How are kx-wt and kx+wt in terms of the direction of the wave. I have been stuck at this or an hour, still can not find a definitive answer.
In the textbook, it said a wave in the form $y(x, t) = A\cos(\omega t + \beta x + \varphi)$ propagates along negative $x$ direction and $y(x, t) = A\cos(\omega t - \beta x + \varphi)$ propagates along positive $x$ direction. This statement looks really confusing because when it says the wave is propagating along $\pm$ x direction, to my understanding, we can drop the time term and ignore the initial phase $\varphi$ while analyzing the direction, i.e. $y(x, 0) = A\cos(\pm\beta x)$, however, because of the symmetry of the cosine function, $\cos(\beta x)\equiv \cos(-\beta x)$, so how can we determine the direction of propagation from that?  I know my reasoning must be incorrect but I don't know how to determine the direction. So if we don't go over the math, how to figure out the direction of propagation from the physical point of view? Why $-\beta x$ corresponding to the propagation on positive x direction but not the opposite?
Would a monk with the Tavern Brawler feat have improvised weapons count as monk weapons?  It states in the monk's Martial Arts feature (PHB, p. 78) that:     At 1st level, your practice of martial arts gives you mastery of combat styles that use unarmed strikes and monk weapons, which are short swords and any simple melee weapons that don’t have the two-handed or heavy property.   The Tavern Brawler feat (PHB, p. 170) states that:     You are proficient with improvised weapons and unarmed strikes.   And the rules for improvised weapons (PHB, p. 147-148) state:     In many cases, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club. At the DM’s option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus. An object that bears no resemblance to a weapon deals 1d4 damage.    If the improvised weapon does not resemble a weapon, is simple, and is not heavy or two-handed, could a monk replace the d4 dice? For example if the monk is level 6 and has a unarmed attack of a d6, with a d6 dice instead? Do electrons have shape? According to :     The electron has no known substructure. Hence, it is defined or assumed to be a point particle with a point charge and no spatial extent.   Does point particle mean the particle should not have a shape, surface area or volume?  But when I searched Google for the "electron shape" I got many results (like  and ) that says electrons are round in shape. What is the best way to convert an array to a hash in Ruby In Ruby, given an array in one of the following forms...  [apple, 1, banana, 2] [[apple, 1], [banana, 2]]   ...what is the best way to convert this into a hash in the form of...  {apple =&gt; 1, banana =&gt; 2} Prove continuous functions $f$ and $g$ are equal I just bought a  game (Forza 4) for my Xbox 360 (bought in the US). When I tried to play the game, it wouldn't load because of region issue. Is there a legal way to solve this (no crack)? Old Sci-Fi book with skin crystalizing disease and characters with psi powers Let $a_{n+1}=\sqrt{a_1+a_2+\cdots+a_n}$ .Prove that $ \lim\limits_{n \rightarrow \infty} \frac{a_n}{n}=\frac{1}{2}$ A sequence $a_n$ is defined as $a_1=1$ and $a_{n+1}=\sqrt{a_1+a_2+\cdots+a_n}$ .Prove that $ \lim\limits_{n \rightarrow \infty} \frac{a_n}{n}=\frac{1}{2}$  I have no idea how to approach this. But I have a feeling that Cesaro's lemma may come in handy When would you use the Builder Pattern? What are some common, real world examples of using the Builder Pattern?  What does it buy you?  Why not just use a Factory Pattern? Facebook message notification showing blank message Since I updated the system to 10.11.5, the Facebook message notification is not showing message text.   Is this a bug in this software upgrade of OSX? Any class of a nonempty $X$ generates a base for a topology on $X$. Theorem Any class $\mathcal{A}$ of subsets of a nonempty set $X$ is the subbase for a unique topology $\tau$ on $X$. That is to say, the finite intersections of members of $\mathcal{A}$ form a base for the topology $\tau$ on $X$. Example Consider $X=\{a,b,c,d\}$ Let $\mathcal{A}=\{\{a,b\},\{b,c\},\{d\}\}$ The finite intersection of members of $\mathcal{A}$ give us: $$\mathcal{B}=\{\{a,b\},\{b,c\},\{d\},\{b\},\varnothing,X\}$$  How is it true that $X \in \mathcal{B}$ ? By definition of subbase, the finite intersection of members of $\mathcal{A}$ are the members of some base $\mathcal{B}$. But for $a_i \in \mathcal{A}, \displaystyle\bigcap a_i \neq X$ for any finite intersection of $a_i$'s. Look: $\{a,b\}\cap\{b,c\}\cap \{d\} \neq \{a,b,c,d\}$ nor do any other intersection of elements of $\mathcal{B}$ result in $X$. Anybody know what's going on here? Why is $X \in \mathcal{B}$? Regarding the sum $\sum_{p \ \text{prime}} \sin p$ I'm very confident that  $$\sum_{p \ \text{prime}} \sin p $$  diverges. Of course, it suffices to show that there are arbitrarily large primes which are not in the set $\bigcup_{n \geq 1} (\pi n - \epsilon, \pi n + \epsilon)$ for sufficiently small $\epsilon$. More strongly, it seems that $\sin p$ for prime $p$ is dense in $[-1,1]$.   This problem doesn't seem that hard though. Here's something that (to me) seems harder.      If $p_n$ is the nth prime, what is $$\limsup_{n \to +\infty} \sum_{p \ \text{prime} \leq p_n} \sin p?$$   What is $$\sup_{n \in \mathbb{N}} \sum_{p \ \text{prime} \leq p_n} \sin p? $$   Of course, we can ask analogous questions for $\inf$.   I'm happy with partial answers or ideas. For example, merely an upper bound. Alternating sum of binomial coefficients multiplied by (1/k+1) I'm trying to prove that  $$\sum_{k=0}^n {n \choose k} (-1)^k \frac{1}{k+1} = \frac{1}{n+1}$$  So far I've tried induction (which doesn't really work at all), using well known facts such as  $$\sum_{k=0}^n {n \choose k} (-1)^k = 0$$  and trying to apply identities like  $${n \choose k} = {n-1 \choose k-1} + {n-1 \choose k}$$  Would anyone be able to point me towards the right method? Should I be looking to apply an identity or is there a method I'm missing? How to filter IPv4 and IPv6 addresses? Struct padding in C++ Cannot reach public WIFI login page in Ubuntu 18.04 Captive Portal Issue Regard the diff equation:  $mϕ′′+aϕ′+(mg/L)ϕ=0$  $ϕ(0)=0.1$  $ϕ′(0)=0$  where $m=0.1,L=1,a=2,$  1) Rewrite the second order diff equation as a system of first order linear equations.  2) What is the matrix that comes from the rewritten equations? Dependency problem with mysql-server-core-5.5 When I start the Ubuntu software centre, it says I cannot do anything until the package catalog is repaired. However, repairing fails.  I ran sudo apt-get -f install and found the problem to be:  mysql-server-5.5 depends on mysql-server-core-5.5 (= 5.5.24-0ubuntu0.12.04.1); however:   Version of mysql-server-core-5.5 on system is 5.5.28-0ubuntu0.12.04.2.   So, the question is, how do I install that version and resolve the dependency problem? Can the weak force create a bound state? My understanding is that the exchange of Z bosons could yield an attractive or repulsive "force" between two fermions. For most combinations the electromagnetic or strong interactions will take place.  However, if we have two neutrinos, or a neutrino and a charged lepton, only the weak interaction comes into play. Are these possible bound states? Xml validation using XSD schema The following code helps me validate an XML file with an XSD schema.  XmlReaderSettings settings = new XmlReaderSettings(); settings.Schemas.Add(null, xsdFilePath); settings.ValidationType = ValidationType.Schema; settings.ValidationEventHandler += new System.Xml.Schema.ValidationEventHandler(settings_ValidationEventHandler); XmlDocument document = new XmlDocument(); document.Load(xmlFilePath); XmlReader rdr = XmlReader.Create(new StringReader(document.InnerXml), settings);  while (rdr.Read()) {  } isValid = true;   The ValidationEventHandler also tells me what the errors are, but doesn't tell me on 'where' or 'on which line' they are located. Is there any way to get the line number where the XML fails to be validated? Apostrophes for possession: nouns My best friend of the moment - 'Essential Grammar in Use' for elementary students by Raymond Murphy - says that 'we normally use -'s for people.'     I stayed at my sister's house. (not the house of my sister).   Then, bellow in the same unit (64) says that 'we use of... for things, places, etc.'     Look at the roof of that building. (not that building's roof)   After that I thought:  Finally!!! I understand the rules for apostrophes!!!  But soon I lost all that naif excitement when I saw other written examples on the internet that were not following those 'rules'.     I've learnt that we can apply apostrophes to ...     one year's pay   and, finally, amazingly enough ... to , which means ANY kind of nouns (I deduce uncountable nouns are also part of the possibilities). This demolishes what is said in the grammar. So, it seems that eventually:     That building's roof -> is valid   So, now I need your help.    1) Can you tell me the rules concerning the apostrophe to show possession in respect to its usage with nouns? Can we really apply to all kind of nouns? e.g. teeth's, vehicle's, Paula's, forest's, manager's, Ireland's, etc     Scotland's worst drivers   Metadata improves information's accessibility       2) Is there a sense that justifies what is written in the grammar: 'we normally use -'s for people'? or it does not make sense at all?