text:
\section{Introduction}

The purpose of this paper is to study the \emph{Littlewood-Offord} and \emph{inverse Littlewood-Offord} problems regarding concentration of random walks in torsion-free abelian groups.  We recall some notation from \cite{TVcond}.

\begin{definition}[Concentration probabilities]
Let $G = (G,+)$ be an additive group (e.g. the integers $\Z$, the complex numbers $\C$, or a vector space $\R^m$).
Let $\bv = (v_1,\ldots,v_n)$ be a multiset of $n$ elements of $G$ (allowing
repetitions). For any $0 \leq \mu \leq 1$, we define the \emph{lazy random walk} $S^{\mu}(\bv)$ with steps $\bv$ and density $\mu$ to be the $G$-valued random variable
$$S^{\mu}(\bv)  := v_1 \eta^{\mu}_1 + \dots + v_n \eta^{\mu}_n $$
\noindent where the
$\eta^{\mu}_i$'s  are iid copies of the (lazy coin flip) random
variable $\eta^{\mu}$
  which equals $0$ with probability $1-\mu$ and
$\pm 1$ with probability $\mu/2$ each.  We define the \emph{concentration probability}
$\P_\mu(\bv)$ to be the quantity
\begin{equation} \label{equa:defofPmu}  \P_\mu(\bv) :=
\max_{a \in G} \P( S^{\mu} (\bv) = a).\end{equation}
\end{definition}

\begin{remark} We are interested in the regime when $0 < \mu \leq 1$ is fixed and $n$ is large.  The most interesting case is perhaps when $\mu=1$. In this case $\eta$ is the Bernoulli random variable (fair coin flip), and $\P_1(\bv)$ is the maximum multiplicity among the $2^n$ signed sums $\pm v_1 \pm \ldots \pm v_n$, divided by $2^n$.
Such probabilities appear in many situations in combinatorics and the theory of random structures, for instance in understanding the singularity probability of discrete random matrices (see e.g. \cite{KKS}, \cite{TVdet}, \cite{TVsing}, \cite{RV}, \cite{TVcir}).
\end{remark}

We will assume throughout this paper that $G$ is \emph{torsion-free}, thus $nx \neq 0$ whenever $x \in G$ is non-zero and $n$ is a non-zero integer. In this case we can usually reduce to the model case $G=\Z$ by means of Freiman isomorphisms (see \cite[Lemma 5.25]{TVbook}).

Broadly speaking, we expect $\P_\mu(\bv)$ to be large if and only if $\bv$ has significant additive structure.  To explore this phenomenon, we ask the following two general (and closely related) questions:

\begin{itemize}
\item (Forward Littlewood-Offord problem) Given additive structural hypotheses on $v_1,\ldots,v_n$, what bounds can one give for $\P_\mu(\bv)$?
\item (Inverse Littlewood-Offord problem) Given bounds on $\P_\mu(\bv)$, what can one say about the additive structure of the $v_1,\ldots,v_n$?
\end{itemize}

Let us now recall some previous results on these problems; further discussion may be found in \cite[Chapter 5]{TVbook}.  For simplicity we take $\mu=1$.  With no assumptions on $\bv = (v_1,\ldots,v_n)$, we easily obtain the inequalities
$$ 2^{-n} \leq \P_1(\bv) \leq 1$$
with the upper bound being attained precisely when all the $v_i$ are zero, and the lower bound attained precisely when the the $v_i$ are \emph{dissociated} (which means that all the $2^n$ partial sums $\sum_{i \in A} v_i$ with $A \subset \{1,\ldots,n\}$ are distinct).  These two cases represent extreme additive structure and extreme lack of additive structure respectively.

Throughout this paper we adopt the following asymptotic notation:

\begin{definition}[Asymptotic notation]  The asymptotic notation $X = O(Y)$, $X \ll Y$, $Y = \Omega(X)$, or $Y \gg X$ denotes the bound $X \leq CY$ for all $n \geq C$ and some absolute constant $C$; we also use $X = \Theta(Y)$ for $X \ll Y \ll X$.  Subscripting such as $O_d(Y)$ means that the implied constants $C$ in the asymptotic notation are allowed to depend on $d$.
\end{definition}

Littlewood and Offord \cite{LO} and then Erd\H{o}s \cite{Erd1} were able to improve the upper bound assuming that some of the $v_i$ were non-zero.  In particular, from the results in \cite{Erd1} one obtains the inequality
\begin{equation}\label{erdos-lo}
\P_1(\bv) \ll n^{-1/2}
\end{equation}
if all of the $v_i$ are non-zero (Littlewood and Offord obtained the slightly weaker bound $\P_1(\bv) \ll n^{-1/2} \log n$).   This bound is sharp: if $v_1=\ldots=v_n$, one easily verifies that $\P_1(\bv) \gg n^{-1/2}$ (and in fact this example gives the precise maximum value of $\P_1(\bv)$.

The above result is phrased as a forward Littlewood-Offord result, but can be easily rephrased as an inverse theorem:

\begin{theorem}[Erd\H{o}s' inverse Littlewood-Offord theorem]\label{eilo}  Let $\bv = (v_1,\ldots,v_n)$ be an $n$-tuple in a torsion-free additive group $G$.  Suppose that $\P_1(\bv) \gg k^{-1/2}$ for some $k \geq 1$.  Then all but $O(k)$ of the $v_1,\ldots,v_n$ are zero.
\end{theorem}

One can improve the upper bounds further by excluding the above counter-example.   Indeed, from the work of Erd\H{o}s and Moser \cite{erdmos} and then S\'ark\"ozy and Szemer\'edi \cite{SS}, the bound
\begin{equation}\label{ss-lo}
\P_1(\bv) \ll n^{-3/2}
\end{equation}
was established if all the $v_i$ were distinct (the earlier paper \cite{erdmos} establishes the slightly weaker bound $\P_1(\bv) \ll n^{-3/2} \log n$).
Again, this result is sharp, since if one takes $v_1,\ldots,v_n$ to be a proper arithmetic progression, one easily verifies that $\P_1(\bv) \gg n^{-3/2}$.  Later, Stanley \cite{Stan}, using algebraic methods, gave a very explicit bound for the optimal value of $\P_1(\bv)$.

The higher dimensional version of the problem, in which $G$ is a vector space $\R^m$, has also attracted attention.
Without the assumption that the $v_i$'s are different, the best
result was obtained by Frankl and F\"uredi in \cite{FF}, following earlier results by Katona \cite{Kat},
Kleitman \cite{Kle}, Griggs, Lagarias, Odlyzko and Shearer
\cite{GLOS} and many others. However, the techniques used in these
papers did not seem strong enough to recover \eqref{ss-lo}.  On the other hand, Hal\'asz \cite{Hal}, using harmonic analysis methods, managed to generalise \eqref{ss-lo}, proving even stronger bounds upon forbidding more additive correlations among the $v_i$'s.

\begin{theorem}[Halasz inequality]\label{theorem:halasz}\cite{Hal}, \cite[Exercise 7.2.8]{TVbook} Let $\bv = (v_1,\ldots,v_n)$ be an $n$-tuple in a torsion-free additive group $G$.  Let $l \geq 1$ be an integer and let $0 < \mu \leq 1$.
Let $R_l$ be the number of solutions of the equation
$$\epsilon_1 v_{i_1} + \dots  +\epsilon_{2l} v_{i_{2l}} =0 $$
\noindent where $\epsilon_i \in \{-1,1\}$ and $i_1, \dots, i_{2l}$ are
(not necessarily different) elements of $\{1,2, \dots, n \}$. Then
$$\P_{\mu} (\bv) \ll_{l,\mu} n^{-2l-1/2} R_l. $$
\end{theorem}

It is easy to see that the $l=1$ case of Theorem \ref{theorem:halasz} implies the bound \eqref{ss-lo}.

\subsection{Main results}

Theorem \ref{theorem:halasz} states, roughly speaking, that if $\P_\mu(\bv)$ is large, then there is a large amount of additive structure (in the form of short additive relations) between the $v_i$.  Now we consider a slightly different type of additive structure, namely containment in a (symmetric) \emph{generalized arithmetic progression} (or GAP); we recall this concept in Section \ref{gap-appendix}.  It is not hard to show that if all the $v_i$ are contained in a GAP of bounded rank and controlled size, then the concentration probability $\P_\mu(\bv)$ is large.  More precisely, one has

\begin{proposition}[Forward Littlewood-Offord theorem]\label{filo}  Let $Q$ be a symmetric GAP in an additive group $G$ with rank $d$, and let $\bv = (v_1,\ldots,v_n)$ be such that $v_1,\ldots,v_n \in Q$.  Let $0 < \mu \leq 1$.  Then we have
$$ \P_\mu(\bv) \gg_d |Q_{\sqrt{\mu n}}|^{-1} \gg_d (1 + \mu n)^{-d/2} |Q|^{-1},$$
where the dilate $Q_t$ of $Q$ is defined in Section  \ref{gap-appendix}.
\end{proposition}

We prove this easy result in Section  \ref{gap-appendix}; it reflects the intuition that a lazy random walk with steps in $Q$ should mostly take values in the dilate $Q_{O(\sqrt{\mu n})}$.  Note that this result incorporates the examples used to demonstrate that \eqref{erdos-lo} and \eqref{ss-lo} are sharp.  See also \cite[Theorem 6.6]{TVcir} for a more complicated result in a similar spirit.

We now turn to the question of whether a converse to Proposition \ref{filo} exists.  In \cite{TVcond}, the authors showed

\begin{theorem}[Weak Inverse Theorem]\label{theorem:weak} Let $A, \eps > 0$ and $0 < \mu \leq 1$, and let $\bv = (v_1,\ldots,v_n)$ be an $n$-tuple in a torsion-free additive group $G$ be such that
$$\P_{\mu} (v) \ge n^{-A}. $$
Then there exists a proper symmetric GAP $Q$ of rank $d$ for some $d=O_{A,\eps}(1)$, of volume $O_{A,\mu,\eps}(n^B)$ for some $B = O_{A,\eps}(1)$, which contains all but $O_{A,\mu,\eps}(n^{1-\eps})$ elements of $\bv$ (counting multiplicity).
\end{theorem}

The reason we call Theorem \ref{theorem:weak} a \emph{weak} inverse theorem because the dependence of $B$ on $A$ is not optimal ($B$ is roughly $2A^2$).  The first main result of this paper is to obtain a sharper converse to Proposition \ref{filo}, in which $B$ is taken to be $A-\frac{d}{2}+\eps$:

\begin{theorem}[Strong Inverse Theorem] \label{theorem:strong}
Let $A, \eps > 0$, and let $\bv = (v_1,\ldots,v_n)$ be an $n$-tuple in a torsion-free additive group $G$ be such that
\begin{equation}\label{pmuv}
\P_{\mu} (v) \ge n^{-A}.
\end{equation}
Then there exists a proper symmetric GAP $Q$ of rank $d \leq 2A$ of volume $O_{A, \mu, \eps}(n^{A - \frac{d}{2} + O_A(\eps)})$, which contains all but $O_{A,\mu,\eps}(n^{1-\eps} )$ elements of $\bv$ (counting multiplicity).
\end{theorem}

Comparing this with Proposition \ref{filo} we see that except for epsilons, the exponent $A-\frac{d}{2}+O_A(\eps)$ here cannot be improved.

Theorem \ref{theorem:strong} will be deduced as the special case of the following stronger result.

\begin{theorem}[General Strong Inverse Theorem] \label{theorem:stronggeneral}
Let $d \geq 1$ be an integer  and let $0 < \eps, \mu < 1$ be
constants.  Then there is a constant $C_{0}=C_{0} (d, \eps, \mu)$ such
that the following holds for all sufficiently large $n$ and $k$
with $1 \leq k < \sqrt n$. Suppose that $\bv= (v_{1}, \dots, v_{n})$ is an $n$-tuple in a torsion-free additive group $G$ that satisfies
\begin{equation}\label{pi2}
\P_{\mu}(\bv) \ge C_{0}k^{-d}.
\end{equation}
Then there exists a proper symmetric GAP $Q$ of rank at most $d-1$ and volume
\begin{equation}\label{qde}
\operatorname{vol}(Q)  \le    \P_{\mu}(\bv)^{-1} k^{\eps}
\end{equation}
such that $Q_{1/k}$ contains all but at most $O_{d, \mu, \eps} (k^2
\log k)$ of the $v_1,\ldots,v_n$.  Furthermore, there is a positive integer $C=
C(d, \mu, \eps)$ such that the steps of $Q$ lie in $\{ v_1/C, \ldots, v_n/C \}$.
\end{theorem}

Let us see how this theorem implies Theorem \ref{theorem:strong}.

\begin{proof}[Proof of Theorem \ref{theorem:strong} assuming Theorem \ref{theorem:stronggeneral}]  Let $A, \mu, \eps, \bv, n, G$ be as in Theorem \ref{theorem:strong}.  By shrinking $\eps$ if necessary we may assume that $\eps$ is small depending on $A$.  We may assume that $n$ is large depending on $A,\mu,\eps$ as the claim is trivial otherwise. Let $d$ be the first integer larger than $2A$, and let $C_0$ be as in Theorem \ref{theorem:stronggeneral}.   For $\eps$ small and $n$ large, we see from \eqref{pmuv} that \eqref{pi2} holds for $k := n^{1/2 - \eps}$.  By Theorem \ref{theorem:stronggeneral}, we obtain a proper symmetric GAP $Q$ of rank $r$ at most $d-1$ and volume $O( n^{A+\eps} )$ such that $Q_{1/k}$ contains all but $O( n^{1-\eps} )$ of the $v_1,\ldots,v_n$.  Observe that any dimension of $Q$ that is less than $k$ does not contribute anything to $Q_{1/k}$, so by deleting these steps (and reducing the rank $r$ of $Q$) we may assume that all dimensions of $Q$ are at least as large as $k$.  Then $Q_{1/k}$ is a proper symmetric GAP of rank at most $2A$ and volume $O_{A,\mu,\eps}( k^{-r} |Q| ) = O_{A,\mu,\eps}( n^{A - r/2 + O_A(\eps)} )$, and the claim follows.
\end{proof}

\subsection{Applications}

We now give some applications of Theorem \ref{theorem:strong} and Theorem \ref{theorem:stronggeneral}.  We first observe that these theorems can recover the classical bounds \eqref{erdos-lo}, \eqref{ss-lo} except for epsilon losses:

\begin{proposition}\label{class}  Let $\bv = (v_1,\ldots,v_n)$ be an $n$-tuple in a torsion-free additive group $G$, and let $\eps > 0$ and $0 < \mu \leq 1$.
\begin{itemize}
\item[(i)] If all the $v_i$ are non-zero, then $\P_\mu(\bv) \ll_{\mu,\eps} n^{-1/2+\eps}$.
\item[(ii)] If all the $v_i$ are distinct, then $\P_\mu(\bv) \ll_{\mu,\eps} n^{-3/2+\eps}$.
\end{itemize}
\end{proposition}

\begin{proof} We may assume that $n$ is large compared to $\mu,\eps$, as the claim is trivial otherwise.

We first prove (i).  Suppose for contradiction that $\P_\mu(\bv) \geq n^{-1/2+\eps}$.  Applying Theorem \ref{theorem:strong} with $A := 1/2-\eps$ we see that there exists a symmetric GAP $Q$ of rank at most $1-2\eps$ which contains all but $O_{\mu,\eps}(n^{1-\eps})$ of the $v_1,\ldots,v_n$.  But rank has to be an integer, thus $Q$ has rank zero and is therefore just $\{0\}$.  Thus at least one of the $v_i$ is zero, a contradiction.

Now we prove (ii).  Suppose for contradiction that $\P_\mu(\bv) \geq n^{-3/2+\eps}$.  Applying Theorem \ref{theorem:strong} with $A := 3/2-\eps$ (and $\eps$ replaced by a smaller quantity $\eps'$ depending only on $\eps$) we see that there exists a symmetric GAP $Q$ of rank $d$ at most $3-2\eps$ and volume $O_{\mu,\eps}( n^{3/2 - d/2 - \eps'} )$ which contains all but $O_{\mu,\eps}(n^{1-\eps'})$ of the $v_1,\ldots,v_n$.  Since the $v_i$ are all distinct, this forces $|Q| \gg n$, which forces $d=0$, which forces more than one of the $v_i$ to be zero, a contradiction.
\end{proof}

In a similar spirit, we obtain the following variant of Theorem \ref{theorem:halasz}, which essentially asserts that equality in Theorem \ref{theorem:halasz} is only attained when the $v_i$ lie in a symmetric arithmetic progression (i.e. a symmetric rank $1$ GAP):

\begin{proposition} \label{theorem:newhalasz}  Let $n,\bv,G,\mu,l,R_l$ be as in Theorem \ref{theorem:halasz}, and let $0 < \delta, \eps < 1/2$.  Then
one of the following statements hold:
\begin{itemize}
\item $\P_{\mu} (\bv) \ll_{l, \mu, \eps, \delta} n^{-2l-1/2-\delta} R_l$.
\item All but at most $n^{1-\eps}$ of the $v_i$ lie in an symmetric arithmetic
progression of length at most $n^{2l+\delta +\eps} R_l^{-1}$.
\end{itemize}
\end{proposition}

Note that by combining this proposition with Proposition \ref{filo} and taking $\delta=\eps$ we obtain Theorem \ref{theorem:halasz} up to epsilon losses.

\begin{proof}  By shrinking $\eps$ if necessary, we may assume $\eps$ is small depending on $l, \delta$.  We may assume that $n$ is large depending on $l,\mu,\eps,\delta$, since the claim is trivial otherwise.  Finally, we may assume that
$$ \P_{\mu} (\bv) \geq n^{-2l-1/2-\delta} R_l$$
since we are clearly done otherwise.

Applying Theorem \ref{theorem:stronggeneral} with $k := n^{1/2-\eps}$ and $d = O_l(1)$ we obtain a proper symmetric GAP $Q$ of rank $r = O_l(1)$ and volume
$$ \operatorname{vol}(Q) \ll_{\mu,l,\delta,\eps} n^{2l + 1/2 + \delta + \eps} R_l^{-1} $$
such that $Q_{1/k}$ contains all but at most $n^{1-\eps}$ of the $v_i$.   Arguing as in the proof of Theorem \ref{theorem:strong}, we may assume that all dimensions of $Q$ are at least $k$.

If $r \leq 1$ then we are done,  as $Q_{1/k}$ is an arithmetic progression having the right length (one can adjust the constant $\eps$). Now
 assume for contradiction that $r \geq 2$.  Then (if $\eps$ is small enough) we conclude
$$ |Q_{1/k}| \ll_l k^{-2} \operatorname{vol}(Q) \ll_l n^{2l - \eps} R_l^{-1}.$$
By relabeling we may assume that the $v_1,\ldots,v_{\lfloor n/2 \rfloor}$ (say) lie in $Q_{1/k}$.  Consider the $\Theta_l(n^l)$ sums formed by taking $l$ of these $v_1,\ldots,v_{\lfloor n/2 \rfloor}$; these lie in $Q_{l/k}$, which has cardinality $O_l( n^{2l-\eps} R_l^{-1} )$.  Applying the Cauchy-Schwarz inequality, we conclude that the number of solutions to
$$ v_{i_1} + \ldots + v_{i_l} = v_{i_{l+1}} + \ldots + v_{i_{2l}}$$
with $i_1,\ldots,i_{2l} \in \{1,\ldots,\lfloor n/2\rfloor\}$, is $\gg_l n^{2l} / (n^{2l-\eps} R_l^{-1}) = n^{-\eps} R_l$.  On the other hand, this number is clearly bounded above by $R_l$, giving the required contradiction.
\end{proof}

The rest of the paper is organized as follows. In the next two sections, we recall and prove several lemmas. The proof of Theorem \ref{theorem:stronggeneral} will be presented in the last two sections of the paper.

\section{Generalized arithmetic progressions}\label{gap-appendix}

In this section we recall the concept of a generalized arithmetic progression (GAP) and their basic properties.
A detailed treatment of this topic can be found in \cite[Chapter 3]{TVbook}.  We will restrict our attention to symmetric GAPs.

\begin{definition}[GAPs]  Let $G$ be an additive group.  A \emph{symmetric generalized arithmetic progression} in $G$, or \emph{symmetric GAP} for short, is a quadruplet $\Q = (Q, N, v, d)$, where the \emph{rank} $\rank(\Q) = d$ is a non-negative integer, the \emph{dimensions} $N = (N_1,\ldots,N_d)$ are a $d$-tuple of positive reals, the \emph{steps} $v = (v_1,\ldots,v_d)$ are a $d$-tuple of elements of $G$, and $Q \subset G$ is the set
$$ Q = \{ \sum_{i=1}^d n_i v_i: n_i \in [-N_i,N_i] \forall i =1,\ldots,d\},$$
where $[a,b]$ denotes the set of integers between $a$ and $b$ inclusive.
We shall often abuse notation and write $Q$ for $\Q$.  For any $t > 0$, we define the \emph{dilate} $\Q_t$ of $\Q$ to be the GAP $\Q_t := (Q_t, t N, v, d)$ formed by dilating all the dimensions by $t$.   We say that $\Q$ is \emph{proper} if all the elements $n_1 v_1 + \ldots + n_d v_d$ for $n_i \in [-N_i,N_i]$ are distinct.   We say that $Q$ is \emph{$t$-proper} if $tQ$ is proper.

We define the \emph{volume} of $\Q$ to be $\operatorname{vol}(\Q) := \prod_{i=1}^d (2 \lfloor N_i \rfloor + 1)$.  Note that $|Q| \leq \operatorname{vol}(\Q)$, with equality if and only if $\Q$ is proper.
\end{definition}

If $Q$ is a GAP of rank $d$, a simple covering argument (see \cite[Lemma 3.10]{TVbook}) shows the doubling bounds
\begin{equation}\label{double}
|Q_t| \ll_d (1+t)^d |Q|
\end{equation}
for all $t > 0$.

\begin{proof}[Proof of Proposition \ref{filo}]  Let $w_1,\ldots,w_d$ be the steps of $Q$, let $N_1,\ldots,N_d$ be the dimensions, and let $\phi: \Z^d \to G$ be the homomorphism $\phi(a_1,\ldots,a_d) := a_1 w_1 + \ldots + a_d w_d$.  By hypothesis, we can write $v_i = \sum_{j=1}^d c_{ij} w_j$ for some integers $-N_j \leq c_{ij} \leq N_j$.  Then we have $S_\mu(\bv) = \phi( x )$, where $x = (x_1,\ldots,x_d) \in \Z^d$ is the random variable whose coefficients are given by
$$ x_j := \sum_{i=1}^n \eta_i c_{ij}.$$
A simple computation shows that each $x_j$ has mean zero and variance $O( N_j^2 \mu n )$, and so
$$ \E \sum_{j=1}^d |x_j|^2 / N_j^2 \ll_d \mu n.$$
By Markov's inequality, we thus conclude that
$$ \sum_{j=1}^d |x_j|^2 / N_j^2 \ll_d \mu n$$
with probability at least $1/2$ (say).  This implies that $S_\mu(\bv) \in Q_{O(\sqrt{\mu n})}$ with probability at least $1/2$, and so by the pigeonhole principle
$$ |\P_\mu(\bv)| \gg 1 / |Q_{O(\sqrt{\mu n})}|$$
and the claim follows from \eqref{double}.
\end{proof}

One can easily pass from GAPs to proper GAPs by the following lemma:

\begin{lemma} [Embedding Lemma] \cite{TVjohn} \label{lemma:embedding}
Let $Q$ be a symmetric GAP of rank $d$ in a torsion-free additive group $G$, and let $t$ be a positive constant. Then
there is a $t$-proper symmetric GAP $Q'$ of rank at most $d$ such that $Q
\subset Q' \subset Q_{O_{d,t}(1)}$ and $|Q'| \ll_{d,t} |Q|$.   If $Q$ was not already $t$-proper, one can take $Q'$ to have rank at most $d-1$.
\end{lemma}

\begin{proof} See \cite[Theorem 1.11]{TVjohn}.
\end{proof}

Recall from the homomorphism theorems that if $H, K$ are two finite subgroups of an abelian group $G$, then $|H| |K| = |H+K| |H \cap K|$.  We now establish the analogous conclusion for GAPs (cf. \cite[Exercise 2.4.7]{TVbook}):

\begin{lemma}[Intersection lemma]\label{lemma:sandwich}
Let $P$ and $Q$ be symmetric GAPs in an additive group $G$ of
rank at most $d$, then
\begin{equation}\label{pqp}
 |P\cap Q| |Q+P| = \Theta_d( |P| |Q| ).
\end{equation}
Here of course $Q+P := \{ q+p: q \in Q, p \in P \}$ denotes the sumset of $Q$ and $P$.
\end{lemma}

\begin{proof}  We recall the Ruzsa triangle inequality
$$|A-C||B| \le |A-B| |B-C|$$
for finite non-empty sets $A,B,C \subset G$ (see e.g. \cite[Lemma 2.6]{TVbook}); this follows from the fact that any element $a-c $ with $a \in A$
and $c \in C$ has at least $b$ representations of the form $a-c = (a-b)+ (b-c) $ with $b \in B$.  Applying this with $A=P, C=Q, B=P \cap Q$ we obtain
$$|P-Q||P \cap Q| \le | P- (P\cap Q)| | (P\cap Q) -Q | \leq |2P| |2Q|$$
where we use the symmetry of $P, Q$.  But from \eqref{double} we have $|2P| \ll_d |P|$, $|2Q| \ll_d |Q|$, which gives the upper bound in \eqref{pqp}.

Now we turn to the lower bound. By reducing $d$ if necessary, we can assume that the
dimensions of both $P$ and $Q$ are divisible by two, thus $P = P_{1/2} - P_{1/2}$ and $Q = Q_{1/2} - Q_{1/2}$.  Now we recall the inequality
$$ |A| |B| \le |(A-A) \cap (B-B)| |A+B|$$
for finite non-empty $A,B \subset G$ (cf. \cite[Corollary 2.10]{TVbook}), which follows by combining the identity
$$|A||B| = |\{(a,b)|a \in A, b \in B \}|=  \sum_{x \in A+B} |\{(a,b)| a \in A, b \in B, a+b = x\} | $$
with the inequality
$$ |\{(a,b)| a \in A, b \in B, a+b = x\} | \leq |(A-A) \cap (B-B)|$$
for all $x \in G$ (which follows from the observation that if $a,a' \in A$ and $b,b' \in B$ are such that $a+b=a'+b'=x$, then $a-a'=b-b'$ lies in $(A-A) \cap (B-B)$).  Applying this inequality with $A=P_{1/2}$ and $B=Q_{1/2}$ and using \eqref{double}, one obtains the claim.
\end{proof}

\section{Arithmetic on words} \label{fourier}

In this section, we recall some tools developed  earlier in
\cite{TVcond}, which were used to prove Theorem \ref{theorem:weak}
and will be useful here as well.

For our purpose, it  is convenient to think of $\bv = (v_1, \dots, v_n)$ as a word,
obtained by concatenating the $v_i$:
$$\bv = v_1 v_2 \dots v_n. $$
This will allow us to perform several operations such as
concatenating, truncating and repeating. For instance, if $\bv
=v_1\dots v_n$ and $\bw= w_1\dots w_m$, then
$$\P_{\mu} (\bv \bw) = \max _{a \in Z} \Big(\sum_{i=1}^n \eta_i^{\mu} v_i
+ \sum_{j=1}^m \eta_{n+j} ^{\mu} w_j=a \Big)$$ \noindent where
$\eta_{k}^{\mu}, 1\le k \le n+m$ are  i.i.d copies of $\eta^{\mu}$.
Furthermore, we use $\bv^{[k]}$ to denote the concatenation of $k$
copies of $\bv$.

We will need to generalize the concentration probabilities $\P_\mu(\bv)$ as follows. For finite non-empty set $Q \subset G$, define
\begin{equation}\label{pbq-def}
\P_\mu(\bv;Q) := \sup_{a \in G} \P( S^\mu(\bv) = a + q - q' )
\end{equation}
where $q, q'$ are independently chosen uniformly at random from $Q$.  Note that $\P_\mu(\bv;Q) = \P_\mu(\bv)$ if $Q$ is a singleton set.

Since $\P( a+q-q'=x) \leq 1/|Q|$ for any fixed $a,q', x$, a simple conditioning argument reveals the crude bound
\begin{equation}\label{pbound}
 \P_\mu(\bv;Q)  \leq 1/|Q|.
\end{equation}

We have the following basic properties of the $\P_\mu(\bv)$ and $\P_\mu(\bv;Q)$:

\begin{lemma}\label{lemma:properties} Let $\bv = v_1\ldots v_n$ be a word $v_1,\ldots,v_n$ in a torsion-free additive group $G$, and let $Q \subset G$ be a finite non-empty set.  Then the following properties
hold.
\begin{itemize}

\item[(i)] $\P_\mu(\bv;Q)$ is  invariant under permutations of $\bv$.

\item[(ii)] For any words  $\bv, \bw$

\begin{equation} \notag \P_\mu(\bv \bw;Q) \leq \P_\mu(\bv;Q).
\end{equation}

\item[(iii)] For any $0 < \mu \le 1$, any $0 < \mu' \leq \mu/4$, and any word $\bv$,
\begin{equation} \notag \label{14} \P_{\mu}(\bv;Q) \leq \P_{\mu'}(\bv;Q).
\end{equation}

\item[(iv)]  For any number $0 < \mu \leq 1/2$ and any word $\bv$,
\begin{equation} \notag \label{pmub-2} \P_\mu(\bv;Q) \leq \P_{\mu/k}(\bv^{[k]};Q).
\end{equation}

\item[(v)] For any number $0 < \mu \leq 1/2$ and any words $\bv, \bw_1,
\dots, \bw_m$ we have
\begin{equation} \notag \label{pmub-3}
\P_\mu(\bv \bw_1 \ldots \bw_m;Q) \leq \left( \prod_{j=1}^m \P_\mu(\bv
\bw_j^{[m]};Q) \right)^{1/m}.
\end{equation}

\item[(vi)]  For any number $0 < \mu \leq 1/2$ and any words $\bv,
\bw_1, \dots, \bw_m$, there is an index $1\le j \le m$ such that
\begin{equation}\notag \label{pmub-3a}
\P_\mu(\bv \bw_1 \ldots \bw_m;Q) \leq \P_\mu(\bv \bw_j^{[m]};Q).
\end{equation}
\end{itemize}
\end{lemma}

\begin{proof}  When $G=\Z$ and $Q$ is a singleton, this is \cite[Lemma 5.1]{TVcond}.  When $G=\Z$ and $Q$ is not a singleton, the claim can be established by repeating the proof of \cite[Lemma 5.1]{TVcond}, using the Fourier identity
$$ \P( S^\mu(\bv) = a + q - q' ) = \int_{0}^{1 } e(-a t) |\E (e(qt))|^{2 } \prod_{i=1}^{n} (1-\mu + \mu \cos 2\pi v_{i }t ) \,\, dt$$
in place of
$$ \P( S^\mu(\bv) = a ) = \int_{0}^{1 } e(-a t) \prod_{i=1}^{n} (1-\mu + \mu \cos 2\pi v_{i }t ) \,\, dt;$$
we omit the details.
(Here and later, $e(x)$ denotes $ \exp(2\pi \sqrt{-1} x) $.) Finally, the generalization to arbitrary torsion-free $G$ can be accomplished by using Freiman isomorphisms (see \cite[Lemma 5.25]{TVbook}).
\end{proof}

Note that for fixed $0 < \mu < 1$, a random walk $S^\mu( v^{[k^2]} )$ is roughly uniformly distributed on the progression $[-k,k] v := \{ j v: j \in \Z, -k \leq j \leq k \}$, thanks to the central limit theorem.  (Here $v^{[k^2]}$ is the word $v$ repeated $k^2$ times.) The following lemma can be viewed as a formalization of this intuition.

\begin{proposition}[Comparison of random walks]\label{cor:prob3}  Let $0 < \mu \leq 1/2$, let $\bv = (v_1,\ldots,v_n)$ be a tuple in a torsion-free additive group $G$, let $v_0 \in G$, and let $k \geq 1$.  Let $Q$ be a symmetric GAP in $\Z$ of rank $d$.  Then
$$ \P_\mu( \bv v_0^{[k^2]}; Q ) \ll_{\mu,d} \P_\mu( \bv; Q + [-k,k]v_0 ).$$
\end{proposition}

\begin{proof}  Fix $\mu,d$; we allow all implied constants to depend on these quantities.
By definition,
$$ \P_\mu( \bv v_0^{[k^2]}; Q ) = \P( S^\mu(\bv) + X v_0 = q-q' )$$
where $q,q'$ are independent random variables uniformly distributed in $Q$, and $X := \sum_{i=1}^{k^2} \xi_i^\mu$.  A direct computation using Stirling's formula shows that
$$ \P( X = m ) \ll k^{-1} \exp( - \Omega(|m|/k) )$$
for all $m \in \Z$, thus
$$ \P_\mu( \bv v_0^{[k^2]}; Q ) \ll k^{-1} \sum_{m \in \Z} \exp( - \Omega(|m|/k) ) \P( S^\mu(\bv) + m v_0 = q-q' ).$$
This implies that
$$ \P_\mu( \bv v_0^{[k^2]}; Q ) \ll k^{-1} \sum_{m \in \Z} \exp( - \Omega(|m|/k) ) \P( S^\mu(\bv) + m v_0 = q-q'+jv_0-j'v_0 )$$
where $j, j'$ are drawn uniformly at random from $[-k,k]$, independently of each other and of $q, q'$.
It therefore suffices to show that
$$ \P( S^\mu(\bv) = a+q-q'+jv_0-j'v_0 ) \ll \P_\mu( \bv; Q + [-k,k]v_0 )$$
for all $a \in G$.

The random variable $q+jv_0$ is supported in $Q + [-k,k]v_0$.  If it were distributed uniformly in this set, we would be done.  It is not quite uniform, nevertheless we can compare it to the uniform distribution as follows.  Given any $x \in Q + [-k,k]v_0$, we have
$$ \P( q+jv_0 = x ) = \frac{1}{|Q| |[-k,k]v_0|} |Q \cap (x - [-k,k]v_0)|.$$
Since $|A| \leq |A-A|$, we see that
$$ |Q \cap (x - [-k,k]v_0)| \leq |(Q-Q) \cap ([-2k,2k]v_0)|$$
and so by Lemma \ref{lemma:sandwich} and \eqref{double}
$$ |Q \cap (x - [-k,k]v_0)| \ll \frac{|Q| |[-k,k]v_0|}{|Q + [-k,k]v_0|}$$
and thus
$$ \P( q+jv_0 = x ) \ll \frac{1}{|Q + [-k,k]v_0|}.$$
Thus the probability distribution of $q+jv_0$ is majorized by a constant multiple of the uniform distribution on $Q+[-k,k]v_0$, and the claim follows.
\end{proof}

\section{The algorithm} \label{algorithm}

We begin the proof of Theorem \ref{theorem:stronggeneral}.  By Lemma \ref{lemma:properties} we may assume that $\mu \leq 1/2$.
Fix $d, \eps, \mu, n, k, \bv, G$ as in that theorem; we assume that $n, k$ are sufficiently large depending on $d,\eps,\mu$.  We let $K \geq 1$ be a large number depending on $d,\eps,\mu$, and then let $C_0$ be an even larger number depending on $d,\eps,\mu,K$.  We assume that \eqref{pi2} holds.

In this section, we describe an algorithm which takes $\bv$ as  input and produces, as output, a symmetric GAP $Q$ as claimed by Theorem \ref{theorem:stronggeneral}.  A key concept is that of a \emph{bad} element with respect to a symmetric GAP.

\begin{definition}[Bad element]  Let $K \geq 1$, $x \in G$, and let $Q$ be a symmetric GAP in $G$.  We say that $x$ is \emph{bad} with respect to a symmetric GAP $Q$ if
$$|Q  + [-k,k] x | \ge K |Q|$$
and \emph{good} otherwise.
\end{definition}

We will also need the generalized concentration probabilities $\P_\mu(\bv;Q)$ defined in \eqref{pbq-def}.  We now consider the following algorithm that generates words $\bv^i$ and symmetric GAPs $Q_i$ for various $i=0,1,2,\ldots$:

\vskip2mm

{\bf Step 0.} Set  $\bv^{0}= \bv, Q_{0}:= \{0\}$.

{\bf Step $i+1$.} Count the number of elements of $\bv^i$  which are
bad with respect to $Q_i$.

{\it Case 1.}  If this number is less than $k^2$  then STOP.

{\it Case 2.}  If this number is at least $k^2$, we can assume
(without loss of generality) that the last $k^{2}$ coordinates of
$\bv^{i}$ are bad. Let $\bv^{i+1}$ be the vector obtained from
$\bv^{i}$ by truncating these bad coordinates. By Lemma
\ref{lemma:properties}(vi), there is some value $v_0$ among the bad coordinates such
that
$$P_{\mu} (\bv^{i+1} v_0 ^{[k^{2}]}; Q_{i}) \ge  \P_{\mu}(\bv^{i}; Q_{i}). $$
Set $r_i := \operatorname{rank}(Q_i)$ and $Q_{i+1}' : = Q_{i} + [-k,k] v_{0}$, thus $Q'_{i+1}$ is a GAP with rank $r_i+1$. If $Q'_{i+1}$ is proper,
then set $Q_{i+1}:= Q'_{i+1}$. If it is not proper, then use Lemma
\ref{lemma:embedding} to embed it into a proper symmetric GAP of
rank at most $r_i$ and volume $O_{r_i}(|Q_{i+1}'|)$;. Call this
proper GAP $Q_{i+1}$.   CONTINUE  to Step $i+2$.

\vskip2mm

Notice that by the algorithm the $Q^{i}$ are symmetric GAPs at every step.

\section{Analysis of the algorithm} \label{analysis1}

For each $i$ that occurs in the algorithm, we define the rank
$$ r_i := \operatorname{rank}(Q_i)$$
and the potential
$$ F_i := |Q_i| \P_\mu(\bv^i; Q_i).$$
Initially we have
\begin{equation}\label{r0f0}
 r_0 = 0; \quad F_0 = \P_\mu(\bv) \geq C_0 k^{-d}; \quad |Q_0| = 1.
\end{equation}

We now record how $r_i$, $F_i$, and $Q_i$ evolve with the algorithm.
We say that Step $i+1$  is \emph{proper} if $Q'_{i+1}$ is proper.

\begin{lemma}\label{grow}  Let Step $i+1$ be a step that occurs in the algorithm.
\begin{itemize}
\item[(i)] We have $r_{i+1}=r_i+1$ if Step $i+1$ is proper, and $r_{i+1} \leq r_i$ otherwise.
\item[(ii)] We have $|Q_{i+1}| \geq K |Q_i|$.  If Step $i+1$ is proper, we can improve this to $|Q_{i+1}| \geq k |Q_i|$.
\item[(iii)] We have $F_{i+1} \gg_{r_i,\mu} K F_i$.  If Step $i+1$ is proper, we can improve this to $F_{i+1} \gg_{r_i,\mu} k F_i$.
\end{itemize}
\end{lemma}

\begin{proof} The first two claims are clear from construction.  To prove the third claim, we observe from Proposition \ref{cor:prob3} that
\begin{equation}\label{pmuq}
\P_{\mu} (\bv^{i+1}; Q_{i+1} ) \gg_{r_i,\mu} \P_{\mu} (\bv^{i} ; Q_i);
\end{equation}
the claim (iii) now follows from (ii).
\end{proof}

\begin{corollary}\label{term} The algorithm has at most $d-1$ proper steps, and terminates in $O( d \log_K k )$ steps.
\end{corollary}

\begin{proof}  Suppose for contradiction that there were at least $d$ proper steps.  Let $1 \leq i_1 < \ldots < i_{d}$ be the first $d$ proper steps.  By Lemma \ref{grow}(i) and \eqref{r0f0}, the ranks $r_i$ are bounded by $d$ for all $i \leq i_d$.  From Lemma \ref{grow}(iii), we have $F_{i_d} \geq_{d,\mu} k^d F_0$ if $K$ is large enough; on the other hand, from \eqref{pbound} we have $F_{i_d} \leq 1$.  This contradicts \eqref{r0f0} if $C_0$ is large enough.

Now that there are at most $d-1$ proper steps, $r_i \leq d-1$ for all $i$.  By Lemma \ref{grow}(iii), we thus have $F_{i+1} \geq \sqrt{K} F_i$ for all $i$ if $K$ is large enough.  On the other hand, from \eqref{pbound} we have $F_i \leq 1$ for all $i$.  Applying \eqref{r0f0}, we conclude that the algorithm terminates in $O(d \log_K k)$ steps as claimed.
\end{proof}

Let $\bv^{T}$ and $Q_{T}$ be the vector and GAP at the stopping
time $T = O(d \log_K k)$.

\begin{lemma} $Q_T$ has rank at most $d-1$ and
$$ |Q_T| \leq k^{\eps/2} \P_\mu(\bv)^{-1}$$
\end{lemma}

\begin{proof} The rank bound follows from \eqref{r0f0}, Lemma \ref{grow}(i) and Corollary \ref{term}.

As proved above, $Q_T$ has rank at most $d-1$. We next prove
that it has small cardinality.  Iterating \eqref{pmuq} starting from \eqref{r0f0}, we see that
$$ \P_{\mu} (\bv^{T}; Q_{T}) \geq \Omega_{d,\mu}(1)^T \P_\mu(\bv).$$
Combining this with \eqref{pbound} and the bound $T = O(d \log_K k)$ we conclude
$$ |Q_T| \leq \exp( O_{d,\mu}(\log_K k) ) \P_\mu(\bv)^{-1}$$
and the claim follows by taking $K$ sufficiently large.
\end{proof}

By construction, all but $O(d k^2 \log_K k) = O( k^2 \log k )$ of the $v_1,\ldots,v_n$ are good relative to $Q_T$.  To exploit this we use

\begin{lemma} \label{lemma:good}  Suppose that $x \in G$ is good relative to a symmetric GAP $Q$ of rank $r$.  Then there exists a proper symmetric GAP $Q'$ of rank at most $r$ containing $Q$ and volume $|Q'| \ll_{K,r} |Q|$ such that $Cx \in Q'_{C/k}$, where $C \geq 1$ is an integer depending only on $K$ and $r$.
\end{lemma}

\begin{proof}  The $|Q| |[-k,k]|$ sums $q+jx$ with $q \in Q$ and $j \in [-k,k]$ lie in the set $|Q + [-k,k]x|$, which has cardinality at most $K|Q|$ by hypothesis.  By Cauchy-Schwarz, we conclude that there are $\gg_K |Q| k^2$ quadruplets $(q,q',j,j') \in Q \times Q \times [-k,k] \times [-k,k]$ such that $q+jx=q'+j'x$.  By the pigeonhole principle, we conclude that the set $A := \{ j \in [-2k,2k]: jx \in Q-Q \}$ has cardinality $|A| \gg_K k$.   Applying a result of S\'ark\H{o}zy \cite{sar} (see also \cite{lev}, \cite{sv}, or \cite[Chapter 12]{TVbook}) we conclude that there exists a positive integer $K_1 = O_K(1)$ such that the iterated sumset $K_1 A$ contains an arithmetic progression of length $l = \Theta_K(k)$ and positive integer step $a = O_K(1)$.  We conclude that $[-l,l]ax \in Q_{4K_1}$.

At present, $l$, $a$, and $K_1$ are all dependent on $x$.  But $K_1, a$ are bounded by $O_K(1)$, and $l$ is bounded from below by $\Omega_K(k)$.  Thus, by taking the gcd over all possible values of $a$, one may assume that $l, a, K_1$ are independent of $x$.

By Lemma \ref{lemma:embedding}, we can place $Q_{4K_1}$ inside a $2$-proper symmetric GAP $Q'$ of rank at most $r$ and volume $O_{K,r}(|Q|)$, thus $[-l,l]ax \in Q'$.  Write $N'_1,\ldots,N'_{r'}$ for the dimensions of $Q'$.  Since $Q'$ is $2$-proper, the obvious map $\phi: [-N'_1,N'_1] \times \ldots \times [-N'_r,N'_r] \to Q'$ is a Freiman isomorphism of order $2$ (see \cite[Section 5.3]{TVbook}), and $\phi^{-1}([-l,l]ax) = [-l,l]\phi^{-1}(ax)$ is also an arithmetic progression.  From this we see that
$$ \phi^{-1}(ax) \in [-N'_1/l,N'_1/l] \times \ldots \times [-N'_r/l,N'_r/l]$$
and thus
$$ ax \in Q'_{1/l}$$
and the claim follows.
\end{proof}

Through the proof of  this lemma, we see that all but at most $O(k^2 \log k)$ of the $v_i$ are such that $Cv_i \in (Q_T)_{C/k}$.  By Lemma \ref{lemma:embedding}, we may place $(Q_T)_C$ inside a proper symmetric GAP $Q$ of rank at most $d-1$ and volume
$$|Q| \le k^{\eps} \P_{\mu} (\bv)^{-1}. $$
Since $(Q_T)_C \subset Q'$, we have $(Q_T)_{C/k} \subset Q_{1/k}$, and Theorem \ref{theorem:stronggeneral} follows.

{\it Acknowledgement.} We would like to thank the referees for their  careful reading and useful remarks.

redpajama_set_name: RedPajamaArXiv


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The entire history of the United States, particularly the Western lore, were all about gambling, and in particular all about poker. Even though for many years playing poker was not considered something that nice people would want to do, the fact remains that many presidents of the United States going back to the Revolutionary war, and soon thereafter, and for all the years since, all played poker.
Poker played well is not a gamble, but a game of skill and knowledge. Although there is a significant element of luck, such is present in all aspects of human life, and not relegated only to poker. Success in business, or the success in your job, or your career, or whatever it is that is your primary endeavor and the source of income, is always, and has always been, a combination of knowledge and skill. The more knowledge you have, the better equipped you are to be successful. The better your skills in transferring that knowledge into action, the equally better your chances of success. In all these endeavors, luck will always be present, both good and bad.
Sometimes no matter how knowledgeable or how skilled you are, bad luck will simply happen to you and you have to learn how to overcome it. Doing that is also part of your knowledge and your skills, and how well you do it also directly determines your success. On the other side of the coin, equally so good luck will also visit you many times during your life, no matter what your endeavor might be. So it is for poker, where there will be many times during your poker playing career where it will appear that you can do no wrong. Those will be the great and wonderful times, because everything that you do will always work out your advantage, all hands you play will always be winners, and all your draws will come to be and win you the pot. And so it is also in life. Poker is a microcosm of life, and it equally reflects all that is good and bad in human nature.
Not all people in the world are good and not all poker players will be nice people. Not all people in the world will be equally successful, and not all poker player will be equally as profitable as others. How well you handle your life will be reflected in how well you handle your poker playing success. If you have learned how to discipline yourself in life, you will not have trouble disciplining yourself in the play of poker. If, on the other hand, you don't know what personal discipline is, then you will have lots of trouble in poker no matter how much you learn. Learning the theory of poker, or the theory of games, is alone not enough. In fact, learning the theory of anything is alone not enough. In order for the theory to have any practical meaning, it has to have what is called "practical application". This practical application is the skill by which a human person is able to transcend the theory into the actual doing of it, commonly called "putting the theory into practice". In my books I have often called this "the workability principle".
What this simply means is the ability to be able to take the knowledge and actually apply it to real-world situations successfully. Many people have the ability to be able to apply knowledge into real-world situations, but very few of them have the ability to be able to do it successfully on a consistent basis. Being a consistently successful and profitable poker player is very difficult, and you should have no illusions about it. Yes, there will be times when you begin to play poker and you will get enormously lucky, something that is very often called "beginners luck". This happens with startling regularity, and I have witnessed this amongst novice players many, many times. But such luck is fleeting, and if you do not learn to tampered it with knowledge and skill, your overall end-result experience is bound to be unlucky.
The world of card rooms and casino poker rooms is full of graveyards of the hopeful who came equipped with a little knowledge, a little skill, a lot of hope, and a great deal of beginners luck, only to discover that such luck is fickle, and that a little knowledge and little skill is not enough, and is in fact very dangerous. Such players very often experience enormous highs as they glory in their initial success, and quite as often erroneously attribute their initial success to a misperceived sense of their own skills and abilities, as well as their own sense of greatness and invincibility. Such are signs of early danger, and if the player does not understand them he or she is heading for an inevitable disaster. In fact, any poker player who allows himself or herself to experience such great highs opens himself or herself up to equally deep lows.
We will continue with this discussion in the next issue.


text:
Asian Football | January 15, 2020
Mohun Bagan fans start petition to stop potential merger with ATK
Harigovind
Just a day after reports emerged of Indian I-League club Mohun Bagan's potential merger with Indian Super League giants ATK, Bagan's fans have started an online petition in an attempt to stop the deal.
The online petition has been created in Change.org, one of the largest portals of its kind in the world with over 240 million users.
"ATK is going to take over an 80% stake in Mohun Bagan, after which theoretically ATK will seize to exist, but practically it is MB who will die eventually. No club's fans want to share the gallery space with their fellow rivals from the same city." the post reads.
"Both the clubs and their fans have created a different identity for themselves in spite of several hardships and struggle just for the sake of their mother club and the love for the sports. We the fans of both the clubs protest against this merger which is being done only for the sake of business of the people in command and demand the security of our hard-earned identity. We want both the clubs to be a separate entity and as rivals."
"By signing this petition, we want to deliver a message to the officials of both the clubs that we don't want this merger or rather this takeover. Only fans have the power to change. Say no to the merger," it further adds.
It is The Bridge who reported that they can confirm that Mohun Bagan are all set to join hands with ATK. This latest move from Bagan is apparently a part of their plan to enter the ISL next season, following the All India Football Federation (AIFF)'s plans to introduce a two-tier league system with ISL as the top division and the I-League as the second division.
2020 AFC U23 Championship:...
2020 AFC U23 Championship: Korea...
LIVE: AFC U23 Championship 2020 –...
Iran clubs to pull out of AFC...
Tanjong Pagar United to replace...
Home Football Asian Football Mohun Bagan fans start petition to stop potential merger with ATK

redpajama_set_name: RedPajamaCommonCrawl


Salvatore Anthony "Torey" Lovullo (born July 25, 1965) is an American professional baseball coach and former Major League infielder and minor league manager. In his playing days he was listed at 6 feet 1 inch (1.85 m) tall and 185 pounds (84 kg) and was a switch hitter who threw right-handed. As of 2016, Lovullo is the bench coach of the Boston Red Sox, after serving as Boston's interim manager for the final seven weeks of the 2015 season when his boss, John Farrell, stepped aside for successful treatment for lymphoma. Lovullo compiled a win-loss record of 28–20 (.583) from August 14, 2015, through the final day of the season, October 4.
A native of Santa Monica, California, Lovullo is an alumnus of the University of California, Los Angeles; his father, Sam, was a producer on the television show Hee Haw. He was drafted by the Detroit Tigers in the fifth round of the 1987 Major League Baseball Draft and made his Major League debut with the Tigers on September 10, 1988. He also played in the Majors for the New York Yankees, California Angels, Seattle Mariners, Oakland Athletics and Cleveland Indians, and appeared in his final big-league game on October 3, 1999, for the Philadelphia Phillies. After his MLB career ended, Lovullo spent one season in Japan as a member of the Yakult Swallows in 2000.
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The following video clip is extracted from the film by Christianne van Wijk entitled 'The Mind Matrix'. As to the merit or truthfulness of the content, each viewer must decide for themselves, but what I am interested in is the explanation given by the British activist Ian R Crane – a once very rich man who used to work in the oil industry. He gave-up his large salary and easy life when in 1991, he discovered the bodies of young Iraqi Conscripts shot in the back of the head (by US Forces after surrender) and left in fox-holes near the burning oil fields. Although the media was fed the lie that the retreating Iraqi Army had set fire to the oil fields, Ian R Crane witnessed US Special Forces carrying-out this destructive act. After the well-being of him and his family was threatened by the US Military, Ian R Crane started living in a tent, protesting against the State and big business, and educating the public. In this clip he praises Vladimir Putin for keeping Russia out of the Rothschild banking empire, and the international banking system run by the US. Putin has built up a strong economy OUTSIDE of the international capitalist system, and so this cannot be tolerated by the US because it is a parallel form of capitalism with leftist leanings. Ian R Crane aptly explains that the Ukraine is a fabricated anti-Russia project designed to eventually destroy Russia's independent economy and subjugate the Russian people to US and EU dictates.


Welcome to Americas Best Value Inn and Suites Carbondale, conveniently located less than a mile from South Illinois University. We are also near Rend Lake, African American Museum of Southern Illinois, John A Logan College Museum, Grand Avenue Mall, Stone Creek Golf Club, and McLeod Theater. Enjoy a delicious meal at one of the local restaurants including Hunan Village, Sonic Drive-In, and Fazoli's.
Our helpful 24-hour front desk staff at our pet friendly Americas Best Value Inn and Suites Carbondale strives to make our hotel your home away from home. Start your day right with our free continental breakfast and a free newspaper available in the lobby. Each guest room features free wireless high-speed Internet access, cable TV with HBO, and free local calls. At Americas Best Value Inn and Suites Carbondale, We've Got You Covered from the time you hit the road until your head hits the pillow.


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December 14, 21, 28 and February 8, 22
Fat Bike Invasions
Fat Bike Series
The Loppet Foundation :: Events :: Fat Bike Invasion
Dates: December 14, 21, 28 and February 8 and 22.
The Fat Bike Invasion events are geared towards getting people outdoors in the winter while creating a social environment. We strive to connect people and encourage participants to cheer on other categories. These events are the winter continuation of Wirth on Wednesdays Mountain Bike Series.
Headlamps and tail lights (and of course helmets) are mandatory.
For further questions contact Jeremy Sartain at 612-414-5224 or via email at Sartain.Jeremy@gmail.com.
Race fee of $20 per event. On-site registration will open at 4:00pm each night.
PLEASE NOTE: Relay teams for February 8 must register BOTH participants. If paying together, please select TWO tickets and fill in information for both participants.
Categories are subject to change (additions or combinations) depending on need. We encourage constructive feedback each night!
Schedule / Race Categories during December dates:
Juniors (30 minutes) — 5:30pm
Novice Male (30 minutes) — 6:00pm
Novice Female (30 minutes) — 6:01pm
Short Distance Male (30 minutes) — 6:30pm
Short Distance Female (30 minutes) — 6:31pm
Open Fat-Bike Male (60 minutes) — 7:00pm
Open Fat-Bike Female (60 minutes) — 7:00pm
Schedule / Race Categories during February dates:
February 8: Nordic Ski / Fat Bike Relays
Short Distance Relay Male (30 minutes) — 5:30pm
Short Distance Relay Female (30 minutes) — 5:31pm
Open Relay Male (60 minutes) — 6:15pm
Open Relay Female (60 minutes) — 6:16pm
* We will assist with pairing Nordic skiers with fat bikers. Skiers will start and then alternate each lap with the fat bikers.
*A junior category may be added if demand exists.
February 22: Nordic Skiing vs. Fat Biking
Let's put to the test who is fastest… Nordic skiers or fat bikers! Both athletic disciplines will compete on the same course, and at the end of the night the average time for the top 5 males and females in each category will be calculated to determine the fastest sport. Racers can choose to compete in one or both disciplines.
Short Distance Nordic Ski Male (30 minutes) — 5:30pm
Short Distance Nordic Ski Female (30 minutes) — 5:31pm
Open Nordic Ski Male (60 minutes) — 5:32pm
Open Nordic Ski Female (60 minutes) — 5:33pm
Short Distance Fat Bike Male (30 minutes) — 6:45pm
Short Distance Fat Bike Female (30 minutes) — 6:46pm
Open Fat Bike Male (60 minutes) — 7pm
Open Fat Bike Female (60 minutes) — 7:01pm
* Juniors can choose short or open categories and will have their own awards.
All participants receive a raffle ticket for amazing swag prizes at the end of each night. Small prizes are awarded to the top 3 finishers in each category for the December races and the February 8 race after each race. The February 22 race will have a trophy for each category that will be passed to the winners (Fat Bikers or Nordic Skiers) each year.


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Review My Singing
Don't let me down - please critique!
By dbass
October 7, 2012 in Review My Singing
dbass
I'm new to this forum, and I'm also new to singing. I've been playing bass for a long time, now I would like to learn to sing, but I realize how hard it is. I started to take some lessons, and I'm trying to work hard.
I posted this song... a cover from The Beatles, Don't Let me Down..
I would appreciate to receive comments, critiques to my singing, because I know how hard it will be to become a good singer and I am open to listen to critiques and humbly work hard on my singing.
So please comment!
Thanks everybody
http://soundcloud.com/dbass1/dont-let-me-down-the-beatles
That is a great song and I have always wanted to sing it. That and "Oh Darling." But my voice isn't for that I don't think. Your version wasn't bad, but it needs work to make it really sing This is typical of many new singers posting here. You have the right tone, stay on pitch and have the voice potential but poor technique. It is like someone who can fight very well and in a bar can win every fight he gets into. However if he got into a professional boxing ring he would lose. Why? Technique. His punches would be too wide, his head may be exposed, his footwork poor. There are sort of rules for singing. Singing without them might get you by with some listeners, but it isn't right.
I can hear a pronunciation problem also, that I know is because English isn't your first language. It doesn't matter because it detracts from the song and it makes it airy. For example near the beginning "Nobody ever really loved me." You pronounce ever as hever. That's giving an H sound that shouldn't be there as well as a "breath." Small details like this make a difference. I too am learning all these things so it is very common
Good effort. You sound like a young John Lennon, and I don't think that it's just because of the song. You might want to ease up on the consonants, just a little. Make the 'd' sound a little softer. Or, maybe it was the mix. I've changed the entire perception of consonants and even note onsets just what settings on compressor and eq, in post-recording editing.
Most important, you've got the right feel for this song.
Thank you very much Tommy for you constructive critique. I feel I have lots of problems with sound emission, technique is poor and that's what I will be focusing on. Now I'm seriously approaching to singing as a proper "instrument", just as I did with bass guitar, but it takes years of practice, I know.
English is not my first language, that's true I'm italian, however I will try to pay more attention to these pronunciation nuances, but I don't think it is a big problem..I am more concerned about my singing technique, and I would appreciate to know from you in which moments of the song you sensed specific technical mistakes (emission, "throat-singing", out of pitch, etc..), because of the lack of experience it's still hard for me to detect all the mistakes I do, but surely you can.
Thanks ronws I will try to soften up on the consonants, I don't think it is a mixing problem, it's still my imprecise english pronunciation. I really appreciate what you said about the feel for the song, to me is as equally fundamental as technique.
Mind you I am not an instructor but only a student and I'm really not the technical type But what I hear is this. On the verse at 31 seconds in until 56 secs, it sounds thin and weak, very gravel sounding at points and not rasp for effect but more like a vocal fry. I think it may be a support issue. Then on the chorus you are slightly cutting "let" it becomes "don't leme down" (slightly) which rushes the phrase. phrasing is everything. This type of thing goes on through out the song repeating on all verses and chorus.
When you say "Don't you know it's gonna last" the "don't" gets that vocal fry again. That phrases is a bit low in the original song but it has support and enough airflow to not give it that vocal fry sound you are. Then the phrase wobbles a bit and isn't as controlled at 133 - 145, the last word there at around 145 the note drops.
I think it would be a good idea to learn good breath support and voice placement so you don't sink down to the throat. Your pronunciation isn't bad and I know you think it isn't that big of a deal but it is. If you add an "h" sound to the beginning of words and phrases (as an example) it ruins the phrase and at times makes a song too breathy. Also depending on the emotion or feel of a phrase or word, it changes it. How you attack (or don't) consonants is important to a note. Onsets.
Also a lot of singing and "singing the notes" (not just hitting them) depends on vowel placement and modification as well as enunciation and pronunciation. Very important. Again, it's the small details that count. In my opinion you have to pay attention to the small details. If you get the small things correct and second nature then the big things become easy. Fundamentals my friend. It takes many small things to make a big thing.
One of my saying used to be "You can't move a mountain, but you can move a stone. Move enough stones and you have moved a mountain."
Again....These are just my thoughts, and I am really not the guy who really knows about much of this. I'm learning too. Always learning as it is a continuous path and a forever changing one.
Not bad though....keep working
KillerKu
I like it. It's raw. If you're comfortable, I don't know if I would change it 'drastically' from this style.
Polish, tweaks, basically breath support and placing it slightly out of the throat as Tommy said, while keeping a kind of similar flavor would basically polish you off, imo.
Your tone tends to sometimes thin out into kind of weak fry. I think he's spot on there.


Chamomilla pellets are indicated for Irritability, Colic, Frantic, Sensitive, Teething, Sleeplessness.
Chamomilla is a homeopathic remedy commonly used for irritability, colic, teething (including the diarrhea that often accompanies teething), sleeplessness. The irritable, cranky mood is a key indication for this remedy. Very sensitive to pain.
The remedy is made from a daisy-like plant that is widespread all over Europe.


It doesn't matter your musical bent, if you appreciate a supreme experience I highly recommend spending Monday evening at the Orpheum Theatre. What I experienced last night was profoundly beautiful. I've heard Mozart's Requiem on countless recordings and will always be intrigued by its story – Mozart died giving birth to this piece of music. Literally, he died before could complete it on his deathbed.
Controversy and debate will always keep this composition aloft in the academic psyche as the debate rages on how Mozart would have finished the work, which now ends with patched quilt his student was left to stitch together from ideas and sketches orphaned by his master who feverishly wrote to his last breath.
Literally, there are hundreds – if not thousands – of recorded versions of this work and each with a uniquely interpretation. The ensemble can range from hundreds of musicians and orchestra members to stripped down versions with merely dozens of singers and instruments. There are now multiple endings as scholarly composers try to outrank each other or supplant the man who was there at Mozart's bedside as this piece of music lifted from his dying body like a ghost exorcised.
Of course you see the irony, a man who was renown for impious indulgences (myths powerfully reinforced by the 1984 film Amadeus) is confronted by mortality and scoring the holy text that may be his last chance for an invitation into heaven.
The contrition and desperate urgency confronts you when you hear 200 voices rise and swell in a unison with cries of sorry and joy. These voices are pushed to even greater heights by the instruments of Mozart's genius – an orchestra burgeoning with strings, brass, woodwinds, organ and tympani.
Male and female voices, of the Vancouver Bach Choir overflow make shift rafters on the Orpheum stage supported by the sweetness of the Vancouver Symphony Orchestra conducted by maestro Bramwell Tovey. Mozart's own voice is represented by four superb singers: Kathleen Brett, soprano; Elizabeth Turnbull, mezzo-soprano; Benjamin Butterfield, tenor and John Relyea, baritone.
I attended a concert but was witness to a connection beyond genre or style. My experience – it's obvious, I relished it – and I encourage you to have your own. The same ensemble performs again at the Orpheum Monday night and it's not often this piece is performed and even more rare to this magnitude. There are great savings on walk-ups tickets at the VSO on Mondays. Go the VSO box office an hour or so before show time. If you do attend please share your experience through a post it here.


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German Headquarter Vattenfall
Hohenzollern Campus
Trier Galerie
Flensburg Gallerie
Schönhauser Allee Arcaden
HermannQuartier
Beisheim-Center
Steigenberger Hotel
Prager Platz
Residential Park Staaken
Prager Platz, Berlin
Gross surface area:
40.000 qm (Gesamtkomplex)
Conference rooms:
Shopping arcade,
Fitness club ELIXIA
Erich Kästner and an urban oasis
Prager Platz is a square located in the heart of Berlin. Erich Kästner lived here from 1927 to 1931 and wrote his famous novel »Emil and the Detectives«. Take a 15-minute stroll to Kurfürstendamm, along streets with grand old facades, and you could almost imagine yourself transported back to old Berlin at the turn of the century. In his books, Kästner's fictional character Emil also walked along these streets.
In the 1920s, Prager Platz was an important cultural area in the western part of Berlin. It was part of the Bayerisches Viertel, a quarter that promised a high quality of life, and was commonly referred to as the »Jewish Switzerland«. Many Jews lived here, including artists and intellectuals. The original historic buildings on the square were destroyed during World War II. Reconstruction work began in 1986, following the lines of the original historical structures.
The final reconstruction phase was implemented by TRIGON Immobilien. Within just a two-year period, under the company's direction, an entire mixed-use complex was built with a gross surface area of some 40,000 square meters,
including a Ramada Hotel, a shopping arcade – the »Prager Passage« – and the »Elixia« fitness and spa center with a 25-m swimming pool. This ensemble filled the last vacant lot on the square.
Prager Platz is a prime example of a successful urban development project that combines modern architecture and historic elements. The urbanity of the new and meticulously renovated buildings on the oval-shaped square is balanced by large green areas, which create a park-like landscape. The low-traffic streets around the square help to create an oasis in the middle of this large city. Thanks in part to the close proximity of Kurfürstendamm, Berlin's most famous shopping boulevard, this is a highly attractive location.
RAMADA HOTEL
The Ramada Hotel has 184 elegant double rooms and suites, along with six conference rooms with state-of-the-art equipment, air conditioning, ample daylight, and access to two large roof terraces. The restaurant serves modern and down-to-earth cuisine in a relaxed atmosphere.
The adjacent »Elixia« fitness club and spa offers an opportunity to work out and/or relax and unwind.
Contact, About Us and Privacy
TRIGON Immobilien is a recognized leading developer for major real estate projects and asset management. It specializes in the development of shopping centers, hotels, commercial projects and residential housing construction.
TRIGON Immobilien Holding GmbH
D 14199 Berlin
Tel +49 (30) 254 11 100
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CONCEPT, WEB DESIGN & PROGRAMMING
incorporate berlin GmbH & CO. KG


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10+ Best Social Media Management Softwares 2023
Sprinklr Modern Engagement
Statusbrew
Khoros Marketing
Loomly Social Media
AgoraPulse
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Sendible
The Best Social Media Management Tools
Buffer for straightforward social media scheduling. Hootsuite for all-in-one social media scheduling, monitoring, and analytics. Sprout Social for team-based social media management. Iconosquare for managing Instagram business accounts. Sendible for lead generation.
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Zoho Social 4.6 Sridhar Vembu $8.01 1996 $500 M 10+ Chennai 1777+ Reviews 40000000+
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Hootsuite 4.8 Ryan Holmes $16.83 2008 $393.8 M 1504+ Canada 1784+ Reviews 17000000+
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Sendible is a social media management platform that is tailored to suit the needs of all businesses. The app aims to help users manage multiple social media pages, including Google+, Facebook, Twitter, Pinterest, and LinkedIn. With Sendible, users can track social media engagement in real-time and create bulk posts for their followers. Sendible ensures to make social media management a lot easier on its user's end. Established in 2009, Sendible was designed to elevate social media management across every platform. Sendible was programmed to allow users to plan, post, and measure social media content, making the platform the ideal social media management tool for various users. With over 15,000+ agencies, marketers, and brand managers that leaned on Sensible's social media management capabilities, the platform becomes a part of the best social media management software today, together with other platforms like Agorapulse, Hootsuite, Sprinklr Modern Engagement, Loomly Social Media, Khronos Marketing, and Buffer.
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Zoho Social is an advanced cloud-based social media management software programmed by the Zoho Corporation. Zoho Social aims to help companies manage their social media pages by monitoring keywords and scheduling their posts. With Zoho Social, users can also have access to its high-end social media management features, such as content sharing, bulk scheduling, team collaboration, unified messaging, and so much more. Taking back to its history, Zoho Social was primarily released by Zoho Corporation in 2015 to help growing businesses build their social media presence. Given that the 21st-century setting revolves around social media, especially for business marketing, Zoho Social emerge to leverage social media processes in one platform. Since its release, Zoho Social becomes the main platform for thousands of agencies and individual users around the globe, including leading organizations like WunderTRE, Widobizz, dotConverse, Securolytics, and Tabletop Records. Zoho Social also becomes the primary competitor for other social media management platforms, such as Sprinklr Modern Engagement, Loomly Social Media, Khronos Marketing, Sendible, Agorapulse, Hootsuite, Buffer, and Sendible.
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Buffer is one of the most trusted social media management software in the market today. Programmed to leverage social media management, Buffer emerge to simplify social media management leading to streamlined engagement and workflow. With the platform's wide array of tools for social media publishing, engagement, analytics, and team collaboration, the Buffer app guarantees the best services every social media marketer and team deserve. Established in 2010, Buffer emerge through the idea of Joel Gascoigne in Birmingham, United Kingdom. With a goal to deliver simple publishing, analytics, and engagement service for social media, Buffer rapidly captures multiple users since its establishment. As of today, more than 75,000+ customers backed on Buffer's social media management capabilities, which includes renowned organizations like Shopify, Spotify, Huckberry, DressUp, Food52, and Burrow. With the platform's continuous popularity among users, Buffer was also branded as today's one of the best social media management platforms, competing against other leading software like Khronos Marketing, Loomly, Agorapulse, Zoho Social, Statusbrew, and Sprinklr Modern Engagement.
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When we talk about social media management software, it is quite impossible that one cannot recognize Hootsuite. Hootsuite is recognized as one of the most popular and most used social media management software in today's market. This software perfectly caters to every social media management and project on every platform such as Facebook, Instagram, Twitter, YouTube, and more. Regardless of what's the size of the business, guaranteed that Hootsuite can provide the best social media management solutions. Established in 2008 by Ryan Holmes, Hootsuite provides social media management that organizes social media accounts and networks. Since its establishment, Hootsuite instantly becomes the most preferred social media management software, thanks to its easy-to-use capabilities. Today, with over 17 million customers that trust Hootsuite, the platform was recognized as one of the best social media management software to use. Hootsuite also becomes the primary competitor for other social media management providers like Buffer, Sprout Social, Zoho Social, Sendible, Agorapulse, Loomly, and Khoros Marketing.


The California Institute for Regenerative Medicine (CIRM) is a major funding agency for stem cell research in the Golden State. Since voters approved the establishment of the agency in 2004, the CIRM has spent billions on research and facilities with the aim of making California the stem cell capital of the US. Now, in a move to advance that research mission even further, the agency has announced awards of $32M to investigators and stem cell companies to create a biobank of diseased cell lines for the use of researchers around the world. Called the Human Induced Pluripotent Stem Cell (hiPSC) Initiative Awards, the project will generate and ensure the availability of high quality disease-specific hiPSC resources for disease modeling, target discovery and drug discovery and development for prevalent, genetically complex diseases.


I seem to be having a bit of an IKEA theme going on at the moment.
Following on from the Dream Kitchen site there's a nice new piece called The Complete Bedroom.
It uses the same kind of multi-angle video technique that the other versions use, though this one is simpler. It has some nice quick-cut segments as you switch between each bedroom to contrast the hectic lives of those we are observing with the relaxing bedroom zone.
In terms of interactivity it's very simple and the charm is really down to the filming and music, which is what ad guys are good at. But I think it's a good blend of digital and traditional approaches.
Hope your dream bedroom has memory foam beds inside it:)) Never forget about that because such a bed can make you sleep as a baby and we all seek that, don't we?
Okay, so this is a spam comment, but I'm replying anyway.
actually cheap rubbish. My next mattress will be a proper sprung one.


Research indicates that who and how a leader is "BEing" has a greater impact on performance than what a leader does. Whether you're an entrepreneur, solopreneur, or an executive / manager in a company, you are in a leadership and decision-making role.
So who and how should a leader "BE"? What defines the mind of a leader? What results should high performance teams deliver?
As a leader, Your business is the vehicle through which your purpose is delivered to the world. Every successful brand and business became that way because of a unique alignment between the vision for what is possible and the operations of the business. Apple, Zappos, Wal-Mart, Microsoft, Amazon, Google, Salesforce, Starbucks, Mars, Inc., and many others were all vision-aligned around providing something special and impactful to the world. These companies were not started as a means just to make money. Money was the result of alignment and authentic service.
We call businesses that experience such alignment a "Being Based Business". The reason is the concern of the founders was who they and their business was being in the world first, and monetization second. As a result of this aligned perspective, these businesses experienced flow and unparalleled growth.
NeuroBe works with small and mid-sized companies to ensure alignment between Intention, Vision, Leadership, and Operations. Using neuroscience mapping and principles we work from our proprietary curriculum to design and develop high performance leaders, deconstruct and reconstruct the ethos and pathos of your business, and ensure that your vision, mission and operations are in perfect alignment with the founding intention of the company. We then go further, ensuring that your business maps to a strategic actionable plan for the next year of operations.
The "aligned by Labor Day" program is exclusively for Solopreneurs, small business owners and mid-sized companies (under 500 people) wishing to access and Implement the Being Based Business program. Those who sign up before June 30, 2019 will receive a $1000 discount off of the monthly rate of $1499/mth. This means that for only $499/mth you will receive the same 4 hours/mth of one-on-one Being Based Business coaching.
Note: Minimum commitment is 3-months, and then a month to month plan is available at the same $499/mth rate.
Whether it is for the development of a new brand, an advertising campaign, or massaging for marketing and promotional collateral, or a political campaign, get the advantage by designing perception through the integration of neuroscience and behavioral psychology principles into your marketing and messaging efforts.
NeuroMarketing is the use and integration of neuroscience and behavioral psychology research into your marketing strategy. This includes everything from branding, customer / voter psychographic and demographic ID, and messaging (all marketing collateral, digital identity, promotional / GOTV tactics). Get the upper hand by intentionally designing your marketing to activate your customer, versus hoping that by copying the mere tactics of what everyone is also doing, you will somehow stand out. You will never stand out when stuck in and doing the same things as the crowd!
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A Design-Thinking process that provides clients access points to tackle the a) known, b) known unknown, c) unknown unknown dynamics limiting success. Design your business strategy to align with your original intention, and the outcomes you seek in the world.
DNA identifies challenges and opportunities within your current business strategy, operational model and/or marketing strategy through a process of deconstruction/reconstructon. It develops and connects ideas, where ideas are treated as models to be tested and explored, not judged or condemned. It then accelerates action towards the execution of the design to integrate it into your businesses operational strategy, and other essential systems.


Hey everyone. I am feeling much better thanks! 💫 I think I will be pushing #FortuneFriday to next week. This way everyone can take a break since I know some people have probably spent a lot of money because of the holidays.


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Army runner to compete at U.S. Junior Nationals
West Point — Training runs through the woods would not be unusual for distance runners entered at the U.S. Junior National championships, but it's unlikely any of them also had to sling a rifle, as David Weart did.
Weart, who will be a sophomore this fall at Army, spent the past week at Camp Buckner working on his land-navigation skills.
His task of orienteering involved being handed a map, a compass and being sent into the woods to find 10 forest signs inside of four hours.
Battle fatigues and army boots were the dress requirements, complemented by survival gear and an M-16 rifle.
"It's not what you want to be doing before a big race," Weart said before his training began.
Yesterday was a rare off day for Weart and he hopes he gets his legs back during his cross-country flight. The James I. O'Neill graduate and former state champion will race in the 5,000-meter run at 8:46 p.m. WDT today — 11:46 p.m. here — at the Home Depot Center in Carson, Calif.
"It's not the best week leading up to a big race like that," Weart said, "but I am excited for the opportunity to go out to California. It will be a lot of fun, and I hope to be in the top eight to be a junior all-America, and try to improve on my (personal record)."
Weart qualified for the Junior Nationals (reserved for athletes age 19 and younger) with a 15-minute finish at an indoor meet in Boston. He dropped that time to 14:57.92 during a second-place finish at the Indiana Relays during the outdoor season.
"He took his time off between seasons, more like during spring break," Army coach Jerry Quiller said. "He is ready to go and zero in on this thing and then train right through the summer for cross country. He's on a roll, physically fresh, healthy and in great shape."
There were plenty of health issues during the season. Shortly after his qualifying run in Boston, Weart missed a month of action with bronchitis and a foot injury.
His outdoor season started slowly and he couldn't muster any quality runs until six weeks ago, too late to do anything at the Patriot League meet or to qualify for the IC4A championships.
The indoor setback was actually his second since last fall. Weart got off to a surprisingly strong start in his rookie year, becoming the only freshman to make the cross-country travel squad.
He handled the large jump from 5-kilometer high school runs to 8-kilometers quite well, but the rigors of freshman life at West Point caught up to him in the middle of cross-country season.
"Running was a lot tougher," Weart said. "We were increasing the mileage, there was an increased academic load, I was constantly tired, it was tough staying motivated. I guess living here at West Point I had a false impression of what to expect."
Quiller said Weart has bounced back well from his recent health problems, and his recent three-week training period at his parents' new home in Charleston, S.C., has brought him to the best shape of his life.
"I am definitely race-ready," Weart said.
Weart's classmate, John Mickowski, runs today in the 1,500-meter run. The Mukwonago, Wis., resident broke the four-minute barrier twice in three spring races, clocking 3:57.22 for a fifth-place finish against Navy.


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Set Includes: 1½-qt., 2-qt. & 5-qt. Dutch ovens with lids, strainers & pouring spouts, 9½" frying pan & 5 serving utensils.


If there is only one place to go in Athens, I'd head straight to Anfiotika.
Don't get me wrong. Athens is quite an amazing city. The Acropolis is mindblowing. The food amazing and the coffee incredible. Athens, in a way, reminds me a lot of Hanoi–that bustling, chaotic city with too much heat and too much traffic. Just like how I cannot quite identify my feeling for Hanoi, be it love, fear, hatred, confusion–I am not sure how I felt about Athens. Athens is so different from the Western European cities I have been to; it possesses a totally different vibes from Mediterranean destinations I have visited. I tell everyone I know to visit Athens because, heck, it's cheap, and I usually stop there.
But if there is one place that makes me want to go back to Athens, again and again and again, it must be Anafiotika.
As I mentioned earlier, taking in the view of Lisbon from one of its many miradouros is hands down the ultimate Lisbon experience. But don't ever think only from a high altitude can you enjoy Lisbon's distinctive beauty, a walk through the streets of this fine city, especially in Alfama–the oldest neighborhood of Lisbon, is nothing sort of an incredible experience.
Finally, finally I got around to write about Lisbon–the last city I visited during my winter break. I booked roundtrip tickets to Lisbon on a whim, simply because the prices on easyJet were tempting and because I wanted to fill my free days with as much traveling as possible (come to think of it, during my 5 week winter break, I only spent 3 days at home!). I did not expect much, to be honest. All that I had known of Lisbon and Portugal were 1) my favorite pastry in the world aka pastel de nata aka egg tart came from Lisbon so I was sure in for a treat and 2) Portugal is not Spain. Confident with my immeasurable knowledge of the world, I hopped on a late flight and arrived in Lisbon at 11pm on a fine winter night.
Warning: heavy flow of photos ahead.
I'm sure we all have heard about the fabled Keukenhof of the Netherlands, otherwise known as 'the garden of Europe', where, every Spring, around 7 million bulbs of flowers vie with each other demanding we miserable ugly human beings' attention and admiration. The legend is true. But, oh, the legend you hear is nothing compared to what you can actually see in Keukenhof in real life.
As an expat living in the Netherlands (or anywhere in Europe for that matter), not making a pilgrimage to Keukenhof from mid-March to mid-May every year is a crime. Simple as that. I am all for a clean record, so, yesterday, amid a shitstorm of papers, assignments and other unnamed tasks, a friend and I still braced ourselves for an adventure into the real heart of this windmill-tulip-clog-cheese-dike country.
Prepare yourselves for the best of cherry blossoms at Kyunghee University because boy do they bloom and beam right to the depth of your soul.
Continuing with my own edition of spring destinations in Seoul, this time I've chosen two places that I think are what make this flower admiring season a bit different from your experiences (if you're not from Korea of course). Introduce to you Deoksugung palace & Changgyeonggung palace.