How to install mysql connector
Integrating MySQL with Python in Windows
How to solve a difference equation? So I have the difference equation $4y_{k+1} = 2y_k$.  I need to solve this equation, first with the initial condition $y_0 = 0$ and then with the initial condition $y_0 = 2$.  The professor wrote on the answer sheet that applying iteratively the recurrence relation it is easy to check that:  $y_k= (1/2)^k y_0 $.  Since I missed the lecture where he explained this, can somebody explain how to come about with this question?  I imagine that once I get the equation, then the initial condition $y_0 = 0$ gives us the particular solution $y_k = 0$. Thus the other initial condition $y_0 = 2$ is verified by the solution $y_k = 2(1/2)^k$.  Any help on how to get to the equation $y_k= (1/2)^k y_0$ will be appreciated. How to show that $\log (\frac{2a}{1-a^2}+\frac{2b}{1-b^2}+\frac{2c}{1-c^2})= \log\frac{2a}{1-a^2}+ \log \frac{2b}{1-b^2}+ \log \frac{2c}{1-c^2}$ If $\log (a +b +c) =\log a+\log b+\log c$ then show that $$\log \left(\frac{2a}{1-a^2}+\frac{2b}{1-b^2}+\frac{2c}{1-c^2}\right)= \log\frac{2a}{1-a^2}+ \log \frac{2b}{1-b^2}+ \log \frac{2c}{1-c^2}$$  Trial: If put $a=\frac{2a}{1-a^2}$ and the similar we are done. Can we do this in this way? Otherwise how to do. Please help. Is there a more complete way of proving this theorem (that I devised)? OR does this theorem already exist and have a name? Last night, I was working on re-solidifying my understanding of parametric equations, and came up with a neat trick for evaluating certain definite integrals. I'm not sure if it exists as a formal theorem, though I refuse to believe I'm the first to notice such a relation. The 'theorem' is a useful trick to evaluate the definite integral of a continuous function over an interval where the function is either strictly increasing or strictly decreasing, given that the inverse of the function is 'nicer' to work with than the function itself. My idea goes as follows:  If $y = f(x)$ is a continuous function over the interval $I$, and either:  $f'(x) &gt; 0$   OR   $f'(x) &lt; 0$   for all values of $x \epsilon I$, then:  $\int_{a}^{b}{f(x)dx} = bf(b) - af(a) - \int_{f(a)}^{f(b)}{f^{-1}(x)dx}$  where $a,b \epsilon I$  Quasi-proof:  Clearly, if a function is strictly increasing or decreasing, it is injective. This is clear because no two values in the domain can correspond to a single value in the range. I feel intuitively that it must also be surjective (though I'm not entirely sure how to prove this part). Assuming it is both injective and surjective over $I$, the function must have an inverse.   If we were to draw two rectangles out of our bounds, they will have areas:  $af(a)$ and $bf(b)$, so that   $bf(b) - af(a)$ will be the area of the region we are observing.  This area is equivalent to the sum of the areas connecting the curve $f(x)$ to the $x$-axis and to the $y$-axis. Thus:  $bf(b) - af(a) = \int_{a}^{b}{f(x)dx} + \int_{f(a)}^{f(b)}{f^{-1}(x)dx}$  Solving for $\int_{a}^{b}{f(x)dx}$, we get:  $\int_{a}^{b}{f(x)dx} = bf(b) - af(a) - \int_{f(a)}^{f(b)}{f^{-1}(x)dx}$  A particularly illuminating example is evaluating:  $\int_{a}^{b}{ln(x)dx}$  $ln(x)$ is strictly increasing over any interval, so the 'theorem' applies.  $\int_{a}^{b}{ln(x)dx} = bln(b) - aln(a) - \int_{ln(a)}^{ln(b)}{e^xdx}$  $\int_{a}^{b}{ln(x)dx} = ln(b^b/a^a) - (e^{ln(b)} - e^{ln(a)})$  $\int_{a}^{b}{ln(x)dx} = ln(b^b/a^a) + a - b$  and so we've evaluated the integral without the need of integrating the function itself (by method of integration by parts), but rather by integrating its inverse, which is trivial to evaluate.!  Response to Jason DeVito's answer: [This image was taken from   () Before upgrading to 14.04 when I clicked on a file in the dash, it opened the file.   This still happens on my 12.04 machine.  In 14.04 when I click on a file in dash search results, it offers me 3 options: 'Email', 'Open' and 'Show in Folder'.  How can I get back the previous behaviour - i.e. to immediately open the files on the first click? I have installed GCC compiler by installing the build-essential package.  After the installation I wrote a simple C program. I tried to run it with the following command:  gcc First.c ./a.out   but I'm getting a bash: ./a.out: Permission denied message. I don't know what to do now. My Ubuntu 16.04 keeps disconnecting from the Wi-Fi (eduroam). Why? I haven't noticed any pattern and I'm sure the Wi-Fi works perfectly because I have no problems with the same computer using Windows 10. The Wi-Fi doesn't turn off though, it simply disconnects from the network and reconnects after about a couple of minutes.  I ran this and got:  lspci -knn | grep Net -A2 03:00.0 Network controller [0280]: Qualcomm Atheros AR9462 Wireless Network Adapter [168c:0034] (rev 01) Subsystem: AzureWave AR9462 Wireless Network Adapter [1a3b:2234] Kernel driver in use: ath9k Kernel modules: ath9k   Also running dmesg  Among the many messages the last one is:  [ 3818.806656] wlp3s0: Limiting TX power to 5 dBm as advertised by 00:23:eb:39:28:21   Any idea why this is happening?  Thank you. Is "tri-quarterly" a real English word meaning 3 times a year? Is "tri-quarterly" a real English word meaning 3 times a year? Are there any other words that mean 3 times a year? Can someone provide a proof that every quadratic residue, when working in $\mathbb Z_p$, where $p$ is a prime, has exactly two roots? Indeed, there cannot be only one root as for any $a^2$, we know $a$ and $-a$ are both roots. So then we need only show that there cannot be more than two- does this follow from some argument for the number of roots a quadratic can have modulo a prime? I'm not well-acquainted with the theorems pertaining to prime moduli. A good reference on optimal control theory Ok, so I am reading about decision making and I came across this subject. Fortunately it has a  , but the point is I want to see some examples, and learn to solve regular problems of this field. Of course I can go for the book by Pontryagin but then I don't think that would be an educative book. Does anyone has a suggestion? I would be happy if it also covers LQ problems. P.S. I studied math, therefore I appreciate a book with rigorous approach to the subject. I don't think I ever understood the rationale behind this.  I get that the dot product $\mathbf{a} \cdot  \mathbf{b} =\lVert \mathbf{a}\rVert \cdot\lVert \mathbf{b}\rVert \cos\theta$ is derived from the cosine rule. (Do correct me if I'm wrong.)  However, I never really understood why $\mathbf{a} \times  \mathbf{b} =\lVert \mathbf{a}\rVert \cdot\lVert \mathbf{b} \rVert \sin\theta$. Is there a proof for this?  In addition, how does the vector cross product lead to orthogonal vectors?  I remember learning that they do and I know how to solve equations using the respective formulas, but I never got why is it so. Cancel misclicked flags Can an observed event in fact be of zero probability? Can an observed event in fact be of zero probability?  Of course, I know that there exist non-empty events of zero probability. Mu question is the reverse: given that we have observed an event (and we have no other information about it, just the fact that it has been observed), is it possible that the event has in fact zero probability? Or does an observation necessarily mean that the probability is strictly positive?    Example context for the question:  Suppose $x_i$ (countably many) are i.i.d on $[0,1]$, but we do not know the distribution they come from. It may be uniform on $[0,1]$, may be discrete, may be any legitimate distribution (discrete or continuous). Just imagine we have some sort of a machine that shows us one by one a list of randomly drawn numbers between $0$ and $1$. We are comparing the observed numbers one by one to some special number chosen beforehand, for example $\frac12$. Now, given that at some iteration we have observed that special number at least once, does that mean that $\frac12$ has some positive probability under that (unknown) distribution? And if the probability can be zero, can we nevertheless say that we will necessarily observe $\frac12$ again later, if we continue the experiment ad infimum?  Also, disregard the "real world limitations" such as an inability to produce truly uniformly distibuted numbers, or rounding errors or any such thing. I would like to use multiple .bib files for my article, such as file1.bib, file2.bib. However, I would like my references not to be separated (such as &quot;Primary sources&quot;, &quot;Secondary sources&quot;, etc.). These bibliographies just need to be merged into one. One way to do this is to copy-paste these .bib files into one .bib file, and this would solve the problem. However, I would like to keep it split into several .bib files, by topic (file1.bib contains articles on topic1, file2.bib contains articles on topic2, and so on). I tried to do \bibliographystyle{alpha}  \bibliography{file1} \bibliography{file2}  but this just gives me the same bibliography twice! On the old style user pages such as mathoverflow I noticed that the long comment on the right is floating lower today than it did previously (as recently as yesterday). I don't know if we care, but I noticed it. Why are my villagers attempting to breed (they have the red hearts) but then stop breeding suddenly and gray "angry" particles pop up? I set up a villager breeder and set down 4 beds with 2 villagers in the middle and a carrot farmer supplying them with carrots. The first time both of those villagers bred the breeder worked perfectly fine. After that, however, the gray angry particles popped up every time they would try to breed (they would engage in breeding at a normal pace, just could not actually produce a baby). I even killed the one villager they produced in case it took a bed and even then, the two villagers would not breed anymore. Is there something I have done wrong? This seems odd considering they were only able to breed once and even by eliminating their offspring the villagers could not breed again. Edit: version is 1.16.4 As a mod when should I vote to close content that is not flagged? Which is correct?        She was told that it was against the law to steal.      Or        She was told that it is against the law to steal.        Background information: The original context is that I was relating an incident via email, where my colleague told a customer that some of our actions were part of the organisation's policy. The policy is still valid, but the statement I used to relate the incident in my email went like this:    "Mr Jay was informed that it was our policy to do this."   And I received feedback that it should go like this instead:    "Mr Jay was informed that it is our policy to do this." I am writing out some parameters to JSON files using pythontex, but would like to add context information in form of section numbers (for example \thesection or \thesubsubsection, perhaps also the page number or other counters).  Is there a straightforward way to pass the values of the section counters to Python?   Josef A long time ago, I remember hiding some really personal files on my computer. Something that nobody could ever find, so I buried them down pretty darn good.  Now to my problem, I don't remember where.   Do anyone know a way to scan a drive and list all hidden folders and/or files?   I know how to view hidden files in Windows, but as far as I know, there is no way to search for them? I've been looking for a program, but I couldn't find one - and now I'm here, asking the experts. ;)  I'd really appreciate any tips. What is data blending? This term appears frequently in the .  Is blending a specific method in data-mining and statistical learning? I cannot get a relevant result from google.  It seems blending is mixing up outcomes from many models and resulting in a better result. Is there any resource that helps me knowing more about it?