The Functional Equation $ ( x + y ) \big( f ( x ) - f ( y ) \big) = ( x - y ) f ( x + y ) $, need solution have answer
Functional Equation: Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $(x+y)(f(x)-f(y))=(x-y)f(x+y)$
How to make a jquery click event fire only on first click How to execute a script just by double clicking like .EXE files in Windows? For which $n≥2$ does there exist $\sigma\in A_n$ with $|\sigma| &gt; n$? I have a proof that I'd like to be verified. My proof that there exists $\sigma\in A_n$ with $|\sigma| &gt; n$ if and only if $n\geq 8$: For $n&lt;8$, one can manually verify for each $n$ that there is no $\sigma\in A_n$ with order greater than $n$. Suppose $n\geq 8$. Case 1: $n$ is even and $3\not\mid n$. Then the permutation $(1,2,3)(4,\dots,n)$ is even and has order $\operatorname{lcm}(3,n-3)=3n-9\geq n+16-9\geq n + 7$. Case 2: $n$ is even and $3\mid n$. Then $n\geq 12$, so $n-2\geq 8$ is even and is not divisible by 3. Applying Case 1, we see that $(1,2,3)(4,\dots,n-2)\in A_n$ has order at least $n+5$. Case 3: $n$ is odd. Then apply one of the previous two cases with $n-1$ to get an even permutation with order at least $n+4$. $\blacksquare$ Is my proof correct? It doesn't seem very elegant. Is there some slicker proof? I want to make a bootable USB thumb drive to install Windows on my computer. I don't want to use my old DVD to create an image and then upgrade to SP1, so I decided to download an untouched .iso file that already has SP1 in it. My question is this, is there a difference between these two files files?     en_windows_7_home_premium_with_sp1_x64_dvd_620851.iso   vs     en_windows_7_home_premium_with_sp1_x64_dvd_u_676549.iso   What does dvd_u mean, is it somehow different from plain dvd? Also will these work with my retail Home Premium key or do these work only for thoe with TechNet/MSDN accounts? How do they decide Blu-ray/DVD release date? Why didn't Gollum grow old and die after he lost the One Ring to Bilbo? In C#, what is the best way to access a property of the derived class when the generic list contains just the base class.  public class ClassA : BaseClass {    public object PropertyA { get; set; } }  public class ClassB: BaseClass {     public object PropertyB { get; set; } }  public class BaseClass { }  public void Main {     List&lt;BaseClass&gt; MyList = new List&lt;BaseClass&gt;();     ClassA a = new ClassA();     ClassB b = new ClassB();      MyList.Add(a);     MyList.Add(b);      for(int i = 0; i &lt; MyList.Count; i++)     {         //I would like to access PropertyA abd PropertyB from the derived classes             } } Sometimes when I'm making a sauce or soup that needs to be a bit spicy, I have no troubles spicing how I want to.  But if the dinner party is a bit late, or I made the sauce in advance to really draw out all the flavors of the spices, vegetables and meat, sometimes I end up with a way too spicy/hot dish.  You can always pour a bit water and try to dilute it a bit, but that also takes away all of the good flavor. So is there some trick that I'm missing I can use, when this happens to me (other than being more careful or using the spices later)? How can I reuse existing materials, objects, etc from an existing .blend? I have created several designs for a particular project and they are related, but I don't want to have to keep recreating any similar elements, such as shared materials or textures.  How can I use the existing data in a new Blender file? Is it correct to write $\Bbb E[X]$ or $\Bbb E_{\theta}[X]$? Suppose that we observe the discrete random variable $ X = (X_1, \dotsc , X_n)$ with state space S, whose distribution we do not know but we are assuming that its joint p.m.f. belongs to a known family $\{f_θ : θ ∈ Θ\}$. We derive a sufficient statistic $T(X)$.  Is it correct to write $\Bbb E[X]$? If not, why? Would it be more correct to write $\Bbb E_{\theta}[X]$? I am trying to create a low poly face with a face from a .png file. The issue is I already have material placed for hair and skin, as shown in the picture below.   I have created a .png file with a face which I would like to overlay over the skin material on the front of the head.    All examples I searched so far assume I also want the color of the material from the .png texture.  Update:  I added UVMapping, changed view to texture, made sure one face is on the eye but nothing is displayed..... see image... I don't know what is going on. If particles are points, then aren't atoms empty space? Zero dimensional points do not take up space, so then wouldn't everything in the universe be literally empty? Or is there something that I'm missing? Resize multiple layers around their respective anchor points We were discussing Kirchoff's theorem on blackbody radiation in class today and the equation was: $$e_{f}= J(f,T)$$  $e_f$ was defined as power per unit area per unit frequency. What does per unit frequency refer to here? I know what per unit time means but don't understand 'per unit frequency'. Making holes in a mesh How does JavaScript .prototype work? Why is Buoyant Force $V\rho g$? For a submerged object, buoyant force ($F_b$) is defined as:  $$F_b = V_{\text{submerged}} \times \rho \text{ (density)} \times g \text{ (gravitational constant)}$$   Conceptually, the buoyant force equation says that buoyant force exerted is equal to the weight of the volume of water a given object displaced. Why? I went online to   and found the following explanation, but it seems like non-sequitur to me:      Explanation: When an object is removed, the volume that the object occupied will fill with fluid. This volume of fluid must be supported by the pressure of the surrounding liquid since a fluid can not support itself. When no object is present, the net upward force on this volume of fluid must equal to its weight, i.e. the weight of the fluid displaced. When the object is present, this same upward force will act on the object. StackExchange drop down hiding chat, blog and log out, concerns Inductive proof of a formula for Fibonacci numbers May someone help me? I am trying to use induction to prove that the formula for finding the $n$-th term of the Fibonacci sequence is:   $$F_n=\frac{1}{\sqrt{5}}⋅\left(\frac{1+\sqrt{5}}{2}\right)^n-\frac{1}{\sqrt{5}}⋅\left(\frac{1-\sqrt{5}}{2}\right)^n.$$  I tried to put $n=1$ into the equation and prove that if $n=1$ works then $n=2$ works and it should work for any number, but it didn't work. I need to prove that this formula gives the $n$th Fibonacci number. Did I derive a new form of the gamma function?