Consider following problem from Evans  Assume $\Omega$ is connected. A function $u\in H^1(\Omega)$ is a weak solution of $$(1)\quad\left\{\begin{array}{ll} -\Delta u(x)=f &amp; x\in\Omega\\ \frac{\partial u(x)}{\partial n}=0 &amp; x\in\partial\Omega\end{array}\right.$$ if for all $v\in H^1(\Omega)$ there holds $$\int_\Omega\nabla u\cdot\nabla v\,dx=\int_\Omega fv\,dx \,.$$ Suppose that $f\in L^2(\Omega)$. Prove $(1)$ has a weak solution if and only if $$\int_\Omega f\,dx=0.$$  It is quite difficult. Do I need to use Lax Milgram. How would I go about problem?
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.
How to convert an instance of std::string to lower case I want to convert a std::string to lowercase. I am aware of the function tolower(). However, in the past I have had issues with this function and it is hardly ideal anyway as using it with a std::string would require iterating over each character. Is there an alternative which works 100% of the time? Embed vimeo video into the body field After mis-reading this question:  I found myself wondering if there are other non-weapon items such as tools that utilize lightsaber technology.  Example: the lightsaber bread knife For all $a, m, n \in \mathbb{Z}^+$,   $$\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$$ Allow me to get notifications for new Meta posts, like mods do I'm currently helping out on . One of the most valuable things on new beta sites, especially private/early public betas, is Meta participation - see :  Treat Meta Questions Like Gold It’s not unusual for your meta site to receive significantly less traffic than the main site. The problems begin when you meta starts to get ignored by the community and ignored by the moderators. It is particularly disturbing when users reach out in meta and starts talking about improving the site or asking for help… only to be ignored; No follow-up, only deafening silence  Because of this, Diamond mods get a notification in their inbox for each new question on their Meta, helping ensure that each meta post is addressed. However, wouldn't it be great if Diamond mods weren't the only ones that could do this? The post does say that the community needs to be seeing these questions too. As an involved user, I would really like to be notified of these new questions - it would increase the good I could do the site. How about this?:  Of course, it would be opt-in, not opt-out, and per-site. A feature like this would greatly improve the good that enthusiastic site-builders can do for the site, and would, most likely, increase the overall chances of a site's success. Is there a way to install fonts on wine?   I'm looking to install fonts so i can use them to render text differently on various apps. Specifically myriad. Preserving signatures of decorated functions Suppose I have written a decorator that does something very generic. For example, it might convert all arguments to a specific type, perform logging, implement memoization, etc.  Here is an example:  def args_as_ints(f):     def g(*args, **kwargs):         args = [int(x) for x in args]         kwargs = dict((k, int(v)) for k, v in kwargs.items())         return f(*args, **kwargs)     return g  @args_as_ints def funny_function(x, y, z=3):     """Computes x*y + 2*z"""     return x*y + 2*z  &gt;&gt;&gt; funny_function("3", 4.0, z="5") 22   Everything well so far. There is one problem, however. The decorated function does not retain the documentation of the original function:  &gt;&gt;&gt; help(funny_function) Help on function g in module __main__:  g(*args, **kwargs)   Fortunately, there is a workaround:  def args_as_ints(f):     def g(*args, **kwargs):         args = [int(x) for x in args]         kwargs = dict((k, int(v)) for k, v in kwargs.items())         return f(*args, **kwargs)     g.__name__ = f.__name__     g.__doc__ = f.__doc__     return g  @args_as_ints def funny_function(x, y, z=3):     """Computes x*y + 2*z"""     return x*y + 2*z   This time, the function name and documentation are correct:  &gt;&gt;&gt; help(funny_function) Help on function funny_function in module __main__:  funny_function(*args, **kwargs)     Computes x*y + 2*z   But there is still a problem: the function signature is wrong. The information "*args, **kwargs" is next to useless.  What to do? I can think of two simple but flawed workarounds:  1 -- Include the correct signature in the docstring:  def funny_function(x, y, z=3):     """funny_function(x, y, z=3) -- computes x*y + 2*z"""     return x*y + 2*z   This is bad because of the duplication. The signature will still not be shown properly in automatically generated documentation. It's easy to update the function and forget about changing the docstring, or to make a typo. [And yes, I'm aware of the fact that the docstring already duplicates the function body. Please ignore this; funny_function is just a random example.]  2 -- Not use a decorator, or use a special-purpose decorator for every specific signature:  def funny_functions_decorator(f):     def g(x, y, z=3):         return f(int(x), int(y), z=int(z))     g.__name__ = f.__name__     g.__doc__ = f.__doc__     return g   This works fine for a set of functions that have identical signature, but it's useless in general. As I said in the beginning, I want to be able to use decorators entirely generically.  I'm looking for a solution that is fully general, and automatic.  So the question is: is there a way to edit the decorated function signature after it has been created?  Otherwise, can I write a decorator that extracts the function signature and uses that information instead of "*kwargs, **kwargs" when constructing the decorated function? How do I extract that information? How should I construct the decorated function -- with exec?  Any other approaches? For all $x,y∈\Bbb{R}$ define that $ x\equiv y$ if$ x^2=y^2$ For all $x,y\in\Bbb{R}$ define that $x\equiv y$ if $x^2=y^2$ . Then $\equiv$ is an equivalence relation on $\Bbb{R}$ , there are infinitely many equivalence classes, one of them consists of one element and the rest consist of two elements.  Solution  True. To show that $\equiv$ is reflexive we need to show that $\forall x\in\mathbb{R} :x\equiv x$. Let $x\in\mathbb{R} , x\equiv x \mbox{ if } x^{2}=x^{2}$, which is obvious.  $[x] ={y\in\mathbb{R} |x\equiv y} =[0]={y\in\mathbb{R}|0\equiv y}={0}$. Hence $y^{2} =0^{2} =0$ which implies that $y=0$.  Can anyone please give me correct answer to this question. I am confused. I don't understand the difference a ARMA and a GARCH process.. to me there are the same no ?  Here is the (G)ARCH(p, q) process  $$\sigma_t^2 =      \underbrace{     \underbrace{      \alpha_0       + \sum_{i=1}^q \alpha_ir_{t-i}^2}     _{ARCH}        + \sum_{i=1}^p\beta_i\sigma_{t-i}^2}     _{GARCH}$$  And here is the ARMA($p, q$):  $$ X_t = c + \varepsilon_t +  \sum_{i=1}^p \varphi_i X_{t-i} + \sum_{i=1}^q \theta_i \varepsilon_{t-i}.\,$$  Is the ARMA simply an extension of the GARCH, GARCH being used only for returns and with the assumption $r = \sigma\varepsilon$ where $\varepsilon$ follows a strong white process? Rsync filter: copying one pattern only Transit Visa for Thailand Im an Indian citizen. Do I need a transit visa in Thailand while going to Angkor Wat from Kolkata (India), and can I stay in Thailand for two days on this transit visa? How can I make it so that each space has its own isolated dock?  Basically what I'm looking for is:   The dock in one space should not show the icon of an app that was open in another space. If I have the same program open in two different spaces, clicking on the icon in one space should not go to the app in the other space.   Is this possible natively, or with a third party program? Minecraft Survival Mode An acquaintance recently posted an image to FB from his recent trip to the JFK Historic Site asking for an explanation of a visible artifact he states was not visible to the naked eye but was visible through the EVF of his camera and in the final image.    The artifact is a series of bright, nearly-vertical bars and a longer, almost horizontal bar superimposed over the lower part of the door frame and the radiator.  I found a very similar artifact in another image of the same room from a blogger which isn't as compressed:    In both cases the artifact seems to be a vertically inverted image of the post and rail outside the saturated window.  From looking at other images of the room available online I know there is spherical glass ornament topping the table lamp whose shape is visible at the left of the photo.  One response to the OP's question is as follows:     RK: It is lens flare. So, even on an iPhone, the camera lens is made   up of multiple lenses. Each designed to focus the image and make sure   it isn't distorted. Remember a lens bends light.  Sometimes, based on   angle and intensity of the incoming light, some stray light bounces   back and forth while traveling through the various elements that make   up what we think of as a camera lens (some of mine have 16 elements   inside a single lens.) Even the best glass is capable of reflecting   some light instead of passing it through the lens elements to the   sensor.  This is what you are seeing. The railing is overexposed, and   some stray light is bouncing around a bit inside the lens.   My response:     DJ: I don't think it's lens flare or ghosting. Those usually form an   image of starbursts, rings, or circles in a row across the image or in   the shape of the aperture; or a veiling glare or haze. This is forming   an image of the light outside minus objects in its path. It is more   likely a direct reflection/projection off a mirror or the glass in a   picture frame elsewhere in the room onto the opposite wall and the   radiator. There are two other clues which seem to confirm this. First,   the image has a gradiated appearance, it is less distinct in the   relatively bright area on the door frame and brighter in the   relatively darker areas behind and on the radiator. You would expect   lens flare to be more evenly specular. Second, the image representing   the light from below the horizontal rail changes angle on the   different faces of the door frame. Unlike a projection, lens flare is   an image created within the lens, the physical shape of the objects it   superimposes do not change its shape.      OP: If that were the case, would I have been able to see said   reflection in the room without the camera? There was no image I could   see with the naked eye. I notice the image in my phone before I took   the pic and did a double, triple and quadruple take before I took it   so I'm sure.      DJ: It would have been visible at the moment the image was taken, but   it may not have appeared nearly as bright and distinct because your   eyes were accommodating for the bright light coming in the window. The   projection could have easily been obscured by you or anyone around you   changing position or moving through the path of the projected   reflection. Also, it's been caught by other people... (see Image 2)   Notice that, in the linked example, the projection is again distorted   by the surface it falls on.   Anyone else want to chime in with an explanation? What are good books for graduates/undergraduates in Astrophysics? There are no book recommendations for Astrophysics here. I will write my own answer, but I am also interested in what are others' views on the question (I will NOT mark my own answer as the best one). Disable back button in android What is the proof that the total number of subsets of a set is $2^n$? This is a silly question but under what conditions is $a^{xy}=(a^x)^y$ true, given all are complex numbers? How can I correct this code to get an answer? How do I use '&' literally in LaTeX?