text:
\section{Introduction}
\label{Introduction}
Consider a graph $G=(V,E)$ with an initial coloring where each vertex is red or blue. Each red/blue vertex could correspond to an infected/uninfected cell in a brain, a burning/non-burning tree in a forest, a positive/negative individual in a community regarding a reform proposal, or an informed/uninformed processor in a distributed system. Starting from an initial coloring, and in discrete-time rounds, all vertices synchronously update their current color based on a predefined rule as a function of the current coloring of their neighbors. By defining a suitable updating rule, this process can model different basic dynamic phenomena, like infection spreading among cells, fire propagation in a forest, opinion forming regarding an election in a community, or information distribution among processors. As two simple examples, a tree starts burning if at least one of its neighbors is on fire, or a person adopts the most frequent opinion among his/her friends.

Researchers from a wide spectrum of fields, from biology to physics, and with various motivations, have extensively investigated the behavior of such processes. One of the most natural updating rules, whose different variants have attracted a substantial amount of attention, is the \emph{majority rule} where a vertex updates its current color to the most frequent color in its neighborhood.

Here, one of the most studied variants is \emph{majority bootstrap percolation} in which by starting from a random coloring, where each vertex is blue with probability $P_b$ and red with probability $P_r=1-P_b$ independently, in each round a blue vertex switches to red if at least half of its neighbors are red, and a red vertex stays red forever. A considerable amount of effort has been put into the investigation and analysis of majority bootstrap percolation, both theoretically and experimentally, from results by Balogh, Bollobas, and Morris~\cite{balogh2009majority} to the recent paper by Stefansson and Vallier~\cite{stefansson2015majority}. Typical graphs are the $d$-dimensional lattice, the $d$-dimensional hypercube, the binomial random graph, and the random regular graph.

The main motivation behind majority bootstrap percolation is to model monotone dynamic processes like rumor spreading, where an informed individual will always stay informed of the rumor (corresponding to red color, say). However, it does not model non-monotone processes like opinion forming in a community, distributed fault-local mending, and diffusion of two competing technologies over a social network. For this, the following \emph{majority model} is considered: we are given a graph $G=(V,E)$ and an initial random coloring, where each vertex is blue with probability $P_b$ and red otherwise, independently of other vertices. In each round, all vertices simultaneously update their color to the most frequent color in their neighborhood; in case of a tie, a vertex keeps its current color. Since the majority model is a deterministic process on a finite state space, the process must reach a cycle of states after a finite number of rounds. The number of rounds that the process needs to reach the cycle is called the \emph{consensus time} of the process.

Even though different aspects of the majority model like the consensus time and its threshold behavior have been studied both experimentally and theoretically (see Section \ref{prior works} for more details), there is not much known about the behavior of the majority model on the random $d$-regular graph. Majority bootstrap percolation~\cite{balogh2007bootstrap}, rumor spreading~\cite{Fountoulakis2010,mourrat2016phase}, and flooding process~\cite{amini2013flooding} have been studied on random regular graphs, but this graph class is not easy to handle. Even though the behavior of majority bootstrap percolation on the random regular graph had been discussed in several prior works, it took almost two decades until Balogh and Pittel~\cite{balogh2007bootstrap} could analyze the behavior of the process partially. They proved (under some limitations on the size of $d$) that there are two values $P_1$ and $P_2$ such that $P_r\ll P_1$ results in the coexistence of both colors and $P_2\ll P_r$ results in a fully red configuration with high probability\footnote{We shortly write $f(n)\ll g(n)$ for $f(n)=o(g(n))$. For a graph $G=(V,E)$, we say an event happens with high probability (w.h.p.) if its probability is at least $1-o(1)$ as a function of $|V|$.}; however, $P_1\ne P_2$ which leaves a gap in the desired threshold behavior of the process.

In the present paper, we prove that in the majority model on the random $d$-regular graph, and for any constant $\epsilon>0$, $P_b \leq 1/2-\epsilon$ results in final complete occupancy by red color in $\mathcal{O}(\log_d\log n)$ rounds w.h.p. if $d\geq c/\epsilon^2$ for a suitable constant $c$. In words, even a narrow majority takes over the whole graph extremely fast. We should point out that the result probably holds for $d\geq 3$, but our proof techniques do not yield this, since they require sufficient edge density in the underlying graph. We also show that the upper bound of $\mathcal{O}(\log_d\log n)$ is best possible.

A natural context of this result is the \emph{density classification problem}; coming from the theory of cellular automata, this is the problem of finding an updating rule for a given graph $G$ such that for any initial 2-coloring the process reaches a monochromatic configuration by the initial majority color. It turned out that the problem is hard in the sense that even for a cycle, there is no rule which can do the density classification task perfectly~\cite{land1995no}. Our result shows that the majority rule does the density classification task acceptably for almost every $d$-regular graph with $d$ sufficiently large (see Theorem \ref{theorem 5} for the precise meaning of acceptably and sufficiently large).

It is an interesting (and currently unanswered question) which properties of a graph are chiefly responsible for the majority rule being able to almost classify density - or failing to do so. For example, we know that on a torus $T_{\sqrt{n},\sqrt{n}}$ (a $\sqrt{n} \times \sqrt{n}$ lattice with ``wrap-around''), already a very small initial blue density of $P_b\gg 1/n^{1/4}$ prevents red color from taking over, w.h.p.~\cite{gartner2017color}. A plausible explanation is that the torus $T_{\sqrt{n},\sqrt{n}}$ has a very low vertex/edge expansion in comparison to the random regular graph; however, we do not know whether expansion is indeed the right parameter to look at here.

As a concrete application of our main Theorem \ref{theorem 5}, we improve a result and answer an open question by Peleg~\cite{peleg2014immunity}. Motivated by the problem of fault-local mending in distributed systems, he introduced the concept of immunity. An $n$-vertex graph $G$ is $(\alpha,\beta)$-\emph{immune} if a set of $m\leq \beta n$ vertices with a common color can take over at most $\alpha m$ vertices in the next round in the majority model. Peleg proved that there exists a $d$-regular graph that is $(\frac{c_2\log n}{d},\beta)$-immune, for suitable constants $c_1,c_2,\beta>0$ and $d\geq c_1$. He also showed that this result is tight up to a logarithmic factor. We close this logarithmic gap. Peleg also asked whether there exist regular graphs that are immune in the sense that no sub-linear size set of a common color can eventually take over the whole graph. We answer his question positively: w.h.p. the random $d$-regular graph is immune.

The outline of the paper is as follows. After presenting basic definitions and prior research in Sections \ref{notation} and \ref{prior works}, the behavior of the majority model on the random $d$-regular graph is analyzed in Section \ref{majority}; the application to immunity is presented in Section \ref{dynamo}.
\subsection{Notation and Preliminaries}
\label{notation}
For a vertex $v$ in graph $G=(V,E)$ the \emph{neighborhood} of $v$ is defined as $N(v) := \{u\in V: (v,u)\in E\}$. Furthermore for $u,v \in V$, let $d(u,v)$ denote the length of the shortest path between $v,u$ in terms of the number of edges, which is called the \emph{distance} between $v$ and $u$ (for a vertex $v$, we define $d(v,v)=0$). For $v\in V$, $N_i(v):=\{u \in V:d(v,u)\leq i\}$ is the set of vertices in distance at most $i$ from $v$.

A \emph{generation} is a function $g:V\rightarrow\{b,r\}$ where $b$ and $r$ stand for blue and red, respectively. In addition to $g(v)=c$ for a vertex $v\in V$ and $c\in \{b,r\}$, we also write $g|_S=c$ for a set $S\subseteq V$ which means $\forall v\in S$, $g(v)=c$. For a graph $G=(V,E)$ and a random initial generation $g_0$, where $\forall v\in V$ $Pr[g_0(v)=b]=P_b$ and $Pr[g_0(v)=r]=P_r=1-P_b$ independently, assume $\forall i\geq 1$ and $v\in V$, $g_i(v)$ is equal to the color that occurs most frequently in $v$'s neighborhood in $g_{i-1}$, and in case of a tie $g_{i}(v)=g_{i-1}(v)$. This model is called the \emph{majority model}. Without loss of generality, we always assume that $P_b \leq P_r$.

The \emph{random $d$-regular graph} $\mathbb{G}_{n,d}$ is the random graph with a uniform distribution over all $d$-regular graphs on $n$ vertices, say $[n]$ (in this paper, we assume whenever talking about $\mathbb{G}_{n,d}$, $dn$ is even). The definition of the random regular graph is conceptually simple, but it is not easy to use. However, there is an efficient way to generate $\mathbb{G}_{n,d}$ which is called the \emph{configuration model}~\cite{bender1978asymptotic}.

In the configuration model for $V=[n]$, which is to be the vertex set of the graph, we associate the $d$-element set $W_i=\{i\}\times[d]=\{(i,i'): 1 \leq i' \leq d\}$ to vertex $1\leq i\leq n$. Let $W=[n]\times[d]$ be the union of $W_i$s; then a \emph{configuration} is a partition of $W$ into $dn/2$ pairs. These pairs are called the edges of the configuration. The natural projection of the set $W$ onto $V=[n]$ (ignoring the second coordinate) projects each configuration $F$ to a multigraph $\pi(F)$ on $V$. Note that $\pi(F)$ might contain loops and multiple edges. Thus, $\pi(F)$ is not necessarily a simple graph. We define the \emph{random $d$-regular multigraph} $\mathbb{G}^{\ast}_{n,d}$ to be the multigraph $\pi (F)$ obtained from a configuration $F$ chosen uniformly at random among all configurations on $W$. Bender and Canfield~\cite{bender1978asymptotic} proved that if we consider $\mathbb{G}_{n,d}^{*}$ and condition on it being a simple graph, we obtain a random $d$-regular graph on $V$ with uniform distribution over all such graphs. Furthermore, it is known~\cite{janson2011random} that if $Pr(\mathbb{G}^{*}_{n,d}\in A_n) \rightarrow 0$ as $n \rightarrow \infty$ then also $Pr(\mathbb{G}_{n,d} \in A_n) \rightarrow 0$, where $A_n$ is a subset of $d$-regular multigraphs on $V$. This allows us to work with $\mathbb{G}^{*}_{n,d}$ instead of $\mathbb{G}_{n,d}$ itself in our context. To generate a random configuration, it suffices to define an arbitrary ordering on the elements of $W$ and repeatedly match the first unmatched element in this order with another unmatched element uniformly at random. In Lemma \ref{lemma 1}, we utilize a slightly different construction from~\cite{bollobas1982diameter}.

\subsection{Prior Work}
\label{prior works}
Even though a substantial amount of effort has been put into the study of a wide spectrum of the majority-based dynamic processes, our attention here is mostly devoted to the prior work concerning the majority model. However, let us briefly point out a couple of remarkable accomplishments regarding the majority bootstrap percolation, which is arguably the closest model to ours. Aizenmann and Lebowitz~\cite{aizenman1988metastability} proved that in the $d$-dimensional lattice there is a threshold value $P_c$ so that $P_r\ll P_c$ and $P_c\ll P_r$ respectively result in the stable coexistence of both colors and fully red configuration w.h.p. Balogh and Bollobas~\cite{balogh2009majority} investigated the model on the $d$-dimensional hypercube and proved that the process has a phase transition with a sharp threshold. As discussed, the case of the random regular graph was also studied by Balogh and Pittel~\cite{balogh2007bootstrap}.

The majority model was introduced by Spitzer~\cite{spitzer1970interaction} in 1970. Afterwards, the model's behavior was investigated mostly by computer simulations (i.e., Monte-Carlo methods). These computer simulations (see for instance~\cite{de1992isotropic}) suggested that the model shows a threshold behavior on the two-dimensional torus $T_{\sqrt{n},\sqrt{n}}$. To address this observation, it was proven~\cite{gartner2017color} that $P_b\ll n^{-1/4}$ and $P_b \gg n^{-1/4}$ respectively result in red monochromatic generation and the stable coexistence of both colors w.h.p. Furthermore Schonmann~\cite{schonmann1990finite} proved in the biased variant of the majority model, where in case of a tie always red is chosen, and torus $T_{\sqrt{n},\sqrt{n}}$, for $1/\log n\ll P_r$ w.h.p. the process reaches fully red generation.

Since the updating rule is deterministic and there are $2^{|V|}$ possible colorings, the majority process must always reach a cycle of generations. Poljak and Turzik~\cite{poljak1986pre} showed that the number of rounds needed to reach the cycle (i.e., the consensus time) is $\mathcal{O}(|V|^2)$, and Goles and Olivos~\cite{goles1981comportement} proved the length of the cycle is always one or two. Frischknecht, Keller, and Wattenhofer~\cite{frischknecht2013convergence} showed there exists graph $G=(V,E)$ which needs $\Omega(|V|^2/\log^2 |V|)$ rounds to stabilize for some initial coloring in the majority model, which thus leaves only a poly-logarithmic gap. Kasser et al.~\cite{kaaser2015voting} studied a decision variant of the problem; they proved for a given graph $G=(V,E)$ and an integer $k$, it is NP-complete to decide whether there exists an initial coloring for which the consensus time is at least $k$.

Kempe, Kleinberg, and Tardos~\cite{kempe2003maximizing}, motivated from viral marketing, and independently Peleg~\cite{peleg1997local}, motivated from fault-local mending in distributed systems, introduced the concept of dynamic monopoly, a subset of vertices that can take over the whole graph. Afterwards, lots of studies regarding the size of dynamic monopolies and their behavior have been done. To name a few, even though it was conjectured~\cite{peleg1997local} that the size of the smallest dynamic monopoly in the majority model is $\Omega(\sqrt{|V|})$ for a graph $G=(V,E)$, Berger~\cite{berger2001dynamic}, surprisingly, proved there exist graphs with dynamic monopolies of constant size. Furthermore, Flocchini et al.~\cite{flocchini2004dynamic} studied the size of the smallest dynamic monopoly in the two dimensional torus. For more related results regarding dynamic monopolies, the interested reader is referred to a more recent work by Peleg~\cite{peleg2014immunity}.

\section{Majority Model on Random Regular Graphs}
\label{majority}
The three special cases of $d=0,1,2$ are exceptions to many properties of the random $d$-regular graph $\mathbb{G}_{n,d}$. For instance, $\mathbb{G}_{n,d}$ is $d$-connected for $d \geq 3$, but disconnected for $d \leq 2$ w.h.p.~\cite{janson2011random}. In the majority model also these three special cases show a different sort of behavior, which intuitively comes from their disconnectivity. We shortly discuss these cases following two purposes. Firstly, their threshold behavior sounds interesting by its own sake. Secondly, as a warm-up it probably helps the reader to have a better understanding of the majority model before going through our main results and proof techniques concerning the density classification in Section \ref{density classifier} and dynamic monopolies in Section \ref{dynamo}.

A $0$-regular graph is an empty graph with $n$ vertices, and a $1$-regular graph is the same as a perfect matching. We argue that in both cases $P_b \ll 1/n$ results in red monochromatic generation and $P_b \gg 1/n$ results in the coexistence of both colors w.h.p. (recall we assume $P_b\leq P_r$). Let random variable $X$ denote the number of blue vertices in the initial generation. $\mathbb{E}[X]=nP_b=o(1)$ for $P_b \ll 1/n$, and by Markov's inequality~\cite{feller1968introduction} w.h.p. $g_0|_V=r$. If $1/n \ll P_b$, then $\mathbb{E}[X]=\omega(1)$. Since X is the sum of $n$ independent Bernoulli random variables, Chernoff bound~\cite{feller1968introduction} implies that w.h.p. there exists a blue vertex in the initial generation, which guarantees the survival of blue color in both cases.

We show the random $2$-regular graph $\mathbb{G}_{n,2}$ also has a phase transition, but at $1/\sqrt{n}$ instead of $1/n$. Actually more strongly, we prove that for any $n$-vertex $2$-regular graph, $P_b\ll 1/\sqrt{n}$ and $1/\sqrt{n}\ll P_b$ w.h.p. result in fully red generation and the stable coexistence of both colors, respectively. Notice a $2$-regular graph is the union of cycles of length at least $3$.

\begin{theorem}
In the majority model and an $n$-vertex $2$-regular graph $G=(V,E)$, $P_b\ll 1/\sqrt{n}$ results in red monochromatic generation and $1/\sqrt{n}\ll P_b$ outputs the stable coexistence of both colors w.h.p.
\end{theorem}
\begin{proof}
In a generation $g$, define a blue (red) edge to be an edge whose both endpoints are blue (red). Consider an arbitrary edge set $E'\subset E$ which contains linearly many disjoint edges (a maximum matching, say), and let random variable $X_1$ denote the number of blue edges of $E'$ in $g_0$. For $1/\sqrt{n}\ll P_b$, $\mathbb{E}[X_1]=\omega(1)$; thus, by Chernoff bound there is a blue edge in $g_0$ w.h.p. which guarantees the survival of blue color.

If $P_b \ll 1/\sqrt{n}$, then $\mathbb{E}[X_2]=o(1)$, where the random variable $X_2$ denotes the number of blue edges in $g_0$, which by Markov's inequality implies there is no blue edge in $g_0$ with high probability. If in a cycle there is no blue edge and there is at least a red edge, then the cycle gets red monochromatic after at most $n/2$ rounds because the red edge grows from both sides in each round until it covers the whole cycle. Thus, it only remains to show that each cycle contains a red edge with high probability. An odd cycle always contains a monochromatic edge (red in our case). Then, let $X_3$ denote the number of even cycles which contain no monochromatic edge; i.e., the vertices are red and blue one by one. Define $n_i$ to be the number of cycles of length $i$. We have

\[
\mathbb{E}[X_3]\leq \sum_{2 \leq i \leq \lfloor n/2\rfloor} n_{2i}\cdot 2P_b^{i}(1-P_b)^{i}\leq 2P_b^{2}\sum_{2 \leq i \leq \lfloor n/2 \rfloor} n_{2i} = o(1)
\]
where we used $P_b\ll 1/\sqrt{n}$ and the fact that there are at most linearly many cycles. Therefore, w.h.p. there is no cycle without a red edge. \qed
\end{proof}
\subsection{Density Classification}
\label{density classifier}
In this section (Theorem \ref{theorem 5}), it is shown that in the $d$-regular random graph $\mathbb{G}_{n,d}$ and the majority model, $P_b\leq 1/2-\epsilon$, for an arbitrarily small constant $\epsilon>0$, results in fully red generation in $\mathcal{O}(\log_d\log n)$ rounds w.h.p. provided that $d\geq c/\epsilon^2$ for a suitable constant $c$. To prove that, first we need to provide Lemmas \ref{lemma 1} and \ref{lemma 4} as the ingredients, which are also interesting and important by their own sake. Specifically, the results in Section \ref{dynamo} concerning dynamic monopolies and immunity are built on Lemma \ref{lemma 4}.

Let say the $k$-neighborhood of a vertex $v$ in a graph $G$ is a tree if the induced subgraph by vertex set $N_k(v)$ is a tree. Roughly speaking, Lemma \ref{lemma 4} explains that for small $d$ and $k$ the expected number of vertices whose $k$-neighborhood is not a tree in $\mathbb{G}_{n,d}$ is small. This local tree-like structure turns out to be very useful in bounding the consensus time of the process.
\begin{lemma}
\label{lemma 1}
In $\mathbb{G}_{n,d}$, the expected number of vertices whose $k$-neighborhood is not a tree is at most $4d^{2k}$.
\end{lemma}
\begin{proof}
Firstly, we assume $k< \log_d(n/2)$ because otherwise the statement is clearly true. Furthermore as discussed in Section \ref{notation}, we work with the random $d$-regular multigraph $\mathbb{G}_{n,d}^{*}$ instead of the random $d$-regular graph $\mathbb{G}_{n,d}$, on vertex set $V=[n]$. We generate a uniformly at random configuration by partitioning $W=\bigcup_{1\leq i\leq n}W_i$ into $dn/2$ pairs as follows, where the $d$-element set $W_i=\{i\}\times [d]$ corresponds to the vertex $1\leq i\leq n$. In step 1, we start from an arbitrary class (we utilize the terms of $d$-element set and class interchangeably), say $W_1$, and match its elements one by one based on an arbitrary predefined order with an unmatched element (from $W_1$ or other classes) uniformly at random. We say a class has been \emph{reached} in step $j\geq 1$ if for the first time in step $j$ one of its elements has been matched. In step $j\geq 2$, we match the unmatched elements of the classes reached in step $j-1$ one by one based on a predefined ordering, say lexicographical order, with unmatched elements uniformly at random. It is possible in some step, no new class is reached. In this case, if all elements are matched, the process is over; otherwise we continue the process from one of the unreached classes, say the one with the smallest index.\footnote{For a more formal description of the construction, please see~\cite{bollobas1982diameter}, and notice since the second element always is chosen randomly, the generated configuration is random.}

Let say in step $j\geq 1$ we match element $x$ with an element $y$, chosen uniformly at random among all yet unmatched elements. One says $xy$ is a \emph{cycle-maker} if $y$ is not the first element matched in its class. The probability that an edge selected in the $j$-th step is a cycle-maker is at most $d^{j}/(n-d^j)$. Thus, the probability that there is a cycle-maker edge in the first $k$ steps is at most $2d^{k}\cdot\max_{1\leq j \leq k}\frac{d^j}{n-d^j}$ which is smaller than $\frac{2d^{2k}}{n-d^k}$. Let $X$ denote the number of vertices whose $k$-neighborhood is not a tree. Then, we have $\mathbb{E}[X]\leq (2nd^{2k})/(n-d^k)$ which is smaller than $ 4d^{2k}$ for $k< \log_d (n/2)$ because $n-d^{k}> n-d^{\log_d (n/2)}= n/2$. \qed
\end{proof}
\begin{corollary}
\label{corollary 3}
In $\mathbb{G}_{n,d}$, the number of vertices whose $(c'\log_d\log_2n)$-neighborhood is not a tree is at most $\log_{2}^{2c'+1}n$ w.h.p., for constant $c'>0$.
\end{corollary}
\begin{proof}
Let $X$ denote the number of vertices whose $(c'\log_d\log_2n)$-neighborhood is not a tree. By Lemma \ref{lemma 1}, $\mathbb{E}[X] \leq 4d^{2c'\log_d\log_2n}=4\log_{2}^{2c'}n$.
By Markov's inequality $Pr[X\geq \log_{2}^{2c'+1}n]\leq 4/\log_2n=o(1)$. \qed
\end{proof}
In a graph $G=(V,E)$ for two (not necessarily disjoint) vertex sets $S$ and $S'$, we say that $S$ \emph{controls} $S'$ if $S$ being monochromatic in some generation implies $S'$ being monochromatic (of the same color) in the next generation in the majority model, irrespective of the colors of other vertices. Clearly in $\mathbb{G}_{n,d}$, $S$ controls $S'$ implies that for every $v\in S'$ at least $\lceil d/2\rceil$ of its neighbors are in $S$.
\begin{lemma}
\label{lemma 4}
In $\mathbb{G}_{n,d}$ on vertex set $V=[n]$ with $d\geq c_1$, w.h.p. there do not exist two vertex sets $S,S'$ such that $S$ controls $S'$, $|S|\leq \frac{n}{c^{\prime \prime}}$, $|S'|=\lceil \frac{10|S|}{d}\rceil$, where $c_1,c^{\prime \prime}$ are sufficiently large constants.
\end{lemma}
This immediately implies that in $\mathbb{G}_{n,d}$ and the majority model, less than $\frac{n}{c^{\prime\prime}}$ blue (red) vertices will die out in $\mathcal{O}(\log_dn)$ rounds, with high probability.
\begin{proof}
We fix two sets $S,S'$ of the given sizes $s$ and $s'$. We show that the probability for $S$ controlling $S'$ is so small that a union bound over all pairs $(S,S')$ yields the desired high probability result. We equivalently work in $\mathbb{G}^{\ast}_{n,d}$ the relevant ``initial'' part of which we generate as follows: we iterate through the pairs in $S'\times[d]$ in some fixed order and match each yet unmatched pair with a random unmatched pair in $V\times[d]$. In order for $S$ to control $S'$, at least $\lceil d/2\rceil$ of the $d$ pairs $(v,i)$ must get matched with pairs in $S\times[d]$, for every $v\in S'$. Overall, at least $\lceil d/2\rceil s'$ of the $ds'$ pairs $S'\times[d]$ get matched with pairs in $S\times[d]$. Such a match is established only when the randomly chosen partner happens to be in $(S\cup S')\times[d]$, and this may actually yield two of the required $\lceil d/2\rceil s'$ pairs. Hence, for $S$ to control $S'$, at least $\ell:=\lceil d/2\rceil s'/2$ of the $L:=ds'$ iterations must be \emph{active}, meaning that they match a yet unmatched pair with a pair in $(S\cup S')\times[d]$. \newcommand{\bb}{\mathbf{b}} For a bit vector $\bb$ of length at most $L$, let $A(\bb)$ denote the event that iteration $i$ is active for exactly the indices where $b_i=1$. Then $Pr[A(b_1,b_2\ldots,b_L)] = \prod_{i=1}^LPr[\mbox{iteration $i$ is active if $b_i=1$}|A(b_1,\ldots,b_{i-1})]$ (the right-hand side is a telescoping product). Now, irrespective of $b_1,\ldots,b_{i-1}$, an iteration is active with probability at most $d(s+s')/(nd-2ds') = (s+s')/(n-2s')\leq 2s/n$. Hence, for a vector $\bb$ with at least $\ell$ ones, $Pr[A(b_1,b_2\ldots,b_L)]\leq (2s/n)^{\ell}$. As there are at most $2^L$ such vectors, the probability that at least $\ell$ iterations are active is at most $2^L(2s/n)^{\ell}\leq 2^{10s}(2s/n)^{\frac{5}{2}s}$. Hence by a union bound, the probability $P$ that there exist such sets $S$ and $S'$ in a random configuration is at most $\sum_{s=1}^{\frac{n}{c^{\prime\prime}}} {n \choose s} {n \choose \lceil \frac{10s}{d}\rceil} 2^{10s}(\frac{2s}{n})^{\frac{5}{2}s}$. Since $d\geq c_1$ for a large constant $c_1$, ${n \choose \lceil \frac{10s}{d}\rceil}\leq{n \choose s}$; thus, applying Stirling's approximation~\cite{feller1968introduction} (i.e., ${n \choose k}\leq (ne/k)^k$) yields $P\leq \sum_{s=1}^{\frac{n}{c^{\prime\prime}}} (\frac{ne}{s})^{2s} 2^{10s}(\frac{2s}{n})^{\frac{5}{2}s}$. Furthermore, since $e^{2s}\cdot2^{10s}\cdot2^{\frac{5}{2}s}\leq (c^{\prime\prime})^{s/4}\leq (n/s)^{s/4}$ for sufficiently large $c^{\prime\prime}$, we have $P \leq \sum_{s=1}^{\frac{n}{c^{\prime\prime}}} (\frac{n}{s})^{2s} (\frac{s}{n})^{\frac{9}{4}s}=\sum_{s=1}^{\frac{n}{c^{\prime\prime}}}(\frac{s}{n})^{\frac{s}{4}}=o(1)$. \qed
\end{proof}
As will be discussed in the proof of Theorem \ref{theorem 5}, for the majority model on $\mathbb{G}_{n,d}$ with $P_b= 1/2-\epsilon$ and $d\geq c/\epsilon^2$ there is a simple argument which shows that the expected density of the blue vertices drops from $1/2-\epsilon$ in $g_0$ to an arbitrarily small constant in $g_1$ if we select the constant $c$ sufficiently large. Therefore, one might want to apply Lemma \ref{lemma 4} to show the process w.h.p. gets red monochromatic in $\mathcal{O}(\log_d n)$ rounds. However, in Theorem \ref{theorem 5} we show actually $\mathcal{O}(\log_d\log n)$ rounds suffice to get red monochromatic with high probability. To prove that, we need the tree structure argued in Lemma \ref{lemma 1}.

\begin{theorem}
\label{theorem 5}
In the majority model and $\mathbb{G}_{n,d}$, by starting from $P_b\leq 1/2-\epsilon$, for an arbitrarily small constant $\epsilon>0$, the process gets red monochromatic in $\mathcal{O}(\log_d\log n)$ rounds w.h.p. provided that $d\geq c/\epsilon^2$ for suitable constant $c$.
\end{theorem}
\begin{proof}
Let say a vertex is in the $j$-th level of a rooted tree if its distance to the root is $j$. Now, consider a tree $T$ rooted at vertex $v$ and of height $k$ so that except the vertices in the $k$-th level (i.e., leaves), all vertices are of degree $d$. We consider the following process on $T$, which we call the \textit{propagation process}, where in the initial configuration all the internal vertices are \emph{inactive} and each leaf is blue with probability $P_b$ and red with probability $1-P_b$ independently. Assume in each round an inactive vertex whose all children are colored adopts color blue if at least $\lfloor (d-1)/2 \rfloor$ of its children are blue and red otherwise. Clearly after $k$ rounds, the root (vertex $v$) is colored with blue or red. Let $P_i$ for $0 \leq i \leq k$ denote the probability that a vertex in the $(k-i)$-th level is blue after round $i$; specifically, $P_k$ is the probability that vertex $v$ is blue at the end of the process. More accurately, $P_0=P_b$ and for $1 \leq i \leq k$ $P_i=\sum_{j=\lfloor (d-1)/2\rfloor}^{d-1}{d-1 \choose j} P_{i-1}^{j}(1-P_{i-1})^{d-1-j}$.

Now, let us get back to the majority model and the random $d$-regular graph $\mathbb{G}_{n,d}$. Consider a vertex $v$ so that the induced subgraph by $N_k(v)$ is a tree $T$. Clearly in $T$, except the vertices in the $k$-th level, all vertices are of degree $d$. Now, we claim the probability that vertex $v$ is blue in generation $g_k$ in the majority model is at most $P_k$, which is equivalent to the probability that the root of $T$ is blue in the propagation process after $k$ rounds, with the same $P_b$. This is true because by starting with the same coloring for the leaves of $T$ (the vertices in distance $k$ from root $v$) if in the $k$-th round in the propagation process the root is red, it is also red in the majority model and generation $g_k$, irrespective of the colors of other vertices. For $k=1$ this is trivially true. Now, do the induction on $k$; if the root is red in the propagation process after $k$-th round, it means that there exist less than $\lfloor (d-1)/2 \rfloor$ blue vertices among root's children in round $k-1$ which implies by the induction hypothesis there are less than $\lfloor (d-1)/2 \rfloor$ blue vertices in $v$'s neighborhood in $g_{k-1}$ in the majority model; then $g_k(v)=r$.

So far, we showed the probability of being blue in $g_k$ for a vertex, whose $k$-neighborhood is a tree, is at most $P_k$ with $P_0=P_b$. Now, we upper-bound the probability $P_k$. Without loss of generality, assume $d$ is odd, and suppose $d'=d-1$. First, let us bound $P_1$; clearly, $P_1\leq \sum_{j=d'/2}^{d'}{d' \choose j} (1/2-\epsilon)^{j}(1/2+\epsilon)^{d'-j}$ which is smaller than
\[ (1/2-\epsilon)^{d'/2}(1/2+\epsilon)^{d'/2}\sum_{j=d'/2}^{d'}{d' \choose j}\leq (1/4-\epsilon^2)^{d'/2}2^{d'}= (1-4\epsilon^2)^{d'/2}.
\]
By applying the estimate $1-x\leq e^{-x}$, we have $P_1\leq e^{-2d'\epsilon^2}$. For $d\geq c_1'\log n$, where $c_1'$ is a large constant, clearly $P_1\leq 1/n^2$ which implies the expected number of blue vertices in $g_1$ is at most $1/n$; i.e., the process gets red monochromatic in one round with high probability. Thus, it only remains to discuss the case of $d\leq c_1'\log n$ for an arbitrarily large constant $c_1'$; in this case, since $P_1\leq e^{-2d'\epsilon^2}$, selecting suitble constant $c$, for $d\geq c/\epsilon^2$, results in $P_1\leq 1/16$. Now, we show $P_i\leq P_{i-1}^{d'/4}$ for $P_{i-1}\leq 1/16$, which yields $P_k\leq 1/n^2$ for $k=c'\log_d\log_2 n$ by selecting constant $c'$ large enough. We know $P_i\leq P_{i-1}^{d'/2}\sum_{j=d'/2}^{d'}{d' \choose j}\leq P_{i-1}^{d'/2}2^{d'}$. Thus, by utilizing $P_{i-1}\leq 1/16$, one has $P_i\leq P_{i-1}^{d'/4}$. Now, let random variable $X_1$ ($X_2$) denote the number of vertices whose $k$-neighborhood for $k=c'\log_d\log_2 n$, is (not) a tree and are blue in $g_k$. We know $\mathbb{E}[X_1]\leq n P_k\leq 1/n$, which implies $X_1=0$ w.h.p. by Markov's inequality. Furthermore, by using Corollary \ref{corollary 3} with high probability $X_2 \leq \log_{2}^{2c'+1}n$. Hence, with high probability the number of blue vertices in $g_k$ is upper bounded by $\log_{2}^{2c'+1}n$. However based on Lemma \ref{lemma 4}, poly-logarithmically many blue vertices die out in $\mathcal{O}(\log_d\log n)$ rounds w.h.p. which finishes the proof. \qed
\end{proof}
Now, we argue that the bound of $\mathcal{O}(\log_d\log n)$ is tight. We prove in $\mathbb{G}_{n,d}$ and for a constant small initial density $P_b$, say $P_b=1/4$, after $k'=\frac{\log_d\log_2n}{2}$ rounds w.h.p. there exist some blue vertices. We claim in $\mathbb{G}_{n,d}$ there are $\sqrt{n}$ vertices, say $u_1,\cdots,u_{\sqrt{n}}$, whose $k'$-neighborhood is pairwise disjoint. Define indicator random variable $x_i$ to be 1 if $g_0|_{N_{k'}(u_i)}=b$. Clearly, $Pr[x_i=1]\geq(1/4)^{2d^{k'}}=1/2^{4\sqrt{\log_2n}}$. Let $X=\sum_{i=1}^{\sqrt{n}}x_i$; then $\mathbb{E}[X]\geq \sqrt{n}/2^{4\sqrt{\log_2n}}=\omega(1)$. By using Chernoff bound, there exists a vertex $v$ so that $g_0|_{N_{k'}(v)}=b$, which implies $g_{k'}(v)=b$. Now, we prove that there exist $\sqrt{n}$ vertices whose $k'$-neighborhood is pairwise disjoint in every $d$-regular graph. For a $d$-regular graph $G$ by starting from the state that all vertices are \emph{unmarked}, recursively we choose an arbitrary unmarked vertex $u$ and add $u$ to set $U$, which is initially empty, and mark all vertices in $N_{2k'}(u)$. Clearly, the vertices in set $U$ have our required disjointness property, and set $U$ will be of size larger than $\sqrt{n}$ at the end because in each step we mark at most poly-logarithmically many vertices while we start with linearly many unmarked vertices.
\subsection{Dynamic Monopoly and Immunity}
\label{dynamo}
In distributed systems, the resolution of inconsistencies by the majority rule is a common tool; the idea is to keep redundant copies of data and perform the majority rule to overcome the damage caused by failures. Motivated from this application, one might be interested in the networks in which no small subset of malicious/failed processors can take over a large part of the network. To address this issue, Peleg~\cite{peleg2014immunity} suggested the concept of immunity. An $n$-vertex graph $G$ is $(\alpha,\beta)$-\emph{immune} if a set of $m\leq \beta n$ vertices with a common color can take over at most $\alpha m$ vertices in the next round. One is interested in graphs which are $(\alpha,\beta)$-immune for constant $\beta$ and small $\alpha$ because roughly speaking these graphs are acceptably tolerant of malicious/failed vertices (processors). Peleg~\cite{peleg2014immunity} proved the following theorem regarding the existence of such graphs.
\begin{theorem}\cite{peleg2014immunity}
\label{theorem 6}
There exist constants $c_1,c_2,\beta>0$ such that for every $d\geq c_1$, there exists a $d$-regular graph $G$ which is $(\frac{c_2\log n}{d},\beta)$-immune.
\end{theorem}
Peleg also argued that this result is tight up to a logarithmic factor, meaning there is a constant $c_2>0$ such that for any constant $\beta>0$, there exist no $(\frac{c_2}{d},\beta)$-immune $d$-regular graph. Now as an immediate implication of Lemma \ref{lemma 4}, we present Corollary \ref{corollary 1} which improves upon Peleg's results by removing the extra logarithmic term. Hence, this result is tight up to a constant.
\begin{corollary}
\label{corollary 1}
There exist constants $c_1,c_2,\beta>0$ such that for every $d\geq c_1$, there exists a $d$-regular graph $G$ which is $(\frac{c_2}{d},\beta)$-immune.
\end{corollary}
Furthermore, we say a graph is \emph{immune} if the smallest dynamic monopoly is of linear size, in terms of the number of vertices. We recall that for a graph $G=(V,E)$, a set $D \subseteq V$ is called a dynamic monopoly whenever the following holds: if in the initial generation all vertices of $D$ are blue (red) then the process reaches the blue (red) monochromatic generation, irrespective of the colors of other vertices. As an open problem, Peleg~\cite{peleg2014immunity} asked that whether there exist regular immune graphs. The existence of immune $d$-regular graphs for $d\gg \log n$ is straightforward from Theorem \ref{theorem 6}, but, the question is unanswered for small $d$, while one is more interested in sparse immune regular graphs from a practical, or even a theoretical, perspective. Again, as an immediate result of Lemma \ref{lemma 4}, we have Corollary \ref{corollary 2} which actually represents a stronger statement.
\begin{corollary}
\label{corollary 2}
$\mathbb{G}_{n,d}$ with $d\geq c_1$ is immune w.h.p. for large constant $c_1$.
\end{corollary}
\section*{Conclusion} We claim our techniques can be applied to analyze the behavior of the majority model on the binomial random graph $\mathbb{G}_{n,p}$. For $p\gg \log n/n$, one can show $P_b\leq 1/2-\epsilon$ results in fully blue generation in one round w.h.p. by using the argument regarding the case of $d\geq c_1'\log n$ in the proof of Theorem \ref{theorem 5}. For $p\ll \log n/n$ the graph contains a red and a blue isolated vertex in $g_0$ w.h.p. for a fixed $P_b>0$ which result in the coexistence of both colors.

Exploring the relation between the behavior of the majority model and the expansion level of the underlying graph can be a prospective research direction. Specifically, it would be interesting to prove that graphs with some certain level of expansion have a density classification behavior similar to $\mathbb{G}_{n,d}$.
\subsubsection*{Acknowledgments.} The authors would like to thank Mohsen Ghaffari for several stimulating conversations and Jozsef Balogh and Nick Wormald for referring to some relevant prior results.
\bibliographystyle{acm}

redpajama_set_name: RedPajamaArXiv


text:
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redpajama_set_name: RedPajamaCommonCrawl


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This Web site contains attorney advertising. Prior results do not guarantee a similar outcome nor should any statements or representations made on this Web site be construed as a guarantee or prediction of outcome or result. The Web site content is for general informational purposes only and is not legal advice nor should it be relied upon as legal advice. Moreover nothing in this Web site is intended to create an attorney-client relationship. Protecting intellectual property assets and proprietary information can be important for moving your business forward and being competitive in the marketplace. Understand your options and develop a plan that fits your business needs.
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Joy to the World! – I Commend Joy!
"What God has done in Christ demands to be praised. It is not enough simply to say what God has done to save us – what he has done needs to be celebrated in song" (Leland Ryken). The past four weeks we have been considering Songs for the Savior as found in Scripture. These last two days before Christmas we are going to use two familiar songs to praise Jesus.
Isaac Watts was born in 1674. He was a very intelligent person who also cared a great deal about spiritual truth and saw the importance of singing and worship. He was not happy with the awful Psalm-singing of his time. When he was challenged to write something better than what was currently sung, he rose to the occasion and started writing hymns. He wrote more than 750 by the end of his life.
Watts wanted people to know the joy of the King and Savior, Jesus, who has come to free us from the curse of sin and judge the world in a right way. "Joy to the World!" is composed from Psalm 98:4 as a way to teach people this truth through song. As we sing that the Lord has come and the Savior reigns, we also sing that he comes to make His blessings flow as far as the curse is found. Watts leads us in rejoicing that Jesus rules the world with truth and grace, and makes the nations prove the glories of His righteousness, and wonders of His love.
Consider: Meditate on the words of "Joy to the World!" that Watts composed (below) in order to help you remember the truth of the Savior's work this Christmas. Jot down what is most important to you in this Christmas carol.
Next Next post: Hark! The Herald Angels Sing!


Rohini is an Indian actress, lyricist, screenwriter, voice actor and director. She has mainly acted in south Indian, notably Malayalam and Tamil films. Having started her acting career at five, she has about 130 south Indian films to her credit. She received National Award of Special mention and Andhra Pradesh State Award for Best Female Actor in 1996 for the film Sthree.
A native of Anakapalli, Andhra Pradesh Rohini spent all her childhood in Chennai. Her father Ravu Naidu was a panchyat officer and mother Saraswathi was a housewife. Her father always wanted to become an actor, though he couldn't be an actor he encouraged his daughter to become an actress and he fulfilled his ambition through his daughter. Her mother died when she was five years old and father remarried later.
She was married to well known actor Raghuvaran in 1996 and they have a son Rishi Varan born in 1998. The couple later got separated and divorced in 2004.
She started her career in 1976 as a child artist. She was first seen as a wide-eyed five year old in Yashoda Krishna, a Telugu film. She is also a popular dubbing artist in the Telugu and Tamil industries. She has voiced six characters in five of Mani Ratnam's films. She lent her voice to actresses like Jyothika (Vettaiyadu Villaiyadu), Aishwarya Rai (Iruvar and Raavanan), Manisha Koirala (Bombay) and Amala (Shiva). She has also dubbed for Girija Shettar in Mani Rathnam's only Telugu movie till date Geethanjali.
She wrote lyrics for the Tamil film Pachaikili Muthucharam, before ending another hiatus by writing all songs in the album, Maalai Pozhudhin Mayakathilaey. Her acting skills were recognized and she was given the Kalaimamani award recently.
She writes episodes for Tamil TV serials. She started writing scripts for TV series way back in 1996, and has adapted the Sahithya Academy Award winning novel Verukku Neer for a tele film in 2005. She hosted the live TV talk show Kelvigal Aayiram for Vijay TV, which she said "dealt with issues close to my heart". She moderated the Tamil programme Azhagiya Tamizh Magal on Kalaignar TV and has worked as the anchor of Raj TV's show, Rohini's Box Office in which she reviewed the latest releases.
Being an activist for AIDS awareness, Rohini has also directed short films for M.G.R. Medical University and Tamil Nadu Aids Control Society. In 2008, Rohini had directed a 50-minute documentary Silent Hues about film industry's child artists, she being one herself. In 2013, she has directed a feature film Appavin Meesai which is yet to be released.
^ a b "Welcome To Amrita TV - Super Dancer Junior 3". Amritatv.com. Archived from the original on 19 October 2013. Retrieved 17 August 2013.
^ "Rohini makes documentary film on child artists". The Hindu. 9 January 2008. Retrieved 17 August 2013.
^ a b c "Documentary on child artists". Indiaglitz. 8 January 2008. Retrieved 17 August 2013.
^ a b c "Metro Plus Hyderabad / Profiles : Actor with substance". The Hindu. 14 September 2010. Retrieved 17 August 2013.
^ "ഈ ജീവിതം മകനുവേണ്ടി..." mangalamvarika.com. Retrieved 15 May 2015.
^ a b c Baradwaj Rangan (25 August 2011). "Roles of a lifetime". The Hindu. Retrieved 17 August 2013.
^ "Rohini's Meesai". Behindwoods. 26 July 2012. Retrieved 17 August 2013.
^ "Girl power, Tamil style!". Rediff. 8 April 2010. Retrieved 17 August 2013.
^ Gupta, Rinku (22 August 2011). "Rohini returns to TV". The New Indian Express. Retrieved 17 August 2013.
^ "Exploring "fragile world" of child actors". The Hindu. 17 February 2008. Retrieved 17 August 2013.
^ "''I cannot do a film like Thuppakki'', Rohini, Appavin Meesai". Behindwoods. 30 July 2013. Retrieved 17 August 2013.


text:
in: Characters, Females, Mission: Impossible III Characters,
Mission: Impossible – Ghost Protocol Characters
Mission: Impossible – Fallout characters
Julia Meade-Hunt
'Mission: Impossible' Data File: Julia Meade
Name: Julia Anne Meade-Hunt
Status: Alive/Active
Role: Ally
Age: 40 (2017)
Date of birth: c. 1977
Birthplace: United States
Residence(s): Kashmir
Nickname(es): Jules
Affiliation: Ethan Hunt (formerly)
Virginia Regional Hospital, Massachusetts General Hospital
Father: Unnamed
Mother: Unnamed
Sibling(s): Sister:
Melissa Meade
Brother:
Rick Meade
Spouse(s): Erik
Ethan Hunt (divorced)
Played by: Michelle Monaghan
First appearance: Mission: Impossible III
Last appearance: Mission: Impossible – Fallout
Dr. Julia Anne "Jules" Meade-Hunt (born c. 1977) is a major character in the Mission: Impossible film series, being a major protagonist in the 2006 film Mission: Impossible III, a cameo character in the 2011 film Mission: Impossible – Ghost Protocol and a supporting character in the 2018 film Mission: Impossible – Fallout. She is a nurse and the former wife of IMF operative Ethan Hunt. She is now married to a man named Erik.
1.1 Courtship with Ethan
1.2 Kidnapping and Rescue
1.3 Presumed Murder in Croatia
1.4 Under Ethan's Protection
1.5 Relationship with Erik and meeting Ethan again
3 Abilities
5.1 Appearances
Courtship with Ethan[]
Ethan meets and falls in love with Julia while on a "heli-boarding" trip at Lake Wanaka. To keep her from finding out about his work with Impossible Mission Force, he tells her, as well as her friends and family (the latter consisting of her siblings, Melissa and Rick, and their widowed mother), that he studies traffic patterns for the Virginia Department of Transportation. Ethan and Julia marry shortly before he set off on a two-day "business trip", actually a search for Owen Davian.
Kidnapping and Rescue[]
Davian eventually kidnapped Julia and threatened to kill her if Ethan didn't hand over the Rabbit's Foot. Ethan manages to find her in Shanghai, but not before Davian plants a micro-explosive into Ethan's head. Ethan helped Julia prepare for the arrival of Davian's men by handing her a Beretta 92F handgun, and instructing her how to reload and shoot with it. He also informed her that she needs to electrocute him to deactivate the explosive, which she reluctantly proceeded to do. Rogue agent John Musgrave found the couple before Ethan could regain consciousness, but Julia shot Musgrave to death, then revived Ethan with CPR.
When Julia asked Ethan how she ended up in China, he finally confessed his involvement with the IMF. He told her that the information is top-secret, she assured him that he could trust her, to which he stipulated. Ethan brought her to the IMF headquarters in Virginia, where she met the other agents including Luther Stickell, Zhen Lei and Declan Gormley, before taking her on their honeymoon.
Presumed Murder in Croatia[]
In Croatia, Ethan and Julia are both together until Ethan realizes that the only way to protect her from IMF's enemies is to fake her death. Agent William Brandt is assigned to watch the family, following Ethan one day as he went for a jog. Brandt assigned two men to watch Julia before he left and after returning, he finds them both unconscious and Julia gone. Days later, Brandt is told that what is left of Julia's body had been found. In apparent retaliation for his wife's murder, Ethan kills six Serbian nationalists linked to the crime and is sent to Rankow Prison as a result lll
Under Ethan's Protection[]
In reality, the IMF invented this cover to allow Ethan to infiltrate the prison, in the hope of gaining information in regards to a Swedish-born Russian terrorist named Kurt Hendricks, code-named Cobalt. Julia, now safe, is kept a secret by Ethan, who has had to keep his distance from her since.
When Ethan and his team visit Seattle after defeating Cobalt, Ethan absolves Brandt of guilt for Julia's death by assuring him that she did not actually get killed in Croatia. This made Brandt the only other IMF agent aware of the complete details of Julia's survival (according to Benjamin Dunn, some of the agents had received word that Julia still lives, but the couple's marriage fell through). Shortly after Brandt left Ethan to sit alone, Julia carries Ethan's attention from across the bay. Ethan and Julia gaze at each other, until Julia went to meet some friends at a nearby restaurant. They get divorced, when Ethan decides not to sit around while the world is in danger. They reunite when Ethan and his team are on a mission. They are still friends and help Ethan's team out.
Relationship with Erik and meeting Ethan again[]
When they meet again, Julia reveals she is now married to Erik. Julia quickly realizes that Ethan is not there for a visit and that something is going on in the field hospital that has inoculated the nearby villages from smallpox.
She finds Luther and assists him in preparing the first nuclear bomb for disarmament. She then leaves to be with her husband after Luther tells her there's nothing more for her to do here.
After the bombs are disabled, she cares for Ethan at the field hospital. Ethan apologizes for everything that has happened, but she says there's nothing to be sorry about. She talks how she's found so much more in life with Ethan's involvement and that she can sleep safely knowing Ethan is watching over the world instead of always over her. Ilsa sees Julia share a tender moment with Ethan, then whispers something to Julia a moment later. After Julia leaves, Ilsa then shares her own tender moment with Ethan without Julia present.
N/A[]
Abilities[]
Skilled Physician: She is a medical doctor
Marksman: She is very good with a gun.
Julia with Ethan in MI:III.
Julia with Luther in MI:Fallout.
Behind the Scenes[]
Julia is portrayed by American actress Michelle Monaghan in Mission: Impossible III, Mission: Impossible – Ghost Protocol and Mission: Impossible – Fallout but Monaghan was credited only in III and Fallout.
Appearances[]
Mission: Impossible III (First appearance)
Mission: Impossible – Fallout
Mission: Impossible III Characters


text:
Liz Allbee and Hans Grüsel - Strategies for Failure/Zuckerkrieg
Sometimes coincidence sure feels like synchronicity. I was asked to help out with a bran(…)pos show in Lincoln last year, which was my introduction to his music, as well as the San Francisco label Resipiscent Records. As you might recall, I dug the hell out of Den of Ordure & Iridescence, and I went on to get way into pretty much everything bran(…)pos-related I could find. And I still spend a lot of time in the "Den." Easily one of the best records of 2013.
As I thumbed through the Resipiscent catalog, I found myself unfamiliar with a lot of the artists, but having heard a few more of their releases, it's clear that I need lots of these wild sounds rattling around in my head. Now I feel like I'm almost destined to hear as much of the Resipiscent label as possible. Their spring release this year was a split between Liz Allbee and Hans Grüsel, both of whom have appeared in their roster before (Allbee's solo debut "Quarry Tones" actually launched the label), and like bran(...)pos, both artists turn in immersive, undefinable, and utterly addictive pieces.
"Strategies for Failure/Zuckerkrieg" presents truly deep listening in the form of a side-length piece each from Allbee and Grüsel. This pair of electroacoustic compositions hang very well together, as both artists share a delirious joy for turning extended-technique acoustic playing and extreme synthesis/deployment-of-effects into almost physically tangible daydreams and daymares. Maybe you can still manage to vacuum your living room with this record on, but there might be psychic consequences. Sit down and listen.
Allbee's side, whose full title is "Strategies for Failure and Relief From Persistent Positive Symptoms," goes through a few distinct movements. The opening section uses some very low sub-bass alternating in a minor-3rd riff, sometimes crashing together to create thick acousmatic combination tones. Over this, Allbee recites a few verses related to freedom and control, alternating with solo lines on a heavily-effected trumpet, delivering very crispy, wasp tones that contrast nicely with the intense bass. The section is almost like an incredibly slowed-down blues riff bouncing around the walls of a rubber room. This gets violently interrupted with synths and effected trumpets around the seven-minute mark, eventually pausing to introduce a new section of intense high-(concert)-G trumpet tones, pushing and pulling around the mic, bending, subtly falling into weird room reverbs in the background. Eventually the trumpet approach heads into extended-technique territory not far from Zorn's "Classic Guide to Strategy" solo work, with liquid sounds and gurgles and game calls. Toward the end of the piece, chimes and bells gradually take over the mix, and layers of muted trumpet and vocals softly appear. Like the introduction, Allbee recites some great lines whose meaning is surely subject to a wide berth of interpretation: as Allbee says, "The words conceal what they mean like a mystery." At the end, powerful subtones gently launch the piece out to sea.
The Allbee piece feels very personal, and listening is a little like overhearing a handful of secrets without knowing the full context of their relevance. In contrast, Grüsel's "Zuckerkrieg (Part 1)" has a more archetypal kind of feeling, evoking deluge myths and perpetually-haunted forests. It starts gently enough, with subtle beats colliding, at times feeling like they're falling in and out of clear metric relationships. But when modular synths and heavily-treated acoustic instruments build into a complex wall of sound, one rapidly feels like Edgard Varèse stumbling out of a promethazine factory. The dense fog lifts, revealing the gentlest section of all, taps and scrapes that sound like the product of contact mics on balsa wood or paper scraps. Chewing and creaking sounds drift back in, and intense field recordings of wind force the mix to an overwhelming mass, obscuring what must be lots of wind instruments buried in effects: trumpets like the bleating of fearful goats, bassoons trapped in amber resin. Complex fields of synth tones form as the wind subsides, mingling with even more horns/reeds awash in effects. Toward the end, it's obvious that heavily-altered strings have been pushing to reveal melodies, and horns have strived to reveal tense chord clusters, but the effects and the surrounding synths continue to terrorize their acoustic sisters (though a few overdubbed trombone swells come through clearly). As "Zukerkrieg" translates loosely to "sugar war," I'm left with impressions of melted, caramelizing goo all over everything in this crazy mix by its sticky end.
On the compositional front, both pieces on this LP seem obviously composed by folks who have taken very serious stock of the resources bestowed on us in 20th. C. academic music circles--the forceful approach to electronics of Xenakis, the sometimes brash densities of Ligeti's sound mass music, just to name a couple--but they also have much more of an emotional impact on me than "the classics." Jesus, I wish somebody would've hipped me to artists like this in music school, as they're every bit as nutritious as "legit" composers, but they're obviously having a much better time. And they're much more connected to culture-at-large, evoking lots of fun/campy/creepy points of reference at every turn. The extramusical aspects of this work--costumes, set design, web/video work, etc--really come through in the music, too. A couple of minutes into either one of these pieces, I felt pretty sure that these folks can hang with composer-types while knowing a lot more about treating true "happening"-style events where people can really get down.
In terms of art and packaging, this split comes in one of those plastic jackets with a reversible poster-style insert, so you can face your favorite cover art outward. I'm leaving the creepy monster art by Bonnie Banks for the Grüsel side out on mine (my copy also has a couple of cut-out gingerbread monsters and a pre-weathered paper thing that you can position how you'd like inside the jacket), but the spooky science-lab art by Alibi & Beins for the Allbee side is great, too. And this is a beautifully-mastered production for vinyl, pressed to very clean vinyl. With lots of quiet passages, the surface noise of vinyl can sometimes be irritating on pieces like these, but this pressing is fantastic. This is a small run of only 250 copies, but you're in luck: Resipiscent still has a copy waiting just for you.
Labels: music reviews
Liz Allbee and Hans Grüsel - Strategies for Failur...
Sontag Shogun - Tale
Spanyurd - OOW


This version adds 2D image nodes or watermarks, improves low-end hardware support and adds many nice new features. Keep reading to know more.
Instead of having a single watermark defined in scene properties, we decided to make watermarks the first-class citizens and let you add as many overlaying images, as needed.
Now if you drop an image into the scene, you'll see Create Watermark in the pop-up menu. Click it to add image as a flat overlaying object. Your old watermarks (if any) will be converted to the new ones automatically.
Read more about watermarks here.
Resource manager has been added, so you can manage external files used by the project.
GLB (Facebook) export option is now deprecated due to the low demand. It is hidden by default, but you can restore it in settings. It will be completely removed in the near future, so it is time to look for replacement. You can use this one for instance.
Metadata - what is node metadata and how to use them in exported scenes.


Getting a custom patio cover will add an extra sense of relaxation to your outdoor area. Some real estate properties have a separate built patio while others are connected to the home directly. The owner s can show their creativity in their different patio covers and their imagination. The possibilities are endless, and as time goes by, people are getting more creative. The patio cover is easy to the eyes, and a lot of people find it to be beautiful. The following are benefits of constructing a custom patio cover.
The patio cover helps to create more living space to the property.The covered area will give a very conducive place to receive and welcome guests to your home. It will not matter if you are hosting a big event or a simple family gathering, the patio will give an area that your guests can be entertained. During the entertaining events, the guests are given a choice to dance or take a rest in the area.This place can as well be the gaming area.
Getting the patio cover is wonderful because it protects you from unbarring climate. Sometimes the extreme weather conditions can get in the way of you enjoying the outdoors.This happens during the rainy seasons or when there is too much sun. In case you have a custom cover, then that does not affect you. You are not affected by the stress of the wet or hot seasons that are being experienced in the area you live in. Your real estate is protected from the wet and extreme heat from getting in.
The custom patio covers shields your furniture from getting destroyed. The custom cover provides protection for your furniture and prevents them from being damaged. Your items will eventually begin to get worn out and look beat down because of the exposure of the bad weather. Under the custom patio cover, the furniture stays for a longer period. eventually you will start to see the benefits.
The patio cover adds value to your property. A good idea is investing in the patio cover prior to you selling the house. This raises the price of the property tremendously before selling the house. The owner is able to enjoy a higher profit. The owner will be able to enjoy the fruits of his labor at an assured rate. The property will be able attract a lot of people because of its beauty and unique feature it receives.


JSC SORBENT is a Russian leader in production of the activated carbons. Our company has the unique equipment, which enables us to make the product show the best correlation with world standards. We produce over 35 types of activated carbons — from the most complex catalysts for respiratory protective devices to sorbents used for spill oil liquidation and soil fertility improvement.
The basic business activity is the production of the products used in every industry, water treatment, and also secures health, environmental control, prevention and liquidation the consequences of man-caused accidents and natural cataclysms.


For businesses and/or individuals with matters pending in Federal Court, there may be an impact on scheduling matters with the Court due to the partial government shutdown. If the party is the United States government, the impact of the shutdown will be determined by the United States Attorney office for that district. The longer the shutdown, the more likely it will result in delay. Of course that may be either "good" or "bad" depending upon the particular strategy for any given case.
"The Administrative Office of the U.S. Courts (AO) now estimates that federal courts can sustain funded operations through Jan. 31, 2019. The Judiciary continues to explore ways to conserve funds so it can sustain paid operations through Feb. 1. No further extensions beyond Feb. 1 will be possible. The Judiciary previously had revised its estimate for exhausting available funds from Jan. 18 to Jan. 25.
Brian P. Williams, a Co-Managing Partner at Kahn, Dees, Donovan & Kahn, LLP, in Evansville, Indiana, has practiced business litigation, economic development, contract and employment litigation, real estate, estate planning and trust litigation for over 30 years. He brings a keen understanding of people and organizations to his legal practice. Brian is an effective advocate for his clients in the court room or at the mediation table, as well as an effective advocate for a new business seeking to locate in this community. Honed from service as a private practitioner, federal magistrate, industrial foundation officer, federal law clerk, and civic leader, Brian's analytical skills empower clients to more swiftly reach durable solutions.


thank you for volunteering to take me around Disneyland yesterday while everyone else was busy. i had an absolute blast and definitely think you are a prime D-Land partner in crime. plus those churros were a good idea, why the heck didn't we get two of 'em?
you know me too well. although I felt bad leaving you in Provo, and was half-tempted to stay; you assured me that you would be perfectly fine while I spent time with my family that I miss so much. why are you so amazing pretty much all the time?
what the fudge. why do you make me nauseous, tired, and woozy while I'm on vacay. STOP IT.
how are you always right??
you make my eyes burn and you make me want lasik sooooooo bad.
yup, i've pretty much been off the radar for the past week or so. it's been a long and hard week, ya know? i'm just havin' a time where i needed to take a break from regular life before i come back to conquer it all. i'll be back in full force soon enough!!
you are a poop for starting so soon. can i graduate already?! this last year feels like it's gonna be a looong time. i'm a smart girl and i know my education is important, so i'll endure! but what the heck am i going to do with my degrees when i graduate?! ohhhh there's another mystery waiting to be solved.
i'll be back sunday night to hug & kiss you. i'm sure you would love alllll the cute things i say about you on the blog if you actually read it. instead of waiting around for me to read it to you. jokes on you bucko!!!
I hear ya when it comes to being off the blog radar lately... I feel like I have lost touch with all my favorite blogs!
haha. My contacts do that to. stinks lasik is so expensive. Hope you had fun in D land!


text:
Bloomfield, a William Carey University graduate, also served as AD at VHS for eight years. Prior to that, he was the head ninth grade football coach for 16 seasons at Gautier.
In the last eight years, Bloomfield has coached four Division I signees, with four of his previous players having goine on to play in the NFL.
Thompson announced in mid-March he was leaving the Fighting Irish for a job closer to his hometown in Illinois. Thompson, who is also a track and field coach, will remain at the school through the end of the semester.
Thompson posted a 5-15 record in two seasons at St. Patrick, but the team's four wins in 2014 were the most by a Fighting Irish squad since 2008.
"We improved significantly from last year to this year," Thompson said at the time. "(The next coach) just needs to keep the kids motivated. What I tried to instill in the kids I think works, which is structure and discipline and making them believe they can win.
"At one point, we won three games in a row and there was a lot of buzz."
Bloomfield is expected to address the Fighting Irish team at 2:30 p.m. and parents at 5:30 p.m.
The Sun Herald is working to update this report.
The untold story of the alleged 1973 Pascagoula alien abduction
Mississippi Coast woman is gaga over Jim Cantore in Pascagoula
Gulf LNG has waited for years for approval from the U.S. Federal Energy Regulatory Commission to be allowed to build a liquified national gas export facility near Pascagoula, Jackson County Mississippi.
MORE BREAKING NEWS ALERTS
What's going on in Moss Point? Mayor Mario King speaks out on lawsuits, 'internal issues'
Margaritaville gets a step closer to new amusement park, first flying bar in U.S.
Human remains found in Gautier identified as missing man
Two now arrested in Gautier gas station killing


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LIL B ATTACKED BY A BOOGIE & CREW BACKSTAGE AT BAY AREA ROLLING LOUD FEST (VIDEO)
October 22, 2017 illegaljamz Video 0
Lil B had another bad incident in the public tonight as he was jumped by A Boogie Wit Da Hoodie and his crew backstage at the Bay Area Rolling Loud fest in California.
The incident apparently happened backstage at the festival at the Shoreline Festival Grounds in Mountain View, CA after Lil B said that the New York rapper sounds like Detroit rapper DeJ Loaf. A video of Lil B being attacked by a bunch of people surfaced online.
Unlike most people, Lil B openly admitted to getting stomped on when he hit the stage later for his set. He also "forgave" them for doing so, saying "it's all love, I'm still here." Watch the footage below.
NEW MUSIC: A BOOGIE WIT DA HOODIE – 'DAY 1'
TALIB KWELI – 'RADIO SILENCE' (ARTWORK & TRACK LIST)


Thank you for supporting me in my campaign to win the election to the Houston City Council – District F. The success of my campaign is driven by those who volunteer their precious time because they believe in what I stand for and have confidence in the skills and expertise I will bring to my work on the Houston City Council – District F.
If you would like to volunteer for my campaign please fill out the following form.
Please specify what you would like to volunteer for and then click the Send button at the bottom of this page.
Other Letter to EditorOther - Specify below in message areaKendall can use my name as an endorser - Specify name as you wish for it to appear below in message area.


- Solution of complex mathematical problems such as, differential equations, minimization/optimization, or high-dimensional sums/integrals.
- Direct simulation of physical processes, as for instance, molecular dynamics or Monte-Carlo simulation of physical/chemical/technical processes.
Consequently, the book is divided into two main parts: Deterministic methods and stochastic methods. Based on concrete problems, the first part discusses numerical differentiation and integration, and the treatment of ordinary differential equations. This is augmented by notes on the numerics of partial differential equations. The second part discusses the generation of random numbers, summarizes the basics of stochastics which is then followed by the introduction of various Monte-Carlo (MC) methods. Specific emphasis is on MARKOV chain MC algorithms. All this is again augmented by numerous applications from physics. The final two chapters on Data Analysis and Stochastic Optimization share the two main topics as a common denominator. The book offers a number of appendices to provide the reader with more detailed information on various topics discussed in the main part. Nevertheless, the reader should be familiar with the most important concepts of statistics and probability theory albeit two appendices have been dedicated to provide a rudimentary discussion.


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Shiva as Bholenath: simple and easy to please...
The Legend of Arunachala Shiva:
Holy Hill of Tiruvannamalai
The Gods Vishnu and Brahma were rocking the universe, quarreling as to who was the superior Deity. To restore equilibrium, Shiva appeared as a flame, and challenged them to find his source. Brahma took the form of a swan, and flew to the sky to see the top of the flame, while Vishnu became the boar Varaha, and sought its base. Neither Brahma nor Vishnu could find the source and were humbled and bowed to Shiva.
Arunachala (Annamalai) or red mountain in Tiruvannamalai, India, is recognized as a lingam, and the embodiment of Shiva as Infinite Light, described in this ancient legend.
Perhaps Shiva (Siva in Sanskrit) represents the flame of Life, without beginning or end… everywhere, always, eternal, forever and ever and ever… transcending Brahma and Vishnu who sustain the rhythms of creation.
Shiva rules tamas guna, the aspect of Mystery manifesting as the experience of matter. The Sanskrit word Shiva means auspicious, and refers to the scintillating divine qualities of our life on earth. The brilliance of the light of seeing, tasting, touching...of consciousness experiencing embodiment. The agony and ecstasy of each moment, as our amazing lives unfold.
Shiva is the light of life...the brilliance of consciousness...that illuminates each instant of our experience. At times we are Shiva taking the form of Vishnu as the the boar Varaha infused with His strength, spelunking the deep dark caves of the infinite possibility of pain and suffering that our human experience offers. Often we become the Swan of creative possibility, of Brahma and his Shakti Saraswati soaring to the sublime ecstatic heights of embodied experience.
We are the cutting edge of the unfolding of infinite possibility expressed in the unique dimension of Earthly experience.
Lingothbhavar: Shiva as The Column of The Infinite light of Consciousness
The Substratum of All Creation: Depicted in the legend of Arunachala Shiva
ॐ ॐ ॐ
Last winter, I arrived in India angry with pie in the sky religiosity that worshiped gods in heavens and stood by helplessly letting the Earth go to Hell. I offered my pain to Dakshinamurthy (Shiva as Guru) and was blessed with the realization that Shiva is the Earth!
To cherish and to serve the Earth is to worship Shiva. The beauty of life that we love so much is Shiva. Our own life, which we cherish so selfishly, is Shiva. Look no further, everything you are seeking, you already have! Aum Namah Shivaya! ॐ ॐ ॐ
SIGNIFICANCE OF LORD SHIVA
by Maha Yogi Paramahamsa Dr. Rupnathji
Shiva is the experiencer and the highest object of experience. Shiva is the goal of Sadhana. There is nothing apart from Shiva. There is nothing other than Shiva. Whatever there is, is Shiva. There is nothing, which is not Shiva. There is no place, which is not Shiva. There is no time, which is not Shiva. To be aware of this is to be aware of Shiva.
Shiva (Sanskrit: Śiva, lit. the auspicious one) is one of the principal deities of Hinduism. He is the supreme god within Shaivism, one of the three most influential denominations in contemporary Hinduism.
Shiva is "the transformer" within the Trimurti, the Hindu trinity that includes Brahma and Vishnu. In Shaivism tradition, Shiva is the Supreme being who creates, protects and transforms the universe. In the goddess tradition of Hinduism called Shaktism, the goddess is described as supreme. A goddess is stated to be the energy and creative power (Shakti) of each, with Parvati the equal complementary partner of Shiva, Lakshmi, Vishnu and Saraswati of Brahma.
At the highest level, Shiva is regarded as formless, limitless, transcendent and unchanging absolute Brahman, and the primal Atman (soul, Self) of the universe. Shiva has many benevolent and fearsome depictions. In benevolent aspects, he is depicted as an omniscient Yogi who lives an ascetic life on Mount Kailash as well as a householder with wife Parvati and his two children, Ganesha and Kartikeya. In his fierce aspects, he is often depicted slaying demons. Shiva is also known as Adiyogi Shiva regarded as the patron god of yoga, meditation and arts.
Shivaism is essentially a nature religion, through ritual, offering alignment and coordination with subtle beings and supernatural forces. Shiva rules magic and the occult. Shiva is the god of the left hand path. The dark, lunar path of Tantriks: of the ritual use of sex, wine, and meat…of intoxication and ecstasy.
Shiva is a protector of those on the fringes of society, of everything chaotic, and dangerous, the unexpected and un-explainable. He is god of the humble (shudras), outcasts and criminals. A protector of animals and trees.
Shiva's son Ganesha, is the lord of the ganas…the nature spirits: gnomes, goblins, gremlins and z, (smaller more focused devas, that abound in every aspect of nature.) Ganesh is universally, invoked first in any ritual, to propitiate the spirits of the earth, before calling forth the heavenly gods.
The main iconographical attributes of Shiva are the third eye on his forehead, the serpent around his neck, the adorning crescent moon, the holy river Ganga flowing from his matted hair, the trishula (trident) as his weapon and the damaru (drum). Shiva is usually worshiped in the form of Lingam. Shiva is a pan-Hindu deity, revered widely across India, Nepal, Sri Lanka and Bali.
Ardhanariswari: Combining both the female and male aspects of creation..
Description of Siva ॐ ॐ ॐ
Unlike Vishnu who is depicted as dark blue, Siva is white in color, except for his neck, which is dark blue. Ancient legends tell us that his neck became dark blue when he swallowed the Halahal poison emanated from the sea in the process of churning the amutham cosmic ocean of all possibility. He took the whole poison by himself to protect the cosmos from destruction and thus the images of him are prevailing in dark blue color here and there.
He leads a life of severe austerities. But in the images we find him tall and well built. His body is usually besmeared with ashes, denoting his frequent rounds to the cremation grounds and highlighting his fondness for Thiruneeru, a white sacred ash. He has three eyes. The third eye rests between his eyebrows. It is the eye of wisdom, by opening which he destroys the evil to maintain the cosmic harmony. His third eye symbolizes the all knowing awareness of the universe.
Though he is described in the scriptures as god of anger, he is, in fact, the god of love and he always bestows all kinds of wealth and power to the devotees and this is the reason for which we generally find him in his cheerful and jovial mood in the images. He is simple and easy to please, equal friend to poor and rich, thief and saint. Often he is depicted with a lot of innocence in his demeanor as Bholenath (Lord of simplicity & simple people.) He is generally shown sitting cross-legged in a yogic posture, with his eyes closed and deep in meditation. When he is shown with his eyes open, his face expresses love and compassion.
Siva is the god of simplicity, exemplary humility and austerity. A tiger skin and an elephant skin serve as his garments. His long matted hair is normally tied into a knot or left flowing. He has four arms. With one he holds his weapon the trident. With another, he holds Damaru, a small drum. The remaining two is held in abhaya mudra (the gesture of reassurance and safety, which dispels fear and accords divine protection) and varada mudra (offering welcome, charity, giving, compassion and sincerity).
The moon adorns his head and represents the movement of time and also his cosmic proportions
🔱 The Trident and the Snakes ॐ ॐ ॐ
The trident is the weapon of Siva with which he destroys evils and restores order. The trident stands for the triple qualities of nature, namely, sattva, rajas and tamas. Siva is the master of maya and thereby master of these three qualities with which he controls the universe.
Shiva is worshiped as sacred pipal and banyon trees and the tri-foliate form of bael leaves is a special offering, symbolizes his trident.
The trident symbolically represents the fact that Siva is the controller of the universe and the controller of all illusion. The trident also represents the three aspects of time, the past, the present and the future.
Siva is master of Time, Kala Bhairava who determines the progress of the universe according to his will. The trident represents the three paths to self-realization, namely the path of knowledge, the path of action and the path of devotion.
With these three paths, Siva transforms his devotees and destroys the evil in them. The trident stands for the three primary evils which need to be destroyed in order to make progress towards the divine. They are the evil of anger, the evil of lust and the evil of pride
The damaru drum denotes his connection with the primal sound AUM, the creation of alphabets, languages, grammar and music. His long matted hair denotes his spiritual life and his great powers.The crescent moon adorns his hair like a silver diadem.
He wears a garland of snakes around his neck. Sometimes we see more snakes; one across his body like a sacred thread and two acting as bracelets around his muscular hands. The snakes symbolically represent his control over desire and sensuality. Sometimes in his ferocious aspects, he is shown wearing a garland of skulls.
There is a deep symbolism hidden behind this. The snake stands for all the evil and demoniacal nature in the world. By wearing the snake around his neck, Lord Siva gives us the assurance that no evil can touch us or destroy us once we surrender to him, seek his protection and worship him with deep devotion.
The snake also stands for the power of kundalini, which is described as a coiled serpent lying dormant in the muladhara chakra of all human beings and descends upwards when one starts ones spiritual journey and becomes increasingly divine oriented. The snake around the neck of Siva conveys the meaning that in him the kundalini not only has arisen.
The snake also stands for all passions and desires. By wearing the snakes around his neck, Lord Siva conveys the message to all his devotees that He has overcome all desires and is in full control of Prakriti, or maya and its various machinations.
Symbolism ॐ ॐ ॐ
Siva symbolically represents the tamasic quality. Because of this, he is called pasupathi, (the lord of the animals). His body color, which is white, denotes his purity (sivam) and association with the snowy mountains. His three eyes represent the sun, the moon and the earth, the three paths of liberation and the triple nature of creation. The third eye is actually the eye of wisdom or occult knowledge. The moon that adorns his head represents the movement of time and also his cosmic proportions. With the moon there, his head becomes the night sky, for which he earned the name Vyomakesa (one who has the sky or space as his hair). The moon also symbolizes his association with the occult and the tantras.
Siva is generally seated in yogic posture with which most Hindus are familiar. However we also come across Siva as nataraja or tandavamurthi in a dance posture. With his hair flying in all directions and hands and feet in dynamic motion, the image of nataraja is a symbol of harmony and rhythm. Among the objects, which are associated with him popularly, apart from the trident and the Damaru, are battle-axe (parasu), rosary (aksamala), pasa (noose), khatvanga (magic wand) and khadga (sword).
Lord Siva is know as Pasupathinath, the lord of all creatures. Being a lord of the animals he has complete control on their behavior. Since a snake is one of the most feared and dangerous animals in the world, the garland of snakes around the neck firmly establish the fact that he is a lover of animals and the animals always remain under his control.
Although universally acknowledged as one of the trinity gods of Hinduism, to His followers, He is Maheswara, representing the Trinity all by Himself, His different aspects manifesting themselves as the creator, preserver and destroyer of the worlds He creates. Symbolically He is worshipped as Lingeswara, which name stands more for His creative powers.
Siva and His Titles ॐ ॐ ॐ
He is known by many names and titles. As a god of anger, He is Rudra, the Red One, feared by one and all. As Kailasapathi, He is the Lord of Kailash, His Abode in the Himalayas. As Purusha, He is Iswara Himself. As the Lord of the beings, He is known as Pasupathinath. As the husband of Uma, the Mother Goddess, He is known as Umapathi or Parvathipathi.
As the bearer of the sacred river Ganges, He is known as Ganagadhar. Because of His matted hair, He is dearly called as Jatadhari, by His ascetic followers. As a perfect being, He is Siddheshwar. With His Trident in His hands, He became popular as heroic and fearless Trisuladhari. As the world teacher, Adi Shankara eulogized him as Dakhshinamurthi in his famous book, "Hymns to Dakhshinamurthi".
Sri Ramana in Dakhshinamurthi posture.
To the connoisseurs of art, He is well known as Nataraja, master of all dance forms. As the perfect Yogi, He is credited with the source of all knowledge concerning the various yogas. Such was His powers and divinity, that Lord Vishnu in His incarnations as Rama and Krishna, worshiped Him with great reverence. He was dearer not only to gods but also to demons.
Many demons like Ravana, Bhasmasur and Surapathma were His ardent followers, to whom He gave several boons out of His unbound love and in the process invited great trouble for Himself and for others. Ever willing to help those who are in distress, He saved the cosmos and all the gods by partaking Halahal during the churning of the oceans.
It is interesting to note that while He is worshiped as he is the guru of Tantrics sastras relating to kama, He is also known for destroying the physical form of Cupid, kama with His third eye when the latter tried to disturb Him during His meditation. Men and women love Him alike for his unsurpassed qualities of love and generosity. Even today many young and unmarried girls in India worship Him and pray to Him seeking a devoted husband like Him.
Ardhanariswara half Shiva-half Parvati
Because of His close association with Vishnu, He is also known as Hari Har. Combining in Himself both the male and female aspects of creation, He earned the popular name of Ardhanariswara.
As rider of the Bull Nandi, He is known to the world as Nandiswara. Loved by His children Ganesha and Skanda, and followed by His dutiful wife.
Aspects of Lord Siva ॐ ॐ ॐ
Siva is known by several names and worshiped in various forms. We are mentioning below some of his most famous aspects:
The Sivalingam is the symbolic of the Supreme Self. It is verily Maheswara Himself, the Highest Self and the Lord of the universe. In this aspect it has three parts. The lower part represents Brahma. The middle part, which is octagonal in shape, represents Vishnu. The upper part, which is cylindrical in shape, represents Rudra and is also called Pujabhaga.
Sivalingam literally means the body of Siva. Next to the symbol of AUM, it is perhaps the most potent, powerful and popular symbol in entire Hinduism. In almost all the Siva temples, worship is generally made to Sivalingas only. In the North part of India, we come across his images in the sanctum sanctorum of any Siva temple. A Sivalingam is usually a round or cylindrical and protruding object. The cylindrical part is held firmly by a circular base.
On the physical plane, the object resembles the male sexual organ, suggestive of the creative power of Siva. The octagonal base resembles that of the female, suggestive of his consort Parvathi. Physically a Sivalingam is a phallic symbol, representing the male and female sexual organs in a state of conjugal bliss.
Mentally it symbolizes the union of mind and body. Spiritually it represents the union between Purusha and Prakriti, the highest principles of the manifest universe.
The Sivalingas are normally found installed in the temples. But many devotees of Siva keep them in their houses and offer regular worship. People are however cautioned not to keep Sivalingas in their houses without offering worship, since they are believed to be powerful sources of divine energy.
Sivalingas are either naturally found or made artificially. Different materials are used in their making, such as clay, gold, crystal, glass, diamonds, precious stones and wood.
The round and smooth stones found in the river beds of the Narmada or the Godavari are considered to be the most ideal for worship. Sometimes Sivlingas are made temporarily with clay or sandal paste and disposed of after worship. Some devotees wear Sivalingas on their bodies or around their necks.
Attributes of Lord Shiva ॐ ॐ ॐ
Shiva's main attributes are :
The trident that represents the three gunas
The snakes that show he is beyond the power of death and poison and also stand for the Kundalini energy.
The sound of Shiva's two-sided Damaru, drum maintains the rhythm of the heartbeat
and creates the sound AUM in its overtones.
Bearer of the sacred river Ganges, whose power is tamed through his dreadlocks.
The moon adorns his head and represents the movement of time and also his cosmic proportions.
The vehicle of Shiva is the white bull called Nandi (the joyful).
Shiva is often seated on a tiger skin or wears a tiger skin, with the tiger representing the mind.
Shiva lives on Mount Kailasa in the Himalayas.
* The above is excerpted and expanded from an article by Maha Yogi Paramahamsa Dr. Rupnathji.
Followers of Shiva have used cannabis as a sacrament for untold thousands of years..
Ganja As Sacrament
Shiva is a name Hindus give to the Mystery of Consciousness which illuminates life on Earth. Followers of Shiva have used cannabis as a sacrament for untold thousands of years. Shivaism is venerated as the oldest continuous form of worship on Earth!
Vanamali Mataji, reminds us: " Shiva is always in a state of bliss due to the fact that he is always in a state of transcendental consciousness which has nothing to do with the intake on any type of hallucinogenic drugs.... there are many other ways of getting that ecstatic state than through psychedelic drugs. Shiva was not a yogi who sat in his Himalayan vastness drunk on Ganja , but he was always and ever immersed in the ecstasy of the inner the Self which is the only lasting bliss. "
Aum Namah Shivaya! ॐ ॐ ॐ
(Siva in Sanskrit) represents the flame of Life/Consciousness, without beginning or end… everywhere, always, eternal, forever and ever and ever...
The Sanskrit word Shiva means auspicious, and refers to the scintillating divine qualities of our life on earth. The brilliance of the gift of seeing, tasting, touching...of consciousness experiencing embodiment. The agony and ecstasy of each moment, as our amazing lives unfold.
Fire is an inter dimensional portal, that potentizes the ultrasonic songs these sacred herbs sing when they are burned. Our lips are Shiva and Shakti dancing to the music beyond sound. As we inhale the sacramental herbs, our consciousness is catapulted to our natural state of Oneness.
Shiva is the Earth. To cherish and to serve the Earth is to worship Shiva. The beauty of life that we love so much is Shiva. Our own life, which we cherish so selfishly, is Shiva. Look no further, everything you are seeking, you already have!
Ganja Mantra
Om Shiva Shankara, Hara Hara Ganja
Om Shiva Shankara, Hara Hara Om
Alek... Bohm!
Maha Mrityunjaya Mantra : Invocation of the Great Conqueror of Death
Om Tryambakam Yajamahe
Sugandhim Pushtivardhanam
Urvarukamiva Bandhanan
Mrityor Mukshiya Maamritat
1: Om, We Worship the Three-Eyed One (Lord Shiva),
2: Who is Fragrant (Spiritual Essence)
and Who Nourishes all beings.
3: May He release our Bondage of Samsara (Worldly Life),
like a cucumber (naturally drops when ripe from the bondage of its vine) ...
4: ...and thus Liberate us from the Fear of Death, by making us realize that we are never separated from our Immortal Nature.
Recommended Reading ॐ ॐ ॐ
Shiva: Stories and Teachings from the Shiva Mahapurana, Vanamali Mataji,
Inner Traditions Books, In India published as: Sri Shiva Lila
Gods of Love and Ecstasy: The Traditions of Shiva,
Alain Danielou, Inner Traditions Books
Shakti,: Realm of the Divine Mother,
Vanamali Mataji, Inner Traditions Books
In India published as: Sri Devi Lila
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