_{R}\right] $ and $\left[ T_{j}-5T_{R},T_{j}+5T_{R}\right] $ centred around the parameter combinations giving the best scores. For *Arabidopsis* optimisations, light parameters controlling the impulse inputs ($L_{1}$ and $L_{3}$ in the 2-loop circuit; $L_{1},L_{3}$ and $L_{4}$ in the 3-loop circuit) were fixed at the values shown in Table \[tab:optimparams\]. These were determined from discrete approximations to the corresponding continuous curves in the DE models.

Each parameter set was then checked for a functioning circadian clock. Firstly, the solution to the free-running model was generated under the appropriate continuous light conditions, using the discretised data as initial conditions. If a limit cycle was obtained with period within $20\%$ of the DE model, this was fed into the model under simulated 12:12 LD cycles. Periodic, entrained oscillations were taken to indicate a viable clock circuit.

Forflexibility under changing environmental conditions. , 424.

Ruoff, P. & Rensing, L. 1996 The temperature-compensated [Goodwin]{} model simulates many circadian clock properties. , 275–285.

Ay, F., Xu, F. & Kahveci, T. 2009 Scalable steady state analysis of [Boolean]{} biological regulatory networks. , e7992.

Hau, L. D. & Kwon, Y. K. 2011 The effects of feedback loops on disease comorbidity in human signaling networks. , 1113–20.

Krawitz, P. & Shmulevich, I. 2007 Basin entropy in [Boolean]{} network ensembles. , 158701.

Ribeiro, A. S. & Kauffman, S. A. 2007 Noisy attractors and ergodic sets in models of gene regulatory networks. , 743–55.

Garg, A., Cara, A. D., Xenarios, I., Mendoza, L. & Micheli, G. D. 2008 Synchronous versus asynchronous modeling of gene regulatory networks. , 1917–25.

Buchler, N., Gerland, U. & Hwa, T. 2003 On schemes of combinatorial transcription logic. , 5136–5141.

Gerstung, M.,


decimal expansion. Scores of 0 and 1 indicate the best and worst fits respectively. Triangles denote LCs for which the Boolean model yields a viable clock. LCs mirroring the activation and inhibition pattern of the corresponding DE models in Figs. \[fig:DECircuitDiags\]C and D are plotted in red. In A, only LCs yielding scores less than 0.75 are shown. Of these, there are 3 LCs consistent with the activation and inhibition pattern of the equivalent DE model from a possible total of 4. From this subset, $G=(10011011)$ emerges as the optimal configuration yielding a viable clock. For this circuit, both LHY and TOC1 can independently inhibit Y while TOC1 is repressed unless LHY is repressed while Y is activated (the corresponding gates are both of the AND type). Inseries (left panels) have been normalised to lie between 0 and 1 in order to facilitate comparison with the Boolean simulations (right panels). Different components within a model are slightly offset from one another so they can be distinguished more easily. The time step used for solving the Boolean models was 0.5h, equal to the data sampling interval.

**Figure 7. Comparing the photoperiodic behaviour of the Boolean and DE versions of each model.** For Boolean models, the phase of each species is taken as the time within the LD cycle of the ON to OFF transition (downward triangles). Analogously, the DE model phases are defined as the times at which species decrease below the thresholds yielding the optimal fit of the corresponding Boolean circuit to data (upward triangles). Shaded


          -                3
  ----------------- -------------- --------------- --------------- -----------------

  :  **[Optimal parameter sets - synthetic data.]{}**

The logic configurations, $\mathbf{G}$, delays, $\protect\tau _{j}$, and discretisation thresholds, $T_{i}$ ($0 <T_{i}<1$), yielding the best fit of each logic model to synthetic time series. The values used for photoperiod simulations - obtained by fitting directly to DE time series - are shown in brackets. For each model, $p_{k}$ indicates the parameter used to simulate light input $L_{k}$ through equation (\[eq:lightfunc\]). These were fixed at the values shown, with $P$ denoting the photoperiod $t_{DUSK}-t_{DAWN}$.

\[tab:optimparams\]

    $\mathbf{G}$     10101011011 ($G_{OPT}$)   10011011011on a list of candidates that Universal is considering, but no offer has been made and it’s not even clear if Black would want to direct this thing. Universal won’t make any official movements until Chris Morgan, who has written every film in the franchise since Tokyo Drift, so this is far from confirmation, but it does let us know what kind of filmmaker the studio is after.

The untitled spinoff will see Johnson and Statham reprising their roles of Hobbs and Shaw, respectively, as the characters team up for an undisclosed mission. Some say that mission involves hunting down Charlize Theron’s Fate of the Furious villain Cypher, but it’s still somewhat early days on this and unclear if she’s involved. Johnson and Statham showcased some terrific chemistry in


---
address:
- '[**P.E.**]{}: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA'
- '[**X.M.**]{}: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA'
author:
- Pavel Etingof
- Xiaoguang Ma
title: Lecture notes on Cherednik algebras
---

Introduction
============

Double affine Hecke algebras, also called Cherednik algebras, were introduced by Cherednik in 1993 as a tool in his proof of Macdonald’s conjectures about orthogonal polynomials for root systems. Since then, it has been realized that Cherednik algebras are of great independent interest; they appeared in many different mathematical contexts and found numerous applications.

The present notes are based on a course on Cherednik algebras given by the first author at MIT in the Fall of 2009. Their goal is to give an introduction to Cherednik algebras, and to review the web of connections between them"a2"-"a3" "a3"-"a4" "a4"-"a5"
"a5"-"a6" "a6"-"a7" "a3"-"a8"}$$

The parabolic subgroups will be Coxeter groups whose Dynkin diagrams are obtained by deleting vertices from the above graph. In particular, the maximal parabolic subgroups are $D_7, A_7, A_1\times A_6, A_2\times A_1\times A_4, A_4\times A_3, D_5\times A_2, E_6\times A_1, E_7$.

Suppose $G'\subset G$ is a parabolic subgroup, and $b \in{\mathfrak{h}}$ is such that $G_{b} = G'$. In this case, we have a natural $G'$-invariant decomposition ${\mathfrak{h}}={\mathfrak{h}}^{G'}\oplus ({{\mathfrak{h}}^{*}}^{G'})^{\perp}$, and $b \in {\mathfrak{h}}^{G'}$. Thus we have a nonempty open set ${\mathfrak{h}}_{{\mathrm{reg}}}^{G'}$ of all $a \in {\mathfrak{h}}^{G'}$ for which $G_{a} = G'$; this set is nonempty because it contains $b$. We also have a $G'$-invariant decomposition ${\mathfrak{h}}^{*}={{\mathfrak{h}}^{*}}^{G'} \oplus ({\mathfrak{h}}^{G'} )^{\perp}$, and we can define the open set ${\mathfrak{h}}_{{\mathrm{reg}}}^{*G'}$ of all $\lambda\in{\mathfrak{h}}^{G'}$ for which $G_{\lambda} = G'$. It


{{\mathcal S}}}\frac{c_s(\alpha_s,\alpha_s)}{\alpha_s}\partial_{\alpha_s^\vee}
+U,$$ where $$U=\delta_c(\Delta_{\mathfrak{h}}\delta_c^{-1})-
\sum_{s\in {{\mathcal S}}}\frac{c_s(c_s+1)(\alpha_s,\alpha_s)}{\alpha_s^2}.$$ Let us compute $U$. We have $$\delta_c(\Delta_{\mathfrak{h}}\delta_c^{-1})=
\sum_{s\in {{\mathcal S}}}\frac{c_s(c_s+1)(\alpha_s,\alpha_s)}{\alpha_s^2}+
\sum_{s\ne u\in {{\mathcal S}}}\frac{c_sc_u(\alpha_s,\alpha_u)}{\alpha_s\alpha_u}.$$ We claim that the last sum $\Sigma$ is actually zero. Indeed, this sum is invariant under the Coxeter group, so $\prod_{s\in {{\mathcal S}}}\alpha_s\cdot \Sigma$ is a regular anti-invariant function of degree $|{{\mathcal S}}|-2$. But the smallest degree of a nonzero anti-invariant is $|{{\mathcal S}}|$, so $\Sigma=0$, $U=0$, and we are done (Proposition \[prop-delta\] and Theorem \[thm-OPsys\] are proved).

A similar method works for any complex reflection group $G$. Namely, the operators $L_{i}=m(P_{i}(D_{y_{1}}, \ldots, D_{y_{r}}))$ form a quantum integrable system. However, if $G$ is not a real reflection group, this system does not have a quadratic Hamiltonian in momentum variables (so it does not have a physical meaning).

Uniqueness of the operators $L_{j}$
-----------------------------------

Theof degree zero, and $L_1^0=H_0$.

Moreover, we can define the operators $L_j(\hbar)$ in ${\mathrm{Rees}}({{\mathcal D}}({\mathfrak{h}}_{{\mathrm{reg}}})^W)$ in the same way as $L_j$, but using the Dunkl operators $D_{y_i}(\hbar)$ instead of $D_{y_i}$. Then $[L_j(\hbar),L_k(\hbar)]=0$, and $L_j(\hbar)|_{\hbar=0}=L_j^0$. This implies that $L_j^0$ Poisson commute: $\lbrace{L_j^0,L_k^0\rbrace}=0$.

Theorem \[thm-classop\] is proved.

As in the quantum situation, Theorem \[thm-classop\] can be generalized to complex reflection groups, giving integrable systems with Hamiltonians which are non-quadratic in momentum variables.

Notes
-----

Section 2.1 follows Section 5.4 of [@E4]; the definition of complex reflection groups and their basic properties can be found in [@GM]; the definition of parabolic subgroups and the notations are borrowed from Section 3.1 of [@BE]; the remaining parts of this section follow Section 6 of [@E4].

The rational Cherednik algebra {#rcera}
==============================

Definition and examples
-----------------------

[\[sec:rca\]]{} Above we have made essential use of the


is an algebra over ${{\mathbb{C}}}[\hbar, c]$.) It has two equivalent definitions similar to the above.

2\. It is more convenient to work with algebras defined by generators and relations than with subalgebras of a given algebra generated by a given set of elements. Therefore, from now on we will use the statement of Proposition \[prop-RCA\] as a definition of the rational Cherednik algebra $H_{c}$. According to Proposition \[prop-RCA\], this algebra comes with a natural embedding $\Theta_c: H_{c}\to {\mathrm{Rees}}({{\mathbb{C}}}G\ltimes
{{\mathcal D}}({\mathfrak{h}}_{{\mathrm{reg}}}))$, defined by the formula $x\to x$, $g\to g$, $y\to D_y(c,\hbar)$. This embedding is called [*the Dunkl operator embedding*]{}.

1\. Let $G={{\mathbb{Z}}}_2$, ${\mathfrak{h}}={{\mathbb{C}}}$. In this case $c$ reduces to one parameter, and the algebra $H_{t,c}$ is generated by elements $x,y,s$ with defining relations $$s^2=1,\ sx=-xs,\ sy=-ys,\ [y,x]=t-2cs.$$

2\. Let $G={\mathfrak{S}}_n$, ${\mathfrak{h}}={{\mathbb{C}}}^n$. Inpositive definite).

\[c2\] We have $$F(k)=b_0^k\prod_i \left(\frac{\Gamma(k+k_i)}{\Gamma(k_i)}\right)^{n_i}.$$

Denote the right hand side by $F_*(k)$ and let $\phi(k)=F(k)/F_*(k)$. Clearly, $\phi(0)=1$. Proposition \[l1\] implies that $\phi(k)$ is a 1-periodic positive function on $[0,\infty)$. Also by the Cauchy-Schwarz inequality, $$F(k)F(k')\ge F((k+k')/2)^2,$$ so $\log F(k)$ is convex for $k\ge 0$. This implies that $\phi=1$, since $(\log F_*(k))''\to 0$ as $k\to +\infty$.

The proof of this corollary is motivated by the standard proof of the following well known characterization of the $\Gamma$ function.

The $\Gamma$ function is determined by three properties:

1.  $\Gamma(x)$ is positive on $[1, +\infty)$ and $\Gamma(1)=1$;

2.  $\Gamma(x+1)=x\Gamma(x)$;

3.  $\log \Gamma(x)$ is a convex function on $[1, +\infty)$.

It is easy to see from the definition $\Gamma(x)=\int_{0}^{\infty} t^{x-1}e^{-t}{\mathrm{d}}t$ that the $\Gamma$ function has properties (i) and (ii); property (iii) follows from this definition and


the Cauchy-Schwarz inequality.

Conversely, suppose we have a function $F(x)$ satisfying the above properties, then we have $F(x)=\phi(x)\Gamma(x)$ for some $1$-periodic function $\phi(x)$ with $\phi(x)>0$. Thus, we have $$(\log F)''=(\log \phi)''+(\log \Gamma)''.$$ Since $\lim_{x\to +\infty}(\log \Gamma)''=0$, $(\log F)''\geq 0$, and $\phi$ is periodic, we have $(\log \phi)''\geq 0$. Since $\int_{n}^{n+1}(\log \phi)''{\mathrm{d}}x=0$, we see that $(\log \phi)''\equiv 0$. So we have $\phi(x)\equiv 1$.

In particular, we see from Corollary \[c2\] and the multiplication formulas for the $\Gamma$ function that part (ii) of Theorem \[thm-MMI\] implies part (i).

It remains to establish (ii).

\[degree\] The polynomial $b$ has degree exactly $|{{\mathcal S}}|$.

By Proposition \[c1\], $b$ is a polynomial of degree at most $|{{\mathcal S}}|$. To see that the degree is precisely $|{{\mathcal S}}|$, let us make the change of variable ${\mathbf x}=k^{1/2}{\mathbf y}$ inoperators for $(G_b,{\mathfrak{h}})$, which, together with functions on the formal neighborhood of $b$ and the group $G_b$, generate the completion of $H_{1,c}(G_b,{\mathfrak{h}})$. This gives a construction of $\theta$ without using explicit formulas.

Also, this argument explains why $\theta$ should be defined by formula (\[eqn:thetay\]) of Theorem \[comp\]. Indeed, what this formula does is just restores the terms with $s\notin G_b$ that have been previously deleted.

The map $\theta$ defines an equivalence of categories $$\theta_{*}: \widehat{H_{1,c}}(G, {\mathfrak{h}})_b-{\mathrm mod}\to
Z(G, G_b, \widehat{H_{1, c}}(G_b,{\mathfrak{h}})_0)-{\mathrm mod}.$$

\[whit\] We have a natural equivalence of categories $$\psi_\lambda: {\mathcal{O}}_c(G, {\mathfrak{h}})_\lambda\to
{\mathcal{O}}_{c}(G_\lambda,{\mathfrak{h}}/{\mathfrak{h}}^{G_\lambda})_0.$$

The category ${\mathcal{O}}_c(G, {\mathfrak{h}})_\lambda$ is the category of modules over $H_{1,c}(G, {\mathfrak{h}})$ which are finitely generated over ${{\mathbb{C}}}[{\mathfrak{h}}]$ and extend by continuity to the completion of the algebra $H_{1,c}(G, {\mathfrak{h}})$ at $\lambda$. So it follows from Theorem \[comp\] that


At the end we used Proposition \[adj1\].

Then we can obtain the following corollary easily.

\[proj\] The functor ${\mathrm{Res}}_b$ maps projective objects to projective ones, and the functor ${\mathrm{Ind}}_b$ maps injective objects to injective ones.

We can also define functors ${\mathrm{res}}_\lambda: {\mathcal{O}}_c(G, {\mathfrak{h}})_0\to
{\mathcal{O}}_{c}(G',{\mathfrak{h}}/{\mathfrak{h}}^{G'})_0$ and ${\mathrm{ind}}_\lambda:
{\mathcal{O}}_{c}(G',{\mathfrak{h}}/{\mathfrak{h}}^{G'})_0\to
{\mathcal{O}}_c(G, {\mathfrak{h}})_0$, attached to $\lambda\in
{\mathfrak{h}}^{*G'}_{{\mathrm{reg}}}$, by $${\mathrm{res}}_\lambda:= \dagger\circ {\mathrm{Res}}_\lambda\circ \dagger,
{\mathrm{ind}}_\lambda:= \dagger\circ {\mathrm{Ind}}_\lambda\circ \dagger,$$ where $\dagger$ is as in Subsection \[sec:duali\].

\[proj1\] The functors ${\mathrm{res}}_\lambda$, ${\mathrm{ind}}_\lambda$ are exact. The functor ${\mathrm{ind}}_\lambda$ is left adjoint to ${\mathrm{res}}_\lambda$. The functor ${\mathrm{ind}}_\lambda$ maps projective objects to projective ones, and the functor ${\mathrm{res}}_\lambda$ injective objects to injective ones.

Easy to see from the definition of the functors and the Theorem \[adj2\].

We also have the following proposition, whose proof is straightforward.

\[Jacc\] We have $${\mathrm{ind}}_\lambda(N)=({\mathcal J}\circ \psi_\lambda^{-1})(N),\quad\text{
and }\quad
{\mathrm{res}}_\lambda(M)=(\psi_\lambda\circ E_\lambda)(\widehat{M}),$$ where $\psi_\lambda$ is Any object of ${\mathcal{O}}_{c}(G, {\mathfrak{h}})_{0}$ is a quotient of a multiple of $M_{c, n}({{\mathbb{C}}}G)$ for large enough $n$.

Directly from the definition and the above theorem.

Now we can finish the proof of Theorem \[GGORt\]. We have shown that for generic $c$, ${\mathrm{KZ}}(M_{c, n}({{\mathbb{C}}}G))\in {\mathrm{Rep}}\mathscr{H}_{q}(G)$. Hence this is true for any $c$, since $M_{c, n}({{\mathbb{C}}}G)$ is a flat family of modules over $H_{c}(G, {\mathfrak{h}})$. Then, ${\mathrm{KZ}}(M)$ is a $\mathscr{H}_{q}(G)$-module for all $M$, since any $M$ is a quotient of $M_{c, n}({{\mathbb{C}}}G)$ and the functor ${\mathrm{KZ}}$ is exact.

\[Broué, Malle, Rouquier, [@BMR]\] Let $q_{j, H}=\exp(t_{j, H})$ where $t_{j, H}$’s are formal parameters. Then $\mathscr{H}_{q}(G)$ is a free module over ${{\mathbb{C}}}[[t_{j, H}]]$ of rank $|G|$.

We have $$\mathscr{H}_{q}(G)/(t)=\mathscr{H}_{1}(G)={{\mathbb{C}}}G.$$ So it remains to show that $\mathscr{H}_{q}(G)$ is free. To show this, it is sufficient


less than $e$ times).

See [@DJ], Theorem 6.3, 6.8.

Now let $M_{c}(\lambda)$ be the Verma module associated to the Specht module for ${\mathfrak{S}}_{n}$ and $L_{c}(\lambda)$ be its irreducible quotient. Then we have the following theorem.

If $c\leq 0$, then ${\mathrm{KZ}}(M_{c}(\lambda))=S_{\lambda}$ and ${\mathrm{KZ}}(L_{c}(\lambda))=D_{\lambda}$.

See Section 6.2 of [@GGOR].

If $c\le 0$, then $\mathrm{Supp}L_{c}(\lambda)={{\mathbb{C}}}^{n}$ if and only if $\lambda$ is $e$-regular. If $c>0$, then $\mathrm{Supp}L_{c}(\lambda)={{\mathbb{C}}}^{n}$ if and only if $\lambda^{\vee}$ is $e$-regular, or equivalently, $\lambda$ is $e$-restricted (i.e., $\lambda_{i}-\lambda_{i+1}<e$ for $i=1, \ldots, n-1$).

Directly from the definition and the above theorem.

Notes
-----

The references for this section are [@GGOR], [@BMR].

Rational Cherednik algebras and Hecke algebras for varieties with group actions
===============================================================================

Twisted differential operators
------------------------------

[\[sec:tdo\]]{}

Let us recall the theory of twisted differential operators (see [@BB], section 2).

Let $X$ be a smooth affine algebraic variety over ${{\mathbb{C}}}$. Given a closed 2-form $\omega$on $X$, the algebra ${{\mathcal D}}_\omega(X)$ of differential operators on $X$ twisted by $\omega$ can be defined as the algebra generated by ${\mathcal{O}}_X$ and “Lie derivatives” ${\mathbf L}_v$, $v\in \mathrm{Vect}(X)$, with defining relations $$f{\mathbf L}_v={\mathbf L}_{fv},\ [{\mathbf L}_v,f]=L_vf,\ [{\mathbf L}_v,{\mathbf L}_w]={\mathbf L}_{[v,w]}+\omega(v,w).$$ This algebra depends only on the cohomology class $[\omega]$ of $\omega$, and equals the algebra ${{\mathcal D}}(X)$ of usual differential operators on $X$ if $[\omega]=0$.

An important special case of twisted differential operators is the algebra of differential operators on a line bundle. Namely, let $L$ be a line bundle on $X$. Since $X$ is affine, $L$ admits an algebraic connection $\nabla$ with curvature $\omega$, which is a closed 2-form on $X$. Then it is easy to show that the algebra ${{\mathcal D}}(X,L)$ of differential operators on


of $\Sigma$, and $a_l,b_l, l=1,\ldots, g$, be the $a$-cycles and $b$-cycles of $\Sigma$. Let $c_j$ be the counterclockwise loops around $P_j$. Then $\Gamma$ is generated by $a_l,b_l,c_j$ with relations $$c_j^{n_j}=1,\ c_1c_2\cdots c_m=\prod_la_lb_la_l^{-1}b_l^{-1}.$$ For each $j$, introduce formal parameters $\tau_{kj}$, $k=1,\ldots, n_j$. Define the Hecke algebra ${\mathscr H}_\tau(\Sigma)$ of $\Sigma$ to be generated over ${{\mathbb{C}}}[[\tau]]$ by the same generators $a_l,b_l,c_j$ with defining relations $$\prod_{k=1}^{n_j}(c_j-{{\mathbf e}}^{2\pi j {{\mathbf i}}/n_j}{{\mathbf e}}^{\tau_{kj}})=0, \
c_1c_2\cdots c_m=\prod_la_lb_la_l^{-1}b_l^{-1}.$$ Thus ${\mathscr H}_\tau(\Sigma)$ is a deformation of ${{\mathbb{C}}}[\Gamma]$.

This deformation is flat if $H$ is a Euclidean plane or a Lobachevsky plane. Indeed, ${\mathscr H}_\tau(\Sigma)={\mathscr H}_\tau(\Gamma,H)$, so the result follows from Theorem \[p2\] and the fact that ${\mathrm{H}}^{2}(H, {{\mathbb{C}}})=0$.

On the other hand, if $H$ is the Riemann sphere (so that the condition ${\mathrm{H}}^{2}(H, {{\mathbb{C}}})=0$ is violated) and $\Gamma\ne 1$theory of Askey-Wilson polynomials.

This is a “multivariate” version of the Hecke algebras attached to Fuchsian groups, defined in the previous subsection. Namely, letting $\Gamma,H$ be as in the previous subsection, and $N\ge 1$, we consider the manifold $X=H^N$ with the action of $\Gamma_N={\mathfrak{S}}_N\ltimes \Gamma^N$. If $H$ is a Euclidean or Lobachevsky plane, then by Theorem \[p2\] ${\mathscr H}_\tau(\Gamma_N,X^N)$ is a flat deformation of the group algebra ${{\mathbb{C}}}[\Gamma_N]$. If $N>1$, this algebra has one more essential parameter than for $N=1$ (corresponding to reflections in ${\mathfrak{S}}_N$). In the Euclidean case, one expects that an appropriate “spherical” subalgebra of this algebra is a quantization of the Hilbert scheme of a del Pezzo surface.

The Knizhnik-Zamolodchikov functor
----------------------------------

[\[sec:KZ\]]{}

In this subsection we will define a global analog of the KZ functor defined in [@GGOR]. This


assume that $R$ is reduced. Let $\{\alpha_{i}\}_{i=1}^{n}\subset R$ be the set of simple roots and $R_{+}$ (respectively $R_{-}$) be the set of positive (respectively negative) roots. The coroots are denoted by $\alpha^{\vee}=2\alpha/(\alpha,\alpha)$. Let $Q^{\vee}=\bigoplus_{i=1}^{n}{{\mathbb{Z}}}\alpha_{i}^{\vee}$ be the coroot lattice and $P^{\vee}=\bigoplus_{i=1}^{n}{{\mathbb{Z}}}\omega^{\vee}_{i}$ the coweight lattice, where $\omega_{i}^{\vee}$’s are the fundamental coweights, i.e., $(\omega^{\vee}_{i}, \alpha_{j})=\delta_{ij}$. Let $\theta$ be the maximal positive root, and assume that the bilinear form is normalized by the condition $(\theta,\theta)=2$. Let ${\overline{W}}$ be the Weyl group which is generated by the reflections $s_{\alpha}$ ($\alpha\in R$).

By definition, the affine root system is $$R^{a}=\{ \tilde{\alpha}= [\alpha,j]\in
{{\mathbb{R}}}^{n}\times{{\mathbb{R}}}|
\text{ where } \alpha\in R, j\in {{\mathbb{Z}}}\}.$$ The set of positive affine roots is $R^{a}_{+}=\{[\alpha,j]\,|\,j\in {{\mathbb{Z}}}_{>0}\}\cup
\{[\alpha,0]\,|\,\alpha\in R_{+}\}$. Define $\alpha_{0}=[-\theta, 1]$. We will identify $\alpha\in R$ with $\tilde{\alpha}=[\alpha,0]\in R^{a}$.

For an arbitrary affine root$A_{+}(W)$ is free as a left module over $R$. We are done.

Notes
-----

Section \[sec:dpez\] follows Section 6 of the paper [@EOR]; Cherednik’s definition of the double affine Hecke algebra of a root system is from Cherednik’s book [@Ch]; Sections \[sec:HaF\] and \[sec:flatF\] follow the paper [@ER]; The other parts of this section follow the paper [@E1].

Symplectic reflection algebras
==============================

The definition of symplectic reflection algebras
------------------------------------------------

Rational Cherednik algebras for finite Coxeter groups are a special case of a wider class of algebras called symplectic reflection algebras. To define them, let $V$ be a finite dimensional symplectic vector space over ${{\mathbb{C}}}$ with a symplectic form $\omega$, and $G$ be a finite group acting symplectically (linearly) on $V$. For simplicity let us assume that $(\wedge^2V^*)^G={{\mathbb{C}}}\omega$ (i.e., $V$ is symplectically irreducible) and that $G$ acts


$A$ are finite dimensional.

1.  The algebra $A$ is said to be quadratic if it is generated over $R$ by $A[1]$, and has defining relations in degree 2.

2.  $A$ is Koszul if all elements of ${\mathrm{Ext}}^i(R,R)$ (where $R$ is the augmentation module over $A$) have grade degree precisely $i$.

<!-- -->

1.  Thus, in a quadratic algebra, $A[2]=A[1]\otimes_R A[1]/E$, where $E$ is the subspace ($R$-subbimodule) of relations.

2.  It is easy to show that a Koszul algebra is quadratic, since the condition to be quadratic is just the Koszulity condition for $i=1,2$.

Now let $A_0$ be a quadratic algebra, $A_0[0]=R$. Let $E_0$ be the space of relations for $A_0$. Let $E\subset A_0[1]\otimes_R A_0[1][[\hbar]]$ be a free (over ${{\mathbb{C}}}[[\hbar]]$) $R$-subbimodule which reduces to $E_0$ modulo $\hbar$ (“deformation of the relations”).on $A_0$ of degree $-2$, which depends linearly on $t,c$. So by the lemma, this bracket has to be of the form $f(t,c)\Pi$, where $\Pi$ is the unique up to scaling Poisson bracket of degree $-2$, and $f$ a homogeneous linear function. Thus the algebra $A={\mathsf{B}}_{t,c}$ is not commutative unless $f(t,c)=0$. On the other hand, if $f(t,c)=0$, and ${\mathsf{B}}_{t,c}$ is not commutative, then, as we’ve shown, $A_0$ has a nonzero Poisson bracket of degree $<-2$. But By Lemma \[poisbr\], there is no such brackets. So ${\mathsf{B}}_{t,c}$ must be commutative if $f(t,c)=0$.

It remains to show that $f(t,c)$ is in fact a nonzero multiple of $t$. First note that $f(1,0)\ne 0$, since ${\mathsf{B}}_{1,0}$ is definitely noncommutative. Next, let us take a point $(t,c)$ such that ${\mathsf{B}}_{t,c}$ is commutative. Look at


isomorphism $\eta_0: A/{\mathfrak{m}} A\to A_0$ where $\mathfrak{m} = \langle\hbar_1,\ldots,\hbar_n\rangle$ is the maximal ideal in ${{\mathbb{C}}}[[\hbar]]$.

When no confusion is possible, we will call $A$ a deformation of $A_0$ (omitting “formal”).

Let us restrict ourselves to one-parameter deformations with parameter $\hbar$. Let us choose an identification $\eta: A\to A_0[[\hbar]]$ as ${{\mathbb{C}}}[[\hbar]]$-modules, such that $\eta=\eta_0$ modulo $\hbar$. Then the product in $A$ is completely determined by the product of elements of $A_0$, which has the form of a “star-product” $$\mu(a,b)=a\ast b=\mu_0(a,b)+\hbar \mu_1(a,b)+\hbar^2 \mu_2(a,b)+\cdots,$$ where $\mu_i: A_0\otimes A_0\to A_0$ are linear maps, and $\mu_0(a,b)=ab$.

### Hochschild cohomology

The main tool in deformation theory of associative algebras is Hochschild cohomology. Let us recall its definition.

Let $A$ be an associative algebra. Let $M$ be a bimodule over $A$. A Hochschild $n$-cochain of $A$ with coefficients in $M$the lemma to $B={\mathrm Weyl}(V)$. For this we need to calculate ${{\mathrm{HH}}}^*(B,Bg)$, where $g$ is any element of $G$. We may assume that $g$ is diagonal in some symplectic basis: $g={\mathrm{diag}}(\lambda_1,\lambda_1^{-1},\ldots,\lambda_n,\lambda_n^{-1})$. Then by the Künneth formula we find that $${{\mathrm{HH}}}^*(B,Bg)=\bigotimes_{i=1}^n {{\mathrm{HH}}}^*(\mathbb A_1,\mathbb A_1g_i),$$ where $\mathbb A_1$ is the Weyl algebra of the $2$-dimensional space, (generated by $x,y$ with defining relation $xy-yx=1$), and $g_i={\mathrm{diag}}(\lambda_i,\lambda_i^{-1})$.

Thus we need to calculate ${{\mathrm{HH}}}^*(B,Bg)$, $B=\mathbb A_1$, $g={\mathrm{diag}}(\lambda,\lambda^{-1})$.

${{\mathrm{HH}}}^*(B,Bg)$ is 1-dimensional, concentrated in degree $0$ if $\lambda=1$ and in degree $2$ otherwise.

If $B=\mathbb A_1$ then $B$ has the following Koszul resolution as a B-bimodule: $$B\otimes B\to B\otimes {{\mathbb{C}}}^2\otimes B\to B\otimes B\to B.$$ Here the first map is given by the formula $$\begin{aligned}
b_1\otimes b_2&\mapsto& b_1\otimes x\otimes yb_2-b_1\otimes y\otimes xb_2
-b_1y\otimes x\otimes b_2+b_1x\otimes y\otimes b_2,\end{aligned}$$ the second map is


which is Cohen-Macaulay, so the result follows from Theorem \[cmac\], (i).

Now, we claim that if $Z,M$ are positively filtered and ${\mathrm{gr}}M$ is a Cohen-Macaulay ${\mathrm{gr}}Z$-module then $M$ is a Cohen-Macaulay $Z$-module. Indeed, let $z_1,\ldots,z_n$ be homogeneous algebraically independent elements of ${\mathrm{gr}}Z$ such that ${\mathrm{gr}}Z$ is a finite module over the subalgebra generated by them. Let $z_1', \ldots, z_n'$ be their liftings to $Z$. Then $z_1', \ldots, z_n'$ are algebraically independent, and the module $M$ over ${{\mathbb{C}}}[z_1', \ldots, z_n']$ is finitely generated and (locally) free since so is the module ${\mathrm{gr}}M$ over ${{\mathbb{C}}}[z_1, \ldots, z_n]$.

Recall now that ${\mathrm{gr}}{\mathsf{H}}_{0, c}{\mathtt{e}}=SV$, ${\mathrm{gr}}{\mathsf{Z}}_{0, c}=(SV)^G$. Thus the $c=0$ case implies the general case, and we are done.

Now let us show that any Azumaya point of ${\mathsf{M}}_c$ is smooth. Let $U$ be an affinethe short exact sequence $$0\to I\to {{\mathbb{C}}}[x, y]\to E\to 0.$$ Let $S^{(n)}{{\mathbb{C}}}^{2}
=\underbrace{{{\mathbb{C}}}^{2}\times\cdots\times{{\mathbb{C}}}^{2}}_{n \text{ times}}
/{\mathfrak{S}}_{n}$, where ${\mathfrak{S}}_{n}$ acts by permutation. We have a natural map ${\mathrm{Hilb}}_{n}({{\mathbb{C}}}^{2})\to S^{(n)}{{\mathbb{C}}}^{2}$ which sends every ideal $I$ to its zero set (with multiplicities). This map is called the [*Hilbert-Chow map*]{}.

1.  ${\mathrm{Hilb}}_{n}({{\mathbb{C}}}^{2})$ is a smooth quasiprojective variety.

2.  The Hilbert-Chow map ${\mathrm{Hilb}}_{n}({{\mathbb{C}}}^{2})\to S^{(n)}{{\mathbb{C}}}^{2}$ is projective. It is a resolution of singularities.

Proof can be found in [@Na].

Now consider the Calogero-Moser space ${{\mathcal C}}_{n}$ defined in Section \[sec:cmspace\].

\[na\] The Hilbert Scheme ${\mathrm{Hilb}}_{n}({{\mathbb{C}}}^{2})$ is $C^{\infty}$-diffeomorphic to ${{\mathcal C}}_{n}$.

\[na1\] More precisely there exists a family of algebraic varieties over $\mathbb{A}_{1}$, say $X_{t}$, $t\in \Bbb A_1$, such that $X_{t}$ is isomorphic to ${{\mathcal C}}_{n}$ if $t\neq 0$ and $X_{0}$ is the Hilbert scheme; and also if we denote by


orbit passing through $\mu$.

The deformed Harish-Chandra homomorphism
----------------------------------------

Finally, we would like to explain how to quantize the Calogero-Moser space ${{\mathcal C}}_n$, using the procedure of quantum Hamiltonian reduction.

Let ${\mathfrak{g}}={\mathfrak{gl}}_n$, $A={{\mathcal D}}({\mathfrak{g}})$ as above. Let $k$ be a complex number, and $W_k$ be the representation of ${\mathfrak{sl}}_n$ on the space of functions of the form $(x_1\cdots x_n)^{k}f(x_1,\ldots,x_n)$, where $f$ is a Laurent polynomial of degree $0$. We regard $W_k$ as a ${\mathfrak{g}}$-module by pulling it back to ${\mathfrak{g}}$ under the natural projection ${\mathfrak{g}}\to {\mathfrak{sl}}_n$. Let $I_k$ be the annihilator of $W_k$ in $U({\mathfrak{g}})$. The ideal $I_k$ is the quantum counterpart of the coadjoint orbit of matrices $T$ such that $T+1$ has rank $1$.

Let $B_k={{\mathcal D}}({\mathfrak{g}})^{{\mathfrak{g}}}/({{\mathcal D}}({\mathfrak{g}})\mu(I_{k}))^{{\mathfrak{g}}}$ where $\mu:U({\mathfrak{g}})\to A$ is the quantum momentum map (the quantum Hamiltonian reduction with respectsituation we have an exact sequence $$1\to \pi_1(X,x)\to \pi_1^{\mathrm orb}(X/G,x)\to G\to 1.$$

[^2]: Here we use the fact that the filtration is bounded from below. In the case of an unbounded filtration, it is possible for a map not to be an isomorphism if its associated graded is an isomorphism. An example of this is the operator of multiplication by $1+t^{-1}$ in the space of Laurent polynomials in $t$, filtered by degree.

[^3]: Note that we don’t have to worry about the existence of a unit in $A$ since a formal deformation of an algebra with unit always has a unit.

[^4]: In spite of the universal property of $A_u$, it may happen that there is an isomorphism between the algebras $A^1$ and $A^2$ corresponding to different paths $\gamma_1(\hbar),\gamma_2(\hbar)$ (of course,


parental rights

to J.M. (born in December 2004) (hereinafter “Child”).1 We affirm.

      Since 2012, Lycoming County Children and Youth Services has been

involved   with   this   family   due   to   Mother’s   neglect   of   the   children,

homelessness, and mental health issues.           Child has two half-sisters who

* Retired Senior Judge assigned to the Superior Court.
1
  At the conclusion of the termination hearing on December 11, 2014, the
Honorable Michael H. Sholley directed Mother and Child’s father, J.B.
(hereinafter “Father”), to submit findings of fact and conclusions of law on or
before January 31, 2015. N.T. Hearing, 12/11/14, at 116-117. Mother and
Father filedmental
           health issues were a primary factor, and not offering any
           other evidence as to the condition and cause of her negative
           behavior, therefore, could not and did not state that the
           conditions and causes of the alleged misbehavior could not or
           would not be remedied as is required under the statute?

        3. Should the [t]rial [c]ourt have denied termination and ruled


incapacity, abuse, neglect or
           refusal cannot or will not be remedied by the parent.

                                     ***

     (b) Other considerations.─The court in terminating the rights
     of a parent shall give primary consideration to the
     developmental, physical and emotional needs and welfare of the
     child. The rights of a parent shall not be terminated solely on
     the basis of environmental factors such as inadequate housing,
2003)

(citations omitted).

      Our Supreme Court set forth our inquiry under section 2511(a)(2) as

follows.

      As stated above, § 2511(a)(2) provides statutory grounds for
      termination of parental rights where it is demonstrated by clear
      and convincing evidence that “[t]he repeated and continued
      incapacity, abuse, neglect or refusal of the parent has caused
      the child to be without essential parental care, control or
      subsistence necessary for his physical or mental well-being and
      the conditions and causes of the incapacity, abuse, neglect or
      refusal cannot or will


convincing evidence in the record that

the repeated and continued incapacity, abuse, neglect or refusal of the

Mother had caused Child to be without essential parental care, control or

subsistence necessary for his physical or mental well-being, and the

conditions and causes of the incapacity, abuse, neglect or refusal cannot or

will not be remedied by the Mother. Trial Court Opinion, 4/27/15, at 2-4.

      This Court has stated that a parent is required to make diligent efforts

towards the reasonably prompt assumption of full parental responsibilities.

In re A.L.D. 797 A.2d 326, 337 (Pa. Super. 2002).         A parent’s vow to

cooperate, after a long period of uncooperativeness regarding the necessity

or availability of services, may properly be rejected as untimely or

disingenuous. Id. at 340.have instructed that the court must also discern the nature

and status of the parent-child bond, with utmost attention to the effect on

the child of permanently severing that bond. See id.

      While Mother did not contest section 2511(b) on her appeal, we will

still review whether termination of parental rights would best serve the

developmental, physical, and emotional needs of Child.         The trial court

found:

      [C]hild has been placed in the pre-adoptive home of [the foster
      parents]. [C]hild is thriving in this home and refers to his foster
      parents as “Mom” and “Dad.” [Foster father] coached [C]hild in
      football,


  Dr. Meacham testified that Child “is doing well in the foster home” and

Child “views the foster family as his family.” N.T. Hearing, 12/11/14 at 13.

Ms. Minnier testified that the foster parents provide excellent care for Child,

and that Child is very comfortable and very happy with them.        Id. at 38.

Ms. Benfer testified that Child has bonded with his foster parents, and that

Child refers to them as “mom and dad.” Id. at 80.

      In the instant case, on the issue of bonding, our review of the record

reveals no evidence of a bond between Mother and Child.         Dr. Meacham

testified that, when he talked to Child about Mother, “Child immediately

becameUnfortunately, when using the following PowerShell command, it erases all other entries in your PATH Environment Variable. Most forums I’ve seen are telling people to do this, good thing I backed up my paths before trying this method, as all my paths got erased.

[Environment]::SetEnvironmentVariable("PATH","C:\Testing","Machine")

Let’s see if we can fix this, first let us exam our current paths we have configured (it’s good to take a backup of this configuration before you make any changes, you can do that with the following PowerShell command:

$env:path | Out-File C:\EnvironmentVariableBackup.txt

Script:

Let’s create a whole script now so we can update the PATH Environment Variable and be able to use it in the future. I saved this to a file named UpdatePathEnvironmentVariable.ps1.

# Separate multiple paths with semi-colon ;
$NewPathsToAdd = "C:\Testing;C\Testing2"
# Save this to a


article of clothing which he purchased had a "full" price tag attached to it. In Patterson's experience, the salesperson always detached the first part of the three-part price tag and then removed the clothing from the hanger prior to wrapping the clothing and completing the sale.
Patterson further asserted that after observing the clothing in defendant's shopping bag, he identified himself to defendant as a police officer and asked defendant "where he had gotten the clothing." Defendant shrugged his shoulders. Patterson then took the shopping bag from defendant, opened it up, and saw that each piece of clothing was on a hanger and had a "full" price tag attached. Patterson identified this clothing at trial as the clothing he had recovered from defendant's possession and which had been ina bag.
Malchin also explained that the code numbers on the price tags indicated that the item would be sold in a particular Carson store and department. Malchin also stated that the price tags contain a season number which indicates what month the item will be sold. Malchin examined the code numbers on the price tags of the clothing which Patterson had recovered. Malchin stated the tags disclosed that the clothing came from the Carson's store located on State Street and was available for sale in early September. Malchin also identified the hangers as property of a Carson's store and stated that he had observed clothing similar to that recovered from defendant on sale in Carson's State Street store on September 27, 1979.
On cross-examination, Malchin again asserted that the


activity shown here should not prevent the introduction of statements [or actions] freely made." (392 F.2d 373, 376.) Since defendant's action was given in response to Patterson's on-the-scene preliminary questioning, Patterson's testimony describing defendant's shoulder shrugging action and the prosecutor's comment upon it did not violate defendant's constitutional rights. People v. Hayes (1971), 133 Ill. App.2d 885, 272 N.E.2d 423. See also People v. Parks (1971), 48 Ill.2d 232, 269 N.E.2d 484; People v. Tolefree (1972), 9 Ill. App.3d 475, 292 N.E.2d 452.

III
Defendant next contends that certain statements made by the prosecutor during closing argument necessitate reversal. The prosecutor remarked that the evidence was uncontradicted and then stated:
"Ladies and gentlemen, if that property was indeed paid for what reasonable explanation, reasonable, I didn't say speculative * * *a bunker, the short sixth is a good birdie opportunity with the biggest difficulty coming in judging the distance of the approach to a green below the level of the fairway. Expect players to use an iron from the tee, followed by a wedge.

7th: Le Dromadaire

A dogleg right which features a tee shot played to a blind, elevated section of a two-part fairway, followed by a tricky approach to the long green which starts narrow but widens dramatically. Out of bounds awaits a wild drive to the right.

8th: Le Green Keeper

A solitary, albeit deep, bunker guards the front of the large green on this par three, where judging the wind is a common difficulty, as is keeping the ball on a seriously sloping green from a tee shot


a par five. Hitting the fairway from the tee is vital.

18th: La Foule

An exciting closing hole, with water all the way down the left – and in front of the green – plus pot bunkers on the right of the fairway. Used to be a par five before regrading. Strategy from the tee is imperative, if only to ease the sense of fear around the approach shot.you have to sample at 2x the highest frequency that is present in your signal. Note - it's not just "that you are interested in", but "that is present". If you sample a high frequency signal with a low frequency sample clock, it will appear "aliased" - it wil show up in your output as something completely different. So before you digitize a signal you have to decide what the frequencies of interest are, and remove all higher frequencies with a filter.  Let's say you are interested in frequencies up to 500 Hz (about 1 octave above middle C on a piano). To give your filter a chance to work, you might choose to cut off all frequencies above 1 kHz (filters "roll off" - i.e. they


ask for specifics of the order’s development. Preview Your Paper When our qualified finishes focusing on your assignment, you're going to get a paper for preview. We have been 100% absolutely sure that our writer will do a terrific task, but sometimes misunderstandings may well take place. In case you discover that the writer didn't give exactly what you expected, ask for a revision, and we could make the corrections. The revisions are Completely totally free! Ask a Support Staff for Assist If it is The very first time you are likely to use our article writing service, you most likely have a lot of questions. To make certain that you will find a full solution to every query, We have now a aid staff which is usuallyother Furl members).

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Your MLA literature review have to be concluded to a very precise list of procedures for formatting whether it is for being acknowledged.

a large perspective from the review is required,, issues look to get viewed over the reduce facet in their examine.

☆ Get into a firm routine for doing all your writing and study. Getting a set time each day along with a target for studying and writing that you'll attain will assist you to make regular development.

If you will discover currently a couple of


iDGLAX iDG-787W LCD LED Video Multimedia Mini Portable Projector has 1500 lumens that make it 50% brighter than ordinary LED projector. It has 800 X 480 native resolution and it is compatible with 1080i/p, has 16:9/4:3 aspect ratio and 1000:1 contrast ratio. Please make sure that this iDGLAX LED projector is not recommended for PPT or business presentation.

You will surely receive good experience of home theater movies nights with the help of this high quality and long lasting video projector from iDGLAX brand. And this is the best budget projector for home theater according to many reviews on popular gadget blogs.

It has the capability to deliver bigger screen size from 30 inches to 100 inches, the best projection distance is approx. 6 to 9 feet with best screenyou in a good mood." "Something tells me it's not going to last." "How'd the press get here so fast?" "Oh, a bunch of 'em were in the audience," "Covering the retirement of this ballerina," "Odessa koza-something." "Kozlova." "Ukrainian, 35, one of 12 living ballerinas" "To be given the title of prima." "It's the equivalent of being a queen." "Yeah, well, her majesty's farewell performance" "Came to a screeching halt when this thing showed up." "Any idea who it belongs to?" "A dancer, obviously." "And judging by the dome of her foot," "A rather gifted one." "5'4", female," "And judging by the skin tone," "I'd say that she was of latin origin -- mexico, perhaps." "Or cuba?" "Meet eva selgas " "Only dancer who failed to show for


made of synthetic materials," "But the carpet in classic cars is made" "Of 100% natural fiber," "Like the wool we found on eva's shoe." "If we find that car " "We may find eva's body." "Hanson:" "Parking authority found it" "Abandoned this morning on a curbside in the bronx." "Interior appears to be clean," "But check this out." "Jo:" "Look at this pool of blood." "Henry:" "The stain is too small." "When eva's killer cut her foot off," "It would have been gushing with blood, unless..." "Unless what?" "Unless she stopped the bleeding." "Eva's killer put her in the trunk" "Believing that she was either" "Unconscious or perhaps even dead." "But she was awake." "[ screaming ] [ sobbing ]" "The rubber tubing has been torn off thispretty girl like you" "Didn't want to get her hands dirty," "So you went to your pal sergei." "Jo:" "Your record is pretty impressive." "Robbery, extortion, aggravated assault." "So, what the hell is a thug like you doing" "Hanging out with a bunch of ballerinas?" "My colleagues and i are investors" "In odessa's career." "We pay for her dance coaches," "Her living expenses " "And in return, you take" "A portion of her earnings." "But all that money was about to go away" "When eva took her job." "You chopped off eva's foot," "And then you left it at that ballet as a warning." "I never touched eva!" "You got close enough for her to pepper-spray you." "You must have been following her for a while," "Waiting for


that perfect moment to kidnap her." "Odessa:" "He was never trying to kidnap her." "He was only watching her." "It's an old trick in russia." "You hire a man to follow from distance," "Leave a few threats." "Suddenly the dancer gets distracted." "I was just trying to make eva slip up." "But i told sergei never to touch her," "Just to follow." "Jo:" "Were you following her the day she went missing?" "Yes." "She left rehearsal around 3:00," "Went to the courtyard, hung around some statue," "And then she left." "Jo:" "And you didn't follow her?" "She already pepper-sprayed me once." "I thought it best to let her go." "Besides, it was raining outside." "Hanson:" "Let me get this straight, henry." "You dragged us all the way backgo." "Besides, detective, don't you have a plane to catch?" "Bring me back a croissant or something." "I just -- i can't believe" "Eva put herself through that much pain." "And for what?" "I once knew someone similar to eva." "Never underestimate what some people will do to " "To live forever?" "[ elevator bell dings ]" "Enjoy paris, jo." "[ sighs ]" "Excited?" "I'm excited." "Paris with you -- very exciting." "I can't believe i'm actually going." "Oh, you're going, all right." "You're going to experience" "The most romantic city in the world" "In best possible way there is to experience it." "Really?" "Courtesy of my 100% foolproof, paris-is-magical itinerary." "I've got every minute planned, every meal, every sip of wine." "[ chuckles ]" "Every minute, huh?"


lacked

authority to sign the entry. In 2010, Lowe filed a motion for a final appealable order. After the

trial court denied his motion, Lowe appealed to this Court, presenting two assignments of error.
                                                  2

                                                II.

                                  ASSIGNMENT OF ERROR I

       “TRIAL JUDGE MARY CACIOPPO LACKED THE LEGAL AUTHORITY TO
       ENDORSE THE FINAL APPEALABLE ORDER AS REQUIRED BY
       ARTICLE IV SECTION (6) OF THE OHIO CONSTITUTION AND R.C.
       2505.02(3) THE SIGNATURE OF THE JUDGE.”


 ASSIGNMENT OF ERROR II

       “TRIAL JUDGE, LYNNE S. CALLAHAN[,] COMMITTED ABUSE OF
       DISCRETION BY NOT PROVIDING A REASON TO SUPPORT HER
       DECISION TO DENY DEFENDANT-APPELLANT’S MOTION FOR FINAL
       APPEALBLE ORDER WHILE KNOWING THAT R.C. 2505.02(3) THE
       SIGNATURE OF THE JUDGE & ARTICLE IV SECTION (6) OF THE OHIO
       CONSTITUTION IS CONTROLLING LAW.”

       {¶3}    In his motion for a final appealable order, Lowe argued that Judge Mary

Cacioppo, a visiting judge assigned by the Chief Justice of the Ohio Supreme Court, lacked

authority to act.has gotten a lot of publicity, I don’t think you want us to treat hers any different than a whole lot of cases that I think people want us to review.”

Apparently unmoved by the governor’s equivocation, the audience began chanting: “What do we want? Clemency! When do we want it? Now!” and “No justice, no peace!”

On Aug. 6, 2004, when she was 16, Brown was picked up by 43-year-old Johnny Mitchell Allen, who planned on paying Brown for sex. Paranoid from drugs and abuse, Brown reportedly became convinced that Allen was going to kill her. She fired one bullet into Allen’s chest, drove his car to a Wal-Mart parking lot and walked back to her hotel to meet “Cut-Throat,” a pimp who been providing her with drugs while


/*=============================================================================
    Copyright (c) 2001-2011 Joel de Guzman
    Copyright (c) 2007 Dan Marsden

    Distributed under the Boost Software License, Version 1.0. (See accompanying
    file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
==============================================================================*/
#if !defined(BOOST_FUSION_NONE_07062005_1128)
#define BOOST_FUSION_NONE_07062005_1128

#include <boost/fusion/support/config.hpp>
#include <boost/fusion/algorithm/query/any.hpp>

namespace boost { namespace fusion
{
    namespace result_of
    {
        template <typename Sequence, typename F>
        struct none
        {
            typedef bool type;
        };
    }

    template <typename Sequence, typename F>
    BOOST_CONSTEXPR BOOST_FUSION_GPU_ENABLED
  Retained lens fragment in the anterior segment as a cause of recurrent anterioruveitis.
A 61-year-old male developed recurrent anterior uveitis over a period of 8 months after an uncomplicated phacoemulsification and posterior chamber intraocular lens implantation surgery. This was subsequently found to be due to a retained lens fragment in the anterior segment, with complete resolution following surgical extraction of the fragment. To our knowledge, this is the first report of recurrent anterior uveitis attributable to a retained lens fragment following uncomplicated cataract surgery, and the diagnosis should be considered in a pseudophakic patient presenting with recurrent episodes of anterior uveitis.


CHAPTER 37: BETWEEN THE HERMITAGE AND THE HOSPITAL OF ST. MARY, ROUNCEVAL (SITES OF NOS. 1 TO 15, WHITEHALL)

The frontage between the hermitage and the Hospital of St. Mary,
Rounceval, seems to have been first built upon about the year 1400. A
document of Westminster Abbey (fn. n1) giving the rents received by the Chapel
of St. Mary in 1392–4 mentions the receipt of 12d. in respect of a certain
vacant place lying between "Rouncevale" and the hermitage. In the
account for 1405–6 (fn. n2) is an item concerning ten shops at "Charryng"
belonging to the sacristan, which appear in the account for 1411–2 (fn. n3) as ten
shops near "Rouncyvall" demised to the sacristan, and in a statement of
receipts for the years following 1413 (fn. n4) the ten shops are specified as "inter
Rouncyvall etto John Wylton and John Ibleton. Subsequently
Robert Gardiner, "sergant of her mats seller," obtained a reversionary grant
for 21 years. (fn. n10) No record of a grant in perpetuity has been found, but his
grandson Richard was distinctly under the impression that Robert was
"lawefullye seised in his Demeane as of fee" in the messuage, when on
26th May, 1565, he granted a 25 years' lease of it to Thomas Putterell,
then in occupation. The property was described as "ioyninge on a Tenement then in the Tenure of one Peter Nicholson on the south parte, and the
Tenement … of William Horsley on the north parte, and adioyninge upon
a Tenement … of John Rede on the east parte, and the street Leadinge
betwene Charinge Crosse and Westm' palac on the west parte." The
property descended to


but from the depositions in the suit it
appears that the author was Robert Tresswell. It shows the whole of the
premises between "Scotland" and Rounceval. The two centre houses
marked "Sir John Parker "and "Mr Gardyner," and shown in ground plan,
are described with some exactness, but the plan was probably not intended to
show the rest of the houses with meticulous accuracy, and while giving a true
general impression cannot be relied upon in details. None of the measurements
(except of the two centre houses) agrees with the facts save that of 143 feet
from the street to the north-east corner of "The Armitage." The latter is
represented with an eastern boundary roughly parallel to the line of the
street, but this is very wide of the mark, for its south-east corner was really
about 220 feetPacke, (fn. n84) 46 feet east to west and 37 feet north to south,
(ii) The Three Feathers, adjoining (i) and containing 66 feet east to west,
10 feet in front and 17 feet in the rear, in the tenure of Joseph Hopkins,
(iii) A messuage next to (ii), 50 feet east to west and 11½ feet in front in
the occupation of … Benson, (iv) A messuage next to (iii) 58 feet east to
west, 11½ feet in front and 7 feet in the rear, formerly in the occupation of
Miles Knight and then of … Hawgood, (v) A messuage on the south side
of (i) 37 feet from east to west and 10 feet north to south, then occupied by
… Davison, and (vi) The Golden Sword next to (v), 25 feet east to


to Browne and Cliffe, and that the fee-farm rents payable by the latter in respect of both
Cutler's and Urshaw's tenements were sold on 16th June, 1658, to Francis Offley and Richard
Redding (P.R.O., E. 307/15/02). This is, however, partly explained by the King's statement
on 18th February, 1675–6, that "many mistakes have been committed and many doubts have
arisen in and about the Sale of Our Fee farm Rents … which have been occasioned partly through
the inadvertency of Our Officers before whom the said Rents stood in charge or of the Clerks
employed in the Sale of the said Rents or by some other waies or meanes." (P.R.O., S.O. 8/11.)

n56. The identification of tenants of this house before 1687 presents difficulty. From that
date, however, for a number of years they can beside of ye aforesaid … Messuage … & conteyning
in lenght from ye said street there on ye West to ye said house now in ye occupacion of … Mr Darling on ye East xxv feet … & in breadth from North to south 10 feet … & consisting of … one
shopp & a small kitchin, a Cellar under neath ye same, a little yard behinde ye said shopp & kitchen,
And above stayers in ye first story one faire Chamber & over ye same one faire garrett."

n87. See indenture dated 28th September, 1752, between Martha Edwin and John Revis
and Samuel Spencer, linen drapers. The ratebook for 1750 shows the two houses as empty,
and that for 1752 gives "Messrs Revis and Spencer" at a much larger house.

n89. Indenture between (1)


November, 1555. (Registers of St. Martin-in-the-Fields, p. 105.)

n103. "It' R' the xviij Daye [of November] for the herse Clothe, the knyll, and the waste of
two tapers, At the buryall of Ambrose gurnard." (Accounts of the Churchwardens of St. Martin-inthe-Fields, p. 170.)

n117. William Wright first appears in the ratebooks in 1607 and continues until 1653.
(i) "Recognizance … of William Wright … tombe maker, in the sum of forty pounds, for the
said William Wright's appearance at the next Session of the Peace … to answer … 'for annoying
the streete nere Charing Crosse with loading of cartes, turning the Judges and all other passengers
into the Channell.' "—4th November, 1613 (Middx. Session Records, II, p. 93). (ii) "Account
by Wm. Weld of disbursements for Sir Wm. Pelham; it includes 10l paid to Mr- headers already sent by (output started at /home3/adamshap/public_html/usdbls/wp-includes/post.php:1468) in /home3/adamshap/public_html/usdbls/wp-includes/option.php on line 787"
"Cannot modify header information - headers already sent by (output started at /home3/adamshap/public_html/usdbls/wp-includes/post.php:1468) in /home3/adamshap/public_html/usdbls/wp-includes/option.php on line 788"

Additional info:
I only receive the errors when uploading my custom post_type plugins (not the theme I developed, just the plugins).
The errors appear on all of my plugins, not any single one.
I know the error lies with the plugins I made, so I need to find out how to fix my plugins, not the wordpress core files.
The first error points to line 1468 of wp-includes/post.php.
The corresponding section of code is as follows:
if ( $args->register_meta_box_cb )
    add_action( 'add_meta_boxes_' . $post_type, $args->register_meta_box_cb, 10, 1 );

$args->labels = get_post_type_labels( $args );
$args->label = $args->labels->name;

$wp_post_types[ $post_type ] = $args;

add_action( 'future_' . $post_type,


         => false,
        'hierarchical'        => false,
        'public'              => true,
        'show_ui'             => true,
        'show_in_menu'        => true,
        'menu_position'       => 5,
        'show_in_admin_bar'   => true,
        'show_in_nav_menus'  Epstein-Barr virus nuclear antigen 2-estrogen receptor fusion proteins transactivate viral and cellular genes and interact with RBP-J kappa in a conditional fashion.
Epstein-Barr virus nuclear antigen 2 (EBNA2) is a transcriptional activator of viral and cellular genes involved in B cell transformation by EBV and is targeted to EBV responsive promoters through interaction with cellular DNA binding proteins such as RBP-J kappa. To develop a conditional system in which the function of EBNA2 can be switched on and off, we have fused the hormone binding domain of the estrogen receptor to the N- or C-terminus of EBNA2. Here we show that after transient or stable transfer of these chimerical EBNA2 genes into human B cell lymphoma lines, transactivation of LMP1, TP1, and TP2 promoter constructs, expression of the cell


font-size: 1.3em;

   }

  .button section {
    display: none;
    position: absolute;
    top: 2em;
    left: 32em;
    padding: 20px;
    margin: 0;
    border: 5px solid #fd0;
    background: white;
    border-radius: 1em;
  }

 .button section > p {
    display:block;
    text-align:center;
  }
 }

 .button .parentPic {
   display: block;
   border-radius: 0.8em;
   max-width: 100%;
   height: auto;
  }

 .button:focus section,
 .button:hover section {
    display: inline-flex;
    top:0px;
  }

 .speech {
     position: relative;
     width: 300px;
 were trailed by a procession of well-known royals, including Queen Beatrix of the Netherlands, Princess Victoria and Prince Daniel of Sweden, Prince Naruhito of Japan, and Britain's Prince Edward - Queen Elizabeth's youngest child - and his wife, Sophie.

Stephanie plans to renounce her Belgian citizenship in order to - one day - become Luxembourg's grand duchess. The tiny country wedged between France, Belgium and Germany is an important financial center and continues to prosper despite Europe's economic trouble.

Stephanie wore a lace Elie Saab dress with a 5-meter-long wedding train during the ceremony, which was conducted in a mixture of French, German and Luxembourgish. It began with a minute's silence to honor her late mother, Countess Alix de Lannoy.

The nuptials gave tiny Luxembourg - a founding member of the


    noInfo: true,
        // enable HMR
        hot: true,
        // embed the webpack-dev-server runtime into the bundle
        inline: true,
        // serve index.html in place of 404 responses to allow HTML5 history
        historyApiFallback: true,
        port: PORT,
        host: HOST
    },
    plugins: [
        new webpack.NoEmitOnErrorsPlugin(),
        new webpack.HotModuleReplacementPlugin(),
             }
        })
    ]
};

in .bablelrc
{

  "presets": ["es2015", "react", "stage-0"],
  "plugins": ["react-html-attrs","transform-runtime", "transform-decorators-legacy", "transform-class-properties", "react-hot-loader/babel"]
}

App entry point
import 'react-hot-loader/patch';
import React from 'react';
import ReactDOM from 'react-dom';
import { AppContainer } from 'react-hot-loader'

import Application  from './components/app';

render((
  <AppContainer>
    <Application />
  </AppContainer>
), document.getElementById('app'));

I am new to react and when i was trying few examples i was getting this issue to thought for a help or suggestion.

In webpack loaders
module.exports = [
    {
      test: /\.es6\.js$/,
      exclude: /node_modules/,
      loader: 'babel-loader'
  },
    {
        test: /\.jsx?$/,


we have $-C\leq -\Delta u\leq C$ in the viscosity sense for some constant $C>0$, while from [@I] we have $-C\leq -\Delta u\leq C$ in the viscosity sense if and only if $-C\leq -\Delta u\leq C$ in the distributional sense. Hence, $-\Delta u \in L^\infty({\mathbb T}^d)$. Elliptic regularity theory ensures that $u\in W^{2,p}$ for any $p>1$ and, hence, $u\in {\mbox{\rm C}}^{1,\sigma}$ for any $0<\sigma<1$. Moreover, since $$-\Delta u +H(x,Du)=\alpha\quad\hbox{ in } {\mathbb T}^d,$$ by the Schauder theory, we have $u\in {\mbox{\rm C}}^{2,\sigma}$ for any $0<\sigma\leq \gamma$.

\(ii) Let $u, v$ be classical solutions of . Pick $R\geqslant \|Du\|_\infty,\,  \|Dv\|_\infty$ and set $C:=\max_{{\mathbb T}^d\times B_{R+2}}|H(x,p)|$. By convexity, we have that $H(x,\cdot)$ is $C$–Lipschitz in $B_R$, for every $x\in{\mathbb T}^d$. Hence, by subtracting for $u$ and $v$, respectively, we get $$0=-\Delta(u-v)+H(x,Du(x))-H(x,Dv(x))
\geq
-\Delta(u-v)-C|Du(x)-Dv(x)|\quad\hbox{inif $\alpha\leq\beta$. Furthermore:

-   $u+k\in{\mathcal H}_\tau(\alpha)$ for every $u\in{\mathcal H}_\tau(\alpha)$ and $k\in{\mathbb R}$;

-   $\Leg_\tau\left({\mathcal H}_\tau(\alpha)\right)\subseteq {\mathcal H}_\tau(\alpha)$.

Next, we show that all $\alpha_\tau$–subsolutions for $\Leg_\tau$ are actually solutions to .

\[prop sub=sol\] Let $(\alpha_\tau,u_\tau)\in{\mathbb R}\times{\mbox{\rm C}}({\mathbb T}^d)$ be a solution of $$\Leg_\tau u_\tau=u_\tau-\tau \alpha_\tau\qquad\hbox{in ${\mathbb T}^d$}.$$ Then ${\mathcal H}_\tau(\alpha_\tau)=\{u_\tau+k\,:\,k\in{\mathbb R}\,\}$. Furthermore, $$\label{def alpha tau}
\alpha_\tau=\min\{{\alpha}:{\mathcal H}_\tau({\alpha})\ne\emptyset\}.$$

Let us pick $u\in{\mathcal H}_\tau(\alpha)$ and argue as in the proof of Theorem \[teo unique alpha tau\] with $u_1:=u$ and $u_2:=u_\tau$. By also using the fact that $u\leq \Leg_\tau u +\tau\alpha$, we end up with $$u(x)-u_\tau(x)\le \int_{{\mathbb T}^d} (u(z)-u_\tau(z))d\eta_y(z)+\tau(\alpha-\alpha_\tau)\qquad\hbox{for all $x\in{\mathbb T}^d$.}$$ By picking as $x$ a maximum point of $u-u_\tau$, we conclude that $\alpha\geq\alpha_\tau$. When $\alpha=\alpha_\tau$, we furthermore get that $u-u_\tau$ is constant.

We conclude this section by deriving the following bounds on


the constant $\alpha_\tau$.

\[prop alpha\_tau\] The following holds: $$\label{eq alpha tau}
-\max_{y\in{\mathbb T}^d}\min_{q\in{\mathbb R}^d} L(y,q) \leq \alpha_\tau \leq -\min_{{\mathbb T}^d\times{\mathbb R}^d} L.$$

Let us set $L_m(x):=\min_{q\in{\mathbb R}^{d}} L(x,q)$. Pick $u\in {\mathcal H}_\tau({\alpha}_\tau)$ and set $v(x):=\max_{{\mathbb T}^d} u$, $w(x):=\min_{{\mathbb T}^d} u$ for all $x\in{\mathbb R}^d$. Then, for all $x\in{\mathbb T}^d$, $$u(x)-\tau\alpha_\tau
=
{\mathcal L}_\tau u(x)
\leq
{\mathcal L}_\tau v(x)
=  \tau L_m(x) +\max_{{\mathbb T}^d} u
\leq \tau \max_{{\mathbb T}^d} L_m+\max_{{\mathbb T}^d} u$$ and $$u(x)-\tau\alpha_\tau
=
{\mathcal L}_\tau u(x)
\geq
{\mathcal L}_\tau w(x)
=
\tau L_m(x) +\min_{{\mathbb T}^d} u
\geq
\tau \min_{{\mathbb T}^d} L_m+\min_{{\mathbb T}^d} u.$$ That implies $-\max_{{\mathbb T}^d} L_m \le \alpha_\tau \leq -\min_{{\mathbb T}^d} L_m$, as it was asserted.

Equi-continuity of the functions $u_\tau$ {#sez equicontinuity}
=========================================

This section is devoted to prove equi-continuity of the functions $\{u_\tau\,:\, \tau\in (0,1)\,\}$, where $u_\tau$ denotes a solution in $C({\mathbb T}^d)$ of the equation $$\label{eq upair $(\alpha,u)\in {\mathbb R}\times{{\mbox{\rm C}}}({\mathbb T}^d)$ with $u(x_0)=0$ such that $u$ is a viscosity solution to $$\label{eq3 viscous HJ}
-\Delta u +H(x,Du)=\alpha\qquad\hbox{in ${\mathbb T}^d$}.$$ In view of Proposition \[prop alpha\_tau\] and Proposition \[equi-con\], of the fact that $u_\tau(x_0)=0$ for all $\tau\in (0,1)$ and of Arzelà-Ascoli Theorem, we have that the set $\{(\alpha_\tau,u_\tau)\mid\tau\in (0,1)\}$ is precompact in ${\mathbb R}\times{{\mbox{\rm C}}}({\mathbb T}^d)$. In order to prove the assertion, it is therefore enough to show that, if the pair $(\alpha, u)$ is the limit of $(\alpha_{\tau_n},u_{\tau_n})$ in ${\mathbb R}\times{{\mbox{\rm C}}}({\mathbb T}^d)$ for some $\tau_n\to 0$, then $u$ is a solution to .

Let us first show that such an $u$ is a viscosity subsolution to . Let $\varphi\in{{\mbox{\rm C}}}^3({\mathbb T}^d)$ be such that $u-\varphi$ has a strict maximum at $x_0$. Then there exists


.

Let us now show that $u$ is a viscosity supersolution to . Let $\varphi\in{{\mbox{\rm C}}}^3({\mathbb T}^d)$ be such that $u-\varphi$ has a strict minimum at $x_0$. Then there exists a sequence of points $(x_n)_n$ converging to $x_0$ in ${\mathbb T}^d$ such that $u_{\tau_n}-\varphi$ has a minimum at $x_n$. Let us set ${\varepsilon}_n:=\min(u_{\tau_n}-\varphi)$ and $\varphi_n:=\varphi+{\varepsilon}_n$. Then $$u_{\tau_n}\geq \varphi_n\quad\hbox{in ${\mathbb T}^d$}\qquad\hbox{and}\qquad u_{\tau_n}(x_n)=\varphi_n(x_n).$$ By arguing analogously, we end up with $$\frac{\varphi(x_n)-\Leg_{\tau_n}\varphi(x_n)}{\tau_n} \geq  \alpha_{\tau_n}$$ For each $n\in{\mathbb N}$, pick a minimizing $q_n\in{\mathbb R}^d$ for $\Leg_{\tau_n}\varphi(x_n)$, so that the previous inequality rereads as $$\label{eq2 asymptotic}
\frac{\varphi(x_n)-\left(\eta^{\tau_n}*\varphi\right)(x_n+\tau_n q_n)}{\tau_n}-L(x_n,-q_n)\geq \alpha_{\tau_n}.$$ By making use of Proposition \[prop argmin L tau\] and of the fact that $|D(\eta^\tau*\varphi)(x_n)|\leq \|D\varphi\|_\infty$, we infer that there exists $R>0$ such that $q_n\in B_R$ for every $n\in{\mathbb N}$. Up to extracting a further// Signature format: 3.0
package androidx.navigation.fragment {

  public final class FragmentKt {
    ctor public FragmentKt();
    method public static androidx.navigation.NavController findNavController(androidx.fragment.app.Fragment);
  }

  public final class FragmentNavArgsLazyKt {
    ctor public FragmentNavArgsLazyKt();
    method @MainThread public static inline <reified Args extends androidx.navigation.NavArgs> androidx.navigation.NavArgsLazy<Args> navArgs(androidx.fragment.app.Fragment);
  }

  public final class FragmentNavigatorDestinationBuilder extends androidx.navigation.NavDestinationBuilder<androidx.navigation.fragment.FragmentNavigator.Destination> {
    ctor public FragmentNavigatorDestinationBuilder(androidx.navigation.fragment.FragmentNavigator navigator, @IdRes int id, kotlin.reflect.KClass<? extends androidx.fragment.app.Fragment> fragmentClass);
    method public androidx.navigation.fragment.FragmentNavigator.Destination build();
  }

  public final class FragmentNavigatorDestinationBuilderKt {
    ctor public FragmentNavigatorDestinationBuilderKt();
    method public static inline <reified F extends androidx.fragment.app.Fragment> void fragment(androidx.navigation.NavGraphBuilder, @IdRes int id);
    method public static inline <reified F extends androidx.fragment.app.Fragment> void fragment(androidx.navigation.NavGraphBuilder,


Field
The disclosed concept pertains generally to circuit breakers used in power transmission and distribution systems, and, more particularly, to a portable diagnostic apparatus including an onboard diagnostic feature for performing diagnostic testing on circuit breakers, such as, without limitation, low or high voltage power circuit breakers.
Background Information
Electrical power transmission and distribution systems typically utilize a plurality of power circuit breakers which include one or more sets of separable contacts for protecting circuits against overcurrent conditions. In the field, a number of such circuit breakers are generally mounted and housed within a non-conductive housing or enclosure, often referred to as a switchgear enclosure. Power connections for the circuit breakers are generally terminated in the rear of the enclosure, and instrumentation and control terminal blocks for the circuit breakers mayonly speed up your progress if you apply them day after day. But first, I want you to envision what life will be like once you reach your goals.

Imagine how you will feel once you’ve achieved each thing on your list, one by one. How will your life be changed? How will the world be changed by it? The greatest thing is it’s going to happen to you. You can achieve absolutely anything as long as you don’t give up.

Are you ready to kick your progress into super gear? Let’s get to it!

Analyze Your Schedule

What does your schedule currently look like? Are there any areas you can cut back on? Maybe you’ve never taken the time to write or even look at your schedule. If you’re serious about


build momentum and are able to achieve amazing results. Tell yourself what goal you’re going to get done and don’t stop until you complete it!

Discipline Yourself

Discipline is a great way to push yourself to success. Let’s face it, sometimes we simply don’t feel like doing the hard work. There are days when the athlete doesn’t want to go to the gym, but the reason they are successful is that they go anyway. That’s the key.

You don’t always have to feel like doing something to get it done. You just have to do the actions to get it done. The good news is: you can train your body to enjoy it. Change your thought patterns and correlate your hard work with the result you’re chasing. When pursuing your dreams,be a little rough..." "You're not an assassin, hiding behind the shrubs.." "You have a right to be angry." "But to test the skills and courage, this is the best way." "Test?" "Actually, this house is... looking for a strong swordsman." "We are looking for swordsmen who can win the match in the "Shogun's presence"."" "Match in the "Shogun's presence"?" "It's a match to be presented to Shogun Ashikaga Yoshitaka-sama." "You know that Kanrei (Shogun deputy) Hosokawa Masamoto-sama has been killed last year." "Masamoto-sama did not have an heir." "So he adopted 3 sons." "That was a huge mistake." "Fighting broke out over the inheritance, and Masamoto-sama was assasinated." "The capital became a battle ground." "In the middle of the fight, one of the adopted sons, Hosokawa Takakuni-sama


Learning Areas of Religious Education, English, Mathematics, Science, Australian History and Geography as well as Personal Development, Health and Physical Education. Students have the opportunity to select two elective courses in Year 9 and can choose again in Year 10. Additional subject choices include: Commerce, Information Software Technology, Building, Graphics Technology, Industrial Tech (Timber), Language, Music and Physical Activity and Sports Studies. In the compulsory Mathematics course there are three levels, which are graded according to students’ abilities.

For the senior years of Year 11 and 12 there is a wide choice of subjects offered, which includes Catholic Studies, Studies of Religion (1 or 2 unit), English (Standard, Advanced and Extension), Mathematics (all four levels are offered), Biology, Chemistry, Physics, Ancient History, Modern History, Geography, Economics, Engineering, Business Studies,from Jermaine Gresham to make up for Sunday’s incident, but it’s definitely not the same.

@PrincipalCara Truly sorry about you guys not getting that ball. I'll make sure you guys receive something greater for compensation. — Jermaine Gresham (@JGresham84) November 16, 2014

Maybe he should have followed the baseball fan rules and thrown back any ball not from your own team.

MORE FROM FOX SPORTS SOUTHWEST:

– Ranking NFL quarterback salaries

– Wake up! Athletes caught sleeping

– Best college football traditions

– Greatest NBA players without a championship ring


a second device would render the connection of the first device unusable
if the Facebook User Access Token is invalidated upon the uninstallation of the iOS app, then, in the case when the app is installed on two devices, uninstalling it from a device would render the connection to the second device unusable

So, logically I assume that the Facebook User Access Token should be portable, that is, after the user uninstalls the iOS app, reinstalls it and tries to connect with Facebook, a function should be usable to gather the Facebook User Access Token, as it should logically be linked to a user, not to a device. Am I right or wrong? If right, how can a Facebook User Access Token be regathered when the user (re)installs the app?the fly height of the slider, which is determined by the ABS of the slider. As areal densities of magnetic recording disks increase, it becomes more important to precisely control the spacing of the read/write heads in relation to the magnetic recording disk, as the spacing may be 10 nanometers or less.
To further control the spacing between the read/write heads and the magnetic recording disk, some sliders include heating elements which are fabricated proximate to the read/write heads. The read/write heads are fabricated from materials that have a different thermal rate of expansion than the body of the slider. When a current is applied to the heating elements, the read/write heads protrude from the ABS of the slider towards the surface of the magnetic recording disk. This protrusion


reduces the spacing between the read/write heads and the magnetic recording disk. The use of heating elements, which is also referred to as thermal fly height control, allows for a more precise control of the spacing between the read/write heads and the magnetic recording disk.
After manufacturing, but before being shipped to an end-user, a magnetic disk drive system is calibrated to generate a desired spacing between the read/write heads and the magnetic recording disk. To calibrate a magnetic disk drive system, a control system applies a motor current to a spindle motor which in turn rotates a spindle connected to the magnetic recording disk. As the magnetic recording disk rotates, the slider flies over the surface of the magnetic recording disk on the air bearing. A thermal flyrare case that rand() returns 0, the "<" test fails, so
  // we get 0.

  // NOTE: cast is necessary to do unsigned arithmetic rather than signed.
  // RAND_MAX is an int so adding 1 won't overflow a uint64_t.
  uint64_t max = (uint64_t)RAND_MAX + 1u;
  uint64_t threshold = p1 * max;
  int i;
  for (i = 0; i < num_bits; ++i) {
    // NOTE: The comparison is <= so that p1 = 1.0 implies that the bit is
    // ALWAYS set.  RAND_MAX is the maximum value returned by rand().
    uint64_t bit = (rand() < threshold);
    result |= (bit << i);
  }
  return result;
}

static PyObject *
func_randbits(PyObject *self,


of picking 1 m and 1 j?
40/91
Two letters picked without replacement from ciccccddabcdx. What is prob of picking 1 c and 1 i?
1/13
Three letters picked without replacement from gcmcwgcwgf. Give prob of picking 1 g, 1 w, and 1 c.
3/20
What is prob of picking 2 z when two letters picked without replacement from {m: 1, z: 3}?
1/2
Three letters picked without replacement from rcgbgli. What is prob of picking 1 b, 1 g, and 1 l?
2/35
Calculate prob of picking 4 m when four letters picked without replacement from ommooomomooomooooo.
1/612
What is prob of picking 2 p when two letters picked without replacement from {p: 2, n: 3, j: 3}?
1/28
Three letters picked without replacement from {q: 7, g: 5, t: 1, m: 6}. What is prob of picking 2 g andthese items don’t locate hired as a result of specific background check and said “I idea that law suit was expunged”. If individuals did certainly go with the steps of filing for the expungement as well as the have information technology approved, in that case the save is all the same available to be see.

Employers become NOT limited to 7 years within a criminal record find. Some states have these own Nice Credit Revealing Act tips that limit what crimes are reportable past 8 years, often based within the position and how much it pays. Through the process of federal L.C.R.A. standards, however, a assurance can wind up reported forever. The “7 year rule” usually delivers to busts that probably did not end product in a complete conviction.


down the expenses incurred on a bargaining unit basis. While plaintiffs are public employees, most of the employees covered by Local 38 work in the private sector in other bargaining units. Plaintiffs contend that the Court should require the Union to specify which expenses relate to their particular bargaining unit and which relate to the private bargaining units.
No authority requires such a breakdown and we decline to extend existing law to create one. Requiring such a breakdown may be extremely impracticable, if not impossible, as well as unduly burdensome. Moreover, the Supreme Court recently held that non-members can be charged for their prorata share of costs associated with otherwise chargeable activities of state and national affiliates, even if those activities were not performed for the "direct benefit" ofchargeable; the latter is not. Id. ___ at ___ - ___, 111 S.Ct. at 1959-61. ("Accordingly, we hold that the State constitutionally may not compel its employees to subsidize legislative lobbying or other political union activities outside the limited context of contract ratification or implementation").[3]
Given this test, the above quoted components of the definition of chargeable activities used by MKA appear overbroad since they count as chargeable any lobbying and public relations activity that concerns "work related matters" or "work related issues." On their face, these two phrases could easily encompass matter or issues beyond those concerning contract ratification and implementation. Cf; Lehnert v. Ferris Faculty Ass'n, 707 F.Supp. 1473, 1482 (W.D.Mich.1988), aff'd in part and reversed in part, ___ U.S. ___, 111 S.Ct. 1950, 114 L.Ed.2d 572


(1991).[4]

B. Retroactivity and Notice

As noted above, plaintiffs also contend that, even if the MKA statement comports with first amendment requirements, we should rule that (1) Local 38 can not retroactively assess agency fees, and (2) that Local 38 failed to provide sufficient notice to plaintiffs concerning their right to object to the $45.10 agency fee and have such objection resolved by arbitration.

*1239 1. Retroactive assessment
When Local 38 provided plaintiffs with a copy of MKA's schedule for the year end 1990, it was accompanied by a letter, dated June 17, 1991, which states that MKA has determined that the proper agency fee for plaintiffs is $45.10 per month (based on a 78.9% chargeability ratio) and that "[t]his amount is to be retroactive to July 1, 1989." Mazzola Decl., Exh.of the Union's disclosure concerns the two components of the chargeability definition discussed above.
Accordingly, and good cause appearing, it is HEREBY ORDERED as follows:
1. Ruling on plaintiffs' motion for summary judgment and Local 38's cross-motion for dissolution of the preliminary injunction is deferred pending the following. Within 14 days of the date of this Order, Local 38 shall submit a supplemental filing which consists of either:
(a) a declaration, under penalty of perjury, from MKA that clarifies that that portion of the chargeability definition which includes:
(1) the costs of "communications with community organizations, civic groups, government agencies and the media respecting the Local's position on work-related matters," and
(2) lobbying and legislative activities with respect to matters concerning employees' work-related issues"
was construed by MKA in a manner consistent with Lehnert


a first position and a second position, an abutment surface carried by the strap in a position intersecting the excursion path of the finger when the strap is in a first position, a limit abutment carried by the strap in a position intersecting the excursion path of the finger when the strap is in the second position, means temporarily holding the strap in the second position and means urging the strap toward the first position. The strap is moved toward the second position by contact of the finger with the abutment surface in response to rotation of the second gearlike member.
Electronic towel dispensers are also known. U.S. Pat. Nos. 3,730,409, 3,971,607, 4,738,176, 4,796,825 and 4,826,262 each disclose electronic towel dispensers. For example, in Ratti, U.S. Pat. No. 3,730,409,to dispense paper so long as the user's hand remains in front of the sensor. Also, if a change in ambient light occurs, prior hands-free dispensers have to be manually reset to adjust to a new light reference.
Therefore, it would be advantageous to provide improved towel dispensers for automatically dispensing a length of towel in response to the movement of an object such as a user's hands. In this manner, a user can avoid contact with viruses or bacteria on the dispenser left by prior users' hands. It would be further advantageous to provide energy-efficient hands-free dispensers which utilize light energy. It would also be advantageous to provide hands-free dispensers which are simple in design, safe and easy to use. It would be even further advantageous to provide


Kinetics of hybridization of mRNA of c-myc oncogene with 111In-labeled antisense oligodeoxynucleotide probes by high-pressure liquid chromatography.
Antisense oligodeoxynucleotides (ASON) were labeled with gamma-emitting 123I, 99mTc and 111In radionuclides. The hybridization kinetics of 111In-labeled ASON probes [phosphorothioate (S) and phosphodiester (O)] with intact leukemic cells (P388) and purified mRNA was studied by gel filtration technique. The 15-mer oligodeoxynucleotide (ON) sequence was synthesized, amino linked and coupled to diethylenetriaminepentaacetic acid (DTPA)-isothiocyanate, and aliquots were lyophilized to make a kit for convenient preparation. 111In radionuclide was chelated to DTPAASON derivatives and free 111In was separated by gel filtration. The probe was incubated with P388 cells and mRNA extract of P388 cells. Hybridization kinetics was studied by measuring the free and mRNA-bound probe separated by the HPLC technique. The distribution of radioactivitya default judgment. So, the judge went through the process of discussing the evidence. The discussion was brief. The plaintiff had presented no contrary evidence. The judge did discuss Plaintiff Mzyk’s rebuttal information from her deposition. But, the judge was necessarily only seeing the rebuttal testimony provide by the defendant. Any information provided by the defendant, we can expect, would be favorable to the defendant. So, of course, the judge found Ms. Mzyk did not make out her case. Her lawsuit was dismissed. It is fortunate for her that the defendant did not seek sanctions. It did make a counter-claim in its Answer that her lawsuit was frivolous. So, clearly it did occur to the employer that it could seek sanctions.

But, it is unlikely a judge would award


substantial and irreparable harm in that he must attempt to defend criminal charges brought in a State in which there has been prejudicial collusion directly affecting plaintiff, whether intentional or inadvertent between the Judicial and Executive branches of the New Jersey State government.  Plaintiff is being made to defend criminal charges which have been obtained, inter alia, as a result of that collusion, and the deprivation of plaintiff's constitutional rights by not too subtle cooperative coercion on the part of the defendants.  Furthermore, in the event of his conviction upon any one of the charges presently pending against him, plaintiff's only recourse would be review by the State Courts and ultimately the New Jersey Supreme Court, which Court he has alleged has been involved in theprosecution of the charges against him."

12
Insofar as this appeal reviews the order dismissing Helfant's appeal for failure to state a claim upon which relief may be granted these factual allegations must be taken as true.

13
The opposing affidavits filed by the state defendants in opposition to Helfant's motion for a preliminary injunction do not dispute any of the historical factual allegations of the Complaint quoted above, except that defendant Hayden avers:

14
"I had no knowledge that Helfant was to appear before the New Jersey Supreme Court until I was called by the Supreme Court on November 6, 1972 and told that Helfant might be a few minutes late for his grand jury appearance."

15
Read in the light most favorable to those defendants, the affidavits do tend to suggest that Helfant's testimony


Chief Justice asked me about the seating arrangements for the Bar Mitzvah and then he asked me who had purchased the liquor for the Bar Mitzvah, whether Mrs. Schusterman was there and whether I had purchased any other gifts for Judge Rauffenbart.  He asked if formal invitations were sent out.  It was basically things pertaining to Abe Schusterman who I had known had testified on the 25th of October, one week before.

43
Q. Now was there any file in the presence of the Chief Justice?

44
A. There was a file in front of the Chief Justice, Mr. Perskie, but it was closed and it was with the same brown folder that was submitted to you by Mr. Hayden in your request with the clasp on the top ofon remand receive accelerated consideration, and the court should enter an order consolidating the hearing on the motion for a preliminary injunction with a trial on the merits.  Rule 65(a)(2) Fed.R.Civ.Proc.

55
At the oral argument on this appeal we asked the attorney for the appellees if the State intended to commence the trial of the indictment, now scheduled for September 10, 1973, while this appeal was sub judice.  We were advised that this was the State's intention.  Accordingly we issued an order enjoining commencement of the prosecution until such time as we could decide the appeal.

56
The order dismissing the complaint will be reversed.  The order denying the motion for a preliminary injunction will be vacated, and the case will be remanded to the district court


          Add(i);
        }

        /// <summary>
        /// Adds a collection of tasks to this AsyncPipeline.
        /// </summary>
        /// <param name="tasks">A collection of tasks to add to this AsyncPipeline.</param>
        public void AddRange<T>(IEnumerable<Task<T>> tasks)
        {
            foreach (var i in tasks)
                Add(i);
               tcs.Task.Wait();
                // the following quantity is an inconsistent value but i don't have a means to geuuuut one in this part of the
                // code because adding the actual add has already been performed from within a continuation.
                pipeSz = Count;
            }
            finally


in.

4. First President to preside over a cut to the credit-rating of the United States.

5. First President to violate the War Powers Act.

6. First President to be held in contempt of court for illegally obstructing oil drilling in the Gulf of Mexico.

7. First President to require all Americans to purchase a product from a third party.

8. First President to spend a trillion dollars on “shovel-ready” jobs when there was no such thing as “shovel-ready” jobs.

9. First President to abrogate bankruptcy law to turn over control of companies to his union supporters.

10. First President to by-pass Congress and implement the Dream Act through executive fiat.

11. First President to order a secret amnesty program that stopped the deportation of illegal immigrants across the U.S., including those with criminal convictions.

12. FirstNobles". In addition, any effect(s) of the monster Special Summoned by this effect that would be activated by Normal Summon or Flip Summon are activated when the monster is Special Summoned by this effect.

Royal Legacy
Normal Trap

This card can only be activated when a "Curran", "Pikeru", or "Royal" monster is destroyed and sent to your Graveyard. Select and activate one of the following:

* Draw a card. Then, if you have any "Curran", "Pikeru", or "Royal" monsters in your Hand, show one of them to your opponent in order to draw another card.
* Special Summon one level 5 or lower "Curran", "Pikeru", or "Royal" monster from your Hand or Deck, ignoring summoning conditions.

Aaand...the way I was planning to get the card game playing to commence would require being on site


accommodated in a similar region of the receptor. The receptor binding site identified in the optimized complexes contained many residues in positions known to affect ligand binding in G-protein coupled receptors. These results also allowed identification of key residues as candidates for point mutations for further assessment and refinement of this model as well as preliminary indications of the requirements for recognition of this receptor.the hay making. Being grounded in reality makes the surreal stuff even more gut punching. Do you think of the book as prose or poetry or both? Does it matter? Were the rhythms of the sentences more important to you than their meanings?
Creating a tension between the real and the surreal is important. The idea I think is that it is not one or the other. Or it is one or the other, but we are not sure which. Maybe it is neither real nor surreal. Maybe it’s something else. I hope something else. I guess that’s the way I think of prose versus poetry, too. I mean, I would categorize the work as prose, but it certainly emphasizes the sonic qualities of language akin to some poetry.


come together to make a
whole picture but it's just the
1838 version is the one that most people are familiar with.
One version references only the
Lord, probably Jesus Christ, which might mean that
God the Father departed after presenting his Son
which would have been too much different from
what happened at Christ's baptism and
transfiguration. Some don't mention Satan's
attempt to keep Joseph from praying. Other
versions reference an angel or angels coming and
that's okay too. Just because something is
mentioned in one version and not the others
doesn't mean that it's a contradiction. It's just
that it gives us a better understanding of everything that happened.
So why is the 1838 version the one
found in the scriptures like Doctrine and Covenants?
So the 1838 version is the one that
is the most complete account, total, of the First
Vision that Josephresults suggest that cordycepin can synergize with other anti-leukemia reagents by targeting LSCs to treat leukemia.

Materials and Methods {#s2}
=====================

Cell Culture and Reagents {#s2a}
-------------------------

U937, K562 and THP1 cells were maintained in RPMI 1640 medium (Gibco, Gaithersburg, MD) and A549, HepG2, SK-HEP-1 and MCF-7 cells were maintained in Dulbecco's modified Eagle's medium (DMEM) (Gibco, Gaithersburg, MD) supplemented with 10% fetal bovine serum (Biological Industries, Kibbutz Beit-Haemek, Israel), 100 units/ml penicillin, 100 µg/ml streptomycin (Gibco, Gaithersburg, MD) at 37°C in a 5% CO~2~ humidified incubator. Cordycepin, adenosine, SB216763 and rapamycin were purchased from Sigma-Aldrich (St. Louis, MO). Compound c and MG-132 were from Merck KGaA, (Darmstadt, Germany). Ly-294002 was obtained from Enzo Life Sciences (New York City, NY) and Wnt-3a was from R&D Systems (Minneapolis, MN).

Cell Proliferation Assay {#s2b}
------------------------

Cell proliferation of


in a time course-dependent manner ([Figure 6D](#pone-0076320-g006){ref-type="fig"}). These results indicate that cordycepin may regulate GSK-3β by inactivating PI3-K/Akt signaling, thereby inducing protein degradation of β-catenin ([Figure 6E](#pone-0076320-g006){ref-type="fig"}).

![PI3-K/Akt signaling is involved in cordycepin-suppressed β-catenin.\
U937 cells were treated with cordycepin (100 µM) combined with/without (A) compound c (1 µM), (B) rapamycin (10 µM) and (C) Ly-294002 (20 µM) for 4 hrs. Expression level of β-catenin, Akt phosphorylated Akt (Ser473) were determined by Western blot analysis. Actin was used as loading control. (D) U937 cells were treated with cordycepin (100 µM) for 5, 60 and 240 min. Expression level of phosphorylated Akt (Ser473), Akt, phosphorylated GSK-3β (Ser9) and GSK-3β were determined by Western blot analysis. Actin was used as loading control. (E) A schematic model illustrating the role of cordycepin insynthesis and cell adhesion through modulating AMPK and mTOR-dependent pathway in fibroblasts [@pone.0076320-Wong1]. However, results from treatments with compound c and rapamycin reveal that AMPK and mTOR pathways are not involved in cordycepin-reduced β-catenin ([Figure 6A and 6B](#pone-0076320-g006){ref-type="fig"}) in leukemia cells. In contrast, Ly-294002 dramatically reduced β-catenin level in leukemia cells ([Figure 6C](#pone-0076320-g006){ref-type="fig"}). Thus, cordycepin may regulate GSK-3β/β-catenin via a PI3-K/Akt dependent mechanism.

Increasing numbers of studies have focused on targeting β-catenin to suppress LSCs. A combination of inhibiting BCR-ABL activity and β-catenin signaling is considered to be a potential therapeutic strategy for CML patients [@pone.0076320-Hu1], [@pone.0076320-Heidel1]. Thus, our findings that cordycepin selectively inhibits β-catenin in leukemia cells provides a novel leukemia therapy. Although the minimal concentration of cordycepin to significantly suppress leukemia cell proliferation *in vitro* is 50


an efficient real-time transport protocol multiplexing method and apparatus for transporting compressed speech between IP telephony gateways. Subbiah includes creating a header for a plurality of data packets, wherein each header provides identification of a user associated with a packet. Then, each header is added to the data packet associated therewith to form mini-IP payloads. The mini-IP payloads are multiplexed into a RTP payload and the RTP payload is transmitted over a single RTP/UDP/IP connection. Thus, Subbiah multiplexes a number of users in to a single RTP/UDP connection. However, Subbiah is applicable only between a pair of nodes.
An IP based Radio Access Network (RAN) or Core Network (CN) requires that the solution proposed by Subbiah be extended so that it can be used for switching the mini packets.I’ll lose that but if I don’t do this then I can do that and so forth. There is no advantage to being fat. Period.

Sorry…

It’s unhealthy, you’ll just be a burden on society, you’ll be miserable because a lot of the people you need to network with won’t want to talk to you, and you’re more likely to be single for the rest of your life.

It’s hard but not that hard to lose weight. There are programs like the this one that can make you lose 21 pounds in 21 days.

There’s no excuse to be fat, son.

Fat people earn less money, have fewer friends, have more health problems, and die sooner.

Physical fitness is very necessary for human beings.

5. Get a boyfriend!

Don’t believe the nonsense that we can be


the people to avail services to be provided by the state government and its agencies and lapses on the part of officials in discharge of their obligations will amount to violation of proposed act," Kumar said.

Referring to his election promise that the state government would ensure confiscation of property of corrupt public servants and run schools there, he said applications were being filed by the prosecution agency in this connection.

Kumar said a Common Minimum Programme for governance would be evolved for the next five years after the winter session of the state legislature that will end on 9th December.

The Chief Minister said he would resume his popular 'Janata Ke Darbar men Mukhyamantri' programme to get a first-hand account of problems facing the common man.

He also pledged to implementWell that title was a big one! Oh my, banana bread; you have been a challenge for us. Vegan baking is not our thing, honestly we are absolutely hopeless at it. It’s a skill we are slowly working on. But we love banana bread so much we were determined to get this one right sooner rather than later. The day has finally come, no more mushy banana bread, no more stale tasting banana bread. Just you wait till you get your winter (or summer) mitts on this!

Boasting a few titles this one is GF, OF, SF & VE – did you like those abbreviations; if you removed the walnuts it will be also NF (nut free for those playing at home). We have been swaying away from the


teaspoon of nutmeg

1/2 teaspoon sea salt

1 1/2 cup mashed over ripe banans

1 cup of pitted dates (medjool would taste the best)

2 tablespoons of diary free milk

3 tablespoons of flax seeds (or flax meal)

3 tablespoons of water

2 teaspoons of vanilla extract

1 teaspoon of apple cider

1/3 chopped walnuts

1 banana to put on top

HOW YOU’LL CREATE IT

Let us begin by preheating your oven to 180 degrees °C. Also soak your dates in hot water for 5 minutes while you do the next step.

Now sift the flours, baking powder & soda in to a mixing bowl and also throw in the cinnamon, nutmeg & salt. Mix with a whisk ensuring it is all lump free.

Now lets add all the remaining ingredients to a blender of your choice (except the walnuts & 1IP.Calendar imports (iCal)

1 post in this topic

I think it would be nice to have a few added features to the iCal feed like the ability (or have it as default) to add in a row switch for each event to make them more readable (on the monthly view) ... and the ability on how each event is displayed (ie colored titles, line breaks, etc).

It would also be sweet if the import could detect changes to previous feeds (such as descriptions) and import any changes that is found rather than just skipping the event.


messages destined for each other thread. */

class linux_message_hub_t : private linux_event_callback_t {
public:
    typedef intrusive_list_t<linux_thread_message_t> msg_list_t;

    linux_message_hub_t(linux_event_queue_t *queue, linux_thread_pool_t *thread_pool,
                        threadnum_t current_thread);

    /* For each thread, transfer messages from our msg_local_list for that thread to our
    msg_global_list for that thread */
    void push_messages();

    /* Schedules the given message to be sent to the given thread by pushing it onto our
    msg_local_list for that thread */
    void store_message_ordered(threadnum_t nthread, linux_thread_message_t *msg);

    // Schedules the given message to be sent tosome of the factors that hinder women from equal participation with men as they try to balance sports and family. Benefits that men get are significantly higher than women’s and generally the incentives do not reflect their disadvantaged position and contribution to the sport. Offering of equitable benefits to women will ensure that they put efforts towards matching men’s performance when they compete alongside or against each other. Golf has proved that women can compete against or alongside men in sports.


which may be utilized without interrupting weighing and packaging operations.
A more specific object is to provide an apparatus and method of the foregoing type which minimize the chance that a package contaminated with a metal test object is inadvertently transported from the weighing and packaging station for shipping to a customer.
Another specific object of the present invention is to provide an apparatus and method of the foregoing type which does not cause a filled test package to be wasted.
Still another specific object of the invention is to provide an apparatus and method of the foregoing type which fully automates the testing operation.HH behavior. 98% considered that this communication should be integrated in patient care and 91% thought it could be routinely performed.

Conclusion
==========

We note a direct benefit for patients who are rather "pleasantly surprised". This communication also takes parts in the individual and collective engagement of caregivers about HH by making them do what they declare. We wish to propose this "easy to implement" approach to a greater number of HCF in May 2013.

Disclosure of interest
======================

None declared