Does increasing the tension on a string also increase the wave speed
Consider a string under tension, for example, a string on a guitar. When a guitar string is plucked, it vibrates at a certain frequency. When the tension on the string is increased by twisting the tuning peg at the end of the neck, the frequency of the string increases. According to , increasing the tension on a string also increases the speed of the wave moving through it. According to  the speed of a wave is higher in a more dense medium.  Does this mean that increasing the tension also increases the density?  This does not seem very logical to me, since density is the "compactness" of a substance, or how close together the individual molecules are. By twisting the tuning peg on a guitar, you rotate the cylinder that the string is fixed to, wrapping it around more tightly. This pulls more of the string around that cylinder, similarly to when you reel in a fishing rod. Since more of the string's molecules are now wrapped around the peg, there should be fewer molecules over the same distance between the bridge and the nut (the two ends of the vibrating portion of the string), therefore bringing the density down.
What should I do if a user posts sensitive information as part of a question or answer? Do I need quote marks when I quote myself? Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable. Let $X$ be an infinite dimensional Banach space. Prove that every basis of $X$ is uncountable.  Can anyone help how can I solve the above problem? Can somebody explain "leveraged debt investment positions" and "exposures" in this context for me, please? "The strong bond performance has led to a surge in leveraged debt investment positions that are ultimately funded by the banks' interbank exposures and WMPs. But nobody really knows how levered these positions are, who has financed them, and by how much," S&amp;P said. "The contagion risks for the financial sector could be high."  source: I'm trying to find the longest consecutive set of composite numbers Hello and I'm quite new to Math SE.  I am trying to find the largest consecutive sequence of composite numbers. The largest I know is:  $$90, 91, 92, 93, 94, 95, 96$$  I can't make this series any longer because $97$ is prime unfortunately.  I can however, see a certain relation, if suppose we take the numbers like (let $a_1, a_2, a_3,...,a_n$denote digits and not multiplication):  $$a_1a_2a_3...a_n1,\ a_1a_2a_3...a_n2,\ a_1a_2a_3...a_n3,\ a_1a_2a_3...a_n4,\ a_1a_2a_3...a_n5,\ a_1a_2a_3...a_n6,\ a_1a_2a_3...a_n7,\ a_1a_2a_3...a_n8,\ a_1a_2a_3...a_n9,\ a_1a_2a_3...(a_n+1)0$$  The entire list of consecutive natural numbers I showed above can be made composite if:   The number formed by digits $a_1a_2a_3...a_n$ should be a multiple of 3 The numbers $a_1a_2a_3...a_n1$ and $a_1a_2a_3...a_n7$ should be composite numbers   If I didn't clearly convey what I'm trying to say, I mean like, say I want the two numbers (eg: ($121$, $127$) or ($151$, $157$) or ($181$, $187$)) to be both composite.  I'm still quite not equipped with enough knowledge to identify if a random large number is prime or not, so I believe you guys at Math SE can help me out. Symbolizing using RGB values from Excel in ArcGIS Desktop? I have an excel file with 2 attribute columns, 1-feature info 2-Number of color (RGB) that the current feature should be symbolize. This file file can be joined with the layer using the "feature info" field which is unique (join by attribute table). So i have as attribute the color number attribute for every polygon feature.  How can  i use this this information to symbolize every feature with their attributed color already imported from the excel sheet? I was reading a  where the author mentions: "The coefficient of variation is primarily a descriptive statistic, but it is amenable to statistical inferences such as null hypothesis testing or confidence intervals. Standard procedures are often very dependent of the normality assumption and current work is exploring alternative procedures which are less dependent on this normality assumption."  The paper is from 2010. So my question is: has there been any recent "advancements" in terms of a coefficient of variation statistic that is not dependent upon the normality assumption? And, is this what one may call a non-parametric CV? $X$,$Y$,$Z$ mutually independent implies $X+Y$ independent of $Z$ Supposing $X$, $Y$ and $Z$ and mutually independent real random variables, how can we prove that $X+Y$ and $Z$ are independent from the definition? If not from the definition, using $\sigma$-algebras?  I know that if $X$ and $Y$ are not independent this doesn't work so I know I have to make use of their independence in my proof but I can't figure out how. And it means that if I write the following, I'm making a mistake, and I can't find it:  $$\mathbb{P}(X+Y \leq k,Z \leq l)=\int_{\mathbb{R}} \mathbb{P}(X=x,Y \leq k-x, Z \leq l) dx=\int_{\mathbb{R}} \mathbb{P}(X=x,Y \leq k-x) \mathbb{P}(Z \leq l) dx = \mathbb{P}(Z \leq l)\int_{\mathbb{R}} \mathbb{P}(X=x,Y \leq k-x)dx= \mathbb{P}(Z \leq l) \mathbb{P}(X+Y \leq k) $$  PS: I have seen the proof here but it much more advanced stuff, kind of a sledgehammer... Split List into Sublists with LINQ Is there any way I can separate a List&lt;SomeObject&gt; into several separate lists of SomeObject, using the item index as the delimiter of each split?  Let me exemplify:  I have a List&lt;SomeObject&gt; and I need a List&lt;List&lt;SomeObject&gt;&gt; or List&lt;SomeObject&gt;[], so that each of these resulting lists will contain a group of 3 items of the original list (sequentially).  eg.:   Original List: [a, g, e, w, p, s, q, f, x, y, i, m, c] Resulting lists: [a, g, e], [w, p, s], [q, f, x], [y, i, m], [c]   I'd also need the resulting lists size to be a parameter of this function. Add space before and after tabular environment The tabular environment does not add any vertical space before or after it. This looks odd, in my view. What is the "correct" or "best" way to add some space? Also, why is there more space after the tabular environment than before it, as seen in the example below?  \documentclass{article} \begin{document} \noindent Here is a line with some text in it.  \begin{tabular}{*4{l}}     Worg &amp; Worg &amp; Worg &amp; Worg\\     Worg &amp; Worg &amp; Worg &amp; Worg\\ \end{tabular}  \noindent Here is another line with some text in it. \end{document} Difference between projection and datum? How do I convince someone that $1+1=2$ may not necessarily be true? Me and my friend were arguing over this &quot;fact&quot; that we all know and hold dear. However, I do know that $1+1=2$ is an axiom. That is why I beg to differ. Neither of us have the required mathematical knowledge to convince each other. And that is why, we decided to turn to Math Stackexchange for help. What would be stack's opinion? Is there a way to call process builder from Apex? $a _{1} = \frac{1}{2}, a _{2} = 1, a _{n}= \frac{1}{2} a_{n-1} + \sqrt{a _{n-2} }$  Show by the induction that it's $\le 4$ and also by the induction that $a_{n+1}-a_n \ge 0$  If it comes to first I don't know what to do and if it comes to second I tried many different ways but I couldn't prove it. Trouble with proving $A$ is an integrally closed domain $\Rightarrow$ $A[t]$ is integrally closed domain How to load Windows 7 from GRUB rescue In 1735 Euler gave the value of $\gamma$ as $0.577218.$ The constant is generally defined as the limit of the difference between the harmonic series and $\log n:~\gamma= \lim_{n\to\infty}\sum_{k=1}^{n}\frac{1}{k}-\log n.$ Euler apparently relished this sort of calculation and must have taken quite a few terms to get such a good approximation.  My question is whether without a computer one can now prove that  $$\gamma &lt;  \sqrt{1/3}$$   with at least some savings in terms of the type of work Euler apparently expended? I don't think there's any point in raising the bar to $\gamma &lt; \ln 2\sqrt{\frac{23}{29}} $  because it seems to be the same sort of question.       My own thought was to compare $\frac{1}{2}\int_0^{1/3}\frac{dx}{\sqrt{x}}$ to something like $(-1)\cdot\int_0^\infty e^{-u}\ln u~du$ but I expect there's a better way. Transforming geometry PostGIS column with unknown geometry type? I run PostgreSQL 9.1 and PostGIS 2.0. I have a table that contains one geometry column.   But that column contains point, polygons lines, etc.   So I set it like so,   ALTER TABLE jake ADD COLUMN jake_geom geometry;  and then I setted some constraints to make it geometry enabled, but not limited on a single geometry type  ALTER TABLE jake   ADD CONSTRAINT enforce_dims_geom CHECK (st_ndims(jake_geom) = 2);  ALTER TABLE jake   ADD CONSTRAINT enforce_srid_geom CHECK (st_srid(jake_geom) = 900913);   Now, I want to reproject the geometries in that column to 4326.   According to , for the transform to work, I need to set a geometry type.  (the TYPE geometry(MultiPolygon,2163) part)  but my column does not have certain geometry type. How do I proceed?  Or I can simply do  ALTER COLUMN geom USING ST_Transform(geom,2163);  avoiding to set a type? What is scientific mind? Below is my code to create the component dynamically - The component is "ui:inputselect"  $A.createComponent(             RecoveryLineItem.Attributes[i].name,{                 "class" : "slds-input ",                 "aura:id" : 'ExampleId',                 "label" : RecoveryLineItem.Attributes[i].label,                 "multiple" : RecoveryLineItem.Attributes[i].multiple             }, ....   ..  console.log(component.find('ExampleId').get("v.value"))    Or console.log(component.find('ExampleId'))  is Giving as 'Undefined"  How to access the aura:id of the dynamically created components?