text:
\section{{\bf Introduction}}

Let $\Gamma$ be a noncocompact Fuchsian group. Then $\Gamma\backslash\mathbf{H}$ is an open Riemann surface, and its compactification $X$ can be seen as a smooth algebraic curve. Fixing a base point $x\in X$, we have an embedding  $X\hookrightarrow J(X)$ into its Jacobian $J(X)$. Manin and Mumford conjectured that for any such embedding, there are only finitely many torsion points in the image. The conjecture was first confirmed by Raynaud \cite{Raynaud1983}.

If $\G$ is a congruence subgroup of $\mathrm{SL}_2(\mathbf{Z})$, Drinfeld \cite{Drinfeld1973}, generalizing a result of Manin \cite{Manin1972}, provided significant examples of these finitely many torsion points on modular curves; the subgroup $C(\G)\subset J(X)$ spanned by the images of the cusps of $\G$ is contained in $J(X)^{\rm tor}$. Their proofs used Hecke operators, limiting the reach of the statement to modular curves. In this note, we observe a new connection between the Manin--Drinfeld theorem and Rademacher symbols, which are ubiquitous invariants appearing in connection to Dedekind sums \cite{Rademacher}, class numbers of real quadratic fields \cite{Meyer1957,Zagier1975}, aspects of Atiyah--Bott--Singer index theory \cite{Atiyah1987}, and linking numbers of knots \cite{GhysICM,DukeImamogluToth2017}. On the basis of this relation, we give a new proof of the Manin--Drinfeld theorem and examine for which other families of curves it might hold.

As is customary, we reframe this discussion in terms of divisors. Recall that a divisor is a formal sum
$$
D = \sum a_i x_i,
$$
where $x_i\in X$, and $a_i$ are integers, only finitely many of which are non-zero. The degree of the divisor $D$ is the sum $\sum a_i$, and divisors of degree zero form a group. Let $D$ be a divisor of degree zero. By the Abel--Jacobi theorem, there is an isomorphism between divisor classes (with respect to linear equivalence) of degree zero and the Jacobian $J(X)$ modulo periods.
By Riemann--Roch, there is a differential $\omega_D$ of the third kind (i.e., a meromorphic differential) whose residual divisor is $D$, and this differential can be made unique; see, e.g., Theorem 5.3 in \cite{LangAbelian}. Then $D$ is torsion if and only if the periods of $\omega_D$ are in $2\pi i\mathbf{Q}$

\begin{Thm}\label{thm:1}
Let $\Gamma$ be a noncocompact Fuchsian group with $h$ inequivalent cusps
 $\frak{a}_1,\dots,\frak{a}_h$. Let $D=\sum m_i (\frak{a}_i)$ be a divisor of degree zero, and let $\omega_D$ be the canonical differential of the third kind associated to the residue divisor $D$. Then for every hyperbolic element $\g\in\G$ of positive trace,
$$
\int_\gamma \omega_D = 2\pi i\left(m_1\Psi_{\frak{a}_1}(\gamma)+\dots+m_h \Psi_{\frak{a}_h}(\gamma)\right),
$$
where $\Psi_\frak{a}$ is the Rademacher symbol on $\Gamma$ for the cusp $\frak{a}$. Moreover, if all Rademacher symbols on $\G$ are rational-valued, then $C(\G)\subset J(X)^{\rm tor}$.
\end{Thm}

We first describe Rademacher symbols as they appear in the classical theory of modular forms and elliptic functions.  Dedekind showed that the transformation-theory of the $\eta$-function
\begin{align}\label{eta}
\eta(z) = q^{\tfrac{1}{24}}\prod_{n\geq1} (1-q^n),
\end{align}
($q=e^{2\pi iz}$), under the linear fractional action of $\g=\left(\begin{smallmatrix} a&b\\ c&d\end{smallmatrix}\right)\in\mathrm{SL}_2(\mathbf{Z})$ on the upper half-plane $\mathbf{H}$  is encoded by the Dedekind symbol
$$
\Phi(\gamma)\ =\ \begin{dcases} \frac{b}{d} & {\rm if }\ c=0,\\
\frac{a+d}{c}-12\mathrm{sign}(c) \cdot s(a,c) & {\rm if }\ c\neq0.
\end{dcases}
$$
where $s(a,c)$ are Dedekind sums --- arithmetic sums depending only on $a/c$, and obeying a reciprocity law that makes them easy to compute via the Euclidean algorithm. The same reciprocity law also allows to show that $\Phi(\g)$ is always an integer. These classical facts are exposed in the monograph of Rademacher's lecture notes on the subject; see \cite{Rademacher}. In his study of Dedekind sums, Rademacher introduced the associated function
\begin{align}\label{Rademacher}
\Psi(\gamma)=\Phi(\gamma)-3\mathrm{sign}(c(a+d)),
\end{align}
which has the remarkable feature of being conjugacy class invariant. In fact, on hyperbolic elements, it can be realized as the period
$$
\int_\g E_2(z)dz = \Psi(\g),
$$
where $E_2$ is Hecke's modified Eisenstein series of weight 2 (see \secref{sec:proof} and \cite{DukeImamogluToth2018} for further historical references).

This is for the classical theory. Over the last 50 years, analogues of these various functions have been constructed for other noncocompact Fuchsian groups. A standard construction builds on extending the celebrated first limit formula of Kronecker, which classically can be used to deduce the transformation-theory of $\eta$ and determine the Dedekind sums. The classical first limit formula of Kronecker expresses the constant term in the Laurent expansion of the non-holomorphic Eisenstein series $E(z,s)$ for $\mathrm{PSL}_2(\mathbf{Z})$ at its pole $s=1$. More generally, to each cusp $\frak{a}$ of a general noncocompact Fuchsian group, one has an associated non-holomorphic Eisenstein series $E_\fa(z,s)$. On the basis of Selberg's study of their analytic properties, one obtains a Kronecker limit formula in this extended setting, and this limit formula can be used to define a generalized Dedekind symbol $\Phi_\fa$. Such constructions have been extensively studied in recent years \cite{Goldstein1973,Burrin2017,JorgensonO'SullivanSmajlovic2020}. Much less attention has been given the associated Rademacher symbol $\Psi_\fa:\G\to\mathbf{R}$, which we define in \secref{sec:Rademacher}. In \lemref{lemma}, we show that $\Psi_\fa$ can again be realized as the hyperbolic period
$$
\int_\g E_{2,\fa}(z)dz = \Psi_\fa(\g).
$$

The Fuchsian Rademacher symbols $\Psi_\fa$ are a priori real-valued. In the face of \thmref{thm:1}, we do not expect Fuchsian Rademacher symbols to be rational-valued in general. In fact, Rohrlich observed that for some non-congruence subgroups of $\Gamma(2)$ appearing in connection to the Fermat curve, $C(\Gamma)$ must be infinite (see p.~198 in \cite{KubertLangBook}) and thus the associated Rademacher symbols can not all be rational.
 Nevertheless, for certain groups, it is possible to show that the image is in fact rational through rigorous arithmetic computations. This is the case for the Rademacher symbol $\Psi_\infty^\G$ when $\G$ is any principal congruence group $\G(N)$ or Hecke congruence group $\G_0(N)$, following the works of \cite{Driencourt1983}, \cite{Takada1986}, and \cite{Vassileva}. By a theorem of Helling \cite{Helling1966}, every noncocompact arithmetic subgroup of $\mathrm{SL}_2(\mathbf{R})$ is conjugate to a subgroup of $\G_0(N)^+$, the group obtained by adding all Atkin--Lehner involutions to $\G_0(N)$, for some squarefree $N$. (See \secref{sec:rationality} for an explicit parametrization.) Moreover, each maximal noncocompact arithmetic group (under inclusion) is a conjugate of $\G_0(N)^+$ for some squarefree $N$. (The converse does not hold; see \cite{Baily1987}.) As noted by Helling, the groups $\G_0(N)^+$ have a unique cusp, which can be taken at the point at $\infty$. The 44 groups $\G_0(N)^+$ of genus 0 famously appear in connection to monstruous moonshine.

\begin{Thm}\label{thm:2}
Rademacher symbols are rational for all genus 0 noncomcompact Fuchsian groups, for all Helling groups $\G_0(N)^+$, and for all congruence subgroups of $\mathrm{SL}_2(\mathbf{Z})$.
\end{Thm}

In particular, we recover a new proof of the Manin--Drinfeld theorem. By a theorem of Takeuchi \cite{Takeuchi1977}, we know that among triangle groups, only finitely many (up to conjugation) are arithmetic; it is interesting to note that the phenomenon described by Manin--Drinfeld does not restrict to arithmetic groups. The proof of \thmref{thm:2} builds on certain relations between Rademacher symbols for $\G_1$ and $\G$, when $\G_1<\G$ is a normal subgroup of finite index; see \propref{proposition}.

\subsection{Acknowledgments.} This note was inspired by the beautiful paper of Murty and Ramakrishnan \cite{MurtyRamakrishnan1987} on a similar connection between the Manin--Drinfeld theorem and generalized Ramanujan sums. The author thanks Jay Jorgenson for many discussions on connections between Manin--Drinfeld and Dedekind symbols of various types, and Asbj\o rn Nordentoft for pointing out a mistake in a previous version of this manuscript.

\section{{\bf Rademacher symbols for cusps}}\label{sec:Rademacher}

The classical monographs of Shimura \cite{Shimura} and Kubota \cite{Kubota} and the more recent book of Iwaniec \cite{Iwaniec} serve as references for facts cited in this section on Fuchsian groups and Eisenstein series.

\subsection{Cusps, Eisenstein series}

Let $\frak{a}$ be a cusp for $\Gamma$, and let $\Gamma_\frak{a}$ be the stabilizer subgroup of $\frak{a}$, i.e., $\Gamma_\frak{a}=\{\gamma\in\Gamma:\gamma\frak{a}=\frak{a}\}$. The group $\Gamma_\frak{a}$ is infinite cyclic. We will denote by $\g_\fa$ the cyclic generator of $\G_\fa$. We say that two cusps $\frak{a}$, $\frak{b}$ are equivalent if $\frak{b}=\gamma\frak{a}$ for some $\gamma\in\Gamma$. If $\frak{a}$ and $\frak{b}$ are equivalent, then $\Gamma_\frak{a}=\Gamma_\frak{b}$.

We may choose a scaling transformation $\sigma_\frak{a}\in\mathrm{PSL}(2,\mathbf{R})$ such that
\begin{align*}
\sigma_\frak{a}(\infty)&=\frak{a}\qquad\text{and}\qquad
\sigma_\frak{a}^{-1}\Gamma_\frak{a}\sigma_\frak{a} = \pm\begin{pmatrix} 1&\mathbf{Z}\\&1\end{pmatrix}.
\end{align*}
Such a choice of $\sigma_\frak{a}$ is only unique up to right multiplication by an element of $\pm\left(\begin{smallmatrix} 1&\mathbf{R}\\0&1\end{smallmatrix}\right)$. Conjugating $\Gamma$ by $\sigma_\frak{a}$ provides a group with a cusp at infinity of width 1, and facilitates computation.

The group $\Gamma$ acts discontinuously on the upper half-plane $\mathbf{H}$ by fractional linear transformation $\pm\left(\begin{smallmatrix} a&b\\ c&d\end{smallmatrix}\right):z\mapsto\tfrac{az+b}{cz+d}$. For each $z\in\mathbf{H}$, the Eisenstein series for the cusp $\frak{a}$ is defined by
$$
E_\frak{a}(z,s) =\sum_{\gamma\in\Gamma_\frak{a}\backslash\Gamma} \mathfrak{Im}(\sigma_\frak{a}^{-1}\gamma z)^s,
$$
where $s$ is a complex parameter controlling the convergence of the infinite series; the series $E_\frak{a}(z,s)$ converges absolutely and uniformly on compact subsets when $\mathfrak{Re}(s)>1$. Beyond this half-plane, Eisenstein series admit a meromorphic continuation to the whole complex plane, as famously proved by Selberg. The definition of  $E_\frak{a}(z,s)$ does not depend on the particular choice of scaling $\sigma_\frak{a}$. Moreover, equivalent cusps yield identical Eisenstein series.

To understand the behavior of the Eisenstein series $E_\frak{a}$ at the (not necessarily distinct) cusp $\frak{b}$ of $\Gamma$, one considers the series $E_\frak{a}(\sigma_\frak{b}z,s)$, for which we have the relation
$$
E_\frak{a}(\sigma_\frak{b}z,s) = E_\frak{a}(\gamma_\frak{b} \sigma_\frak{b}z,s) = E_\frak{a}(\sigma_\frak{b}(z+1),s)
$$
and the resulting Fourier expansion is explicitly given by
$$
E_\fa(\sigma_\fb z,s) = \delta_\fab y^s +\frac{\sqrt\pi \G(s-1/2)}{\G(s)}\varphi_\fab(s) y^{1-s} +\frac{2\pi^s \sqrt{y}}{\G(s)} \sum_{n\neq0} |n|^{s-1/2}K_{s-1/2}(2\pi|n|y)\varphi_\fab(n,s) e(nx),
$$
where $\delta_\frak{ab}=1$ if the cusps $\frak{a}$ and $\frak{b}$ are in the same $\Gamma$-orbit, and 0 otherwise, where $\G(s)$ denotes the classical $\G$-function, where $K_s(z)$ is the $K$-Bessel function (or, modified Bessel function of the second kind), where $e(z)=e^{2\pi i z}$, and where $\varphi_\fab(s)$ and $\varphi_\fab(n,s)$ are the Dirichlet series
\begin{align*}
\varphi_\fab(s) &= \sum_{c>0} c^{-2s} \#\{ d\in[0,c): \left(\begin{smallmatrix} *&*\\ c&d\end{smallmatrix}\right)\in\sigma_\fa^{-1}\G\sigma_\fb\}, \\
\varphi_\fab(n,s) &= \sum_{c>0} c^{-2s} \sum_{\substack{0\leq d<c\\ \left(\begin{smallmatrix} *&*\\ c&d\end{smallmatrix}\right)\in \sigma_\fa^{-1}\G\sigma_\fb}} e\left(n\frac{d}{c}\right).
\end{align*}
These Dirichlet series do not depend on a particular representative for the cusps $\fa$, $\fb$, but $\varphi_\fab(n,s)$ depends on the particular choice of the scaling transformation $\sigma_\fb$ up to a unitary multiplicative factor. Indeed, if we replace $\sigma_\fb$ by $\sigma_\fb n_x$, $n_x=\pm\left(\begin{smallmatrix} 1& x\\ 0&1\end{smallmatrix}\right)$, then $\varphi_\fab(n,s)$ is replaced by $\varphi_\fab(n,s)e(nx)$. Both Dirichlet series are absolutely convergent on $\mathfrak{Re}(s)>1$.

On the vertical line $\mathfrak{Re}(s)=1$, the Eisenstein series are holomorphic, save for a simple pole at $s=1$. Examining the local behavior of its Fourier coefficients, we note that the $K$-Bessel function is entire as a function of $s$ --- in fact, at $s=1$, we have $2(|n|y)^{1/2} K_{1/2}(2\pi|n|y)e(nx) =  e^{-2\pi|n|y}e(nx)$, see \cite[p.~227]{Iwaniec} ---, and that the Dirichlet series $\varphi_\fab(n,s)$ are holomorphic along the vertical line $\mathfrak{Re}(s)=1$. At $s=1$, $\varphi_\fab(n,1)$ is real valued, hence $\varphi_\fab(n,1)=\varphi_\fab(-n,1)$, and satisfies the estimate $|\varphi_\fab(n,1)|\ll n^{1+\varepsilon}$, for any $\varepsilon>0$; see \cite[Theorem 1.1]{JorgensonO'Sullivan2005}. Finally, the Dirichlet series defining $\varphi_\fab(s)$ has a simple pole at $s=1$ of residue $(\pi V)^{-1}$, where $V$ denotes the hyperbolic volume of any fundamental domain for $\G$. Thus $\varphi_\fab(s)$ has a Laurent expansion at $s=1$ of the form
$$
\varphi_\fab(s) = \frac{(\pi V)^{-1}}{s-1} +\sum_{n\geq0} c_n (s-1)^n.
$$

\subsection{Kronecker's first limit formula}

An extension (see \cite[Theorem 3-1]{Goldstein1973}) of Kronecker's classical first limit formula gives the constant coefficient in the Laurent series for $E_\frak{a}(\sigma_\frak{b}z,s)$ at the simple pole at $s=1$. Using the Fourier expansion of the Eisenstein series and letting $s\to1^+$, we immediately obtain the corresponding formal Laurent series at $s=1$, given by
\begin{align*}
\lim_{s\to 1^+} \left(E_\frak{a}(\sigma_\frak{b}z,s) - \frac{V^{-1}}{s-1}\right)\ &=\ c_0 - V^{-1}\log y +\delta_\fab y +\sum_{n>0} \pi \varphi_\fab(n,1) e(nz) + \sum_{n<0} \pi \varphi_\fab(n,1) e(n\overline{z})\\
& =\ c_0 -V^{-1}\log y - \mathfrak{Re}\left(\delta_\fab iz - 2\pi \sum_{n>0} \varphi_\fab(n,1)e(nz)\right).
\end{align*}
We denote the term in parenthesis by $f_{\fa,\sigma_\fb}(z)$ and record that $f_{\fa,\gs_\fb n_x}(z)=f_{\fa,\gs_\fb}(n_x z) -\delta_\fab ix$. As a function of $z$, $f_{\fa,\gs_\fb}$ is holomorphic; this establishes the validity of the limit formula.

Specializing to $\G=\mathrm{PSL}_2(\mathbf{Z})$, we recover the classical first limit formula of Kronecker as follows. We have
\begin{align*}
E(z,s) &= \frac12 \sum_{\substack{c,d\in\mathbf{Z}\\ (c,d)=1}} \frac{y^s}{|cz+d|^{2s}},\qquad \varphi(s) = \sum_{c\geq1} \frac{\phi(c)}{c^{2s}},\\
\varphi(n,s) &= \sum_{c\geq1} c^{-2s} \sum_{\substack{d=1\\ (c,d)=1}}^c e\left(n\frac{d}{c}\right) = \sum_{c\geq1} \frac{1}{c^{2s}}\sum_{\delta|(c,n)}\delta\mu\left(\frac{c}{\delta}\right) = \frac{1}{\zeta(2s)}\sum_{\delta|n} \frac{1}{\delta},
\end{align*}
where $\phi$ is Euler's totient function, and we have used Kluyven's identity for Ramanujan sums in the last line. Using that $\zeta(2)=\tfrac{\pi^2}{6}$,
$$
f(z) = \frac{12}{\pi}\left(\frac{\pi iz}{12}-\sum_{m,n>0} \frac{q^{mn}}{m}\right),
$$
and the term in parenthesis is a branch of the logarithm of Dedekind's $\eta$-function, defined by \eqref{eta}.

\subsection{Dedekind and Rademacher symbols}
We introduce two auxiliary functions on $\G$; the inner automorphism $\tau_{\sigma_\fb}(\gamma)=\sigma_\fb^{-1}\gamma \sigma_\fb$, and the automorphic cocycle $j(\gamma,z)=cz+d$. The latter is a multiplicative cocycle: for any pair $\g_1,\g_2\in\G$, we have $j(\g_1\g_2,z)=j(\g_1,\g_2 z)j(\g_2,z)$. The limit formula derived above implies the relation
$$
\mathfrak{Re} f_{\fa,\gs_\fb}(\tau_\fb(\g)z) = V^{-1} \ln|j(\tau_\fb(\g),z)|^2 + \mathfrak{Re} f_{\fa,\gs_\fb}(z)
$$
for each $\g\in\G$. We choose $\log z$ to be the principal branch of logarithm, i.e., $\log z = \ln|z|+i\arg(z)$ with $\arg(z)\in(-\pi,\pi]$. Then $\log(-j(\g,z)^2)$ is well defined, and $\mathfrak{Re}\left( \mathrm{sign}(c)^2 \log(-j(\g,z)^2)\right) = \ln|j(\g,z)|^2$.
Therefore, the holomorphic function
\begin{align}\label{F}
F_{\frak{a},\gs_\fb}(\g;z): = f_{\frak{a},\gs_\fb}(\tau_{\gs_\fb}(\gamma)z) -f_{\frak{a},\gs_\fb}(z) - V^{-1}\mathrm{sign}(c_{\tau_{\gs_\fb}(\g)})^2 \log\left(-j(\tau_{\gs_\fb}(\g),z)^2\right)
\end{align}
has trivial real part, and thus by the Open Mapping Theorem, it must be a constant function of $z$, i.e., $F_{\fa,\gs_\fb}(\g;z)=F_{\fa,\gs_\fb}(\g)$. As a result, it is also independent of the particular choice of the scaling transformation $\gs_\fb$, since
$$
F_{\fa,\gs_\fb n_x}(\g) = f_{\fa,\gs_b}(\tau_{\gs_\fb}(\g)n_x z) - f_{\fa,\gs_\fb}(n_x z)  -V^{-1}\mathrm{sign}(c_{\tau_{\gs_\fb}(\g)})^2\log\left(-j(\tau_{\gs_\fb}(\g),n_x z)^2\right) = F_{\fa,\gs_\fb}(\g).
$$
We henceforth denote this function $F_\fab$. We can now define the Dedekind and Rademacher symbols attached to the cusp $\fa$. Set
\begin{align*}
\Phi_\frak{ab}(\gamma) &\coloneqq-i F_\frak{ab}(\gamma),\\
 \Phi_\frak{a}(\gamma) &\coloneqq\Phi_\frak{aa}(\gamma),\\
\Psi_\frak{a}(\gamma) &\coloneqq  \Phi_\frak{a}(\gamma) - \pi V^{-1}\mathrm{sign}( c(a+d)),
\end{align*}
with $\left(\begin{smallmatrix} a&b\\ c&d\end{smallmatrix}\right)=\tau_{\gs_\fa}(\g)$.
When $\Gamma=\mathrm{PSL}(2,\mathbf{Z})$, we recover the classical Dedekind and Rademacher symbols $\Phi$ and $\Psi$; see Chapters 4A-C in \cite{Rademacher}. A detailed study of the properties of (a slight modification of) the Dedekind symbols $\Phi_\fab$ appears in \cite{JorgensonO'SullivanSmajlovic2020}. We only record here that, following the analysis in pp.~52--53 of \cite{Rademacher},
\begin{align}\label{qm}
\Phi_\fa(\g_1\g_2) = \Phi_\fa(\g_1) + \Phi_\fa(\g_2) - \pi V^{-1}\mathrm{sign}(c_1 c_2 c_3)
\end{align}
for any $\g_1$, $\g_2\in\G$, with $\left(\begin{smallmatrix} *&*\\ c_1&*\end{smallmatrix}\right)=\tau_{\gs_\fa}(\g_1)$, $\left(\begin{smallmatrix} *&*\\ c_2&*\end{smallmatrix}\right)=\tau_{\gs_\fa}(\g_2)$, and $\left(\begin{smallmatrix} *&*\\ c_3&*\end{smallmatrix}\right)=\tau_{\gs_\fa}(\g_1\g_2)$. The following relations follow immediately:
$\Phi_\fa(I) = \Psi_\fa(I) = 0$, $\Phi_\fa(-\g) = \Phi_\fa(\g)$, $\Psi_\fa(-\g) = \Psi_\fa(\g)$,
$\Phi_\fa(\g^{-1}) = -\Psi_\fa(\g)$, $\Psi_\fa(\g^{-1})=-\Psi_\fa(\g).$

\begin{prop}\label{prop:3}
Rademacher symbols are rational on elliptic and parabolic motions.
\end{prop}

\begin{proof}
Let $\g$ be a parabolic element. Recall that $\G_\fb$ is an infinite cyclic group and denote its generator by $\g_\fb$. Then $\g=\g_\fb^m$ for some $m\in\mathbf{N}$.  We first observe that $\Phi_\fab(\g)= -i\left( f_{\fa,\gs_\fb}(z+m) - f_{\fa,\gs_\fb}(z)\right) = \delta_\fab \cdot m$. Hence $\Phi_\fb$ is an integer on $\G_\fb$. Suppose now that $\fa$ and $\fb$ are inequivalent cusps. Since $E_\fa(\gs_\fa z,s) = E_\fa(\gs_\fb z',s)$ with $z'=\gs_\fb^{-1}\gs_\fa z$, the Kronecker limit formula implies the relation
$$
f_{\fa,\gs_\fb}(z')-f_{\fa,\gs_\fa}(z) = V^{-1}\log\left(-j(\gs_\fb^{-1}\gs_\fa,z)^2\right) + \text{constant}.
$$
Then
\begin{align*}
\Phi_\fa(\g) &= -i\left( f_{\fa,\gs_\fa}(\tau_{\gs_\fa}(\g)z) - f_{\fa,\gs_\fa}(z) - V^{-1}\log\left(-j(\tau_{\gs_\fa}(\g),z)^2\right)\right)\\
&=
-i\left( f_{\fa,\gs_\fb}(z'+m)-V^{-1}\log(-j(\gs_\fb^{-1}\gs_\fa,\tau_{\gs_\fa}(\g_\fb)z)^2\right) - f_{\fa,\gs_\fb}(z')\\
& \qquad  + V^{-1} \log\left(-j(\gs_\fb^{-1}\gs_\fa,z)^2\right) - V^{-1}\log\left(-j(\tau_{\gs_\fa}(\g),z)^2\right))\\
& =
\Phi_\fab(\g) +2iV^{-1}\left( \log j(\gs_\fb^{-1}\gs_\fa,\tau_{\gs_\fa}(\g)z) + \log j(\tau_{\gs_\fa}(\g),z) - \log j(\gs_\fb^{-1}\gs_\fa,z) + i\frac{\pi}{2} A_{\fab}(\g)\right)\\
& =
\delta_\fab\cdot m +2iV^{-1}\left(2\pi i B_\fab(\g)+i\frac{\pi}{2}A_\fab(\g)\right),
\end{align*}
where $A_\fab$ and $B_\fab$ are integer-valued.
Looking at the Gauss--Bonnet formula for $V$ (see, e.g., \cite[p.~46]{Iwaniec}), we have $\pi V^{-1}\in\mathbf{Q}$ and this concludes. Let now $\g$ be an elliptic element of order $m$. Applying \eqref{qm} recursively, we have
\begin{align}\label{recursion}
0 = \Phi_\fa(\g^m) & = m\Phi_\fa(\g) - \pi V^{-1} \sum_{k=1}^{m-1} \mathrm{sign}( c_\g c_{\g^{k}} c_{\g^{k+1}})
\end{align}
and this shows that $\Phi_\fa(\g)\in\mathbf{Q}$. Since $\G$ is finitely generated and the transformation formula \eqref{qm} for $\Phi_\fa$ depends only on the rational constant $\pi V^{-1}$, we conclude that genus 0 noncocompact Fuchsian groups have rational-valued Rademacher symbols.
\end{proof}

For certain congruence subgroups, it is possible to deduce an explicit expression for $\Phi_\fa$, following the method of proof of Dedekind. For example, using the Fourier expansion of the Eisenstein series for $\G(N)$ and the cusp at $\infty$, the function $f:=f_{\infty,\sigma_\infty}$, $(\sigma_\infty=I$), arising from the Kronecker limit formula is explicitly given by \cite[Theorem 1]{Takada1986}
$$
f(z) = iz + 4V^{-1}_{\G(N)}\sum_{j=0}^{N-1}\log \prod_{\substack{m\geq1\\ m\equiv j (N)}} (1-q^{mN})
$$
and (following the method of Dedekind) the corresponding Dedekind symbol is\footnote{ We have the following dictionary between our notation and Takada's: $\pi\alpha_N = V^{-1}_{\G(N)}$, $f= 4\pi\alpha_N \log\eta_N$
, $\Phi(\g) = 4\pi \alpha_N(S_N(\g)-\tfrac14 \mathrm{sign}(c))$ (for $\g=\left(\begin{smallmatrix} *&*\\ c&*\end{smallmatrix}\right)$ with $c\neq0$).} \cite[Theorem 2]{Takada1986}
$$
\Phi^{\G(N)}_\infty(\g) = \begin{dcases} \frac{b}{d} & \text{ if } c=0, \\
\frac{a+d}{c} - \frac{4\pi V_{\G(N)}^{-1}}{|c|N}\sum_{j=1}^{|c|N-1} j C_{N,j} \left(\!\left(\frac{Naj}{c}\right)\!\right) & \text{ if } c\neq0,\end{dcases}
$$
where $(\!(x)\!)= x - \lfloor x\rfloor -\tfrac12$ if $x\not\in\mathbf{Z}$ and 0 otherwise, and
$$
C_{N,j} = \frac{\pi^2}{6}\prod_{p\mid N}(1-p^{-2})\sum_{\substack{a=1\\ (a,N)=1}}^N \sum_{\substack{n\geq1\\ na\equiv 1 (N)}} \frac{\mu(n)}{n^2} \cos\frac{2\pi aj}{N}.
$$

If $N=2$, then $C_{2,j}=\cos(\pi j)\in\{\pm 1\}$ and the Dedekind symbol is rational-valued. If $N>2$, Takada shows that $C_{N,j}$ lies in the cyclotomic field $\mathbf{Q}(\zeta_N,\zeta_{\phi(N)})$, where $\phi(N)$ is the Euler totient function. This implies that $\Phi^{\G(N)}_\infty$ --- and in turns $\Psi^{\G(N)}_\infty$ --- are rational-valued for all $N$'s.

\section{{\bf Proof of Theorem 1}}\label{sec:proof}

For each cusp $\frak{a}$, consider the modified Eisenstein series
\begin{align*}
E_{2,\frak{a}}(z,s) \coloneqq\ 2i\frac{\partial}{\partial z} E_\frak{a}(z,s) = \left(\frac{\partial}{\partial y} +i\frac{\partial}{\partial x}\right) E_\frak{a}(z,s).
\end{align*}
Since the residue of $E_\frak{a}(z,s)$ at the simple pole at $s=1$ is constant in $z$, this modified Eisenstein series is regular at $s=1$, and we set $E_{2,\frak{a}}(z)\coloneqq E_{2,\frak{a}}(z,1)$. Explicitly, we have
\begin{align*}
E_{2,\fa}(z)
&= \lim_{s\to1^+} sy^{s-1} \sum_{\g\in\G_\fa\bk\G} j(\gs_\fa^{-1}\g,z)^{-2}|j(\gs_\fa^{-1}\g,z)|^{-2s+2}.
\end{align*}
Hence this is a real-analytic Eisenstein series of weight 2; for principal congruence groups, this recovers Hecke's construction.
 The Kronecker limit formula for $E_\fa(\gs_\fb z,s)$ yields the Fourier expansion of $E_{2,\fa}(\gs_\fb z)$, which is  given by
\begin{align*}
E_{2,\fa}(\gs_\fb z) \gs_\fb'(z) &= 2i \ddz\left( c_0 - V^{-1}\log y +\delta_\fab y +\sum_{n>0} \pi \varphi_\fab(n,1)e(nz) +\sum_{n<0}\pi \varphi_\fab (n,1)e(n\overline{z})\right)\\
&= -\frac{V^{-1}}{y} +\delta_\fab -4\pi^2\sum_{n>0}n\varphi_\fab(n,1)e(nz)\\
& =
-\frac{V^{-1}}{y}+\frac{1}{i}\frac{d}{dz}f_{\fa,\gs_\fb}(z).
\end{align*}

The relation \eqref{qm} implies that $\Psi_\fa(\g^{-1}) = -\Psi_\fa(\g)$ and $\Psi_\fa(-\g)=\Psi_\fa(\g)$. Thus up to replacing $\g$ by $\pm\g^{\pm1}$, we may assume that for $\g=\left(\begin{smallmatrix} a&b \\c &d\end{smallmatrix}\right)$, we have $c>0$ and $a+d>0$.

\begin{lm}\label{lemma}
Let $\g\in\G$ be a hyperbolic element of positive trace. Then
$$
\int_{\g} E_{2,\fa}(z)dz = \Psi_\fa(\g).
$$
\end{lm}

\begin{proof}
Fix a base point $z_0\in\mathbf{H}$ on the geodesic axis of $\g$. Using the above Fourier expansion,
\begin{align*}
\int_{z_0}^{\g z_0} E_{2,\fa}(z) dz &= \int_{\gs^{-1}_\fa z_0}^{\gs^{-1}_\fa \g z_0} E_{2,\fa}(\gs_\fa z) \gs_\fa'(z) dz\\
&= -V^{-1} \int_{\gs^{-1}_\fa z_0}^{\gs^{-1}_\fa \g z_0} \frac{dz}{y} +\frac{1}{i}\left( f_{\fa,\gs_\fa}(\tau_{\gs_\fa}(\g)\gs_\fa^{-1}z_0)-f_{\fa,\gs_\fa}(\gs_\fa^{-1}z_0)\right).
\end{align*}
For any $g=\left(\begin{smallmatrix} *&*\\c&d\end{smallmatrix}\right)\in\mathrm{SL}_2(\mathbf{R})$, we have
$$
\frac{dz}{y}\circ g - \frac{dz}{y} = \frac{|cz+d|^2-(cz+d)^2}{(cz+d)^2 y}dz = -\frac{2ic}{cz+d} dz = -2i\ d \log j(g,z)
$$
and this shows that
$$
\int_{\gs_\fa^{-1}z_0}^{\tau_{\gs_\fa}(\g)\gs_\fa^{-1}z_0} \frac{dz}{y} = -2i \log j(\tau_{\gs_\fa}(\g),\gs_\fa^{-1}z_0).
$$
On the other hand,
\begin{align*}
f_{\fa,\gs_\fa}(\tau_{\gs_\fa}(\g)\gs_\fa^{-1}z_0)-f_{\fa,\gs_\fa}(\gs_\fa^{-1}z_0) =
i\Phi_\fa(\g)+V^{-1} \mathrm{sign}(c_{\tau_{\gs_\fa}(\g)})^2 \log\left(-j(\tau_{\gs_\fa}(\g),\gs_\fa^{-1}z_0)^2\right)
\end{align*}
Since $\tr(\g)>2$, we have $c_{\tau_{\gs_\fa}(\g)}\neq0$ and
\begin{align*}
\log\left(-j(\tau_{\gs_\fa}(\g),\gs_\fa^{-1}z_0)^2\right) &= 2\log\left(\frac{j(\tau_{\gs_\fa}(\g),\gs_\fa^{-1}z_0)}{i \mathrm{sign}(c_{\tau_{\gs_\fa}(\g)})}\right)= 2\log j(\tau_{\gs_\fa}(\g),\gs_\fa^{-1}z_0)
- i\pi \mathrm{sign}(c_{\tau_{\gs_\fa}(\g)}).
\end{align*}
We conclude that
$$
\int_{z_0}^{\g z_0} E_{2,\fa}(z)dz = \Phi_\fa(\g) - \pi V^{-1}\mathrm{sign}(c_{\tau_{\gs_\fa}(\g)}),
$$
which coincides with the definition of the Rademacher symbol $\Psi_\fa(\g)$ for $\g$ of positive trace.
\end{proof}

Let $\fa_1,\fa_2,\dots,\fa_h$ denote the inequivalent cusps of $\Gamma$. For each integer tuple ${\bf m}=(m_1,\dots, m_h)$ such that
$
\sum m_i = 0,
$
define
$$
E_{2,\fm}(z) \coloneqq \sum_{i=1}^{h} m_i E_{2,\frak{a}_i}(z).
$$
Then $E_{2,\fm}(z)$ has the Fourier expansion
$$
E_{2,\fm}(\sigma_\frak{b} z)\gs'_\fb(z)
= m_\fb -4\pi^2\sum_{n\geq1} n \left( \sum_{i=1}^h m_i\varphi_{\fa_i \fb}(n) \right)e(nz),
$$
from which it can be seen that $E_{2,\fm}(z)$ is a holomorphic modular form of weight 2. In particular, $E_{2,\fm}$ induces a holomorphic differential on $X$ and we conclude our proof of \thmref{thm:1} with the following result of Scholl; see Proposition 2 in \cite{Scholl1986}.

\begin{prop} Let ${\bf m}=(m_1,\dots, m_h)$. In the notation above,
$$
2\pi i E_{2,\fm}(z)dz =\omega_D
$$
which is the canonical differential of the third kind associated to the residue divisor $D=\sum m_i(\frak{a}_i).$
\end{prop}

Let
$$
I_{\bf m}(\g) := 2\pi i \int_{z_0}^{\g z_0} E_{2,{\bf m}}(z)dz
$$
and note that the integral does not depend on the particular choice of $z_0\in\mathbf{H}$. In particular, $I_{{\bf m}}:\G\to\mathbf{R}$ is a group homomorphism and $I_{{\bf m}}(\g)=0$ if $\g$ is elliptic. Considering the Fourier expansion of $E_{2,{\bf m}}$, we see that if $\g=\g_\fb^k$, where $\g_\fb$ denotes the generator of the parabolic subgroup $\G_\fb$, then $I_{\bf m}(\g) = k\cdot m_\fb$. We conclude that the periods of $\omega_D$ are in $2\pi i\mathbf{Q}$ if and only if all Rademacher symbols for $\G$ are rational-valued on hyperbolic elements.

\section{{\bf Proof of Theorem 2}}\label{sec:rationality}

\begin{lm}\label{lm:simplification}
If $\G_1<\G$ is a subgroup of finite index, then each Rademacher symbol $\Psi_\fa$ is rational if its restriction to $\G_1$, i.e., $\Psi_\fa\vert_{\G_1}$ is rational on hyperbolic elements.
\end{lm}

\begin{proof}
By (\ref{qm}), we have that for any $\g_1$, $\g_2\in \G$,
$$
\pi^{-1} V\Psi_\fa(\g_1\g_2) \equiv \pi^{-1} V(\Psi_\fa(\g_1) +\Psi_\fa(\g_2))\qquad (\text{mod }1),
$$
where we recall that $\pi^{-1} V\in\mathbf{Q}$. We write $\G=\cup_\tau \G_1\tau$ as a disjoint union of $\G_1$-cosets. Since $\G_1$ has finite index, there is some $n\in\mathbf{N}$ such that $\tau^n\in\G_1$. By the recursion formula (\ref{recursion}),
$$
\pi^{-1} V \Psi_\fa(\tau^n) \equiv n\pi^{-1} V \Psi_\fa(\tau)\qquad (\text{mod }1).
$$
Each $\g\in\G$ can be written as $\g=\g_1\tau$ for some $\g_1\in\G_1$ and some coset representative $\tau$, so that
$$
n\pi V^{-1} \Psi_\fa(\gamma) \equiv \pi V^{-1}(n\Psi_\fa(\g_1) + \Psi_\fa(\tau^n))\qquad (\text{mod }1).
$$
By \propref{prop:3}, Rademacher symbols are always rational on parabolic and elliptic elements, hence it suffices to check that $\Psi_\fa\vert_{\G_1}$ is rational on hyperbolic elements.
\end{proof}

It immediately follows that Rademacher symbols on genus 0 Fuchsian groups are rational. To obtain new classes of examples among Fuchsian groups of genus $g\geq1$, we build on the following relations along normal covers.

\begin{prop}\label{proposition}
Let $\G_1<\G$ be a normal subgroup of finite index, and let $\fa$ be a cusp of $\G_1$ with stabilizer subgroup $\G_{1,\fa}=\G_1\cap\G_\fa$. We denote the Rademacher symbols for the cusp at $\fa$ on $\G$ and $\G_1$ by $\Psi^\G_\fa$ and $\Psi^{\G_1}_\fa$ respectively. For each hyperbolic element $\gamma\in\Gamma_1$ of positive trace,
$$
\Psi_\fa^\G (\gamma) = \Psi^\G_\fa \vert_{\G_1}(\g) = \sum_{\tau\in\Gamma_1\backslash\Gamma} \Psi_\fa^{\Gamma_1}(\tau\gamma\tau^{-1}),
$$
and this formula does not depend on the particular choice of coset representatives. Moreover, if $\fa$ and $\fb$ are cusps of $\G_1$ that are $\G$-equivalent --- i.e., if there is some $\tau\in\G$ such that $\tau\fa=\fb$ --- then for each hyperbolic element $\g\in\G_1$ of positive trace,
$$
\Psi_\fa^{\G_1}(\gamma) = \Psi_\fb^{\G_1}(\tau\gamma\tau^{-1}).
$$
\end{prop}

\begin{proof}
Writing $\G=\cup_\tau \G_1\tau$ as a finite disjoint union of $\G_1$-cosets, we have the formal equality
$$
\sum_{\g\in\G_\fa\backslash\G} j(\gs_\fa^{-1}\g,z)^{-2} = \sum_\tau \sum_{\g\in\G_{1,\fa}\backslash\G_1} j(\gs_\fa^{-1}\g\tau,z)^{-2} =  \sum_\tau j(\tau,z)^{-2} \sum_{\g\in\G_{1,\fa}\backslash\G_1} j(\gs_\fa^{-1}\g,\tau z)^{-2}.
$$
Notice that this equality does not depend on the particular choice of coset representatives. Hence for each hyperbolic element $\g\in\G_1$ of positive trace,
$$
\Psi^\G_\fa(\g) = \int_\g E^\G_{2,\fa}(z)dz = \sum_\tau \int_\g E^{\G_1}_{2,\fa}(\tau z)\tau'(z) dz = \sum_\tau \int_{\tau\g\tau^{-1}} E^{\G_1}_{2,\fa}(z)dz = \sum_\tau \Psi_\fa^{\G_1}(\tau\g\tau^{-1}).
$$
Suppose that $\tau\fa=\fb$, and set $\delta=\gs_\fb^{-1}\tau\gs_\fa.$ Since $\tau\gs_\fa\infty=\gs_\fb\infty$, we have $\delta=\left(\begin{smallmatrix} d &*\\ 0& d^{-1}\end{smallmatrix}\right)$ for some $d\neq0$. We claim that $d=\pm 1$. We have $\tau^{-1}\G_\fb\tau=\G_\fa$ and hence
$$
B:= \gs_\fa^{-1}\G_\fa \gs_\fa = (\tau\gs_\fa)^{-1}\G_\fb (\tau\gs_\fa) = \delta^{-1}(\gs_\fb^{-1}\G_\fb \gs_\fb) \delta = \delta^{-1}B\delta.
$$
Using that $B$ is generated by $\pm \bsm1 & 1\\0&1\end{smallmatrix}\right)$, a direction computation yields $d=\pm1$. We thus have the formal equality
\begin{align*}
\sum_{\g\in\G_{1,\fa}\backslash\G_1} j(\gs_\fa^{-1}\g,z)^{-2}&= \sum_{\g\in\G_{1,\fa}\backslash\G} j(\delta^{-1}\gs_\fb^{-1}\tau \g ,z)^{-2} =  j(\tau,z)^{-2} \sum_{\g\in\G_{1,\fa}\backslash\G} j(\gs_\fb^{-1}(\tau
\g\tau^{-1}),\tau z)^{-2}\\
& =  j(\tau,z)^{-2}\sum_{\g\in\G_{1,\fb}\backslash\G_1} j(\gs_\fb^{-1}\g,\tau z)^{-2}
\end{align*}
where we used $\tau\G_{1,\fa} \tau^{-1}=\G_{1,\fb}$ and $\tau\G_1\tau^{-1}=\G_1$ for the last equality. Thus
$$
E^{\G_1}_{2,\fa}(z) = E^{\G_1}_{2,\fb}(\tau z) \tau'(z)
$$
and the equality of corresponding Rademacher symbols follows.
\end{proof}

Let $\G$ be a congruence subgroup of level $N$. By \lemref{lm:simplification} and \propref{proposition}, Rademacher symbols on $\G$ are rational-valued if Rademacher symbols on $\G(N)$ are rational-valued on hyperbolic elements. The computations of Takada (see the last paragraph of \secref{sec:Rademacher}) verify this for $\Psi_\infty^{\G(N)}$. Since $\G(N)$ is normal in $\G(1)$ and each cusp of $\G(N)$ is $\G(1)$-equivalent to the cusp at $\infty$, we conclude by the second part of \propref{proposition} that each Rademacher symbol on $\G(N)$ is rational-valued.

The groups $\Gamma_0(N)^+$ are obtained from $\G_0(N)$ by adding all (finitely many) Atkin--Lehner involutions and normalizing to matrices of determinant 1 by taking the quotient by a positive scalar. Explicitly, one has
$$
\Gamma_0(N)^+ = \left\{ e^{-1/2} \begin{pmatrix} a&b\\c&d\end{pmatrix}\in\mathrm{SL}_2(\mathbf{R}) :  a,b,c,d,e\in\mathbf{Z},\  e\mid\mid N,\ e\mid a, d,\ N\mid c\right\},
$$
where $e\mid\mid N$ signifies that $e\mid N$ and $(e,\tfrac{N}{e})=1$. It is easily seen that the groups $\G_0(N)^+$ are commensurable to $\mathrm{SL}_2(\mathbf{Z})$ but not necessarily conjugate to a subgroup of $\mathrm{SL}_2(\mathbf{Z})$, that they have a single (inequivalent) cusp at $\infty$, and that $\G_0(N)^+$ contains $\G_0(N)$ as a normal subgroup of index $2^{\omega(N)}$, where $\omega(N)$ is the number of distinct prime divisors of $N$. Once again, by \lemref{lm:simplification} and \propref{proposition}, the Rademacher symbol on $\G_0(N)^+$ is rational if $\Psi_\infty^{\G_0(N)}$ is rational, which we have just established.

This proves \thmref{thm:2}. It would be very interesting to understand under which conditions the rationality of Rademacher symbols for a Fuchsian group $\G$ passes down to its normal subgroups $\G_1<\G$. For instance, finite-index normal subgroups of Fuchsian triangle groups are precisely the uniformizing groups of Galois Belyi curves, i.e.~curves that admit a Galois Belyi map, see \cite{ClarkVoight2019}, sometimes also called curves with many automorphisms, triangle curves, or quasiplatonic surfaces. There are only finitely many such curves for each given genus, but these include much studied examples such as the Fermat curves (for which Manin--Drinfeld is already established \cite{Rohrlich1977}).

\bibliographystyle{alpha}

redpajama_set_name: RedPajamaArXiv


Michael Kennedy Insurance Group has teamed up with Blue Insurance to provide you with the best Wedding Insurance. Having wedding insurance helps you take the stress out of the weeks and months ahead leading up to your wedding day or civil ceremony. You will never have to worry if your dress or suit hire shop goes out of business, photos are damaged and need to be retaken, your reception or wedding venue is double booked or even if you have to cancel or postpone due to illness or an accident.


text:
Q: Setting up queue in Laravel 5 I am trying to run code after returning a HTTP response. I know that Laravel 5 has support for queues, but I still find them confusing. I am trying to run code after the user has registered which requires the user's password and username. This answer seems interesting, but is not directly applicable to Laravel.

*

*How do I create a job in a queue?

*How can I pass data to the new job?

I know this sounds lazy and all, but I really don't understand the documentation.

A: Setting up queues requires, as the very first step, to choose what driver you'll be using. Because it's the quickest to get running, I'll explain how to begin with the database driver, as it doesn't require any other services to be installed on the server (as it's the case for beanstalkd for example). Here's how to get this set up:
1. Set the QUEUE_DRIVER in your .env file:
QUEUE_DRIVER=database

2. Run this command to generate the migration file for the jobs table, that will be used to store job information:
php artisan queue:table

3. Now run the migration to create the table:
php artisan migrate

A jobs table was created that will store data when jobs are pushed on the queue.

You can push both commands and clojures onto queues. For the sake of brevity, I'll show an example of how to push a closure onto a queue:
$username = Request::input('username');
$password = Request::input('password');

// Do your registration stuff

// Push a job onto the queue
\Queue::push(function($job) use ($username, $password)
{
    // Do the stuff you need here with $username and $password

    // Delete the job from the queue
    $job->delete();
});

The final step to making this work is to run the queue listener. Jobs will not be processed automatically unless the queue listener is running. So run this command:
php artisan queue:listen

There are further steps you can take, such as setting up Supervisor to monitor and restart the queue listener should it crash, but this should be enough to get you started.

A: Generally we pass data on the queue like this -
On the controller we have written -
$this->dispatch(new videoToAudioConvert($video_id))

On the job section you have to write like this -
protected $video_id

public function __contructor($video_id){
  $this->video_id = $video_id
}

public function handle(){
  $this->video_id
}

You can get more idea how to create jobs in queue and how to pass variable from here.

A: what is the requirement data to store in jobs table
i use this to send email with queue and i will Schedule it
i do the first 3 steps.
 Set the QUEUE_DRIVER in your .env file:
QUEUE_DRIVER=database
2. Run this command to generate the migration file for the jobs table, that will be used to store job information:
php artisan queue:table
3. Now run the migration to create the table:
php artisan migrate
A jobs table was created that will store data when jobs are pushed on the queue.

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text:
Q: Rewrite URL in PHP without htaccess The website is running on a web hosting where we don't have access on htaccess file.The web server is Apache. I want to rewrite URL. For example: The original URL:
www.mydomainname.com/en/piecework/piecework.php?piecework_id=11
Expected URL:piecework.mydomainname.com/en/11
How to do it?

A: Sadly without htaccess or apache config access, you won't be able to do this. Bug the heck out of your host or get a new one. Most of the cheapo shared hosts out there offer mod_rewrite from my experience.

A: I've seen people tokenize the query string to accomplish (sort of) what you are talking about, i.e.:
domain.com/foo/bar/11

Is actually interpreted as:
domain.com/foo/bar/11.php?action=foo&query=bar

However you have to create these semantics in the physical directory structure. This is ugly, unruly to manage and much easier accomplished with mod_rewrite.
You probably have mod_rewrite enabled, here's a short tutorial to get you started.

A: This code might be what you looking for:
Usage: phpfile.php/en/11
Note: no mod_rewrite required.
<?php

$DEFAULT_ACTION = 'badAction';

$pathInfo = isset($_SERVER['PATH_INFO']) ? $_SERVER['PATH_INFO'] : '';
$pathParts = explode('/', substr($pathInfo, 1));
$action = isset($pathParts[0]) ? $pathParts[0] : $DEFAULT_ACTION;

if (function_exists($action)) {
    $action($pathParts);
} else {
    badAction();
}

function badAction($parts = null)
{
    print 'The function specified is invalid.';
}

function en($parts)
{
    $name = isset($parts[1]) ? $parts[1] : 'undefined';
    $xml = new SimpleXMLElement('<message>Number: ' . $name . ' </message>');
    header('Content_Type: text/xml');
    print $xml->asXML();
}

?>

ps. I found the code in 'Professional PHP6' ISBN:978-0-470-39509-7

A: You could actually physically make folders/files (so the path being accessed actually physically exists) and then redirect/handle appropriately in those - unwieldy but if it's your only choice....
Another possibility would be seeing if your host will allow you at least a custom 404 error document and then handling everything in there based on the REDIRECT_URL.


text:
Samsung Electronics to acquire US network service provider
By Shim Woo-hyun
Samsung announced Tuesday that it will acquire TeleWorld Solutions, a Virginia-based network service provider.
(Yonhap)
TWS specializes in providing network designs, installations, testing and optimized solutions to mobile service and cable operators as well as original equipment manufacturers.
With the latest deal with TWS, Samsung Electronics is expected to increase its presence in the network infrastructure market as an end-to-end service provider.
The South Korean company has been supplying network equipment to large-sized mobile carriers in the US, including Verizon, AT&T and Sprint.
"The acquisition of TWS will enable us to meet mobile carriers' growing needs for improving their 4G and 5G networks, and eventually create new opportunities to enhance our service capabilities to our customers," said Cheun Kyung-whoon of Samsung EVP.
TWS, a privately owned company, will become a wholly owned subsidiary of Samsung Electronics America, the company said. Financial details of the agreement have not been revealed.
By Shim Woo-hyun (ws@heraldcorp.com)

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At Journey, we really do believe that Growing brings Change. We want to be a part of helping you grow to be more like Jesus, so we are committed to curating resources to help you in your own walk.
Below you'll find books, websites, and other resources that will help you grow in your faith. They are organized in different categories of interest, so look around and check out some of these great resources that will help you on your walk.


text:
Rate the last thing you read
By Batman
November 29, 2005 in Random Thoughts
I finished "Dialogues on Fundamental Questions of Science and Philosophy" by A. Pfeiffer on the tram today. I enjoyed it, and wish people more people would incorporate reason into their arguments. An enlightening discussion between 2 characters that weaves philosophical theories into science with the aim of arriving at the answer to the question of ethics. I saw an echo of my own growing-up mentality with regards to socialism i.e. well it's a good idea but what do we do with those pesky socialists?
edna 70 posts
The Seeker 61 posts
Shawna 61 posts
Jenny 58 posts
night2day
"Dialogues on Fundamental Questions of Science and Philosophy"
Whoa, that's some heavy reading! How much does it weigh on your mind?
I feel my eyes didn't do it enough justice! I've stopped reading anything 'light' because I find it reflects in my thesis writing which is a badthing. But I do need to read stuff that's not thesis-related though!
It's quite a thin book though - ten dialogues, about a dozen pages each. It's also quite interesting. Kantian philosophy applied (or not applied) to animal behaviour. I like these kinds of books for how they don't make a distinction between science and the humanities, because really you can't have one without the other.
ha ha! Your words are true. It seems reading has rubbed off on you. That's good!
Now we just hope that our dear Rad-ji rubs off on us.
Well, she can rub on me anytime!
Farin
well it's a good idea but what do we do with those pesky socialists?
haha, true - and well put
TheLizard
Kantian philosophy applied (or not applied) to animal behaviour.
What? How? That's... strangely awesome.
I was going to ask the same question (only forgot to click 'post')
so you mean that animals follow the categorical imperative?
Actually, I am quite disappointed in how both parties seem to be united in their disagreement against Kant's principles. I quite like the guy
No, both chaps seem to agree that the categorical imperative and pretty much anything Kant said was stuff and nonsense. The apply it to animal behaviour by discussing what makes human actions superior, if anything does. Kant, apparently (go easy on me, I've only read a couple of essays), believed that "action engendered by any directly experienced drive must necessarily be ethically inferior". Of the two talking, one believes that 'the striving for good indeed exists as an original impulse of the human heart', while the other refutes Kant, AND this guy, by saying 'Surely you do not assume that anything which distinguishes man from rest of the living world must needs be considered different in principle from the rest of nature?' - and gives the example of language, to prove how you can't free human beings' uniquer traits from the laws of biology. From this we launch into a discussion about if/how the reward motive guides behaviour which is one of my favourite bits because it's all about Pavlov.
Right at the start, though, things are even cooler. They use Kant here too, but apply his theories to quantum physics. It's quite an old debate actually, but it's what the book starts with - do elementary particles exist objectively? do they have individuality?
Bit tough to condense all this into a post, especially after just one reading, but hopefully it wasn't terribly anticlimactic
Edited November 25, 2011 by Guest
one believes that 'the striving for good indeed exists as an original impulse of the human heart'
Which should leave to question as to what is "good". Good for whom? For himself? Or the community/society/humankind?
Coincidentally (and a bit off topic) just yesterday I saw a discussion by another philosopher, who claimed that one of the worst things that could be done in upbringing or educating small children is to give them rewards for good behaviour. Because eg if you drop something, by nature every 1.5 to 2 year old child would help you picking it up again (after you gave them a prompting look, maybe). But if you'd give them a piece of chocolate every time they'll do it, what would you think will happen if you'd then stop doing it?
I just read Sammy's chili-cookoff story in the "Jokes Relaunched" thread. 10/10 I laughed so hard I snorted.
I believe for Kant it would be for humankind as a whole. But selfish gene theory puts an end to that for both humans and animals. Not that altruism doesn't exist in selfish gene theory, but I'm not sure you'll see species-wide altruism in any animal. I don't even think people are the closest to exhibiting it, bees and ants are more altruistic than we are due to their unusual genetics.
Wow...just...wow!
Things that are said at Songfacts never cease to amaze me.
^^so you don't agree then?
Remember that the quote from Levis' post comes from the guy who disagrees with Kant.
And I admit I'm not so versed on such specific biological theories (maybe I need to read more into that topic), I guess it could make sense from an evolutionary viewpoint. If I contribute to the survival of my species as a whole, even if it doesn't help myself directly, it still can be beneficial for my genes in future generations. Obviously this is especially true when I'm helping close relatives.
But I think you're right that, compared to ants, a general group of human probably wouldn't form a raft to save their queen from drowning
Eusociality
Take a look at the genetics table. It goes a long way towards explaining that sort of behavior.
oh "bee"have! "Where have all the young bees gone?" Gone to honeycomb, every one.
When will they ever learn? x2
Well all this is the ethical question isn't it? What is 'good', does it have an absolute value, can it be defined, can a universally applicable code of ethics be created etc.
The book doesn't answer the question obvs, but neither party manages to cover my POV, either. The person who DOES seem to be on my kind of track is Plato whose Republic I'm reading for the very first time (shhh, don't tell me what happens!). It's full of views that's all too easy to distort because everyone forgets it was written in 380 BC (even I had to look that up) when in actual fact it's remarkable that so much of what he says can be applied to the present state of society. BUT I digress...
Like the Republic, the problem with people's idea of 'reward' is that they see at as a physical concept without taking into account that it might be motivated by an internal mechanism. You don't have to reward a child with candy when s/he helps you with something, a smile and a thank you is positive reinforcement on its own. It's one that's likely to keep occurring when the child grows up and interacts with strangers. In that way human beings ARE trained to reinforce 'good' behaviour so maybe there is an innate set of 'rules' that govern ethical behaviour. IMO altruism, 'good' behaviour etc. are motivated by pursuit of reward/avoidance of punishment.
Game theory spends a lot of time working out the mechanics of altruism, and in people at least the behaviour isn't based on such pure thoughts. Turns out people with damage to that part of the brain which governs their perception of long term consequences are more likely to take less altruistic decisions. Probably reflecting a belief in some form of karma i.e. if you're cooperative, you're likely to receive assistance should you need it in the future.
The saving-lives kind of altruism I look at from the same perspective - avoidance of the guilt that is the result of inaction. I doubt many people rush into a dangerous situation KNOWING they're going to die and without the faintest bit of hope of both sides surviving.
As for insects, I figure it's just conformity to a different set of rules to those that govern human societies.
O, but I do go on
LOL...on you do go.
"But you are Blanche, you are, you are in that chair!"
(Betty Davis replying to Joan Crawford) in "What Ever Happened to Baby Jane?"
Bing Crosby walking through a cow pasture exclaimed and crooned
Lot a do-do
Game theory spends a lot of time working out the mechanics of altruism, and in people at least the behaviour isn't based on such pure thoughts.
Good point - The prisoner's dilemma actually shows that people can actually NOT act altruistic (or just cooperative in this case), even if it's clear that it would be best for the group as a whole to do so.
That's an interesting thought, one I'm not sure whether to agree or disagree to.
On the one hand it's quite a bit cynical, and if everyone would rather not help others, why should they feel guilty for not doing so? Why is it a social(?) or biological(?) norm that there would be guilt to be felt?
On the other hand there ARE these cases that the more people are witnessing a person in despair, the less likely it is for anyone to actually help, maybe because if no one else is helping there's less guilt to feel if an individual doesn't help either, as he's just one of many. (I think that's called bystander phenomenon or so)
So maybe you're spot on with your explanation?
PS. night2day, UncleJoe, what's your take on this?
The person who DOES seem to be on my kind of track is Plato whose Republic I'm reading for the very first time
His politics and ethics are certainly interesting, but wait until you get to his metaphysics. His theory of forms is... I guess I'll just say outdated.
Then again, I find most attempts at metaphysics to be a silly waste of time.
If my memory serves; was it not Plato that said the following?
Everything that deceives may be said to enchant.
Remarkable philosopher he was!
n2d, is your name Marcus, by any chance?


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Sometimes something is so cool you thought it could never exist. So cool that the universe simply wouldn't allow it. So cool that if it ever did exist bad things would happen like the laws of physics would stop, the sun would super nova, fire and brimstone coming down from the skies, rivers and seas boiling, dogs and cats living together… Okay you get the picture. Well here at R Brand we decided to throw caution to the wind and risk all of that.
We proudly present to you the all new, totally cool, fully functional (well as a guitar anyways) Police Box Guitar.
We came up with this baby as our way of commemorating the 50th anniversary of the longest running Sci-fi show in television history. I think you know the one. As you can see from the pictures below its based on the classic British Police box from the 1950's, and sports a beautiful Gallifreyan design on the neck.
PS. This product is not endorsed by or in any way associated with the BBC, Doctor Who, or the TARDIS. Any similarity is purely umm accidental.
WOW yes got one !! Who wouldn't want this great build ? 😉 can't wait to get this in .
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HARIBO United Kingdom
Careers at HARIBO
Contact information & FAQ
The Goldbear answers
Can you tell me about your recruitment process?
Our recruitment process can vary depending on the role.
For our office based opportunities, the recruitment process involves a first stage interview, and in some cases a second stage interview, which may include an aptitude test depending on the role that you are applying for. There may also be occasions where we ask you to do a short presentation on a topic related to the role.
For our operations based opportunities, the first stage of the recruitment process is a series of multiple choice ability tests that are aimed at assessing a candidate in the role and work environment and also predicting job success.
This is then followed by an interview with the Supervisor and/or Department Manager.
Can I apply directly or do I have to do this online?
We welcome applications made online, as we have found that this is the most effective method of ensuring that we have received all applications for consideration. Please complete the online application and submit your CV for the role that you would like to apply for… we don't accept paper applications anymore, so get typing and we will look forward to hearing from you!
Absolutely! If you are interested in joining Haribo then you have the option to "Express Your Interest" on our website. If you head over to our website and follow the links, then you can upload your CV and be considered for opportunities that may arise in the future.
Do you use recruitment agencies?
We have a number of roles which we prefer to recruit working with our external recruitment partners. This is to ensure that the specialist knowledge and skills can easily be found. We already have an agreed list of preferred suppliers that we use.
What training opportunities do you offer?
We pride ourselves on the development offering that we have at Haribo. The development of our colleagues and their progression through the business and departments that they work in is extremely important to us. We have training that is relevant for all our roles and each year development plans are discussed and agreed with our colleagues focussing on increasing their skills and experience in order that they are given opportunity to progress their careers with us.
We are also proud to be able to offer university and apprenticeship based courses working with local colleges, where colleagues develop their skills for future progression.
In the UK we employ circa 750 colleagues. This changes on a regular basis as we continually expand our business. Haribo UK are part of a global organisation, employing over 7,000 colleagues worldwide across 26 countries.
How many departments do you have?
We are a progressive business and all our departments and functions work together to make our fantastic products, which are loved by kids and adults alike. Operations based departments include: Production, Packing, Despatch and Logistics, Engineering, Quality and Security.
Office based departments include: Finance, Marketing, Category, Design, Sales, Human Resources, Import / Export, Demand & Forecasting, Planning, Health & Safety, IT and Procurement.
We have opportunities for a wide range of specialisms and disciplines!
What benefits do Haribo offer?
Haribo offer some great benefits to all their colleagues. These include:
28 days holiday per year (FTE), plus statutory holidays
Annual bonus scheme
Contributory pension scheme
Staff discount in our company shop
Occupational sick pay scheme
Various local benefits offered by external companies
* Please note that some benefits are only applicable for certain roles
What does a typical career path look like at Haribo?
Here at Haribo we love to progress from within where we can, working with dedicated career paths for all roles within the business. We believe in the retention of our colleagues and their knowledge is key to the success of our business. We are proud of the length of service history of our colleagues!
Many of our Supervisors started their careers with us and have successfully worked their way through the organisation to reach their current position. Although we have a broad organisation structure, we have a wide range of opportunities across different disciplines and at different levels in the business for colleagues to consider.
What are your working hours?
Our working hours are:
Operations - 40 hours per week, Monday to Friday working 06:00 to 14:00 and 13:55 to 21:55, alternating weekly.
Overtime is available during our peak times and is offered to colleagues on a voluntary basis.
Office based - 38 hours per week, working 08:00 to 17:00 Monday to Thursday, and 08:00 to 12:00 Friday.
HariboTwitter
© HARIBO Dunhills (Pontefract) PLC | Imprint | Terms of use | Privacy | Code of Conduct | Informant system


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G. Hannelius 02/02/2023
Caylee Cowan at All Time Restaurant in Los Feliz 01/31/2023
Paige Spiranac 02/02/2023
Miley Cyrus Jokingly Calls Billy Ray 'Worst Dad Ever' Over Head Injury When She Was 2
September 3, 2020, 8:00 AM 283 Views
Miley Cyrus has playfully trolled her country singer father. When discussing mental health during a podcast interview, the former star of "Hannah Montana" jokingly called Billy Ray Cyrus "worst dad ever" for causing her to sustain a head injury at only two years old in a "dirt bike" accident.
In the Wednesday, September 2 episode of "The Joe Rogan Experience" podcast, the "Wrecking Ball" hitmaker opened up about scans of her brain and her childhood experience with host Joe Rogan. At some point during the chat, she talked about her 59-year-old father and plainly stated, "I've given him an award for worst dad ever."
Explaining why she made such a comment, the ex-wife of Liam Hemsworth shared a shocking incident that took place when she was young. "I had a head injury when I was, you know, 2 years old. It's bad," she began recalling. "My dad had me – this is really bad, but he can't go to jail, I don't think, because it's a long enough time away."
"He had me in a baby backpack and I was on a dirt bike with my dad," the 27-year-old offered more details on the accident. "And he was riding, and a tree had fallen, and he ducked, I didn't, and I hit my head on the tree. It was bad. So that's what's wrong. Everyone's asked me that for years."
The "Slide Away" singer went on to speculate that the injury affected her personality. "Maybe it knocked me into this identity, or something," she pointed out, claiming that her therapist Dr. Daniel Amen shared a similar idea. "He knows that there's something to that, "she stated. "Dr. Amen, we've talked about that a lot."
"So when I get really overwhelmed," the former Disney darling elaborated, "I also have a tendency, when I know something's really stupid, I just gotta try it, to know that it's stupid, which makes it stupid, 'cause I already knew about it. Sometimes I'm like, is it better to know it's dumb and do it? Or not know it's dumb and do it?"
Though branding Billy Ray the "worst dad ever," Miley unhesitatingly declared that the "Achy Breaky Heart" hitmaker was "also the best." In the interview, she told Joe that she was grateful he allowed her to do a lot of crazy things and made her tough.
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The ambulance services listed below are members of the Professional Ambulance Association of Wisconsin and currently receiving benefits of membership.
With its exclusive focus on emergency medical services (EMS) and the ambulance industry in Wisconsin, PAAW is in an extremely fortunate position. The association brings together leaders and managers from across the state operating in a variety of ambulance service delivery models varying in number of patient contacts per year, population and demographics, billing and collections, staffing and volunteers, and a variety of patient calls.
Not a PAAW member? Refer to >> PAAW member benefits for more details.
Each member ambulance service or group in good standing receives one vote at the annual meeting for board of directors, officers, honorary lifetime members, and is eligible to hold a position on the board of directors or be elected an officer of the association. Organizations that join PAAW as professional members, include: ambulance services, air medical operations, fire departments, medical first responder groups and other eligible groups. Each person holding voting privileges with their respective organization is listed below.


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Q: Reading and Updating data of Oracle DB without License I've got to update the data of an Oracle DB but I'm not the owner and ain't got any Oracle License.
The goal is to explain to my interlocutor how to create a dump from his Oracle DB, to find out how to restore this dump in a DB (a free version of Oracle or something else), update some data in some tables, and then make another dump to send it back to my interlocutor.
So the differents questions I have are:
1- Is it possible to create a dump (maybe in SQL format) without any specifics dependencies to Oracle ?
2- Is there a way to restore this dump in a free lightweight Oracle, or another kind of DB like Postgresql ?
3- Does Oracle, is able to handle any kind of dump et restore it in an Oracle DB or is there any constraints to respect ?
I am very new to Oracle and ain't got, on my personal computer, any possibility to try out the dump/restore by myself; that's why, any help will be appreciated !

A:
1- Is it possible to create a dump (maybe in SQL format) without any specifics dependencies to Oracle ?

Oracle offers Data Pump utilities (export and import) for such purposes. You'd export table (or schema) - result is a ".dmp" file, readable by import utility. You'd then move that file to your own server (see #2 for the rest)

2- Is there a way to restore this dump in a free lightweight Oracle, or another kind of DB like Postgresql ?

On your own server (which could be a laptop; no problem), you'd install a free Oracle Express Edition (XE) database. Currently, the last version is 21c, but some others should still be available in Oracle Technology Network's Download section.

3- Does Oracle, is able to handle any kind of dump et restore it in an Oracle DB or is there any constraints to respect ?

XE database has its limits - from your point of view, the most important is that it can handle up to 12GB of user data. Therefore, if the .dmp file doesn't contain more data than that, you should be able to import it.
Another constraint is the compatibility. Not all exports can be imported into all databases. There's a matrix which shows which versions match; it is available on My Oracle Support site (but you have to have access to it, which you - as you said - don't. Though, generally speaking, "close" Oracle database versions can interchange .dmp files. It would be best if these two database versions match, of course.


R&D Tax Credits Bolton. We provide expert R&D tax credits advice and support to limited businesses of all sizes. R&D Tax Credits are an incentive operated by the government to compensate Ltd. companies participating in trying to push the boundaries of what is possible within their industry.
R&D Tax Credits were first announced in 2000, so in the case your organization has invested in its own advancement over this time and you're yet to claim, you may have now lost tens, if not hundreds of thousands of pounds in lost tax relief since you can only retrieve business spend put down over the last two-open tax years.
How can they help your organization?
Small businesses like yours can get back in between 25-33% of their eligible investment costs back again any time they've been investing in developing their firms products, devices, processes, systems etcetera and may also benefit from decreased corporation tax liabilities moving forward.
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R&D Tax Credits are A top hmrc incentive designed to reimburse companies for working to secure an advance inside their sector.
Wherever your company is functioning in profit, you may expect a £25,000 reduction away from your next tax bill for every £100k of eligible pay out uncovered. When your company has a loss, it could surrender this loss to secure an almost immediate cash rebate from HMRC worth £33,350 for every £100k of qualifying spend identified.
Many organisations are nevertheless not aware that many of their ongoing activities could easily constitute qualified R&D and that after they appreciate as much as 100% of people's earnings are generally included, that the quantities rapidly stack-up to their benefit.
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Any time you and/or your staff (this includes sub-contractors) are attempting to settle technical issues with an undertaking, even if the work doesn't achieve the desired end result you will still possess the ability to claim an apportionment of money used up back again. What is crucial is that you are attempting to achieve an advancement as this will further your understanding, experience and skills going forward.
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Mineral Resource of the Month: Boron
Boron does not occur in nature in an elemental state. It is always found in oxidized states as borate salts. Four mineral forms account for the majority of the economically viable borate deposits throughout the world: two sodium borates, tincal and kernite; a calcium borate, colemanite; and a sodium-calcium borate, ulexite. Borate deposits are formed in areas that are tectonically active with arid climates. Large deposits are found in the Mojave Desert of the United States near Boron, Calif., the Tethyan belt in western Asia and the Andean belt of South America. Most borates are extracted in California and Turkey.
Read more about Mineral Resource of the Month: Boron
December 2018 - Table of Contents
Read more about December 2018 - Table of Contents
Night lights reveal that people move as rivers rise
Flooding is one of the most damaging natural hazards, so it's no surprise that many people resettle farther from rivers after catastrophic flooding. A new study finds that the distance that people move from a given river depends on the degree of flood protection in place, with people rebuilding closer to rivers that offer high levels of structural flood protection, such as levees and dams, than they do to rivers without such protections.
Read more about Night lights reveal that people move as rivers rise
Down to Earth With: Coastal scientist Gary Griggs
Maybe it was the summer camping trips with his family along the Pacific coast, or perhaps surfing off Santa Barbara, Calif., during college, but Gary Griggs always gravitated to the ocean. He turned that love into a career and has spent the last 50 years teaching about the oceans and coasts at the University of California, Santa Cruz (UCSC).
Read more about Down to Earth With: Coastal scientist Gary Griggs
Survey says: U.S. lakes getting murkier
A study of more than 1,000 lakes across the United States between 2007 and 2012 found that the number of murky green and brown lakes surpassed the number of clear blue lakes.
Read more about Survey says: U.S. lakes getting murkier
Geologic Column: Light amid the darkness: Celebrations during the winter solstice
The winter solstice falls on Friday, Dec. 21, marking the arrival of winter and the shortest day of the year in the Northern Hemisphere. For at least 5,000 years, humans have celebrated this celestial event with festivals of light and ceremonies of renewal.
Read more about Geologic Column: Light amid the darkness: Celebrations during the winter solstice
"Critical Minerals" list snubs copper, sparks discussion of criticality
In 2017, the United States relied on imports for more than half the needed supply of 50 mineral commodities that are critical in manufacturing. Some of the commodities come from just a few major suppliers — especially China and Canada — or are produced in tiny quantities in just a few places. Others come from conflict zones. Some are produced only as byproducts of processing ores of major metals, such as copper and zinc. Such complex international supply chains are at risk of being disrupted by a variety of problems, from trade wars and market volatility to natural disasters and terrorism.
Read more about "Critical Minerals" list snubs copper, sparks discussion of criticality
Bubbly discovery may impact volcanic hazard assessment
Programs that manage volcanic hazards use a variety of tools and techniques to monitor impending eruptions. But researchers recently found evidence — in the form of gas bubbles bursting out of Hawaii's Kilauea Volcano — suggesting that scientists should forgo one common method for assessing hazards from basaltic volcanoes: averaging gas composition measurements.
Read more about Bubbly discovery may impact volcanic hazard assessment
Revealing the ghosts of glaciers past
On the ground, it's hard to fathom the effects that glaciers once had on the now nearly ice-free Rocky Mountains, but from the air, the landscape is clearly haunted by the ghosts of glaciers past. In 2015, pilot and photographer Garrett Fisher set out in his antique 1949 Piper Cub aircraft to photograph all the remaining glaciers in the American Rockies.
Read more about Revealing the ghosts of glaciers past
Two new species fill gap in dinosaur family tree
Cretaceous rocks in northwestern China have yielded two new dinosaur species that help fill a 70-million-year gap in dinosaur phylogeny. The new species — Xiyunykus pengi and Bannykus wulatensis — are both alvarezsaurians, an odd group of dinosaurs that, by the Late Cretaceous, had evolved many avian characteristics such as birdlike skulls, tiny teeth and light, slender bodies, as well as unique mole-like single-clawed forearms that were likely useful for digging.
Read more about Two new species fill gap in dinosaur family tree


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Star Tours to be Divided into Two Paths
By: Adam F Manfredo
With the arrival of the new episode, Star wars: The Last Jedi, Star Tours will see a switch similar as when The Force Awakens came out. It will only feature segments that involve the most current trilogy, instead of the random sequences that can take you anywhere in the Star Wars universe. After the the buzz of the The Last Jedi wears off, the attraction will switch to two separate ride experiences that you will have to choose a version to ride (Similar to Mission: Space). One version will feature the episodes 1-6, and the other will feature episodes 7 and 8 (Eventually episode 9, and who knows).
There is no set date on when there will be two separate experiences, but beginning November 17th, The Last Jedi version will be taking place in both Disneyland and Walt Disney World Resort.
Keep dreaming here for more updates!
Source: WDWNT
Previous Secret Sundays: Learn to Draw Disney Characters at Animation Academy
Next From the Vault: Wonders of Life and Cranium Command


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Q: Abstract Binary Search In the article http://community.topcoder.com/tc?module=Static&d1=tutorials&d2=binarySearch,
he says

Careful readers may note that binary search can also be used when a
  predicate yields a series of yes answers followed by a series of no
  answers. This is true and complementing that predicate will satisfy
  the original condition. For simplicity we'll deal only with predicates
  described in the theorem.

I couldn't get what he meant could some one please explain?
Thanks

A: Imagine you're doing a binary search on a set of numbers: for the search to work, you need to put the numbers in order, so that the question "is this number less than the number I'm searching for" gives yeses followed by nos.
Example: searching for number 8 in the sequence [1,1,2,3,5,8,13,21]

*

*is 1 less than 8 ? "yes"

*is 1 less than 8 ? "yes"

*is 2 less than 8 ? "yes"

*is 3 less than 8 ? "yes"

*is 5 less than 8 ? "yes"

*is 8 less than 8 ? "no"

*is 13 less than 8 ? "no"

*is 21 less than 8 ? "no"

This means if you looked at, say the middle number in the sequence, you could tell instantly whether your target number was before or after this mid-point (if you get a 'no' look before, if you get a 'yes' look after). You can then exclude the unwanted half of the series and repeat the process with the remaining half...
This way of halving the search field at each step is the key to binary search, and guarantees you will find the target in O(log n) time.
Looking at the second part of your paragraph:

complementing that predicate will satisfy the original condition

To complement the predicate means to swap 'yes' and 'no', which would give us 'a series of no answers followed by a series of yes answers', which is referred to in the previous paragraph (the original condition).
So in summary, your quote is saying 'yes followed by no' will work equally as well as 'no followed by yes'

A: He is talking about formal logic, and using terms from formal logic.
Directly from the article:
"Behind the cryptic mathematics I am really stating that if you had a yes or no question (the predicate), getting a yes answer for some potential solution x means that you'd also get a yes answer for any element after x. Similarly, if you got a no answer, you'd get a no answer for any element before x. As a consequence, if you were to ask the question for each element in the search space (in order), you would get a series of no answers followed by a series of yes answers."
I think you're going to need to use some elbow grease and buff up on those terms. I'm not sure which part you're having trouble with.


I mentioned yesterday that the painting project has turned into a bigger project then I had originally anticipated. Now Greg wants to remove all the paint from the woodwork in the room. The paint is probably lead-based. We plan on having children in the house so we would like to take the time to remove as much as the paint as possible. I honestly wasn't that freaked out about the paint. Lead paint's biggest danger is one it crumbles or cracks. Children could eat the chips which could cause permanent brain damage. Many people remove lead based paint from old windows. When you constantly open and close the windows small amounts of the paint gets worn away which could then cause the contaminated substance to enter the air. We have new windows so we don't have to worry about that part, yeah!
So to be super safe we are taking the time now to remove the paint all around the house. Below are some photos of the process. We found a pretty little flower detail hidden on the mantel last night (below). I'll post a little info on how the stripping process works this afternoon.
Doesn't sound creepy. I've actually been getting a lot e-mails from people that have lived in marblehead!
i found your blog by doing a google image search for tiny floral wallpaper.
now get ready, this is going to sound creepy.
your house looked SO familiar, i knew it was from somewhere near my hometown of beverly, ma. it wasn't until i saw the "marblehead postcards" sign in a storefront window that i figured it out. i've probably driven by it many times.
your home is beautiful already and i'll be checking your blog to see the progress. it's inspiring me to consider moving home when i'm ready to settle down.
How fun!! Can't wait to see the progress.


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Normally I am a big AVG fan and I have been getting the business anti-virus 10-key licensing for almost a decade now.
AVG Chat Support: Welcome to AVG.
You: Can you do an easy accross-all-platforms change of the email address to something like obfuscated?
You: And forward the complaint for me? The whole idea was that avg@unistellar.com was for official company to company communications between us and AVG. I suppose someone could brute-force guess "avg@" but none of the other addresses are getting this trojan. It is like the perpetrator knew that it was our most trusted anti-trojan relationship, you know what I mean?
You: Are you able to make the email change easier? I have shut down avg@unistellar.com and turned on obfuscated for you.
This entry was posted in Front Page on April 19, 2012 by reskin.


text:
Download Song O Holy Night By Mariah Carey
How Does Catholicism Contradict Scripture May 05, 2016 · Some of Pope's comments contradict teachings of Bible. As the current, worldwide spiritual leader for over 1 billion Catholics, God holds Pope Francis responsible to: "Cry out, spare not, lift up thy voice like a trumpet, and show my people their transgressions. ," just as Isaiah warned the Israelites in his day
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This year, forget "Silent Night" and "O Holy Night" (and literally any other boringly slow song with the word "night" in. 12 'All I Want for Christmas Is You' by Justin Bieber and Mariah Carey.
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The first are Christmas-themed songs without any religious connotation (other than the word Christmas, I guess?). The second are the religious tunes like "O Holy Night" and "What. and bopping along.
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In the city where I live, one of the pop music radio stations shifts to an all. I think of a version of "O Holy Night" released over a decade ago by one of Mariah Carey's early imitators, Christina.
The newest song to crack the top 10 on Pinnacle's "adult contemporary" and "adult top 40" panels was Mariah Carey's "All I Want for Christmas. Patsy's "Grandma Got Run Over by a Reindeer"; and "O.
R&B star John Legend is getting into the Christmas spirit a few months early for his first festive album. The All of Me hitmaker has been busy revamping traditional songs for the holidays, and his.
In 1906, "O Holy Night. Then the song writer Ray Evans told his wife about it, who responded "Are you out of your mind? Do you know what the word tinkle is?" 26. There are three, count them, three.
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ANDREW RICHARDS
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Military Service Questionnaire
The Greatest Generation?
The term 'Greatest Generation' has been used to describe the generation that grew up during the depression and fought through the Second World War. Tom Brokaw's book gives credit to a generation that saw huge changes and overcame adversity.
I would never question that the GI generation, born between the turn of the 20th Century and 1925, do not deserve the title. But In a decade or two, I think, the generation who have had to deal with everything that has happen since 9/11, especially those who have served their country, might be be considered the 'Greatest Generation 2.0.'
I am reading a book written by an Army buddy of mine, Mick Flynn. In Trigger Time, Mick's second book, he details the life of soldiers in Afghanistan during two tours in 2005 and 2007. Mentioned in the book, are several soldiers that I had the pleasure of serving with. When I knew them, they were young guys, just starting their Army careers.
After 9/11, the world changed for them. The fight against terrorism meant that they were committed to fighting in Iraq and Afghanistan. Politics aside - they went and did their job under the most exacting conditions, under the most rigorous scrutiny in this high tech, 24-hour news cycle world. Those guys, that I knew as young soldiers, in Afghanistan and Iraq were hardened war vets and leaders.
Not only has this generation had to fight the Iraq and Afghan wars, for a decade now. But most of them were involved in the Balkan wars, which lasted a decade as well. Although I didn't leave the Army until 2002, I wasn't involved in an operational tour since the first Gulf War. Unlike me, the guys mentioned in Mick's book, were on constant tours throughout the nineties to Bosnia, Kosovo and Macedonia.
There are men serving in my regiment that have known nothing but war - their entire military careers have been one long conflict. In years to come, when the history books are being written, I honestly believe that they will say that this generation was truly great.


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Simon Coates may refer to:

Simon Coates (actor), British actor
Simon Coates (artist), British artist

redpajama_set_name: RedPajamaWikipedia


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This is a picture from Fred McCave's 1984 'The Bulletin'.
Apparently this was in the Admiral Jellicoe.
The 'girls' at the rear were Jack Windell, Doug Scammell, Teddy Cayton, Charlie Neale, John Lawrence, Date Atwell, Cliff Selby and Peter Dawson.
The 'boys' and 'girls' in the front include Jack Fenwick, Pat Wood (Scammell), Vera Bigsby (Lawrence) and Ken Gordon.