I am sort of confused about this. Wave particle duality says that sub atomic particles are waves.  There is something more though. What is the actual meaning of ?
I often hear about the wave-particle duality, and how particles exhibit properties of both particles and waves. However, I wonder, is this actually a duality? At the most fundamental level, we 'know' that everything is made up out of particles, whether those are photons, electrons, or maybe even strings. That light for example, also shows wave-like properties, why does that even matter? Don't we know that everything is made up of particles? In other words, wasn't Young wrong and Newton right, instead of them both being right?
How do I prove that $f(x)f(y)=f(x+y)$ implies that $f(x)=e^{cx}$, assuming f is continuous and not zero? This is part of a homework assignment for a real analysis course taught out of "Baby Rudin."  Just looking for a push in the right direction, not a full-blown solution.  We are to suppose that $f(x)f(y)=f(x+y)$ for all real x and y, and that f is continuous and not zero.  The first part of this question let me assume differentiability as well, and I was able to compose it with the natural log and take the derivative to prove that $f(x)=e^{cx}$ where c is a real constant.  I'm having a little more trouble only assuming continuity; I'm currently trying to prove that f is differentiable at zero, and hence all real numbers.  Is this an approach worth taking? Normed vector spaces over finite fields Normed vector spaces are typically defined over the reals or complex numbers.  Is there any "standard," well-behaved construction that generalizes this to a vector space over a finite field, such as $\Bbb F_2$?  I'm looking for something kind of like the class of $\ell_p$ norms, except designed with finite fields in mind.  Ideally, something that has deep fundamental properties making it well-behaved in the same way that the Euclidean norm is. Can I have a hint on how to construct a ring $A$ such that there are $a, b \in A$ for which $ab = 1$ but $ba \neq 1$, please? It seems that square matrices over a field are out of question because of the determinants, and that implies that no faithful finite-dimensional representation must exist, and my imagination seems to have given up on me :) I often see questions edited into this form:     Lorem ipsum dolor sit amet blah blah blah paragraph paragraph original question.            UPDATE: The Lorem disappeared and left this message: "new information".   I find this format unaesthetic (since Stack Exchange posts are supposed to be useful in the long term and update boundaries are artifacts of the creation rather than the content) and feel that new information should, whenever possible, be incorporated into the original text “as if it had always been that way” if practical, and pointing out an update should be done mostly in comments.  Is this an appropriate writing style, and if so under what circumstances? If not, should it be removed or discouraged? throwing exceptions out of a destructor Most people say never throw an exception out of a destructor - doing so results in undefined behavior. Stroustrup makes the point that "the vector destructor explicitly invokes the destructor for every element. This implies that if an element destructor throws, the vector destruction fails... There is really no good way to protect against exceptions thrown from destructors, so the library makes no guarantees if an element destructor throws" (from Appendix E3.2).   seems to say otherwise - that throwing destructors are more or less okay.  So my question is this - if throwing from a destructor results in undefined behavior, how do you handle errors that occur during a destructor?  If an error occurs during a cleanup operation, do you just ignore it? If it is an error that can potentially be handled up the stack but not right in the destructor, doesn't it make sense to throw an exception out of the destructor?  Obviously these kinds of errors are rare, but possible. How do you greet multiple recipients in an e-mail? How do you greet multiple recipients in an e-mail?  Assuming they're both male, I just use "Sirs", but it seems a bit informal. How to perform a true GIS clip of polygons layer using a polygon layer in R? Can I use 'in' instead of 'at'? -- "I have more books in my house." I am learning English using the Duolingo English course, one of correct solution was      I have more books at my house.   Can I use      I have more books in my house.   If it is OK, what is the difference between both? How to prove that we are living in a 3+1D world? What class of Devil Fruit does the Gomu Gomu no Mi come under? Considering the three types of Devil Fruits that has been explained are 1. Paramecia Type - where the user can get super human abilities and change their body parts into weapons, etc. 2. Zoan type - where the use can turn into a particular animal and inherit its abilities 3. Logia Type - where the user's body inherits a particular element and the body becomes like air which cannot be touched or damaged by non Haki users.  So what exactly is the type of Gomu Gomu no Mi? It cannot be Zoan or Logia. It's very close to a Paramecia but still doesn't have a logical explanation to be of Paramecia type! So is it some special type of Devil Fruit? Double Factorial: Number of possibilities to partition a set of $2n$ items into $n$ pairs I know that the partition of $2n$ items into $n$ pairs has something to do with double factorial, but I am not sure how many possibilities we exactly have.   We can choose such a partition into pairs by first choosing $n$ elements to be in one half and the remaining $n$ in the other half, which gives ${{2n} \choose n}$ possibilities. Then we can take a permutation of both sets and match them accordingly.   This gives $${{2n} \choose n} n! n!=\frac{(2n)! n! n!}{n! n!}=(2n)!$$ possibilities. But with this we overcount, since we do not want the pairs to be ordered. Each of the $n$ pairs is counted twice, wherefore we have to divide by 2 for every pair, hence $\frac{(2n)!}{2^n}$?   Well, another possiblity to do that is just to iteratively pick $2$ elements among the remaining ones, which gives ${{2n} \choose 2} {{2n-2} \choose 2} \dotsb {4 \choose 2}$  I am not sure, but I do not think that this is the same and also, I do not see the connection to double factorial. Would be very glad if you could help me with these 3 possiblities.   Thank you. I have frequently encountered both $\langle a,b \rangle$ and $[a,b]$ as notation for closed . I have mostly encountered $(a,b)$ for open intervals, but I have also seen $]a,b[$. I recall someone calling the notation with $[a,b]$ and $]a,b[$ as French notation.   What are the origins of the two notations?  Is the name French notation correct? Are they used frequently in France? Or were they perhaps prevalent in French mathematical community at some point? (In this  Bourbaki is mentioned in connection with the notation $]a,b[$.)      Since several answerers have already mentioned that they have never seen $\langle a,b \rangle$ to be used for closed intervals, I have tried to look for some occurrences for this. The best I can come up with is the article on , where these notations are called Czech notation and French notation. Using $(a,b)$ and $[a,b]$ is called English notation in that article. (I am from Central Europe, too, so it is perhaps not that surprising that I have seen this notation in lectures.) I also tried to google for  or . Surprisingly, this lead me to a  where this notation is used for open interval. Batch export Photoshop layers to individual PNG files I am a web dev and competent in Fireworks, but not so much in Photoshop.  I have just received a layered PSD file to turn into a web page.  Can anyone tell me the easiest way to export all layers to individual png files?  There are lots of layers and doing this manually seems wrong.  I have seen  but it does seem there should be native functionality for this in PS.  I have access to Photoshop CS4. Any pointers appreciated. Redefining "cited on" string (and others) in biblatex I'm trying to modify the strings biblatex uses to set the "cited on" string used in the bibliography to identify the pages where a given references has been cited.    What I gather from the biblatex documentation (e.g., § 4.9.2.16) is that something like   \renewbibmacro{backrefpage}{&lt;newtext&gt;} \renewbibmacro{backrefpages}{&lt;newtext&gt;}   ought to work, but they have no effect for me (I'd also like to get this working for the "visited on" string, urlseen).  My MWE is unchanged from , with the addition of these (and related lines) just about anywhere. Prove the following inequality:   $$\dfrac{e^x + e^{-x}}2 \le e^{\frac{x^2}{2}}$$   This should be solved using Taylor series.  I tried expanding the left to the 5th degree and the right site to the 3rd degree, but it didn't help.  Any tips? Birthday line to get ticket in a unique setup At a movie theater, the whimsical manager announces that a free ticket will be given to the first person in line whose birthday is the same as someone in line who has already bought a ticket.  You have the option of choosing any position in the line. Assuming that you don't know anyone else's birthday, and that birthdays are uniformly distributed throughout the year (365 days year), what position in line gives you the best chance of getting free ticket? What are the differences between "lay" and "lie"? Why was Padmé made to clean an astromech? Reading  (the Amidala stand-in) I remembered that I always wondered why she ordered the actual queen to scrub R2-D2. There were enough actual servants who could have done it, so why Padmé? If that wasn't enough, after giving the actual queen (Padmé) the order to go away, she tells Panaka to continue with the discussion. How does that make sense?  The significant piece of dialogue:     AMIDALA : Thank you, Artoo Detoo. You have proven to be very loyal...Padme!    ...    AMIDALA : (Cont'd) Clean this droid up the best you can. It deserves our gratitude...(to Panaka) Continue, Captain.   Note that this question is two-fold:   Why was Padmé chosen in the first place? Why would an important meeting with the ship's captain be continued without her? Read Post Data submitted to ASP.Net Form I have a working login form in an asp.net application.  Standard stuff with a username and password text box and a button to process the login.  Works fine.  I have a new requirement to allow the user to input the username and password from a separate plain html page that is not a part of my asp.net application.  I plan on achieving this using standard html - form, input, submit button etc.  The form action will be the URL of my asp.net login page and its method will be POST.    What I want to do in the C# code behind page of the asp.net login form, presumably in the Page_Load event, is to check if the request for the page contains a username and password value being passed in.  If it does then I need to read those values and process the login as if someone had clicked the login button on the asp.net page.  If not then I will display the login form as usual.    How do I check for the existence of, and read, the username and password values in the request for my page. Commas that are sentence flow-stoppers ... Can I leave them out?