text:
Derivation of the sieve
In the 1940's, Selberg found a way to sharpen the already good inequalities given by the Brun sieve, by using a different method. The basic principle is astonishingly simple, but turns out to be fruitful. For any set $A\subset \mathbb{Z}_+$(with $x$ elements) and any set $P$ of primes, we have
\[\begin{eqnarray}S(A,P,z)=\sum_{a\in A\atop (a,P(z)=1)}1\leq \sum_{a\in A}\left(\sum_{d\mid a\atop d\mid P(z)} \lambda_d\right)^2\end{eqnarray}\]
for arbitrary real numbers $\lambda_1,\lambda_2,...$ with $\lambda_1=1$. Simplifying, this expression becomes
\[\begin{eqnarray}\sum_{d_1,d_2\mid P(z)}\lambda_{d_1}\lambda_{d_2}A_{[d_1,d_2]}\end{eqnarray}\]
(where $[\cdot,\cdot]$ denotes the least common multiple), for each term $\lambda_{d_1}\lambda_{d_2}$ is counted as many times as there are elements of $A$ that are divisible by $d_1$ and $d_2$. Writing $|A_d|=\frac{\omega(d)}{d}x+R_d$ as usual, with $\omega(\cdot)$ being multiplicative, we obtain
\[\begin{eqnarray}S(A,P,z)\leq x\sum_{d_1,d_2\mid P(z)}\frac{\omega([d_1,d_2])}{[d_1,d_2]}\lambda_{d_1}\lambda_{d_2}+\sum_{d_1,d_2\mid P(z)}|\lambda_{d_1}\lambda_{d_2}||R_{[d_1,d_2]}|\quad (1).\end{eqnarray}\]
The first term is regarded as the main term, while the second is the error. The basic principle is just that we should minimize the main term as a function of the $\lambda_d$'s, with the constraint $\lambda_1=1$. For convenience, we assume $\lambda_d=0$ for $d>z$. Our task is just to minimize a quadratic form, but setting all partial derivatives equal to zero would lead to a rather unpleasant looking system of equations, so we first diagonalize the form. Using $mn=(m,n)[m,n]$, the form becomes
\[\begin{eqnarray}\sum_{d_1,d_2\mid P(z)}\frac{(d_1,d_2)}{\omega((d_1,d_2))}\frac{\lambda_{d_1}\omega(d_1)}{d_1}\cdot \frac{\lambda_{d_2}\omega(d_2)}{d_2}.\end{eqnarray}\]
We have $\frac{p}{\omega(p)}=1+\frac{\omega(p)}{p-\omega(p)}=1+\frac{1}{g(p)}$, where $g(p)$ is as in earlier posts (it is defined like this for primes and it is extended multiplicatively to square free numbers), so we can rewrite the above expression as
\[\begin{eqnarray}&&\sum_{d_1,d_2\mid P(z)}\prod_{p\mid (d_1,d_2)}\left(1+\frac{1}{g(p)}\right)\frac{\lambda_{d_1}\omega(d_1)}{d_1}\cdot \frac{\lambda_{d_2}\omega(d_2)}{d_2}\\&=&\sum_{d_1,d_2\mid P(z)}\sum_{k\mid(d_1,d_2)}\frac{1}{g(k)}\frac{\lambda_{d_1}\omega(d_1)}{d_1}\cdot \frac{\lambda_{d_2}\omega(d_2)}{d_2}\\&=&\sum_{k\mid P(z)}\frac{1}{g(k)}\left(\sum_{d\mid P(z)\atop d\equiv 0\pmod k}\frac{\omega(d)}{d}\lambda_d\right)^2.\end{eqnarray}\]
Now denote
\[\begin{eqnarray}\xi_k=\sum_{d\mid P(z)\atop d\equiv 0\pmod k}\frac{\omega(d)}{d}\lambda_d;\end{eqnarray}\]
we are going to minimize the diagonal form $\sum_{k\mid P(z)}\frac{1}{g(k)}\xi_k^2$ subject to $\lambda_1=1$ and $\lambda_d=0$ for $d>z$ (for convenience, $\lambda_d$ is defined for every positive integer $d$ but is supported of the numbers up to $z$ dividing $P(z)$). First we should express the condition $\lambda_1=1$ in terms of the $\xi_k$'s.
\[\begin{eqnarray}\xi_1=\sum_{d=1}^{\infty}\frac{\omega(d)}{d}\lambda_d,\quad \xi_2=\sum_{d=1}^{\infty}\frac{\omega(2d)}{2d}\lambda_{2d},...,\end{eqnarray}\]
inclusion and exclusion gives
\[\begin{eqnarray}1=\lambda_1\omega(1)&=&\sum_{n=1}^{\infty}\mu(n)\xi_n=\sum_{n\mid P(z)\atop n\leq z}\mu(n)\xi_n,\end{eqnarray}\]
and more generally
\[\begin{eqnarray}\frac{\omega(d)}{d}\lambda_d&=&\sum_{n=1}^{\infty}\mu(n)\xi_{dn}=\sum_{n\mid P(z)\atop n\leq \frac{z}{d}}\mu(n)\xi_{dn}\quad \quad\quad (2).\end{eqnarray}\]
We can now easily minimize the quadratic form under consideration. By Cauchy-Schwarz,
\[\begin{eqnarray}1=\sum_{n\mid P(z)\atop n\leq z}\mu(n)\xi_n&=&\sum_{n\mid P(z)}\mu(n)\sqrt{g(n)}\cdot \frac{\xi_n}{\sqrt{g(n)}}\\&\leq& \left(\sum_{n\mid P(z)\atop n\leq z}\mu^2(n)g(n) \sum_{n\mid P(z)\atop n\leq z} \frac{1}{g(n)}\xi_n^2\right)^{\frac{1}{2}},\end{eqnarray}\]
\[\begin{eqnarray}\sum_{n\mid P(z)\atop n\leq z} \frac{1}{g(n)}\xi_n^2\geq \left(\sum_{n\mid P(z)\atop n\leq z}\mu^2(n)g(n)\right)^{-1}:=\frac{1}{G(z)}.\end{eqnarray}\]
Equality is attained if and only if $\xi_n=\frac{\mu(n)g(n)}{G(z)}$ for all $n\leq z$ such that $n\mid P(z)$.
After this, we just need to estimate the error term in $(1)$. We make the usual assumption $|R_d|\leq \omega(d)$. It turns out that actually $|\lambda_d|\leq 1$ for all $d$. By $(2),$ the parameters $\lambda_d$ are given by
\[\begin{eqnarray}\lambda_d&=&\frac{d}{\omega(d)}G(z)^{-1}\sum_{n\mid P(z)\atop n\leq \frac{z}{d}}\mu(n)\mu(dn)g(dn)\\&=&\frac{d\mu(d)g(d)}{\omega(d)}G(z)^{-1}\sum_{n\mid P(z)\atop n\leq \frac{z}{d}, (n,d)=1}\mu^2(n)g(n)\\&=&\frac{\mu(d)}{\prod_{p\mid d}\left(1-\frac{\omega(p)}{p}\right)}G(z)^{-1}\sum_{n\mid P(z)\atop n\leq \frac{z}{d}, (n,d)=1}\mu^2(n)g(n)\\&:=&\frac{\mu(d)}{\prod_{p\mid d}\left(1-\frac{\omega(p)}{p}\right)}\cdot\frac{G_d(\frac{z}{d})}{G(z)},\end{eqnarray}\]
where we use the fact that $\mu(nd)=0$ unless $(n,d)=1$.
To see that $\lambda_d$ is bounded by $1$ in modulus, we notice that
\[\begin{eqnarray}G(z)&=&\sum_{\ell\mid d}\sum_{\substack{n\mid P(z)\\n\leq z\\ (n,d)=\ell}}\mu^2(n)g(n)=\sum_{\ell\mid d}\sum_{\substack{m\mid P(z)\\m\leq \frac{z}{\ell}\\ (m,d)=\ell}}\mu^2(m\ell)g(m\ell)\\&=&\sum_{\ell\mid d}\mu^2(\ell)g(\ell)G_{\ell}\left(\frac{z}{\ell}\right)\geq \sum_{\ell\mid d}\mu^2(\ell)g(\ell)G_d\left(\frac{z}{d}\right)\\&=&\prod_{p\mid d}(1+g(p))\cdot G_d\left(\frac{z}{d}\right)=\prod_{p\mid d}\frac{1}{1-\frac{\omega(p)}{p}}\cdot G_d\left(\frac{z}{d}\right),\end{eqnarray}\]
so indeed $|\lambda_d|\leq 1$. Now the error term in $(1)$ is at most
\[\begin{eqnarray}\sum_{d_1,d_2\mid P(z)\atop d_1,d_2\leq z}|R_{[d_1,d_2]}|\quad \quad (3).\end{eqnarray}\]
The number of pairs of integers with $[d_1,d_2]=d$ for a fixed square-free $d$ is $3^{\nu(d)}$. Namely, if $d=p_1...p_{\nu(d)}$, then $d_1=\prod_{i\in S}p_i$ for some $S\subset\{1,...,\nu(d)\}$, so $d_2=\prod_{i\in T}p_i$ for some $T$ containing $\{1,...,\nu(d)\}\setminus S.$ If the size of $S$ is $r$, there are $\binom{\nu(d)}{r}$ choices for $S$ and $2^{\nu(d)-r}$ choices for $T$, so the binomial identity $\sum_{r=0}^{\nu(d)}\binom{\nu(d)}{r}2^r=3^{\nu(d)}$ proves the desired statement.
Hence the error is not greater than
\[\begin{eqnarray}\sum_{d\mid P(z)\atop d\leq z^2}3^{\nu(d)}\omega(d)&\leq& z^2\sum_{d\mid P(z)\atop d \leq z^2}\frac{3^{\nu(d)}\omega(d)}{d}\\&=&z^2\prod_{p\leq z\atop p\in P}\left(1+\frac{3\omega(p)}{p}\right)\\&\leq& z^2\prod_{p\leq z\atop p\in P}\left(1+\frac{\omega(p)}{p}\right)^3 &\leq& z^2\prod_{p\leq z\atop p\in P}\left(1-\frac{\omega(p)}{p}\right)^{-3}\\&:=&\frac{z^2}{W(z)^3}.\end{eqnarray}\]
We have now derived Selberg's sieve.
Theorem 1. Let $A$ be an arbitary set of positive integers with $x$ elements, and $P$ an arbitrary subset of primes. Assume $A_d=\frac{\omega(d)}{d}+R_d$ with $\omega(d)$ multiplicative and $|R_d|\leq \omega(d)$. Then for any $z\geq 2$,
\[\begin{eqnarray}S(A,P,z)\leq \frac{x}{G(z)}+\frac{z^2}{W(z)^3}.\end{eqnarray}\]
Without assuming $|R_d|\leq \omega(d)$, we still have
\[\begin{eqnarray}S(A,P,z)\leq \frac{x}{G(z)}+\sum_{d\leq z^2\atop d\mid P(z)}3^{\nu(d)}|R_d|.\end{eqnarray}\]
Without assuming $|R_d|\leq \omega(d)$, we still have
\[\begin{eqnarray}S(A,P,z)\leq \frac{x}{G(z)}+\sum_{d\leq z^2\atop d\mid P(z)}3^{\nu(d)}|R_d|.\end{eqnarray}\]
Proof. Given above. ■
The inequalities above has many good features. First, the assumptions are minimal. Second, $z$ is not assumed to be large enough and we do not need to adjust any further parameters. Third, and most importantly, it gives sharp results, as we will see below.
Applications to primes in arithmetic progressions
We intend to give good upper bounds for the number of primes on (short) intervals using Selberg's sieve. When sifting the primes in the arithmetic progression $qn+a$, we should take $A=\{n\geq 0:qn+a\in [x-y,x]\}$, $P=\mathbb{P}\setminus\{p:p\mid q\}$ and $\omega(p)=1$ for $p\in P$, $\omega(p)=0$ for other primes. The first thing we have to do is to estimate $G(z)$ in this situation. As $g(p)=\frac{1}{p-1}=\frac{1}{\varphi(p)}$, we have $g(d)=\frac{1}{\varphi(d)}$ for $d$ coprime to primes not in $P$. Therefore
\[\begin{eqnarray}G(z)=H_q(z):=\sum_{d\leq z\atop (d,q)=1}\frac{\mu^2(d)}{\varphi(d)}.\end{eqnarray}\]
Notice that
\[\begin{eqnarray}H_1(z)=\sum_{\ell\mid q}\sum_{d\leq z\atop (d,q)=\ell}\frac{\mu^2(d)}{\varphi(d)}=\sum_{\ell\mid q}\frac{\mu^2(\ell)}{\varphi(\ell)}\sum_{m\leq \frac{z}{\ell}\atop (m,q)=1}\frac{\mu^2(m)}{\varphi(m)}=\sum_{\ell\mid q}\frac{\mu^2(\ell)}{\varphi(\ell)}H_q\left(\frac{z}{\ell}\right),\end{eqnarray}\]
\[\begin{eqnarray}H_q(z)\geq \frac{H_1(z)}{\sum_{\ell\mid q}\frac{\mu^2(\ell)}{\varphi(\ell)}}=H_1(z)\prod_{p\mid q}\left(1-\frac{1}{p}\right).\end{eqnarray}\]
Hence it suffices to estimate $H_1$ from below in order to give an upper bound for $\frac{1}{G(z)}$. It holds that
\[\begin{eqnarray}H_1(z)=\sum_{d\leq z}\frac{\mu^2(d)}{d}\prod_{p\mid d}\left(1+\frac{1}{p}+\frac{1}{p^2}+...\right)\geq \sum_{n\leq z}\frac{1}{n}\geq \log z\end{eqnarray}\]
because every positive integer $n\leq z$ can be written in the form $n=dp_1^{k_1}...p_r^{k_r}$ where $d\leq z$ is square-free and $p_1,...,p_r\mid d$ are primes. We have now
\[\begin{eqnarray}H_q(z)\geq \frac{\varphi(q)}{q}\log z,\quad (5)\end{eqnarray}\]
so we conclude
\[\begin{eqnarray}S(A,P,z)\leq \frac{y}{\varphi(q)\log z}+\frac{z^2}{W(z)^3}.\end{eqnarray}\]
However, the error term can be improved. As $|R_d|\leq 1$ for all $d$, returning to (3) and estimating that error term trivially (using the fact that the number of square-free $d\leq z$ is certainly less than $z-1$), we get
\[\begin{eqnarray}\pi(x;q,a)-\pi(x-y;q,a)\leq \frac{y}{\varphi(q)\log z}+(z-1)^2+\pi(z)+1\leq \frac{y}{\varphi(q)\log z}+z^2.\end{eqnarray}\]
Taking $z=\frac{\sqrt{y}}{\sqrt{\varphi(q)}\log y}$, we get
\[\begin{eqnarray}\pi(x;q,a)-\pi(x-y;q,a)&\leq& \frac{2y}{\varphi(q)\log \frac{y}{q\log^2 y}}+\frac{y}{\varphi(q)\log^2 y}\\&\leq& \frac{2y}{\varphi(q)\log \frac{y}{q\log^2 y}}\left(1+\frac{1}{2\log y}\right).\quad (4)\end{eqnarray}\]
This is the Brun-Titchmarsh inequality, except that the second factor could be removed and $\log \frac{y}{q\log^2 y}$ replaced with $\log\frac{y}{q}$ (in that case, the inequality is due to Montgomery and Vaughan). One can obtain the latter improvement (while increasing the second factor to $1+\frac{8}{\log \frac{y}{q}}$) by estimating the error term $(\sum_{d\leq z,(d,q)=1}\lambda_d)^2$ very accurately using the explicit formula for $\lambda_d$ and rather tedious computations that bound the error by $24\frac{q}{\varphi(q)}\frac{z^2}{\log^2 z}$ for $z\geq e^6$; see Halberstam and Richert. With even more careful estimation of the sum over $\lambda_d$, one can prove the inequality of Montgomery and Vaughan. In any case, we arrived at the following.
Theorem 2. For any fixed $q$, we have
\[\begin{eqnarray}\pi(x;q,a)-\pi(x-y;q,a)\leq \frac{(2+o(1))y}{\varphi(q)\log y}\end{eqnarray}\]
as $y\to \infty$.
We make several remarks about this inequality. First of all, it is off by a factor of two from what we expect and what we have in the prime number theorem. Nevertheless, the Brun-Titchmarsh inequality gives upper bounds that are (conjectured to be) correct up to the factor $2$ on intervals on which the prime number theorem or the Riemann hypothesis say nothing about the number of primes. Indeed, let $\pi(x;q,a)=\frac{1}{\varphi(q)}Li(x)+E(x;q,a)$, where $Li(x)=\int_{2}^{x}\frac{dt}{\log t}$ is the logarithmic integral and the approximation is based on the heuristic that $n$ is a prime congruent to $a\pmod q$ with "probability" $\frac{1}{\varphi(q)\log n}$. Then
\[\begin{eqnarray}\pi(x;q,a)-\pi(x-y;q,a)&=&\frac{1}{\varphi(q)}(Li(x)-Li(x-y))+E(x;q,a)-E(x-y;q,a)\\&\leq& \frac{(1+o(1))y}{\log x}+E(x;q,a)-E(x-y;q,a).\end{eqnarray}\]
Using the error bound $E(x;q,a)\leq\frac{c_{q,k} x}{\log^k x}$ in the prime number theorem for arithmetic progressions, we get
\[\begin{eqnarray}\pi(x;q,a)-\pi(x-y;q,a)\leq \frac{(1+o(1))y}{\log x}+\frac{c_{q,k}x}{\log^k x}.\end{eqnarray}\]
because we can expect no cancellation in the two errors. Despite being a good bound on long intervals, as soon as $y=o\left(\frac{x}{\log^{k}x}\right)$, the bound becomes worse than the trivial one. Assuming the generalized Riemann hypothesis , we have
\[\begin{eqnarray}\pi(x;q,a)-\pi(x-y;q,a)\leq \frac{(1+o(1))y}{\log x}+c_q\sqrt{x}\log x,\end{eqnarray}\]
and even this is useless on intervals of length $y\ll \sqrt{x}$. Therefore, the theorem above and its refinements are very powerful tools on short intervals. In addition, formulas such as $(4)$ or the inequality of Montgormery and Vaughan are completely explicit unlike the prime number theorem (there are explicit versions of the prime number theorem though, for instance by Rosser and Schoenfeld, but proving them is more complicated than proving the explicit inequalities mentioned above. For primes in arithmetic progressions the possibility of an exceptional Siegel zero of an $L$-function complicates estimations a lot). It is also good to note that the constant $2$ in the Brun-Titchmarsh inequality has not been improved, and in fact cannot be without proving that there are no Siegel zeros.
Upper bounds for Goldbach's problem and linear equations in primes
Next we consider applications of Selberg's sieve to the number of representations of an integer as the sum of two primes and the number of primes $p$ such that $q=ap+b$ is also prime. Such primes include the twin primes and the Sophie Germain primes. The refined Brun's sieve from this previous post would already give bounds for the number of these integers, but with an unspecified constant factor (or a rather large constant if one makes everything explicit). There are however a couple of good reasons to do the necessary work with Selberg's sieve. One is that the estimates we get are only off by the factor $2$ from the conjectured ones, as in the Brun-Titchmarsh inequality, and improving the constant further would require proving that there are no Siegel zeros. Another reason is that the proof is a nice application of the Bombieri-Vinogradov theorem, and assuming it the proof is not very complicated.
To consider the number of representations of an even integer $N$ as the sum of two primes, set $A=\{N-p:p\leq N,p\in \mathbb{P}\}$, $P=P_N:=\mathbb{P}\setminus\{p:p\mid N\}$. The reason we do not sift all the integers in $A'=\{n\leq x:n(N-n)\}$ by the primes up to $\sqrt{N}$ and coprime to $N$ is that for this set $A'$ the function $\omega(d)$ would be $2^{\nu(d)}$ for $d$ coprime to $N$, while for the set $A$ that we choose it is much smaller, namely $\frac{d}{\varphi(d)}$.
We are interested in the primes in $A$, and we are not sifting with the prime divisors of $N$ as $A_p=1$ for $p\mid N$. Let us write $|A_d|=\frac{1}{\varphi(d)}\pi(N)+E(N,N,d)$ for $d$ square-free and coprime to $N$. Here $E(x;q,a)$ has the same meaning as above. The most efficient proofs of the prime number theorem in arithmetic progressions give essentially $E(x;q,a)\ll_q x\exp(-(\log x)^{\frac{1}{3}})$ (a bit better), but still the error $E(N,N, d)$ is not even close to being bounded by $\omega(d)=\frac{d}{\varphi(d)}$ since the error depends crucially on $N$. The Bombieri-Vinogradov theorem is therefore required to establish that the error is in fact small when we average over $d\leq z$.
We should first compute $G(z)$. Note that $g(p)=\frac{\frac{p}{p-1}}{p-\frac{p}{p-1}}=\frac{1}{p-2}=\frac{1}{p-1}(1+g(p))$, so
\[\begin{eqnarray}G(z)&=&\sum_{n\leq z\atop (n,N)=1}\mu^2(n)g(n)=\sum_{n\leq z\atop (n,N)=1}\frac{\mu^2(n)}{\varphi(n)}\prod_{p\mid N}(1+g(p))\\&=&\sum_{n\leq z\atop (n,N)=1}\frac{\mu^2(n)}{\varphi(n)}\sum_{d\mid N}g(d)\\&=&\sum_{d\leq z\atop (d,N)=1}g(d)\sum_{\substack{n\leq z\\(n,N)=1\\ d\equiv 0\pmod N}}\frac{\mu^2(n)}{\varphi(n)}\\&=&\sum_{d\leq z\atop (d,N)=1}\frac{\mu^2(d)}{\varphi(d)}g(d)\sum_{k\leq \frac{z}{d}\atop (k,dN)=1}\frac{\mu^2(k)}{\varphi(k)}\\&\geq&\prod_{p\mid N}\left(1-\frac{1}{p}\right) \sum_{d\leq z\atop (d,N)=1}\frac{\mu^2(d)}{\varphi(d)}g(d)\prod_{p\mid d}\left(1-\frac{1}{p}\right)\log\frac{z}{d}\\&=&\prod_{p\mid N}\left(1-\frac{1}{p}\right)\left(\sum_{d\leq z \atop (d,N)=1}\frac{\mu^2(d)g(d)}{d}\log z-\sum_{d\leq z \atop (d,N)=1}\frac{\mu^2(d)g(d)}{d}\log d\right)\end{eqnarray}\]
where the inequality follows from formula $(5)$. The expression above is further equal to
\[\begin{eqnarray}\prod_{p\mid N}\left(1-\frac{1}{p}\right)\left(\sum_{d=1\atop (d,N)=1}^{\infty}\frac{\mu^2(d)g(d)}{d}\log z-\sum_{d=1\atop (d,N)=1}^{\infty}\frac{\mu^2(d)g(d)}{d}\log d+O\left(\frac{\log z}{z}\right)\right)\end{eqnarray}\]
We simplify the sums as follows:
\[\begin{eqnarray}\sum_{d=1\atop (d,N)=1}^{\infty}\frac{\mu^2(d)g(d)}{d}&=&\prod_{p\nmid N}\left(1+\frac{g(p)}{p}\right)\\&=&\prod_{p\nmid N}\left(1+\frac{1}{p(p-2)}\right),\\\sum_{d=1\atop (d,N)=1}^{\infty}\frac{\mu^2(d)g(d)}{d}\log d&=&\sum_{d=1\atop (d,N)=1}^{\infty}\frac{\mu^2(d)g(d)}{d}\sum_{p\mid d}\log p\\&=&\sum_{p\nmid N}\log p\sum_{\substack{d=1\\(d,N)=1\\d\equiv 0\pmod p}}^{\infty}\frac{\mu^2(d)g(d)}{d}\\&=&\sum_{p\nmid N}\log p\cdot\frac{g(p)}{p}\sum_{k=1\atop (k,pN)=1}^{\infty}\frac{\mu^2(k)g(k)}{k}\\&=&\prod_{p\nmid N}\left(1+\frac{1}{p(p-2)}\right)\sum_{p\nmid N}\frac{\log p}{p(p-2)}\left(1+\frac{1}{p(p-2)}\right)^{-1}\\&=&\prod_{p\nmid N}\left(1+\frac{1}{p(p-2)}\right)\sum_{p\nmid N}\frac{\log p}{(p-1)^2}\end{eqnarray}\]
Putting this all together yields
\[\begin{eqnarray}&&S(A,P,z)\\&\leq& \pi(N)\prod_{p\nmid N}\left(1+\frac{1}{p(p-2)}\right)^{-1}\prod_{p\mid N}\left(1-\frac{1}{p}\right)^{-1}\left(\log z-\sum_{p}\frac{\log p}{(p-1)^2}\right)^{-1}\left(1+O(z^{-1})\right)\\&+&\sum_{d\leq z^2\atop (d,N)=1}\mu^2(d)3^{\nu(d)}|E(N;N,d)|,\quad\quad (5)\\\end{eqnarray}\]
by Selberg's sieve (Theorem 1). The two products can easily be rewritten as
\[\begin{eqnarray}\left(\frac{1}{2}\prod_{p>2}\left(1+\frac{1}{p(p-2)}\right)\prod_{2<p\mid N}\frac{p-2}{p-1}\right)^{-1}=2\prod_{p>2}\left(1-\frac{1}{(p-1)^2}\right)\prod_{2<p\mid N}\frac{p-1}{p-2},\quad (6)\end{eqnarray}\]
\[\begin{eqnarray}&S(A,P,z)\leq 2\mathcal{G}\prod_{2<p\mid N}\frac{p-1}{p-2}\cdot\frac{N}{\log N \log z}\left(1+O\left(\frac{1}{\log z}\right)\right)\\&+\sum_{d\leq z^2\atop (d,N)=1}\mu^2(d)3^{\nu(d)}|E(N;d,N)|,\end{eqnarray}\]
where $\mathcal{G}$ is the first product in $(6)$, and we used $(\log z-c)^{-1}=(\log z)^{-1}(1+O((\log z)^{-1}))$ and $\pi(N)=\frac{N}{\log N}(1+O(\frac{1}{\log N}))$. Now it remains to estimate the error term. By Cauchy-Schwarz,
\[\begin{eqnarray}\sum_{d\leq z^2\atop (d,N)=1}\mu^2(d)3^{\nu(d)}|E(N;d,N)|&\leq& \sum_{d\leq z^2\atop (d,N)=1}\sqrt{d}\cdot3^{\nu(d)}\cdot \frac{|E(N;d,N)|}{\sqrt{d}}\\&\leq& \left(\sum_{d\leq z^2}d\cdot 6^{\nu(d)}\right)^{\frac{1}{2}}\left(\sum_{d\leq z^2\atop (d,N)=1}\frac{|E(N;d,N)|^2}{d}\right)^{\frac{1}{2}}\\&\leq& \left(\sum_{d\leq z^2}d\cdot 6^{\nu(d)}\right)^{\frac{1}{2}}\left(\sum_{d\leq z^2\atop (d,N)=1}|E(N;d,N)|\right)^{\frac{1}{2}}.\end{eqnarray}\]
What the Bombieri-Vinogradov theorem says is that for every positive $M$ there exists a $c(M)$ such that
\[\begin{eqnarray}\sum_{1\leq q\leq \frac{\sqrt{x}}{\log^{c(M)}x}}\max_{y\leq x}\max_{a\in \mathbb{Z}_q^{\times}}|E(y;q,a)|\ll_{M} \frac{x}{\log^M x}.\end{eqnarray}\]
This inequality beats the bound following from the Brun-Titchmarsch inequality by an arbitrary power of $\log x$. The statement would follow from the generalized Riemann hypothesis, as then we had pointwise $|E(x;q,a)|\ll \sqrt{x}\log x\log q$, but the averaged version is something that can actually be proved and hence use as a replacement of the GRH in several situations. We choose $z^2=\frac{N}{\log^{c(12)}N}$, whence the error is bounded by
\[\begin{eqnarray}\left(\sum_{d\leq z^2}d\cdot 6^{\nu(d)}\right)^{\frac{1}{2}}\left(\sum_{d\leq z^2\atop (d,N)=1}|E(N;d,N)|\right)^{\frac{1}{2}}&\ll& \left(\sum_{d\leq z^2} 3^{\nu(d)}\right)^{\frac{1}{2}}\cdot \frac{\sqrt{N}}{\log^6 N}\\\end{eqnarray}\]
We finally calculate
\[\begin{eqnarray}\sum_{d\leq y}6^{\nu(d)}&\leq&y\sum_{d\leq y}\frac{6^{\nu(d)}}{d}\\&\leq& y\prod_{p\leq y}\left(1+\frac{6}{p}\right)\\
&\leq & y\prod_{p\leq y}\left(1+\frac{1}{p}\right)^6\\
&\ll& y\log^6 y.\end{eqnarray}\]
&\leq & y\prod_{p\leq y}\left(1+\frac{1}{p}\right)^6\\
&\ll& y\log^6 y.\end{eqnarray}\]
and setting $y=\frac{N}{\log^{c(12)}N},$ the error is at most
\[\begin{eqnarray}\ll N(\log N)^{3-\frac{1}{2}c(12)-6}\ll \frac{N}{\log^3 N}.\end{eqnarray}\]
Thus we obtain the following.
Theorem 3. If $N$ is an even positive integer, we have
\[\begin{eqnarray}|\{(p,p'):N=p+p'\}|\leq 8\mathcal{G}\prod_{2<p\mid N}\frac{p-1}{p-2}\cdot\frac{N^2}{\log^2 N}\cdot\left(1+O\left(\frac{\log \log N}{\log N}\right)\right).\end{eqnarray}\]
Proof. We choose $z^2=\frac{\sqrt{N}}{\log^{c(12)}N}$. Then $S(A,P,z)\geq |\{(p,p'):N=p+p'\}|+O(\sqrt{N})$. The error term in Selberg's sieve becomes irrelevant, and we have
Proof. We choose $z^2=\frac{\sqrt{N}}{\log^{c(12)}N}$. Then $S(A,P,z)\geq |\{(p,p'):N=p+p'\}|+O(\sqrt{N})$. The error term in Selberg's sieve becomes irrelevant, and we have
\[\begin{eqnarray}\frac{1}{\log \frac{\sqrt[4]{N}}{\log^{\frac{1}{4}c(12)}N}}=\frac{4}{\log N}\left(1+O\left(\frac{\log \log N}{\log N}\right)\right);\end{eqnarray}\]
the calim is proved. ■
This bound is twice what the result should be, since one expects that in $(5)$ the error term is negligible even if $z$ is very close to $x$, say $z=x^{1-\varepsilon}$ for any $\varepsilon>0$. Indeed, the claim that the Bombieri-Vinogradov estimate holds for the very long range $1\leq q\leq x^{1-\varepsilon}$ is the Elliott-Halberstam conjecture, which seems to be out of reach as the Bombieri-Vinogradov theorem has not been proved even for averages over $1\leq q\leq \sqrt{x}$. However, Zhang was able to prove the desired average estimate for $1\leq q\leq x^{\frac{1}{2}+\delta}$, where $\delta>0$ is small enough, but with the condition that the average is over square-free and smooth $q$, that is those $q$ with no ''large'' prime factors. This already was enough to prove the existence of infinitely many gaps between primes.
We turn to the problem of estimating the number of primes of the form $ap+b$, where $p$ is prime and $a$ and $b$ are coprime. We choose $A=\{p\leq N: ap+b\in \mathbb{P}\}$ and $P=\mathbb{P}\setminus\{p:p\mid ab\}$. Then $\omega(p)=\frac{p}{p-1}$ for $p$ coprime to $ab$. The exact same argument as for the Golbach problem shows
\[\begin{eqnarray}&S(A,P,z)\leq 2\mathcal{G}\prod_{2<p\mid N}\frac{p-1}{p-2}\cdot\frac{N}{\log N \log z}\left(1+O\left(\frac{1}{\log z}\right)\right)\\&+\sum_{d\leq z^2\atop (d,N)=1}\mu^2(d)3^{\nu(d)}\left|E(N;d,-ba^{-1}\hspace{-0.3cm}\pmod d)\right|,\end{eqnarray}\]
and we get with the same argument as before that the error sum is $\ll \frac{N}{\log^3 N}$, independently of $a$ and $b$. Therefore we obtain the following theorem.
Theorem 4. For any integers $a>0$ and $b$ that are coprime we have
\[\begin{eqnarray}|\{p\leq N:ap+b\in \mathbb{P}\}|\leq 8\mathcal{G}\prod_{2<p\mid ab}\frac{p-1}{p-2}\cdot\frac{N}{\log^2 N}\left(1+O\left(\frac{\log \log N}{\log N}\right)\right).\end{eqnarray}\]
In particular, we get (supposedly) very good estimates for the number of twin primes and the number of Sophie Germain primes, that is primes $p$ such that $2p+1$ is also prime. The Sophie Germain primes are for example important as secret keys in the RSA cryptosystem, since if $q=2p+1$ is a prime, then attacks for finding the secret key that are based on finding small primes $q-1$ will fail.
We have already seen several successful applications of the Selberg sieve, but now we want to show that the estimate $S(A,P,z)\leq \frac{x}{G(z)}+\sum_{d\leq z^2}3^{\nu(d)}|R_d|$ given by the Selberg sieve is $\ll xW(z)$, that is what we expect up to a constant, with mild assumptions. For later purposes, we estimate instead of $G(z)$ the more general quantity
\[\begin{eqnarray}G(x,z)=\sum_{d\leq x\atop d\mid d P(z)}g(d).\end{eqnarray}\]We prove the following.
Theorem 5. Suppose
\[\begin{eqnarray}\sum_{y\leq p<z}\frac{\omega(p)\log p}{p}\leq \kappa \log \frac{z}{y}+A\quad (7)\end{eqnarray}\]
for all $2\leq y<z\leq x$ for some constants $\kappa,A$, and $\frac{\omega(p)}{p}\leq \eta<1$ for all primes $p$. For any $\lambda>0$ and $z<x$, it holds that
\[\begin{eqnarray}&&\frac{1}{G(x,z)}\leq W(z)\left(1+O\left(\exp\left(-\lambda \frac{\log x}{\log z}+\left(\frac{2\kappa}{\lambda}+\frac{C}{\log z}\right)e^{\lambda}\right)\right)\right).\end{eqnarray}\]
Assuming additionally that $|R_d|\ll \omega(d)$, noticing $G(z)=G(z,z)$, and setting $\lambda=1$, Selberg's sieve gives for $z\leq x$ the result
\[\begin{eqnarray}S(A,P,z)\ll xW(z)+O\left(\frac{z^2}{W(z)^3}\right).\end{eqnarray}\]
We start with a couple of remarks. First of all, it suffices to prove the claim about $G(x,z)$; the latter claim follows then directly from the Selberg sieve. Second, the condition $(7)$ says that $\omega(p)$ is bounded on average by $\kappa$ in a weak sense (since by Mertens' theorem, $\omega(p)=\kappa$ for all primes $p$ satisfies $(7)$); conditions like this arise often in sieve theoretic results, for example because $W(z)$ can be estimated using partial summation if we know a bound for $(7)$; a pointwise bound is not necessary for estimating $W(z)$. If the left-hand side of $(7)$ is also at least $\kappa \log \frac{z}{y}-A$, the number $\kappa$ is called the sieve dimension, and it plays an important role when it comes to applying various sieves. Third, the situation of Theorems 3 and 4 is a good example where $\kappa=1$, so $(7)$ is on average at most $1$, while the best pointwise bound for $\omega(p)=\frac{p}{p-1}$is strictly greater than $1$. In order to see that $W(z)$ is controlled by $(7)$, we note the following lemma.
Lemma 6. For $2\leq y<z$, supposing $(7)$, we have
\[\begin{eqnarray}\frac{W(z)}{W(y)}\ll \left(\frac{\log z}{\log y}\right)^{\kappa}\end{eqnarray}.\]
Proof. Using $1+t\leq e^t$, we see that
\[\begin{eqnarray}\frac{W(z)}{W(y)}\leq \prod_{y<p\leq z}(1+g(p))\leq \exp\left(\sum_{y<p\leq z}g(p)\right).\end{eqnarray}\] The rest is just a rather boring calculation applying partial summation. ■
We proceed to the proof of Theorem 5.
Proof. Notice that
With Theorem 5, we can give an upper bound of the correct order of magnitude for $S(A,P,z)$ given almost any reasonable $A$ and $P$. In particular, we could bound from above the number of primes up to $x$ represented by a polynomial. Such applications usually reduce to a routine check of the condition $(7)$, and we do not address them $-$ also because this post is already rather lengthy.
A general upper bound estimate for the Selberg sieve
\[\begin{eqnarray}G(x,z)=\sum_{d\leq x\atop d\mid d P(z)}g(d).\end{eqnarray}\]We prove the following.
Theorem 5. Suppose
\[\begin{eqnarray}\sum_{y\leq p<z}\frac{\omega(p)\log p}{p}\leq \kappa \log \frac{z}{y}+A\quad (7)\end{eqnarray}\]
for all $2\leq y<z\leq x$ for some constants $\kappa,A$, and $\frac{\omega(p)}{p}\leq \eta<1$ for all primes $p$. For any $\lambda>0$ and $z<x$, it holds that
\[\begin{eqnarray}&&\frac{1}{G(x,z)}\leq W(z)\left(1+O\left(\exp\left(-\lambda \frac{\log x}{\log z}+\left(\frac{2\kappa}{\lambda}+\frac{C}{\log z}\right)e^{\lambda}\right)\right)\right).\end{eqnarray}\]
Assuming additionally that $|R_d|\ll \omega(d)$, noticing $G(z)=G(z,z)$, and setting $\lambda=1$, Selberg's sieve gives for $z\leq x$ the result
\[\begin{eqnarray}S(A,P,z)\ll xW(z)+O\left(\frac{z^2}{W(z)^3}\right).\end{eqnarray}\]
We start with a couple of remarks. First of all, it suffices to prove the claim about $G(x,z)$; the latter claim follows then directly from the Selberg sieve. Second, the condition $(7)$ says that $\omega(p)$ is bounded on average by $\kappa$ in a weak sense (since by Mertens' theorem, $\omega(p)=\kappa$ for all primes $p$ satisfies $(7)$); conditions like this arise often in sieve theoretic results, for example because $W(z)$ can be estimated using partial summation if we know a bound for $(7)$; a pointwise bound is not necessary for estimating $W(z)$. If the left-hand side of $(7)$ is also at least $\kappa \log \frac{z}{y}-A$, the number $\kappa$ is called the sieve dimension, and it plays an important role when it comes to applying various sieves. Third, the situation of Theorems 3 and 4 is a good example where $\kappa=1$, so $(7)$ is on average at most $1$, while the best pointwise bound for $\omega(p)=\frac{p}{p-1}$is strictly greater than $1$. In order to see that $W(z)$ is controlled by $(7)$, we note the following lemma.
Lemma 6. For $2\leq y<z$, supposing $(7)$, we have
\[\begin{eqnarray}\frac{W(z)}{W(y)}\ll \left(\frac{\log z}{\log y}\right)^{\kappa}\end{eqnarray}.\]
Proof. Using $1+t\leq e^t$, we see that
\[\begin{eqnarray}\frac{W(z)}{W(y)}\leq \prod_{y<p\leq z}(1+g(p))\leq \exp\left(\sum_{y<p\leq z}g(p)\right).\end{eqnarray}\] The rest is just a rather boring calculation applying partial summation. ■
We proceed to the proof of Theorem 5.
Proof. Notice that
\[\begin{eqnarray}\frac{1}{W(z)}-G(x,z)&=&\prod_{p\leq z\atop p\in P}\frac{p}{p-\omega(p)}-G(x,z)\\&=&\prod_{p\leq z \atop p \in P}(1+g(p))-G(x,z)\\&=&\sum_{d>x}g(d)\end{eqnarray}\]
To estimate this, we use Rankin's trick: we replace the summation over $d>x$ by a summation over all $d\geq 1$, weighing each term by $\left(\frac{d}{x}\right)^{1-s}$, where $s<1$ is to be optimized later. We obtain
\[\begin{eqnarray}\sum_{d>x}g(d)\leq x^{s-1}\sum_{d=1}^{\infty} \frac{dg(d)}{d^{s}}\leq x^{s-1}\prod_{p\leq z}\left(1+\frac{\omega(p)}{p^s(1-\frac{\omega(p)}{p})}\right)\end{eqnarray}\]
\[\begin{eqnarray}1-W(z)G(x,z)&\leq& x^{s-1}\prod_{p\leq z}\left(1-\frac{\omega(p)}{p}+\frac{\omega(p)}{p^s}\right)\\&\leq& \exp\left((s-1)\log x+\sum_{p\leq z}\omega(p)\left(p^{-s}-p^{-1}\right)\right).\end{eqnarray}\]
Since $p^{-s}-p^{-1}=p^{-1}(\exp((1-s)\log p)-1)$, the Taylor series of the exponential gives
\[\begin{eqnarray}\sum_{p\leq z}\omega(p)(p^{-s}-p^{-1})=\sum_{n=1}^{\infty}\sum_{p\leq z}\frac{\omega(p)(1-s)^n\log^n p}{p\cdot n!}\quad (8).\end{eqnarray}\]
If we denote $F(t)=\sum_ {p\leq z}\frac{\omega(p)\log p}{p}$, then by partial summation
\[\begin{eqnarray}\sum_{p\leq z}\frac{\log^n p}{p}&=&F(z)\log^{n-1}z-(n-1)\int_{1}^z \frac{F(t)\log^{n-2}t}{t}dt\\&=&F(z)\log^{n-1}z+(n-1)\int_1^z \frac{(F(z)-F(t))\log^{n-2}t}{t}dt\\&-&(n-1)F(z)\int_{1}^z \frac{\log^{n-2}t}{t}dt\\&=&F(z)\left(\log^{n-1}z-(n-1)\int_1^z \frac{\log^{n-2}t}{t}dt\right)+(n-1)\int_1^z \frac{(F(z)-F(t))\log^{n-2}t}{t}dt\\&\leq& (n-1) \int_{1}^z\frac{\kappa \log \frac{z}{t}\cdot \log^{n-2}t}{t}dt+A(n-1)\int_{1}^z \frac{\log^{n-2}t}{t}dt\\&=&\kappa(n-1)\log z \int_{1}^z \frac{\log^{n-2}t}{t}dt-\kappa(n-1)\int_{1}^z \frac{\log^{n-1}t}{t}dt+O(\log^{n-1}z)\\&=&\left(\kappa-\kappa\frac{n-1}{n}\right)\log^n z+O(\log^{n-1}z)\\&=&\frac{\kappa}{n}\log^n z+O(\log^{n-1}z).\end{eqnarray}\]
Hence $(8)$ is at most
\[\begin{eqnarray}\sum_{n=1}^{\infty}\frac{((1-s)\log z)^n}{n!}\left(\frac{\kappa}{n}+\frac{A}{\log z}\right)\leq 2\kappa\sum_{n=1}^{\infty}\frac{((1-s)\log z)^n}{(n+1)!}+\frac{A}{\log z}\sum_{n=1}^{\infty}\frac{((1-s)\log z)^n }{n!}.\end{eqnarray}\]
Setting $1-s=\frac{\lambda}{\log z}$, this is further bounded by $2\kappa \frac{e^{\lambda}}{\lambda}+\frac{A}{\log z}e^{\lambda}$. We conclude
\[\begin{eqnarray}1-W(z)G(x,z)\leq \exp\left(-\lambda \frac{\log x}{\log z}+2\kappa\frac{e^{\lambda}}{\lambda}+\frac{A}{\log z}e^{\lambda}\right)\quad (9).\end{eqnarray}\]
Denoting the right-hand side of $(9)$ by $L$, we deduce
\[\begin{eqnarray}\frac{1}{G(x,z)}\leq W(z)\left(1+\frac{L}{W(z)G(x,z)}\right),\end{eqnarray}\]
so it reamains to show $W(z)G(x,z)\gg 1$. Choose $c \in (0,1)$ such that $L=\frac{1}{2}$ for $x=z^c$ and $\lambda=1$. Then $W(z^c)G(z,z^c)\geq\frac{1}{2}$, so
\[\begin{eqnarray}\frac{1}{W(z)G(x,z)}= \frac{W(z^c)}{W(z)}\cdot \frac{1}{W(z^c)G(x,z)}\leq \frac{W(z^c)}{W(z)}\cdot \frac{1}{W(z^c)G(z,z^c)}\ll 1\end{eqnarray}\]
because $\frac{W(y)}{W(z)}\ll \left(\frac{\log^{\kappa}y}{\log^{\kappa} z}\right)$. This completes the proof. ■
With Theorem 5, we can give an upper bound of the correct order of magnitude for $S(A,P,z)$ given almost any reasonable $A$ and $P$. In particular, we could bound from above the number of primes up to $x$ represented by a polynomial. Such applications usually reduce to a routine check of the condition $(7)$, and we do not address them $-$ also because this post is already rather lengthy.


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3D camera in smart phones is an emerging technologies that provide enhanced picture quality and videos to end users. Currently, smart phones are integrated with 3D cameras that can sense movements, while taking pictures and enhance the picture quality. Smartphone 3D camera gains popularity among the young generation due to its attractive features such as real sensing of the object, HD clarity and improved performance. Increase in adoption of smartphone is the key driver for the growth of the smartphone 3D camera market. Whereas, high cost of smartphone 3D camera (a new technology) and limited compatibility restrict the market growth.
The global smart phone 3D camera market is segmented based on technology, resolution, and geography. The technology segment is divided into stereoscopy and time-of-flight. Stereoscopic camera technology offers precision, reliability, and high quality while capturing the exact 3D image of any object. Time-of-flight 3D camera technology can accurately sense the movement of objects, facial expressions and emotions and gives optimum picture clarity even in dim light conditions. The smartphone 3D camera is available in three resolution namely below 8 MP, 8–16 MP and above 16 MP. Megapixel is a unit of graphic resolution of the camera that plays an important role. Picture quality is dependent upon the mega pixels for instance camera with high mega pixel (MP) will have the better picture quality and clarity. The global market is studied across four geographical regions, which include North America, Europe, Asia-Pacific, and Latin America, Middle East & Africa (LAMEA).
The key players operating in this market include Toshiba Corporation, Sharp Corporation, Sony Corporation, Microsoft Corporation, Infineon Technologies AG, SoftKinetic Systems S.A., PMD Technologies AG, and Pelican Imaging.
- The study provides an in-depth analysis of the 3D camera smart phone market with current and future trends to elucidate the imminent investment pockets in the market
- Current and future trends are outlined to determine the overall attractiveness and to single out profitable trends to gain a stronger foothold in the market
- The report provides information regarding key drivers, restraints, and opportunities with impact analysis
- Quantitative analysis of the current market and estimations from 2015 to 2022 is provided to highlight the financial appetency of the market
- Porter's Five Forces model analysis and SWOT analysis of the industry illustrate the potency of the buyers and suppliers of the market
- Value chain analysis in the report provides a clear understanding of the roles of stakeholders involved in the value chain
- The market is segmented based on technology, resolution, and geography
By Technology
- Stereoscopy
- Time-of-Flight
By Resolution
- Below 8 MP
- Above 16 MP
By Geography
- North America
- Europe
- Asia-Pacific

subset: commoncrawl


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What a beautiful wedding this was! Danny and Vi did not miss any detail when planning their special day. From the fresh hand-rolled cigar station, to the decked out S'MORE BAR. For entertainment the couple brought in Poreotics, a former winner on America's Best Dance Crew (ABDC). Needless to say, the party vibes for this crowd was through the roof!
Photographer: Ether and Smith
Cake & S'more Bar: Intricakes


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Balela is an opportunity for local crafters to shine. To create and build online retail markets in order to put some cash in their pockets.
- Stage Full Product Ready
- Industry Internet / Web Services
- Location Zakariyya Park, Johannesburg, GP, South Africa
- Currency USD
- Founded November 2012
- Employees 1
- Website balela.co.za
Company Summary
Balela is the fulfilment of a dream – to give small business owners, and crafters an opportunity to grow their enterprises by selling their products online. We've done this by creating a secure transaction space where you can interact with potential buyers and safely conduct financial dealings. Create your online store for FREE, all we ask in return is a 6.5% commission on all paid orders.
Bradley Charles CordenMr
I have over 13 years of online experience. I have a passion for online, it's what gets me up in the mornings, and to bed late in the evenings.
I have over 8 years experience in e-commerce development and system integration into third party system. I have developed systems for both the Local and International markets.
I'm an ideas man, always thinking of business ideas and their growth potential. I will strive to be the next Mark Zuckerberg.


text:
They're Off for the 17th Adirondack Marathon/
Bathed in glorious sunshine, a field of around 400 runners took off at 9am in downtown Schroon Lake for the 17th Annual ADK Marathon.
The runners -- dressed in an array of dazzling neon colors – were cheered on by hundreds of enthusiastic spectators who lined Main Street.
With cooler temperatures, perfect for a marathon, organizers expect some records may be broken today.
Before the race, there was a party atmosphere along Main Street. Runners of all shapes and sizes were stretching in a multitude of poses, adjusting their iPods, while their friends and families looked on.
This year's race featured four runners who have participated in all 17 marathons: Joseph and Mary Healt of Forestport, NY, Laura Scarpinato of Milton, Vermont and Dan Kress of Churchville, NY.
Race Organizer Joel Friedman welcomed te runners, from as far away as New Zealand, Broome, from Western Australia Down Under, (Aussie, Aussie, Aussie, Oy!, Oy! Oy!), Brazil, Costa Rica, Slovakia, the United Kingdom, Germany, Canada, Hong Kong and Finland.
The starting pistol was fired by Patty Christian from Witherbee's Carriage House, one of the event's major sponsors.
The first of the runners are expected to cross the finish line between 11:30 am and 12 pm.


text:
Pregnant with potential meanings, the title of the xx's immersive third album gradually reveals itself over the course of 10 new songs. It is not accusative – "I see you, stealing that guitar sound" – but more in keeping with the ongoing themes of this minimal, intimate band; more consolatory. "I see you", this album says, "as you are."
Prefaced by a series of more outgoing tracks, I See You has been touted as the xx's least insular album thus far. If Jamie Smith's solo outing of 2015, In Colour, constituted a successful experiment with shades beyond the minimal monochrome that originally defined the band, then I See You finds them a little gaudier of palette. Using samples for the first time, they have tweaked their sound in myriad ways, while still retaining the sense of proximity within spaciousness for which they are famous. You could say it was more commercial, if the mainstream hadn't been biting the xx's style relentlessly. ("Justin Bieber is doing tropical house," noted Smith in a recent interview.)
The first song opens with a Caribbean horn fanfare, the shock of which will have you running to the track-listing to see if you haven't accidentally hit play on a different band. Sinuous and garagey, Dangerous sounds like it could have come from Smith's solo outing, and finds singers Oliver Sim and Romy Madley Croft sharing lines such as "I couldn't care less/ If they call us reckless."
Likewise, the bittersweet Say Something Loving is – in xx money, at least – an out-and-out pop song, with a nod to Sade's The Sweetest Taboo ("Don't let it slip away!"). Smith's signature steel pan arpeggio sound is there, but – pace, Bieber – transcribed on to keyboards. As ever, Madley Croft and Sim – childhood best friends – enact a loving boy-girl relationship in which sex is absent but present, just directed offstage. Lips is a particularly sensual outing for this most glacial of bands. Throughout the album runs a subplot in which one sense takes precedence: that of seeing and being seen.
The three-way relationships that sustain the band were sorely tested over the course of recording. Sims, in particular, has stepped back from partying as a result. ("Does the night chase me?" he wonders on Replica).
Located at the heart of the album, Performance finds Madley Croft "putting on a show" and "playing hide and seek" while Smith's scything keyboard violin sound etches zigzags above her head. The album closer, a gospelly piano-led duet called Test Me, is more abrupt. A ghostly analogue curlicue tracks a sonar bloop. "You look", murmurs Madley Croft, "but you never see."


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Press the arrow and listen to the "ding !" when the till opens :-)
Press a number to see it appear, then release all pressed numbers by pushing the "A". When you press the arrow, the till opens with a nice "ding!" (see video at the bottom of this page).
This toy is old, and the drawer sometimes opens half way only. The system needs a bit of oil inside.
You can see a short video of this toy here :
Weight of this toy (without bubble wrap and box) : 640 g = 1.41 lb
Reference of the item : AM(NT)-TG-070
This item is now for sale on Etsy :-)


text:
Various events were organized to for the students of Foundation Stage to celebrate English Week between 7th and 11th September, 2014. They participated in 'Group Recitation' , an event to enhance their speaking and communication skills. They were taught a poem in class which they recited before their peers with actions and voice modulation. Their confidence and clarity in diction was commendable.
In addition, the students of FS1 were shown videos of various stories related to the topics being done in class. The students of FS2 had a 'show and tell competition' where they spoke a few sentences on their favourite toy. The competition helped enhance their communication, speaking and listening skills and develop their self confidence. Each participant was given a certificate.


text:
Toronto Real Estate Broker Gunned Down in Targeted Attack
It was a busy night at a downtown Toronto steakhouse Saturday when a gunman strode through the crowded restaurant looking for one man in particular, then opened fire among the tables of diners at Michael's on Simcoe.
Simon Giannini, a 54-year-old well-known real estate broker with Royal LePage, was shot multiple times and was later pronounced dead in hospital. Toronto Police are now looking for a suspect in what they say is a targeted shooting, though friends and colleagues are stunned as to what could have motivated the attack.
"He had a heart of gold. He was an amazing dad and he was ever ready to help people," said Nishad Lakhani, a colleague of Mr. Giannini's in the real estate industry. "He was a mentor. If I had a question or I needed help, he was always willing."
Ms. Lakhani said her office is shocked by the news. "Right now, everybody's just at a loss for words," she said.
The shooter entered Michael's, an upscale steakhouse in the downtown Entertainment District, just before 9 p.m., wearing a hooded jacket and jogging pants. Witnesses reported hearing at least four shots fired in the restaurant before the gunman fled on foot and got into a white SUV.
"The hood [on his jacket] was up and his face was covered," Toronto Police Detective Shannon Dawson said of the shooter. "It appears that it was targeted."
Paramedics performed CPR on Mr. Giannini outside the restaurant before taking him by ambulance to hospital. About 150 diners were said to be in the restaurant at the time, including a man Mr. Giannini was dining with, who was unharmed. Det. Dawson said police are now looking to talk to as many of those customers as possible to piece together any details of what happened.
"It was a busy restaurant," Det. Dawson said. "We think many of [the customers] may have fled very quickly and perhaps those people haven't had a chance to speak to us, and we'd like them to take this time to come speak to us to provide anything they witnessed."
In particular, police are hoping someone outside may have caught the shooter's escape on a cellphone or dashboard camera.
"We have had several witnesses come forward and provide statements, but we'd like further information to come forward," Det. Dawson said.
As a real estate broker in Toronto, Mr. Giannini enjoyed a high profile in the city, authoring a book on home buying and hosting an online show that also once ran as a paid program on Corus Radio. The married father of two boys was not previously known to police, Det. Dawson said.
It is the second time in two years the steakhouse has been the scene of a targeted shooting. On Sept. 20, 2015, two men wearing masks strode into the restaurant just before 10 p.m. and opened fire on two customers, with about 40 people dining nearby. The shooters then fled. Terry Nguyen, 29, and a 28-year-old woman were both seriously injured but survived.


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Wwe Raw Laredo Tickets
Monday Night Raw, WWE's longest running television show, is now coming to Laredo, Texas this September. Expect to see the best WWE superstars in action and witness exciting, entertaining wrestling matches that will enthrall fans for the duration of the show. Superstars expected to appear in the show include John Cena, Seth Rollins, Randy Orton, Sheamus, Roman Reigns and Bray Wyatt, among a host of other names. To catch all the action live form the Laredo Energy Arena, get your WWE Raw Laredo tickets now.
Wwe-Raw in Other Cities
Laredo has hosted several WWE events in the past so the fans should know exactly what to expect. The Laredo Energy Arena should be packed to the rafters with thousands of fans cheering on their favorite stars. The experience really is much better live, than what is seen on television.
WWE Raw
WWE Raw first aired on January 11, 1993 and is now in its twenty third season. Having aired more than eleven hundred episodes, the show is one of the longest running weekly television shows in the world.
The show usually follows a set pattern, which several matches on the night's card, followed by a hyped up main event, in which more high profile stars participate. Apart from the wrestling aspect, the show features interviews and segments which help establish story lines or move rivalries forward.
The Raw after Night of Champions
The WWE Raw show after every pay-per-view event is a huge deal. So Laredo's Monday Night Raw show following the Night of Champions pay-per-view will be extremely exciting for the fans. Story lines which came to their conclusion at Night of Champions will be over, while new exciting feuds and other story lines will begin.
Night of Champions usually sees several titles change hands, so expect to see newly crowned champions to make their first appearances with their new titles. Some of the losers from the previous night are also sure to make some trouble, so this particular edition of WWE Raw should be an action packed show.
Similar Events
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What is actually definitely an 'entrepreneur' from a commercial industry agency in the present day? Are you undoubtedly them? Interesting questions are they not. In my view everybody more of all of these salespeople are generally forward looking and revolutionary.
Tip: Investors usually get that phrase Sheet is negotiable. Eventhough it comes printed on official letterhead, you don't need to take how they give your organization.
We take presctiption the verge of an inflation spill over. You'll find nothing laid out by the teachings using the Ludwig Von Mises Austrian School of Economics features failed to occur. This discusses the approaching phase 2 which states that monetary inflation must accelerate to sustain growing bubbles. It also must accelerate to sustain these bubbles that break anyway along with the inflation must continue to accelerate to try to prevent key system from collapsing.
Once an individual has a house under contract, it's high time for you to identify your potential customer. The best thing you can do is build up a buyers list prior to have a home. Find out where the buyers want to live, then go find your dream house in that area. Is actually possible to much more readily found a house for a buyer as opposed to choose a buyer with a house.
So when the commercial market is just a little bit difficult for you, anyone think things can or should be better, alternate to look at personal business strategy plan. Here are some good ideas ,.
In publicity, we be compelled to think with the the media needs – not end up receiving. It's exactly the same at networking events. Whenever learn a person are in conversation with a purchased by scott plank in 2016 and 2017 designer, you need to to get more information about what his / her needs are an individual can detect whether or not you will work together.
It is smarter, which makes it a less risky solution to get engaged in real estate investing. Within a nut shell, this is possible to let someone else pay the mortgage, build equity and flip the exact property when industry industry in one area reached its sweet spot. Can also a significantly less stressful way of turning a nice profit (even before shifting to sell).
The concept behind it is that after a period of years, records are lost or misplaced and memory isn't as accurate. Once the statute of limitations has run, the irs can't focus on you for additional commercial real estate investing taxes, an individual can't carry out the IRS for additional refunds, perhaps.
Shelter your income. The IRS allows you take 2 deductions that lower tax liability: the mortgage interest deduction and cost recovery (depreciation) deduction. Which means that reduce your earnings taxes from your interest and cost recovery you've taken. To be able to more money left with your checking concern.
By leveraging a team of people, you can create a irreversible income likewise let continue arrive in no the time that you in oneself. Just about every industry has a primary sales company with form of compensation plan. Find a company with an extended track record of success, find a knowledgeable mentor, and take massive action using your recession-proof business.


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In hopes of extending the growing season, my husband and I constructed an easy to build hoop house. I have a raised border around my Square Foot Garden and this made anchoring the hoop house down fairly easy. Any sized hoop house can be built atop a raised bed. I am not including any written directions; we didn't have any to follow, but I am including a list of the materials we used, and a step-by-step photo diary of the construction.
We built the hoop house with ease in lifting a priority; the top will open up and lay on its side much like a door opening. This solves the problem of creating an opening in the side of the hoop house and also give me easy access for planting, maintaining and harvesting. I'm hoping to grow salad greens through the colder months of the year.
Our hoop house covers a 4 x 6 garden plot. Happy Hooping!
7 Lengths 1/2″ PVC Pipe
4 PVC elbows (size to fit pipe lengths)
10 PVC tees (size to fit pipe lengths)
3 conduit hangers (I think we might end up using more of these around bottom edge)
10′ x 20′ roll of 3 or 4 mil. plastic sheeting
Roll of packaging tape


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It's Scholastic book fair time! The Ecole Kenwood Book Fair will be held February 23-27, 2015 in the library. The Book Fair mostly will be open during school hours, but will also be available the evening of International Pot Luck (Friday, 2/27). WE NEED YOUR HELP to staff the Book Fair during the times listed below, including during the Potluck. Volunteers are needed to help set up and break down, but mainly to assist students and teachers during visits to the book fair to buy books. If you would like to volunteer please click HERE. Merci!
If you have questions, please contact Megan McGonigle at email@example.com.


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Society has a problem with female nudity when it's not packaged for the consumption of male entertainment. Then it becomes confusing – E. Badu
It is highly unlikely that you'll ever find me walking down U Street butt ass naked to prove that I agree with Erykah Badu. But I do. In fact I'll take it one step further, I believe that our society has a problem with female sexuality PERIOD when it's not packaged for the consumption of male entertainment.
As women, how often do we challenge ourselves to define sexy, not in the context of what men think but instead in the context of how we feel? We spend thousands of dollars and countless hours trying to recreate this image of "sexy" that was originally created for men by men. And if that weren't enough we've allowed ourselves to be conditioned to believe that being consumed by men is in some way empowering to women. To suggest otherwise is often quickly dismissed as feminist rhetoric by both men and women alike.
This question of defining sexy on our own terms was the topic of a conversation a few months back during a Q & A session after a performance of Giselle Mason's play, Women, Sex & Desire: Sometime You Feel Like a Ho, Sometime You Don't, I asked her about the short video shown during intermission that featured the all female cast in various forms of dress engaged in random activites. She explained that the piece was a montage of home videos recorded by the cast doing things that made them feel sexy. The biggest challenge to that piece, she admitted, was getting the actresses to express themselves without concern for what they believe men or society at large find sexy about them, to take control of their own image and sexuality.
That got me to thinking about how I define sexy. Can I honestly say that my definition of sexy is not influenced by what the media or the man in my life considers it to be? A decade ago the answer would be undoubtedly "no". Back then I truly believed that sexy could only exist within the context of what "he" thought. I belived that we both deserved to enjoy certain parts of our relationship but mine was for him and his was for me. If he thought high-heels were sexy, then I wore high heels. If he thought the color red was sexy, red it was, even if I didn't really like red all that much.
Thankfully as we get older and slightly more mature we begin to realize that the greatest gift a woman can give herself is the freedom to take control of her own sexuality by embodying it in a way that makes HER feel good. That's not to say that we shouldn't desire to be desired, of course we should, that's a natural part of being a woman but instead of trying to re-invent ourselves everytime a new guy with his own tastes (and hang-ups) comes along, we should celebrate our authentic expression of sexuality in a way that will attract the right man – the one who digs us just the way we are.
In the meantime, we should consider what Erykah is trying to say. Whether you agree with her tactics or not, you can't deny that by taking control of her body and the way that it was presented to the world her actions were in stark contrast to the countless women who are being held captive by an industry that wouldn't dream of giving them that choice.
So, I ask you, What does sexy mean to you?


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Overnight Camp Scholarship
Once again, the Temple Board, WRJ/Sisterhood and Temple Brotherhood have set aside some money for campers who are attending Jewish camps this summer. The camper MUST meet the following criteria:
1) Applicant must be a member in good standing of the Temple.
2) Applicant must be enrolled in and meet attendance requirements for the Temple Religious School during the year of the request.
3) Request must be for attendance at a URJ camp, or any other non-denominational Jewish camp (i.e., JCC run camp)
4) Must be an overnight camp.
5) Applicant must submit request in writing by March 30, 2018.


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"Cape Cod National Seashore"
Exif: F Number: 11, Exposure Bias Value: 0.00, ExposureTime: 1/30 seconds, Flash: did not fire., ISO: 160, White balance: Auto white balance, FocalLength: 50.00 mm, Model: NIKON D200
Uploaded on October 12, 2011
Copyright for this gallery photo belongs solely to Mike R. Howe. Images may not be copied, downloaded, or used in any way without the expressed, written permission of the photographer.
Log in to follow or message this photographer or report this photo.


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As the market for trendy smartphones heads toward maturity, the next consumer "must have" appears to be a wearable device. Often designed for use with a smartphone, using its short and long-range gateway connectivity capabilities to cloud-based applications, the demand for wearables is forecast to intensify rapidly...
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Today was a short day of work, as I too had to prepare for our contest versus the Philly Soul. I prepared film evaluation using the best and most used plays from Philly's games this 2015 season. They are undefeated and we are currently 2-1, this game will provide us with the division lead. We will face Philly two more times this season, with the possibility of a fourth matchup if we both make the playoffs.
Game Day!!!
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It has been one crazy hyped-up ride since my article on "Shenmue & Dreamcast Intertwined Fate" which turned out to be most viewed article on shenmue500K website never imagined it would hit so many Shenmue & Sega fans harts.
Shenmue fans got their wish and it seems there are many Sega fans who want their dream realized too: that is simply for Sega to come back into hardware and have a console for their titles.
Sega Games belong on a Sega System. It doesn't feel right when we see "Sonic" team up with "Mario" or playing "Sonic Adventure 2" on the GameCube or Phantasy Star Universe on 360 or Phantasy Star portable 2 on PSP or playing a watered down mobile version of Crazy Taxi on a mobile device then not being able to play Phantasy Star online 2 on a PC because there has been no Western release; it's almost soul shattering.
What makes it harder is the bitter yet awesome rivalry that existed between Sega and Nintendo back in the 90s, with those golden years behind us, I think a lot of Sega fans felt lost and fractured having to play games on systems where the Sega brand doesn't really fit. This damaged Sega the most in profitability and image of the company; especially how their game IP's have been roaming the waste lands of other consoles.
It's no wonder Sega had to focus on milking Sonic to death and abandon other IP's.
Sega's approach to gaming now feels fragmented when we see Sega games on rival consoles; remember the tag line in the 90s "Genesis Does What Nintendon't".
Now look at Sonic the Hedgehog teaming up Mario in winter Olympics. When classic outsells your cash cows: It wasn't long ago that "Streets of Rage Two / Bare Knuckle 2" was released on Nintendo 3DS, funnily enough this turned out to be the most profitable game for Sega. "Streets of Rage II / Bare Knuckle 2" series has been popular since the early 90s.
It's also one of the IPs that hasn't seen a sequel for very long time. When we can see a game like "Streets of Rage II" outselling "Sonic" or "Miku" and the amazing response and overwhelming support from the fans to get Shenmue 3 funded, surely you would think it would be a massive wake up for Sega!
It's clear as day in my eyes that Sega has been missing a trick or two to make themselves more profitable as a brand and better their company image.
Only recently Sega has been asking everyone to take part in a survey and with all this commotion surrounding Sega at the moment, it would seem they want to hear back from us.
I see this as a good move by Sega, and proof of how out of touch they have let themselves become since the demise of the Dreamcast. With the rise of social media they have only just realized what their own fans can contribute to them and to the gaming world.
Is Sega listening? Most certainly. Could we see more engagement from Sega in the future towards their fans? Well, it depends on Sega and their parent company Sega Sammy Holdings and how they plan to conduct themselves and their future business operations.
Take a look at what Dreamcast fans have been working on Recently.
Dreamcast Console Designed by Bruno Ferrari
Charis Charavopoulos Dreamcast Logo Design
It's time to bring the Sega brand Home
in one form or another…
Hey guys I would like to remind you we now have Official Shenmue500K APP for Android Phone Users, to make it even easier for you to keep track Shenmue 3 News.
Please visit this link to Download 500K APP:
If you would like to share your own Shenmue experience or you have worked on your own fan art or a project. Please get in touch Use our Contact Form
In order for Shenmue 3 to be the best game we could want, remember to make a PayPal Donation at Shenmue 3 Slacker Backer Campaign if you missed out during the Shenmue 3 Kickstarter – (https://shenmue.link/order/) It's a great way to show support and become part of gaming history there are still stretch goals to be reached.
Created this website for the 3rd Shenmue500K Group this website will also play role in supporting shenmue.link.
Will also bring you all the latest – (Shenmue III News, Updates from the Community and many Interviews from Yu Suzuki and partners and maybe even odd article from myself) – Regarding Shenmue 3 Development over the next 2 years.


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An executive's office is usually an office and a room for meeting purposes at the same time, and sophisticated interior design is often called for. The ideal lighting solution must satisfy the demands of qualitative design and also energy efficiency. This challenge can be mastered especially well with LED lighting, the compact design of the light source enables new design flexibility for innovative luminaires and ensures low energy consumption. Accent light can be targeted and different scenes adapted to create lighting perfectly suited to each task.
Neximo free-standing luminaire
The stylish Neximo free-standing luminaire unites a sensational form with optimal light quality and distribution to achieve a pleasant working atmosphere.
Athenik Ligra LED
The high quality Athenik Ligra LED luminaire has a fine appearance and extremely compact construction. A wide diversity of attachments makes it the ideal solution for nearly all interiors.
Belviso LED
The Belviso LED is a recessed or semi-recessed luminaire for innovative, work zone-oriented lighting. The luminaire family has a high recognition value with the exclusive light frame as a defining element. A special version adapts colour temperature to the day-night cycle of people.
Neximo LED
The Neximo H is a suspended LED luminaire for prestigious office lighting, especially the work zone-oriented lighting of single or double workstations.
Trilux Lighting Efficiencies for Executive Offices
The combination of Neximo LED free standing lamps and Athenik Ligra LED downlights offers a good alternative to an obsolete lighting installation. Energy savings of up to 59% can be achieved.
|Executive's Office
|Obsolete Installation
|New Installation
|Recessed Luminaire
4 x 18 W (VVG)
|AthenikL C05 MR LED 2000 nw 01 ET
Neximo S1 LED 6200 nw 01 ETD
|Power Consumption per luminaire
|Number of luminaires in the space
|6 pieces
|AthenikL : 5 pieces
Neximo : 1 piece
|Total power consumption
|Kilowatt hours p.a.
|1386 kWh
|Energy saving
|Saved kilowatt hours p.a.
|767 kWh
|CO2 savings p.a.
|0.46 tons


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Building A Strong Defense Of Your Interests
Criminal charges of any kind are a direct threat to your personal freedom and your future opportunities. Whether you are facing misdemeanor or felony charges, securing experienced defense help is crucial in your effort to fight the allegations against you.
At Lucido & Manzella, P.C., our lawyers offer tenacious advocacy and knowledgeable defense counsel for Michigan clients facing a range of criminal charges. Our team has successfully defended thousands of clients in the Detroit metro area, and we are prepared to put our experience and resources to work for you.
Our Experienced Team Is On Your Side
A criminal conviction can leave you with a permanent mark on your record and even time behind bars. Our lawyers understand what is at stake for you, and we will take immediate action to shield your interests, assert your rights and start working on a strong defense strategy. Our record of success in criminal defense cases includes matters pertaining to the following:
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Our attorneys take every case seriously. If you are facing civil infractions, we will strive to minimize the effect on your auto insurance and your driver's license. For both misdemeanor and felony offenses, our lawyers will protect your rights and vigorously fight the charges against you.
From the very beginning of your case, we work to develop a defense strategy that suits your needs and objectives. We will walk with you through every step, from the beginning of the investigative stage and throughout the entire court procedure. Our trial experience and ability to effectively navigate complex criminal defense cases can benefit you during this difficult time.
Do Not Wait – Learn About Your Defense Options Today
We offer free initial case evaluations to individuals throughout the Detroit area. Contact our conveniently located Clinton Township office at 586-842-0198 or email us for a free evaluation of your case.


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Conscious Community Lectures
- Presented By: A.R.E. Bookstore & Gift Shop
- Recurring monthly on the 4th Wednesday
- Visitor Center Main Auditorium
- 215 67th St, Virginia Beach, VA 23451
- 757-428-3588
- Time: 10:30 AM to 12:00 PM
- Price: Free!
Sponsored by the A.R.E. Bookstore & Gift Shop, Conscious Community Lectures are free and open to the public. Held once a month (varies, dependent on holidays and other factors), these in-depth programs delve into mind, body, spirit topics.
Verify the upcoming lecture speaker and topic by calling 757-428-3588 or visiting https://www.edgarcayce.org/events/event-listings/bookstore-sponsored/conscious-community-lectures/.
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Hello readers, and welcome to the latest (in more than one sense of the word) issue of the Tribune! Apologies for the wait - we've decided to go with wild dogs as this week's excuse. Or possibly rabid squirrels.
I'll let your imagination do the rest.
Breaking News!!!
In case you haven't already heard, our member of the month for February is Troy!! So if you haven't already, head on over and congratulate him!
Our Love and Pirates writing contest is nearing the closing date for submissions - as is the new banner contest! So get those last minute submissions in, people!!!
Aaaand rating your mature and private threads correctly has never been so simple with the new sticky threads in both forums, detailing the rating rules!
The Storyteller
Since we had no submissions this week (come on people...) I took a stroll through the creative writing forum, and decided to share with you this piece from Dani Cojo's Prose/Poetry that seemed very fitting for the month of love;
"What Is Love" - Dani Cojo
What is love?
Is it when your eyes cant stay off that one person across the room, and your mind swirls with worries and questions because you dont want to stare for too long, but you just cant tear your gaze away?
Or when your heart beats far too fast for your body to handle, until it feels like its going to explode into a million ruby shards because he spoke to you for the first time?
Or is it when your lungs refuse to function, and your breath is taken away just by his looks, or his voice, his personality, or a beautiful and harmonious combination of the three?
Or that moment when your entire being is trembling and goosebumps are dancing along your skin just from being closer than you ever would have imagined?
Is it when he holds your hand, and you can feel your palm growing warmer where your skin connects?
Or when he steals your first kiss and claims your lips as his own in a completely innocent and romantic way that causes your knees to quake and your heart to flutter?
Is it when the line between friendship, romance, lust, and every other fond feeling you have is blurred together into a mosh posh of emotion you cant hold back?
Maybe, its something much more deep than how your body is reacting.
Maybe its when you want to spend every moment with that person because you feel so at peace with him?
Or when hes able to draw you from your shell, exposing you to the wonders of life outside of your comfort zone; allowing you to thrive and experiment with optimistic and care free activities by supporting you with his own charming and outgoing charisma.
Is it when you can share every detail of your life, your thoughts, your heart, without worry of being shut down or embarrassed because you know in the end hell understand?
Or even when he doesnt understand, hell do everything he can to make you feel comfortable and beautiful and perfect in every way?
Is love when you feel like youve lost a piece of yourself when he walks away, because he truly is your better half?
Or is it when he comes back and your face aches from the laughter and smiles that instantly escape your lips when you draw him close to yourself in a hug?
Is it when you want to show him off and introduce him to your family, because you honestly want him to be with you for the rest of your life, if not more?
Maybe its that moment when you think how much you love having that him around; when your entire core is screaming to finally just give in to the emotion.
When your thoughts of him revolve around those three important and life changing words, and youre too afraid to say them because you fear rejection, or going too fast and burning out, or mistaking true love for something else, or a terrifying combination of the three?
What is love?
And when is it time to give into it?


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Impeachment worth the wait in unruly Zimbabwe
News, Noah Feldman, Published on 22/11/2017
» When you get rid of your dictator, is it important to follow the rules? That delicate question is dominating the transition-in-progress in Zimbabwe, where long-time President Robert Mugabe has refused to step down despite the demands of the public, the army and his own political party.
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Post #83
Here we are in the middle of winter again and I'm reviewing Iced Earth. For the 3rd straight year, I managed to review this band while temps in my neck of the woods are well below freezing. I believe a tradition has been set...
This time I'm focusing on the band's most recent studio album, the conclusion of their 2-part concept album, Something Wicked Part II-The Crucible of Man. Surprisingly, much has transpired within the band between this album and it's predecessor. The most significant change is that Matt Barlow has returned to the band as lead vocalist, and replaced Tim 'Ripper' Owens (who had replaced Matt prior to The Glorious Burden).
It is interesting that this change occurred between two halves of a 2-part concept album. Barlow's return restores the band's classic formula from their late-90's era, a cherished reunion in the eyes of many fans. At the same time, this leaves many hypothetical questions in the minds of Iced Earth followers: What might the "Something Wicked" double-album sound like if Tim Owens had sung on both parts? Or if Barlow had? Fans of Iced Earth will probably be able to debate this issue for quite a while, but it is as is it.
Something Wicked Part II: The Crucible of Man continues the plot from Framing Armageddon and the Something Wicked 3-part epic track from Something Wicked This Way Comes. Iced Earth blends melodic clean guitar with rapid-fire thrash riffs throughout this album much like they did on Framing Armageddon. Barlow's vocal range may be a bit more confined to the mid-range than Ripper's was on the first half of this concept album, but it still does not subtract from the overall experience.
The concept of an alien race that seems poised to pass judgment on the human race is developed a bit more. Most of the lyrics focus on the nature of humanity; laced with examples of deceit and hypocrisy. There is a track, "Something Wicked (Part 3)" that musically continues themes from the previous album, as do several other tracks. A melancholy instrumental Epilogue closes the album... Overall, a successful musical endeavor, if a little heavy on its ambition.
My picks from Something Wicked Part II:
-Behold The Wicked Child
-A Gift or a Curse? (This is a slower tempo track. Minor-key clean guitar throughout. A great deal of the concept's plot is developed with this song.)
-Crown of the Fallen
-The Dimension Gauntlet (More of a thrasher here, but again more plot points are revealed.)
-I Walk Alone
-Crucify the King (This one really sounds like classic Barlow-era Iced Earth!)


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If you're reading this blog, you're probably already well acquainted with how awesome music can be. Maybe you're a music therapist, or know and love one. You might be an activity director who has seen your residents sing along to Frank Sinatra long after they're lost their ability to speak. Or maybe you know the feeling of getting in the car after a long day and hearing your favorite song come on the radio. No matter what, nearly everyone has some kind of connection with music. That's why I want to share this next story with you. If you love music, you'll understand.
It's Monday after a weekend of pouring rain. The sun is trying to peek through the clouds. I get called in for an impromptu music therapy session at a day program for adults with mental health needs. I and another music therapist normally have a group here on Tuesday and Thursdays but they requested an extra session this week and I'm glad to step in. When I walk in the door I see a woman beelining towards the exit, staff not far behind. Exit-seeking behavior is a common symptom in dementia, and the staff at this day program are well-versed in redirection, but nothing seems to be working. I make eye contact with the director and she says, "Do you want to go to music?" The woman looks at the director, looks at me, and says, "I don't think I can." I approach the woman with my hand extended and introduce myself. She takes my hand and tells me her name. Then I ask, "Do you like rock & roll?" The woman's eyes light up. Like the sun peeking through the clouds, her affect begins to brighten. She starts humming and moving her shoulders up and down in time, and enthusiastically agrees to walk to my group with me. We hum "The Twist" by Chubby Checker as we go.


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For special applications you need a purpose built zip fastener. The zip fasteners shown below have been specifically designed to meet the demands of the end use.
S 60 shoe closed end
Welded top tape end (= 0 mm)
Flat wire tops stops
Special flat spiral from Opti S-Technology
Spiral with 8 woven in warp threads for increased resistance to wear.
Low profile components
S 60 shoe closed end with user friendly "Stitchline""
Ultrasonically welded plastic top and bottom stops
"Stitch Line" to help insertion.
S60 shoe closed end "coated"
Hot melt adhesive coating for welding zip fastener into shoe
S 60 shoe closed end "curved"
Pre-set curve to match the shoe profileSupplied in pairs (1x left and 1x right)


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Generates a real matrix from a specified set of eigenvalues.
An array that contains the eigenvalues from which you want to create matrix. eigenvalues must be real or conjugate pairs.
Default: [0]{}
Real matrix whose eigenvalues are specified by eigenvalues.
A value that represents any error or warning that occurs when this node executes.


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Wolf Gourmet Blender + Matcha Mint Chip Smoothie
I use a blender every day, so it's important to me to have a good one. Sometimes I use a blender for simple things, like making my matcha frothy, but other times I'll use it to make things like homemade hummus or vegan sauces using cashews. In these cases, it's nice to have a blender that not only works well, but looks good too!
Full disclosure, I was sent this Wolf Gourmet blender to review, but all opinions expressed here are my own and not influenced in any way by the brand! I was truly impressed when I tried out this blender. First of all, it looks nice and sleek, and it's a good size so that it fits on the countertop without taking up too much room. It feels very sturdy, and the stainless steel looks sophisticated in my kitchen.
Second, I love the pre-programmed settings. You can just turn the knob to smoothie, and the blender does all the work on its own when it comes to the settings. I like this because I am very hands off with my cooking — I like to multitask and do many things at once.
The blender works well for something simple likes smoothies, but it is definitely high speed, so making things like nut butters or hummus, which often require food processors, are totally possible using this blender. You can also make soups, which is always a plus.
I love how the blender shows the measurements on the side (essential), and there's even an emulsion option for when you're making dressings and other marinades. All in all, this blender fits my needs, and it's definitely the best blender I have used so far (no lie). In my opinion, it definitely compares to the Vitamix, but I also happen to like the aesthetics of this blender as well.
If you want to purchase this blender, you can get it from either Williams Sonoma or Sur La Table.
Matcha Mint Chip Smoothie Recipe
Ingredients (serves 1)
- 2 cups non-dairy milk of choice (I suggest almond milk or oat milk)
- 2 teaspoons matcha
- 1 teaspoon pure maple syrup
- 1 banana
- 4-5 mint leaves
- 2 tablespoons cacao nibs
- 4-5 cubes of ice
1. Add ingredients to the blender, in order.
2. Set the blender to the smoothie setting, and blend. Serve immediately.


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"Incredible Safari" – 23 January 2015
One more amazing stay. Of course, we hope to come back soon and see all of you again. Safaris were incredible and Adam is a fantastic singer! :) Say "thank you" will never be enough, but whatever... THANK YOU!!!!!!! Love all of you... Tournut family


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A record of Holocene sea-ice variability off West Greenland and its potential forcing factors
- Source:
- Palaeogeography, palaeoclimatology, palaeoecology 2017 v.475 pp. 115-124
- 0031-0182
- Subject:
- Retaria, cooling, ice, paleoecology, sediments, Arctic region, Greenland, Greenland Sea
- Abstract:
- We present a reconstruction of Holocene sea-ice variability from sediment core GA306-GC3, from the Holsteinsborg Dyb off West Greenland, which provides an index of palaeoceanographic and palaeoenvironmental conditions within this climatically sensitive region during the last 6700yr. The reconstructed sea-ice record, combined with previously published proxy data, suggests that relatively warm conditions with reduced sea-ice extent prior to 5000cal.yrBP were associated with the Holocene Thermal Maximum. Subsequent cooling and extensive sea-ice cover between ca. 5000 and 4000cal.yrBP was followed by even colder conditions and persistent sea-ice cover during the Neoglacial cooling phase, particularly after ca. 1500cal.yrBP. There is a positive correlation between West Greenland sea-ice cover and solar activity over the past 5000yr, but the correlation is much weaker prior to 5000cal.yrBP. In addition, there is a strong link between West Greenland sea ice and changes in the abundance of arctic benthic foraminifera species, related to different water masses of the West Greenland Current during the entire interval, even prior to 5000cal.yrBP. Our findings indicate that sea-ice variability off West Greenland was driven not only by solar activity, but also by ocean circulation (the strength of cold Polar water from the East Greenland Current and warm Atlantic water from the Irminger Current).
- Agid:
- 5654373
- https://dx.doi.org/10.1016/j.palaeo.2017.03.022


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The upper department recently started new electives! We found out what they are all about!
Picnic Games are simple outdoor games. The students also learn outdoor manners. On April 25th, the students played badminton. On May 9th, the people that were in Picnic Games played a game called Cangium. The object of the game was to throw the frisbee through the slot or into the bin. You have to try to get to twenty one points to win. These are the fun activities kids in Picnic Games got to experience.
In the elective Home Economics, students learn different skills such as sewing. On the first day of the new electives, they sewed buttons to felt sheets to practice. Carly Diago said it is a fun class.
When the people in Newsletter chose their stories they wanted to write about, they interviewed people and received everyone's opinion on their new elective. For example, the people mentioned above in "Picnic Games," and "Home Economics," were interviewed for facts and opinions. Kai Goodman (6th Grade), said, "Newsletter is very fun, but very hard."
Tyler Munoz and Alex Chung, Navigator News Reporters
Mon, May 21, 2018 @ 9:38 AM
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Professional Home & Commercial Inspections (704) 797-1103
Whether you're buying or selling a home, having a professional home inspection from a leading home inspection company will give you peace of mind and assist you in a fair and smooth transaction.
House2Home Inspections, LLC is the premier home inspection company in your area. We have been trusted by home buyers to help them make informed buying decisions. Our home inspections typically range from 3-4 hours long depending on the size of the house and you are invited to follow along with your inspector as they perform their inspection so you can see what we see and answer any questions you may have. We work for you, giving you the information you need to make an informed decision regarding the prospective home. You will receive a clear, concise, easy-to-read Inspection Report with color digital pictures e-mailed to you the very next day (day of inspection if necessary!) Saturday, Sunday and after hours appointments available- whatever works best for you.
We take great pride in every inspection we perform and your referrals is what we strive to achieve- every time! We will meet you and your agent at the home at the end of the inspection and go over any defects at no extra charge!
Additional inspection services offered- Pool/Spa, Boat Dock & Seawall, Water Quality, Radon, Termite & Mold
Realtors love us because.........we use Supra eKey® technology!
Call us today at (704) 797-1103 to book your home inspection
10% discount for current or retired military- THANK YOU!
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Caleb Swanigan had 18 points and 13 rebounds on Sunday to help No. 20 Purdue win at home over No. 13 Wisconsin, 66-55, and end the Badgers' nine-game winning streak.
The Boilermakers (14-3, 3-1 Big Ten) have won nine of 10 over all and three straight against the Badgers.
Ethan Happ had 17 points and Nigel Hayes had 10 for the Badgers (13-3, 2-1).
Wisconsin trailed, 40-36, with 14 minutes 35 seconds to go when D'Mitrik Trice missed an open layup. The Boilermakers responded with a 12-0 run and led by at least 11 the rest of the way.
MISSISSIPPI STATE BEATS LADY VOLS Blair Schaefer made a go-ahead 3-pointer with 2:39 left as No. 4 Mississippi State capitalized on its superior depth to win, 74-64, at Tennessee (10-5, 2-1 Southeastern Conference) and stay undefeated. Mississippi State (17-0, 3-0) is one of only three remaining unbeaten Division I women's teams, along with No. 1 Connecticut and No. 18 Virginia Tech.
Bulgarian Ends His Drought in Australia
Grigor Dimitrov of Bulgaria ended a long wait between ATP titles, beating third-seeded Kei Nishikori, 6-2, 2-6, 6-3, to win the final of the Brisbane International in Australia and claim his first tournament victory since 2014. Dimitrov lost the Brisbane International final to Andy Murray in 2013. He won three titles in 2014 and reached the Wimbledon semifinals as he rose to a career-high ranking of No. 8.
Shiffrin Wins Despite Debris on Course
Not even a broken gate rolling down the course could stop the Olympic slalom champion Mikaela Shiffrin from winning a women's World Cup race in Maribor, Slovenia. In light snowfall, she was leading the field by 0.19 of a second when a gate that she had just passed correctly broke and flew down the hill. Shiffrin skied over it but managed to stay on the course. She fell 0.09 of a second off the lead but accelerated in the bottom section to take the win, 0.19 of a second ahead of Wendy Holdener of Switzerland.
76ers Pull Away From Nets in 4th Quarter
Joel Embiid scored 20 points, and the Philadelphia 76ers earned their 10th victory, matching their total from last season and beating the host Nets, 105-95. The 76ers broke it open with a 17-2 surge in the fourth quarter to win the matchup between the N.B.A.'s two worst teams. Brook Lopez scored 26 points for the Nets, who have lost six in a row.
Chelsea Easily Advances in F.A. Cup
Chelsea cruised into the fourth round of the F.A. Cup with a 4-1 victory at home over Peterborough United, even though the team's captain, John Terry, received a red card. Terry, who was handed a rare start, was given the red card for a foul on Lee Angol, but Chelsea progressed thanks to two goals by Pedro and goals from Michy Batshuayi and Willian.
Tottenham Hotspur joined Chelsea in the fourth round by winning at home over Aston Villa, 2-0, but Liverpool faces a replay at fourth-tier Plymouth Argyle after a lineup of second-stringers was held to a 0-0 draw at Anfield.
MESSI RESCUES BARCELONA Lionel Messi saved Barcelona from its second straight defeat by scoring on a 90th-minute free kick in a 1-1 draw at Villarreal in the Spanish league. Three days after losing to Athletic Bilbao, 2-1, in a first-leg match in the Copa del Rey, Barcelona had failed to score until Messi produced a superb strike from the edge of the penalty area.
California Fires Coach After Four Seasons
In a surprise move, California fired Coach Sonny Dykes after four seasons that produced just one bowl appearance. Dykes finished his tenure with a 19-30 record, including 10-26 in the Pacific-12. The offensive coordinator Jake Spavital will serve as interim coach.
American Bounces Back in Hawaii
Justin Thomas of the United States bounced back at just the right time to hold off Hideki Matsuyama of Japan and win the SBS Tournament of Champions in Kapalua, Hawaii. Thomas closed with a four-under-par 69 and ended Matsuyama's bid for a fourth straight victory.


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Well, we've made it this far. Winter is almost behind us and I can see spring arriving soon (maybe just not in the next few days). We've been super busy in the store preparing for our new stock and can't wait to share it with you!
We're thrilled to announce our custom Grace Tea Towels and Dish Cloths, as well as our retro pop-up sponges that are now available in the store! These new products are welcomed additions to our existing collection of Grace decor items, and will join the ranks of our beloved Grace Paper Placemats and Cheese Papers.
As well, we're happy to report we've RESTOCKED our supply of Falcon Enamelware that had been extremely well received. Enamelware is such a fun and versatile trend that is well suited for both city and cottage!
Oh, and we can't forget — SOHO Coco nuts and seeds confectionaries have arrived as well! We're proud to be the exclusive retailer of this product in Winnipeg — but hurry! They're insanely delicious and sure to go fast!
Finally, we are currently developing a brand new online shop that we are super excited about. We have to extend a huge thanks to Michael Roberts of Duality Photo who has been working hard to ensure we have beautiful images of our products for the e-shop. We're hoping to launch this new feature in the early summer of 2018.
With care,


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Get in Touch
If you'd like to connect with other aces in the SF Bay Area, check us out on facebook here!
You can also contact us directly at firstname.lastname@example.com.
Staying in the Loop
One easy way to make sure you're up to date on all the latest happenings, including meetup announcements, outreach events, and calls for volunteers, is to join our meetup.com group here and set your notification options to receive communications from event organizers.
Alternatively, if you'd like to receive fewer messages, or if you would rather not make an account on meetup.com, you can also join our general mailing list here.
(All major announcements about things like pride, UnConferences, outreach opportunities, and calls for volunteers are sent to all lists, but the meetup.com mailing list will also include additional reminders about monthly social meetups and other minor events.)


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Data Driven Instructional Decisions - Inviting Teachers into the Discussion
Marlys WeaverStoesz
According to Paul Bambrick-Santoya, author of Leverage Leadership, data analysis meetings between leaders and teachers are the highest-leverage time a leader can spend. During which time it is critical that teachers analyze their own assessment data. But even for school leaders who have learned to analyze data closely, supporting teachers to do the same can be challenging.
So, what needs to happen before a leader can ask the sort of targeted questions that lead to data-driven instructional decisions?
Know your teachers. The type of support and questions will depend on the self-reflection and analysis capacity of the teachers.
Analyze the teacher's results before the meeting to ensure you have a good idea of the root cause and how teachers might address the problems.
If needed, get help with content expertise.
Informed by your own analysis of the student data, arrive to the meeting with a few specific strategies that will be effective. Keep those in your "back pocket" and lead by asking questions like…
- What did the students need to be able to do to get that question right? How was this more that what they are able to do with you in class?
- What's so interesting is that they did really well on question number ____ but struggled with question number ___ on the same standard. Why do you think that is?
- Let's look at question number ____. What do you think that the students are doing wrong here?
After analysis meetings like these, teachers have a more developed capacity to analyze their students' output and a clearer understanding of what steps to take to improve student learning.


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ASSORT is a 6 letter word that starts with A.
\As*sort"\, v. t. [imp. & p. p. {Assorted}; p. pr. & vb.
n. {Assorting}.] [F. assortir; ? (L. ad) + sortir to cast or
draw lots, to obtain by lot, L. sortiri, fr. sors, sortis,
lot. See {Sort}.]
1. To separate and distribute into classes, as things of a
like kind, nature, or quality, or which are suited to a
like purpose; to classify; as, to assort goods.
Note: [Rarely applied to persons.]
They appear . . . no ways assorted to those with
whom they must associate. --Burke.
2. To furnish with, or make up of, various sorts or a variety
of goods; as, to assort a cargo.
\As*sort"\, v. i.
To agree; to be in accordance; to be adapted; to suit; to
fall into a class or place. --Mitford.