Calculation & derivation of de-Broglie wavelength
Calculating de Broglie wavelength
What is the best approach for solving the equations of the form      $p^x = x^q$ ; where $p$ and $q$ are some constants.   Basically, I wanted to solve $2^x = x^2$ to analyze algorithms with respective growth functions. Why does an atom remain uncharged after emission of an alpha particle? When an alpha particle is emitted, two protons and two neutrons leave the nucleus but the electrons remain the same in number. Why does the atom remain uncharged although it appears it should have a net charge of $-2e$? Can players on a PC and an Xbox play together in any way? I.e. can Xbox users connect to a (modfied?) PC server, or vice versa? Make links posted by users open in a new window Most of the time when I follow a user's link, I just want to see their site, look it over, then return to answer their question (not leave SO completely and not come back). Could we make the links users post open in a new window please?  EDIT: I originally wanted this implemented because it's what some other sites did, but I now definitely prefer SO's way. If I want to open a new tab, I middle-click/control-click (or whatever); if not, I just click. I'm in control of what happens. Would never want to go back now. Is it possible to make my laptop's screen act as a second monitor for my desktop computer? How many integers between $1$ and $10^4$ contain exactly one $8$ and one $9$? I have this problem: How many integers between $1$ and $10^4$ contain exactly one $8$ and one $9$?, but i dont know if $18, 19, 28, 29...$ serve, else it could be $8$ y $9 = 2$ numbers, help please. Everything was working fine for a long time. But suddenly the device is getting slower than usual (e.g. apps react slowly when starting an action -- like the RSS reader takes "ages" to open an article). Also, several apps start crashing: force-closes as well as unexpected restarts. Sometimes this even involves some automatic reboot of the device.  Is there anything I can do, besides doing a factory-reset? Preferably it should work on un-rooted devices.  Remark: not all described problems may occur together, so on one device e.g. there might be no automatic reboot involved. Question remains the same, nonetheless. Why None is the smallest in python? Say an airplane flies at Mach 1. The doppler effect causes the front of the sound emitted by the plane to reach infinite frequency:    $f = \left ( \frac {343+0}{343-343} \right ) f_0 = ∞$  If the plane suddenly stopped, and there was an observer in front of the sound waves to receive the sound, what would the bunched up waves sound like to the observer? Would they cancel each other out and cause no sound? Assume $f:\mathbb R\rightarrow  {\mathbb R} $ is Lebesgue integrable, prove $\sum_{n\in\mathbb N } f(x+n)$ converges for almost everywhere x. This is a question discussed with friends. Thanks Prove that $T$ is an orthogonal projection Let $T$ be a linear operator on a finite-dimensional inner product space $V$. Suppose that $T$ is a projection such that $\|T(x)\| \le \|x\|$ for $x \in V$. Prove that $T$ is an orthogonal projection.    I can't understand well. The definition of orthogonal operator is $\|T(x)\| = \|x\|$. But why that $\|T(x)\| \le \|x\|$ for $x \in V$ means orthogonal projection? JavaScript equivalent to printf/String.Format I'm looking for a good JavaScript equivalent of the C/PHP printf() or for C#/Java programmers, String.Format() (IFormatProvider for .NET).  My basic requirement is a thousand separator format for numbers for now, but something that handles lots of combinations (including dates) would be good.  I realize Microsoft's  library provides a version of String.Format(), but we don't want the entire overhead of that framework. Calculationg base-resistor between microcontroller and transistor I'm curious, when connecting my 2N2222 transistor's base to a PWM output pin on my Arduino, I know the Arduino can deliver a maximum of 40mA current. The 2N2222 only requires around 5 to 10 mA on it's base to switch the collector and emitter. Now, when calculating the base-resistor under "normal" conditions where I'm merely switching it directly from Vcc, the following applies:   However, when putting in a 500 ohm or 1K resistor in (1K for 5mA) directly it works great. But putting it in between the Arduino and the transistor, the transistor doesn't switch.  I suspect thus, that the resistance is too high of the base-resistor. Therefore, we may very well not need a base-resistor.  But, according to spec, the Arduino can deliver 40mA. So, am I correct in assuming the folling: 1) I should calculate the perceived resistance that the Arduino is giving, if it goes high to 5Volt and only gives 40mA, and then 2) Subtract that from the 500ohm resistor needed to make the transistor operate optimally.  Example:    So, I'm theorising that I have to then put in a 375 ohm resistor in to make the total resistance between the Arduino and the base resistor 500 ohm.  Somehow this doesn't feel right. Keep in mind, I'm not an electronic engineer, so I might have the cat by the tail here :P Why does Windows only show about 3.5 GB of my 4 GB of RAM? I recently upgraded my computer's RAM to 4&nbsp;GB. My 32-bit Windows installation shows only 3574&nbsp;MB of the memory. How can I make Windows use the full amount of RAM? Martingale formulation of Bellman's Optimality Principle From David Williams' Probability with Martingales     Related question:   What I tried: For no 2, if $\ln Z_n - n \alpha$ is a supermartingale, then for $m &lt; n$, $$E[\ln Z_n - n \alpha | \mathscr F_m] \le \ln Z_m - m \alpha$$ $$ \to E[\ln Z_N - N \alpha | \mathscr F_0] \le \ln Z_0 - 0 \alpha$$ $$ \to E[\ln Z_N - N \alpha] \le \ln Z_0$$ $$ \to E[\ln \frac{Z_N}{Z_0}] \le N \alpha$$ Is that right?  For no 1, for $m &lt; n$, $$E[\ln Z_n - n \alpha | \mathscr F_m] = E[\ln Z_n | \mathscr F_m] - n \alpha$$ $$ = E[\ln \frac{Z_n}{Z_0} | \mathscr F_m] - n \alpha + \ln Z_0$$ $$ = E[\ln \frac{Z_n}{Z_{n-1}} | \mathscr F_m] + ... + E[\ln \frac{Z_{m+1}}{Z_{m}} | \mathscr F_m]$$ $$+ \ln \frac{Z_m}{Z_{m-1}} + ... + \ln \frac{Z_{1}}{Z_{0}} - n \alpha + \ln Z_0$$ Note that $$Z_n - Z_{n-1} \le |Z_n - Z_{n-1}| \le C_n |\epsilon_n - \epsilon_{n-1}| \le 2C_n \le 2Z_{n-1}$$ Also, I think $$\ln \frac{Z_n}{Z_{n-1}} \le |Z_n - Z_{n-1}|$$ Hence we have $$E[\ln \frac{Z_n}{Z_{n-1}} | \mathscr F_m] + ... + E[\ln \frac{Z_{m+1}}{Z_{m}} | \mathscr F_m]$$ $$+ \ln Z_m - n \alpha$$ $$\le E[2Z_{n-1} | \mathscr F_m] + ... + E[2Z_{m} | \mathscr F_m]$$ $$+ \ln Z_m - n \alpha$$ $$= 2(E[Z_{n-1} | \mathscr F_m] + ... + E[Z_{m} | \mathscr F_m])$$ $$+ \ln Z_m - n \alpha$$ One thing to do would be to show that $$2E[Z_{n-1} | \mathscr F_m] + ... + 2E[Z_{m} | \mathscr F_m] \le (n-m) \alpha$$ possibly by showing that $2E[Z_{\{\cdot\}} | \mathscr F_m] \le \alpha$ How would I do that? Something else: $$E[\ln \frac{Z_n}{Z_m} | \mathscr F_m] + \ln Z_m - n \alpha$$ $$ \le E[|Z_n - Z_m| | \mathscr F_m] + \ln Z_m - n \alpha$$ $$ \le E[2Z_m | \mathscr F_m] + \ln Z_m - n \alpha$$ $$ \le 2Z_m + \ln Z_m - n \alpha$$ Now $2Z_m \le (n-m) \alpha?$ We have $$2Z_m = 2\sum_{k=1}^{m} C_k(\epsilon_k - \epsilon_{k-1})$$ $$\le 2\sum_{k=1}^{m} Z_{k-1}$$ I'm not quite sure how to show that $$2\sum_{k=1}^{m} Z_{k-1} \le (n-m)\alpha$$ if that's even true. How can/else can I approach this problem?  For no 3, no idea. Any hints? I think it has something to do with a stopping time. Might we have to use the ff lemma (red box):    ? Alternative definition of hyperbolic cosine without relying on exponential function Ordinary trigonometric functions are defined independently of exponential function, and then shown to be related to it by Euler's formula.   Can one define hyperbolic cosine so that the formula $$\cosh{x}=\dfrac{e^x+e^{-x}}{2}$$ becomes something to be proven? Activate Rigid Body Physics at certain time? I've been trying to make a wrecking ball smash through a figure of sorts, not really sure what exactly but I made an example. The problem I'm having is that the figure does indeed fall apart, but I would like it to stay put together until the ball hits it. I achieved this by using Cell Fracture on the figure and just using Rigid Body. Again, I just need a way for the figure to stay still until hit by the ball, and then have all of its pieces come apart. Etymology of "cut someone some slack" Teenagers. All the literature tells you one thing and one thing only – that  whatever they are doing, give them a break, cut them some slack, it's normal.      From the novel, Apple Tree Yard   I'm curious about the origins of . I know slacks are American English for trousers or pants, and a slacker is a lazy person. But what is a slack and how does one "cut" it? What is a good word for “best example”? What is a good word for “best example”? I would like to describe a film as the greatest example of its features. how to plot two graphs in a single sheet? I am new to Latex. Cann some one tell me how can I plot two graphs in a single page and How can resize the legend. I have attached the picture how the legend looks. I need it out from the plot and how can i have gridded lines in the plot. Please help me. My code is as follows: \documentclass[12pt, a4paper]{report} \usepackage{filecontents,pgfplots}  \begin{filecontents}{mydata.dat} m0_FEED, AT 773.15K, AT 823.15K, AT 873.15K, AT 923.15K, AT 973.15K 0.005, 1.30E+04, 12959.71536,12959.71536,12959.71536,12959.71536 0.007, 1.81E+04, 18145.78478,18145.78478,18145.78478,18145.784 0.009, 2.33E+04, 23333.9541,23333.9541,23333.9541,23333.9541 0.011, 2.85E+04, 28524.71381,28524.71381,28524.71381,28524.71381 0.013, 3.37E+04, 33718.48576,33718.48576,33718.48576,33718.48576 0.015, 3.89E+04, 38915.61947,38915.61947,38915.61947,38915.61947 0.017, 4.41E+04, 44116.39602,44116.39603,44116.39603,44116.39603 0.019, 4.93E+04, 49321.02848,49321.02848,49321.02878,49321.02849 0.021, 5.45E+04, 54529.67237,54529.67238,54529.67238,54529.67238 \end{filecontents}  \begin{document}   \pgfplotstableread[col sep=comma]{mydata.dat}{\mydata} \begin{tikzpicture}[scale=1.8] \begin{axis}[minor tick num=1, xlabel=Degrees, ylabel = temperature] \addplot [black,very thick] table [x={m0_FEED}, y={AT 773.15K}] {\mydata}; \addplot [red,very thick] table [x={m0_FEED}, y={AT 823.15K}] {\mydata}; \addplot [blue,very thick] table [x={m0_FEED}, y={AT 873.15K}] {\mydata}; \addplot [green,very thick] table [x={m0_FEED}, y={AT 923.15K}] {\mydata}; \addplot [dashed,yellow,very thick] table [x={m0_FEED}, y={AT 973.15K}] {\mydata}; \legend{AT773.15K, AT823.15K, AT873.15K, AT923.15K, AT973.15K} \end{axis} \end{tikzpicture}  \begin{filecontents}{mydatax.dat} m0_FEED AT 773.15K  AT 823.15K  AT 873.15K  AT 923.15K  AT 973.15K 0.005   4.06E+03    4601.511738 5147.966831 5703.120845 6267.035737 0.007   5.69E+03    6439.241858 7204.278989 7981.494608 8625.923 0.009   7.31E+03    8274.205759 9257.824928 10257.10215 11272.14896 0.011   8.92E+03    10105.75535 11307.95656 12529.29539 13769.90815 0.013   1.05E+04    11933.33165 13354.1149  14797.51533 16263.69405 0.015   1.21E+04    13756.46929 15395.83457 17061.29661 18753.04129 0.017   1.37E+04    15574.79159 17432.7389  19320.26254 21237.57317 0.019   1.53E+04    17388.009   19464.53834 21574.12348 23717.00014 0.021   1.69E+04    19195.90581 21491.01717 23822.664   26191.10653 \end{filecontents}  \pgfplotstableread{mydatax.dat}{\mydatax} \begin{tikzpicture}[scale=2] \begin{axis}[minor tick num=1, xlabel=Degrees, ylabel = temperature] \addplot [black,very thick] table [x={m0_FEED}, y={AT 773.15K}] {\mydatax}; \addplot [red,very thick] table [x={m0_FEED}, y={AT 823.15K}] {\mydatax}; \addplot [blue,very thick] table [x={m0_FEED}, y={AT 873.15K}] {\mydatax}; \addplot [green,very thick] table [x={m0_FEED}, y={AT 923.15K}] {\mydatax}; \addplot [dashed,yellow,very thick] table [x={m0_FEED}, y={AT 973.15K}] {\mydatax}; \legend{AT773.15K, AT823.15K, AT873.15K, AT923.15K, AT973.15K} \end{axis} \end{tikzpicture}   \end{document}