I've met the symbol ||- many times in modal logic textbooks, especially in epistemic logic books. But I cannot find any info about how to type or read it. Usually, it means &quot;semantical entailment&quot; or &quot;modelling&quot; in modal logic. But in first-order logic, semantical entailment is symbolised as |=. By contrast, we also use |- to express &quot;syntactical entailment&quot; or &quot;proving&quot;. Can anyone help me?
I know what my symbol or character looks like, but I don't know what the command is or which math alphabet it came from. How do I go about finding this out?
Torsion elements of a group aren't necessarily a subgroup In a question from an exam in an undergraduate group theory course, we were asked to prove or disprove that the set of all Torsion elements of a group is necessarily a subgroup.  I knew that the set of Torsion elements is closed under the inverse operation, but was later told that it is not closed under multiplication, therefore disproving the claim. However, I couldn't find any examples of a group $G$ and two elements $a,b$ such that both $a$ and $b$ have a finite order, but $ab$ doesn't. I know that for this to happen $G$ must be an infinite and non-Abelian group, but still couldn't find a valid example.   What are some examples of groups/elements fulfilling the aforementioned property? In both the movies and comics Cap gains his peak human strength via the supersoldier serum and a does of vita rays, resulting in the rapid development of his super buff body:     While the most obvious real world analogue would be the use of steroids, these would still require that the person taking them train extensively and significantly increase their protein intake in order to gain muscle mass. While there are frequent scenes of Steve training in both mediums I've yet to see any scenes of him eating a 12-egg omelette or downing protein shakes. Are there any in-world explanations that would account for the increased muscle mass? Extra-dimensional energy conversion? Problem regarding radius of convergence of $\sum n!x^{n^2}$. $$\sum_{n=0}^\infty n!x^{n^2}$$   Can anyone tell me what will be the radius of convergence for this power series. According to me it is $0$.   If you can please show me how to go about it if I am wrong. PHP Echo Line Breaks Labels disappear while zooming Can hypercomputation compute the impossible? There are things which are illogical/logically impossible (like saying that 2+2=4 and 2+2=5. Without changing anything in the axioms of mathematics or logic, this would be a contradiction and would be inconsistent and illogical/logically impossible.   There are other types of logic systems apart from classical logic, like paraconsistent logics or even trivialism, that allow these contradictions to occur, prove them as right and work with them.   We can make a paraconsistent or trivialist system and work with it. For example, with trivialism, in theory, we would be able to derive and state everything we would want (since literally everything, even including illogical/logically impossible inconsistencies and contradictions), but we as humans (or as brains), are limited and can't conceive everything we want (at least to what I know). Therefore, no matter how many trivialist models we create and how much time we would work with them, we would never find or conceive many illogical/logically things because they are just that: impossible. There are impossible things to describe and conceive. For example, Russel's set is the set of all sets that do not contain themselves. If Russel's set contains itself, then it cannot contain itself, since it only contains sets that don't contain themselves. But if Russel's set does not contain itself, then it must contain itself, since it contains all sets that don't contain themselves.  There are quite a few logic bombs like this. You cannot ever compute the contents of Russel's set, and there are more formal, mathematical ways to present it. All of them have in common that you can't actually compute what the set is, whether you do it by hand, in your head, or on a computer. It's just a statement that cannot be fully logically processed.  If you take every possible state the human brain can be in, none of them include the computation of Russel's Set's contents. That is, not only can the contents not be computed, they cannot even be represented. No stimulus can cause us to comprehend Russel's Set, since such comprehension is not possible to begin with. It does not have a solution. Even if we try to solve it using trivialism, we would just be able to write a solution that does not make sense and prove it has sense and it is the real solution, but we could not be able to have a solution that would make sense "outside" the realm of trivialism (for example in classical logic), even though, using trivialism, we could prove that such solution would have sense in whatever context and logic system.   But what about hypercomputational machines (for example oracle-like machines)? I've read about some models of hypercomputation which are compatible with paraconsistent or trivialism logics. I've also read there are some models of hypercomputation (particularly those oracle-like models which use a black box) where, essentially, the hypercomputer is an algorithm that cannot exist. It might be because such an algorithm is fundamentally forbidden by logic itself (which is hided in a black box). Would any of these be capable of computing/"conceiving" these impossible things I wrote before? Do you know of anything that would help? Creating Ellipse in WGS84, but with metric parameters I have the function  --Ellipse(x,y,rx,ry,rotation,#of segments in 1/4 of ellipse) SELECT ST_Translate( ST_Rotate( ST_Scale( ST_Buffer(ST_Point(0,0), 0.5, $6), $3, $4), $5), $1, $2)   I have points in WGS84 (x,y), radiuses in meters (rx, ry), rotation angle in degrees. And I need a new WGS84 geometry as a result.  As you can see I can't call this function without conversions between WGS84 and some metric SRID. But my points aren't in a definite region. So a SRID which is optimal for one point, throws an error for another point during conversion.  Is there a common way to get an optimal metric projection (SRID) for a WGS84 point? Or at least to use some rough, but world-wide universal metric projection? Or maybe there's a trick for this case? When is JavaScript's eval() not evil? I'm writing some JavaScript code to parse user-entered functions (for spreadsheet-like functionality). Having parsed the formula I could convert it into JavaScript and run eval() on it to yield the result.  However, I've always shied away from using eval() if I can avoid it because it's evil (and, rightly or wrongly, I've always thought it is even more evil in JavaScript, because the code to be evaluated might be changed by the user).  So, when it is OK to use it? If M+N and M$\cap$N are finitely generated modules, so are M and N. If the universe is expanding, what is it expanding into? If the universe is expanding, what is it expanding into? When the big bang happened where did it occur?  When the big bang happened how did it occur?  Where did the energy come from? Energy can not be created or destroyed does that mean, energy has existed before the universe was here? Using the affine linearized polynomial $f(x)=x^p-x-b$ is irreducible over $\mathbb{F}_q$ if and only if $\operatorname{Tr}_{\mathbb{F}_q/\mathbb{F}_p} \neq 0$. But here $\mathbb{F}_3$ is not a field extension of $\mathbb{F}_5$.  Also we have that: Let $a \in \mathbb{F}_q$ and let $p$ be the characteristic of $\mathbb{F}_q$. Then the trinomial $x^P - x - a$ is irreducible in $\mathbb{F}_q[x]$ if and only if it has no root in $\mathbb{F}_q$. You can think this of the following as a 3d QFT where we try to calculate the self-energy of two fields. $I$ is a this external self-energy and let us assume it does not depend on the loop momenta which I denote a $\vec{x}=(x,y)$. Consider the following propagator  $$ P(\vec{r}) = \frac{x-ia}{2i^2[\,(x-ia)^2+(y-ib)^2+(z-ic)^2]\,} \,\, \gamma_i. $$ Then we want to calculate $$ I(x) = \int \frac{\,dx \,dy\, dz}{(2\pi)^3} \text{Tr} \left( \frac{x-ia}{2i^2[\,(x-ia)^2+(y-ib)^2+(z-ic)^2]\,}\gamma_x \gamma_x  \right), $$ where Tr$(\gamma_i \gamma_j)=2\delta_{ij}$. Therefore  I am trying to resolve the following integral where $x,y,z \in \mathbb{R}$ and $a,b,c$ are constants (internal self-energies if you wish) in $\mathbb{R}$.  Then, after simplifying the propagator $P(x,y)$ we end up with the following expression for the $x$ component:  $$ I = \frac{1}{2i^2} \int \frac{dxdy}{(2\pi)^2} (x-ia) \int \frac{dz}{2\pi} \frac{1}{[\,(x-ia)^2+(y-ib)^2+(z-ic)^2]\,} $$ but it can be shown that  $$ \int \frac{dq}{(q+iv)^2+w^2} = \frac{\pi}{|w|}\theta(|w|-|v|).  $$ (e.g. see here ) Thus we get that for $|(y-ib)^2+(x-ia)^2|&gt;|a|$, $\theta=1$ (for this is what we are interested about) and then we are left with $$ I = \frac{1}{4i^2} \int \frac{dxdy}{(2\pi)^2} \frac{(x-ia)}{|(x-ia)^2+(y-ib)^2|}.   $$ The question is wether I can make the transformation $$ (x-ia) = q \cos \theta  $$ $$ (y-ib) = q \sin \theta $$ in polar coordinates. Then $dx dy = qdqd\theta$ and the previous integral becomes, assuming $|q|&gt;0$, $$ I = \int_0^{\infty} dq \int_0^{2\pi} d\theta \frac{q \cos \theta}{|q|} =0. $$ Is this correct? I am confused since one can find complex poles in the $I$. If not, what would the correct way to resolve this be? I have a suspicion that this integral should indeed give zero but how can we find this? Ubuntu 12.10 "Turn screen off when inactive for: Never" still turns off Do we need a global test before post hoc tests? I often hear that post hoc tests after an ANOVA can only be used if the ANOVA itself was significant.    However, post hoc tests adjust $p$-values to keep the global type I error rate at 5%, don't they?  So why do we need the global test first?  If we don't need a global test is the terminology "post hoc" correct? Or are there multiple kinds of post hoc tests, some assuming a significant global test result and others without that assumption? Configure tab-switch hotkey in gnome-terminal I used to use Alt+number to switch between gnome-terminal tabs. In my recent 19.10 installation, this does not work. When I press Alt+1, instead of switching tab, the command line shows this: (arg: 1)   and if I press Alt+2, it becomes: (arg: 12)   I checked the options in gnome-terminal, it is set to use Alt+number to switch between tabs. I would like to know which application is hijacking the hotkey, and how to fix it. PS. I do NOT want to use Ctrl+PgUp/PgDn for this. Thanks. Is it possible to unwrap materials made in Vertex Paint? I painted an group of objects in Blender using the Vertex Paint and was wondering if it would be possible to unwrap the mutliple materials(when the objects are joined together using Ctrl+J)? Do I have to paint it all over again? What hardware does Ubuntu Touch support? There is a common story that airline ticket prices rise significantly if you buy them less than two weeks before departure.  And that any time before that will yield standard prices.  Is that advice valid for flying during holiday seasons?  As an example: if I want to fly coast to coast USA on December 7th, must I buy my ticket ASAP, or would I get a similar price buying on November 21st? Updating to latest Blender version I currently have version 2.74. How do I update to version 2.76? I do not see any links in the UI to update my version of Blender I'd like to create a PDF report with knitr with LaTeX-code in RStudio with biblatex for citations. However, the in-text-citations are always undefined and no bibliography is printed. The same bibliography works perfectly with a pure .tex-document stored in the same folder and containing the same citations, same path to the .bib-file etc. However, as for all references (also with hyperref) I have to compile twice. I thought this might be the problem with knitr (that the compilation is run only once), but the hyperref-references to figures work well. Any idea what might go wrong with the bibliography when using knitr and how I can solve the problem?  MWE:  \documentclass{article}  \usepackage[style=authoryear-comp, maxnames=1, bibencoding=utf8, backend=biber]{biblatex} \bibliography{ZoteroOutput.bib}  \begin{document}  According to the literature \parencite{gonzalez2003} we can assume...  \printbibliography  \end{document}   From the log-file:  Package biblatex Warning: Please (re)run Biber on the file: (biblatex)                mwe (biblatex)                and rerun LaTeX afterwards.