It seems that a hidden variables theory could reproduce Bell's experiments results, preserving locality, as long as it accepts that the superposition state(or at least a faux superposition) existed at $t=0$ of the creation of entangled pair. In this case, the question "where and how were the particles previous to the measurement?" makes sense, but only up to the moment they were defined to be opposites. It could be that, while movement over time for a entangled particle is also described by a wave function, its spin for each vector remains constant over time (when considering the same direction measured), only being uncertain at t=0. There's even a global vertical direction that could be responsible for this: gravity.  Bell, in fact, accepts that $\vec\lambda\cdot\vec p &gt;0$ could properly describe the possibly states for a single particle in a local hidden variable theory. Why can't it be that when the entangled pair is created, this measurement is ran over in a particle, which defines every possible vector for this particle (respecting the probability), then the result is transferred and reversed in the other particle? That's all this local hidden theory would need to accept. In this case, the particles only communicate at $t=0$, and never again.
Bell's inequality theorem, along with experimental evidence, shows that we cannot have both realism and locality.  While I don't fully understand it, Leggett's inequality takes this a step further and shows that we can't even have non-local realism theories.  Apparently there are some hidden variable theories that get around this by having measurements be contextual.  I've heard there are even inequalities telling us how much quantum mechanics does or doesn't require contextuality, but I had trouble finding information on this.  This is all confusing to me, and it would be helpful if someone could explain precisely (mathematically?) what is meant by: realism, locality (I assume I understand this one), and contextuality.  What combinations of realism, locality, and contextuality can we rule out using inequality theorems (assuming we have experimental data)?
Any examples of life forms on Earth that could have survived on Mars or Moon? Do we know of any living forms on Earth that could continue to live on Mars? Best way to return multiple values from a function? The canonical way to return multiple values in languages that support it is often . Option: Using a tuple Consider this trivial example: def f(x):   y0 = x + 1   y1 = x * 3   y2 = y0 ** y3   return (y0, y1, y2)  However, this quickly gets problematic as the number of values returned increases. What if you want to return four or five values? Sure, you could keep tupling them, but it gets easy to forget which value is where. It's also rather ugly to unpack them wherever you want to receive them. Option: Using a dictionary The next logical step seems to be to introduce some sort of 'record notation'. In Python, the obvious way to do this is by means of a dict. Consider the following: def g(x):   y0 = x + 1   y1 = x * 3   y2 = y0 ** y3   return {'y0': y0, 'y1': y1 ,'y2': y2}  (Just to be clear, y0, y1, and y2 are just meant as abstract identifiers. As pointed out, in practice you'd use meaningful identifiers.) Now, we have a mechanism whereby we can project out a particular member of the returned object. For example, result['y0']  Option: Using a class However, there is another option. We could instead return a specialized structure. I've framed this in the context of Python, but I'm sure it applies to other languages as well. Indeed, if you were working in C this might very well be your only option. Here goes: class ReturnValue:   def __init__(self, y0, y1, y2):      self.y0 = y0      self.y1 = y1      self.y2 = y2  def g(x):   y0 = x + 1   y1 = x * 3   y2 = y0 ** y3   return ReturnValue(y0, y1, y2)  In Python the previous two are perhaps very similar in terms of plumbing - after all { y0, y1, y2 } just end up being entries in the internal __dict__ of the ReturnValue. There is one additional feature provided by Python though for tiny objects, the __slots__ attribute. The class could be expressed as: class ReturnValue(object):   __slots__ = [&quot;y0&quot;, &quot;y1&quot;, &quot;y2&quot;]   def __init__(self, y0, y1, y2):      self.y0 = y0      self.y1 = y1      self.y2 = y2  From the :  The __slots__ declaration takes a sequence of instance variables and reserves just enough space in each instance to hold a value for each variable. Space is saved because __dict__ is not created for each instance.  Option: Using a  (Python 3.7+) Using Python 3.7's new dataclasses, return a class with automatically added special methods, typing and other useful tools: @dataclass class Returnvalue:     y0: int     y1: float     y3: int  def total_cost(x):     y0 = x + 1     y1 = x * 3     y2 = y0 ** y3     return ReturnValue(y0, y1, y2)  Option: Using a list Another suggestion which I'd overlooked comes from Bill the Lizard: def h(x):   result = [x + 1]   result.append(x * 3)   result.append(y0 ** y3)   return result  This is my least favorite method though. I suppose I'm tainted by exposure to Haskell, but the idea of mixed-type lists has always felt uncomfortable to me. In this particular example the list is -not- mixed type, but it conceivably could be. A list used in this way really doesn't gain anything with respect to the tuple as far as I can tell. The only real difference between lists and tuples in Python is that lists are , whereas tuples are not. I personally tend to carry over the conventions from functional programming: use lists for any number of elements of the same type, and tuples for a fixed number of elements of predetermined types. Question After the lengthy preamble, comes the inevitable question. Which method (do you think) is best? Is the .minecraft folder hidden on a Mac? I have a friend who is playing with me on a server we've created. Unfortunately, he is on a  Mac whereas I am on a Windows machine. I told him that I should be able to compile and send a .minecraft folder to him so that we can both play. Unfortunately, he can't find his .minecraft folder where it is supposed to be. Is it hidden and how can I unhide it? $$\tan(\theta_1+\theta_2+\cdots+\theta_n)=\frac{S_1-S_3+S_5+\cdots}{1-S_2+S_4+\cdots}$$  where  $S_i$ denotes the sum of product of tangent of angles taken $i$ at a time.     For example,$$\tan(\theta_1+\theta_2)=\frac{S_1}{1-S_2}=\frac{\tan\theta_1+\tan\theta_2} {1-\tan\theta_1\tan\theta_2}$$   (This formula is given  in my textbook with no derivation or background)  How to derive this? A ring without the Invariant Basis Number property I was reviewing my homework and it seems I overlooked something crucial while proving some ring has no Invariant Basis Number property. This is exercise VI.1.12 in Aluffi's Algebra: Chapter 0  The setup: $V$ is a $k$-vector space and let $R = \mathrm{End}_{k}(V)$.   Prove that $\mathrm{End}_{k}(V\oplus V) \cong R^4$ as an $R$-module Prove that $R$ doesn't satisfy the IBN property if $V = k^{\oplus \mathbb N}$.   For the first, I used to the fact that $V \oplus V$ is both the product and coproduct (in $k$-Vect) of $V$ with itself to get the isomorphism. What I just realized is I only showed that the two are isomorphic as groups not $R$-modules. So what would be the $R$-module structure on $\mathrm{End}_{k}(V \oplus V)$?   For the second, I used the fact that $V = k^{\oplus \mathbb N}$ implies $V \cong V \oplus V$ which in turn implies $R = \mathrm{End}_{k}(V) \cong \mathrm{End}_{k}(V \oplus V)$. Again, I just realized that I only showed the latter two are isomorphic as groups.   It may be obvious (and maybe why my professor let it pass?) but I can't come up with a good $R$-module structure that makes the two group isomorphisms $R$-linear.  Edit:  Explicitly, these are the isomorphisms I'm dealing with. Let $\pi_j, i_j$ be the natural projection/inclusion maps of the $j$-th factor resp. and $\psi: k^{\oplus \mathbb N} \oplus k^{\oplus \mathbb N} \to k^{\oplus \mathbb N}$ the isomorphism given by $\psi(e_i, 0)=e_{2i-1}$ and $\psi(0, e_i)=e_{2i}$.  Then the first isomorphism $\mathrm{End}_k(V \oplus V)\to R^4$ is given by $\varphi \mapsto (\pi_1\varphi i_1,\pi_2\varphi i_1,\pi_1\varphi i_2,\pi_2\varphi i_2)$  The second isomorphism $R \to \mathrm{End}_k(V \oplus V)$ is given by $\alpha \mapsto \psi^{-1} \alpha \psi$  The composition doesn't seem to be $R$-linear if I use the obvious $R$-module structure on $R$ and $R^4$. How to scale and shift voltage signal? I want to scale and shift my voltage signal by below relation:  $$ V_{out} = \frac {10} {12} \cdot V_{in} - 10 $$  Indeed my primary voltage is from 0v to 24v and I want to put it from -10v to +10v.  I try to use the below circuit but its gain is always greater than 1.    Image from SLOA097 TI.  Does anyone know how I can achieve my goal? I have ±12v and 5v power lines.  EDIT: According to the innovative solution by scorpdaddy, I thought it would be valuable to edit my question:   High input impedance is required to decrease load effects. How can it be achieved? How can I achieve the more and more accuracy? Does every Abelian group admit a ring structure? Given some Abelian group $(G, +)$, does there always exist a binary operation $*$ such that $(G, +, *)$ is a ring? That is, $*$ is associative and distributive:  \begin{align*} &amp;a * (b * c) = (a*b) * c \\ &amp;a * (b + c) = a * b + a * c \\ &amp;(a + b) * c = a * c + b * c \\ \end{align*}  We also might have multiplicative identity $1 \in G$, with $a * 1 = 1 * a = a$ for any $a \in G$. Multiplication may or may not be commutative.  Depending on the definition, the answer could be no in the case of the group with one element: then $1 = 0$. But the trivial ring is not a very interesting case. For cyclic groups the statement is certainly true, since $(\mathbb{Z}_n, +, \cdot)$ and $(\mathbb{Z}, +, \cdot)$ are both rings. What about in general? Is there some procedure to give arbitrary abelian groups ring structure? When is it best to write "can't" versus writing "cannot"? Are they interchangeable in every situation? How to change the list of file types which Safari will open immediately after download? I would like to change Safari Safe List of type files which open immediately after download. Any ideas how? What exactly happens when a Weeping Angel is quantum locked? Proving that if $A^n = 0$, then $I - A$ is invertible and $(I - A)^{-1} = I + A + \cdots + A^{n-1}$ Can we move "Start a bounty" to a more intuitive location? What is the optimum button size of touch screen applications? There was a not so recent blog post about the ideal button size of touch screen control sizes (sorry there is no link, the website is now a spam trap) that was based on the  form the MIT touch lab that gives some measures of the size and sensitivity of the finger. If found that averages for finger pads are between 10–14mm and fingertips are 8–10mm. This would make 10mm x 10mm a good minimum touch target size.  As the author of the blog pointed out, this ideal contact size is much larger than the suggested size for Windows and Apple devices that is in their design guides. I am wondering why there is such a difference, and whether it makes more sense to increase the size of controls on touch screen applications, or if this is not really a usability issue at all? Also, it seems that many mobile websites and apps seem to completely ignore these principles. I am sure the screen size poses a limitation to how big the interface controls can be, so is this simply a trade-off between being able to fit in as much information versus how easy it is to interact with the control? It feels as if the stylus is not something that is in much use. A similar question has been raised on the minimum/smallest size that you can get away with, but I would consider this to be a different question to what the optimum size should be, because the smallest size accommodates the minimum usability requirement, whereas the optimum size caters for the best usability requirement. Archived link of the blog (original domain shows spam)  And here is another reference it to (just to show that it is not the problem with the link but the website () UPDATE #1: another new reference that has a very precise figure of a minimum of 44 x 44 points, not sure how it is derived exactly but interesting (). Read Section 2. Provide Large Clickable Elementss. UPDATE #2: another reference, this time from  citing touch target size to 9 mm square or greater (48x48 pixels on a 135 PPI display at a 1.0x scaling plateau). They also suggest avoid using touch targets that are less than 7 mm square. UPDATE #3: something from the  that I referenced recently citing 44 x 44 points. UPDATE #4: NN/g has published an article on  and the information/conclusion remains unchanged. UPDATE #5 A really nice write-up on the need to enhance the  of common UI elements, citing the  that the &quot;minimum target size for touch or mouse should be 44×44 pixel&quot; (2.5.5 - Target Size - Level AAA). What are the advantages of Light Application objects I was hoping that the introduction of "Light Application object" would do something about the data storage of junction objects, but it doesn't seem to do anything about it.  The  seems limited to why I would use a "Light Application object", what are the advantages? I have heard a few times that one way of describing quantum computers is that they essentially use the computing power of their counterparts in alternate realities that they access through superposition. My first question is, of course,    is this actually an accurate description of how quantum computers work, or just a misrepresentation?   Also, if it were to be assumed true and taken literally, then presumably all the possible outcomes of any given computation done would be experienced by each of these alternate realities. I have a few questions regarding the implications of this:   Would it be correct to say that our universe essentially "spawns" these alternate realities to use for each computation, or that they all exist simultaneously in a higher dimension and only come into contact via the initiation of superposition by each computer at the same time? Would it be possible to create contact between universes via this connection? I'd expect that, if it was, it would be entirely random, and so not only would not cause any information transfer, it also would be completely undetectable, because the deviations would be within the expected probabilities of error rates - but is there any way to circumvent this? Show the title character limit when writing a title I just ran into this while writing .  [I originally wrote:     you need a minimum of X reputation to comment on questions that are not your own, and on answers that are not to your own questions   But had to shorten it to:     you need a minimum of X reputation to comment]   First, it didn't tell me that I'd gone over the character limit for the title until after I had already tabbed out of the title field.  Second, it didn't tell me how much I'd gone over or anything. I had to copy-paste the text I'd written into my text editor in order to get some real-time feedback on the exact character count, to help figure out how to best shorten it. For a physics major, what are the best books/references on Greens functions for self-studying?  My mathematical background is on the level of Mathematical Methods in the physical sciences by Mary Boas. Find the limit $ \lim_{x \to \infty} \frac{e^{x}}{(1+\frac{1}{x})^{x^{2}}} $ Find the limit $$ \lim_{x \to \infty} \frac{e^{x}}{(1+\frac{1}{x})^{x^{2}}}. $$  I know that the limit is of indeterminate form type $\frac{\infty}{\infty}$, but it seems using L'Hopital's rule directly does not help here.  Do I somehow use the fact that $\lim_{x \to \infty} (1+\frac{1}{x})^{x}=e$? !! Do not use the future tense after unless   [✓]  I won't go unless you go   [✘] [I won't go] unless you will go. [✘]   Why not? Even if the above is prescriptive, what may be some reasons? In my separate question , I quote: unless = if not. If not can precede future tense; so why can't unless?  Footnote:  motivated this. Blender Cycles Bake Not Looking Like Object in Rendered View I have a couch in Blender that looks like this in rendered view.    I baked it with a combined texture and this is what it came out to be.    This is what it looks like added to the mesh.    Why does the view with the baked image not look exactly like the rendered view?  Does it have something to do with my mesh and faces?  This is my mesh.    This has never happened to me before as I baked a couple things before this which came out perfectly.