text:
\section{Introduction}
The term ``Plancherel measure'' for the infinite symmetric group was suggested by me and introduced in our joint paper with S.~V.~Kerov~\cite{VK-77}. Later, this notion proved to be extremely important not only in the original context. {\it In the asymptotic representation theory} of inductive families of groups and algebras it plays the same role as Haar measure in the theory of locally compact groups. In the present paper, we also illustrate this by examples, in particular, by showing how diverse are situations where it appears. But the motivation behind the choice of the name for this measure does not fully agree with the conventional use of this term; this question is discussed below. This difference is especially clear in the combinatorial definition of the Plancherel measure as a measure on lattice numberings.

With regard to the theory of symmetric groups, it is natural to compare the Plancherel measure with the Poisson--Dirichlet measure, which arises, as shown in our previous papers~\cite{VSh1, VSh2}, in the study of limit distributions of conjugacy classes (i.e., cycle lengths of random permutations). We describe completely different situations in which this measure arises implicitly or explicitly. To understand the overall picture of asymptotic theory, it is of importance to consider the Plancherel measure and its properties as an example of a distinguished central measure on the path space of a graded graph, or as an example of a special Gibbs measure of maximal entropy in constructions of statistical physics.

We begin with a theorem having the most beautiful and clear statement and the most difficult known direct proof. Namely, Edrei's theorem on the uniqueness of the exponential function as a totally positive entire nonconstant function without zeros, see~\cite{Edr,W,ASM}, early papers by Schoenberg~\cite{Sch}, papers by Krein and Gantmacher~\cite{GK}, the book~\cite{Kar}, etc.

Then we turn to the main part of the paper, where we describe the Plancherel measure from different viewpoints and consider the uniqueness theorem for this measure as a nondegenerate central measure on the path space of the Young graph and as a nondegenerate homogeneous monotone random numbering of the two-dimensional lattice~${\Bbb Z}_+^2$.

The interest to the Plancherel measure is due to the fact that it  became the source of uniting such different facts from the theory of entire functions, representation theory, combinatorics, and the theory of invariant measures. That is why we discuss in detail the group nature of Plancherel measures for locally compact groups, both in the most general context and in the case of locally finite groups and, in particular, the infinite symmetric group. Here, the main uniqueness theorem looks as a theorem on nondegenerate ergodic central measures on the path space of the Young graph.

We treat separately the combinatorial definition of the Plancherel measure as the limit of Plancherel measures of finite groups; in particular, in the case of symmetric groups this is a measure on lattice numberings.

It is important to note that, in contrast to the finite case and the usual understanding of Plancherel measures, this measure can be defined meaningfully not on the space of classes of irreducible representations (Young diagrams in the case of symmetric groups), but only on the space of tableaux, i.e., sequences of representations.

The uniqueness problem is the deepest and most difficult question; it is convenient to state it in a general form, as a purely combinatorial problem, for instance, as a theorem on numberings of multidimensional lattices. It would not be an overstatement to compare the existence and uniqueness theorem for numberings of the two-dimensional lattice with  the existence and uniqueness theorem for Haar measure.

The third uniqueness theorem is a rephrasing of the previous one in terms of representations of the infinite symmetric group and their restrictions to finite subgroups. Namely, it states that the regular representation of the group~$\frak{S}_{\Bbb N}$ is distinguished among all representations with character. Here, the focus moves to the asymptotics of the first row of a random Young diagram, whose role in the asymptotic representation theory of symmetric groups  is well known.

In this paper, we do not discuss numerous proofs of these theorems (their list is given below), leaving such a discussion to another occasion, partly because the author does not think that we know the ``proof from the Book'' (in the sense of P.~Erd\"os) for any of these theorems.

The paper is an extended version of my talk at the conference {\it Algebraic Topology, Combinatorics, and Mathematical Physics} (Moscow, May 2018) dedicated to the 75th anniversary of V.~M.~Buchstaber.

\section{A nonvanishing totally positive entire function is the exponential function}

We begin with  the most clear uniqueness theorem, essentially belonging to classical complex analysis.

\begin{theorem}[A.~Edrei, M.~Assein, A.~Whitney, I~Schoenberg \cite{ASM,Edr,W}]
Let $\varphi(z)$ be an entire function on the complex line~$\Bbb C$,
 $$
\varphi(z)=\sum_{n=0}^{\infty}c_n z^n, \qquad c_{-n}=0 \quad \text{for all}\quad n>0,
$$
satisfying the following conditions.

{\rm 1)} \emph{Total positivity:} for every $n\in {\Bbb Z}$, for every collection of indices $\{i_1,\dots, i_n; j_1, \dots, j_n\}$ where $i_s,j_s \in {\Bbb Z}$,
$$
\det ||c_{i_s-j_t}||_{s,t=1,\dots, n} \geq 0.
$$

{\rm 2)} \emph{Nondegeneracy:} the function does not vanish, $\varphi(z)\ne 0$;  and (to simplify the statement) \emph{normalization:}
$\varphi(0)=\varphi'(0)=1$.

Then
 $$\varphi(z)=\exp z.
 $$

If, keeping the total positivity condition, we assume that  $\varphi(z)$ is only meromorphic (i.e., allow it to have poles) and waive the nonvanishing condition and the normalization, then the general form of such a function is as follows:
  $$
  \varphi(z)=z^m e^{\gamma z}\prod_{n=1}^{\infty}\frac{1+\alpha_n z}{1-\beta_n z},
 $$
where $\alpha_1\geq \alpha_2 \geq \dots \geq 0$; $\beta_1\geq \beta_2 \geq \dots \geq 0$; $\gamma \geq 0$;
   $\sum_n \alpha_n +\sum_n \beta_n +\gamma =1$; $m\in {\Bbb N}_+$.
 \end{theorem}

There is a similar theorem for Laurent series, but we will not consider it. The notion of total positivity appeared in the 1930s in works by Schoenberg and von Neumann (on the geometry of Hilbert spaces), and simultaneously in pre-war papers by M.~G.~Krein and F.~Gantmacher~\cite{GK} in connection with functional analytic methods in the study of differential equations. It also arises in statistics and the theory of convex sets (the so-called Chebyshev curves), see S.~Karlin's book~\cite{Kar}. In recent years, the interest to total positivity has considerably increased in connection with algebraic and representation-theoretic problems (papers by G.~Lusztig~\cite{Lus}, A.~Zelevinsky and S.~Fomin~\cite{FZ}, and many others). The recent paper~\cite{BGl} by V.~M.~Buchstaber and A.~Glutsyuk also uses this notion. But here the notion is greatly extended: one considers not Toeplitz matrices as above, but arbitrary matrices. From this point of view, it would be interesting to consider this notion for arbitrary groups (rather than only~$\Bbb Z$ and $\Bbb R$), see below.

In E.~Thoma's paper~\cite{T}, total positivity appeared as follows (to the author's knowledge, this connection was known already to M.~G.~Krein). Consider a positive definite (in the ordinary sense) normalized central function~$\chi(\cdot)$ on the infinite symmetric group~${\frak S}_{\Bbb N}$, i.e., a character of this group. Assume that it is indecomposable, i.e., cannot be written as a nontrivial convex combination of other characters, and restrict this character (regarded as a function on the group) to the conjugacy classes of cycles, i.e., to the semigroup~$\Bbb Z_+$.\footnote{A character $\chi(\cdot)$ takes the same value~$\chi(n)$ at all cycles of length~$n$, hence it produces a function on~$\Bbb Z$ supported by~${\Bbb Z}_+$.} Then the function
    $$ \exp{\sum_{n\geq 1} \frac{\chi(n)}{n}z^n}$$
of a variable~$z$,  i.e., the exponential of the generating function of these values on cycles,  is a totally positive function in the above sense. Here it is important to keep in mind the multiplicativity property of indecomposable characters (see~\cite{VK}): the values of such a character on an arbitrary conjugacy class is the product of its values on the corresponding cycles, i.e., the values on cycles uniquely determine the character. Thus, the positive definiteness of an indecomposable character (regarded as a function on the group~${\frak S}_{\Bbb N}$) can be restated in terms of its restriction to cycles as the total positivity. It is this observation and the multiplicativity property of indecomposable characters (which we do not need now) that allowed E.~Thoma to use the previous theorem to give for the first time a formula for the characters of~${\frak S}_{\Bbb N}$. Thoma did not know about Edrei's theorem, but used a close theorem by Bieberbach, which, together with his own considerations, gave an equivalent result. Thoma's paper contains no ``representation-theoretic'' interpretation of the parameters and no realization of representations.

Later, a ``representation-theoretic'' proof of the theorem was given in~\cite{VK}, as well as an interpretation of the parameters and a realization of the factor representations. By now, there are already five different proofs (see below), but  still no purely combinatorial proof, which should clarify important properties of general lattices.

In this connection, we pose the following problem, which, apparently, has not been studied. As mentioned above, instead of an entire or meromorphic function, or even a Laurent series, one can consider the sequence of its Taylor (Laurent) coefficients as a function on~$\Bbb Z$, and call this sequence a positive definite function on the group~$\Bbb Z$. This justifies a general definition of a totally positive function on a locally compact, or even arbitrary, group or even semigroup with given linear order. For group $\Bbb Z$, or $\Bbb R$ this is usual linear order; for general countable group this is a numeration of he group  using integers $\Bbb Z$ or $\Bbb N$. For example the semigroup ${\Bbb Z}^2_+$ a numeration is defined by infinite Young tableau which cover
all semigroup.

Namely, in the definition of {\it total positive definiteness} of function on the group or semigroup, we require the nonnegativity of all, and not only principal, minors with given order of variables (elements of the group or semigroup). This means that for any two finite subsets of the elements of the group with the same number of elements,
$\{g_i\}_{i=1,\dots n}, \{h_j\}_{j=1,\dots n}$, with have monotonic numeration with respect to the given linear order we require non-negatively ofthe minor $\|f(g_i\cdot h^{-1}_j)\|_{i,j}$.
Evidently the answer depends on the orders.

There appears a series of questions which are not clear even for $\Bbb Z$ with standard order. For instance, what unitary representations and what vectors in these representations correspond (in the sense of the GNS construction) to totally positive functions? These representations are most likely reducible, and these vectors have some additional properties. For the group~$\Bbb Z$, such a representation is regular, but is  is not known what property distinguishes the vector states corresponding to totally positive matrix elements; this is an interesting question close to the problem of central measures.

\section{Plancherel measure}

The definition of the Plancherel measure for the infinite symmetric group as a measure on the space of infinite Young tableaux, which was given in our paper~\cite{VK-77} (see~\cite{VK-81,VK-85} and also~\cite{Shepp}), became established in the literature, but it is  long  in need of a more complete justification. Moreover, in the context under consideration, the term ``Plancherel measure'' itself can be disputed, since, in a sense, it does not coincide with the canonical understanding of this term. That is why we make several general comments on the considerations we had in mind, on numerous links to concrete studies of traditional Plancherel measures (for example, to the 1960s papers by  Gelfand's school on Plancherel measures for some semisimple groups, which became classical), on the relation to general representation theory. Then we return to our main case of interest, that of locally finite groups and, in particular, the infinite symmetric group. In short, we use this term, first, because the Plancherel measure in our sense is the limit of Plancherel measures for finite subgroups; and, second, because for the limiting group  it can be regarded as the most natural candidate for such a measure in the classical sense; recall that the group is not of  type~I, so the notion itself is not uniquely defined. The corresponding uniqueness theorem should be understood with regard to these considerations.

\subsection{Plancherel measure in the general context}

By a Plancherel measure one usually means a measure on the space of equivalence classes of irreducible representations of a group.

For a {\it commutative locally compact group}, the Plancherel measure is the Haar measure on the group of characters. For a {\it finite group}, the probability measure on the space of equivalence classes of irreducible complex representations whose value at each representation~$\pi$ is proportional to the squared dimension of~$\pi$  is distinguished and should be called the Plancherel measure by analogy with the classical Plancherel measure.

For a {\it nondiscrete compact group}, the Plancherel measure is $\sigma$-finite, and its weights, more exactly, the ratios of the measures of two irreducible reprentations, are still equal to the ratio of their squared dimensions. The duality in this case is well known starting from works by Tanaka and Krein.

In all these cases, there is an analog of Parceval's identity (or the Plan\-che\-rel isometry) for the group~$\Bbb R$, i.e., an isometry between the group algebra
 ${{\Bbb C}[G]\sim l^2[G]}$ with the Hilbert structure determined by the inner product corresponding to the Haar measure (the uniform measure on~$G$) and the space~$C(G)$  of compex-valued functions on~$G$ with the Hilbert structure generated by the Plancherel measure on the space of equivalence classes of representations.

This isometry can be expressed as the following equality of inner products:
$$\langle f_1,f_2\rangle_{\rm Ha} = \langle R(f_1),R(f_2)\rangle_{\rm Pl};$$
here $f_1,f_2 \in {\Bbb C}[G]$ and $R$ is the right regular representation of the algebra on itself,  $\langle\cdot,\cdot\rangle_{\rm Ha}$ is the inner product corresponding to the Haar measure on the group~$G$, and  $\langle\cdot,\cdot\rangle_{\rm Pl}$ is the inner product defined as the normalized direct sum of the inner products in each of the minimal primary subrepresentations, whose spaces are the matrix algebras~$M_k({\Bbb C})$ with the ordinary inner product $\langle A,D\rangle=\operatorname{Tr}\{AB^*\}$.

For type~I groups, the definition of the Plancherel measure reduces to the same formula, which uses the pairing of operator functions corresponding to the representation; such a definition is essentially given in Dixmier's book~\cite{Dix} for type~I groups (in his terminology, GCR groups), which exhausts the problem of defining the Plancherel measure for these groups.

But there is another important circumstance, which was discussed in the old literature. By the decomposition of the regular representation one should mean the {\it central decomposition of the regular representation}. It is uniquely defined: this is the decomposition into factor representations, and for type~I groups they are multiples of irreducible representations  (primary). Hence, it is natural to regard the Plancherel measure as a measure on primary representations, which does not change matters much, since there arise only multiplicities of irreducible representations. Therefore, the uniqueness of the Plancherel measure in this sense is ensured by an agreement about the choice of additional normalizations depending on the relation between multiplicities.

Let us add that for nonamenable groups, representations that are not weakly contained in the regular representation (e.g., complementary series representations for semisimple groups) do not belong to the support of the Plancherel measure and are not related to it in any sense.

Only the central decomposition of a representation, i.e., the decomposition with respect to the center of the von Neumann algebra generated by the image of the group algebra under the representation, is canonical; this is a decomposition into factor representations (and not in irreducible representations). If the von~Neumann algebra generated by a representation is of type~II or~III, then its factor representations are no longer primary, i.e., multiples of irreducibles. The central decomposition is a necessary step in harmonic analysis, but, of course, it does not give complete information on the representation.

The difficulty with the definition of the Plancherel measure for groups not of type~I is due to the fact that if we want to obtain a measure on the decomposition of a representation into irreducible representations, i.e., on the dual object of a locally compact group, then we must have a Borel structure on this object. But such a structure is not available, so we are forced to choose a part of the set of equivalence classes of irreducible representations. This choice, according to the general theory, is performed by choosing a maximal commutative self-adjoint subalgebra in the commutant of the algebra generated by the representation.

But if we take such a maximal commutative subalgebra in the commutant of the algebra generated by a factor representation, then its  decomposition into irreducible representations depends substantially on the choice of this subalgebra. It is known that these decompositions can even be disjoint for two different maximal subalgebras. This question is still poorly studied.

   \subsection{Plancherel measure for locally finite groups}

Recall that the infinite symmetric group is not a group of type~I, but, on the contrary,  is rather a typical group of type~II or~III (or
 NCGR according to Dixmier~\cite{Dix}).

In the framework of asymptotic representation theory, a natural definition of the Plancherel measure is based on considerations quite different from those described above, namely, approximative considerations. Indeed, the analysis of finite groups leads to the conclusion that in the case of an inductive limit of finite groups (yielding a group not of type~I), the natural candidate for the Plancherel group is the inductive limit of the Plancherel measures on the prelimit groups. Moreover, this measure should be defined not on the space of equivalence classes of all irreducible representations, which does not exist in a reasonable sense, but on another object, which will be discussed below.

Let a group $G_{\infty}$ be defined as an increasing limit of finite groups:
   $$G_{\infty}=\mbox{lim-ind}_n G_n.$$
The dual spaces ${\hat G}_n$ (the sets of equivalence classes of irreducible representations of~$G_n$) form an inverse spectrum, whose limit could be regarded as the dual object to the inductive limit of groups:
 $$\hat G= \mbox{lim-proj}_n {\hat G}_n.$$
However, according to the general theory, points of the projective limit are coherent (with respect to the projections) sequences of points of prelimit spaces, i.e., in terms of our constructions, sequences of representations~$\pi_n$ of~$G_n$ such that $\pi_n$ is an irreducible representation of the group~$G_n$ occurring as a component in the representation induced by the representation~$\pi_{n-1}$ of the group~$G_{n-1}$. In terms of symmetric groups, these are tableaux. Thus, we arrive at the conclusion that the Plancherel measure of the limiting group is, possibly, a measure defined on the space of sequences of partitions. The collection of Plancherel measures on the spaces of representations of
$G_n$  can be regarded as a coherent sequence of points of the corresponding simplices, and the limit point  is a point of  the projective limit of simplices, which, in turn, is also a Choquet simplex (see~\cite{VPr}). We will show that the sequence of Plancherel measures does indeed determine a point of the projective limit, which will be called the Plancherel measure of the limiting group~$G_{\infty}$.

 \begin{lemma}
Let $G$ be a finite group and $H\subset G$ be a subgroup of~$G$, let $\hat G$ and~$\hat H$ be the spaces of equivalence classes of irreducible representations of these groups, and let $\mu_G$ and~$\mu_H$ be the Plancherel measures (see the definition above) on~$\hat G$ and~$\hat H$, respectively. Denote by~$\operatorname{Ind}$ and~$\operatorname{Proj}$ the operations of induction and restriction of irreducible representations, and let $\widehat{\operatorname{Ind}}$ and
~$\widehat{\operatorname{Proj}}$ be the corresponding maps on the simplices~$\operatorname{Simp}(G)$ and~$\operatorname{Simp}(H)$ of probability measures:
 $$\widehat{\operatorname{Ind}}:\operatorname{Simp}(H)\rightarrow \operatorname{Simp}(G), \quad \widehat{\operatorname{Proj}}:\operatorname{Simp}(G)\rightarrow \operatorname{Simp}(H). $$
Then $\widehat{\operatorname{Proj}}\,\mu_G=\mu_H$.
 \end{lemma}
  \begin{proof}
The equality we need to prove is equivalent to the following one:
  $$\operatorname{Proj}(\mu_G)\{\lambda\}=\sum_{\Lambda\supset \lambda}\frac{\{\dim{\Lambda}\}^2}{\operatorname{ord}(G)} =\frac{\{\dim \lambda\}^2}{\operatorname{ord}(H)}=\mu_H(\lambda),$$
which follows immediately from the properties of induced representations. It seems that this relation has not yet appeared in the literature, but for necessary information on induction, see~\cite{Serr}.
    \end{proof}

 \begin{definition}
 A locally finite graded graph is said to be a Plancherel graph if for every vertex~$\lambda$ of level~$n$, the following relation holds:
$$ \frac{\dim(\lambda)}{\sum_{\Lambda:\Lambda \succ \lambda}\dim(\Lambda)}=\frac{d_n}{d_{n+1}},$$
where $d_n=\sum_{\lambda} \dim{(\lambda)}^2$ is the sum over all vertices of level~$n$, i.e., the ratio depends only on the number of the level and is equal to the ratio of the sums of squared dimensions at the corresponding levels.
 \end{definition}

We have proved the following theorem.

\begin{theorem}[theorem-definition]
The Bratteli diagram of a locally finite group $G=\bigcup_n G_n$ is a Plancherel graph. The Plancherel measures on its levels are coherent and determine (by definition) a measure on the space of paths (tableaux), called the Plancherel measure of~$G$.
\end{theorem}

In view of the importance of this notion, we will again discuss it in detail for the symmetric group. The fact that the limiting measure is a measure on sequences of representations (tableaux) poses new questions about this Plancherel measure. For example, a remarkable fact is that it is a Markov measure, like all central measures. We will present the transition and cotransition measures of the corresponding Markov chain. Apparently, there is a necessity to make a list of various properties of graded branching graphs of general locally finite groups, which so far have been studied only for some special groups.

It is of interest to give examples of Plancherel Bratteli graphs different from those arising from groups and to understand to what extent the Plancherel property  is close to being a characterization of Bratteli diagrams of groups. Another question is what Bratteli diagrams are simultaneously Plancherel graphs and differential posets in the sense of Stanley and Fomin~\cite{St,Fo}.

It is easy to check that for commutative locally finite groups,  the new definition of the Plancherel measure coincides with the standard one: it is the Haar measure on the compact group of characters.

So, we associate a Plancherel measure with a locally finite group. One can easily check that this measure does not change if we pass from an inductive sequence of groups~$\{G_n\}$ to a subsequence~$\{G_{n_k}\}$. Thus, the measure is determined by the inductive limit itself. But, of course, an essential modification of the system of embeddings of finite groups in the definition of the inductive limit may change the measure.

Here we have a vast area for various group-theoretic and representation-theoretic questions.

Note that an attempt to define a reasonable probability measure on the set of irreducible representations, e.g., on the set of infinite Young diagrams, is hardly meaningful; the reason is not only in the absence of a separable Borel structure on this set for groups not of type~I, but also in the fact that the notion of a class of measures that ``decompose'' representations of this type itself has little sense.

A convenient tool for the study of asymptotic properties of the Plancherel measure is the ``grand ensemble,'' i.e., the union of all sets of finite diagrams, on which one can define the ``Poissonization'' of the Plancherel measures of finite groups, see~\cite{V96,BOO}.

The above considerations can be applied to a more general case, that of semidirect products of commutative $C^*$-algebras and their automorphism groups. Recall that, in particular, the group  $C^*$-algebra of a locally finite group, being a so-called AF-algebra, is the semidirect product of a commutative (``Gelfand--Tsetlin'') subalgebra  and the group algebra of some (``adic'') group. In this case, the Plancherel measure is defined in a natural way as the invariant measure on the spectrum of this commutative subalgebra, and thus as the spectral measure of the regular representation. Here we can say again that such a definition depends on the realization of the algebra as a semidirect product.

The study of the Plancherel measure for the infinite symmetric group shows to what extent this notion is useful in the study of groups and related combinatorial and other objects.

Below we give an independent, purely combinatorial, definition of the Plancherel measure for the infinite symmetric group. But first we present the main formulas for this measure.

\subsection{The Plancherel measure for the infinite symmetric group}

Recall that the branching of representations of a locally finite group allows one to construct a graph (in the general case, a multigraph) which is a graded graph (Bratteli diagram); a measure on the path space of such a graph is defined in a natural way as the inverse limit of  a coherent system of measures on initial segments of paths. From the probabilistic point of view, we originally have only measures on levels, i.e., know the one-dimensional distributions of the future Markov process, and we must check that there exists a system of transition probabilities that determines a central measure with these one-dimensional distributions (the centrality of a measure is equivalent to its invariance under inner automorphisms).

Consider the infinite Young graph~$\textbf{Y}$, i.e., the graded graph of all finite Young diagrams~$\lambda$, or finite ideals of the lattice~${\Bbb Z}_+^2$. A Young tableau $t=\{t_k\}_{k=1}^n$ with $n$ cells is a monotone sequence of $n$ diagrams, or a path in the Young graph that starts from the one-cell diagram. The space~ ${\cal T}(\textbf{Y})$ of all infinite paths in the Young graph, i.e., the space of infinite Young tableaux, is the inverse limit of the sequence of spaces of finite Young tableaux with respect to the natural projections $P_{n+1}:{\cal T}_{n+1}\rightarrow {\cal T}_n$, $n=1,2\dots$, forgetting the last cell:
$$
{\cal T}(\textbf{Y})=\lim_n ({\cal T}_n, P_n).
$$
This is a Cantor-like compact set equipped with a Borel structure and a tail filtration~$\{\xi_n\}_n$. By definition, two tableaux $t \in {\cal T}_n$ and $t' \in {\cal T}_n$ lie in the same element (class) of the partition~$\xi_n$ if their diagrams coincide for $k>n$, i.e., $t_k=t'_k$ for $k=n+1,n+2, \dots $.

The Plancherel measure on the $n$th level $\textbf{Y}_n$, $n=1,2 \dots$, of the graph~$\textbf{Y}$ is the probability measure~$\mu_n$ defined by the formula
\begin{equation}\label{pl}
    \mu(\lambda)=\frac{\dim(\lambda)^2}{n!}.
\end{equation}
In this and subsequent formulas, $\dim (\lambda)$ is the dimension of the representation~$\pi_{\lambda}$ of the group~$S_n$  canonically associated with the diagram~${\lambda}$; its combinatorial interpretation is as follows: this is the number of paths leading to the diagram~$\lambda$ from the initial vertex~$\emptyset$. This dimension can be found by the Frobenius formula, or the well-known hook-length formula.

By definition, the Plancherel measure~$\mu^n$ on the set~${\cal T}_n$ of all Young tableaux with $n$ cells is a measure whose restrictions to the sets of tableaux with the same diagram~$\lambda$ are uniform, hence it is given by the formula
 $$
 \mu^n(t)=\frac{\dim(\lambda)}{n!},\qquad t\in {\cal T}_n
 $$
   (recall that the number of tableaux with given diagram~$\lambda$ is equal to~$\dim(\lambda)$).

   \begin{lemma}
The collection of measures $\{\mu^n\}_{n=1}^{\infty}$ on the sets of finite tableaux is coherent with respect to the projections~$P_n$; in other words,
    $$\sum_{t' \in {\cal T}_{n+1}: P_{n+1}(t')=t} \mu^{n+1}(t')=\mu^n(t).$$
    \end{lemma}

    \begin{proof}
Upon substituting the values~$\mu^n(t)$ from the previous formula, the desired relation becomes equivalent to the following relation for dimensions of diagrams:
  \begin{equation}\label{ind}
    \sum_{\Lambda:\Lambda\gtrdot \lambda} \dim(\Lambda)=(n+1)\dim(\lambda),
\end{equation}
where $\Lambda$ and $\lambda$ lie at the $(n+1)$th and $n$th level of the Young graph, respectively, and differ by one cell.

The last formula immediately follows from the representation-theoretic meaning of the Young graph as the graph of induced representation. Namely, counting the dimension of such a representation, we have
\begin{equation}\label{ind}
 \sum_{\Lambda\gtrdot \lambda}\pi_{\Lambda}= \operatorname{Ind}_{S_n}^{S_{n+1}}\pi_{\lambda}.
\end{equation}
\end{proof}

   \begin{corollary}
The family of cylinder measures ${\mu^n}$ on the space~{\cal T} of infinite tableaux determines a well-defined Borel probability measure $\mu^{\infty}\equiv {\rm Pl}$, called the Plancherel measure. It is central and nondegenerate.
   \end{corollary}

    \begin{proof}
The existence of this measure follows from the coherence of the cylinder measures with respect to the projections (Kolmogorov's theorem). The centrality is equivalent to the uniformness of the measures on the sets of tableaux with given diagram. The nondegeneracy can be verified directly, since all proper infinite ideals of the lattice~${\Bbb Z}_+^2$ are unions of finitely many rows or columns of the lattice, and the density of every such ideal is equal to zero.
    \end{proof}

This fundamental measure was introduced and named ``Plancherel measure'' in~\cite{VK-77}. The behavior of typical diagrams with respect to the Plancherel measure~$\mu_n$ as $n\to\infty$ is very well studied since the late 1970s; namely, we know the limit shape of diagrams, the asymptotics of the lengths of rows, their fluctuations, and many other things.

For further work with the Plancherel measure, it is convenient to use the language of the theory of Markov chains. Note that it is natural to regard the space~$\cal T$ of Young tableaux as a nonstationary Markov compactum. By construction, ${\rm Pl}$ is a Markov measure (a measure corresponding to a Markov chain) on this space; the state space of this Markov chain at time~$n$ is~${\cal T}_n$, its  \textbf{transition probabilities} are equal to
\begin{equation}
     \operatorname{Prob}\{t_{n+1}=\Lambda|t_n=\lambda\}=
     \frac{\dim (\Lambda)}{(n+1)\dim (\lambda)},
\end{equation}
and its \textbf{cotransition probabilities} (which express the centrality of~${\rm Pl}$) are equal to
\begin{equation}
    \operatorname{Prob}\{t_n=\lambda|t_{n+1}\}=\Lambda\}=\frac{\dim(\lambda)}
     {\dim(\Lambda)}.
\end{equation}

This remarkable Markov chain was already studied in~\cite{VK,Ker}; its new properties (in particular, quasi-stationarity) will be considered later. Here we do not discuss an interesting relation between the Plancherel measure and its generalizations and the notion of a model of representations in the sense of I.~M.~Gelfand~\cite{Gel}. For finite symmetric groups, a model was suggested in~\cite{Klya}. For the infinite symmetric group, this is the representation in the $L^2$ space over the Plancherel measure; the latter plays the role of the ``Koopman representation'' corresponding to the regular representation regarded as a von~Neumann representation. However, we call it the basic representation and not the model, because there can be other candidates for a model. In more detail we consider this question in a paper under preparation.

     \section{Random monotone numberings and the combinatorial meaning of the Plancherel measure}

There is another useful and important interpretation of tableaux and measures on the space of tableaux, in particular, the Plancherel measure, in terms of  random numberings of partially ordered sets (posets). First, we will give general definitions for arbitrary locally finite posets~$P$ with minimal element  $\{\emptyset\} \in P$, and then consider our main example of the lattice~$\Bbb Z_+^2$ with the natural partial order.

A monotone numbering of a poset~$P$ (the main example is $P={\Bbb Z}_+^2$) is a map~$\phi:{\Bbb N}\to P$ from the set of positive integers to~$P$ satisfying the conditions
$$\phi(0)=\emptyset,\qquad \phi(n)\succcurlyeq \phi(m) \Rightarrow  n>m.$$

If $\phi$ is a monotone numbering, then for every~$n$ the set $\{\phi(1),\phi(2),\dots, \phi(n)\}$ is a finite ideal in~$P$ (a Young diagram for the lattice~$\Bbb Z_+^2$).

The graph (Hasse diagram) of all finite ideals of~$P$ ordered by inclusion will be denoted by~$\Gamma(P)$ (for the lattice~$\Bbb Z_+^2$, this is the Young graph). A monotone numbering is an infinite connected path (or tableau) in the graph~$\Gamma(P)$.

The density of an infinite set $I \subset P$ with respect to a numbering ${\phi:{\Bbb N}\rightarrow P}$ is the limit
$$
\lim_n\frac{1}{n}\left|\{i: i<n; \phi(i)\in I\}\right|.
$$

The set of all numberings is a Cantor-like compact set~$\cal N$. A random numbering is, by definition, a Borel probability measure on~$\cal N$. It is determined by a sequence of random elements of~$P$, i.e., a random process $\{\xi_n\}_{n \in \Bbb N}$ with values in~$P$.

A random numbering is said to be {\it nondegenerate} if every proper (possibly, infinite) ideal $I \ne P$ has zero density with probability~1.

A random numbering  with values in~$P$ is said to be a {\it Markov} numbering if the corresponding random process is, in a natural sense, a (nonhomogeneous) Markov chain whose state space at time~$n$ is the set of ideals (diagrams in the case of~${\Bbb Z}^2_+$) with $n$ elements.

A random numbering $\phi$ is said to be {\it central} if for every~$n$ the conditional measure on the set of finite ideals with $n$ elements given that the values $\phi(k)$ with $k>n$ are fixed is uniform with probability~$1$.

A random central numbering is said to be {\it ergodic} if the action of the group of all finite renumberings of the random path (which is well defined by centrality) on the path space equipped with this measure is ergodic.

One can easily see that a central numbering, regarded as a measure on the path space of the Young graph, is exactly a central measure. To describe all (ergodic) central measures on a path space is one of the main problems in the whole theory of graded graphs.

The following theorem is one of the deepest facts of the whole asymptotic representation theory of symmetric groups; this is the second uniqueness theorem in our original list.

\begin{theorem}
There is a unique nondegenerate central random numbering of the lattice~$\Bbb Z_+^2$; the corresponding measure on the space of numberings (paths, or tableaux) is the Plancherel measure. It is Markov and ergodic. Another reformulation: the system of cotransition probabilities of the Plancherel measure determines a unique nondegenerate Markov measure on the space of tableaux.
\end{theorem}

This uniqueness theorem is the most difficult part of the general Thoma theorem~\cite{T} on the characters of the infinite symmetric group, which we discussed in the first section of the paper. By now, there are several proofs of this theorem, but still no purely combinatorial proof. Here are the corresponding references:  \cite{VK,O1,OO1,BG,P}.

The proof closest to a purely combinatorial one is given in~\cite{P}; it uses the following interpretation of the problem suggested by me: it suffices to prove the following combinatorial-probabilistic lemma, which is so far a corollary of the theorem, but is in fact equivalent to the theorem.

\begin{lemma}
For every $k\in {\Bbb N}$ and every $\epsilon>0$, there exists $\delta>0$ and $N\in {\Bbb N}$ such that for every
$n>N$ and for every Young diagram $\lambda \vdash n$ with $\lambda \subset[1,\delta\cdot n]\times[1,\delta\cdot n]$,
$$\rho_k(d_k^{\lambda}(\cdot), {\rm Pl}_k(\cdot))< \epsilon;$$
here $\rho_k$ is an arbitrary metric on the space of probability measures on the set of Young tableaux with $k$ cells, while $d_k^{\lambda}(\cdot)$ and ${\rm Pl}_k(\cdot)$ are the probability distributions on these sets of Young tableaux induced by the Young diagram~$\lambda$ and the Plancherel measure~${\rm Pl}$, respectively.
\end{lemma}

The proof in~\cite{P} uses the theory of symmetric functions and, in a sense, is a continuation of the paper~\cite{OO1} on shifted Schur functions. We will return to the discussion of this proof elsewhere.

Another property of the Plancherel measure in terms of random numberings looks as the quasi-invariance under a lattice shift, more exactly, under passing from a given numbering to the induced one, obtained by omitting the elements of the first row, or the first column, or both. It is easy to prove that such an induction sends the Plancherel measure to an equivalent one. Hence, another equivalent formulation of the uniqueness theorem is as follows.

\begin{proposition}\label{prop1}
The Plancherel measure is the unique central quasi-in\-vari\-ant measure.
\end{proposition}

\subsection{General locally finite posets}

All definitions from the previous section make sense for arbitrary locally finite posets, i.e., arbitrary distributive lattices that are Hasse diagrams of lattices of finite ideals of posets. We mean the notions of monotone numberings, random numberings, and the properties of measures on the space of numberings of a poset (or the space of tableaux, the path space of the graph of ideals): centrality, nondegeneracy, quasi-invariance, Markov property, ergodicity, etc.

The existence of at least one monotone nondegenerate central numbering for an arbitrary poset can usually be easily proved by compactness considerations.

However, the uniqueness of a central nondegenerate measure holds not for all posets.

\begin{definition}
A countable locally finite poset is said to be \textbf{rigid} if a central nondegenerate Markov monotone numbering on this poset exists and is unique.
\end{definition}

The above theorem says that the poset~${\Bbb Z}^2_+$ is rigid.

\begin{problem}
Are the posets ${\Bbb Z}^d_+$ for $d>2$ rigid?
\end{problem}

A direct proof of the uniqueness theorem in any of the suggested formulations, e.g., of Proposition~\ref{prop1}, would reveal deep properties of numberingss of rigid lattices. This would mean a considerable progress in the asymptotic theory of posets and in the theory of traces of  $C^*$-algebras.

Apparently, an example of a nonrigid poset is the poset with the set of elements~${\Bbb Z}^d_+$ and the partial order defined as follows:
$$(x,y)\succ (u,v) \mbox{ if } y>v, \quad (x,0)\succ (u,0) \mbox{ if } x>u,$$
and $(x,y)$ and $(u,v)$ are incomparable if $x\ne u$ and $y$ or $v$ is not zero.

The deformation of the order structure on~${\Bbb Z}_+^2$ that turns rigidity into nonrigidity reminds of percolation phenomena.

All interpretations and properties of the Plancherel measure, including those discussed in the previous sections, are closely intertwined. We will consider another invariance property, but state it in a more general situation.

It is not difficult to show that every central measure on the path space of a graded graph is Markov, but not stationary, because the state space  depends on~$n$. Nevertheless, it has the property of invariance with respect to a ``quasi-shift,'' which we will call the {\it quasi-stationarity} of a nonstationary Markov chain. For the Young graph, this property is a generalization of the combinatorial notion of the Sch\"utzenberger (jeu de taquin) transformation, which in the case of central measures and the Plancherel measure for the lattice~${\Bbb Z}^2_+$ was studied in \cite{VKS,RS,S}. Here we will give a general definition.

We define a (deterministic) map from the path space of a distributive lattice to itself, calling it the transfer. For the Young lattice, this is an infinite analog of the Sch\"utzenberger transformation. It is most convenient to describe the transfer in terms of numberings.

Consider an arbitrary numbering (sequence of elements) $t_1=\emptyset, t_2, t_3, \dots$ of a poset, and associate with it a new numbering $t'_1=\emptyset, t'_2, t'_3, \dots$ as follows: if $t_{n+1}\succ t_n$, then $t'_n=t_n$; and if $t_n$ and $t_{n+1}$ are not comparable, then $t'_n=t_{n+1}$, $n=2,3 \dots$. This is a natural generalization of the ordinary shift; distributivity guarantees that all 2-intervals in the graph are either segments or rhombi. This definition makes sense for more general graphs than Hasse diagrams of distributive lattices, namely, we need only the existence of a ``translation'' of edges for adjacent vertices.

Thus, we can consider the class of Markov chains for which there is a well-defined shift (transfer) in the path space that shifts paths in time in a generalized sense.

\begin{definition}
Consider a Markov (in general, nonstationary) compactum with a well-defined transfer, e.g., the path space of the Hasse diagram of a distributive lattice. A Markov measure on this compactum is said to be quasi-stationary if it is invariant under the transfer.
\end{definition}

\begin{problem}
Every central measure on the Markov compactum of paths in the Hasse diagram of a distributive lattice is quasi-stationary.
\end{problem}

For the  Young graph, this is true and follows from the main result of~\cite{VKS}. Most likely, quasi-stationarity also holds for an arbitrary distributive lattice. The Sch\"utzenberger transformation, i.e., the transfer for the Young graph, has been recently considered in~\cite{RS,S} for other reasons. The main result is that the Sch\"utzenberger transformation on the path space equipped with any central ergodic measure (Thoma measure) is metrically isomorphic to a Bernoulli shift. We will consider these problems in another paper. All these considerations reveal a close relation between this circle of problems and ergodic theory.

\section{The third uniqueness theorem: representations of the infinite symmetric group}

In this section, we essentially consider a corollary of the previous theorems for the representation theory of symmetric groups. In its first form, this corollary is equivalent to the uniqueness theorem and, possibly, has its own independent proof. The second formulation (with a bound) contains an important complement to the uniqueness theorem concerning properties of the Plancherel measure. We state our theorem as a uniqueness theorem for the regular representation of the infinite symmetric group, more exactly, characterize this representation in natural terms.

We will consider traceable factor representations of the group~${\frak S}_{\Bbb N}$, i.e.,  normalized indecomposable characters of~${\frak S}_{\Bbb N}$,  already discussed above. Recall that a character is a function~$\chi(\cdot)$ on~${\frak S}_{\Bbb N}$ satisfying the conditions
$$\chi(e)=1,\quad \chi(gh)=\chi(hg), \quad
||\chi(g_i \cdot g_j^{-1})||_{i,j=1}^n \geq 0 $$
for every $n \in {\Bbb N}$ and any $g_1, \dots g_n \in G$.

Indecomposability means that $\chi(\cdot)$  is an extreme point of the simplex of all characters.

Every indecomposable character~$\chi$ generates a representation~$\pi_{\chi}$ of the group, which is either finite-dimensional, or infinite-dimensional in a von~Neumann factor of type~II$_1$.

A nontautological realization of such representations was given in~\cite{VK-81}, but we will not use it.

The link between the theorem on the exponential function (Section~1) and that on the regular representation and its character (see below) is given by the correspondence
 $$ \exp (z)\dashleftarrow\dashrightarrow \chi_e =\delta_e(g),$$
or, in terms of generating functions,
 $$F_{\chi}(z)=\exp{\sum_n\frac{\chi(g_n)}{n}z^n},$$
 where $g_n\in S_{\infty}$ is a cycle of length $n$; this is a totally positive function.

The most important factor representation of~${\frak S}_{\Bbb N}$ is the (say, left) regular representation. Denote it by~$\Pi$. As mentioned above, it is this representation that corresponds to the Plancherel measure, and the corresponding character is $\chi(g)=\delta_e(g)$, the $\delta$-function at the group identity. It turns out that the regular representation~$\Pi$ and, consequently, its character (i.e., the trace in a von~Neumann factor of type~II$_1$) can be axiomatized as follows.

We will consider restrictions of representations and, in particular, the representation $\Pi$ to finite subgroups ${\frak S}_n$ and define the following sequence:
$$Y_{\chi}(n)\equiv \max \{k<n: \; \exists S_k \subset S_n ,\; \exists h\in H_{\pi_\chi}: \forall g\in S_k \quad \pi_{\chi}(g)h=\pm h \}.$$
In other words, we fix the greatest rank of a subgroup for which the restriction of~$\pi_\chi$ is either
the trivial or the sign representation: $\chi\big|_{S_k}=1$ or  $\chi\big|_{S_k}=\operatorname{sgn}(\cdot)$.

\begin{theorem}[uniqueness and characterization of the regular representation]

{\rm1)} There is a unique factor representation for which   $Y_{\chi}(n)=o(n)$, namely, the regular representation.

{\rm 2)} The regular representation is the unique factor representation for which $Y_{\chi}(n)=O(\sqrt n)$.
\end{theorem}

The second claim follows from the theorem of~\cite{VK-85} which says that the growth of~$Y_{\chi}(n)$ for the regular representation is exactly $Y_{\chi}(n)=2\sqrt n$.

Thus, $Y_n$ grows either linearly, or as $2\sqrt n$, and the latter occurs only in the case of the regular representation~$\Pi$ and the character~$\chi(g)=\delta_e(g)$.

In this form,  the theorem is stronger than the uniqueness theorems, since it implies the total positivity theorem from Section~1 and the uniqueness theorems for the Plancherel measure, but the strengthening is achieved by an additional bound on the growth of the first row of the Young diagram.

redpajama_set_name: RedPajamaArXiv


Weekly show about future cars and future car technology. This week news about: Tesla pushes autopilot software, Mitsubishi i-Miev gets recalled, Volkswagen shifts to plug-in cars, Volvo tells Feds to figure out autonomous drive standards, HUMMER Tax Breaks for Tesla Model X owners, Aston Martin Commits to EV, Toyota Prius gets AWD, Zero unveils new 2016 Motorcycles, Doc Brown supports Hydrogen.
i-Miev Recall – Mitsubishi recalls i-Miev for airbag problems.
VW's Electrified Push– VW pushes towards electrification after dieselgate.
Federalize autonomous cars – Volvo asks Feds to standardize autonomous car legislation.
HUMMER Tax break – How Tesla's Model X SUV is eligible for the Federal business tax credit made famous by gas-guzzlers.
Electric Aston – James Bond's favourite brand is going electric, and diesels are out, says Aston Martin CEO Andy Palmer.
Prius Updates – More details come out concerning the upcoming 2016 Toyota Prius Hybrid.


There has been plenty of news about the Heartbleed bug this week. TidBITS did a great job summing it up. It appears something we all took for granted as really secure (Open SSL) really wasn't. As users that means we've potentially been compromised at a lot of websites. I say "potentially" because there is really no way to log incursions due to the nature of this bug. That's a little terrifying. So what should you be doing this weekend?
First take a look at this handy list from Mashable. If any of your vendors and online accounts show up as compromised AND fixed (that second part is important), log in and reset your password. If the site is compromised but not fixed yet, don't log in. In that case, don't touch it until it is fixed.
You all know how I'll be updating my passwords, with 1Password, which was not compromised. As an aside, someone at Macworld/iWorld asked me why I always change my major passwords (banking, iTunes, Amazon, Dropbox, Paypal) twice a year. Things like this are why (although in fairness this bug is so bad that wouldn't have saved me either).


Establish employment agreements. Ensure that your employees are forbidden from revealing any restricted records, formulas, or intellectual property.
Apply for trademarks, patents & copyrights. Document your ownership and obtain legal protection to safeguard your ideas and your brand.
Secure your information. Restrict access to paper files and apply password protection and back up methods to protect important electronic records.
Sign confidentiality agreements. Keep company information from becoming public knowledge by establishing nondisclosure agreements.
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Regardless of the size of their company, responsible business owners protect their most valuable assets. incorporate.com can help you incorporate or form an LLC online, or by phone, in minutes.
Not sure if incorporating is right for you?
Learn more about forming a corporation or LLC or check out another service.


text:
Loss of final high school season has seniors feeling a wide range of emotions
Edwin Stanton @edwinstantonu2
It's been less than a month since the sports world was turned upside down. The coronavirus pandemic forced the cancellation of all college spring seasons, major pro seasons, including NBA, NHL and MLB as well as PGA, LPGA and USTA tennis events.
Athletics on the high school level were not spared either. Gov. Kay Ivey announced the closing of schools on March 26, effectively ending all athletics for the remainder of the semester.
For college athletes the cancellation had a silver lining. The NCAA granted an extra year of eligibility to 2019 seniors who want it. For high school seniors, however, there is no do-over.
The Tuscaloosa News talked with nine seniors from the Tuscaloosa area to give them a chance to express their feelings on the matter.
They feel sadness, disappointment, a sense of loss and a feeling of not having closure for athletic careers that have been a huge part of their high school lives. They also feel understanding, knowing the health of the country is more important than risking it all to play sports.
Colbi Snider Hillcrest tennis
Snider and the Patriots finished sixth at the state tournament last season and had high hopes of getting back there this season.
It's very upsetting, the fact of closing school and not getting to finish academics and senior year. Plus, on top of it, not knowing that (four) weeks ago was my last tennis match. To know I won't be walking those halls again as a high school student is just devastating.
We had a lot of younger girls on our team. Being an influence on them had a big impact on my life. To have the opportunity to show them and be a leader and a role model. I'm really going to miss wearing the 'H' across my chest. I can always wear an 'H' but it won't mean as much as it did when I was in high school.
Every other year Hillcrest has done well in sectionals and made it to state. We wanted to get to state and keep that going. We wanted to get the younger girls who haven't had that experience an opportunity to go to state.
I'm very competitive and love sports and being outside. My coaches really helped me grow as a person and a player. I'm going to miss my tennis family.
Preston Askew, Hillcrest baseball
Askew is a pitcher for the Patriots, who advanced to the Class 6A semifinals last season and in 2018.
It was very unfortunate, especially when you have been working for four years to get to the moment where you are the leaders of the team. We wanted to make a run of our own, considering what we have done the last few years.
We were looking to keep the legacy going. We were disappointed but we also knew that a lot of people have it worse off, with what's going on (with the coronavirus). We were kind of fortunate that the only thing we are worrying about is not getting to play our final season.
It still hurts. Most of us have been playing baseball for 10 or 12 years or so and it's all led up toward this and it got taken out from beneath us.
It was tough knowing that was the last time I'd ever take the field with my teammates again. Knowing that was most likely the last time I'll ever play baseball. Not knowing the fact it was my last time out there and I've got to make the most if it - it really hurt.
Some of the best memories of my life have come from baseball. Knowing that I'll never get to play alongside them again really hurts. You need to play every game like it's your last. I didn't know it was my last game. It leaves me with a feeling of, what if I had done this, or what if I had done that.
Andrew Richardson, Northridge soccer
Richardson, The Tuscaloosa News' 2019 Spring Sports MVP for soccer, was one of the leaders for the Jaguars for this season's team (5-4-1).
I was in shock. I didn't want to believe it. I was hoping that we would have some sort of miracle, I guess, and continue our season. But it got canceled. I was pretty devastated because I've been playing with my friends since fourth grade. We've played together for 10 years. For it to go away suddenly - it really sucks.
We didn't really get a last game feeling. We just figured there would be more games ahead of us.
Fortunately I can go on to play college soccer (Belhaven University in Jackson Mississippi), but a lot of my friends won't get to.
I loved the atmosphere and feeling of it. It always felt good to go out and play with your teammates, who you have worked all preseason with to get in shape. We are all pretty close and had pretty good chemistry.
My favorite part was everyone having the same goal and working toward that goal. When we score or when we win everyone celebrating and showing emotion. I really enjoyed that.
Ana Caroline Sullivan, Northridge tennis
Sullivan was the lone senior for the Jaguars and was part of last season's Class 6A, No. 3 doubles state championship team with Mary Calloway Rodgers, and she was runner-up at the state tournament in No. 4 singles. Sullivan and the Jaguars won the state title in 2018 and she was the No. 6 singles champ.
When I found out about the schools being closed for the remainder of the year I was devastated because I knew that would mean no longer having a tennis season. That's something I was looking forward to tremendously because I have been on the team for the past four years.
While I've been playing for Northridge we've won a state title and I've won two individual titles. I was hoping to make that three with my doubles partner (Rodgers). To have that cut short is so sad. I've expressed that to the rest of the girls on the team and told them how much I love them and loved being able to grow as a person by being on this team.
This is the last time I'll ever play for a team. The thought of never wearing a Northridge uniform again is sad. The last match I didn't even know it was my last time on the court. Had I known it was going to be the last time we played together it would have been so different.
I've loved getting to lead these girls. They are some of the most amazing girls I've ever gotten to work with. To get to lead them and get to know them and grow closer to them was amazing.
M'Kay Gidley, Hillcrest softball
Gidley has been a member of the varsity team for four years and is a Shelton State signee.
There was a good amount of tears when I heard the news the season was canceled. We were all upset about it.
This Hillcrest High School softball team was a special team. We had great team chemistry and the sky was the limit. Our main goal was make it to state, if not win state. We had the capabilities to make it far in the championships of late April and May.
Our goals were to win regular season area so we could host the area tournament, then go to regionals and make it back to Montgomery in May for the Elite 8. This team was starting to gel as teams do after playing several weeks. We trusted each other and knew we had each other's backs. It was just a great team.
To know how good we were and how great we could have been hurts. Like I said, a lot of tears were shed. We loved representing Hillcrest High School and wearing that script H on our jersey. To realize we will never represent HHS again tears us apart.
There was no closure. It was like our season and our senior year was ripped out from under us. No warning. Just gone. We did win our final game. We beat TCHS 8-4. All seniors got to play in this game. Hillcrest varsity had not beat County High since before we were freshmen— so we ended on a positive.
But still— to not have senior night, to not have the opportunity to play through the championships, and to not take the field one more time has been very hard to come to grips with.
Four of us will continue the game we love in college, but we never will stand together to represent Hillcrest High School again. That hurts.
Gracie Vanderford, Hillcrest softball
Vanderford has been a member of the varsity team for four years and is a Shelton State signee.
At first I was very upset I wouldn't get to play with my teammates again. This team was really close. We had been working so hard since the summer and to have it taken away from us was tough.
No softball team at Hillcrest had ever won state and we wanted to be that first team to do that. Especially this team because we were so close. I really did think that we were going to be able to do it.
We wanted to be able to say that we were the team that ran Tuscaloosa. There are a lot of good teams we compete with. We wanted to be able to say that we run this town and we are the big team in Tuscaloosa.
I'm going to miss the school spirit. We had a lot of support from folks cheering us on in center field (bleachers). It was like one big family. We would go to the baseball games and the baseball players would come to our games. Everyone was there for each other.
I'm trying to get the younger players to realize that you never know when your last game will be and to play like every game is your last.
Going into our last game (with Tuscaloosa County) there was just a weird feeling in the air and I told the team that we need to play this game like it's our last because it very well could be. That was one of the best games we played all year.
Edwin Rainey, Northridge track and field
Rainey won the Class 6A long jump and triple jump state title for the 2020 Indoor season. He was runner-up in the triple jump last season in indoor. He finished third in the triple jump and long jump in last year's Outdoor state meet. He signed to run track for Samford University.
It's always hard when you put in that hard work and can't see it through. I feel Indoor season is a warmup for Outdoor and to have my last one taken away from me really hurt. At least I ended on top.
They extended spring break and I thought, 'Ok that's cool.' Then my mom came into my room and said, 'Did you hear, they canceled spring sports.' I was like, 'Wow, that really sucks.' I thank God I got a chance to compete at all and I signed. If that was not the case I would really be in the dumps right now.
It wasn't the way I wanted it to end at Northridge but that's just the way life is.
I'm going to miss my teammates. They say it's like family but it truly is like a second family. You are all working for the same goals and you are around each other all the time and you go to the meet to compete for them.
I love building relationships with my opponents. I've grown close to them. Track is a different kind of sport. We are all friends here. There are some good rivalries there.
Marion Mann, ACA cross country/track
Mann has been on the varsity for six years and was part of American Christian Academy's 2017 state championship cross country team and the 2017 and 18 Class 3A Outdoor state title teams and the 2017 and 2018 Class 1A-3A Indoor state title teams.
At first I thought this coronavirus thing was going to go away in just like a few weeks but now that it's here and all across the United States, it was just shocking to me that I couldn't finish my senior season.
I've been doing this for six years but it really hits you hard when it's your last year and you can't go (to the state championships) and it's all out of your control.
We had a track practice that Thursday night and we began to hear the news about probably all the track meets being canceled. After that we just stood around and talked about all our memories of being on the team. So we did have a little closure there.
The underclassmen on our team is a really strong group coming up. So I just told them stick through it and get through this. I know they will win a state championship. I know God has a plan for this team and the future.
John Bridges, ACA cross country/track
Bridges has been on the varsity for six years and was part of American Christian Academy's 2017 state championship cross country team and the 2017 and 18 Class 3A Outdoor state title teams and the 2017 and 2018 Class 1A-3A Indoor state title teams.
I trained for a while as if (the cancellation) wasn't going to happen, but then Gov. Ivey shut down the schools. I don't think I was prepared. After a while I understood that it needed to be done and it's for the benefit of everyone else in the world. As long as I keep that in my mind, then no way is my not having my last season comparable to the safety of everyone's health.
It's still sad and I'm upset about it, but I understand the reasoning behind it and I've come to peace with that fact.
I realized this was going to be it with the last practice we ever had. Coach (Charles) Tiller had a workout scheduled but he canceled the workout. That's when I knew this was probably going to be it. We just did a relay type of Olympic thing, just hung out and talked and enjoyed each other one last time.
I told (the underclassmen) that I've enjoyed every single moment I've had with them and I would never trade it for the world. I told them to take this time for themselves and continue to grow as people and just be safe and be smart. But know that it's not over yet.
Most of all I'm going to miss the people we ran against. I'm not the only senior that had their season canceled. But getting out on the track with people you don't know, all of a sudden you feel like you have known them for a while. And having your teammates cheer you on as you are finishing that last lap - it's the best thing ever.
Reach Edwin Stanton at 205-722-0226, edwin.stanton@tuscaloosanews.com or via Twitter, @edwinstantonu2

redpajama_set_name: RedPajamaCommonCrawl


text:
In 2006 Catastrophic Theatre artistic director Jason Nodler created Speeding Motorcycle, a hit musical drawing from cult phenomenon Daniel Johnston's words and music, which received praise from The New York Times, American Theatre and No Depression magazine. It premiered at Infernal Bridegroom Productions in Houston and received a second production at Austin's Zachary Scott Theatre Center.
"One play wasn't enough," according to Nodler. "Daniel is so prolific and his body of work is so remarkable I feel like I could make ten of these things."
Daniel's early years in Austin are the stuff of legend. He arrived knowing no one in town and set about making music in a practice room at The University of Texas, where the play is set.
Where Speeding Motorcycle focused largely on unrequited love, Life is Happy and Sad concerns itself with the creative impulse and the importance of friendship.
Matt Brownlie, well known to local music enthusiasts as the frontman for bands Bring Back the Guns and Groceries, plays young Daniel. A live rock band, Roky Moon and Bolt (Best New Band, Houston Press), doubles in the other roles. Local legend Joe Mathlete (The Mathletes, Speeding Motorcycle) makes a key cameo as well.
Nodler created the new play during a residency at New Hampshire's prestigious MacDowell Colony, made possible by a fellowship from The National Endowment for the Arts. He is working closely with Daniel's best friend David Thornberry (also a character in the play) to create what he refers to as a "docu-dream play."
"Think Beckett's Krapp's Last Tape but with Daniel Johnston songs," Nodler says.
Johnston has received a lot of attention in recent years. He was the focus of an award-winning documentary in 2006 (The Devil and Daniel Johnston) and a featured artist in that year's Whitney Biennial. He is the subject of an upcoming motion picture based on his life and his work is featured in a new iPhone video game. On October 6, he will release his first solo album in seven years.
Johnston famously suffers from an extreme case of manic depression.
"When most people think of Daniel Johnston, the first thing they focus on is his mental health condition," according to Nodler. "I am focused on how his sensitivity makes him similar to the rest of us, not different. His highs and lows are recognizable to us all and his willingness to share this sort of nakedly honest work, in the face of the difficulties he's endured, is an inspiration. This play concerns itself with the deep sadness born of loneliness and isolation and the encouragement he derives from his special friendship with David."
The form could not be more different from Speeding Motorcycle, but the result is the same. "Audiences will laugh, they will be moved and they will leave with Daniel's songs and, more importantly, his inspirational story, stuck in their heads for many months to come," says Nodler.
Speeding Motorcycle sold out two runs in Houston and dozens of people were turned away at the door. Get your tickets now for Life is Happy and Sad by calling the Catastrophic offices at 713-522-2723.
LIFE IS HAPPY AND SAD previews Wednesday, December 2 and plays December 3, 4, 5, 9, 10, 11, 12, 16, 17, 18 and 19 at DiverseWorks ArtSpace, 1117 East Freeway. All performances begin at 8pm. Click for more information on DiverseWorks.
$50 preview admission includes refreshments and a private party with the cast and crew. All other performances are Pay-What-You-Can.


text:
Hurricane Rick is already a Major Cat 4 storm
Hurricane Rick forms in the Pacific
New Orleans seeks storm preparation helps - from CUBA
Mayor's surprise visit to Cuba (WVUE)
New Orleans mayor Ray Nagin has chosen to learn about disaster management from Cuba's Defense Committee. This is a clear case of ideology taking precidence over safety.
Why would the mayor of a mainland city need to seek guidance from an island nation?
If Cuba has effective methods for evacuating people out of harms' way, and I hope they do for their people's sake, then that is totally irrelevant with regards to a city such as New Orleans. An island nation such as Cuba, the Bahamas, Dominican Republic and others have unique evacuation challenges that must be addressed. The authorities on island nations have to find a way to shift their populations from the path of danger when there is no place to go.
People who live near the shore have to be moved inland but heavy rains in the mountains can produce mudslides and flash floods. Frequently the cone of uncertainty is wider that the island itself so how do you get out of the way.
If Ray Nagin wanted to learn effective storm preparation methods that could be applied to his city, then he should go to FLORIDA, not CUBA.
In 2004, Florida was battered by four storms in a row. In 2005 when New Orleans was hit by Katrina, Florida was battered by another three storms including Wilma, the storngest storm on record based on central pressure. Nagin needs to ask Jeb Bush what works.
Another option is for Nagin to visit his neighbor Haley Barbour in Mississippi. Mississippi was successful in evacuating their coast before New Orleans even began their evacuation.
Could it be that Ray Nagin is too proud to talk to his Republican neighbors? Does ideology prevent a Democrat from learning a thing or two from a Republican?
Of course the Mississippi's coastal cities are no where near the size of New Orleans in area or population. Lyda Ann Thomas, the mayor of Galveston knows about protecting a coastal city. The evacuation preceeding Hurricane Rita is frequently criticized due to the overwhelming traffic jam stretching over 100 miles from Houston. The facts are that Galveston Island was empty days before Rita came ashore and Galveston County was evacuated 24 hours before landfall. Additionally, the problems that were discovered were recognized and changes were made as quickly as possible. When Hurricane Ike made a direct hit, the evacuation went smoothly. The great loss of life on Bolivar Peninsula, as in Katrina, were people who did not leave.
Cooperation between Mayor Thomas, Galveston Emergency Management, Houston mayor Bill White and Governor Perry showed how competent officials can work together. Oh, by the way, Mayors Thomas and White are Democrats while Perry is a Republican. Party and ideology can be set aside when safety is set as the priority.
The five states along the Gulf Coast have plenty of resources Mayor Nagin could have chosen to work with to improve the safety of New Orleans.
Trackback to: Dumb Ox Daily News, Woman Honor Thyself


As I've said before, there are certain films on my list that are not out and out Great War Films. They may not have one single frame of war stuff in them but for some reason or another they count as Great War Films, at least in my head anyway. My film today is one of these films. It's set in the years after the war and deals with what happened to some of the heroes once the war machines wheels stopped turning.
The Great Waldo Pepper (1975) is the story of the aviators who survived the war in the air and returned to civilian life needing to scratch a living with the only skills they had. It's the story of the Barnstorming era. In the years after the Great War there was a glut of cheap aircraft sold off by the peacetime government and a raft of experienced pilots in need of a living. Add in a lack of regulation to this triumvirate and you have a situation where national heroes of old, the legends of the skies were reduced to performing stunts for paying crowds in flying circuses that toured the US.
As the years passed the crowds and pay packets dwindled, as did the barnstormers as one by one they spannered themselves in flaming wrecks or when their livers exploded. By the time the Government intervened with the introduction of civil aviation regulations the few remaining battered and bruised flyers were forced to move on to other ventures or take jobs within the newly regulated industry.
By all accounts this was a bit of a vanity/pet project for Director George Roy Hill (Butch Cassidy and the Sundance Kid, The Sting, Slapshot). As a pilot himself he was infatuated with the flying heroes of the Great War, the early pioneers of flight and their plight as the years passed. Robert Redford stars as the eponymous Waldo Pepper, a nearly man with natural flying talent. He never made it into battle, but still regales the crowds with made up tales of how he fought alongside and against some of the greats. Susan Sarandon co-stars in a bitsy role that never really gains momentum and Bo Svenson rounds out the cast with possibly the stand out performance.
The story uses made up names and composite characters but Director Hill makes it clear from the outset that this is a story very much based in the real world. The opening credits feature images of some of the eras leading pilots including Ormer Locklear and German Ernst Udet including their dates of birth and death. Locklear died flying a Hollywood stunt in 1920 and Udet committed suicide in 1941 after a falling out with the Nazi party. Interesting fact about Locklear is that his girlfriend of the time, young starlet Viola Dana witnessed his fiery crash in a late night film shoot. She was invited to the premier of the Great Waldo Pepper by Director Hill in a gesture that shows his earnestness and his passion for the era.
As we should expect, there are a few deaths during the course of the film. Firstly Sarandon drops off a wing into the never never after freezing during a wing walk and then just a few screen minutes later Waldo's brother crashes his plane and burns to death. Both of these deaths come out of the blue, yes the stunts they are attempting at the time are dangerous, but this is Hollywood and I was expecting happy endings. The deaths come out of the blue and are jarring, just like in real life, I suppose. Everything is going okay and then bang out of nowhere everything changes. These moments cause the story to jump in unexpected directions, again just like life.
There are plenty of aerial stunts throughout the film including an impressive plane to plane transfer by a wing walker. A plane barrels, feet from the ground down a high street and a stuntman transfers from a car to a rope ladder hung from a plane. Good stuff throughout and a hark back to the era. You don't see stunts like this on film these days. Also worthy of mention are the planes. What amazing things they were and we see glimpses of their fragility and their wood, string and hinge controls.
The final scenes attempt to pull the story to some kind of a conclusion and is the only part of the film that I feel is a bit laboured. Waldo moves to Hollywood and meets his hero Ernst Kessler (a thinly veiled Ernst Udet). They fly together to film a scene for a film. 2 other pilots have already died during the production. Once in the air they quickly abandon the scene and set off on a dog fight. Waldo keen to test himself against his hero and Kessler reminiscing and wallowing in a chivalric past. They are young again and reliving the thrill of the fight. I shan't ruin the ending for you but I will say I felt a bit let down by it. I was expecting more of a crescendo. Again, probably just like real life.


text:
Tattooed Jesus Billboards Flood Lubbock
By Sonia Smith
Lubbock, a city where churches outnumber tattoo parlors 25 to one, has some new billboards that combine the love of religion and ink. The billboards feature a hirsute Christ with words including "outcast," "jealous," and "faithless" tattooed on his body and outstretched arms.
An anonymous group has erected 59 of the billboards in a fifty-to-sixty mile radius of Lubbock, according to a spokesperson for the group, speaking to Texas Monthly on condition of anonymity. The billboards went up on September 30 and are set to stay up for four weeks. The billboards direct people to visit jesustattoo.org, where they can watch videos and learn more about Christ.
"This idea was a creation of a group of people that who were humbled by the love of Jesus and they just wanted to share it with everyone, both churched and unchurched, in a creative way," the spokesperson explained.
The group is being coy about its identity. On their website they refer to themselves as "a small group of people humbled by the love of Jesus. We are not a church. We are not selling anything. We encourage you to tell as many people as possible. That's it." The domain name was registered through a proxy service, making it harder to trace, but this appears to be a well-financed marketing campign designed to go viral. We wouldn't be surprised if there were a tech-savvy megachurch somewhere behind the effort.
Of course, the idea of renting out billboards to get your message out is older than the Interstate Highway System. But this project seems more polished than most, which can be attributed to the fact that a Lubbock-based advertising agency volunteered to handle the creative aspects of the campaign.
Reaction in Lubbock's religious community has been mixed. Davis Wilson, a pastor at Lubbock's Southcrest Baptist Church, told ABC News that he liked the campaign. "I thought that it was cleverly done because, basically, it's a visual of Jesus taking the sins of people and covering them and taking them from an outcast or something and giving them a new start, which is what the gospel is about," he said. "I looked it up, and I said … this is perfect because it just draws people in here." The pentecostals, however, are unimpressed: Pastor Mark Herridge of Pentecostal Lighthouse Church in Livingston opined to the Christian News Network that he found the project "Utterly detestable. …Tattoos have always been associated with paganism. To lift them up as an evangelistic tool is to completely miss the point of why Jesus saved us," he said.
The group's spokesperson said that most of the feedback they've heard about the project is positive. But that's not the point. "We welcome any kind of feedback," she said. "Anything that gets people talking about the greatest love story of all time."


text:
using System;
using System.IO;
using System.Text;

using BizHawk.Common;

namespace BizHawk.Client.Common
{
	public partial class BkmMovie
	{
		private int _preloadFramecount; // Not a a reliable number, used for preloading (when no log has yet been loaded), this is only for quick stat compilation for dialogs such as play movie

		public void SaveAs(string path)
		{
			Filename = path;
			if (!Loaded)
			{
				return;
			}

			var directoryInfo = new FileInfo(Filename).Directory;
			if (directoryInfo != null)
			{
				Directory.CreateDirectory(directoryInfo.FullName);
			}

			Write(Filename);
		}

		public void Save()
		{
			if (!Loaded || string.IsNullOrWhiteSpace(Filename))
			{
				return;
			}

			SaveAs(Filename);
			_changes = false;
		}

		public void SaveBackup()
		{
			if (!Loaded || string.IsNullOrWhiteSpace(Filename))
			{
				return;
			}

			var backupName = Filename;
			backupName = backupName.Insert(Filename.LastIndexOf("."), $".{DateTime.Now:yyyy-MM-dd HH.mm.ss}");
			backupName = Path.Combine(Global.Config.PathEntries["Global", "Movie backups"].Path, Path.GetFileName(backupName));

			var directoryInfo = new FileInfo(backupName).Directory;
			if (directoryInfo != null)
			{
				Directory.CreateDirectory(directoryInfo.FullName);
			}

			Write(backupName);
		}

		public bool Load(bool preload)
		{
			var file = new FileInfo(Filename);

			if (file.Exists == false)
			{
				Loaded = false;
				return false;
			}

			Header.Clear();
			_log.Clear();

			using (var sr = file.OpenText())
			{
				string line;

				while ((line = sr.ReadLine()) != null)
				{
					if (line == "")
					{
						continue;
					}

					if (line.Contains("LoopOffset"))
					{
						try
						{
							_loopOffset = int.Parse(line.Split(new[] { ' ' }, 2)[1]);
						}
						catch (Exception)
						{
							continue;
						}
					}
					else if (Header.ParseLineFromFile(line))
					{
						continue;
					}
					else if (line.StartsWith("|"))
					{
						_log.Add(line);
					}
					else
					{
						Header.Comments.Add(line);
					}
				}
			}

			if (Header.SavestateBinaryBase64Blob != null)
			{
				BinarySavestate = Convert.FromBase64String(Header.SavestateBinaryBase64Blob);
			}

			Loaded = true;
			_changes = false;
			return true;
		}

		/// <summary>
		/// Load Header information only for displaying file information in dialogs such as play movie
		/// TODO - consider not loading the SavestateBinaryBase64Blob key?
		/// </summary>
		public bool PreLoadHeaderAndLength(HawkFile hawkFile)
		{
			Loaded = false;
			var file = new FileInfo(hawkFile.CanonicalFullPath);

			if (file.Exists == false)
			{
				return false;
			}

			Header.Clear();
			_log.Clear();

			var origStreamPosn = hawkFile.GetStream().Position;
			hawkFile.GetStream().Position = 0; // Reset to start

			// No using block because we're sharing the stream and need to give it back undisposed.
			var sr = new StreamReader(hawkFile.GetStream());

			for (;;)
			{
				// read to first space (key/value delimeter), or pipe, or EOF
				int first = sr.Read();

				if (first == -1)
				{
					break;
				} // EOF

				if (first == '|') // pipe: begin input log
				{
					// NOTE - this code is a bit convoluted due to its predating the basic outline of the parser which was upgraded in may 2014
					var line = '|' + sr.ReadLine();

					// how many bytes are left, total?
					long remain = sr.BaseStream.Length - sr.BaseStream.Position;

					// try to find out whether we use \r\n or \n
					// but only look for 1K characters.
					bool usesR = false;
					for (int i = 0; i < 1024; i++)
					{
						int c = sr.Read();
						if (c == -1)
						{
							break;
						}

						if (c == '\r')
						{
							usesR = true;
							break;
						}

						if (c == '\n')
						{
							break;
						}
					}

					int lineLen = line.Length + 1; // account for \n
					if (usesR)
					{
						lineLen++; // account for \r
					}

					_preloadFramecount = (int)(remain / lineLen); // length is remaining bytes / length per line
					_preloadFramecount++; // account for the current line
					break;
				}
				else
				{
					// a header line. finish reading key token, to make sure it isn't one of the FORBIDDEN keys
					var sbLine = new StringBuilder();
					sbLine.Append((char)first);
					for (;;)
					{
						int c = sr.Read();
						if (c == -1 || c == '\n' || c == ' ')
						{
							break;
						}

						sbLine.Append((char)c);
					}

					var line = sbLine.ToString();

					// ignore these suckers, theyre way too big for preloading. seriously, we will get out of memory errors.
					var skip = line == HeaderKeys.SAVESTATEBINARYBASE64BLOB;

					if (skip)
					{
						// skip remainder of the line
						sr.DiscardBufferedData();
						var stream = sr.BaseStream;
						for (;;)
						{
							int c = stream.ReadByte();
							if (c == -1 || c == '\n')
							{
								break;
							}
						}

						// proceed to next line
						continue;
					}

					var remainder = sr.ReadLine();
					sbLine.Append(' ');
					sbLine.Append(remainder);
					line = sbLine.ToString();

					if (string.IsNullOrWhiteSpace(line) || Header.ParseLineFromFile(line))
					{
						continue;
					}

					Header.Comments.Add(line);
				}
			}

			hawkFile.GetStream().Position = origStreamPosn;

			return true;
		}

		private void Write(string fn)
		{
			Header.SavestateBinaryBase64Blob = BinarySavestate != null
				? Convert.ToBase64String(BinarySavestate)
				: null;

			using (var fs = new FileStream(fn, FileMode.Create, FileAccess.Write, FileShare.Read))
			{
				using (var sw = new StreamWriter(fs))
				{
					sw.Write(Header.ToString());

					// TODO: clean this up
					if (_loopOffset.HasValue)
					{
						sw.WriteLine($"LoopOffset {_loopOffset}");
					}

					foreach (var input in _log)
					{
						sw.WriteLine(input);
					}
				}
			}

			_changes = false;
		}
	}
}

redpajama_set_name: RedPajamaGithub


Chris Paul is looking like Chris Paul again as Houston soars up the Western Conference standings. Right now, that's all that matters to the Rockets.
Carmelo Anthony's brief stop with the Houston Rockets is over, league sources told ESPN.
Carmelo Anthony hasn't been the answer for the Rockets so far this season. In news that no one could have possibly seen coming, it appears Carmelo Anthony and the Houston Rockets might be re-evaluating their relationship after a slow start to the season for both parties. With Anthony out for the Rockets' game against the San Antonio Spurs on Saturday, the team and player are engaged in ongoing talks on how they "might still be able to proceed together for the rest of the season," according to ESPN's Adrian Wojnarowski.
Restricted free agent Clint Capela is returning to the Rockets on a five-year, $ 90 million deal, sources told ESPN's Adrian Wojnarowski.
A huge Game 7 could come down to a lot of little details. Zach Lowe breaks down what to look for in Monday's decisive game.
Will Harden's Rockets beat Warriors without Chris Paul?
With Chris Paul out, the Rockets will have to rely even more on James Harden. But will it be enough?
James Harden scored 20 first-half points and Houston is in complete control against Utah in the second half.
James Harden agreed with coach Mike D'Antoni's assessment that Rockets are out of rhythm. Said Harden: "It's been a long time" since the team was really clicking.
Gerald Green, who was ejected Sunday for shoving the Timberwolves' Gorgui Dieng while coming to the defense of Chris Paul, was fined $ 25,000. Paul said Sunday that he'd pay any fine as a result of the incident.
Scouting Zhou Qi: Can Rockets rookie contribute to contender?
Rogue One: A Star Wars Story has scored the second-best December opening in history and one of the top debuts in history with $ 155m (£124m) in estimated ticket sales over the weekend.
Rockets and Revolution offers a multifaceted study of the race toward space in the first half of the twentieth century, examining how the Russian, European, and American pioneers competed against one another in the early years to acquire the fundamentals of rocket science, engineer simple rockets, and ultimately prepare the path for human spaceflight. Between 1903 and 1953, Russia matured in radical and dramatic ways as the tensions and expectations of the Russian revolution drew it both westward and spaceward. European and American industrial capacities became the models to imitate and to surpass. The burden was always on Soviet Russia to catch up-enough to achieve a number of remarkable "firsts" in these years, from the first national rocket society to the first comprehensive surveys of spaceflight. Russia rose to the challenges of its Western rivals time and again, transcending the arenas of science and technology and adapting rocket science to popular culture, science fiction, political ideology, and military programs. While that race seemed well on its way to achieving the goal of space travel and exploring life on other planets, during the second half of the twentieth century these scientific advances turned back on humankind with the development of the intercontinental ballistic missile and the coming of the Cold War.
Houston Rockets general manager Daryl Morey said Sunday night he thinks his team can win the NBA title.


Sooke is located a few miles west of Victoria on the southern tip of Vancouver Island. It is actually positioned south of the 49th Parallel, the common border between Canada and the United Sates of America.
The future home of the Residences on Sooke Harbour, is located at 1820 Maple Avenue, Sooke, BC.


Do you want to reduce stress and gain vitality with pure essentials oils?
Join me on the beach as we walk through the top essential oils that can help with physical and emotional healing.
This is a class you can't miss! Enjoy life with MORE ENERGY, LESS STRESS AND MORE VITALITY with an oil-infused lifestyle. Get your personal reading with an iTovi scan.
iTovi is a pocket-sized scanner that combines temperature, pressure sensor, and Bluetooth technologies to generate personalized wellness evaluations. The reading is a non-diagnostic device that shows you which essential oils and supplements your body has the strongest response to at the time of the scan. Using the recommended products displayed on your scan report may be beneficial for your overall wellness.
Sand, Air, Sun and Water of the beach, alone, holds proven medical benefits that help with relaxing the mind and body. The beach air is packed with healthy negative ions that boost our ability to absorb oxygen. These negative ions balance the serotonin level, a body chemical that affects mood and stress in our body, making us more relaxed, yet alert and energized. Imagine the possibilities when you are awakened and empowered with essential oils!
If you want to improve your wellness with a personalized reading for managing your BODY, MIND AND SPIRIT; this is your opportunity to create a wellness protocol to reduce stress and balance your body.
You will have the opportunity to try recommended oils and learn about the emotional and physical benefits of integrating oils into your daily routines. Join me on fb for wellness coaching on detoxing chemicals from food personal-care products.
Folow and Subscribe to my channel. Let me be you light. Return to nature to eturn to your natural self!


TOEFL stands for "TEST OF ENGLISH AS A FOREIGN LANGUAGE". TOEFL is a test of English language.
The TOEFL iBT™ test, administered in an internet-based format, contains 100 percent academic content created by working with institutions to identify the English-language demands faced by international students.
It simulates actual tasks from classrooms — from comprehending a lecture to participating in discussions and extracurricular activities.
Our researchers recorded examples of classroom teaching, study groups and student interactions at several institutions. They collected textbooks and packets from students' courses and compiled their findings into a language database called a "corpus." The language used in the TOEFL iBT test closely reflects what is used in everyday academic settings.
This ensures your applicants are equipped with the skills students need in a higher education classroom. Students who score well on the TOEFL iBT test are prepared for success at your institution and to go further in their careers.
The TOEFL iBT test uses integrated tasks to measure all four skills students need to communicate: reading, listening, speaking and writing. Students may read a passage, listen to a lecture, assimilate what they have learned and then speak or write just as they do in a classroom.
How is the TOEFL test marked?
TOEFL is scored on a 900 score, with each band corresponding to a specified competence in English.
IELTS /TOEFL is recognized in 121 countries by more than 6000 institutions, government departments, professional associations and other organizations. IELTS is internationally owned and globally recognized assessment of English language proficiency.
When will I receive my IELTS/TOEFL test results?
You will receive your IELTS/TOEFLTest Report Form in the mail 13 days after your test date. Some test centers also provide SMS alerts and an Online Results Service. Keep your IELTS Test Report Form in a secure place as you only receive one copy.
Does Rezina Training & Overseas Education Consultants provide IELTS/TOEFL coaching?