Mathematical proof that Fermions are half-integer spin particles, and half-integer spin particles are Fermions See:        It was said that all fermions were half integer spin particles and all half spin particles were fermions. The Griffiths's testbook said that it could be proved in relativistic quantum and was taken as axiom in non-relativistic theory.  My question was that:   Could you give me the complete proof that Fermions are half integer spin particles, and half integer spin particles are Fermions? Is there a way to prove it without relativistic condition?
Why bosons have integer spin and fermions have half-integer ones? Due the fact that the fermions are the "block particles" and the bosons are the "carriers" I just came out with the question that, why the "block particle" have half-integer spin and the "carriers" have an integer spin?
Change ViewPager animation duration when sliding programmatically Other ways to say "I have a bad hunch" I'm looking for ways to say "I'm having a bad hunch", or more like a bad feeling about something upcoming. The gut-wrenching feeling that something bad will happen. Is Cursed Child canon to the original story? From what I am hearing the stage-play called The Cursed Child is based on the characters from the Harry Potter series. It is set 19 years from the end of Deathly Hallows (i.e. When Harry's children are starting their time at Hogwarts).   I have also seen the J.K. Rowling has had at least some involvement and been on the set and a part of rehearsals, etc.   My question is has there been a statement given relating to the canon of the stage play to original series? If so, would any conflicting information be considered a retcon?  Related:  - but not dupe because this asking about in-universe information. I have a question about the following problem from a Putnam review:  Let $n\in \mathbb{N}$. Find how many pairs of natural numbers $(x, y)\in \mathbb{N}\times \mathbb{N}$ solve $$ \frac{xy}{x+y}=n. $$  I have found some solutions for all $n$, such as $(2n, 2n)$ and $(n+1, n(n+1))$, but I feel as though this is the wrong approach, as the question is only asking for the number of solutions to the equation.  I don't really want a complete solution, but any hints would be greatly appreciated. I am trying to read a custom (non-standard) CSS property, set in a stylesheet (not the inline style attribute) and get its value. Take this CSS for example:  #someElement {   foo: 'bar'; }   I have managed to get its value with the currentStyle property in IE7:  var element = document.getElementById('someElement'); var val = element.currentStyle.foo;   But currentStyle is MS-specific. So I tried getComputedStyle() in Firefox 3 and Safari 3:  var val = getComputedStyle(element,null).foo;   ...and it returns undefined. Does anyone know a cross-browser way of retreiving a custom CSS property value?  (As you might have noticed, this isn't valid CSS. But it should work as long as the value follows the correct syntax. A better property name would be "-myNameSpace-foo" or something.) homeomorphism of cantor set extends to the plane? Suppose C is a Cantor set in the Euclidean plane, or even in R^3. Suppose h is a homeomorphism of C onto itself. Can h be extended to a homeomorphism of the whole space? What about if h preserves the ´end points´ of the Cantor set? Seems that the latter is a necessary condition but I don´t know how to prove it. I want to view the fonts that are installed on my system. Specifically, I want the client-side (fontconfig) fonts, not the old-style X11 bitmap fonts on the server (for which xfontsel does an adequate job). I guess what I'm looking for would be called a font viewer or font gallery. I want to see a bit of sample text for each font, ideally configurable sample text.  Being able to filter by font name, available character sets and other characteristics would be a plus.  A search of packages in Debian wheezy found gnome-font-viewer and kfontview, but they can only display a single font. I'd rather avoid depending on software that isn't widely available, including on the current Debian stable and Ubuntu LTS. Click to hide in the Hero content is not aligned correctly for the non joined communities The "Click to hide" icon in the hero content (welcoming message) for the non joined communities are not aligned correctly for most of the private beta and some other communities.  A few examples are:        Can the icon be aligned to the right hand side with correct padding? I'm not quite sure how the total attack bonus interacts with BAB.   Can you make more attacks than shown by your BAB (e.g. +11/+6/+1 BAB allows you to make 3 attacks) by having a high enough DEX, STR or other attack bonus modifiers? Assuming I have +11/+6/+1 BAB, and a spell gives me +6 attack bonus, would that make me able to attack as if I had +17/+12/+7/+2 BAB, giving me an extra attack?)   Can you make multiple attacks even if you have one of your stats drained, making your attack bonus negative, or less than 5?      It says in the SRD that the formula for attack bonus in melee is this:       Base attack bonus + Strength modifier + size modifier   I've also been told that you can make multiple attacks in one turn, but each attack gains a -5 penalty to attack bonus. Correct formula for LED current-limiting resistor? Does magic missile hit silent image's illusory creature? How to set vertex weights using Blender's Python API? Looking at the API docs I found , but It cannot be called in Edit Mode. When I try to call it, I always get:     RuntimeError: VertexGroup.add(): cannot be called while object is in edit mode.   The buttons in the vertex group list use bpy.ops.objects.vertex_group_assign(), but to work with it you have to manipulate selections, this is not nice to do from a script.    Why I try to use VertexGroup.add() in Edit Mode? My script works this way:  Human weight paint a bone, "Arm_1.L", "Arm_1.L" shares vertices with "Arm_2.L" in this case they share half the vertices (4 bone arm rig, trick to have nice deform in certain movements). When script run, all selected vertices that exist in both groups (Arm_1.L and Arm_2.L) will copy the weights of Arm_1.L into Arm_2.L but inverted. This way, I only need to weight paint "Arm_1.L" and "Arm_3.L", the script will fill the weights for "Arm_2.L".  import bpy  # ensure we are in edit mode bpy.ops.object.mode_set(mode="EDIT")  # configurable parameters src_group = "Arm_03.L" dst_group = "Arm_02.L"  obj = bpy.context.edit_object src_group_index = obj.vertex_groups[src_group].index dst_group_index = obj.vertex_groups[dst_group].index  mesh = obj.data sel_verts = [v for v in mesh.vertices if v.select] #sel_verts = mesh.vertices  # SEGFAULT in this line bpy.ops.object.mode_set(mode="OBJECT")  print("Working")  for v in sel_verts:     v1in = False     v2in = False     for g in v.groups:         if g.group == src_group_index:             v1in = True         if g.group == dst_group_index:             v2in = True     if v1in and v2in:         print("True")         obj.vertex_groups[dst_group].add([v.index], 1.0 - obj.vertex_groups[src_group].weight(v.index), 'REPLACE')  print("Finished") Using QGIS 3 graphical modeler I created a model with a variety of input parameters (> 10). When starting the model, a nice user interface is generated. But the arrangement of the GUI elements is totally confused and apparently unpredictable during creating the model. It does neither seem to depend on the arrangement of input parameters in the graphical modeler nor on the chronological order in which the input parameters are added to the model.   Is there a way that let's me arrange the GUI elements in the order I desire? Where should I start learning? There are various topics to learn like Objects, Classes, Apex, Visualforce, SOSL, SOQL, Triggers, etc. There are so many options out there to start learning. I couldn't decide what topic to learn first, and then step by step get proficiency in all aspects of Saleforce. Navigation bar is too long on progressbar theme, can it wrap? My navigation bar at the top of each slide (that links to each section) has too many sections. I would like to keep the bar and keep each section but I would like to wrap it i.e. have it on two lines.  Is that possible?  I am using compress and I am using  as a theme. $\sin(n)$ subsequence limits set We were given a challenge by our calculus professor and I've been stuck on it for a while now.   Show that the set of subsequence limits of $A_n=\sin(n)$ is $[-1, 1]$ (another way to phrase this would be: $\forall r\in [-1,1]$ there exists a subsequence of $A_n=\sin(n)$ that converges to $r$).  What would be a good way to start? Let $q \in \mathbb C$, $|q|=1$ and $q^n \neq 1, \forall n \in \mathbb N$. Show that $\{q^n:  n \in \mathbb N\}$ is dense in $S^1$.  My attempt: As $(q^n)$ is limited, there is subsequence $(q^{n_j})$ convergent. So given $\epsilon &gt; 0, \exists j_0 \in \mathbb N, \quad |q^{n_j} - q^{n_k}| &lt; \epsilon, \quad \forall j,k \geq j_0$. Supose that $n_j &gt; n_k$, then  $|q^{n_j} - q^{n_k}| = |q^{n_k}||q^{n_j - n_k} - 1| = |q^{n_j - n_k} - 1| &lt; \epsilon, \quad \forall j, k \geq j_0$. So, there is a subsequence of $(q^n)$ that converges to $1$.  Using this fact, how can I ensure that for all $w \in S^1$ there is a subsequence of $(q^n)$ that converges to  $w$? What do the supercharges in extended supersymmetry do? What do the supercharges in  do?  In ${\cal N}=1$ supersymmetry there are a certain number of fermions and and equal number of bosons. You can transform all fermions to the bosons (and vice versa) in a 1 to 1 fashion using a single supercharge, $Q$.  So what happens when you have, for example, ${\cal N}=2$ supersymmetry with 8 supercharges? Since $Q$ is a generator of supersymmetry transformations, is it a linear combination of these supercharges that act on the particles? In which case could one particle be acted on by two separate linear combinations of $Q$? Or is it strictly one linear combination of $Q$ per fermion/boson?  Also, what does ${\cal N}$ mean physically? What difference does ${\cal N}=2$ have to ${\cal N}=1$ other than more supercharges? Or is that the only difference between the two theories? If $a \cdot a = 1$ then $a = 1$ or $a = -1$ Convert a string to an enum in C#