Is muon a point particle?
What is spin as it relates to subatomic particles?
I'm having some troubles understanding the solution (going from step $2$ to step $3$) of this linear congruence equation. Here's the problem with the solution:   Solve the congruence $2x + 11 \equiv 7 \pmod 3$.   First, we reduce all the coefficients $\pmod 3$:  $$2x + 2 \equiv 1 \pmod 3. \tag{1}$$  Next, we add $1$ to both sides, using the fact that $2 + 1 \equiv 0 \pmod 3$:  $$2x \equiv 2 \pmod 3. \tag{2}$$  Finally, we multiply both sides by $2$, using the fact that $2 · 2 = 4 \equiv 1 \pmod 3$:  $$x \equiv 1 \pmod 3. \tag{3}$$  That is, any number in the set $\{\ldots, −5, −2, 1, 4, \ldots\}$ will solve the original congruence.    Like I said my problem is the step between $(2)$ and $(3)$. If I multiply both sides of $(2)$ by $2$ I get:  $$4x \equiv 4 \pmod 3.$$  Now, I understand that $4$ is congruent to $1 \pmod 3$ so the RHS "simplifies" to $1 \pmod 3$ but I don't understand why the LHS "simplifies" to $x$. Can someone help me with this? $(a_n+a_{n+1})$ is convergent implies $(a_n/n)$ converges to $0$ Let $(a_n)$ be a sequence such that $(a_n+a_{n+1})$ is convergent. Prove that $(a_n/n)$ converges to $0$.   $(a_n+a_{n+1})$ convergent to $L$ means that for all $\epsilon$, there exists $N$ such that for all $n\geq N$, $|L-a_n-a_{n+1}|&lt;\epsilon$, or in other words $L-\epsilon&lt;a_n+a_{n+1}&lt;L+\epsilon$. How to proceed from here? Minimal polynomials and cyclic subspaces I'm trying to make my way through two problems in Curtis's Linear Algebra, chapter 25.  One of the two problems is this one, #5:     Prove that $V$ is cyclic relative to a linear transformation $T \in \mathcal{L}(V)$ if and only if the minimal polynomial of $T$ is equal to the characteristic polynomial.   Part of the issue is that I'm not really sure what it's asking. The statement that $V$ is cyclic seems to suggest that the entire vector space (say it has dimension $n$) is generated by some vector $v \in V$, along with $Tv, T^2v, \ldots T^{n-1}v$; Curtis's definition of a cyclic subspace (as opposed to a space, though I can't see why there should be a difference) seems to reinforce this:     A $T$-invariant subspace $V_1$ of $V$ is called cyclic relative to $T$ if $V_1 \neq 0$ and there exists a vector $v_1 \in V_1$ such that $V_1$ is generated by $\{ v_1, Tv_1, T^2v_1, \ldots T^kv_1 \}$ for some $k$.   However, I've seen several examples that seem to directly contradict this interpretation. For example, we recently had a take-home midterm that required we find elementary divisors, rational canonical form, etc., of a $6\times 6$ matrix; a classmate who received a perfect score on her test found that her matrix's characteristic polynomial was equal to its minimal polynomial &#151; it was $(x+8)^2 (x+14)^2 (x^2+61x+80)$ &#151; and yet the vector space could be written as $\left&lt; v_6 \right&gt; \oplus \left&lt; v_4 \right&gt; \oplus \left&lt; -v_2+v_3+v_6 \right&gt;$ &#151; i.e., as the direct sum of three cyclic subspaces, not as one cyclic subspace.  So that suggests that "$V$ is cyclic" might mean the direct sum of cyclic subspaces. However, while working on the exercise that follows (#6), I came across this:  $$S= \left(\begin{array}{ccc} -1 &amp; 0 &amp; 0 &amp; 0\\ 0 &amp; 1 &amp; 0 &amp; 0\\ 0 &amp; 0 &amp; 1 &amp; 0\\ 0 &amp; 0 &amp; 0 &amp; 2 \end{array}\right)$$  Here, the characteristic polynomial is not equal to the minimal polynomial, which is $m(x) = (x-1)(x+1)(x+2)$. The eigenspace associated with $\lambda = 1$ has dimension $2$. And so it seems as though, if $V$ has basis $\{ v_1, v_2, v_3, v_4 \}$, we can write $V = \left&lt; v_1 \right&gt; \oplus \left&lt; v_2 \right&gt; \oplus \left&lt; v_3 \right&gt; \oplus \left&lt; v_4 \right&gt; $. This would seem to contradict my second interpretation of the original question.  Obviously I'm missing something. Could someone fill me in on what it might be? (I've tried to be thorough, but I'm also trying not to make this ridiculously long; if I've left out necessary information, I'll be happy to append it.) MATLAB superclass can not be found on the MATLAB's search path Is it possible to keep publishing under my professional (maiden) name, different from my married legal name? A reward becomes a penalty if I am working to build a reinforcement agent with DQN. The agent would be able to place buy and sell orders for a day trading purpose. I am facing a little problem with that project. The question is "how to tell the agent to maximize the profit and avoid the transaction where the profit is less than 100$".  I want to maximize the profit inside a trading day and avoid to place the pair (limit buy order, limit sell order) if the profit on that transaction is less than 100$. The idea here is to avoid the little noisy movements. Instead, I prefer long beautiful profitable movements. Be aware that I thought using the "Profit &amp; Loss" as the reward.  "I want the minimal profit per transaction to be 100$" ==> It seems this is not something that is enforceable. I can train the agent to maximize profit per transaction, but how that profit is cannot be ensured.  At the beginning, I wanted to tell the agent, if the profit of a transaction is 50 dollars, I will remove 100 dollars, then It becomes a penalty of 50 dollars for the agent. I thought it was a great way to tell the agent to not place a limit buy order if you are not sure it will give us a minimal profit of 100$. It seems that all I would be doing there is simply shifting the value of the reward. The agent only cares about maximizing the sum of rewards and not taking care of individual transactions.   How to tell the agent to maximize the profit and avoid the transaction where the profit is less than 100$? With that strategy, what guarantee that the agent will never make a buy/sell decision that results in less than 100 dollars profit? Does the sum of reward - # transaction * 100 can be a solution? Finding a set of continuous functions with a certain property 2 I need help finding the set of continuous functions $f : \Bbb R \to \Bbb R$ such that for all $x \in \Bbb R$, the following integral converges:  $$\int_0^1 \frac {f(x+t) - f(x)} {t^2} \ \mathrm dt$$  I think it might be the set of constant functions but i havent been able to prove it :(   I was thinking that you can use the stone weiestrass theorem considering the set of continuous functions on a closed  interval(non trivial) ,and a subset which contains the set of continuous functions whose integral above diverges in some point in that interval along with  with the set of constant functions. So in order to solve the problem i need only to prove that if two functions do not meet the condition of the problem then their product does not as well .  I hope you can provide some insight and thank you . How well does jQuery support backward compatibility? We have had issues with Mootools not being very backward compatible specifically in the area of drag and drop functionality. I was wondering if anyone has had any similar problems with jQuery not being backward compatible. We are starting to use it quite heavily and are thinking about upgrading to a newer version to start using several plugins that require it. Will we have any issues if we get rid of the older version? How to see python symbolic names of objects on cursor hover of menu items in Blender 2.8 in Blender 2.79 one could hover over a UI element and see the symbolic python name of said element. How would one accomplish the same with Blender >= 2.8? Which types of particles are affected by the wave-particle duality? I would like to have \alph{counter} for values larger than 26. The following MnWE gives me the error ! LaTeX Error: Counter too large.  \documentclass{article} \begin{document}  \begin{enumerate} \def\theenumi{\alph{enumi}} \setcounter{enumi}{24} \item twenty-five \item twenty-six \item twenty-seven \end{enumerate}  \end{document}   I would like it to behave like one of these:  ..., x, y, z, aa, bb, cc, ..., xx, yy, zz, aaa, bbb, ccc, ... (preferred) ..., x, y, z, aa, ab, ac, ..., ax, ay, az, ba,  bb,  bc,  ... (can be if nothing else works)   A solution without LaTeX3 extensions would be awesome as I cannot run LaTeX3 extensions. fontawesome and pdflatex When using pdflatex and the fontawesome package the below error shows up.   !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! ! Fatal fontspec error: "cannot-use-pdftex" !  ! The fontspec package requires either XeTeX or LuaTeX to function. !  ! You must change your typesetting engine to, e.g., "xelatex" or "lualatex" ! instead of plain "latex" or "pdflatex". !  ! See the fontspec documentation for further information. !  ! For immediate help type H &lt;return&gt;. !...............................................     Wish to keep on using "pdflatex" instead of "xelatex" or "lualatex".   In fontawesome Version 4.4.0 manual it says "When using the (pdf)(LA)TEX engine, the fontawesome package doesn’t require any external package".  Tried solutions proposed for this question () without success.   MWE:  \documentclass{article} \usepackage{fontawesome} \begin{document} {\faAdjust} \end{document} How do high traffic sites service more than 65535 TCP connections? Book a return flight a year in advance Is it possible to buy an airline ticket that has me returning to my country after a year being abroad? I can't seem to book a flight that far in advance online. Removing spaces between "words" in math mode A general explanation of rim, tire and tube specifications Can I use the "Scan" button with SANE? I have an HP Deskjet with integrated scanner. It has a "Scan" button on its top, but this button does not interact with the xsane program on my Debian machine: To get a page scanned, I have to initiate the scanning process from xsane.  Is there a setting (or a scan program) that allows me to press the scan button and cause the program to scan a page? Cell reference with colour formatting Is it possible to reference a cell in Google Sheets so that cell where it gets displayed also displays it using the same text and cell colour formatting?  =A1   Will only reference cell's value. But if that particular cell has red background and white text I'd like that to be copied as well.  I'm leaning toward existing formula solutions rather than scripts. If applicable of course. Where are the outliner toggles in 2.8? The selection toggle arrows in the Outliner disappeared. Is there some preference in the settings that has to be turned on? Below is an example from I.N. Herstein:       Let  $R=\Bigg\{\left( \begin{array}{ccc} a &amp; b \\ -b &amp; a \end{array} \right)\Bigg|a,b\in \mathbb R\Bigg\}$ and let $\mathbb C$ be the field of complex numbers. Define $\psi :R\to \mathbb C$ by $\psi\left( \begin{array}{ccc} a &amp; b \\ -b &amp; a \end{array} \right)=a+bi.$       It is asked to prove that $\psi$ is a ring isomorphism of $R$ onto $\mathbb C$ .    can anyone help me with some hint how to prove the ring isomorphism.