Show that $\sum_n \frac{1}{a_n}\lt90$
Sum of reciprocals of numbers with certain terms omitted
Is  (i.e. repulsive gravity) possible in string theory?  I have read some articles about scientists making assumptions about the existence of anti-gravity, but is it possible in string theory? For some reason I can't unhide my objects anymore in this file. Is it a bug? Publishing a conference version of a journal paper at the same time ? (copyright issues) Slave duplicate counter Is it possible to create a counter that is a "slave" to another (master) counter, but also an exact duplicate?  The \@addtoreset{&lt;slave&gt;}{&lt;master&gt;} resets &lt;slave&gt; whenever &lt;master&gt; is incremented (similar functionality is offered by ), but I don't know about anything incrementing one counter (&lt;slave&gt;) automatically whenever another counter (&lt;master&gt;) is incremented.  The idea is to have a macro (say) \dupcntr{&lt;slave&gt;}{&lt;master&gt;} as an analogue to \@addtoreset. The application for this might be where one is interested in renaming a counter used/provided by a package into something more meaningful for the user (say, &lt;slave&gt;), without having to reference &lt;master&gt;. Or, if the original counter contains @ that needs to be escaped every time it's used, it is sometimes more convenient to have a "more user friendly" counter.  I know one way around is to use a macro that returns the value of &lt;master&gt;, but is it possible to perform the same manipulation with a macro than one could with the counter? Manipulation here refers to the printing of the counter (via \the&lt;slave&gt;) and perhaps it's representation (like \arabic, \alph, ...). Deriving Schrodinger equation from Klein-Gordon QFT with the definition $\psi(\textbf{x},t)\equiv \langle 0|\phi_0(\textbf{x},t)|\psi\rangle$ In the book "Quantum Field theory and the Standard Model" by Matthew Schwartz, , the position space wavefunction is defined as   $$\psi(x)=\langle 0|\phi(x)|\psi\rangle, \tag{2.82+2.83}$$  where $|\psi\rangle$ is any state in the Fock space. Then he uses the equations (i) $\partial_t^2\phi_0=(\nabla^2-m^2)\phi_0$ (i.e., the Klein-Gordon equation for the free massive scalar field $\phi_0(\textbf{x},t)$) and (ii) $[H,\phi_0]=-i\partial_t\phi_0$ to derive Eq. 2.85, in following 3 steps:  $$i\langle 0|\partial_t\phi_0(\textbf{x},t)|\psi\rangle=\langle 0|\int \frac{d^3\textbf{p}}{(2\pi)^3}\frac{\sqrt{\textbf{p}^2+m^2}}{\sqrt{2\omega_\textbf{p}}}(a_p e^{-ip\cdot x}-a^\dagger_p e^{ip\cdot x})|\psi\rangle\\ =\langle 0|\sqrt{m^2-\nabla^2}\phi_0(\textbf{x},t)|\psi\rangle.\tag{2.85}$$  This equation is used to successfully derive the Schrodinger equation in quantum mechanics for the state $\psi(\textbf{x},t)\equiv \langle 0|\phi_0(\textbf{x},t)|\psi\rangle$.   The first equality follows from differentiation of $\phi_0(x)$ w.r.t $t$. How does the second equality follow from the first? How are the inputs (i) and (ii) are utilized to derive the second equality from the first? Is it true that the operator $$i\partial_t\sim\sqrt{m^2-\nabla^2}~?$$ If yes, what is the use of doing the middle step? Forgotten the pattern and phone not showing the "forgot pattern" button. I upgraded my Moto G2 to Android Lollipop using over-the-air upgrade. It was awesome but I set a pattern lock which I forgot. Usually after 5 wrong attempts it shows the "forgot password" button but it is not showing that button even after 100 wrong attempts. How to unlock the phone? Proof of upper-tail inequality for standard normal distribution $X \sim \mathcal{N}(0,1)$, then to show that for $x &gt; 0$, $$ \mathbb{P}(X&gt;x) \leq \frac{\exp(-x^2/2)}{x \sqrt{2 \pi}} \&gt;. $$ Let $U\subset \mathbb{C}$ be a nonempty connected open set such that for every $z\in U$, $\overline z\in U$.  Let $f$ be analytic on $U$. Suppose $f(x)\in\mathbb R$ for every $x\in U\cap\mathbb R$. Prove that $f(\overline{z})=\overline{f(z)}$ for any $z \in U$.    By definition, I know that $f$ analytic on $U$ means that for every $z_0 \in U$, there exists $r&gt;0$ and a sequence of complex numbers $\left(a_n\right)_{n=0}^\infty$ such that $f(z)=\sum_{n=0}^\infty a_n\left(z-z_0\right)^n$ on the disc $D(z_0,r)\subseteq U$. I see that $f$ takes points without imaginary components to other points without imaginary components. But I don't see how this implies a symmetry that $f(\overline{z})=\overline{f(z)}$ for $z \in \mathbb C \setminus \mathbb R$. It seems like a very strong conclusion and I'm not sure how I would prove it.  I have also shown in the preceding question that $U\cap\mathbb R$ contains an open interval, however I am not sure if that detail is meant to be helpful to this question. Solve the equation $x^2+\frac{9x^2}{(x+3)^2}=27$ Java2D Performance Issues I'm having performance oddities with Java2D. I know of the sun.java2d.opengl VM parameter to enable 3D acceleration for 2D, but even using that has some weird issues.   Here are results of tests I ran:  Drawing a 25x18 map with 32x32 pixel tiles on a JComponent Image 1 = .bmp format, Image 2 = A .png format  Without -Dsun.java2d.opengl=true  120 FPS using .BMP image 1 13 FPS using .PNG image 2  With -Dsun.java2d.opengl=true  12 FPS using .BMP image 1 700 FPS using .PNG image 2  Without acceleration, I'm assuming some kind of transformation is taking place with every drawImage() I do in software, and is pulling down the FPS considerably in the case of .PNG. Why though, with acceleration, would the results switch (and PNG actually performs incredibly faster)?! Craziness!  .BMP Image 1 is translated to an image type of TYPE_INT_RGB. .PNG Image 2 is translated to an image type of TYPE_CUSTOM. In order to get consistent speed with and without opengl acceleration, I have to create a new BufferedImage with an image type of TYPE_INT_ARGB, and draw Image 1 or Image 2 to this new image.   Here are the results running with that:  Without -Dsun.java2d.opengl=true  120 FPS using .BMP image 1 120 FPS using .PNG image 2  With -Dsun.java2d.opengl=true  700 FPS using .BMP image 1 700 FPS using .PNG image 2  My real question is, can I assume that TYPE_INT_ARGB will be the native image type for all systems and platforms? I'm assuming this value could be different. Is there some way for me to get the native value so that I can always create new BufferedImages for maximum performance?   Thanks in advance... How to get world-space matrix of any pose bone? Is there a command for getting world-space matrix of pose bones, same as obj.matrix_world for objects? I found pose_bone.matrix, but it gives weird result for me. Find $ P(Z>X+Y)$ where $X,Y,Z \sim U(0,1)$ independently I'm trying to follow a line in a derivation for $P(Z&gt;X+Y)$ where $X,Y,Z$ are independent continuous random variables distributed uniformly on $(0,1)$.  I've already derived the pdf of $X+Y$ using the convolution theorem, but there's a line in the answer that says:  $P(Z&gt;X+Y) = \mathbb{E}[\ P(Z&gt;X+Y\ |\  X+Y )\  ]$ where $\mathbb{E}$ is the expectation.  I'm not familiar with this result. Could anyone give a pointer to a similar result if one exists?  Thanks. php String Concatenation, Performance In languages like Java and C#, strings are immutable and it can be computationally expensive to build a string one character at a time.  In said languages, there are library classes to reduce this cost such as C# System.Text.StringBuilder and Java java.lang.StringBuilder.  Does php (4 or 5; I'm interested in both) share this limitation?  If so, are there similar solutions to the problem available? Proving that $\arctan(x)+\arctan(1/x)=\pm \pi/2$, could this line of reasoning possibly be correct? How to mount a virtual hard disk? Is it possible to mount a virtual hard disk (VHD, HDD, VDI, VMDK) on Ubuntu? How can this be done? I want to apply a macro to all the cells of a column with the cell content as the input to the macro. Is this possible?  For example if the cell contents have the word TEST, the macro should take TEST as its argument. How to catch and print the full exception traceback without halting/exiting the program? When I open a window to air out the room, how does the smell disperse? Let's say I'm in a room with some kind of noxious stink, possibly of flatulent nature.  The quickest way to right the world that comes to mind is to open a window.  When I open a window, how do the stank particles leave the room? Fluent and Query Expression — Is there any benefit(s) of one over other? How to return only the Date from a SQL Server DateTime datatype SELECT GETDATE()   Returns: 2008-09-22 15:24:13.790  I want that date part without the time part: 2008-09-22 00:00:00.000  How can I get that?