I'm actually lost regarding inductions
Proof by induction that $2 + 4 + 6 + \cdots + 2n = n(n+1)$
Can I have two UK visas simultaneously? I have a visitor UK visa valid until October 2017.  I also want to apply for a Tier 4 student visa. My studies in the UK start in September 2017.  If I apply for the student visa in June, will they simply issue an overlapping student visa, or will they cancel my visitor visa and I will not be able to come to the UK until September?  I am worried because in a similar situation with Schengen visas, my existing visa was canceled. If I'm not wrong, torque is perpendicular to both the radius and force i. e. It is along the axis of rotation. Questions that arise are- why do we consider the length between the axis/point of rotation while calculating torque? More importantly why is torque a cross product? How did Princess Leia survive this? In the Last Jedi movie, there is a scene where      her spaceship is taken down, Princess Leia is thrown into outer space, and seen floating there, frozen.    However, she manages to survive this, while any ordinary human being couldn't.   How did she manage this? Is this done by using the Force? On freedom in Sartre's existentialism I am reading "Existenstialism is a humanism", the text of the famous conference by Sartre in which he explains his own version of existentialism. I think is full of logical inconsistencies, but maybe it's just because I don't understand it or I don't know enough about existentialism. Anyway, my question is about a specific passage. After having "demonstrated" that there is no way one can derive an ethics outside of Man (God does not exist and reason is insufficient), he asks himself according to what principle a person should behave and he says:     "For I declare that freedom, in respect of concrete circumstances, can   have no other end and aim but itself; and when once a man has seen   that values depend upon himself, in that state of forsakenness he can   will only one thing, and that is freedom as the foundation of all   values. That does not mean that he wills it in the abstract: it simply   means that the actions of men of good faith have, as their ultimate   significance, the quest of freedom itself as such."   So it seems Sartre says that when deciding a course of action, people will always choose "freedom," (and also, elsewhere, that "freedom" is always a good thing). That is for him the only principle, the only moral that governs or should (because we can do wrong judgments about what is freedom) govern our action.  My questions are:  1) do I understand correctly Sartre's thinking?  2) He says: a) freedom is absolute; b) the values that man chooses are also absolute because no God or reason can tell him what to do, c) therefore man will choose freedom. Is there any logic in this statement? If I replace freedom with X where X is "pleasure" or "justice" or "pain" or "zero temperature", wouldn't I "demonstrate" by this logic that man will choose X?  3) assuming what I said makes any sense, how can existentialism ever find a rule of moral conduct? and if it does, what is it and how can be demonstrated? and if it cannot, why is or has been important at all as a philosophy?  Thanks! Conditional expectation on more than one sigma-algebra Modern ice-age/plans for weapon hidden in genome Defrosting meat/fish in warm water I have found warm water to be a great way to quickly defrost meat and fish as opposed to just letting it sit outside the freezer, which is what my mom does.  But I wonder if there are any negative side effects to the method, as in, for example, compromised taste or compromised food safety.  I do understand that the water shouldn't be too hot as to avoid actually cooking the frozen food. Zero dimensional points do not take up space, so then wouldn't everything in the universe be literally empty? Or is there something that I'm missing? Purpose of @ Symbol Before Strings? Which is correct: "what if there was" or "what if there were"? Possible Duplicate:          Is this correct grammar?     What if there was a Stack Overflow on…    Shouldn't it be "what if there were a Stack Overflow on…"? Maximum of $|\sin(z)|$ as $\{z: |z| \leq 1 \} $  So according to the Maximum Principle, the maximum is when $|z|=1$. I tried using the fact that $\sin z= \dfrac {e^{iz}-e^{-iz}}{2i}$, but didn't know how to continue from there. Setting $z=e^{it}$ doesn't help much.  I saw this question was answered before, however the answer uses hyperbolic functions ($\sinh, \cosh$) which I haven't actually studied.  Thanks in advance for any assistance! Meaning of the word "axiom" Thus, axioms are propositions we assume to be true and we use them in an axiomatic theory as premises to infer conclusions, which are called "theorems" of this theory.  For example, we can use the  to prove theorems of arithmetic.  This is one meaning of the word "axiom". But I recognized that the word "axiom" is also used in quite different contexts.  For example, a group is defined to be an algebraic structure consisting of a set $G$, an operation $G\times G\to G: (a, b)\mapsto ab$, an element $1\in G$ and a mapping $G\to G: a\mapsto a^{-1}$ such that the following conditions, the so-called group axioms, are satisfied:   $\forall a, b, c\in G.\ (ab)c=a(bc)$, $\forall a\in G.\ 1a=a=a1$ and $\forall a\in G.\ aa^{-1}=1=a^{-1}a$.      Why are these conditions (that an algebraic structure has to satisfy to be called a group) called axioms? What have these conditions to do with the word "axiom" in the sense specified above? I am really asking about this modern use of the word "axiom" in mathematical jargon. It would be very interesting to see how the modern use of the word "axiom" historically developed from the original meaning.   Now, let me give more details why it appears to me that the word is being used in two different meanings:  As peter.petrov did, one can argue that group theory is about the conclusions one can draw from the group axioms just as arithmetic is about the conclusions one can draw from the Peano axioms. But in my opinion there is a big difference: while arithmetic is really about natural numbers, the successor operation, addition, multiplication and the "less than" relation, group theory is not just about group elements, the group operation, the identity element and the inverse function. Group theory is rather about models of the group axioms. Thus: The axioms of group theory are not the group axioms, the axioms of group theory are the axioms of set theory.  Theorems of arithmetic can be formalized as sentences over the signature (a. k. a. language) $\{0, s, +, \cdot\}$, while theorems of group theory cannot always be formalized as sentences over the signature $\{\cdot, 1, ^{-1}\}$. Let me give an example: A typical theorem of arithmetic is the case $n=4$ of Fermat's last theorem. It can be formalized as follows over the signature $\{0, s, +, \cdot\}$: $$\neg\exists x\exists y\exists z(x\not = 0\land y\not = 0\land z\not = 0\quad\land\quad x\cdot x\cdot x\cdot x + y\cdot y\cdot y\cdot y = z\cdot z\cdot z\cdot z).$$ A typical theorem of group theory is  which states that for any finite group G, the order of every subgroup H of G divides the order of G. I think that one cannot formalize this theorem as a sentence over the "group theoretic" signature $\{\cdot, 1, ^{-1}\}$; or can one? How do you combine two jQuery search results? eg:  var $allFoos = $('.foo'),     $allBars = $('.bar')     $allFoosAndBars = $allFoos + $allBars;   Obviously, I just made up that last line, but I hope it makes it sorta clear what I mean. To be clear, the example is greatly simplified, and it could be any arbitrary sets i'm talking about, so $('.foo, .bar') is not what I'm after. How can I reliably get the operating system's name? Database schema documentation for the public data dump and SEDE The only place I am aware of is the . I have the table below and i want it to read through columns A-B and then sum the values that match their names in columns C-D.  So that in a column elsewhere of a list of names i have the sum of their values from this table. For example Ed features in both columns with a value of 1 and 3, so would return a value of 4. Struggling to get this to work nicely. My actual table consists of 4 columns names and two sets of 4 columns with numbers.    A       B       C   D Steve   Jeff    4   1 James   Calum   1   1 Alex    Andy    2   1 Peter   Ed      0   3 Owen    Harry   1   2 Jeff    Steve   4   1 James   Calum   1   1 Andy    Alex    2   1 Peter   Owen    0   3 Ed      Harry   1   2 "India has" or "India have" What does "if not" mean in the given sentence Let there be given this sentence (which came from an ):     The contest has become personalised, if not bitter.   Then what does the phrase if not mean?   Seeking after is a general guide or rule of such usage. Why are local variables not initialized in Java? Was there any reason why the designers of Java felt that local variables should not be given a default value? Seriously, if instance variables can be given a default value, then why can't we do the same for local variables?  And it also leads to problems as explained in :     Well this rule is most frustrating when trying to close a resource in a finally block. If I instantiate the resource inside a try, but try to close it within the finally, I get this error. If I move the instantiation outside the try, I get another error stating that a it must be within a try.      Very frustrating.