Solve the initial value problem: $\frac{dy}{dx} = e^{x+y}$, given $y(0)=0$.
Find the general solution of $y'= a^{x + y}$ where y is the function
I had the following two problems:     Find a counterexample for $f_*(A \cap B) \supseteq f_*(A) \cap f_*(B)$ and $ f_*(A-B) \subseteq f_*(A) -f_*(B).$ Where $f_*(X)$ is the image of $X$ under $f$ for some function $f:A \rightarrow B $ and some subset $X \subseteq A$.   I came with the following answer:  Let $ f: \mathbb{Z} \rightarrow \mathbb{Z} $, where $f: x\mapsto 0$; let $A=\{1,2\}$, and $B=\{3,4\}$.     For $f_*(A \cap B) \supseteq f_*(A) \cap f_*(B)$, $$ f_*(A \cap B) = f_*\{ \varnothing\} = \varnothing \hspace{1cm} \text{and} \hspace{1cm} f_*(A) \cap f_*(B)=\{0\} \cap \{0\} = \{0\}. $$ It follows that $ \varnothing \nsupseteq \{0\} $. For $ f_*(A-B) \subseteq f_*(A) -f_*(B)$, $$f_*(A-B)=f_*(\{1,2\})=\{0\} \hspace{1cm} \text{and} \hspace{1cm} f_*(A) - f_*(B) = \{0\} - \{0\} = \varnothing$$ It follows that $ \{0\}  \nsubseteq \varnothing $.   Now (assuming my answer is correct), let's say I want to find more counterexamples or maybe a more general answer (e.g. all the conditions under which the inclusion fails). Other than by trial and error or just intuition, how would I do that?   I know this is a broad question, so I'm specifically asking for methods or advice on how to build counterexamples in elementary set theory (for example, tips on where to begin or rules that apply in general that may be unobvious). Documents with typical LaTeX errors I am giving a LaTeX course and have the idea to let the students find and solve problems within a few documents. Basically they shall learn how to create minimal examples and read error messages.  However, currently I have no document with 'typical' errors around.  If someone has such examples from him- or herself or within this forum I would like to have the code or link. It does not matter if its short or long. It should just be enough to let the students play with it for some minutes. There is a very bad problem with the "Blank screen" function in Ubuntu 17.10. I refer to the setting accessed via Settings->Power->Blank Screen. This seems to actually perform a screen lock, since I need to enter my password after un-blanking the screen. (However, given the behavior described below it's not clear exactly what this function actually does - maybe it just logs you out? maybe it reboots the machine?)  The problem is that a simple screen lock, which one would expect to lock the screen without affecting the state of any applications, in fact closes all applications that are open when the screen lock activates. In other words, if you have the screen lock set to 5 minutes, and you walk away from your machine to get a sandwich or whatever, when you come back and unlock the machine all your apps have closed without saving their state. So far, the only work-around I have found is to entirely disable screen locking in Ubuntu 17.10. My question is, how can this be fixed so that screen locking works properly - that is: the screen locks on schedule after a period of inactivity, and when unlocked all applications that were running are still running?  I am asking a new question for the following reasons:  1) The existing answers to (and comments on)  do not address the issue, and in fact misrepresent the actual problem in ways that make me think the comment writers have not understood what is happening;  2) I cannot usefully address that question because I do not have sufficient reputation to either upvote the question or comment on it, and I cannot answer the question because I have no idea why this problem occurs;  3) I will probably never achieve a reputation of 15, much less 50, on StackExchange. Which is fine, that's not among my life goals.  4) But this question is important and deserves to be addressed. It is a drastically unacceptable bug. Who is / was the "Lord of the Rings"? Who is the Lord, as referred to in the title? Frodo? Sauron? Someone else? Or is it more of an abstract title, not referring to a single person, like "He who wields the One Ring"? What geometric object is given by this equation? What are preimage resistance and collision resistance, and how can the lack thereof be exploited? What is "preimage resistance", and how can the lack thereof be exploited? How is this different from collision resistance, and are there any known preimage attacks that would be considered feasible? Have iPhone 4 with iOS 5.1.1 Trying to update to 6.1.2 (not the latest) via downloaded IPSW but unfortunately getting error:     iphone could not be restored device isn't eligible for the requested   build   After searching for fix all over the internet found some suggestion like putting # in hosts file in line gs.apple.com but not helped.  Have you any idea why I'm getting this error and can't restore iphone? Can you retrain the Magic Initiate spell using the Spellcasting feature? got me thinking about the mechanics of the Magic Initiate feat when you have class levels in the class into which you were inducted.    Among the classes you can pick with this feat are the Bard, Sorcerer, and Warlock, which know a limited number of spells and all have the same statement under Spells Known.     Additionally, when you gain a level in this class, you can choose one of the [class] spells you know and replace it with another spell from the [class] spell list , which must also be of a level for which you have spell slots.   The term "learning" is used both to describe new spells acquired on level-up and the spell acquired through the feat. This would seem to indicate your Magic Initiate spell counts among the spells you know, and is thus eligible for retraining if you learned it as a spell of your class.  An odd change in wording occurs during the Magic Initiate feat description (emphasis mine):     In addition, choose one 1st-level spell from the same list. You learn that spell and can cast it at its lowest level.   This generalization in describing the learned spell's level seems to imply that there exists a scenario where it would not be appropriate to restrict the spell to being cast as a 1st-level spell.  Could one of the charisma casters retrain Magic Initiate into a higher level spell, which they could then cast once per long rest without using a slot?  This would turn an otherwise unexciting feat into, effectively, a Mystic Arcanum for Bards and Sorcerers. This seems too powerful to be intentional,  but the feat description appears to have gone out of its way to permit it. Cancelling upvote on comment? As mentioned on :     For something that can be done so   easily by accident, there should be   the ability to undo mistakes.   I know that I have clicked several times on a comment up-vote without actually wanting to.  I would suggest being able to cancel the up-vote (maybe only for a couple of minutes or even less than a minute, no problem, as long as I can undo a miss-click). Change the size of a math symbol How can I get something like in the figure, where G in the middle is larger than normal letters in math mode? Prove that $\sum_{k=0}^{m}\frac{\binom{m}{k}}{\binom{n}{k}}=\frac{n+1}{n+1-m}$ Prove that $\sum_{k=0}^{m}\dfrac{\binom{m}{k}}{\binom{n}{k}}=\dfrac{n+1}{n+1-m}$.  We know, n > m. From the right side. we have $\dfrac{n+1}{n+1-m}=\dfrac{1}{1-\dfrac{m}{n+1}}$. since n > m. $0&lt;\dfrac{m}{n+1}&lt;1$. Then $\dfrac{1}{1-\dfrac{m}{n+1}}=1+\dfrac{m}{n+1}+(\dfrac{m}{n+1})^2+...$ I don't know what to do next? Did I choose the right path? Derivation of Maxwell's equations from field tensor lagrangian I've started reading Peskin and Schroeder on my own time, and I'm a bit confused about how to obtain Maxwell's equations from the (source-free) lagrangian density $L = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$ (where $F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu$ is the field tensor).  Substituting in for the definition of the field tensor yields $L = -\frac{1}{2}[(\partial_\mu A_\nu)(\partial^\mu A^\nu) - (\partial_\mu A_\nu)(\partial^\nu A^\mu)]$.  I know I should be using $A^\mu$ as the dynamical variable in the Euler-Lagrange equations, which become $\frac{\partial L}{\partial A_\mu} - \partial_\mu\frac{\partial L}{\partial(\partial_\mu A_\nu)} = - \partial_\mu\frac{\partial L}{\partial(\partial_\mu A_\nu)}$, but I'm confused about how to proceed from here.  I know I should end up with $\partial_\mu F^{\mu\nu} = 0$, but I don't quite see why.  Since $\mu$ and $\nu$ are dummy indices, I should be able to change them: how do the indices in the lagrangian relate to the indices in the derivatives in the Euler-Lagrange equations? Textbook approach to modeling non-proportional hazards in the Cox model In Cox models with time varying coefficients, the effect of covariates on the hazard is allowed to change through time. In cases where a coefficient has a linear relationship with time, I am aware of at least two text books ( and an Appendix to ) that advocate for the following model, first in R code:  coxph(Surv(time, death) ~ disease + disease:time)   And now in mathematical notation:  $$ h(t|X) = h_0(t) \exp(\beta_1X + \beta_2X_T) $$ Where $h(t|X)$ is the instantaneous hazard rate at time $t$ given covariates $X$; $h_0(t)$ is the baseline hazard rate; and $T$ is the censoring or event times. $\beta_1$ represents the log hazard ratio for individuals with $T=0$ (which is sort of nonsensical since you can't have $T=0$ in the data), while $\beta_2$ is harder to interpret... $\beta_2$ shows how the log hazard ratio depends on censoring/event times, but that is circular because event times are also the response variable (this point is made in Terry Therneau’s vignette “”).  According to Therneau's vignette, to correctly specify a time varying coefficient, you need to use the following model, first in R code:  coxph(Surv(time, death) ~ disease + tt(disease), tt=function(x,t,...){x*t})   And now in mathematical notation: $$ h(t|X) = h_0(t) \exp(\beta_1X + \beta_2X_T) $$  The difference is that the time interaction is with $t$ in the model rather than with the event times $T$ in the data. The resulting parameters are easy to interpret: they are the intercept and slope of the linear relationship between the log hazard ratio of $X$ and time. This approach seems like the proper way to model a time varying effect.  Can anyone justify or advocate for the first approach to modeling non-proportional hazards in the Cox model? If it is really incorrect (as I suspect it is), what are the practical consequences of using this approach? Prove $f(S \cup T) = f(S) \cup f(T)$ how can I add an overlapped anchor in both overlapped paths automatically How I can add an anchor point for each intersected selected paths in illustrator automatically?  I know smart Guide and add anchor point tool can do the job but it will add just one anchor in the front path only. If I use pathfinder outline or the scissors my paths get cut and I have to re-join them again one by one. Bias and variance in leave-one-out vs K-fold cross validation How do  methods compare in terms of model variance and bias?   My question is partly motivated by this thread: . The answer there suggests that models learned with leave-one-out cross-validation have higher variance than those learned with regular $K$-fold cross-validation, making leave-one-out CV a worse choice.  However, my intuition tells me that in leave-one-out CV one should see relatively lower variance between models than in the $K$-fold CV, since we are only shifting one data point across folds and therefore the training sets between folds overlap substantially.  Or going in the other direction, if $K$ is low in the $K$-fold CV, the training sets would be quite different across folds, and the resulting models are more likely to be different (hence higher variance).  If the above argument is right, why would models learned with leave-one-out CV have higher variance? Is there any intuition or meaning regarding Cauchy-Riemann equations? Holomorphic functions are one of the most beautiful objects in mathematics. However, Cauchy-Riemann equations are a little bit of a mystery for me. What does that mean intuitively when a function satisfyes those equations? I'd like to create a LaTeX document that when rendered into PDF, has forms that can be filled out using Adobe Reader or other such programs. Then I'd like to be able to extract the data. I deliberately would like to avoid using Acrobat for all the usual reasons (non-free, need different versions for different platforms etc).   Can this be done ? How to prove sum of powers property of roots of unity? How does Lucio's aura work vertically? If I play Lucio, can I heal the teammates located at a different height? (roof, upstairs, hovering Pharah... )  In other word is that aura strictly horizontal? Or does it work more like a bubble like Winston's barrier?  If it's horizontal, does it require Line Of Sight or not? (will I heal a nearby ally right behind a wall?)