exists $\delta$ such that $f(x)$ monotonically increasing in $(p-\delta,p)$ and monotonically decreasing in $(p,p+\delta)$? I was trying to solve Prob. 15, Chap. 4 from Rudin : &quot;Every continuous open mapping of R into R is monotonic&quot;, and I thought it would be pretty easy to solve it if the proposition above holds, but couldn't find any previous answers mentioning it.
This is a homework problem for my real analysis class. It is problem 13.1 from Mattuck's Introdution to Analysis. The question reads:       Suppose F(x) is continuous on some open interval I and c is a maximum point inside this interval. Is it true that f(x) must be increasing immediately to the left of c and decreasing immediately to the right of c? Proof or counterexample. (Note: A constant function is considered to be both increasing and decreasing.)      My friend and I have been attempting to find a counterexample. We figured a function that begins oscillating infinitely many times near c but converges to c would work. We're having difficulty constructing such a function out of elementary functions so we think we will have to define it in another way, piece-wise using receding intervals. I have no clue how this would look. Any advice would be greatly appreciated.
Meta-analysis of means and medians in R? I've been asked to review the literature for all studies that estimate the mean and median Blood Lead Level (BLL) in a certain country, and then perform a meta-analysis to come up with one overall value for the mean and median level.   Is such a thing typically done? I'm not interested in any "effect size," just a univariate measure.   I'd like to create a forest plot and do some of the standard tests of heterogeneity across studies. Is there a package and function in R that will allow me to do a meta-analysis of mean and median BLLs?  Just to clarify, this would be a separate meta-analysis for means, and a separate meta-analysis for medians. How can I ssh to a remote server with an automated password and no key? Chrome not asking to remember passwords I want to do something like String.Format("[{0}, {1}, {2}]", 1, 2, 3) which returns:  [1, 2, 3]   How do I do this in Python? Polynomial $P(a)=b,P(b)=c,P(c)=a$ Let $a,b,c$ be $3$ distinct integers, and let $P$ be a polynomial with integer coefficients.Show that in this case the conditions $$P(a)=b,P(b)=c,P(c)=a$$ cannot be satisfied simultaneously.  Any hint would be appreciated. What is the easiest way to duplicate an activerecord record? I want to make a copy of an ActiveRecord object, changing a single field in the process (in addition to the id).  What is the simplest way to accomplish this? I realize I could create a new record, and then iterate over each of the fields copying the data field-by-field - but I figured there must be an easier way to do this. Perhaps something like this:  new_record = Record.copy(:id) What is the difference between sourcing ('.' or 'source') and executing a file in bash? I was just wondering why Fresnel Lenses are not widely used in the production of solar electricity. Their use there would mean that you could produce heat within a fraction of a second, up to a few minutes and run a turbine to produce electricity.  Though it is used in welding, I am not sure what are the problems in producing electricity, as stated by :     New applications have appeared in solar energy, where Fresnel lenses can concentrate sunlight (with a ratio of almost 500:1) onto solar cells. How to use notify-send with crontab? I really read many tutorials, but crontab doesn't start notify send.  What I've tried:   etc/crontab  added at the end:  15 * * * * notify-send 'crontab' 'test'  with crontab interface:  crontab -e   added at the end:  15 * * * * notify-send 'crontab' 'test' Overview:  Lots of games with RPG-like statistics allow for character "buffs", ranging from simple "Deal 25% extra damage" to more complicated things like "Deal 15 damage back to attackers when hit."  The specifics of each type of buff aren't really relevant.  I'm looking for a (presumably object-oriented) way to handle arbitrary buffs.  Details:  In my particular case, I have multiple characters in a turn-based battle environment, so I envisioned buffs being tied to events like "OnTurnStart", "OnReceiveDamage", etc.  Perhaps each buff is a subclass of a main Buff abstract class, where only the relevant events are overloaded.  Then each character could have a vector of buffs currently applied.  Does this solution make sense?  I can certainly see dozens of event types being necessary, it feels like making a new subclass for each buff is overkill, and it doesn't seem to allow for any buff "interactions".  That is, if I wanted to implement a cap on damage boosts so that even if you had 10 different buffs which all give 25% extra damage, you only do 100% extra instead of 250% extra.  And there's more complicated situations that ideally I could control.  I'm sure everyone can come up with examples of how more sophisticated buffs can potentially interact with each other in a way that as a game developer I may not want.  As a relatively inexperienced C++ programmer (I generally have used C in embedded systems), I feel like my solution is simplistic and probably doesn't take full advantage of the object-oriented language.  Thoughts?  Has anyone here designed a fairly robust buff system before?  Edit: Regarding Answer(s):  I selected an answer primarily based on good detail and a solid answer to the question I asked, but reading the responses gave me some more insight.  Perhaps unsurprisingly, the different systems or tweaked systems seem to apply better to certain situations.  What system works best for my game will depend on the types, variance, and number of buffs I intend to be able to apply.  For a game like Diablo 3 (mentioned below), where nearly any bit of equipment can change a buff's strength, the buffs are just character stats system seems like a good idea whenever possible.  For the turn-based situation I'm in, the event-based approach may be more suitable.  In any case, I'm still hoping someone comes along with a fancy "OO" magic bullet which will allow for me to apply a +2 move distance per turn buff, a deal 50% of damage taken back to the attacker buff, and a automatically teleport to a nearby tile when attacked from 3 or more tiles away buff in a single system without turning a +5 strength buff into its own subclass.  I think the closest thing is the answer I marked, but the floor is still open.  Thanks to everyone for the input. Determining a matrix from its characteristic polynomial Let $A\in\mathcal{M}_{n}(K)$, where $K$ is a field. Then, we can obtain the characteristic polynomial of $A$ by simply taking $p(\lambda)=\det(A-\lambda I_n)$, which give us something like $$p(\lambda) = (-1)^n\lambda^n + (-1)^{n-1}(\text{tr } A)\lambda^{n-1} +  \cdots + \det A$$ Now, how can we obtain the matrix $A$ knowing the characteristic polynomial? How to install clang++? I'm having trouble getting clang++ to work as I compile my code. Specifically, I'm getting a make: clang++: Command not found error.  I've run sudo apt-get install llvm, and also sudo apt-get install build-essential and sudo apt-get update. What do I have to do to get clang++ installed? Traffic (views, visits) is not correctly registered on Site Analytics or Area 51 In the ,  can monitor some things like the total traffic on the site. However, it seems to be broken at the moment, all figures are almost zero. This is how it looks for Meta Stack Exchange (other sites show similar pictures): pgfplots -> contour gnuplot with lualatex instead of pdflatex Can a wizard choose where to place their soul piece when creating a horcrux? What exactly is tested when a fire extinguisher is tagged? I am just curious: the obvious is that the  pressure mass   can be inspected /measured to ensure that the device has not been discharged. Is there anything else that is measured \ checked? Is time dilation an illusion? Why does Type I error always occur in a NHST? Or, is NHST too weak to tell the truth? Recently I have noticed that Type I error of a test is too much persistent. Even if all the values are almost equal (variance is extremely low) and tested to that value (true mean), still there is nearly 5% Type I error of the test. I am frightened and curious to rethink, are the people who have banned (or skeptic about) NHST (null hypothesis significance testing) truly correct? Let's have a look at the following r code:       n=10000 t1err=0 set.seed(1235) for (i in 1:n){    x=rnorm(100, 10, 20)    if (((t.test(x, mu=10))$p.value)&lt;=0.05) (t1err=t1err+1)  } cat("Type I error rate in percentage is", (t1err/n)*100,"%")   It will produce roughly 5% as Type I error. Seems ok. But let's see this:      n=10000   t1err=0   set.seed(1235)   for (i in 1:n){      x=rnorm(100, 10, 0.000001)      if (((t.test(x, mu=10))$p.value)&lt;=0.05) (t1err=t1err+1)    }   cat("Type I error rate in percentage is", (t1err/n)*100,"%")   Each values are extremely close to 10. Still it produces about 5% Type I error. What conclusion we may draw from it? Or, am I missing something? Access network share from within VBScript eg FileSystemObject Emmy Noether's theorem in simpler terms I'd like to understand  and its contents as to what it implies in a bit simpler terms. I am familiar with mathematics unto Calculus 1,2,3 and some linear algebra and group theory. I am familiar with that for each symmetry there is a conservation law.    Now is there a one-one correspondence between these two always? What about laws like conservation of charge?  And conservation of energy?  And conservation of mass?  Considering our view of nature was changed as Special Relativity was introduced and thus, what earlier the theorem would have asserted as mass and energy to be originating from different symmetries then how did we reconcile these two as just one symmetry?