prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What are the stability conditions for a solution of $-\Delta u = f(u)$?

Read the following text and answer briefly.

Context 1: \section{Introduction and main results}


In this note we are interested in the existence versus non-existence of stable sub- and super-solutions of equations of the form
\begin{equation} \label{eq1}
-div( \omega_1(x) \nabla u ) = \omega_2(x) f(u) \qquad \mbox{in $ {\mathbb{R}}^N$,}
\end{equation} where $f(u)$ is one of the following non-linearities: $e^u$,  $ u^p$ where $ p>1$ and $ -u^{-p}$ where $ p>0$   We assume that $ \omega_1(x)$ and $ \omega_2(x)$, which we call \emph{weights},  are smooth positive functions (we allow $ \omega_2$ to be zero at say a point) and which satisfy various growth conditions at $ \infty$     Recall that we say that a solution $ u $ of $ -\Delta u = f(u)$ in $ {\mathbb{R}}^N$ is stable provided
\[ \int f'(u) \psi^2 \le \int | \nabla \psi|^2, \qquad \forall \psi \in C_c^2,\] where $ C_c^2$ is the set of $ C^2$ functions defined on $ {\mathbb{R}}^N$ with compact support   Note that the stability of $u$ is just saying that the second variation at $u$ of the energy associated with the equation is non-negative      In our setting this becomes:  We say a $C^2$ sub/super-solution $u$ of (\ref{eq1}) is \emph{stable} provided
\begin{equation} \label{stable}
\int \omega_2 f'(u) \psi^2 \le \int \omega_1 | \nabla \psi|^2 \qquad \forall \psi \in C_c^2 
\end{equation}
One should note that (\ref{eq1}) can be re-written as
\begin{equation*}
- \Delta u  + \nabla \gamma(x) \cdot \nabla u ={ \omega_2}/{\omega_1}\ f(u) \qquad \text{ in $ \mathbb{R}^N$},
\end{equation*}
where
$\gamma = - \log( \omega_1)$ and on occasion we shall take this point of view 


\begin{remark} \label{triv} Note that  if $ \omega_1$ has enough integrability then it is immediate that if $u$ is a stable solution  of (\ref{eq1}) we have $ \int \omega_2 f'(u) =0 $  (provided $f$ is increasing)    To see this let $ 0 \le \psi \le 1$ be supported in a ball of radius $2R$ centered at the origin ($B_{2R}$) with $ \psi =1$ on $ B_R$ and such that $  | \nabla \psi | \le \frac{C}{R}$ where $ C>0$ is independent of $ R$    Putting this $ \psi$ into $ (\ref{stable})$ one obtains
\[ \int_{B_R} \omega_2 f'(u) \le \frac{C}{R^2} \int_{R < |x| <2R} \omega_1,\] and so if the right hand side goes to zero as $ R \rightarrow \infty$ we have the desired result 

\end{remark}





The existence versus non-existence of stable solutions of  $ -\Delta u =  f(u)$ in $ {\mathbb{R}}^N$ or $ -\Delta u = g(x) f(u)$ in $ {\mathbb{R}}^N$ is now quite well understood, see  \cite{dancer1, farina1, egg, zz, f2, f3, wei, ces, e1, e2}   We remark that some of these results are examining the case where $ \Delta $ is replaced with $ \Delta_p$ (the $p$-Laplacian) and also in many cases the authors are interested in finite Morse index solutions or solutions which are stable outside a compact set 
  Much of the interest in these Liouville type theorems  stems from the fact that the non-existence of a stable solution is related to the existence of a priori estimates for stable solutions of a related equation on a bounded domain 




       In \cite{Ni}  equations similar to  $ -\Delta u = |x|^\alpha u^p$  where examined on the unit ball in $ {\mathbb{R}}^N$ with zero Dirichlet boundary conditions   There it was shown that for $ \alpha >0$ that  one can obtain positive solutions for $ p $ supercritical with respect to Sobolev embedding and so one can view that the term $ |x|^\alpha$ is restoring some compactness       A similar feature happens for equations of the form
\[ -\Delta u = |x|^\alpha f(u) \qquad \mbox{in $ {\mathbb{R}}^N$};\]    the value of $ \alpha$ can vastly alter the existence versus non-existence of a stable solution,  see \cite{e1, ces, e2, zz, egg} 

We now come to our main results and for this we need to define a few quantities:

\begin{eqnarray*}
I_G&:=& R^{-4t-2} \int_{ R < |x|<2R} \frac{ \omega_1^{2t+1}}{\omega_2^{2t}}dx , \\
 J_G&:=& R^{-2t-1} \int_{R < |x| <2R} \frac{| \nabla \omega_1|^{2t+1} }{\omega_2^{2t}} dx  ,\\I_L&:=& R^\frac{-2(2t+p-1)}{p-1}  \int_{R<|x|<2R }{ \left(   \frac{w_1^{p+2t-1}}{w_2^{2t}}    \right)^{\frac{1}{p-1} }  }  dx,\\ J_L&:= &R^{-\frac{p+2t-1}{p-1} }  \int_{R<|x|<2R }{ \left(   \frac{|\nabla w_1|^{p+2t-1}}{w_2^{2t}}     \right)^{\frac{1}{p-1} }  }  dx,\\
I_M  &:=&   R^{-2\frac{p+2t+1}{p+1} }  \int_{R<|x|<2R }{ \left(   \frac{w_1^{p+2t+1}}{w_2^{2t}}    \right)^{\frac{1}{p+1} }  } \ dx, \\
J_M  &:= &  R^{-\frac{p+2t+1}{p+1} }  \int_{R<|x|<2R }{ \left(   \frac{|\nabla w_1|^{p+2t+1}}{w_2^{2t}}     \right)^{\frac{1}{p+1} }  }  dx 
\end{eqnarray*}


The three equations we examine are
\[ -div( \omega_1 \nabla u ) = \omega_2 e^u \qquad \mbox{ in $ {\mathbb{R}}^N$ } \quad  (G), \]
\[  -div( \omega_1 \nabla u ) = \omega_2 u^p \qquad \mbox{ in $ {\mathbb{R}}^N$ } \quad  (L), \]
\[ -div( \omega_1 \nabla u ) = - \omega_2 u^{-p} \qquad \mbox{ in $ {\mathbb{R}}^N$ } \quad  (M),\]  and where we restrict $(L)$ to the case $ p>1$ and $(M)$ to $ p>0$     By solution we always mean a $C^2$ solution    We now come to our main results in terms of abstract $ \omega_1 $ and $ \omega_2$     We remark that our approach to non-existence of stable solutions is the approach due to Farina, see \cite{f2,f3,farina1} 

\begin{thm} \label{main_non_exist} \begin{enumerate}


\item   There is no  stable sub-solution of $(G)$ if $ I_G, J_G \rightarrow 0$ as $ R \rightarrow \infty$   for some   $0<t<2$ 
 \item    There is no positive   stable sub-solution (super-solution) of $(L)$  if  $ I_L,J_L \rightarrow 0$ as $ R \rightarrow \infty$   for some   $p- \sqrt{p(p-1)}   < t<p+\sqrt{p(p-1)} $  ($0<t<\frac{1}{2}$) 
\item   There is no positive   stable super-solution of (M) if  $ I_M,J_M \rightarrow 0$ as $ R \rightarrow \infty$  for some $0<t<p+\sqrt{p(p+1)}$ 

\end{enumerate}

\end{thm}     If we assume that $ \omega_1$ has some monotonicity we can do  better    We will assume that the monotonicity conditions is satisfied for big $x$  but really all ones needs is for it to be satisfied on a suitable sequence of annuli 

\begin{thm} \label{mono}  \begin{enumerate} \item There is no  stable sub-solution of $(G)$ with $  \nabla \omega_1(x) \cdot  x \le 0$ for big $x$ if  $ I_G \rightarrow 0$ as $ R \rightarrow \infty$   for some   $0<t<2$ 

\item There is no positive  stable sub-solution of  $(L)$ provided  $  I_L \rightarrow 0$ as $ R \rightarrow \infty$ for either:
\begin{itemize}
\item  some $  1 \le t < p + \sqrt{p(p-1)}$ and $  \nabla \omega_1(x) \cdot x \le 0$ for big $x$,  or  \\

\item some $ p - \sqrt{p(p-1)} < t \le 1$ and $ \nabla \omega_1(x) \cdot x \ge 0$ for big $ x$ 
\end{itemize}
 There is no positive super-solution of $(L)$ provided $ I_L \rightarrow 0$ as $ R \rightarrow \infty$ for some $ 0 < t < \frac{1}{2}$ and $ \nabla \omega_1(x) \cdot x \le 0$ for big $x$ 



\item  There is no positive stable super-solution of $(M)$ provided  $ I_M \rightarrow 0$ as $ R \rightarrow \infty$  for some  $0<t<p+\sqrt{p(p+1)}$ 

\end{enumerate}

\end{thm}


\begin{cor}  \label{thing}  Suppose $ \omega_1 \le C  \omega_2$  for big $ x$, $ \omega_2 \in L^\infty$,  $ \nabla \omega_1(x) \cdot  x \le 0$ for big $ x$ 
\begin{enumerate} \item There is no stable sub-solution of $(G)$ if $ N \le 9$ 

\item  There is no positive stable sub-solution of $(L)$ if  $$N<2+\frac{4}{p-1} \left( p+\sqrt{p(p-1)}  \right) $$

\item  There is no positive stable super-solution of $(M)$ if $$N<2+\frac{4}{p+1} \left( p+\sqrt{p(p+1)}  \right) $$

\end{enumerate}

\end{cor}


If one takes $ \omega_1=\omega_2=1$ in the above corollary, the results obtained for $(G)$ and  $(L)$,  and for some values of $p$ in $(M)$, are optimal, see \cite{f2,f3,zz} 





We now drop all monotonicity conditions on $ \omega_1$ 

\begin{cor} \label{po} Suppose  $ \omega_1 \le C \omega_2$ for big $x$, $ \omega_2 \in L^\infty$, $ | \nabla \omega_1| \le C \omega_2$ for big $x$ 
\begin{enumerate} \item  There is no stable sub-solution of $(G)$ if $ N \le 4$ 

\item  There is no positive stable sub-solution of $(L)$ if $$N<1+\frac{2}{p-1} \left( p+\sqrt{p(p-1)}  \right) $$

\item There is no positive super-solution of $(M)$ if $$N<1+\frac{2}{p+1} \left( p+\sqrt{p(p+1)}  \right) $$

\end{enumerate}

\end{cor}

Some of the conditions on $ \omega_i$ in Corollary \ref{po} seem somewhat artificial   If we shift over to the advection equation (and we take $ \omega_1=\omega_2$  for simplicity)
\[ -\Delta u + \nabla \gamma \cdot \nabla u = f(u), \] the conditions on $ \gamma$ become: $ \gamma$ is bounded from below and has a bounded gradient 



In what follows we examine the case where $ \omega_1(x) = (|x|^2 +1)^\frac{\alpha}{2}$ and $ \omega_2(x)= g(x) (|x|^2 +1)^\frac{\beta}{2}$,  where $ g(x) $ is positive except at say a point, smooth and where $ \lim_{|x| \rightarrow \infty} g(x) = C \in (0,\infty)$      For this class of weights we can essentially obtain optimal results 


\begin{thm} \label{alpha_beta}   Take $ \omega_1 $ and $ \omega_2$ as above 
\begin{enumerate}

\item If $ N+ \alpha - 2 <0$ then there is no stable sub-solution for $(G)$, $(L)$ (here we require it to be positive) and in the case of $(M)$ there is no positive  stable  super-solution  This case is the trivial case, see Remark \ref{triv}   \\



\textbf{Assumption:} For the remaining cases we assume that $ N + \alpha -2 > 0$ 

  \item If  $N+\alpha-2<4(\beta-\alpha+2)$ then there is no  stable sub-solution for $ (G)$ 

\item If $N+\alpha-2<\frac{ 2(\beta-\alpha+2)   }{p-1} \left( p+\sqrt{p(p-1)}  \right)$ then there is  no positive stable sub-solution of $(L)$ 

\item If $N+\alpha-2<\frac{2(\beta-\alpha+2)   }{p+1} \left( p+\sqrt{p(p+1)}  \right)$ then there is no positive stable super-solution of $(M)$ 

\item Further more 2,3,4 are optimal in the sense if $ N + \alpha -2 > 0$ and the remaining inequality is not satisfied (and in addition we assume we don't have equality in the inequality) then we can find a suitable function $ g(x)$ which satisfies the above properties and a stable sub/super-solution $u$ for the appropriate equation 

\end{enumerate}

\end{thm}

\begin{remark}  Many of the above results can be extended to the case of equality in either the $ N + \alpha - 2 \ge 0$ and also the other inequality which depends on the equation we are  examining   We omit the details because one cannot prove the results in a unified way 

\end{remark}






















In showing that an explicit solution is stable we will need the   weighted Hardy inequality given in \cite{craig} 
\begin{lemma} \label{Har}
Suppose $ E>0$ is a smooth function   Then one has
\[ (\tau-\frac{1}{2})^2 \int E^{2\tau-2} | \nabla E|^2 \phi^2 + (\frac{1}{2}-\tau) \int (-\Delta E) E^{2\tau-1} \phi^2 \le \int E^{2\tau} | \nabla \phi|^2,\] for all $ \phi \in C_c^\infty({\mathbb{R}}^N)$ and $ \tau \in {\mathbb{R}}$ 
\end{lemma}  By picking an appropriate function $E$ this gives,

\begin{cor} \label{Hardy}
For all $ \phi \in C_c^\infty$ and $ t , \alpha \in {\mathbb{R}}$  We have
  \begin{eqnarray*}
\int (1+|x|^2)^\frac{\alpha}{2} |\nabla\phi|^2 &\ge& (t+\frac{\alpha}{2})^2 \int |x|^2 (1+|x|^2)^{-2+\frac{\alpha}{2}}\phi^2\\
&&+(t+\frac{\alpha}{2})\int  (N-2(t+1)   \frac{|x|^2}{1+|x|^2}) (1+|x|^2)^{-1+\frac{\alpha} {2}} \phi^2 
\end{eqnarray*}
 \end{cor}























\section{Proof of main results}

\textbf{ Proof of Theorem \ref{main_non_exist} }   (1)  Suppose $ u$ is a stable sub-solution of $(G)$ with $ I_G,J_G \rightarrow 0$ as $ R \rightarrow \infty$ and  let $ 0 \le \phi \le 1$ denote a smooth compactly supported function   Put $ \psi:= e^{tu} \phi$ into (\ref{stable}), where $ 0 <t<2$, to arrive at
\begin{eqnarray*}
\int \omega_2 e^{(2t+1)u} \phi^2 &\le & t^2 \int \omega_1 e^{2tu} | \nabla u|^2 \phi^2 \\
&& +\int \omega_1 e^{2tu}|\nabla \phi|^2 + 2 t \int \omega_1 e^{2tu} \phi \nabla u \cdot \nabla \phi 
\end{eqnarray*}  Now multiply $(G)$ by $ e^{2tu} \phi^2$ and integrate by parts to arrive at
\[ 2t \int \omega_1 e^{2tu} | \nabla u|^2 \phi^2 \le \int \omega_2 e^{(2t+1) u} \phi^2 - 2 \int \omega_1 e^{2tu} \phi \nabla u \cdot \nabla \phi,\]
and now if one equates like terms they arrive at
\begin{eqnarray} \label{start}
\frac{(2-t)}{2} \int \omega_2 e^{(2t+1) u} \phi^2  & \le & \int \omega_1 e^{2tu} \left( | \nabla \phi |^2 - \frac{ \Delta \phi}{2} \right) dx \nonumber \\
&& - \frac{1}{2} \int e^{2tu} \phi \nabla \omega_1 \cdot \nabla \phi 
\end{eqnarray}   Now substitute $ \phi^m$ into this inequality for $ \phi$ where $ m $ is a big integer to obtain
\begin{eqnarray} \label{start_1}
\frac{(2-t)}{2} \int \omega_2 e^{(2t+1) u} \phi^{2m}  & \le & C_m \int \omega_1 e^{2tu} \phi^{2m-2}  \left( | \nabla \phi |^2 + \phi |\Delta \phi|  \right) dx \nonumber \\
&& - D_m \int e^{2tu} \phi^{2m-1} \nabla \omega_1 \cdot \nabla \phi
\end{eqnarray} where $ C_m$ and $ D_m$ are positive constants just depending on $m$    We now estimate the  terms on the right but we mention that when ones assume the appropriate monotonicity on $ \omega_1$ it is the last integral on the right which one is able to drop 

\begin{eqnarray*}
\int \omega_1 e^{2tu} \phi^{2m-2} | \nabla \phi|^2 & = & \int \omega_2^\frac{2t}{2t+1} e^{2tu} \phi^{2m-2}  \frac{ \omega_1 }{\omega_2^\frac{2t}{2t+1}} | \nabla \phi|^2  \\
& \le &  \left( \int \omega_2 e^{(2t+1) u} \phi^{(2m-2) \frac{(2t+1)}{2t}} dx \right)^\frac{2t}{2t+1}\\ &&\left( \int \frac{ \omega_1 ^{2t+1}}{\omega_2^{2t}} | \nabla \phi |^{2(2t+1)} \right)^\frac{1}{2t+1} 
\end{eqnarray*}
Now, for fixed $ 0 <t<2$ we can take $ m $ big enough so $ (2m-2) \frac{(2t+1)}{2t} \ge 2m $ and since $ 0 \le \phi \le 1$ this allows us to replace the power on $ \phi$ in the first term on the right with $2m$   and hence we obtain
 \begin{equation} \label{three}
 \int \omega_1 e^{2tu} \phi^{2m-2} | \nabla \phi|^2  \le \left( \int \omega_2 e^{(2t+1) u} \phi^{2m} dx \right)^\frac{2t}{2t+1} \left( \int \frac{ \omega_1 ^{2t+1}}{\omega_2^{2t}} | \nabla \phi |^{2(2t+1)} \right)^\frac{1}{2t+1} 
 \end{equation}    We now take the test functions $ \phi$ to be such that $ 0 \le  \phi \le 1$ with $ \phi $ supported in the ball $ B_{2R}$ with $ \phi = 1 $ on $ B_R$ and $ | \nabla \phi | \le \frac{C}{R}$ where $ C>0$ is independent of $ R$    Putting this choice of $ \phi$ we obtain
 \begin{equation} \label{four}
 \int \omega_1 e^{2tu} \phi^{2m-2} | \nabla \phi |^2 \le \left( \int \omega_2 e^{(2t+1)u} \phi^{2m} \right)^\frac{2t}{2t+1} I_G^\frac{1}{2t+1} 
 \end{equation}  One similarly shows that
 \[ \int \omega_1 e^{2tu} \phi^{2m-1} | \Delta \phi| \le \left( \int \omega_2 e^{(2t+1)u} \phi^{2m} \right)^\frac{2t}{2t+1} I_G^\frac{1}{2t+1} \]
 So, combining the results we obtain

 \begin{eqnarray} \label{last} \nonumber \frac{(2-t)}{2} \int \omega_2 e^{(2t+1) u} \phi^{2m} &\le& C_m \left( \int \omega_2 e^{(2t+1) u} \phi^{2m} dx \right)^\frac{2t}{2t+1} I_G^\frac{1}{2t+1}\\
 &&- D_m \int e^{2tu} \phi^{2m-1}  \nabla \omega_1 \cdot \nabla \phi 
 \end{eqnarray}
 We now estimate this last term   A similar argument using H\"{o}lder's inequality shows that
 \[ \int e^{2tu} \phi^{2m-1} | \nabla \omega_1| | \nabla \phi| \le \left(  \int \omega_2 \phi^{2m} e^{(2t+1) u} dx \right)^\frac{2t}{2t+1} J_G^\frac{1}{2t+1}  \] Combining the results gives that
\begin{equation} \label{last}
(2-t) \left( \int \omega_2 e^{(2t+1) u} \phi^{2m} dx \right)^\frac{1}{2t+1} \le I_G^\frac{1}{2t+1} + J_G^\frac{1}{2t+1},
\end{equation} and now we send $ R \rightarrow \infty$ and use the fact that $ I_G, J_G \rightarrow 0$ as $ R \rightarrow \infty$ to see that
\[ \int \omega_2 e^{(2t+1) u} =0, \] which is clearly a contradiction    Hence there is no stable sub-solution of $(G)$ 

(2)    Suppose that $u >0$ is a stable sub-solution (super-solution) of $(L)$   Then a similar calculation as in (1) shows that for  $ p - \sqrt{p(p-1)} <t < p + \sqrt{p(p-1)}$,  $( 0 <t<\frac{1}{2})$ one has

\begin{eqnarray}   \label{shit}
(p  -\frac{t^2}{2t-1}   )\int \omega_2 u^{2t+p-1} \phi^{2m} & \le & D_m \int \omega_1 u^{2t} \phi^{2(m-1)} (|\nabla\phi|^2  +\phi |\Delta \phi |) \nonumber \\
&& +C_m \frac{(1-t)}{2(2t-1)} \int u^{2t} \phi^{2m-1}\nabla \omega_1 \cdot  \nabla \phi 
 \end{eqnarray}    One now applies H\"{o}lder's argument as in (1) but the terms $ I_L$ and $J_L$ will appear on the right hand side of the resulting
 equation       This shift from a sub-solution to a super-solution depending on whether $ t >\frac{1}{2}$ or $ t < \frac{1}{2}$ is a result from the sign change of $ 2t-1$ at $ t = \frac{1}{2}$    We leave the details for the reader 


(3)   This case is also similar to (1) and (2) 


\hfill $ \Box$

 \textbf{Proof of Theorem \ref{mono} }   (1)   Again we suppose there is a stable sub-solution $u$ of $(G)$   Our starting point  is (\ref{start_1}) and we wish to be able to drop the term
 \[ - D_m \int e^{2tu} \phi^{2m-1} \nabla \omega_1 \cdot \nabla \phi, \] from (\ref{start_1})     We can choose $ \phi$ as in the proof of Theorem \ref{main_non_exist} but also such that $ \nabla \phi(x) = - C(x) x$ where $ C(x) \ge 0$      So if we assume that $ \nabla \omega_1 \cdot x \le 0$ for big $x$ then we see that this last term is non-positive and hence we can drop the term   The the proof is as before but now we only require that $ \lim_{R \rightarrow \infty} I_G=0$ 

 (2)      Suppose that $ u >0$ is a stable sub-solution of $(L)$  and so (\ref{shit}) holds for all $  p - \sqrt{p(p-1)} <t< p + \sqrt{p(p-1)}$    Now we wish to use monotonicity to drop the term from (\ref{shit}) involving the term $ \nabla \omega_1 \cdot \nabla \phi$       $ \phi$ is chosen the same as in (1)  but here one notes that the co-efficient for this term changes sign at $ t=1$ and hence by restriction $t$ to the appropriate side of 1 (along with the above condition on $t$ and $\omega_1$) we can drop the last term depending on which monotonicity we have and hence to obtain a contraction we only require that $ \lim_{R \rightarrow \infty} I_L =0$    The result for the non-existence of a stable super-solution is similar be here one restricts $ 0 < t < \frac{1}{2}$ 


(3)   The proof here is similar to (1) and (2) and we omit the details 




 \hfill $\Box$













\textbf{Proof of Corollary \ref{thing} }   We suppose  that $ \omega_1 \le C  \omega_2$  for big $ x$, $ \omega_2 \in L^\infty$,  $ \nabla \omega_1(x) \cdot  x \le 0$ for big $ x$      \\
(1)     Since $ \nabla \omega_1 \cdot x \le 0$ for big $x$ we can apply Theorem \ref{mono} to show the non-existence of a stable solution to $(G)$    Note that  with the above assumptions on $ \omega_i$ we have that
\[ I_G \le \frac{C R^N}{R^{4t+2}} \]  For $ N \le 9$  we can take $ 0 <t<2$  but close enough to $2$ so the right hand side goes to zero as $ R \rightarrow \infty$ 

Both (2) and (3) also follow directly from applying Theorem \ref{mono}    Note that one can say more about (2) by taking the multiple cases as listed in Theorem \ref{mono} but we have choice to leave this to the reader 


\hfill $ \Box$


\textbf{Proof of Corollary \ref{po} }   Since we have no monotonicity conditions now we will need both $I$ and $J$ to go to zero to show the non-existence of a stable solution    Again the results are obtained immediately by applying Theorem \ref{main_non_exist}  and we prefer to omit the details 


\hfill $\Box$
















\textbf{Proof of Theorem \ref{alpha_beta} }  (1)   If $ N + \alpha -2 <0$ then using Remark \ref{triv}  one easily sees there is no stable sub-solution of $(G)$ and $(L)$ (positive for $(L)$) or a positive stable super-solution of $(M)$    So we now assume that $ N + \alpha -2 > 0$      Note that the monotonicity of $ \omega_1$ changes when $ \alpha $ changes sign and hence one would think that we need to consider separate cases if we hope to utilize the monotonicity results    But a computation shows that in fact $ I$ and $J$ are just multiples of each other in all three cases so it suffices to show, say, that $ \lim_{R \rightarrow \infty} I =0$  \\
(2)    Note that for $ R >1$ one has
\begin{eqnarray*}
I_G & \le  & \frac{C}{R^{4t+2}} \int_{R <|x| < 2R} |x|^{ \alpha (2t+1) - 2t \beta} \\
& \le  &  \frac{C}{R^{4t+2}}  R^{N + \alpha (2t+1) - 2t \beta},
\end{eqnarray*} and so to show the non-existence we want to find some $ 0 <t<2$ such that
$  4t+2 > N  + \alpha(2t+1) - 2 t \beta$,   which is equivalent to  $ 2t ( \beta - \alpha +2) > (N + \alpha -2)$     Now recall that we are assuming that $ 0 < N + \alpha -2 < 4 ( \beta - \alpha +2) $ and hence we have the desired result by taking $  t <2$ but sufficiently close 
The proof of the non-existence results for
(3) and (4) are similar and we omit the details    \\
(5)   We now assume that $N+\alpha-2>0$   In showing the existence of stable sub/super-solutions we need to consider  $ \beta - \alpha + 2 <0$ and $ \beta - \alpha +2 >0$ separately 

\begin{itemize} \item $(\beta - \alpha + 2 <0)$  Here we take $ u(x)=0$ in the case of $(G)$ and $ u=1$ in the case of $(L)$ and $(M)$  In addition we take $ g(x)=\E$   It is clear that in all cases $u$ is the appropriate sub or super-solution   The only thing one needs to check is the stability     In all cases this reduces to trying to show that we have
\[ \sigma \int       (1+|x|^2)^{\frac{\alpha}{2}  -1}    \phi^2 \le \int      (1+|x|^2)^{\frac{\alpha}{2}}   | \nabla\phi  |^2,\]  for all $ \phi \in C_c^\infty$ where  $ \sigma $ is some small positive constant; its either $ \E$ or $ p \E$ depending on which equation were are examining 
To show this we use the result from Corollary \ref{Hardy} and we drop a few positive terms to arrive at
\begin{equation*}
\int (1+|x|^2)^\frac{\alpha}{2} |\nabla\phi|^2\ge (t+\frac{\alpha}{2})\int \left (N-2(t+1)   \frac{|x|^2}{1+|x|^2}\right) (1+|x|^2)^{-1+\frac{\alpha} {2}}
\end{equation*} which holds for all $ \phi \in C_c^\infty$ and $ t,\alpha \in {\mathbb{R}}$ 
  Now, since $N+\alpha-2>0$, we can choose $t$ such that $-\frac{\alpha}{2}<t<\frac{n-2}{2}$   So, the integrand function in the right hand side is positive and since for small enough $\sigma$ we have
  \begin{equation*}
\sigma \le  (t+\frac{\alpha}{2})(N-2(t+1)   \frac{|x|^2}{1+|x|^2})  \ \ \ \text {for all} \ \ x\in \mathbb{R}^N
\end{equation*}
 we get stability 



\item ($\beta-\alpha+2>0$) In the case of $(G)$ we take   $u(x)=-\frac{\beta-\alpha+2}{2} \ln(1+|x|^2)$ and $g(x):= (\beta-\alpha+2)(N+(\alpha-2)\frac{|x|^2}{1+|x|^2})$  By a computation one sees that $u$ is a sub-solution of $(G)$ and hence we need now to only show the stability, which amounts to showing that
\begin{equation*}
\int \frac{g(x)\psi^2}{(1+|x|^{2   })^{-\frac{\alpha}{2}+1}}\le \int\frac{|\nabla\psi|^2}{    (1+|x|^2)^{-\frac{\alpha}{2}}     },
\end{equation*} for all $ \psi \in C_c^\infty$   To show this we use  Corollary \ref{Hardy}   So  we  need to choose an appropriate $t$ in   $-\frac{\alpha}{2}\le t\le\frac{N-2}{2}$  such that for all $x\in {\mathbb{R}}^N$ we have
 \begin{eqnarray*}
 (\beta-\alpha+2)\left(    N+  (\alpha-2)\frac{|x|^2}{1+|x|^2}\right)         &\le& (t+\frac{\alpha}{2})^2 \frac{ |x|^2 }{(1+|x|^2}\\
&&+(t+\frac{\alpha}{2}) \left(N-2(t+1)   \frac{|x|^2}{1+|x|^2}\right) 
\end{eqnarray*}
With a  simple calculation one sees we need just to have
   \begin{eqnarray*}
 (\beta-\alpha+2)&\le& (t+\frac{\alpha}{2}) \\
  (\beta-\alpha+2)     \left(    N+  \alpha-2\right)      &   \le&  (t+\frac{\alpha}{2}) \left(N-t-2+\frac{\alpha}{2}) \right) 
 \end{eqnarray*}     If one takes $ t= \frac{N-2}{2}$ in the case where $ N \neq 2$ and $ t $ close to zero in the case for $ N=2$ one easily sees the above inequalities both hold, after considering all the constraints on $ \alpha,\beta$ and $N$ 

 We now consider the case of $(L)$   Here one takes $g(x):=\frac {\beta-\alpha+2}{p-1}(    N+  (\alpha-2-\frac{\beta-\alpha+2}{p-1})
\frac{|x|^2}{1+|x|^2})$ and $ u(x)=(1+|x|^2)^{ -\frac  {\beta-\alpha+2}{2(p-1)} }$    Using essentially the same approach as in $(G)$ one shows that $u$ is a stable sub-solution of $(L)$ with this choice of $g$    \\
For the case of $(M)$ we take   $u(x)=(1+|x|^2)^{ \frac  {\beta-\alpha+2}{2(p+1)}   }$ and $g(x):=\frac {\beta-\alpha+2}{p+1}(    N+  (\alpha-2+\frac{\beta-\alpha+2}{p+1})
\frac{|x|^2}{1+|x|^2})$ 



\end{itemize}


\hfill  $  \Box$









Context 2: Paper Info

Title: Nuclear Liquid-Gas Transition in the Strong Coupling Regime of Lattice QCD
Publish Date: 28 Mar 2023
Author List: J Kim (from Institute for Advanced Simulation (IAS-4), Forschungszentrum Jülich), P Pattanaik (from Fakultät für Physik, Bielefeld University), W Unger (from Fakultät für Physik, Bielefeld University)

Figure

FIG  1 Typical 2-dimension configuration at β = 1 0, at non-zero quark mass, temperature, chemical potential The black dots are monomers, the blue lines are dimers, the red arrows are baryon loop segments (or triplets g b + f b = ±3 if adjacent to a non-trivial plaquette), and the green squares are plaquette occupations ±1 The actual configurations are 3+1-dimensional 
FIG 2 Chiral susceptibility on a 2 4 volume for various quark masses, as a function of the bare anisotropy γ (with aT = γ 2 /2), analytic results from enumeration compared to numerical data from simulations via the worm algorithm 
FIG 3 Various observables in the µB-T plane on a 2 4 volume at amq = 0 1 The back-bending of the first order transition at temperatures below aT = 0 5 in all observables is an artifact of the small volume, and vanishes in the thermodynamic limit The temperature aT = 1/2 corresponds to the isotropic lattice here 
FIG  4  The chiral condensate (left) and the baryon density (right) for quark mass m = 1 5 as a function of the chemical potential and for various temperatures 
FIG  7  ∆f at amq = 0 2 as a function of chemical potential and β the on a 6 3 × 4 lattice
FIG  8  Baryon mass from ∆E as a function of the quark mass amq, and contributions from different dual variables: monomers, dimers and baryon segments 
FIG  9  Baryon density for volume 4 3 × 8 in the full µB − mq plane, illustrating the strong quark mass dependence of the onset to nuclear matter 
FIG  10 Baryonic observables on various volumes in the first order region amq = 1 5 Vertical bands indicate the mean and error of the nuclear transition 
FIG  12  Left: Extrapolation of the pseudo-critical values of µB for the various volumes into the thermodynamic limit Right: Critical baryon chemical potential for different quark masses The first order transition region is shown in blue, the crossover region is shown in red and the range for critical end point is marked in black 
FIG  17  Nuclear interaction scaled with baryon mass As the quark mass increases, it tends to zero 
FIG  18  Critical baryon chemical potential and baryon mass from different approaches 
Parameters for the Monte Carlo runs to determine the nuclear transition at strong coupling, with statistics after thermalization 

abstract

The nuclear liquid-gas transition from a gas of hadrons to a nuclear phase cannot be determined numerically from conventional lattice QCD due to the severe sign problem at large values of the baryon chemical potential  In the strong coupling regime of lattice QCD with staggered quarks, the dual formulation is suitable to address the nuclear liquid gas transition 
We determine this first order transition at low temperatures and as a function of the quark mass and the inverse gauge coupling β  We also determine the baryon mass and discuss the nuclear interactions as a function of the quark mass, and compare to mean field results  It is known from experiments that at low temperatures, there is a phase transition between dilute hadron gas and dense nuclear matter as the baryon chemical potential increases 
This transition is of first order and terminates at about T c = 16 MeV in a critical end point  The value of the chemical potential µ 1st B at zero temperature is given roughly by the baryon mass m B , where the difference of µ 1st B −m B is due to nuclear interactions  For a review on nuclear interactions see  
As the nuclear force between baryons to form nuclear matter is due to the residual strong interactions between quarks and gluons, it should be accurately described by QCD  We choose to study the nuclear transition and nuclear interaction via lattice QCD , with its Lagrangian being a function of the quark mass and the inverse gauge coupling 
In order to understand the nature of the transition, it is helpful to study its dependence on these parameters  However, at finite baryon density, lattice QCD has the infamous sign problem which does not allow us to perform direct Monte Carlo simulations on the lattice  Various methods have been proposed to overcome the numerical sign problem, but they are either limited to µ B /T 3 or can not yet address full QCD in 3+1 dimensions in the whole µ B − T plane , in particular the nuclear transition is out of reach 
An alternative method is to study lattice QCD via the strong coupling expansion  There are two established effective theories for lattice QCD based on this: (1) the 3-dim  effective theory for Wilson fermions in terms of Polyakov loops, arising from a joint strong coupling and hopping parameter expansion , the dual representation for staggered fermions in 3+1 dimensions, with dual degrees of freedom describing mesons and baryons 
Both effective theories have their limitations: is limited to rather heavy quarks (but is valid for large values of β) whereas ( ) is limited to the strong coupling regime β 1 (but is valid for any quark mass)  We study lattice QCD in the dual formulation, both at infinite bare gauge coupling, β = 0, and at leading order of the strong coupling expansion in the regime β < 1, which is far from the continuum limit 
But since strong coupling lattice QCD shares important features with QCD, such as confinement, and chiral symmetry breaking and its restoration at the chiral transition temperature, and a nuclear liquid gas transition, we may get insights into the mechanisms, in particular as the dual variables give more information in terms of its world lines, as compared to the usual fermion determinant that depends on the gauge variables 
To establish a region of overlap of both effective theories, we have chosen to perform the Monte Carlo simulations in the dual formulation extending to rather large quark masses  This paper is organized as follows: in the first part we explain the dual formulation in the strong coupling regime, in the second part we provide analytic results based on exact enumeration and mean field theory, in the third part we explain the setup of our Monte Carlo simulations and present result on the m q -and β-dependence of the nuclear transition 
Since the strong coupling regime does not have a well defined lattice spacing, we also determine the baryon mass am B to set the parameters of the grand-canonical partition function, aT and aµ B , in units of am B   We conclude by discussing the resulting nuclear interactions, and compare our findings with other results 

Staggered action of strong coupling QCD and its dual representation

In the strong coupling regime, the gauge integration is performed first, followed by the Grassmann integration to obtain a dual formulation  This was pioneered for the strong coupling limit in and has been extended by one of us to include gauge corrections   The sign problem is mild in the strong coupling limit and still under control for β < 1, where we can apply sign reweighting 
The dual degrees of freedom are color-singlet mesons and baryons, which are point-like in the strong coupling limit, and become extended about a lattice spacing by incorporating leading order gauge corrections  The partition function of lattice QCD is given by where DU is the Haar measure, U ∈ SU(3) are the gauge fields on the lattice links (x, μ) and { χx , χ x } are the unrooted staggered fermions at the lattice sites x 
The gauge action S G [U] is given by the Wilson plaquette action and the staggered fermion action S F [ χ, χ, U] is: where the gauge action depends on the inverse gauge coupling β = 2Nc g 2 and the fermion action depends on the quark chemical potential aµ q which favors quarks in the positive temporal direction, and the bare quark mass am q  
First we consider the strong coupling limit where the inverse gauge coupling β=0 and hence the gauge action S G [U] drops out from the partition function in this limit  The gauge integration is over terms depending only on the individual links (x, μ) so the partition function factorizes into a product of one-link integrals and we can write it as:
with z(x, μ) the one-link gauge integral that can be eval-uated from invariant integration, as discussed in , where we write the one-link integral in terms of new hadronic variables: Only terms of the form (M (x)M (y)) k x, μ (with k x,μ called dimers which count the number of meson hoppings) and B(y)B(x) and B(x)B(y) (called baryon links) are present in the solution of the one-link integral 
The sites x and y = x + μ are adjacent lattice sites  It remains to perform the Grassmann integral of the fermion fields χ, χ  This requires to expand the exponential containing the quark mass in Eq  (4) (left), which results in the terms (2am q M (x)) nx (with n x called monomers)  To obtain non-vanishing results, at every site, the 2N c Grassman variables χ x,i and χx,i have to appear exactly once, resulting in the Grassmann constraint (GC):
where n x is the number of monomers, k x,μ is the number of dimers and the baryons form self-avoiding loops x,μ , which due to the constraint cannot coexist with monomers or dimers  With this, we obtain an exact rewriting of the partition function Eq  ( ) for N c = 3, in terms of integer-valued dual degrees of freedom {n, k, }:
where the sum over valid configurations has to respect the constraint (GC)  The first term in the partition function is the contribution from dimers and the second term is the contribution from monomers  The weight factor w( ) for each baryon loop depends on the baryon chemical potential µ B = 3µ q and induces a sign factor σ( ) which depends on the geometry of :
Here, ω is the winding number of the loop   The total sign factor σ( ) ∈ {±1} is explicitly calculated for every configuration  We apply sign reweighting as the dual formulation has a mild sign problem: baryons are non-relativistic and usually have loop geometries that have a positive signs  The dual partition function of the strong coupling limit is simulated with the worm algorithm (see Section III A) and the sign problem is essentially solved in this limit 

Extension to finite β

The leading order gauge corrections O(β) to the strong coupling limit are obtained by expanding the Wilson gauge action Eq  ( ) before integrating out the gauge links  A formal expression is obtained by changing the order of integration (first gauge links, then Grassmann-valued fermions) within the QCD partition function:
With this the O (β) partition function is The challenge in computing Z (1) is to address the SU(N c ) integrals that receive contributions from the elementary plaquette U P   Link integration no longer factorizes, however the tr[U P ] can be decomposed before integration: Integrals of the type J ij with two open color indices -as compared to link integration at strong coupling -have been derived from generating functions
for either J = 0 or for G = U(N c )   The SU(3) result was discussed in , in terms of the dual variables, neglecting rotation and reflection symmetries, there are 19 distinct diagrams to be considered  The resulting partition function, valid to O(β), is with q P ∈ {0, ±1}, and the site weights w x → ŵx , bond weights w b → ŵb and baryon loop weights w → ŵ receive modifications compared to the strong coupling limit Eq  ( ) for sites and bonds adjacent to an excited plaquette q P = 1 
The weights are given in , and are rederived for any gauge group in   The configurations {n, k, , q p } must satisfy at each site x the constraint inherited from Grassmann integration: which is the modified version of Eq  ( ) with q x = 1 if located at the corner of an excited plaquette q p = 0, otherwise q x = 0 
A more general expression that we obtained via group theory and is valid to higher orders of the strong coupling expansion is discussed in terms of tensor networks   A typical 2-dimensional configuration that arises at β = 1 in the Monte Carlo simulations is given in Fig    Note that if a baryon loop enters a non-trivial plaquette, one quark is separated from the two other quarks, resulting in the baryon being extended object, rather being point-like in the strong coupling limit 
The O(β) partition function has been used in the chiral limit to study the full µ B − T plane via reweighting from the strong coupling ensemble  Whereas the second order chiral transition for small values of the aµ B decreased up to the tri-critical point, the first order nuclear transition was invariant: aµ 1st B 1 78(1) at zero temperature has no β-dependence 
For the ratio T (µ B = 0)/µ 1st B (T 0) we found the values 0 787 for β = 0 and 0 529 β = 1, which should be compared to T c / 0 165 for full QCD   However, since reweighting cannot be fully trusted across a first order boundary, direct simulations at nonzero β are necessary  The Monte Carlo technique to update plaquette variables is discussed in Section III A 
In this section, we provide analytic results from exact enumeration for small volumes, and mean field results based on the 1/d expansion, valid in the thermodynamic limit
Context 3: Hey folks! Here is the shiny new Changelog thread  We're including the archived patch notes from the old forums, so that they are preserved for anyone that would like to reference back to them  We will continue to update this thread with new notes as the patches are released 
Chat is now accessible from the quest board, upgrade screen, and many other menus 
Tapping on objects and menus may reveal helpful hints about that object 
Team PI is now colored red if lower than recommended for the quest 
Many text fixes and consistency improvements 
• A new Basic Catalyst found in Special Events is used in every recipe!
Several heroes have received improvements to their base stats 
The abilities of all Champions have increased in effectiveness 
A new Critical Boost buff has been introduced 
Iron Fist and Spiderman now have the ability to Armor Break with their Critical Hits 
Deadpool’s ability to Regenerate is more powerful, but only triggers once per fight 
Scarlet Witch now has a chance to trigger Nullify off of any Critical Hit 
Juggernaut and Rhino now have a layer of Armor 
Punisher and Winter Solider now may also trigger Fury in addition to Bleed 
Colossus now further increases his base Armor with the Armor Up ability 
Thor and Ronan no longer Armor Break; instead, base stats and Stun durations have improved 
We reduced the effectiveness of the Revive items in order to give away more as rewards 
A bonus of 50% for using ISO-8 matching your Champion’s Class can now be previewed on the Upgrade screen 
It’s now possible sell Champions in exchange for ISO-8 and Gold  The amount received increases proportionately to the Rank and Level of the sold Champion 
-You can now skip dialogue on the quest map by pressing ‘SKIP’ 
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-Various game balance and cosmetic improvements to the available quests 
• PVP energy has been replaced with Hero Stamina  Each Hero has their own Stamina values, meaning the more Heroes you have the more you can play in PVP 
• Each Hero has 1 Stamina and takes 2 hours to recharge 
We have removed the Next Quest button for a much more favorable and flavorful approach to teaching and informing people about Marvel : Contest of Champions  In the Main Menu(Bottom Right Corner) you will now see an image of the Collector showing you what the best or recommended actions that you should preform  This can be anything from opening Crystals, Continuing a Quest, Ranking Up Champions if the difficulty is too hard, Tips where to obtain items, and Playing Versus/Arenas 
• Adjusted the PI calculation for Power Burn and Power Drain abilities to improve accuracy 
• Significantly increased the Power Burn multiplier as well as the amount of Power burned  Prior to this change, Vision's Special Attack damage output was far below the curve  Vision's Special Damage is distinct from other heroes in that the dependency on opponents' Power levels cause the damage dealt to be highly variable, and sometimes quite low; however, when striking an opponent with high Power levels, Vision has the potential to deal very high amounts of direct, Armor-ignoring damage 
• Slightly adjusted the Armor Break trigger to be less punishing to opponents with the Armor Up ability without sacrificing PI or damage output 
• Slightly increased his base Health and, in turn, the amount of Health recovered by Regeneration  This improvement is reflected by an increase to PI of about 1% 
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• Slightly reduced the frequency of Nullify for basic attacks, but slightly increased the chance a Special Attack is critical  Chaotic Bombardment no longer has a chance to critical, and instead has a 100% chance to Nullify the target  This is less punishing to opponents with beneficial effects, while providing a more reliable source of Nullify  Overall, her PI has decreased by about 2% 
• Decreased base Health and Attack by 2% each to bring his PI in line with other Champions without compromising Special Attack effectiveness 
• Slightly increased base Health by 2% to bring his PI in line with other Champions  This change may result in a PI increase of up to 1% 
• Fixed a bug with her Bleed ability scaling incorrectly  This has no effect on PI 
• User's on iPhone 4 devices will no longer encounter a progression blocker after fighting Iron Man in the tutorial 
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• The Crystal Vault has received another polish pass and should now feel much more responsive, thank you for all your feedback on this feature!
• Many more minor bug fixes were included in this update 
• Special Attack 1 base damage increased by +25% Attack Rating 
• Heavy Attack base Power gained reduced to 63 points 
We recently improved the functionality of Heavy Attacks, so they’re easier to use  Their base Power has been reduced to normal levels – previously, they generated Power at a higher rate to compensate for their difficult execution  Special Attacks have been adjusted to give the unlucky recipients more of a fighting chance  These changes bring these attacks in line with existing damage-to-power ratios 
*NOTE: Special Attacks only generate Power for the target struck, not for the user; this prevents infinite loops and helps serve as a comeback mechanic 
Versus Crystal prizes have been adjusted due to the Champion Stamina changes 
Arena Crystal prizes have been increased to help balance the adjustments to the Versus Crystal 
Payouts have significantly increased when receiving a duplicate Champion with a Star rating of two or more  The boosted amount increases based on Star rating  We apologize for any inconvenience caused by delivering each reward individually, and are working to get a fix to you as soon as possible  In the meantime, using the “Skip” button avoids the inconvenience 
• We fixed a bug where finding a new match could cost a player Units 
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*NOTE: This caused some players' progress to reset for a brief time, but that issue should now be corrected 
• Event Quest difficulty has been adjusted to match Catalyst availability 
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Mutant Champions are now effective against Skill Champions 
• The high Special Attack damage and regenerative abilities of Mutant Champions are effective against Skill Champions, which typically rely on Bleed damage from their weaponry  We think of this relationship as if the X-Gene grants Mutant Champions superpowers that evolved to be stronger than Champions that are merely “Skilled” 
Skill Champions are now effective against Science Champions 
• While scientists fiddle in their cute little laboratories to create flasks full of serums to turn even frail young men into super-soldiers, Skill Champions were just born that way baby  Often donning sharp weaponry to make their opponents Bleed, Skill Champions enjoy watching the high base attributes of Science Champions just melt away 
Cosmic Champions are now effective against Tech Champions 
• Tech Champions construct durable robots and thick suits of Armor to outlast their opponents in battles of tank-the-nuke   which gives Cosmic Champions extra time to build up stacks of beneficial effects to overrun Tech Champions using their peculiar alien enhancements 
• Tech Champions are still effective against Mutant Champions 
Tech Champions typically excel at Armor, Resistance, and Power manipulation, which is effective against the high Special Attack damage of Mutant Champions  Think of the robotic Sentinels adapting for tactical advantages in the war against Mutantkind!
Science Champions are still effective against Mystic Champions 
• Science Champions – a Class of behemoths like Hulk and super-soldiers like Captain America – typically have above average base attributes like Health, Attack, and Armor  These raw stats cannot be affected by pesky Mystics and their removal abilities: Nullify and Purge 
Mystic Champions are still effective against Cosmic Champions 
• Cosmic Champions explore strange new beneficial effects to seek out new power and new abilities, to boldly take their attributes where no class has gone before  Well, not if Mystic Champions – who are fully capable of stripping Cosmic Champions of their beneficial effects – have anything to say about it! Maybe it’s the Mystic Agenda to protect the secrets of the universe?
These changes ensure that having a Class Bonus always gives you the advantage it promises, as it now also reflects ability trends for a particular Class  Please keep in mind that these are generalizations, and some Champions abilities may not always strictly align with these relationships  Learn more about Champions’ abilities by viewing their profiles and tapping on features for detailed information 
• When you attack someone, you charge up their Power in addition to yours  This meant they would reach a full three bars while you only reached one and a half  We've reduced the amount defenders receive such that you'll be at two bars when they're at three  This change maintains the underdog functionality to give defenders a chance to comeback while being less punishing to players earning high Combos



ground_truth: $\int f'(u) \psi^2 \le \int | \nabla \psi|^2, \forall \psi \in C_c^2$.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the main focus of the research paper?

Read the following text and answer briefly.

Context 2:  The main purpose is to compare our Monte Carlo results to these analytic predictions 

Exact enumeration

To establish that our Monte Carlo simulations indeed sample the partition functions Eq  ( ) and Eq  ( ), we have obtained analytic results on a 2 4 volume at strong coupling, and at finite beta in two dimensions on a 4 × 4 volume, comparing O (β) and O β 2 truncations  Our strategy to obtain an exact enumeration of the partition function Z is to enumerate plaquette configurations first, then fixing the fermion fluxes which together with the gauge fluxes that are induced by the plaquettes form a singlet, a triplet or anti-triplet, i e  on a given bond b, g b + f b ∈ {−3, 0, 3}, and last we perform the monomerdimer enumeration on the available sites not saturated by fermions yet by a depth-first algorithm  
At strong coupling, with no plaquettes, g b = 0 and f b are baryonic fluxes  All observables that can be written in terms of derivatives of log(z), such as the baryon density, the chiral condensate, the energy density, and also the average sign, are shown in Fig 

Expectations from mean field theory

Another analytical method to study strong coupling lattice QCD is the mean field approach, where the partition function is expanded in 1 d (d is the spatial dimension) and then a Hubbard-Stratonovich transformation performed   After this procedure, the free energy is a function of temperature T , the chiral condensate σ and chemical potential µ B :
here E[m] is one-dimensional quark excitation energy which is a function of the quark mass m = am q   For N c = 3 and d = 3 we determined the minimum of the free energy with respect to the chiral condensate  This gives us the equilibrium chiral condensate as a function of (T, m, µ B )  The chiral condensate and the baryon density as a function of the baryon chemical potential in lattice units aµ B and for various temperatures at quark mass m = 1 5 is shown in Fig    We have determined the critical temperature to be aT c = 0 23 , which is characterized by an infinite slope of the chiral condensate 
For lower temperatures, there is a clear discontinuity of the chiral con-densate, separating the low density phase from the high density phase  For temperatures above and in the vicinity of aT c the chiral condensate and baryon density has no discontinuity but rapidly changes, corresponding to a crossover transition 
With this method, the phase diagram is plotted for different quark masses in Fig    The second order phase transition in the chiral limit is plotted in solid blue line, the dotted lines show the first order phase transition for different quark masses and the solid red line indicates the critical end point for the different quark masses 
Mean field theory also gives an expression for the pion mass am π and the baryon mass am B : The mean field baryon mass for N c = 3, d = 3 is also plotted in red in Fig    Whereas the baryon mass is around N c in the chiral limit (am B 3 12 for N c = 3), it approximately doubles at m = 3 5 (am B 6 28) which corresponds to the pion mass am π = 4 45, i e  m π /m B = 0 708 
Hence, at around bare mass m = 3 5, the valence quark mass of the baryon corresponds roughly to 1/3 of the chiral limit value of the baryon mass  The first Monte Carlo simulations that could extend in the µ B − T plane was the MDP algorithm , but it required the introduction of the worm algorithm to make substantial progress 
First studies of the worm algorithm applied to the strong coupling limit QCD (with gauge group U(3)) are , and for gauge group SU   Monte Carlo simulations to extend the worm to incorporate leading order corrections were first proposed in   We will shortly review the setup of or Monte Carlo strategy for the nuclear transition, with an emphasis on the challenges to address large quark masses 

Strong Coupling

Without any further resummation, there is a mild sign problem in the dual formulation of lattice QCD in the strong coupling limit  When the average sign σ is not too small (close to zero), it implies that most of the configurations have a positive weight thus allowing us to perform sign reweighting strategies 
In Fig  , ∆f is plotted as a function of the baryon chemical potential and the quark masses  It is seen that ∆f is close to zero for most cases except near the critical chemical potential and for small quark masses, but never exceeds 5 × 10 −4   Hence sign reweighting can be performed in the full parameter space 
The result that the sign problem becomes even milder when increasing the mass is related to the fact that larger critical chemical potentials result in a larger fraction of static baryons (spatial baryon hoppings become rare)  FIG    ∆F at strong coupling as a function of chemical potential and quark mass on a 6 3 × 8 
The sign problem becomes milder as the quark mass increases 

Finite β

All runs at finite β have been obtained for N τ = 4, which corresponds to a moderately low temperature aT = 0 25 compared to the value of the chiral transition aT 1 54  Those simulations were too expensive to attempt N τ = 8 runs, in particular as a higher statistics was required  The spatial volumes are 4 3 , 6 3 and 8 3  
For β values are from 0 0 to 1 0 with step size 0 1, and for am q values from 0 00 to 1 00 with step size 0 01  The values of aµ were chosen close to the nuclear transition, the scanning range is shifted to large values as am q increases  At small quark masses the scanning range is from aµ = 0 4 to 1 0 and for the large quark masses, it is from 0 6 to 1 2 with step size 0 01 
The statistics used for are 15 × 10 4 measurements and between measurement, 40 × N 3 s worm updates 

Residual sign problem

Although it is possible to resum the sign problem at strong coupling with a resummation of baryon and pion world lines, this is not possible when including gauge corrections  In order to compare both sign problems, we kept the original dual formulation to monitor the severity of the sign problem  This is done via the relation
between the average sign σ and the difference of the free energy density ∆f between the full ensemble f and of the sign-quenched ensemble f ||  

Nuclear interactions

We have found that aµ 1st B is very different from the baryon mass  This must be due to strong attractive interactions of nucleons  In contrast to continuum physics, in the strong coupling limit there is no pion exchange due to the Grassmann constraint  Instead, nucleons are point like and hard core repulsive 
However, the pion bath, which is modified by the presence of static baryons, results in an attractive interaction  In , this has been analyzed in the chiral limit using the snake algorithm, and it has been found that the attractive force is of entropic origin  Here, we do not quantify the nuclear interaction via the nuclear potential, but via the difference between critical baryon chemical potential and baryon mass, in units baryon mass, as shown in Fig  , given the am B as measured in Section III C 
This compares better to the 3dim  effective theory  The nuclear interaction is maximal and more than 40% in the chiral limit, which is related to pions being massless: the modification of the pion bath is maximal  We clearly find that the nuclear interaction decreases drastically and almost linearly until it almost approaches zero at about am q = 2 0, corresponding to a pion mass am π = 3 36, see Section II B  The large error bars for larger quark masses, that are due to the subtraction of almost same magnitudes, makes it difficult to extract a non-zero nuclear interaction at the largest quark masses 
In this work, we have determined the baryon mass and the nuclear transition via Monte Carlo: the worm algorithm based on the dual formulation, at finite β equipped with additional updates  All those numerical results and various analytic expressions are summarized in Fig    We find that as the quark mass becomes large, spatial mesons hoppings (i e 
spatial dimers) become rare, which makes this 3+1-dimensional system closer to 1dim  QCD   Also, both the baryon mass and the baryon chemical potential obtained in our dual representation, i e  for staggered fermions, approaches the baryon mass of the 3-dim  effective theory which is based on Wilson fermions 
Another comparison that summarizes the validity of the mean field approach discussed in Section II B is shown in Fig    It is evident that mean field theory has strong deviations for small quark masses, but this discrepancy becomes smaller for larger quark masses  The extension of the study of the nuclear transition to finite inverse gauge coupling β is summarized in Fig  , which shows the β-dependence of aµ c B for various quark masses 
For all quark masses ranging from am q = 0 to am q = 1 0, there is only a very weak β-dependence, confirming the expectation from mean field theory   This works was restricted to isotropic lattices ξ = a/a t = 1, i e  we performed simulations at fixed temperature  Non-isotropic lattices are necessary to vary the temperature at fixed values of β 
This requires to include two bare anisotropies, γ for the fermionic action and γ G for the gauge action  Finite β has only been studied by us in the chiral limit   Clearly, it is interesting to study the location of the nuclear critical point also including higher order gauge corrections and at finite quark mass 
Simulations including O(β 2 ) are under preparation 
Context 3: 
New damage types for attacks now play a larger role in the abilities of Champions  For example, some heroes power-up by successfully blocking magical damage, while others’ abilities may harm anyone that makes physical contact with them 
New Resistances and Immunities have found their way to the Battlerealm  Some heroes are completely immune to specific status effects based on either lore from the comics or logic  For example, the android Vision has no blood, and is therefore fully immune to Bleed conditions  We’ve also strengthened the effectiveness of certain status effects, so be careful who you choose to bring into battle! Could you guess who might be immune to the new “Poison” condition?
• Poison: Inflicts damage over time and reduces healing and regeneration effectiveness 
• Unstoppable: A buff to shrug off the impact from attacks, but still take the damage 
• Weakness: A debuff that reduces Attack attributes 
• Heal Block: Fully prevents the target from gaining health in any way 
• Power Lock: Seals the target, preventing them from gaining any Power 
• When fighting, you may notice that many status effects are now able to stack  This also changes how certain beneficial “buffs” and detrimental “debuffs” interact with one another  For example, it's now possible to have both Armor Up and Armor Break effects on you simultaneously  Let the tug-o-war begin, and may the strongest effects win!
• Black Bolt's Corkscrew: +25% damage, but at the cost of minor recoil damage 
• Punisher's “Wrath” has been replaced by "Payback"  Payback deals additional damage based on the total damage dealt to Frank 
• Colossus' “Unbreakable” now deal bonus damage based on his armor level at the time of activation 
• All of Black Panther’s special attacks now deal bonus damage based on the number of Bleeds on the target 
• Spider-Man’s Web-Slinger now has a chance to inflict Weakness 
• Vision’s Physical Disruption: Added a minor Power Burn effect due to “his” use of his Infrared Beam  “He” also now purges all status effects while phasing through the ground 
• Scarlet Witch: Increased the Critical Hit Chance for Hex Bolt and Hex Sphere 
• Many knockback effects have been adjusted to improve consistency 
We’ve tested the Signature Abilities quite extensively before releasing them, but there have been a few abilities that we have been keeping an eye on  We’ve compared our notes with the feedback you’ve been sending us and are making some balance changes to them  Thanks for your feedback!
Slightly reduced the frequency and duration of Juggernaut’s “Unstoppable” ability 
• He was indeed a bit too   unstoppable  We’ve toned down the frequency this ability triggers, as well as reduced the duration it’s active for when it does trigger  We feel Juggernaut is still a powerful Champion despite these revisions  Take care!
Slightly reduced the starting values of Wolverine's “Cellular Regeneration” 
• We found that Cellular Regeneration was too strong at lower levels where fewer counters to Regeneration exist 
Re-scaled Gamora's “Assassination” to start higher but scale slower 
• At lower levels, Special Attacks were used too infrequently, giving this powerful ability little visibility  We’ve adjusted the scaling to better match Special Attack usage at all levels 
Increased the frequency that Black Bolt’s “Provocation” triggers 
• Due to the varying Critical Hit rates across all Champions, in some cases Provocation would trigger rarely or not at all within a fight  We’ve increased the frequency to ensure you’ll see it every match – but especially so against opponents with high Critical Hit rates 
We’ll continue to follow the effect of these new abilities on gameplay  Please keep your feedback coming!
Hey everyone! We have been hard at work on improving the game and have prepared a big update inspired in part by your great community feedback  Please keep letting us know what you think!
• Fixed many Dash, Medium, Heavy and Special Attacks missing or failing to execute 
• Added Alliances and a new Alliance Crystal 
• Rocket Raccoon and Unstoppable Colossus join The Contest 
• Temporary Boosts to Attack, Health, and XP are now available from the Alliance Crystal 
• Rewards for completing and exploring Chapters and Acts  Earn a guaranteed 3-Star hero crystal for each fully explored Act! This is retroactive, just complete any quest to claim them 
• A new Fight Menu combines The Arenas, Story Quests and Event Quest menus 
• Updated Summoner Profiles with new information  Inspect other players’ Profiles and brag about your achievements!
• A list of blocked users has been added to Chat windows  The option to unblock these users is found in this new menu  The power is in your hands now!
• We fixed Dash and Medium Attack issues for many heroes that sometimes missed or did not activate 
• We fixed issues to Drax and Colossus Light and Medium Attacks where they would not connect 
• Fixed an issue where the camera would stop moving after a level 3 special sequence 
• Fixed an issue where the player’s heavy attack would get stuck in charge even after the player has released input 
• Fixed a rare bug where Champions were still able to deal damage after they died, resulting in tied fights 
Form Alliances with your Friends!
What is better than playing? Playing with your friends! Create a new Alliance or join an existing one through the new Alliance Menu 
• Invite other players to your Alliance 
• Search for an Alliance by name or join a Recommended Alliance 
• Receive rewards for entering your first Alliance 
• Alliance News Feed  The news feed celebrates your Alliance member’s achievements 
• Alliance Chat  Chat with other members of your Alliance in a private channel all to yourself 
• Help Allies  Players can ask for help when out of Energy or Stamina  Alliance members help each other as much as they can to earn Loyalty Points  Loyalty points have a daily limit to how many can be earned 
• Alliance Crystal  Access a new Alliance Crystal while part of any Alliance  Use new Loyalty Points for purchasing Alliance Crystals 
He may start out slow, but watch out for his immense power at high ranks!
• Adjusted the range of many Heavy Attacks, including Hulk and Drax, to ensure they correctly connect with enemies 
• Many Special Attacks, including those for Wolverine, Iron Fist, Winter Soldier, Punisher, Black Panther, and many others have had their range adjusted to ensure they correctly connect with enemies even if activated immediately after a combo that knocked the enemy back 
• Payback and Unbreakable now display their maximum potential damage bonus 
• Added detailed descriptions for Bleed Immunity and Poison Immunity 
• Gamora: We’ve adjusted the scaling of her base Special Attack damage to ensure they scale up more similarly to other heroes  This also makes Gamora more reliant on her high Bleed damage, and improves the chances of opponents able to deal with her high Bleed 
Vital Strike and Jade Assassin damage decreased by 10% 
Godslayer damage increased by 10% 
• Magik: Rewind is a game-changer for Magik that allows her to go up against foes like Gamora and Rewind off big Critical Hits and Bleed damage; however, the frequency of Rewind triggering was too low to be there when she needed it 
Increased the likelihood Rewind triggers by +20% at all levels 
Rewind now heals over one second instead of instantly 
Fixed a bug allowing Magik to break out of an enemy combo using Rewind  It now only removes Status Effects 
• Hulk: Given the riskiness of losing Health in certain game modes, Hulk’s anger-management provided too little help too late in the game  We’ve increased the Attack boost to ensure he’s appropriately scary in all game modes – as long as he’s angry!
Increased Hulk Rage by +20% Attack at all ability levels 
Arc Overload no longer causes Armor Break when it expires 
• Vision: Added Poison Immunity to our robot friend 
Arena tuning is an ongoing process  The team is continually making adjustments to Arenas to improve the experience 
Ultron has infected The Contest!
Many new Champions join the battle against Ultron 
Quest through the new Ultron’s Assault Event 
Wield new power with Summoner Masteries 
Grow your Friend’s List with the new Social Hub 
Team up with your Alliance in new Events, Arenas, and more!
Filter and sort your Stash 
Fights have been optimized for performance improvements on all devices 
Users can now filter through the items in their Stash 
Fixed several issues where Hero Rating would fluctuate 
Fixed a bug with Rhino and Juggernaut having 11-20% more Armor than intended 
Fixed a bug with Rocket Raccoon’s Dash attack being slower than intended 
Added a confirmation popup when spending Units on stamina recharges and unlocking arenas 
Regeneration no longer displays green Health values if you’re at full Health 
Several new improvements to how status effects are displayed 
AI opponents are no longer able to perform one unavoidable attack in response to a Special Attack 3 
A new and improved look for all Health Potions in the Battlerealm 
All Revive Potions now revive your Champions with +10% more Health 
We’re adding so many new Champions, they could form their own Alliance!
Some of your favourite heroes of the Marvel Cinematic Universe join The Contest!
Summoner Mastery is on the horizon!
Masteries provide beneficial effects for your Champions 
Access Masteries through your Summoner Profile 
Earn Mastery Points when you level up 
Choose your Masteries wisely and strategically customize your benefits 
Recover your points to try a new specialization as often as you’d like 
Keep an eye on in-game messaging for more information 
The daily loyalty limit has been set to refresh at 08:00UTC for all players 
A timer has been added to show when the daily loyalty limit resets 
Loyalty balance is now displayed in the Alliance menus 
Ask for Versus help with a single tap on the ‘Help’ icon in Team Select 
New Alliance Events are coming very soon!
Work together with your Alliance to complete objectives and receive rewards!
Muster your might, Alliance Arenas will soon open their gates!
Competing in Alliance Arenas shares your points across your whole Alliance; work together to reach milestones and top ranks!
Work together to amass a huge score, and defeat your competition in classic Arena combat! No slackers here either - if you don’t contribute to win the competition, you’re not eligible for the goods!
All social features (Chat, Mail, and Friends) can now be accessed through the new Social Hub 
Search for and add friends, and send private messages to Summoners on your Friends List 
Redesigned chat and mail screens 
Take on other Summoners’ top Champions for bragging rights and prizes in 1-on-1 Duels!
A new series of special Ultron quests are available, starting with the first Chapter  Fight back against Ultron’s infection alongside the Summoner, and team up with some of Marvel’s finest! New quests unlock each week!
The Spider-Man Champion gate has been removed from Act 1, Chapter 1, Quest 5 
• Fixed an issue where chat snapped to the most recent message 
• Fixed several issues where Hero Rating would fluctuate 
• Various improvements to the Summoner Mastery screens and descriptions 
• Increased the ISO8 awarded by duplicate 2-Star Champions 
Quest through the new single-player campaign, Ant-Man’s Adventure!
In addition to Ant-Man and Yellowjacket feuding throughout the Battlerealm, additional new Champions will be joining The Contest!
Access more Masteries in the new Utility Mastery tree!
Please note, these changes may result in a loss of Hero Rating as incorrect effects are restored back to normal levels 
Improved and polished combat mechanics to reduce the amount of stutters and lost input 
Fixed and optimized rendering related issues with Metal enabled devices



ground_truth: Nuclear liquid-gas transition in lattice QCD.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: How can players skip dialogue on the quest map?

Read the following text and answer briefly.

Context 3: 
Team up with Ant-Man, and put a stop to Yellowjacket’s mysterious mission!
All Alliance Quests only last for a specified amount of time, defeat the boss with your Alliance before it expires!
New Prestige System - A dynamic difficulty and score setting that adjusts as you and your Alliance succeed in harder quests  The better you do and the tougher your Alliance is, the higher the prestige  The higher the prestige, the better the rewards!
Choose your teams carefully as Champions within Alliance Quests cannot be used in other Story or Event Quests 
Act 4 has been released! Play Chapter 1 now!
Summoner level maximum has been increased to level 60!
5-Star Champions are coΩming to The Contest! These are the most powerful Champions yet!
Additional improvements have been made to the UI, Versus Arenas, Synergy Bonuses, the Stash & Items Store 
Act 4 - Chapter 1 released!
New challenges - more path variation and features to challenge the strongest Summoners!
Greater challenge means greater rewards! Earn 4 Star Crystals and Mastery Points!
The Summoner Level cap has been increased by ten levels to level 60!
Champion Items will be coming soon! These allow you to apply items and buffs to a specific Champion, keep an eye out for updates on these new Champion Items!
Synergy Bonuses have updated iconography and the calculation has been updated to a distinct, additive bonus - What you see is what you get!
Alliance class distribution is now displayed on team select - Choose the right class!
Your Catalysts now have their own inventory, and will no longer appear in the Upgrade Item inventory 
The Stash is now separated into three tabs: Catalysts, Rewards and ISO, allowing you to sort and view your Stash much faster!
The UI flow for both Quests and Arenas have been greatly improved  You can now skip through fight victory and reward animations!
Here is the rundown of patch 5 1 0, filled with various bug fixes and optimizations  The important ones to note are below 
New Champions, new theme, and a new arena!
To celebrate our one year anniversary AND the holidays, we’ll be running a special event quest! Battle through the history of The Contest, and test your mettle against familiar faces both old and new!
A special reward will be available to those who master every quest!
Our Anniversary Celebration will be happening very soon; stay tuned for more info!
More Act 4 quests are coming very soon!
Opponents in Story Quests now have the ability to use their Special 3 attack! Note that we are not changing previous quest opponents to have this special attack (Act 1-3, Proving Grounds, Realm of Legends will not change); this will be in effect starting with the soon-to-be-released Act 4 content 
As with our previous major build releases (3 0’s Ultron, 4 0’s Ant Man, and 5 0’s Battlerealm), the Contest has been reskinned with a new theme!
The Road to Knowhere map is here! Fight in a new level inspired by Guardians of the Galaxy!
A new button in your Alliance Chat to take you directly to Alliance Quests!
You can now collect Catalyst Fragments in Event Quests, Proving Grounds, and Alliance Quests; these can be pieced together into a Catalyst!
Selling Items is now a thing! Sell any items in your inventory for gold!
Level 3 and Level 4 Health Potions have arrived! These are powerful instruments to help you tackle all the new Act 4 content!
Over 400 bugs were fixed in this patch!
This patch is a fix for the missing Champions during the Special 3 animation on Android devices 
This issue occurred during our upload process to the Google Play Store  This was an odd edge case scenario that we could not have caught during our internal tests, as it began appearing once we uploaded to the Google Play Store  This hotfix will be out by tomorrow, and will put Android at version 6 0 1  As this issue does not occur on iOS devices, iOS will remain at version 6 0 
3:30pm PST: We have started slow-rolling this patch out to Android devices, beginning with about 20% of users  We expect this to be available for 100% of users within the next 24 hours 
We have a few new Champions that you will see within the next couple of months (including one of my personal favorites)!
Over 200 total bugs squashed in this patch!
An artifact left over from the early days of the contest was Black Panther’s ability to gain a Critical Hit Rate boost during Special 3 attacks  As many might know, Critical Hits aren’t possible during a Special 3 anymore, making this effect   unhelpful  We’ve switched it out with a new ability to stack up even more Bleed effects on the opponent based on how many Bleeds are already active 
Example: The opponent has 4 stacks (instances) of Bleed on them when you launch a Special 3  With this new ability, you have a chance to add an additional 0 - 4 more stacks (instances) of Bleed 
Previously, a bug existed that allowed champions with Evade to continue to dodge Black Widow’s attacks, even if her Signature Ability was maxed out  This issue has been fixed 
Captain America WW2 has started to become outpaced by his non-WW2 counterpart and while we want the two to feel different and each have their own specific uses, we also want to ensure they are kept within range of each other in terms of balance  To accomplish this, we’ve given WW2 Cap the ability to Stun with his Special 1 and Special 3 attacks, but kept his Bleed on Special 2 the same, giving him options during combat against non-bleeding champions 
A bug that prevented Daredevil from triggering Armor Breaks from Heavy Attacks has been fixed and is now working as intended 
Against non-bleeding champions: Critical Hits have a chance to Armor Break on Special Attacks 
Increase range of Signature to 25% from 20% 
Many players found Elektra’s signature ability lacked enough opportunities to use it  To remedy this, we’ve increased the range from 20% to 25%  Additionally, to help make Elektra unique from other skill champions, we’ve given her the ability to deal with naturally Bleed Immune champions  Note: This Armor Break only applies to champions naturally immune to bleed, such as Colossus and Ultron, and not to champions granted Bleed Immunity from Local or Link Nodes 
Guillotine’s Bleed effect used to have a chance to activate from any given attack, meaning that it had to be kept quite weak to compensate for the frequency of triggers  We’ve made the switch to have her Bleed behave closer to existing champions, and in doing so have boosted the strength of the Bleed and have allowed it to stack 
Norman Osborn overloads the Arc Reactor in his chest if Health drops below 10%, granting a large burst of power, with (18% - 48% ) Armor, Regeneration, and Power Gain  After that, his suit burns out and cannot trigger Armor Up, Armor Break or Stun and loses all base Armor 
Many players didn’t like Iron Patriot’s old signature ability, feeling that due to the lack of Regeneration, it was considerably weaker than Iron Man’s  While we agreed, we didn’t want to just copy and paste his signature ability, but rather give him his own unique twist on the ability  This “all or nothing version” feels more like Norman Osborn, pushing his suit to the limit to get a larger boost but at the cost of damaging the suit  The addition of Power Gain allows Iron Patriot a large attack before the suit burns out, if timed correctly 
Heavy Attacks: 90% chance to Stagger the enemy for 8 seconds  A Staggered enemy cannot gain their next beneficial effect 
All versions of Juggernaut, even those who haven’t been awakened, now gain the 2 second Unstoppable ability at the start of the fight when they hit Rank 2 
We wanted to add some new functionality to Juggernaut, while also keeping him true to his Mystic class assignment  To accomplish this, we added this “buff smasher” effect which keeps an opponent from gaining their next beneficial effect  Additionally, we wanted to make non-awakened versions of Juggernaut more fun to play, without adding more power to the awakened variations  As a result, we gave all versions of Juggernaut the ability to become Unstoppable at the start of the fight 
While many players liked the new functionality of Star-Lord’s Element Gun effect, they found it to be a little too random, specifically when it would Heal Block a champion incapable of Healing  We’ve now added in some contingencies that will make Heal Block appear less unless the opposing champion shows that he / she can Heal during the fight  This includes both activated healing effects, such as Wolverine or Ultron’s Heal, or passive healing effects gained from Masteries, such as Salve or Willpower 
It’s been a bit weird that Bucky wasn’t friends with his most famous friend  Well, he is now  This affects 3 Star and above versions 
We’ve increased the overall speed of this attack, allowing quick players to use this ability after a four or five hit combo 
It seems the Marvel’s have gotten tired of their beams being dodged so easily and have decided to angle it a little better, increasing the overall range of the attack and making it harder to dodge away from  We’ve also increased the speed of both special attacks to allow them to better flow into combat 
In order to allow this attack to better flow in combat, we’ve shaved off a few frames from the beginning, allowing players to chain this attack into 4 and 5 hit combos 
Alliance Wars have arrived! It’s Alliance versus Alliance in a war for Battlerealm supremacy!
Enter the NEW Loyalty Store to buy Alliance Potions, Mastery items, or other EXCLUSIVE items 
Gain Power back from Special Attacks, enhance or defend against Special Attacks, OR gain a temporary Arena Point Boosts with hoards of new Summoner Boost items!
Additional changes and improvements are listed below 
This patch will be released February 24th 
A new area of the Battlerealm has been opened! Compete with your Alliance-mates for pride, glory, and PRIZES!
Matchmake to find a rival Alliance, then combine strategy and teamwork to dominate them 
Setup the ultimate defensive team to fortify your Battlerealm, then take your offensive team on the assault!
Watch your War Rating skyrocket as your Alliance works together to defeat rivals!
Load up on Crystal Shards, Loyalty, and brand new exclusive rewards!
Note that this will be slow-rolled to Alliances in phases, similar to the introduction of Alliance Quests (to ensure server stability and gather your feedback on the new mode)  Expect tuning changes throughout these phases, as well as into Season 1 
Use Loyalty instead of Units to obtain items for Alliance Quests & Wars!
Items will rotate daily, similar to how the Mastery cores in the current Store change 
Store contents will be randomly chosen from a pool of categories/items; a select few items will be persistent and always be available for purchase 
A 5-Star version of Unstoppable Colossus will be available in the Loyalty Store (keep in mind, this is an expensive Champion due to his exclusivity; this will require winning quite a few Alliance Wars and saving up!) 
This is accessible from the “Store” section of the pop-down menu, and will be available at a later date after the initial 7 0 launch; there will be advance notice through forums and in-game before we release the Loyalty Store 
New Summoner Boosts have arrived in the Loyalty Store; NEW Boost types, purchasable with Loyalty Points 
Class specific Boosts, such as Mystic Champions restoring power after using Special Attacks 2 and 3, or Skill Champions boosting their Special Attack Damage 
Defensive Boosts, where your Champions take reduced incoming Special 3 Attack Damage 
Gain a temporary Arena Point boost with new Arena Boost items!
Fixed an issue where, after Parrying certain Champion’s Special Attacks, your Champion would be stuck in a blocking state until the Special Attack finished 
Fixed an issue where 90s Cyclops’ Armor Breaks would not remove Armor Ups 
Fixed an issue with Scarlet Witch’s Signature Ability proc rate (previously, the % chance displayed did not match in-game functionality; this is now fixed) 
(Netflix) Daredevil’s Heavy Attack now has a chance to apply 2 stacks of Armor Break, instead of the previous 1 stack 
When spending Battlechips to enter an Arena (such as the Tier 4 Basic or Alpha Catalyst Arena), there is now a confirmation popup 
The Alliance Crystal now has a purchase limit that resets daily 
Permanently increased the Alliance Crystal’s points in Summoner Advancement (from 30 to 300) 
Updates to Champion Special Attack animations, flow, and timing 
7 0 1 will be released within the next few days 
A celebration message is sent to the War Room when an Alliance War battlegroup is cleared 
Players can now tap directly on another node icon while the tile info popup is open (previously, the popup had to be closed before selecting another node) 
Alliance’s reward tier position is now highlighted in the Alliance War tier breakdown 
In Attack Phase, players can view the score breakdown for both the battlegroup and overall 
The “Place Your Defenders” text now disappears much faster after tapping on the screen 
Mail messages now display the date they were sent 
It should be much harder to accidentally tap the Units Store when closing a screen 
Players can tap to skip the point animation in Versus mode again 
Resolved an issue with Class Masteries (specifically Mystic Dispersion) not functioning 
The Juggernaut issue with his linked nodes not appearing in Act 4, Chapter 3, Quest 3 (4 3 3) has been fixed 
Fixed a crash that occurs when a player who is not in an Alliance enters Alliance Wars through an outside link 
Fixed a text issue where Alliance War specific descriptions would appear on the Alliance Quest “Select a Battlegroup” screen 
Resolved ~20 various rare crashes and additional minor issues in different game modes 
Fixed and optimized performance on the new Samsung S7 
Fixed an Unknown Error that occurred rarely after a device was woken after going to sleep 
Improved Performance(Frames Per Second) tracking per fight to help diagnose hitches/pauses/lag spikes during gameplay 
Improved gesture tracking(Swipe, Tap, Hold) during low performance moments in combat 
Fixed a rare crash that would sometimes occur when receiving a phone call while in combat 
Tuned and updated many Champion Special Attack animations to improve timing and combat flow  Please see the expanded forum post HERE for a full list 
Fixed She-Hulk’s Special Attacks being marked as a projectile (allowing Daredevil to evade them) 
Fixed an issue where the player would be stuck in place after parrying Captain America’s Special 1 
Fixed an issue where chaining 2 medium attacks into Old Man Logan’s Special 2 would cause the first 2 strikes to miss opponents 
Fixed an issue with Daredevil or Spider-man missing with a dash attack if Vision charges a heavy attack during the dash 
Fixed an issue where some hidden information in Alliance Wars was visible 
Fixed a display issue where Defender Placement percentage was not displaying all placed Alliance members 
Resolved minor issue with the total Alliance’s score being displayed on the War Progress widget (now only displays the score of the battlegroup being viewed) 
Multiple minor Alliance War issues have also been fixed in this patch 
Fixed a display issue where Shard amounts provided by defeating a boss displayed as double 
Fixed a display issue where opponent PI values would display differently between the map, prefight screen, and in combat 
Boss power is now correctly displayed after removing Global and Linked boosts 
Fixed an issue where a player in Alliance Quests would lose input ability on the quest board after sleeping the device 
Fixed an issue where a player enters Alliance Quests and gets stuck after viewing the linked node or buff node tutorial 
Fixed an issue where sending an Alliance invite to a player would cause the “Add Friend” button to become greyed out 
Fixed a text issue that appears when viewing Featured Hero information from the Home Screen 
Join The Iron or fight for The Blue with new events, quests, Champions, and special Shards; inspired by Marvel’s Captain America: Civil War!
Solo Events: constantly-evolving events that vary in length, requirements, and prizes!
Compare statistics against other players and Alliances with the new Leaderboards!




ground_truth: Players can skip dialogue on the quest map by pressing the 'SKIP' button.