I am writing an article related to Computer Vision but to complete the maths under the technique I should proof the following equality.   $$\frac{\mathrm d}{\mathrm dx}(h*f) = \left(\frac{\mathrm d}{\mathrm d x}h \right)*f$$  I'm not mathematician and I'm not used to convolution analysis. So, How can I proof it?
Assume that $f(x),g(x)$ are positive and are in $L^1$. Moreover, they are differentiable and their derivative is integrable. Let $h(x)=f(x)*g(x)$, the convolution of $f$ and $g$. Does the derivative of $h(x)$ exist? If yes, how can we prove that $$ \frac{d}{dx}(f(x)*g(x)) = \left(\frac{d}{dx}f(x)\right)*g(x)$$  Thanks
Find all continuous functions satisfying $\int_0^xf=(f(x))^2+C$ for some constant $C \neq 0$. Find all continuous functions $f$ satisfying   $$\int_0^xf=(f(x))^2+C$$ for some constant $C \neq 0$, assuming that $f$ has at most one $0$.  I have a question about the solution to this problem. It says that   Clearly $f^2$ is differentiable everywhere, its derivative at $x$ is $f(x)$ (I'm assuming that this is because the left hand side is differentiable, the right hand side must be differentiable as well). So $f$ is differentiable at $x$ whenever $f(x) \neq 0$, and  $$f(x)=2f(x)f'(x),$$ so $f'(x) = \frac{1}{2}$ at such points. Thus, $f= \frac{1}{2}x+b$ for some $b$ on any interval where it is non-zero; if $f$ has a zero, this gives two possible solutions, with two possible values of $b$, but since $f$ is assumed continuous, they must be the same. So we need $$\int_0^x(\frac{1}{2}t+b)dt=(\frac{1}{2}x+b)^2+C$$  so we must have $b=\sqrt{-C}$ or $-\sqrt{-C}$, leading to two solutions for $C \lt 0$.  I don't understand the bolded statements. First, how does the differentiability of $f^2$ guarantee that $f$ is differentiable, and why only at $x$ where $f(x) \neq 0$?  Next, why are there two possible solutions if $f$ has a zero, but since $f$ is continuous they must be the same?  Finally, does the last statement mean that if $C \gt 0$, then there is no solution, since we're restricted to real numbers here?  I would appreciate it if anyone answers my questions. What is a good replacement for a cable end cap? I was adjusting the tension of my front derailleur cable and managed to pop off the cable end cap. The cable is now quite frayed at the end and I think it would be difficult to attach another cap.  What would work as a suitable replacement for a cap? I've read that perhaps superglue or duct tape could work? Antisymmetric Relations $ \int_1^2\int_1^2 \int_1^2 \int_1^2 \frac{x_1+x_2+x_3-x_4}{x_1+x_2+x_3+x_4}dx_1dx_2dx_3dx_4 $ Contact from wizpert.com via stackoverflow? Well today as I was going home I got a strange email, in my private inbox (email associated with S.O.) asking me if I had solved a question on S.O. that I didn't ask. To be more precise I had given an answer.   After reading:       and several other email related issues on S.O. I decided to let you all know about this because probably I am not the only one.   So the mail subject was:     Re: Deleted facebook account - Cannot access my apps?   which refers to   And the body (masked last name of sender):      Hi Jimmy,      Just wondering if you ever got this question answered.  If not, maybe   a brief chat with a html wizpert could help: link   .      Good luck getting this resolved!      Cheers,      Michael      ------------------------- Michael W%^%^GF Founder, CEO @ Wizpert   Now I know that my email address is not visible and I read about the MD5 regarding gravatar but I don't think this is the case. I thought about some methods and sure it's possible via metadata, github, etc for someone to get my mail, but I do not dare to think that this has reached an automated level, so that some service could easily get the email of S.O. members.   I didn't get annoyed by the message, since the CEO took the time to contact me :-p but someone else could get annoyed by these kinds of messages, since they come in the important inbox of gmail.   What is more strange is that Wizpert seems to be a nice site and also is active in the Careers 2.0 if I am not mistaken.   Another thing that crossed my mind is that since Wizpert needs unanswered questions to promote their services, possibly this is an action of Wizpert trying to attract people into their services.   Well that's it, I hope I did good to let you know (maybe I did't include the correct tags). FICA was not withheld from my paycheck Hello personal finance,  I was filing my taxes for 2016 and I just found out that I transitioned from Non-Resident-Alien to Resident Alien around September last year by passing substantial presence test. After doing some research, I have found out that I qualify as a resident alien for the whole tax year (from January 2016).   Unfortunately, my employer did not know about this change in status, so for all of 2016 no Social Security or Medicare taxes were withheld.   I have found out that I need to correct this, and I need help figuring out a few things.    Should I have a W-2 re-issued? I spoke to two accountants and one said I should, the other said I shouldn't. I don't know which one to believe.  Who pays for the FICA I should have paid last year? Is it my employer's problem to fix or mine?  Should I file the taxes with the old or the new W-2?  As best as I understand it, the new W-2 will be written as if I made this contributions. So there will be a discrepancy between what I actually earned and what the W-2 says. How should I handle this discrepancy? Declare it as extra income? Or claim it back in case I earned less than I should have? I tried many ways to make a bigger screen in VirtualBox because I do not like the small size of the guest window? Is there is a way to make it big? I'm studying the Darboux definition of integrability, which I completely explained . There's an exercise that asks me to prove that the Darboux Integrability is equivalent to Riemann Integrability, but this Riemann integral is defined as the following:  It first defines a 'pointed' partition (I don't know how to say it in ensligh) by the following: a 'pointed' partition $[a,b]$ is a pair $P^*=(E,ξ)$, where $P=\{t_0, t_1, \cdots, t_n\}$ is a partition of $[a,b]$ and $ξ = (ξ_1, \cdots, ξ_n)$ is a list of $n$ chosen numbers such that $t_{i-1}\le ξ_i\le t_i$ for each $i=1,\cdots ,n$.  Now, the Riemann Integral is defined as:  $$\sum(f,P^*) = \sum_{i=1}^n f(ξ_i)(t_i-t_{i-1})$$  (I didn't understand the notation for the left hand side of the equation, by the way)  Finally, I'm asked to prove the following: given $f,g:[a,b]\to \mathbb{R}$ integrable functions, for the entire partition $P=\{t_0, \cdots, t_n\}$ of $[a,b]$ let $P^* = (P,ξ)$ and $P^{\#} = (P, η)$ be pointed partitions of $P$, then:  $$\lim_{|P|\to 0}\sum f(ξ_i)g(η_i)(t_i-t_{i-1}) = \int_a^b f(x)g(x) \ dx$$  I guess here I need to prove that the Riemann Integral of the product of two functions if the darboux integral of $f(x)g(x)$, but it seems too obvious, I just need to verify that $f,g$ are integrable, then their product is too, isn't it? I'm pretty sure this should be a hard question. Is there another interpretation that I'm missing? Show that the series $$\sum\limits_{n=2}^{\infty} \frac {(n^3+1)^{1/3}-n}{\log n}$$ converges.   I showed it using Abel's theorem and limit comparison test. Any other simpler method? Lots of new batteries (for mobile devices, MP3 players, etc) have connectors with 3 pins. I would like to know what is the purpose of this and how should I use these three pins?    They are usually marked as (+) plus, (-) minus, and T. Domain Controller etiquette after a crash between Server 2003 & Server 2008 I have 2 domain controllers on a non-Internet connected network running Windows Server 2008 R2. They running RAID 5 on a 8 Hard Drive server. Sometime last week the Boot Sector became corrupt on DC1 and I was unable to recover the server. I am not going to have to rebuild the server. I have been reading about the changes from Server 2003 to Server 2008. DC1 was the main domain controller and now that it won't but it is all falling to DC2. I know that in previous versions of Windows Server you had to make on the primary and one the secondary. Since DC2 does not see DC1 anymore is there anything I need to do to DC2 as it is now the one in control? I am thinking some kind of meta data clean up but I can't find any hard information on what procedure to do. I understand as of Server 2008 domain controllers are on an equal footing and provide replication to each other so they should be the same correct? Does the closedness of a subgroup $H$ of $G$ imply that for $h\in H, g\in G \setminus H: hg \not \in H$?  I'm new to abstract algebra and this would appear to be a useful result. However, by definition, a subgroup is nearly closed, and I don't think this directly implies that for any element $g$ from $G$ not in $H$, the composition $gh \not \in H$.  Is this in fact a true statement? Embedding videos and animations I'm aware that it's possible to embed some kinds of videos and animations in LaTeX files (using the beamer or prosper classes) but I couldn't locate good documentation on this. My web searches returned a list of scattered forum questions and answers.   Could anybody give a link to some help or documentation page that systematically deals with the various ways of embedding videos and animations, the file types that work, etc.? barchart with pgfplot - missing bar I am using PGFplot to make a barchart. There are 4 entries in my csv file and unfortunately I have only 3 bars in my figure.   my LaTeX code:  \begin{figure}     \centering     \begin{tikzpicture}          \begin{axis}[             x tick label style={/pgf/number format/1000 sep=},                 xlabel=Sequence,             ylabel=Heigth,                 ymin=0.7,             enlargelimits=0.05,             legend style={at={(0.5,-0.15)},                 anchor=north,legend columns=-1},             ybar interval=0.7,             width=\textwidth,             ]             \addplot table [col sep=comma,trim cells=true,y=height] {numericaltests/data/test.csv};              \addplot[red,sharp plot,update limits=false]                 coordinates {(-1,1.00) (34,1.00)}                 node[above] at (axis cs:15,1.00) {};             \legend{Ca2Fe2As2}         \end{axis}     \end{tikzpicture}     \caption{Sequence}     \label{fig:sequenceHeight} \end{figure}   my test.csv file:  sequence,height 1,2.0 2,1.5 3,1.8 4,2.4   Barchart: Age replacement policy for hard disks and SSDs at servers I'm planing an age replacement policy for our storage and servers. Most of them are for DBs and some for images (static content) so yes, they have an huge I/O everytime.  Also, we use Samsung 840 Pro SSDs for the RAID Controllers (PERC H700i) as CacheCade.  Are you guys managing the replacement of old hard drives and solid state drives? How to handle a lost KeyStore password in Android? How do you select the fastest mirror from the command line? I want to update my sources.list file with the fastest server from the command line in a fresh Ubuntu Server install. I know this is trivially easy with the GUI, but there doesn't seem to be a simple way to do it from from the command line? Why does my object not show up? Pairwise independence of Random variables does not imply indendence Show by a counterexample that for a family $(X_i)_{i\in I}$ of random variables the independence of all pairs $(X_i,X_j)$ with $i,j\in I, i\neq j$ does not imply the independence of the family (It is enough to consider $\mbox{card}(I)=3$.).      Hello, I am searching for a simple example. The professor gave the hint that one might consider random variables which only can take two different values, but I do not know exactly what is meant.  By the way:  We call the family $(X_i)_{i\in I}$ of random variables independent, if $(\sigma(X_i))_{i\in I}$ is independent, whereat we call a family $(\mathcal{E}_i)_{i\in I}$ with $\mathcal{E}_i\subset\mathcal{A}$ independent, if for any finite subset $I_0\subset I$ and any choice $E_i\in\mathcal{E}_i, i\in I_0$, it is $$ \mathbb{P}(\bigcap_{i\in I_0}E_i)=\prod_{i\in I_0}\mathbb{P}(E_i). $$ Calculating ratio between "rows" in ArcGIS field calculator I have a problem similar to  but that is for QGIS whereas I have to perform it in ArcGIS. I have to calculate the Ratio between rows (sequentially, i.e., row1/row2, row2/row3 and so on) of a specific field based on the ID field and have to assign the results to a separate field. Here is the code and the error message:  def CalcRatio(ID,Frequency):     vDict = {}     with arcpy.da.SearchCursor(&quot;Test&quot;,['Frequency'],'ID = {}'.format(ID)) as sCur:     for Row in sCur:     vDict[Row[0]]=Row[1] # Dict[id] =Frequency     try: value = ( vDict[1] / vDict[2])     except: value = -1     return value  Message: ERROR 000539: Error running expression: CalcRatio( 1 , 50680) Traceback (most recent call last): File &quot;&lt;expression&gt;&quot;, line 1, in &lt;module&gt; File &quot;&lt;string&gt;&quot;, line 5, in CalcRatio IndexError: tuple index out of range  Here is the data OID Id  Frequency 0   1   50680 1   2   49740 2   3   48620 3   4   48300 4   5   48004 5   6   45600 6   7   44384 7   8   42720 8   9   41890 9   10  39700 10  11  38530 11  12  38207 12  13  38106 13  14  37303 14  15  37205 15  16  36527 16  17  35301 17  18  31674 18  19  28890 19  20  24612