Can a sorcerer cast the same spell twice if enough spell slots are available?
Can a spell be prepared once and cast multiple times?
The existence of the empty set is an axiom of ZFC or not? I found in the Wolfram MathWorld page of the  that this is one of the Zermelo-Fraenkel Axioms, however on the page about these  I read that it is an axiom that can be deduced from the Axiom of Subsets and the Axiom of Foundation (or Axiom of Regularity), so, the existence of the Empty Set is an axiom of ZFC or not? We hold an L1 visa (Citizens of India) and are planning a cruise to the Bahamas this summer. Are there any Visa requirements to enter Bahamas. From Awodey: In any category $\mathsf{C}$,  an object $0$ is initial if for any object $C$ there is a unique morphism $0 \to C$,  an object $1$ is terminal if for any object $C$ there is a unique morphism $C \to 1$.  First I was puzzled by the fact that we can do mappings from a zero set to any other set in a category $\mathsf{Sets}$. But then after I accepted it conceptually, I do not understand why there is no unique morphism to an empty set from every object in a category $\mathsf{Sets}$. Is this because we cannot map a non-empty set to an empty set (since our mapping is total) although we can map an empty set to a non-empty set?  Please, note, I am familiar with the proposition saying: Initial (terminal) objects are unique up to isomorphism. Why use sucrose rather than glucose in an injectable vaccine? What is a wizard that makes them non-Muggle? Or, more directly, what ancestry does the witch/wizard gene follow? Is there a patriarchal wizard (or matriarchal witch) that Muggles have no part in? Elves perhaps? Stop reading here.  Purpose of the Question/History/Don't Read There seem to be solid consequences associated with being Muggle, or Muggle-related. A dominant theme in the novels is discrimination against the non-gifted race, but, when read into, the wizard bloodline doesn't appear all that well defined. JKR indicates that wizard-ness is genetic, or inherited:  &quot;Muggle-borns will have a witch or wizard somewhere on their family tree, in some cases many, many generations back. The gene resurfaces in some unexpected places.&quot;  By the above, it stands that a family that produces a squib (non-magical born of magical parents) must be a half-blooded family in some way, since half-blooded only requires one non-magical ancestor. Of course, we are dealing in magic genetics. Any other form of genetics would make this inherited, seemingly recessive trait (or squibs couldn't exist), never appear at all. Unless they were XMen. If it is so important, what does the wizard-bloodline follow? Other Mythos In other literature a wizard can be nearly anything, but magical powers are typically inherited as well as learned. Merlin of the Arthurian Legend (also mentioned in HP) is Cambian, meaning he is born of demonic intervention; this explains his source of enchantment. Gandalf of Middle Earth is a Maiar, as are all Wizards of Middle Earth. Wizards in Harry Potter seem to follow a bloodline, and it has been mentioned (in the question Gabe linked) that genetics have a heavy play in it, meaning that squibs and parents of Muggle-born wizards should have identical genetics. This bloodline must come from somewhere. Magical Creatures In Harry Potter it is common for magical creatures to &quot;mingle&quot; with humans, these beings are typically somewhat magical. Elves have been described to have very strong magic, but are for some unreasonable reason subservient to humans. I may be mistaken, but believe it was hinted that elves and the wizarding lineage had intersected at one time. Maybe I'm imagining this. Genocidal Allusions Voldemort hates muggles, and half-bloods, and whatever else is non-wizard. The presence of this theme, suggests that this wizardly bloodline must be at least defined if the main antagonist is so obsessed with keeping that mysterious bloodline pure. Since Slytherin himself was said to be a supremacist (circa a really long time ago) it suggests that this bloodline should have existed long before he was around, especially if he founded a school with people he was not related to, but were also magical. I'm wondering if there's any log which stores information about a particular program such as when the program has started and closed manually on Windows such as MS Word, or Chrome. Any ideas? Form action doesn't submit hard-coded variable Induction on a sum The left hand side has terms involving  $\binom{n}{m}= \dfrac{n!}{(n-m)!m!}$    $$1+\dfrac{1}{2}\binom{n}{1} +\frac{1}{3}\binom{n}{2}+...........+\frac{1}{n+1}\binom{n}{n} = \dfrac{2^{n+1}-1}{n+1}$$  I've done induction and proved $P(0)$ holds and also $P(1),P(2)$ holds (Just in case)  Now I've found $P(K+1)$ so the sum on the left is going to equal $\dfrac{2^{n+1}-1}{n+1}$ + $\dfrac{1}{n+2}$  and the right hand side is going to equal $\dfrac{2^{n+2}-1}{n+2}$  My problem is coming with the algebra though. I can't get those sides to equal. I can't get rid of the $n+1$ in the denominator.   Anyone have any ideas for me? They'd be much appreciated!  P.S. Someone gave me a hint and said that you can use the binomial theorem to solve this. Is there always an equilibrium point in space for gravity? For instance, for a two-body system, there will always be a point in-between the two in which the forces of gravity completely cancel, and if I were there then I would experience no net force. I believe there is also a point, or a region, for three body systems as well. Assuming we treat the stars or planets as simply point-masses, will there always be, for any number of objects, a point, or a region, of equilibrium? Or no? class and private variables Is there anyway to get tuple operations in Python to work like this:  &gt;&gt;&gt; a = (1,2,3) &gt;&gt;&gt; b = (3,2,1) &gt;&gt;&gt; a + b (4,4,4)   instead of:  &gt;&gt;&gt; a = (1,2,3) &gt;&gt;&gt; b = (3,2,1) &gt;&gt;&gt; a + b (1,2,3,3,2,1)   I know it works like that because the __add__ and __mul__ methods are defined to work like that. So the only way would be to redefine them? Control Cycles/Eevee material nodes and material properties using python? Converting an array of objects to an array of their primitive types If you have an array of Java objects which have a primitive type (for example Byte, Integer, Char, etc).  Is there a neat way I can convert it into an array of the primitive type?  In particular can this be done without having to create a new array and loop through the contents.  So for example, given  Integer[] array   what is the neatest way to convert this into  int[] intArray   Unfortunately, this is something we have to do quite frequently when interfacing between Hibernate and some third party libraries over which we have no control.  It seems this would be a quite common operation so I would be surprised if there's no shortcut.  Thanks for your help! If $G$ has no proper subgroup, prove that $G$ is cyclic of order $p$, where $p$ is a prime number.     I know that since $G $is a group with no proper subgroups, $g \in G$ is not just the identity. I don't know where to go from there. How can I produce the text Word_one_two in LaTeX?  I tried:  Samp\_Dist\_Corr   But, it doesn't quite look right. Also, I want it in the typewriter font, so actually, I'm doing:  \texttt{Samp\_Dist\_Corr}   I find it looks a bit like the underscore is merging in to the bottom of the "D", but maybe it's just because of the typewriter D? I can't seem to get these particular faces to take the UV Mapping. I selected the entire object, entered edit mode, hit U>Unwrap.   The rest of the mesh takes the uv wrapping, but not those faces for some reason.    It looks normal in solid mode.     Does anyone know what could be causing this? Ordered Pair Formal Definition I understand what an ordered pair is, but I'm having trouble the formal Kuratowski definition which says that $\langle a,b \rangle = \{\{a\},\{a,b\}\}$. What about this definition imposes order on the pair? Multicollinearity when individual regressions are significant, but VIFs are low I have 6 variables ($x_{1}...x_{6}$) that I am using to predict $y$. When performing my data analysis, I first tried a multiple linear regression. From this, only two variables were significant. However, when I ran a linear regression comparing each variable individually to $y$, all but one were significant ($p$ anywhere from less than 0.01 to less than 0.001). It was suggested that this was due to multicollinearity.  My initial research on this suggests checking for multicollinearity by using . I downloaded the appropriate package from R, and ended up with the resulting VIFs: 3.35, 3.59, 2.64, 2.24, and 5.56. According to various sources online, the point you should be worried about multicollinearity with your VIFs is either at 4 or 5.   I am now stumped about what this means for my data. Do I or do I not have a multicollinearity problem? If I do, then how should I proceed? (I cannot collect more data, and the variables are parts of a model that aren't obviously related) If I don't have this problem, then what should I be taking from my data, particularly the fact that these variables are highly significant individually, but not significant at all when combined.  Edit: Some questions have been asked regarding the dataset, and so I would like to expand...  In this particular case, we're looking to understand how specific social cues (gesture, gaze, etc) affect the likelihood of someone producing some other cue. We would like our model to include all significant attributes, so I am uncomfortable removing some that seem redundant.  There are not any hypotheses with this right now. Rather, the problem is unstudied, and we are looking to gain a better understanding of what attributes are important. As far as I can tell, these attributes should be relatively independent of one another (you can't just say gaze and gestures are the same, or one the subset of another). It would be nice to be able to report p values for everything, since we would like other researchers to understand what has been looked at.  Edit 2: Since it came up somewhere below, my $n$ is 24. Would it be possible to create a wand more powerful than the Elder Wand? Retrieve Voice Memos from device? iOS seems to refuse to allow me to email "long" voice memos, and I'm unaware of any way to extract them from iTunes backups of my phone. How can I retrieve entire voice memos that aren't really short?