I have almost solved the question but I am stuck in the last part we know $X_1 , X_2,......,X_{(n-1)}$ and $Y_1, Y_2 ,.....Y_{(n-1)}$ can be filled only in one way, so the question reduces to filling the $(n-1) \times (m-1)$ table with $±1$ with out any condition i.e $2^{(n-1)(m-1)}$ but what about the green ( right bottom) box how can we fill that box?
How many $m\times n$ tables are there if they are subject to the following two conditions?    In each cell of the table we have $1$ or $-1$. The product of all the cells in any given row and the product of all cells in any given column equals $-1$.    I already know if Parity of $m$ and $n$ are different there is no such a table. what is the answer in general?
Total order on complex numbers How to parse hex values into a uint? Relation between rank and number of distinct eigenvalues of a matrix Let $T : V\to V$ be a linear transformation such that $\dim\operatorname{Range}(T)=k\leq n$, where $n=\dim V$. Show that $T$ can have at most $k+1$ distinct eigenvalues.   I can realize that the rank  will correspond to the number of non-zero eigenvalues (counted up to multiplicity) and the nullity will correspond to the 0 eigenvalue (counted up to multiplicity), but I cannot design an analytical proof of this.  Thanks for any help . When discussing a treaty or international agreement, which is correct?  "There are 100 states party to the treaty." (for example, as used )  Or:  "There are 100 state parties to the treaty." (for example, as used )   Or: "There are 100 states parties to the treaty."  (for example, as used ) I'm new to Linux but not to app development. I come from a RealBasic on a Mac background.   I've accepted that I need to learn a new language to develop for Linux (since RB won't run on 64 bit distros and is a cloed proprietary language) but I want to make sure I pick the right one.   I've been looking at Python and was all set to go until I read that PyGTK is becoming obsolete with the introduction of GTK 3. Is this true? The Ubuntu developer website stills seems to recommend it with Glade for the UI but I want to make sure that I don't learn something that's on the way out! I have a problem with my touchpad; I can't scroll down with 2 fingers and I don't know why. I have searched on Google about my problem and I've found some answers but none of them don't work for me.  One solution I found was to go in System Settings -&gt; Mouse &amp; Touchpad -&gt; Touchpad and check the Two-finger scrolling, but I don't have that option.    Another solution was to try to reinstall, which also didn't work. Yet another solution was to go in Ubuntu Software Center, type synaptic and install the Synaptic Package Manager but that didn't work, either.      What can I do to be able to scroll with two fingers? Let $p$ be prime number ($p\gt2$) and $a,b\in\mathbb Z$ ,$a+b\neq0$ ,$\gcd(a,b)=1$ how to  prove that $$\gcd\left(a+b,\frac{a^p+b^p}{a+b}\right)=1~~\text{or}~~ p$$  Thanks in advance . When you pass a collection like list, array to another function in python, does it make a copy of it, or is it just a pointer? Unknown Syntax Error in Code So, I am following a , and it is using Python code to create an audio visualizer in Blender. Now, I've always been a coding geek, but I don't actually know Python. I've taken a few shots at it but overall, I don't know Python, however, I can read and understand the code because I know basic programming concepts. Now, I'm following this tutorial as closely as I can (the tutorial does not provide code to copy and paste) and I am getting a syntax error. As far as I can tell, there is no syntax error. The line of code (specified by the error message) appears perfect. This line of code is supposed to set the 3D cursor to the active object's origin. Here's the catch: if I run that exact same line of code in the console, it executes just fine. So here's the error message I get:  File "C:\Me\Blender\Blender Projects\audio visualisation.blend\AudioVisualizer", line 17 bpy.context.scene.cursor_location = bpy.context.active_object.location    ^ SyntaxError: invalid syntax location: &lt;unknown location&gt;:-1   It tells me to look in the console for error message, but the console doesn't output anything.    Thanks!  EDIT: Alright. I've tried all the examples and all the procedures that have been suggested by you guys, but it keeps feeding me errors. According to the console, one of the problems it is spitting out is in C:\Program Files\Blender Foundation\Blender\2.74\scripts\modules\bpy\ops.py. This makes me think there is a bug in the API that is causing one of the many errors. EDIT: Here are the error messages I get for my code. Original Code:  Error:   File "C:\Gannon\Blender\Blender Projects\audio visualisation.blend\Audi o Visualizer", line 35     bpy.ops.graph.sound_bake(filepath="C:\Gannon\Blender\Sounds\Music\Batty McFa dden.mp3", low=i*step high=i*step+step                           ^ SyntaxError: invalid syntax    gandalf3's Code and zeffii's Code are essentially the same with alternative formatting which changes the line numbers where the error occurs.  Error: Unsupported audio format  Traceback (most recent call last):   File "C:\Gannon\Blender\Blender Projects\audio visualisation.blend\Audio Visua lizer", line 42, in &lt;module&gt;   File "C:\Gannon\Blender\Blender Projects\audio visualisation.blend\Audio Visua lizer", line 35, in spiral   File "C:\Program Files\Blender Foundation\Blender\2.74\scripts\modules\bpy\ops .py", line 189, in __call__     ret = op_call(self.idname_py(), None, kw) RuntimeError: Error: Unsupported audio format Can “due to” and “because of ” be used interchangeably? Is it fine to use due to in place of because of ? How about the other way around? Are any of these sentences ungrammatical?   He was lost because of the storm. He was lost due to the storm. He lost his way due to the storm. He  lost his way because of the storm. Find all solutions of the equality $y^2=x^3+23$ for integers $x,y$   I guess, that x cannot be even. because we can apply mod 4 test  say $x=2k$ then $y^2=8k^3+23\equiv3\mod(4)$   but, this is not possible, since the square of an integer is congruent to $0$ or $1$ $\mod (4)$  so $x$ is of the form $4k+3$ or $4k+1$  $x=4k+3$ is also not possible for the same reason but for the last case the test fails.  any hints ? How to implement a single instance Java application? Sometime I see many application such as msn, windows media player etc that are single instance applications (when user executes while application is running a new application instance will not created).  In C#, I use Mutex class for this but I don't know how to do this in Java. Technique for finding the centerline of a curved object? I'm curious to know if anyone knows of any effective techniques for finding the center line of a curved object.  In the screenshot below you'll see a crude version of what I mean.     Is there any way to get that line to perfectly trace the center, or is it easiest to just eyeball it?  Photoshop or Illustrator techniques are acceptable. Scattering, Perturbation and asymptotic states in LSZ reduction formula I was following Schwarz's book on quantum field theory. There he defines the asymptotic momentum eigenstates $|i\rangle\equiv |k_1 k_2\rangle$ and $|f\rangle\equiv |k_3 k_4\rangle$ in the S-matrix element $\langle f|S|i\rangle$ as the eigenstates of the full Hamiltonian i.e., $H=H_0+H_{int}$. Therefore, the states $|i\rangle=|k_1 k_2\rangle$ is defined as $$|k_1 k_2\rangle=a_{k_1}^{\dagger}(-\infty) a_{k_2}^{\dagger}(-\infty)|\Omega\rangle$$ where $|\Omega\rangle$ is the vacuum of the full interacting theory. Then the LSZ reduction formula connects the S-matrix element $\langle f|S|i\rangle$ to the Green' functions of the interaction theory defined as $$G^{(n)}(x_1,x_2,...x_n)=\langle \Omega|T[\phi(x_1)\phi(x_2)...\phi(x_n)]|\Omega\rangle.$$ Here are a few doubts. Doubt 1 When the particles are far away, the interaction can be considered to be adiabatically switched off. Therefore, at $t=\pm\infty$ the states are really free particle states and should have been written as $$|k_1 k_2\rangle=a_{k_1}^{\dagger}(-\infty) a_{k_2}^{\dagger}(-\infty)|0\rangle$$ and $$|k_3 k_4\rangle=a_{k_3}^{\dagger}(+\infty) a_{k_4}^{\dagger}(+\infty)|0\rangle$$ where $|0\rangle$ is the vacuum of the free theory.  I do not understand why these the states $|i\rangle$ and $|f\rangle$ are derived from $|\Omega\rangle$ instead of $|0\rangle$. Doubt 2 The initial and final states were derived from the vacuum of the interacting theory $|\Omega\rangle$. According to my understanding, this suggests that the states $|i\rangle\equiv |k_1 k_2\rangle$ and $|f\rangle\equiv |k_3 k_4\rangle$  are eigenstates of the full Hamiltonian $H$. Since then there is no perturbation, there should not be any scattering or transition at all.  More references Even Peskin &amp; Schroeder, Bjorken &amp; Drell, Srednicki take the same approach as Schwartz; they too define the external momentum eigenstates to the eigenstate of the full Hamiltonian $H$. However, if the system was initially in a stationary state why should it undergo a transition in absence of any perturbation? equal size numerator and denominator I am trying to size the numerator and denominator of the below equation so that they are equal in size.  I have not had much luck suggestions are appreciated.  \begin{equation} \sum_{i=1}^{n} \frac{\frac{cf_{n}}{(1+i)^n}} {\displaystyle\sum_{i=1}^{n}\frac{cf_n}{(1+i)^n}} , n_{i} \end{equation} .NET offers a generic list container whose performance is almost identical (see Performance of Arrays vs. Lists question). However they are quite different in initialization.   Arrays are very easy to initialize with a default value, and by definition they already have certain size:  string[] Ar = new string[10];   Which allows one to safely assign random items, say:  Ar[5]="hello";   with list things are more tricky. I can see two ways of doing the same initialization, neither of which is what you would call elegant:  List&lt;string&gt; L = new List&lt;string&gt;(10); for (int i=0;i&lt;10;i++) L.Add(null);   or  string[] Ar = new string[10]; List&lt;string&gt; L = new List&lt;string&gt;(Ar);   What would be a cleaner way?  EDIT: The answers so far refer to capacity, which is something else than pre-populating a list. For example, on a list just created with a capacity of 10, one cannot do L[2]="somevalue"  EDIT 2: People wonder why I want to use lists this way, as it is not the way they are intended to be used. I can see two reasons:   One could quite convincingly argue that lists are the "next generation" arrays, adding flexibility with almost no penalty. Therefore one should use them by default. I'm pointing out they might not be as easy to initialize. What I'm currently writing is a base class offering default functionality as part of a bigger framework. In the default functionality I offer, the size of the List is known in advanced and therefore I could have used an array. However, I want to offer any base class the chance to dynamically extend it and therefore I opt for a list. I absolutely hate it when I watch a video on YouTube and it automatically loads another video page after it's done playing.  This is especially annoying when I'm in the middle of typing a comment as it will cause me to lose what I have typed.  I'd like the video to simply stop and for no further action to take place once it is finished playing.  Is there any way to accomplish this? How to clean an E: drive (new Recovery drive since Windows 10 1803) on HP laptop Is there a way to replicate Nikon D3100 live view on my laptop screen? I own a Nikon D3100 and I need a way to see the live view of my camera on my laptop screen.  There must definitely be a way (I am not just aware of it). I have searched all around the web sites... I could not get a solution.  The Nikon Camera Control Pro 2 does not support D3100, and "Camera Control 5.1" connects and shoots remotely from a laptop but does not show live-view nor shoot in RAW.  There is a way of connecting my camera to a TV or monitor directly via an HDMI cable and getting the live view, but I want it displayed to my laptop screen. Where is the BIOS stored? From :     BIOS software is stored on a non-volatile ROM chip on the motherboard. ... In modern computer systems, the BIOS contents are stored on a flash memory chip so that the contents can be rewritten without removing the chip from the motherboard. This allows BIOS software to be easily upgraded to add new features or fix bugs, but can make the computer vulnerable to BIOS rootkits.   Since ROM means Read-Only Memory,  why can the BIOS contents be rewritten?  Does the "flash memory chip"  mean the same as the "non-volatile ROM", both meaning where BIOS is stored?