Is each particle associated with an unique string and where exactly is the string located, if it's vibration produces the appearance of a particle? What I mean is if I see an electron, does it have an unique string corresponding to it and where is this string physically (inside the electron or somewhere else)?
I understand that the vibration modes of an elementary string determines the identity of its particle. When I first heard this, I visualized a stiff circular steel band-like entity vibrating with 2, 3, 4, 5,... nodes or antinodes, the number determining the particle. This was further supported when I asked a well-known physicist how one gets from equations to physical strings, and he said he recognized the harmonic nature of the solutions.  Bu then I noticed that physicists seem to accept the image of a soft spaghetti-like string flopping about with no well-defined vibrational modes. I asked another physicist about this, and his answer was simply that a string has multiple modes, not necessarily in a harmonic progression.  So my questions are 1) How do you visualize the vibrations of a string, and 2) how are they related to particle identity?
Suppose I am given a morphism of schemes $f: X \longrightarrow Y$ and suppose that for any point $y \in Y$, $\kappa(y)$ denotes the residue field. I would like to show that the fibered product $$X \times_{Y} \text{Spec}(\kappa(y))$$ is homeomorphic to the preimage $f^{-1}(y)$ in the induced topology.  The following is my attempt so far:  The question is obviously affine on the target, so without loss of generality we reduce to the case of a morphism $f: X \longrightarrow \text{Spec }A$. Then we have  $$ f^{-1}(\text{Spec }A) = \bigcup_{i \in I} \text{Spec }B_{i} $$ via morphisms $$ \phi_{i}: A \longrightarrow B_{i}. $$ We can then write, \begin{align*} f^{-1}(\text{Spec } A) \times_{\text{Spec } A}\text{Spec } \kappa(y) &amp;= \bigcup_{i \in I} \left( \text{Spec } B_{i} \times_{\text{Spec }A} \text{Spec }\kappa(y) \right) \\  &amp;= \bigcup_{i \in I} \text{Spec }  \left( B_{i} \otimes_{A} \kappa(y)  \right). \end{align*} Now let $\mathfrak{p}$ be the prime ideal of $A$ corresponding to the point $y \in \text{Spec } A$. Then $\kappa(y) = (A/ \mathfrak{p})_{\mathfrak{p}}$. Then for any of the $B$ (dropping the index for brevity), we have \begin{align*} B\otimes_{A} (A/ \mathfrak{p})_{\mathfrak{p}} &amp;\simeq B \otimes_{A} A / \mathfrak{p} \otimes_{A} A_{\mathfrak{p}} \\ &amp;\simeq B_{\mathfrak{p}} \otimes_{A} A / \mathfrak{p} \\ &amp; \simeq B_{\mathfrak{p}} / \mathfrak{p}B_{\mathfrak{p}} \end{align*}  Now the prime ideals in $B_{\mathfrak{p}} / \mathfrak{p}B_{\mathfrak{p}}$ correspond precisely to the prime ideals of $B_{\mathfrak{p}}$ containing $\mathfrak{p}B_{\mathfrak{p}}$. Further, the prime ideals of $B_{\mathfrak{p}}$ correspond precisely to the prime ideals of $B$ contained in $\phi(\mathfrak{p})B$. That is, we have \begin{align} \text{Spec }B_{\mathfrak{p}} &amp;= \{  \mathfrak{q} \subseteq B :  \phi^{-1}(\mathfrak{q})  \subseteq \mathfrak{p}  \} \\ &amp;=  \{ \mathfrak{q} \subseteq B : f(\mathfrak{\mathfrak{q}}) \subseteq \mathfrak{p}      \}. \end{align} Note that in the very last line we have abused notation slightly: There is not really any notion of being "inside" $\mathfrak{p}$ when it is being treated as a point in the scheme. So really we mean that $f(\mathfrak{q})$ is a point in the scheme whose corresponding prime ideal is a subset of $\mathfrak{p}$. Now as we said above, the prime ideals of $B_{\mathfrak{p}}/\mathfrak{p}B_{\mathfrak{p}}$ are the following $$ \text{Spec }(B_{\mathfrak{p}}/\mathfrak{p}B_{\mathfrak{p}}) = \{ \mathfrak{q} \subseteq B : f(\mathfrak{\mathfrak{q}}) \subseteq \mathfrak{p}      \text{ and } ???  \} $$ This is the point at which I get stuck. Ideally I would like to make some argument from the statement "Now the prime ideals in $B_{\mathfrak{p}}/\mathfrak{p}B_{\mathfrak{p}}$ correspond precisely to the prime ideals of $B_{\mathfrak{p}}$ containing $\mathfrak{p}B_{\mathfrak{p}}$" to conclude that the question marks are the condition $\mathfrak{p} \subseteq f(\mathfrak{q})$. Could someone point me in the right direction as to how to get this condition?   Is the proof up until this point correct? Different sources online seem to do this many different ways, so it's difficult for me to check. Do I need to have antivirus software installed? Why do we need to discard one dummy variable? Why is surface charge distribution uniform for a conducting sphere? Can't it be arbitrary? If the charge $q_1$ has to repel the charge $q_2$, the electric field has to go inside the conductor which contradicts the fact that electric field inside conductors is zero. Then why do the charges distribute themselves in a particular manner? Why can't they be distributed over a small place on the conductor? What are the different endings in Dishonored? In Dishonored the game as I am escaping the prison, I got a prompt that warned me:      Using stealth and nonlethal approach has benefits. Fewer rats and   weepers, some people react favorably, and the final outcome is not as   dark.   What are all the different endings? And how much chaos yields them? And what is the optimal ending? Why is triple-DES using three different keys vulnerable to a meet-in-the-middle-attack? Among other sources,  states that triple DES using three seperate keys (k1, k2, k3) is vulnerable to meet-in-the-middle-attacks, while triple DES using only two keys (k1, k2, k1) is not.  Why is this attack possible if all the keys are different, but not if the first and the third are the same? the numer of partial function between two sets in combinatoric way I am not experienced in Number Theory but what I know is that some results of this filed are applicable in other areas, e.g. algebra. For sure FLT made (and makes) people be interested in Number Theory leading to the development of new methods which can be themselves applied not only for the proof of FLT (like financial problems motivated somehow development of stochastic analysis). I am interested - if there are applications or implications of FLT itself?  More precisely: if the fact "for each $n\geq3$ there are no integer solutions of $a^n+b^n=c^n$" leads to solutions of problems which are not in the field of Number Theory?   I would specify that I wonder about some problems which are already formulated: since FLT is known for more then 300 years I am pretty sure that there were formulated hypothesis which follow from FLT directly (if there are such hypothesis not in the field of Number Theory). Are the following sentences grammatical or not? If not, does substituting a to-infinite make them grammatical? (Optional: designate if the sentence expresses purpose, reason, aim, or function)  1 I have decided to purchase a jet for getting to NYC and back more quickly. 2 I have decided to buy a jet for having a quicker way to get to NYC and back.  This is my only reason and aim; this will be the jet's only purpose and function.  3 I hate hairdryers, so I am buying a space heater for drying my hair more naturally, and that is my only aim &amp; purpose &amp; reason and that will be its only function 4 For having a lasting shine in your hair, buy ShineOn Shampoo. 5 My neighbor and I have decided to cooperate more for having a better life together.   Research done: ELL  (Note the  here is a restatement of Cambridge Dictionary's ) ELL  ELL  BBC English  (Currently the best site I've found.) ELU  All of which have answers that I find  (i) incomplete (ii) contradictory (iii) terribly confusing, if not (iv) wrong. To sort through the haze, I am focusing on one issue. Background Some sources say that you need to use a to-infinitive for expressing purpose, aim, function, reason, etc. I find these to be rather hazy distinctions, given that the ODO defines  as The reason for which something is done or created or for which something exists and the same work lists as : aim, reason, function, etc. It is my assertion that at least some of the following sentences demonstrate that one can use the -ing form (be it a gerund, verbal noun, participle) to express individual purpose (aim, reason, function, intention, whatever you want to call it). Do you agree, or not? Why (not)?  In shorter terms, are the sentences grammatical? or do we have to substitute a to-infinitive (including in order to + infinitive) to make them grammatical? (Optional: designate if the sentence expresses purpose, reason, aim, function, etc.) Show that $A$ is an invertible matrix if $\left(A+I\right)^3=0$ and find $A^{-1}$ MySQL Case problem I am trying to run a SQL code (I use MySQL). What I want to do is to set DATE_REACT a value when the first condition is satisfied, and a different value when the first condition is not satisfied. The problem is that the first condition is executed always, even when it shouldn't. What I am doing wrong?  UPDATE BILLS SET STATE = 2, DATE_REACT = CASE  WHEN (DATE_EM &lt;= CURTIME() &lt;= ADDTIME(DATE_EM, '00:02:00')) THEN ADDTIME(CURTIME(), '00:13:00') ELSE ADDTIME(DATE_REACT, '00:13:00') END WHERE STATE = 1; Money making in Runescape What is a word that means "someone who pretends to be your friend but is actually your enemy?" What do you call someone who pretends to be your friend but is actually your enemy?  A friend suggested spy for me, but that does not nearly describe the word I need for an English project. The character is very good at manipulating how people see him. He's a good liar. And pretends to be your friend but actually is your enemy. If $x,y,z\gt 0$ and $xyz=1$ Then minimum value of $\frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}$ If $x,y,z\gt 0$ and $xyz=1$ Then find the minimum value of $\displaystyle \frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}$   $\bf{My\; Try::}$Using Titu's Lemma $$\frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}\ge \frac{(x+y+z)^2}{2(x+y+z)} = \frac{x+y+z}{2}\ge 3\frac{\sqrt[3]{xyz}}{2} = \frac{3}{2}$$  and equality holds when $$x=y=z=1$$  My question is how can we solve it without the above lemma, like using Jensen's Inequality or other inequality.  Please explain me.  Thanks How to get straight quotation marks? How do I get straight quotation marks, i.e., " and ' in TeX?  (No, \verb changes the font and this is unacceptable.)  [Edit]  I forgot to mention that I’m using XeLaTeX, and fontspec is a necessity.  The first two proposed answers did work, without fontspec. This is similar to , except it's a bug report, not a feature request.  Currently, the nominee's comments on his/her own post are not highlights, unless you click "add / show more comments"  When the page is loaded it looks like:    AfterI click "add / show more comments", it becomes:    Could we have the blue highlighting on the post owner's comments there all the time? It becomes difficult to see the nominees responses to questions, especially on nominations where there aren't enough comments to enable the "add / show more comments" features, like Matt's. given the 2 PDE  $$ \Delta u-au_{tt}+u_{t}=0$$  and $$ \Delta u + Du*Df=0 $$  here $ \Delta u $ is the Laplacian $ Du= grad(u) $ is the gradient and * means scalar product $u_{t} = \frac{\partial u}{\partial t}$  my doubt is what term should i include to get the linear part of the equations $ u_{t} $ and $Du*Df $ from the Euler Lagrange equations in $ R^{n} $ thanks How can I stop fish from sticking to a stainless steel pan? I think Anne Burrell lied to me. She said fish will always stick to a stainless pan. This is true. No matter how well I lubricate my pan, my fish will always adhere like cement. She also said that the pan will release the fish "when it's ready." Maybe meaning it will eventually become unstuck when the fish is done cooking? Well, I've tried that, and it never works. No matter how long I wait, it never becomes unstuck. Most recently I gave it as much time as I could. When I finally chiseled it off the pan, tearing it apart in the process, I saw that it was starting to burn on that side. I've started going back to non-stick until I get this figured out.  Are there any tips? Am I heating the pan wrong? Does it help to slide the fish around a bit after putting it in the pan? Help ending a proof using binomial theorem: $\sum_{k=0}^n {n \choose k}\cdot\frac{(-1)^k}{k+1} = \frac{1}{n+1}$ Simple question but I am confused.  What is the boundary of $S^2\times\mathbb{R}$? Is it just $S^2$?  What would be the general way to evaluate the boundary of a product manifold?  Thanks for the replies!