---
abstract: 'In the present article, real number representations that are generalizations of classical positive and alternating representations of numbers, are introduced and investigated. The main metric relation, properties of cylinder sets are proved. The theorem on the representation of real numbers from a certain interval is formulated.'
address: |
     45 Shchukina St.\
    Vinnytsia\
    21012\
    Ukraine
author:
- Symon Serbenyuk
title: Generalizations of certain representations of real numbers
---

Let $\mathbb N_B$ be a fixed subset of positive integers, $B=(b_n)$ be a fixed increasing sequence of all elements of $\mathbb N_B$, $\rho_0=0$, $$\rho_n=\begin{cases}
1&\text{if $n\in \mathbb N_B$}\\
2&\text{if  $n\notin \mathbb N_B$,}
\end{cases}$$ and $$\tilde Q^{'} _{\mathbb N_B}=
\begin{pmatrix}
(-1)^{\rho_1}q_{0,1}& (-1)^{\rho_1+\rho_2}q_{0,2} &\ldots & (-1)^{\rho_{n-1}+\rho_n}q_{0,n}&\ldots\\
(-1)^{\rho_1}q_{1,1}&  (-1)^{\rho_1+\rho_2}q_{1,2} &\ldots & (-1)^{\rho_{n-1}+\rho_n}q_{1,n}&\ldots\\
\vdots& \vdots &\ddots & \vdots&\ldots\\
(-1)^{\rho_1}q_{m_{1}-1,1}& (-1)^{\rho_1+\rho_2}q_{m_{2}-2,2} &\ldots & (-1)^{\rho_{n-1}+\rho_n}q_{m_{n},n} &\ldots\\
(-1)^{\rho_1}q_{m_{1},1}  &    (-1)^{\rho_1+\rho_2}q_{m_{2}-1,2}        & \dots &                        &  \dots\\
                         &  (-1)^{\rho_1+\rho_2}q_{m_{2},2}                       &  \dots &                & \dots
\end{pmatrix}$$ be a fixed matrix. Here $n=1,2,\dots ,$ $m_n\in\mathbb N\cup \{0,+\infty\}$ ($\mathbb N$ is the set of all positive integers), and numbers $q_{i,n}$ ($i=\overline{0,m_n}$, i.e., $i\in \{0,1,2, \ldots , m_n\}$) satisfy the following system of conditions: $$\left\{
\begin{aligned}
1^{\circ}. ~q_{i,n}>0 ~\text{for all} ~  i=\overline{0,m_n} ~\text{and} ~n=1,2,\dots \\
2^{\circ}.  \   \    \ \  \ \  \ \  \  \  \  \   \  \ \   \ \  \ \  \ \sum^{m_n}_{i=0} {q_{i,n}}=1~ \text{for any } n\in\mathbb N\\
3^{\circ}.   \    \  \   \  \  \   \   \  \ \prod^{\infty} _{j=1} {q_{i_j,j}}=0~ \text{for any sequence} ~(i_n).\\
\end{aligned}
\right.$$ That is the matrix $\tilde Q^{'} _{\mathbb N_B}$ is a matrix whose the nth column (for an arbitrary positive integer $n$) is finite or infinite and the column summing to $(-1)^{\rho_{n-1}+\rho_{n}}$, i.e., each column summing to $1$ or $-1$. Different columns can contain different numbers of elements (i.e., there exists a finite number of elements if the condition $m_n<\infty$ holds and there exists an infinite number of elements if $m_n=\infty$ is true).

Since different representations of real numbers are widely used in the theory of functions with a complicated local structure, theory of dynamical systems, fractal theory, set theory, etc. (e.g., see –, ), in the present article, a representation of real numbers by the following series is introduced: $$\label{eq: the main series}
(-1)^{\rho_1}a_{i_1,1}+\sum^{\infty} _{n=2}{\left((-1)^{\rho_n}a_{i_n,n}\prod^{n-1} _{j=1} {q_{i_j,j}}\right)},$$ where $$a_{i_n,n}=\begin{cases}
\sum^{i_n-1} _{i=0} {q_{i,n}}&\text{if $i_n\ne 0$ }\\
0&\text{if $i_n=0$.}
\end{cases}$$

We say that an expansion of a number $x$ in series is *the sign-variable $\tilde Q^{'} _{\mathbb N_B}$-expansion of $x$* and write $\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{i_1i_2...i_n...}$. The last notation is called *the $\tilde Q^{'} _{\mathbb N_B}$-representation of $x$* or *the quasy-nega-$\tilde Q$-representation of $x$*. The set $\{0,1, \dots , m_n\}$ is an alphabet for the symbol $i_n$ in the $\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{i_1i_2...i_n...}$.

We can model expansion  by the following way. Suppose that $\rho_0=0$ and given the matrix $\tilde Q^{'} _{\mathbb N_B}$. Then $$\sum^{i_1-1} _{i=0}{\left((-1)^{\rho_1}q_{i,1}\right)}+\sum^{\infty} _{n=2}{\left(\left(\sum^{i_n-1} _{i=0}{((-1)^{\rho_{n-1}+\rho_n}q_{i,n})}\right)\left(\prod^{n-1} _{j=1}{(-1)^{\rho_{j-1}+\rho_j}q_{i_j,j}}\right)\right)}$$ $$=(-1)^{\rho_1}a_{i_1,1}$$ $$+\sum^{\infty} _{n=2}{\left(\left(\sum^{i_n-1} _{i=0}{(-1)^{\rho_{n-1}+\rho_n}q_{i,n}}\right)\left((-1)^{\rho_{0}+\rho_1}q_{i_1,1}\cdots(-1)^{\rho_{n-3}+\rho_{n-2}}q_{i_{n-2},n-2}(-1)^{\rho_{n-2}+\rho_{n-1}}q_{i_{n-1},n-1}\right)\right)}$$ $$=(-1)^{\rho_1}a_{i_1,1}+\sum^{\infty} _{n=2}{\left((-1)^{2\rho_1+2\rho_2+\dots+2\rho_{n-1}+\rho_n}a_{i_n,n}\prod^{n-1} _{j=1}{q_{i_j,j}}\right)}=(-1)^{\rho_1}a_{i_1,1}+\sum^{\infty} _{n=2}{\left((-1)^{\rho_n}a_{i_n,n}\prod^{n-1} _{j=1} {q_{i_j,j}}\right)}.$$

It is obvious that the last series is absolutely convergent and its sum belongs to $[a^{'}, a^{''}]$, where $$a^{'}=\inf\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{i_1i_2...i_n...}=(-1)^{\rho_1}\check a_{i_1,1}+\sum^{\infty} _{n=2}{\left((-1)^{\rho_n}\check a_{i_n,n}\prod^{n-1} _{j=1} {\check q_{i_j,j}}\right)}=(-1)^{\rho_1}\check a_{i_1,1}-\sum _{1<n\in\mathbb N_B}{\left(a_{m_n,n}\prod^{n-1} _{j=1} {\check q_{i_j,j}}\right)},
$$ $$a^{''}=\sup\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{i_1i_2...i_n...}=(-1)^{\rho_1}\hat a_{i_1,1}+\sum^{\infty} _{n=2}{\left((-1)^{\rho_n}\hat a_{i_n,n}\prod^{n-1} _{j=1} {\hat q_{i_j,j}}\right)}=(-1)^{\rho_1}\hat a_{i_1,1}+\sum_{1<n\notin\mathbb N_B}{\left(a_{m_n,n}\prod^{n-1} _{j=1} {\hat q_{i_j,j}}\right)},
$$ where $$\check a_{i_n,n}=\begin{cases}
a_{m_n,n}&\text{if $n \in \mathbb N_B$}\\
a_{0,n}&\text{if $n \notin \mathbb N_B$,}
\end{cases}
~~~~~~~\check q_{i_n,n}=\begin{cases}
q_{m_n,n}&\text{if $n \in \mathbb N_B$}\\
q_{0,n}&\text{if $n \notin \mathbb N_B$,}
\end{cases}$$ and $$\hat a_{i_n,n}=\begin{cases}
a_{0,n}&\text{if $n \in \mathbb N_B$}\\
a_{m_n,n}&\text{if $n \notin \mathbb N_B$,}
\end{cases}
~~~~~~~\hat q_{i_n,n}=\begin{cases}
q_{0,n}&\text{if $n \in \mathbb N_B$}\\
q_{m_n,n}&\text{if $n \notin \mathbb N_B$.}
\end{cases}$$

Let $s>1$ be a fixed positive integer and $Q\equiv (q_n)$ be a fixed sequence of positive integers such that $q_n>1$. By $[x]$ denote the integer part of $x$. It is easy to see that the $\tilde Q^{'} _{\mathbb N_B}$-representation is:

-   the nega-$\tilde Q$-representation [@Serbenyuk2016] whenever $\mathbb N_B$ is the set of all odd numbers: $$x=\Delta^{(-\tilde Q)} _{i_1i_2...i_n...}\equiv -a_{i_1,1}+\sum^{\infty} _{n=2}{\left((-1)^na_{i_n,n}\prod^{n-1} _{j=1}{q_{i_j,j}}\right)};$$

-   the representation by positive [@Cantor1] Cantor series whenever $\mathbb N_B=\varnothing$ and $q_{i,n}=\frac{1}{q_n}$ for any $n \in \mathbb N$, $i=\overline{0,q_n-1}$: $$x=\Delta^{Q} _{i_1i_2...i_n...}\equiv \frac{i_1}{q_1}+\frac{i_2}{q_1q_2}+\dots +\frac{i_n}{q_1q_2\dots q_n}+\dots,$$ i.e., in this case, the matrix $\tilde Q^{'} _{\mathbb N_B}$ is following

    $$\begin{pmatrix}
    \frac{1}{q_1}& \frac{1}{q_2} &\ldots & \frac{1}{q_n}&\ldots\\
    \frac{1}{q_1}&  \frac{1}{q_2} &\ldots & \frac{1}{q_n}&\ldots\\
    \vdots& \vdots &\ddots & \vdots&\ldots\\
    \frac{1}{q_1}& \frac{1}{q_2} &\ldots & \frac{1}{q_n} &\ldots\\
    \frac{1}{q_1} &    \frac{1}{q_2}        & \dots &                        &  \dots\\
                             &  \frac{1}{q_2}                       &  \dots &                & \dots
    \end{pmatrix};$$

-   the representation by alternating [@Serbenyuk2017] Cantor series whenever $\mathbb N_B$ is the set of all odd numbers and $q_{i,n}=\frac{1}{q_n}$ for any $n \in \mathbb N$, $i=\overline{0,q_n-1}$: $$x=\Delta^{-Q} _{i_1i_2...i_n...}\equiv\frac{i_1}{-q_1}+\frac{i_2}{(-q_1)(-q_2)}+\dots +\frac{i_n}{(-q_1)(-q_2)\dots (-q_n)}+\dots,$$ where

    $$\tilde Q^{'} _{\mathbb N_B}=\begin{pmatrix}
    -\frac{1}{q_1}& -\frac{1}{q_2} &\ldots & -\frac{1}{q_n}&\ldots\\
    -\frac{1}{q_1}&  -\frac{1}{q_2} &\ldots & -\frac{1}{q_n}&\ldots\\
    \vdots& \vdots &\ddots & \vdots&\ldots\\
    -\frac{1}{q_1}&-\frac{1}{q_2} &\ldots & -\frac{1}{q_n} &\ldots\\
    -\frac{1}{q_1} &    -\frac{1}{q_2}        & \dots &                        &  \dots\\
                             &  -\frac{1}{q_2}                       &  \dots &                & \dots
    \end{pmatrix};$$

-   the s-adic representation (see ) whenever $\mathbb N_B=\varnothing$ and $q_{i,n}=\frac{1}{s}$ for all $n\in \mathbb N$, $i=\overline{0, s-1}$: $$x=\Delta^{s} _{i_1i_2...i_n...}\equiv\sum^{\infty} _{n=1} {\frac{i_n}{s^n}},$$ i.e., in this case, our matrix is following $$\tilde Q^{'} _{\mathbb N_B}=\begin{pmatrix}
    \frac{1}{s}& \frac{1}{s} &\ldots & \frac{1}{s}&\ldots\\
    \frac{1}{s}&  \frac{1}{s} &\ldots & \frac{1}{s}&\ldots\\
    \vdots& \vdots &\ddots & \vdots&\ldots\\
    \frac{1}{s}& \frac{1}{s} &\ldots & \frac{1}{s} &\ldots\\

    \end{pmatrix};$$

-   the $nega-s-adic-representation$ () whenever $\mathbb N_B$ is the set of all odd numbers and $q_{i,n}=\frac{1}{s}$ for all $n\in \mathbb N$, $i=\overline{0, s-1}$: $$x=\Delta^{-s} _{i_1i_2...i_n...}\equiv\sum^{\infty} _{n=1} {\frac{(-1)^{n}i_n}{s^n}},$$ where $$\tilde Q^{'} _{\mathbb N_B}=\begin{pmatrix}
    -\frac{1}{s}& -\frac{1}{s} &\ldots & -\frac{1}{s}&\ldots\\
    -\frac{1}{s}&  -\frac{1}{s} &\ldots & -\frac{1}{s}&\ldots\\
    \vdots& \vdots &\ddots & \vdots&\ldots\\
    -\frac{1}{s}& -\frac{1}{s} &\ldots & -\frac{1}{s} &\ldots\\

    \end{pmatrix}.$$

Suppose that $q_{i,n}=\frac{1}{s}$ for all $n\in \mathbb N$, $i=\overline{0, s-1}$, and $s>1$ is a fixed positive integer. Then we get the following expansion of real numbers $$x=\Delta^{(\pm s, \mathbb N_B)} _{\alpha_1\alpha_2...\alpha_n...}\equiv \sum^{\infty} _{n=1} {\frac{(-1)^{\rho_n}\alpha_n}{s^n}}.$$ The last representation described by the author of the present article in [@S.Serbenyuk], is a generalization of the classical s-adic and nega-s-adic representations.

The term “nega" is used in this article since certain expansions of real numbers are numeral systems with a negative base.

Consider other examples of the matrix $\tilde Q^{'} _{\mathbb N_B}$.

-   Suppose that the condition $m_n=\infty$ holds for all $n\in \mathbb N$ and the set $\mathbb N_B$ is the set $$\{n: n=4k+1, n=4k+2 \}, \text{where}~ k=1,2,3, \dots.$$ Let us consider the matrix $$\tilde Q^{'} _{\mathbb N_B}=
    \begin{pmatrix}
    -\frac{1}{2}& \frac{2}{3} &    -\frac{3}{4} &\ldots & (-1)^n\frac{n}{n+1}&\ldots\\
    -\frac{1}{4}&  \frac{2}{9} &    -\frac{3}{16} &     \ldots & (-1)^n\frac{n}{(n+1)^2}&\ldots\\
    -\frac{1}{8}&   \frac{2}{27}&    -\frac{3}{64}&      \ldots & (-1)^n\frac{n}{(n+1)^3}&\ldots\\
    \vdots& \vdots &\ddots & \vdots&\ldots &\ldots\\
    -\frac{1}{2^i}  &    \frac{2}{3^i}&       -\frac{3}{4^i} & \dots &               (-1)^n\frac{n}{(n+1)^i}         &  \dots\\
      \vdots& \vdots &\ddots & \vdots&\ldots  & \ldots
    \end{pmatrix}.$$

-   Suppose the condition $\mathbb N_B=\varnothing$ holds and the following conditions are true $$q_{i,n}=\begin{cases}
    \frac{1}{2}&\text{whenever $n=1$ and $m_1=1$}\\
    \frac{1}{n}&\text{whenever $n$ is odd, $n>1$, and $m_n=n-1$}\\
    \frac{2^{i-1} (n+1)}{(n+3)^i}&\text{whenever $n$ is even, $m_n=\infty$, and $i=1,2, \dots$}.
    \end{cases}$$ Then we obtain the following matrix $$\tilde Q^{'} _{\mathbb N_B}=\begin{pmatrix}
    \frac{1}{2}& \frac{3}{5} & \frac{1}{3}& \frac{5}{7}&\ldots \\
    \frac{1}{2}&  \frac{6}{25} & \frac{1}{3}& \frac{10}{49}&\ldots \\
    & \frac{12}{125}& \frac{1}{3}& \frac{20}{7^3}& \ldots& \\
    & \vdots & & \vdots & \vdots \\
    & \frac{3\cdot 2^{i-1}}{5^i} & & \frac{5\cdot 2^{i-1}}{7^i}&\ldots  \\

                             &  \vdots       &              &  \vdots &          \vdots
    \end{pmatrix}.$$

Let us consider the $\tilde Q^{'} _{\mathbb N_B}$-expansion.

*A cylinder $\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_n}$* of rank $n$ with base $c_1\ldots c_n$ is a set $\{x: x=\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1...c_ni_{n+1}i_{n+2}...}\}$, where $c_1, c_2, \dots, c_n$ are fixed.

Suppose that $$\tilde a_{m_n,n}=\begin{cases}
a_{m_n,n}&\text{if $n \in \mathbb N_B$}\\
a_{0,n}&\text{if $n \notin \mathbb N_B$,}
\end{cases}
~~~~~~~\tilde q_{m_n,n}=\begin{cases}
q_{m_n,n}&\text{if $n \in \mathbb N_B$}\\
q_{0,n}&\text{if $n \notin \mathbb N_B$,}
\end{cases}$$ $$\tilde a_{0,n}=\begin{cases}
a_{0,n}&\text{if $n \in \mathbb N_B$}\\
a_{m_n,n}&\text{if $n \notin \mathbb N_B$,}
\end{cases}\tilde  q_{0,n}=\begin{cases}
q_{0,n}&\text{if $n \in \mathbb N_B$}\\
q_{m_n,n}&\text{if $n \notin \mathbb N_B$.}
\end{cases}$$

Cylinders $\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_n}$ have the following properties: \[lm: 1\]

1.  a cylinder $\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_n}$ is a closed interval;

2.  $$\inf\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_n}=(-1)^{\rho_1}a_{c_1,1}+\sum^{n} _{k=2}{\left((-1)^{\rho_k}a_{c_k,k}\prod^{k-1} _{j=1}{q_{c_j,j}}\right)}$$ $$-\left(\prod^{n} _{j=1}{q_{c_j,j}}\right)\left(\tilde a_{m_{n+1},n+1}+\sum^{\infty} _{t=n+2}{\left(\tilde a_{m_{t},t}\prod^{t-1} _{r=n+1}{\tilde q_{m_r,r}}\right)}\right),$$ $$\sup\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_n}=(-1)^{\rho_1}a_{c_1,1}
     +\sum^{n} _{k=2}{\left((-1)^{\rho_k}a_{c_k,k}\prod^{k-1} _{j=1}{q_{c_j,j}}\right)}$$ $$+\left(\prod^{n} _{j=1}{q_{c_j,j}}\right)\left(\tilde a_{0,n+1}+\sum^{\infty} _{t=n+2}{\left(\tilde a_{0,t}\prod^{t-1} _{r=n+1}{\tilde q_{0,r}}\right)}\right);$$

3.  the main metric relation is following $$\frac{\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_nc_{n+1}}}{\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_n}}$$ $$=q_{c_{n+1},n+1}\frac{\tilde a_{m_{n+2},n+2}+\sum^{\infty} _{t=n+3}{\left(\tilde a_{m_{t},t}\prod^{t-1} _{r=n+2}{\tilde q_{m_r,r}}\right)}+\tilde a_{0,n+2}+\sum^{\infty} _{t=n+3}{\left(\tilde a_{0,t}\prod^{t-1} _{r=n+2}{\tilde q_{0,r}}\right)}}{\tilde a_{m_{n+1},n+1}+\sum^{\infty} _{t=n+2}{\left(\tilde a_{m_{t},t}\prod^{t-1} _{r=n+1}{\tilde q_{m_r,r}}\right)}+\tilde a_{0,n+1}+\sum^{\infty} _{t=n+2}{\left(\tilde a_{0,t}\prod^{t-1} _{r=n+1}{\tilde q_{0,r}}\right)}}.$$

Let us prove that *the first property is true*. Let $n\notin \mathbb N_B$ and $x\in \Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_n}$, i.e., $$x=(-1)^{\rho_1}a_{c_1,1}+\sum^{n} _{k=2}{\left((-1)^{\rho_k}a_{c_k,k}\prod^{k-1} _{j=1}{q_{c_j,j}}\right)}$$ $$+\left(\prod^{n} _{j=1}{q_{c_j,j}}\right)\left((-1)^{\rho_{n+1}}a_{i_{n+1},n+1}+\sum^{\infty} _{t=n+2}{\left((-1)^{\rho_t}a_{i_t,t}\prod^{t-1} _{r=n+1}{q_{i_r,r}}\right)}\right).$$ Then $$x^{'}=(-1)^{\rho_1}a_{c_1,1}+\sum^{n} _{k=2}{\left((-1)^{\rho_k}a_{c_k,k}\prod^{k-1} _{j=1}{q_{c_j,j}}\right)}$$ $$-\left(\prod^{n} _{j=1}{q_{c_j,j}}\right)\left(\tilde a_{m_{n+1},n+1}+\sum^{\infty} _{t=n+2}{\left(\tilde a_{m_{t},t}\prod^{t-1} _{r=n+1}{\tilde q_{m_r,r}}\right)}\right)\le x\le (-1)^{\rho_1}a_{c_1,1}$$ $$+\sum^{n} _{k=2}{\left((-1)^{\rho_k}a_{c_k,k}\prod^{k-1} _{j=1}{q_{c_j,j}}\right)}+\left(\prod^{n} _{j=1}{q_{c_j,j}}\right)\left(\tilde a_{0,n+1}+\sum^{\infty} _{t=n+2}{\left(\tilde a_{0,t}\prod^{t-1} _{r=n+1}{\tilde q_{0,r}}\right)}\right)=x^{''}.$$

Hence $x\in [x^{'}, x^{''}]$ and $[x^{'}, x^{''}] \supseteq \Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_n}$.

Since the equalities $$x^{'}=(-1)^{\rho_1}a_{c_1,1}+\sum^{n} _{k=2}{\left((-1)^{\rho_k}a_{c_k,k}\prod^{k-1} _{j=1}{q_{c_j,j}}\right)}$$ $$+\left(\prod^{n} _{j=1}{q_{c_j,j}}\right)\inf{\left\{(-1)^{\rho_{n+1}}a_{i_{n+1},n+1}+\sum^{\infty} _{t=n+2}{\left((-1)^{\rho_t}a_{i_t,t}\prod^{t-1} _{r=n+1}{q_{i_r,r}}\right)}\right\}},$$ $$x^{''}=(-1)^{\rho_1}a_{c_1,1}+\sum^{n} _{k=2}{\left((-1)^{\rho_k}a_{c_k,k}\prod^{k-1} _{j=1}{q_{c_j,j}}\right)}$$ $$+\left(\prod^{n} _{j=1}{q_{c_j,j}}\right)\sup{\left\{(-1)^{\rho_{n+1}}a_{i_{n+1},n+1}+\sum^{\infty} _{t=n+2}{\left((-1)^{\rho_t}a_{i_t,t}\prod^{t-1} _{r=n+1}{q_{i_r,r}}\right)}\right\}},$$ hold, we have $ x \in\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_n}$ and $x^{'},x^{''}\in\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_n}$.

It is obvious that the first property is true when $n\in \mathbb N_B$.

*The second property* follows from the proof of the first property.

*The third property* follows from the second property.

Let us consider the mutual placement of $\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}c}$ and $\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}[c+1]}$.

Suppose that $\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}c}\bigcap\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}[c+1]}$ is not the empty or one-element set; then we have the following:

-   if cylinders $\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}c}$ and $\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}[c+1]}$ are *“left-to-right”* situated, then $$\kappa_1=\sup {\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}c}}-\inf {\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}[c+1]}}> 0;$$

-   if cylinders $\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}c}$ and $\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}[c+1]}$ are *“right-to-left”* situated, then $$\kappa_2=\sup{\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}[c+1]}} -\inf {\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}c}}> 0.$$

In addition, in the first case, we get $$\kappa_1<\kappa_2=|\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}c}|+|\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}[c+1]}|-|\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}c}\cap\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}[c+1]}|=W.$$ In the second case, $\kappa_2<\kappa_1=W$.

Suppose that $\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}c}\bigcap\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}[c+1]}$ is the empty or one-element set; then we have the following:

-   if cylinders $\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}c}$ and $\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}[c+1]}$ are *“left-to-right”* situated, then $$\nu_1=\inf {\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}[c+1]}}-\sup {\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}c}}=-\kappa_1\ge 0;$$

-   if cylinders $\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}c}$ and $\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}[c+1]}$ are *“right-to-left”* situated, then $$\nu_2=\inf{\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}c}} -\sup {\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}[c+1]}}=-\kappa_2\ge 0.$$

Also, in the first case, we have $$\nu_1>\nu_2= V=-|\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}c}|-|\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}[c+1]}|-\varpi,$$ where $\varpi$ is the Lebesgue measure of the interval situated between $\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}c}$ and $\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}[c+1]}$. In the second case, $V=\nu_1<\nu_2$.

Let us consider the difference $$\kappa_1\equiv\sup{\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}c}}-\inf{\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}[c+1]}}=a_{c,n}(-1)^{\rho_n}\prod^{n-1} _{j=1}{q_{c_j,j}}$$ $$+q_{c,n}\left(\prod^{n-1} _{j=1}{q_{c_j,j}}\right)\left(\sum^{\infty} _{n<t\notin\mathbb N_B}{\left(\hat a_{i_t,t}\prod^{t-1} _{r=n+1}{\hat q_{i_r,r}}\right)}\right)-a_{c+1,n}(-1)^{\rho_n}\prod^{n-1} _{j=1}{q_{c_j,j}}$$ $$+q_{c+1,n}\left(\prod^{n-1} _{j=1}{q_{c_j,j}}\right)\left(\sum_{n<t\in\mathbb N_B}{\left(\check a_{i_{t},t}\prod^{t-1} _{r=n+1}{\check q_{i_r,r}}\right)}\right)=\left(\prod^{n-1} _{j=1}{q_{c_j,j}}\right)$$ $$\times\left(-q_{c,n}(-1)^{\rho_n}+q_{c,n}\sum_{n<t\notin\mathbb N_B}{\left( a_{m_t,t}\prod^{t-1} _{r=n+1}{\hat q_{i_r,r}}\right)}+q_{c+1,n}\sum_{n<t\in\mathbb N_B}{\left( a_{m_{t},t}\prod^{t-1} _{r=n+1}{\check q_{i_r,r}}\right)}
\right).$$

Using $$\omega_1=\sum^{\infty} _{n<t\notin\mathbb N_B}{\left( a_{m_t,t}\prod^{t-1} _{r=n+1}{\hat q_{i_r,r}}\right)}~~~\text{and}~~~
 \omega_2= \sum_{n<t\in\mathbb N_B}{\left(a_{m_{t},t}\prod^{t-1} _{r=n+1}{\check q_{i_r,r}}\right)},$$ we get $$\kappa_1=(q_{c,n}(-1)^{1+\rho_n}+q_{c,n}\omega_1+q_{c+1,n}\omega_2)q_{c_1,1}q_{c_2,2}...q_{c_{n-1},n-1}.$$

From the last expression, we have the following:

-   $\kappa_1>0$ whenever $\rho_n=1$, i.e., $n\in\mathbb N_B$;

-   if $\rho_n=2$, i.e., $n\notin\mathbb N_B$, then:

$$\text{if}~ (1-\omega_1)q_{c,n}=q_{c+1,n}\omega_2, ~\text{then}~ \kappa_1=0;$$ $$\text{if}~ (1-\omega_1)q_{c,n}>q_{c+1,n}\omega_2, ~\text{then}~ \kappa_1<0;$$ $$\text{if}~ (1-\omega_1)q_{c,n}<q_{c+1,n}\omega_2, ~\text{then}~ \kappa_1>0.$$ Really, since $0\le\omega_1\le 1$ and $0\le\omega_2\le 1$, for $n\notin\mathbb N_B$ we have $$-q_{c,n}\le\frac{\kappa_1}{q_{c_1,1}q_{c_2,2}\dots q_{c_{n-1},n-1}}\le q_{c+1,n}.$$

Let us consider the difference $$\kappa_2\equiv\sup{\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}[c+1]}}-\inf{\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}c}}=(-1)^{\rho_n}a_{c+1,n}\prod^{n-1} _{j=1}{q_{c_j,j}}$$ $$+q_{c+1,n}\left(\prod^{n-1} _{j=1}{q_{c_j,j}}\right)\left(\sum_{n<t\notin \mathbb N_B}{\left( a_{m_t,t}\prod^{t-1} _{r=n+1}{\hat q_{i_r,r}}\right)}\right)-a_{c,n}(-1)^{\rho_n}\prod^{n-1} _{j=1}{q_{c_j,j}}$$ $$+q_{c,n}\left(\prod^{n-1} _{j=1}{q_{c_j,j}}\right)\left(\sum_{n<t\in\mathbb N_B}{\left(a_{m_{t},t}\prod^{t-1} _{r=n+1}{\check q_{i_r,r}}\right)}\right)$$ $$=(q_{c,n}(-1)^{\rho_n}+q_{c+1,n}\omega_1+q_{c,n}\omega_2)\left(\prod^{n-1} _{j=1}{q_{c_j,j}}\right).$$

Hence, we obtain the following:

-   $\kappa_2>0$ whenever $\rho_n=2$, i.e., $n\notin\mathbb N_B$;

-   if $\rho_n=1$, i.e., $n\in\mathbb N_B$, then:

$$\text{if}~ (1-\omega_2)q_{c,n}=q_{c+1,n}\omega_1, ~\text{then}~ \kappa_2=0;$$ $$\text{if}~ (1-\omega_2)q_{c,n}>q_{c+1,n}\omega_1, ~\text{then}~ \kappa_2<0;$$ $$\text{if}~ (1-\omega_2)q_{c,n}<q_{c+1,n}\omega_1, ~\text{then}~ \kappa_2>0.$$ So, $$-q_{c,n}\le\frac{\kappa_2}{q_{c_1,1}q_{c_2,2}...q_{c_{n-1},n-1}}\le q_{c+1,n}.$$

Finally, we obtain the following results:

-   If $\rho_n=1$, then cylinders $\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}c}$ and $\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}[c+1]}$ are right-to-left situated. However the set $\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}c}\bigcap\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}[c+1]}$ is the empty set or the one-element set or an interval. It depends on the matrix $\tilde Q^{'} _{\mathbb N_B}$.

-   If $\rho_n=2$, then cylinders $\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}c}$ and $\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}[c+1]}$ are left-to-right situated. However the set $\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}c}\bigcap\Delta^{\tilde Q^{'} _{\mathbb N_B}} _{c_1c_2...c_{n-1}[c+1]}$ is one of the following sets (that depends on $\tilde Q^{'} _{\mathbb N_B}$): the empty set, the one-element set, an interval.

Since Lemma \[lm: 1\] is true, the following statement holds.

For an arbitrary number $x\in[a^{'} _0, a^{''} _0]$ there exists a sequence $(i_n)$, $i_n\in N^0 _{m_n}\equiv~\{0,1,\dots ,m_n\}$, such that $$x=(-1)^{\rho_1}a_{i_1,1}+\sum^{\infty} _{n=2}{\left((-1)^{\rho_n}a_{i_n,n}\prod^{n-1} _{j=1} {q_{i_j,j}}\right)}$$ whenever for all $n \in \mathbb N$ the following system of conditions holds: $$\left\{
\begin{aligned}
q_{i_n,n}\left(1-\sum_{n<t\in\mathbb N_B}{\left(a_{m_{t},t}\prod^{t-1} _{r=n+1}{\check q_{i_r,r}}\right)}\right)\le q_{i_n+1,n}\left(\sum_{n<t\notin\mathbb N_B}{\left( a_{m_t,t}\prod^{t-1} _{r=n+1}{\hat q_{i_r,r}}\right)}\right)~\text{for}~n\in\mathbb N_B\\
q_{i_n,n}\left(1-\sum_{n<t\notin\mathbb N_B}{\left( a_{m_t,t}\prod^{t-1} _{r=n+1}{\hat q_{i_r,r}}\right)}\right)\le q_{i_n+1,n}\left( \sum_{n<t\in\mathbb N_B}{\left(a_{m_{t},t}\prod^{t-1} _{r=n+1}{\check q_{i_r,r}}\right)}\right)~\text{for}~n\notin\mathbb N_B.
\end{aligned}
\right.$$

[9]{}

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am a newcomer to rock climbing, I've climbed at the gym a bit and done a climbing course (I'd like to climb more and meet more people to climb with). My question: is it worth joining the Mountain Club of South Africa? What are the benefits of joining the club?

The MCSA organises monthly rock meets. These are held at a variety of venues. this is an ideal way of meeting other climbers and being exposed to the large number of crags we have scattered around. The MCSA is DEFINATELY worth joining if the above appeals to you. There is a very old, and conservative element to the club. If you can deal with this and engage with likeminded keen climbers then you are MCSA for away!

For many years I avoided the mountain club because of their reputation for being a bunch of old bureaucratic farts that never did anything for climbers. I realised recently that will never change until we young 'uns get in there and change it. Hence I picked up my membership forms a few weeks ago. There is definitely room for improvement in the way of the MCSA helping climbers, but I think you can get your money's worth out of the club's relatively small subscription. I am happy to see they are now helping sponsor bolting costs for new routes. Perhaps they will do more for access in the future.

The MCSA has been regarded as a bit conservative as the above people have stated. However, this is a volunteer organisation and the only people involved are some more senior people. We as young climbers should see the MCSA as a representative body that will take care of our climbing interests. But this requires effort from all the climbers and it seems that a lot of youth are too selfish to put in some time doing the not-as-fun-as-climbing things. My recommendation to any person, young or old, is join and put some effort in and reap the benefits. PS the CT MCSA rocksubcom has an average age of 30 and we are keen for people to lower that average age even more :-)

What planet are you guys from? This may be a Cape issue but in Joburg all the best climbers are MCSA members. Only wannabee's with no safety standards would avoid becoming members. Climbing meets are arranged for all types (Trad & sport, beginners' & youth)on a regular basis. Maybe if you only wanted to do sport you could get by not joining, but then you wouldn't have access to some of the sport stuff. The best routes in the area are MCSA access area's and it's only people who want to use the land without paying for access that would get any value out of not joining. So maybe the question is whether there is any point in not joining. Sure it takes a while to get to know people and how things work but that goes for any organisation. Your attitude will determine how much worth you get from being a member. Also these are the guys- the rescue section- who will come and haul your ass out of trouble when you need them- which is a good reason to support them. Bureacracy be damned, lets do some climbing.

Not sure about Joburg, but here in the Cape it seems like it is harder to join the MCSA than getting offered a free rack of cams. It should be about the climbing but its not, too much paperwork and rules. Another reason no one here in the Cape joins is that most of the areas don't require any membership to climb. Not knocking the idea of the MCSA rather some of the rules and regulations surrounding it. Times have changed, so should the MCSA.

Quite agreeh Anon! In Europe its very simple. Fill in your details. Pay your susbscription and you're in. Included is also rescue insurance for any eventuality in the mountains. Trust me, a heli-rescue in Switzerland can really blow your pocketmoney. I hope the MCSA can move towards this model. But maybe thats just a Cape gripe.

In response to Grant Rens: I don't think your comment about all the best climbers in Joburg being MCSA members is entirely accurate. I will agree on the issue of rescue though. The rescue service that is on standby for all us climbers is a good enough reason to support the MCSA.

Depends where you are. From my experience the Jhb MCSA offers real benefits and is defenitely worthwhile joining. It provides oppurtunities to meet climbers and go climning through regular and frequent meets as well weekly socials. It also offers preferential access to Magaliesberg crags and some others. It is more of a climbing club than a hiking club. CT on the other hand is more of hiking club than a climbing club, although matters are improving through the recent efforts of the Rock Committee. The CT MCSA defenitely cannot be relied upon to the same extent as a vehicle for frequent climbing meets. It however does offer an oppurtunity to meet climbers and hopefully through this get to climb more. However, it will require more patience & effort. My tip would be to go hang out at the MCSA bouldering wall on tuesday and/or thursday evenings. On the plus side you are more likely to meet climbers at the MCSA who are intersted in doing outdoor stuff than at a commercial gym. So on balance the CT MCSA is probably worthwhile joining, although it far from where it should be as a rock climbing club.

The MCSA also helps pay for bolts (at least the JHB section does) that help make routes for you to climb, so there alone is a reason to join. You also get 10% off at some climbing shops so if you're looking to make some big purchases the saving you make there will cover your annual fee.

Seriously though, for what are we paying MCSA fees? What do you actually get out of it? In the past, yes, you could get free permits or reduced fee admissions for various parks/areas, etc. Now all you get is an ID badge that serves no real purpose. I think the mountain club should start to think more about adding value.

What? I can't believe I'm reading this. Without the efforts of the MCSA there would be no climbing allowed at Rocklands. At all. And you ask what the benefit is? If I have to spell it out now: the MCSA makes very real contributions to ensuring access at climbing venues. By becoming a member, you are supporting a respected organisation that has and uses its influence when it comes to negotiating access (often very generously for all climbers, not just members). By not becoming a member you are being short-sighted; by saying there are no benefits to being a member you are being stupid.

Let's say you are new to climbing and don't know any MCSA members. You can't join!! What kind of club is that? Pushing away potential members because you decide who is worthy? The MCSA is slowly coming right in several areas, but I think they still have alot of work to do, especially removing themseleves from the past. We are in 2004, not 1980.

Anon your comments are totally without foundation. If you apply to join you will be invited to attend rock meets and socials where you can easily meet and get people to propose and second your application. Or simply ask around at your local gym or crag, its not that hard to find people to help your cause. The paperwork has actually been drastically reduced in recent years and the number of signatures required to back your application has also been drastically reduced. There has to be some form of screening to ensure that prospective members are actually interested in the outdoors and are likely to uphold club policies of caring for the environment. The MCSA does more than you can imagine to support climbing. Without them bolting would most likely still be banned in the Western Cape, access to several sensitive areas would be impossible without their efforts. In addition they have a network of club huts for accomodation, in Du toits kloof for example the only places to stay are the hotel at a few hundred a night or the MCSA hut at less than one twentieth of the price. They have an awesome library of climbing books which Im sure most members get endless hours of entertainment and useful information from. There are even some really cool young people amongst the members, and there will be more if you and others join. To top all of this off they help sponsor bolts and even own a drill you can use! (with proper instruction) what more can you ask for? Quit complaining and sign up!

Believe it or not i agree with about 90% of what you say. Just that last part. Let me quote: \"There are even some really cool young people amongst the members, and there will be more if you and others join.\" Why aren't there many more youngsters? Because it's so complicated to join. Good or bad young climbers don't want the hassle of seeking signatures and going on meets. This does not mean they are not dedicated climbers, they are simply full of new ideas and doing their own thing. The old guys like their rules and procedures and I understand that they are an integral part of climbing, but youngsters don't like to sit around discussing policies, that's just the way it is. But let's say they get sent a newsletter every month, addressed to them they might take some interest. Just some thoughts.

hi,I agree with anon. the olde school loves rules.personally,I feel in some way though that its cool for the club to at least require that your basic competency gets tested. It's really for the good of everyone. So what if you have to go on a few meets to get going. If you go into a bank, and you want a credit card, then why do you not complain when they want your dna sample? All the club is doing is protecting itself and rightly so. If you are just starting out climbing, why would you want to turn down valuable mentorship from older, wiser more experienced climbers? I was taught by older climbers from MCSA when I was young, and I can tell you that before I started going on meets, I was nowhere. I really dont understand why so many people are pissed off with this club. Sure, during the dark years in this country, they stopped coloured climbers from joining, which I detest. But that was then, and now, we have a club that is open arms, active and want to get to know you and make you a better person. If they really are such bastards, then how come even if you are NOT a member, they will go out of their way to rescue you? It's just people, with hearts, like you and I. Just join. you dont have to goto every weekly meeting, in fact the older guys would be far happier looking at the photos of your amazing adventures, because, like it or not, you'll get further with a reputable club behind you, which has money to spend on expeditions and gear.

Maybe with all the money that has been spent on paying the phone bills of the internet connections involved in this forum conversation, we could have all become members of the MCSA? I think it costs R20, in nominal terms, to apply, and once you are a member you have the right to vote or raise a potentially influential opinion about the rules and regulations.

this makes no sense to me? some of us actually have broadband in this place with lions roaming the streets. If money is a factor then maybe you should consider a different passtime. Lets just consider that a rope now costs R1500 and shoes are the same...most people are complaining about the rules and regulations which is a joke. Learn to be anarchic elsewhere. The rules of the club are set out to create a positive outcome and all I'm reading is that guys want to fill out a form, get sent a card and then you are a member. Tell me rather, why should this massive club, who own huge portions of land, just let anyone who can come up with a few bucks roam freely without at least being able to teach their ethics and ideals. Every time I go to a non mcsa crag I'm appalled by bottles, stompies, piss stench...and why then when I goto a place like magalies do I not see this? I'm glad they have the balls to keep the riff raff out.

About a year ago we got the Cape Town section to relax its entrance procedure. You no longer need proposers and seconders, or to go on meets, etc. You just fill in the application form and pay your money and you're in. In a couple of months, if you haven't done anything very bad (like burning down a club hut) your application is ratified and made permanent. We also created a new, youth membership category which caters for ages up to 25 with a much lower subscription (R120 pa - little more than the cost of a quickdraw).

I believe that you get a lot of value out of your membership. I think very few people realise just how much effort some club members put into various arenas. There is a huge amount of negotiation with various authorities that has resulted in the preservation of the access that we have. The Cederberg Sport Climbing Management Plan, the path building meets in the TMNP, the money that the MCSA has paid for contractors to fix paths, etc, are examples of this. If it wasn't for this we would not be allowed to climb at Rocklands, Trutjieskraal, the various Peninsula crags, etc at all. Who else is doing this or would do so if the club stopped? The fact that a lot of this effort doesn't produce results is not so much a reflection on the MCSA but on the fact that the authorities are full of it. On top of that there's a huge amount of effort into conservation, search and rescue, etc. These may not affect you directly (and in the case of SAR hopefully never) but they sure as hell are to your benefit as well as that of the entire country.

Lastly, it irks me whenever I hear people asking what the club does for them. It's a club not a business. Members get out what they put in. If there aren't anough climbing meets for your liking it's because you aren't in there helping to organise them. Why is it that so many climbers are so selfish and expect always to get something out and never to put anything in themselves? So join, increase the number of climbers in the club and help to increase the value that the club provides.

why join, they are just a bunch of regulars who think they own the hills, with memebers like hairy uncle, would you join? NO, dont join them. It's a waste of time and money and really, it's just the blind leading the blind.

Dingo please don't join the club. Its probally better for everyone.
I have some questions for you:
Have you ever belonged to the MCSA?
Have you ever belonged to any club?
Have you any clue on what the MCSA has done for climbing?
I suggest you go make yourself a strong cup of coffee and WAKE UP!

While not a member myself the effort undertaken by some of the Cape guys (and specifically girl) to arrange access to Rocklands has been really significant. I think the level of effort varies per province though (not sure if you would see the same efforts in Joburg).

I think securing access to Rocklands alone is enough.

j

PS. Grant Rens - your comments regarding all the best joburg climbers are MCSA is the biggest load of sh*te I have ever heard. Maybe you are speaking off a low base but with the exception of a few....Flex, Pedley, Eric, Mark Millar, Mark Seuring and Alard the majority are extremely average. Nothing wrong with being average though.

Currently: In the Eastern Cape...almost all progress & development of climbing is undertaken by MCSA members or people in the process of becoming members. \"Almost\" indicates that there may be some by non members, but its quite unlikely.

Although not that much is done in the name of MCSA. Its done by privatly by people who are also MCSA members.

MCSA is for safe little wannabees who need someone to change their nappy. get out there by yourself and climb whatever you want. MCSA needs to be changed to HCSA - Hiking Club of SA, 'cos that's all they are...a bunch of berg hugging, long bearded, conservative poncey hikers, and even then, I'm mocking real hikers. There really is NO need to join.

So you're telling me that the guys who climbed Trango recently are a bunch of novice hikers? And how about Cerro Torre - I guess thats another easy day out in the hills? And most of the bolting at Boven, especially Wonderland - those bloody hikers again! And expeditions to climb new peaks in China? And new multi pitch sport routes in the berg? And how about single handedly saving Magaliesberg from big developers by buying all the pristine kloofs? Ya, those hikers sure get on my tits as well

Don't let bitter people put you off the mountain club! The mountain club is like most societies - full of a range of diverse people. Yes there are berg hugging, long bearded, conservative poncey hikers in the mountain club. There is undoubtedly someone in the mountain club that you will dislike and who will irritate you beyond words. Dingo has, after all, has previously claimed (in other posts on this site) to be a member.
There are also some really cool people in the mountain club. Quite apart from the guys who go on expeditions, bolt climbs for the rest of us, negotiate access to and help conserve crags etc etc, there are a whole bunch of regular climbers looking to hang out with like minded people. There is undoubtedly someone in the mountain club that you will like and that you wouldn't have met otherwise. A couple of weeks ago I went on an mcsa meet to Swinburne with a group of very nice people. They were young, cool and decidedly un-poncey. I had a great weekend.
I met a number of new people on that meet. Two of them have come climbing with me and my friends since and will again. Join the mountain club by all means. It is a great way to meet people.

All that is gold does not glitter; not all those that wander are lost.

There are a couple of options regarding the way in which you approach the whole MCSA thing:
1) you don't join
2) you don't join and bitch and moan about the way in which the club is run.
3) you join and enjoy the benefits (you can decide on those benefits based on what's offered)
4) you join and bitch and moan that you're feel you're getting nothing out of being a member or whine about the way in which the club is run.
5) you join and although you don't agree with certain issues you get involved and onto the committee so you can slowly start to change the 'issues' or at least voice your opinion
6) pretend you're Swiss and remain neutral about the whole issue as a member or as a non-member


Internet of Things

Internet of Things(IoT) is an interesting topic in computer science. The idea behind IoT is, enabling everyday objects in the Internet through RFID tags, sensors, mobile devices …etc. There is an interesting study made by Latitude Labs with kids and their expectations from Internet. Sum results from study;


FiTR's NYC Neighborhood Dining Guide

On the Upper East Side, where can you drink a digestif out of this crazy glass, which seems bent on demonstrating a scientific principle?

Four years ago Fork in the Road inaugurated the Our 10 Best guides, and we've never looked back. They've proved so popular that readers have asked for some sort of index to the series. Well, here it is: a reverse-chronological list of the neighborhood-based guides, most in Manhattan and Brooklyn. Click on each headline and be magically transported to the corresponding 10 Best list. Sometimes, you even get 11, or 13. Note that the most elderly of these guides is around three years old, so check to make sure a particular place is still open before you go. And look for future indexes to guides based on a single foodstuff or dining concept.


Jason Ravnsborg said that as he passed by the scene of the crash, he stopped to look for the deer he purportedly hit the previous night but instead found the body of Joe Boever. He did not call 911 but drove to the county sheriff's home to report it.


AR-15 Most Popular Rifle in the U.S.

In recent months, we’ve seen the AR-15 in the news more often than usual. It makes sense, for such a polarizing item. For millions of Americans, this is a rifle that elicits a specific range of emotions. The AR-15 is a hot-topic item, with conversations including availability, regulations, and outlawing.

The AR-15 is related to the Armalite AR-15, a gas-operated rifle adopted from the 1960s. One family, “The Black Rifle” has scores of cousins and configurations.

Why is AR-15 so popular?

The success of this rifle can be traced to its engineering and U.S. Patent Law. At its most simple, an AR-15 is a fairly basic machine that takes advantage of simple physics. Take a tube, add an accelerant, and then put a projectile in front of that force. This pushes it out of the tube at high speeds.

For as long as rifles have been around, it wasn’t until the end of the 1700s that these items came with interchangeable parts.

Before the late 1700s, every rifle was a one-of-a-kind possession, and manufacturers made each one individually. If it happened to break, the owner would need to take it to a smith who made the parts necessary to fix it.

Now a rifle could be modified, repaired, and completely built with interchangeable parts. More advancements included firing faster, insertable magazines containing multiple rounds and improvements in design made rifles lighter and easier to use. And the AR-15 lead this movement.

A Further Look

As a result, the engineers shifted their attention from shot quality to shot quantity. In 1956, in response to the need for speed, E.M Stoner applied for a patent that describes a simple method for siphoning expanding gases from accelerating black powder to operate the reloading mechanism of a military rifle. This opened the door for better-performing and longer-lasting automatic items for military combat.

American military studies completed during World War II and the Korean War discovered the following: when armed with an automatic rifle firing as long as the trigger was squeezed, soldiers would shoot at a large area to suppress the enemy rather than aiming at specific targets.

The U.S. patent law comes into play once in the public domain. Companies now could take advantage of this firing system, and the interchangeable parts.

People can even 3D-print parts to build a black rifle — someone did in 2013 and fired over 600 rounds!

An owner could swap in different uppers for varmint, big game hunting or target practice. Or different lowers to accommodate various magazines or stocks. The lowers are what actually legally constitutes the rifle. That’s where the serial number is, and it contains the action as well. These interchangeable parts can alter the type and caliber of shot, the type of magazine, and can even accommodate a scope.

The National Shooting Sports Foundation estimates 5 to 10 million AR-platform rifles are in circulation in the U.S. but no one knows exactly how many.

AR-15: The More You Know

When it comes to rifle ownership, understanding where you’re coming from will help you know where you’re going. Hopefully, today’s article will give you the key points you need to understand the rifle industry.

Visit Ormond Arms to pick out your 80% lower AR 15 rifle build kit, today, and carry on this proud tradition of rifle ownership. If you are still not sure about AR 15 Riffles, be sure to learn about AR 80 Lower.

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StarkFeed: The News You're Looking For.


Show Notes

An interview with Kelley Baumgartner an adoptee who found her birthmother via a public social media poster search. You’ve likely seen her Facebook post where she held up a poster with personal details in an attempt to be able to connect with her biological family. She shares about the effects this has had on her reunion, and about her relationship with her adoptive parents.

Show Notes

Topics We Discussed

Kelley’s social media search via poster on Facebook went viral with over 200,000 shares

Adoptees On

The podcast where adoptees discuss the adoption experience. Adult adoptees share stories of search, reunion, and secondary rejection. Adoptees On also curates recommended resources to encourage and educate the adoption community about adoptee issues.


Photo by Aaron Burden on Unsplash

I would like to tell you about a shift in the format of educational resources that I think would be greatly beneficial. Beneficial in itself and beneficial because it would allow wider improvements to our education system. This format I would like to talk about is not new and has already proved its worth in many other areas, the most notable example of which is encyclopedias. So let’s start by talking a bit about encyclopedias…

The ambition of the Encyclopédie

Do you know what an encyclopedia is? It is a book in which all available human knowledge is methodically presented, with the aim of disseminating this knowledge to the public.

The one that is often considered the first of encyclopedias, the Encyclopédie ou Dictionnaire raisonné des sciences, des arts et des métiers, is also one of the most famous. Major work of the 18thcentury, it was written by a group of philosophers at the heart of the cultural movement of the Enlightenment.

These philosophers had the will to make knowledge available to all. They wanted to advance Humanity through knowledge, education and erudition. The Encyclopédie is the result of this desire to gather all available knowledge in order to spread it more widely.

Thanks to encyclopedias, people who know nothing about a subject can learn from the greatest experts in the field, just by buying a book. The expert, on the other hand, just has to write his article once, and then an incredible number of people can access his knowledge; which was previously “locked up” in the natural person of the author or of a handful of connoisseurs. Knowledge is therefore widely disseminated.

The defects of the printed format

Although spreading of knowledge is one of the best ways to advance society, and despite the undoubtedly positive impact of encyclopedias, they are not free of defects. The printed format in particular poses various problems.

First, the public it reaches is by nature quite restrained, since it is already sufficiently privileged and cultured to acquire one, and to be able to afford it.

Second, paper encyclopedias are anything but easy to consult. Massive and cumbersome, we rarely have them with us when we need an information. Also, even if we have one at hand, finding this information is long and tedious. Often, the discomfort of looking up the information outweighs the curiosity or the desire to access it. This lack of ease of use means that people access less to knowledge and culture, which is in no way in the interests of society.

Then, the editorial model implied by the paper format is also far from optimal. In terms of quality and comprehensiveness of the content produced, the few hundred experts mobilized to write a traditional encyclopedia are largely overtaken by a collaborative writing process involving millions of people. Such collaborative writing allows everyone to contribute their own knowledge. In addition, all kinds of errors, inaccuracies or poor formulations can be rectified simply by allowing readers to report them, or better yet, correct them.

Finally, the printed format renders the information frozen. Once a book is published, no further updates or corrections are possible. Content in paper format is doomed to obsolescence. In contrast, content published as a wiki is constantly evolving. Wikis not only allow to correct potential mistakes, but also to update and continually improve the content. All this while making the improvements available to everyone instantly, automatically and for free.

A new format

So, encyclopedias are a brilliant concept, but they are limited in their outreach and in their quality by the paper format. In the early 2000s, a new format for publishing written content emerged, and it has been used with great success for encyclopedias. A format that solves all the problems we just outlined. This format is the wiki. But what exactly is a wiki?

A wiki is a format that allows collaborative editing of content. In practice, it is a website that allows its visitors to edit and improve its content. For example, Wikipedia is an encyclopedia in the form of a wiki. And it effectively solves all the problems we listed:

The free publication on the Web and under free license makes knowledge accessible to the entire world. Wikipedia not only made it easier to consult encyclopedias for those who were already using them. Wikipedia made available an enormous amount of knowledge to a part of the world and of the population that wouldn’t otherwise have had access to it — and this release of knowledge leads to incredible things.

Knowledge is easily and instantly accessible. We can search, share and reuse the content as we see fit.

Wikipedia allows the whole world to contribute its own knowledge, and its content is constantly being improved. The editorial model is greatly improved and the information is no longer set in stone.

Ultimately, Wikipedia and wikis in general are knowledge-spreading machines. And that’s a great thing, it’s something we need more and more, and it’s one of the foundations of the knowledge economy. As Idriss Aberkane puts it, “The knowledge-based economy is based on flows, not stocks. So we have to make sure that knowledge circulates faster and better. Today we produce more knowledge than we can circulate. Humanity produces knowledge exponentially. The global quantity of knowledge doubles every seven to nine years, so the first challenge is to circulate it more quickly.”

And wikis make knowledge circulate better and faster, allowing everyone to share their own knowledge and consult that of others. All this for free, freely, instantly and anywhere in the world.

But the mere fact of having the content in the form of a wiki is not enough to get all these benefits. The economic model, the values of Wikipedia and above all its openness are also necessary. To diffuse knowledge, a wiki open to only a few people and whose access is paid is of no interest whatsoever.

Wikis to spread other types of knowledge

Encyclopedias are not really what I wanted to talk about here, but I found the parallel interesting to make in order to bring in the actual subject of this article: school lessons.

Just like encyclopedias, they are books intended to pass on knowledge. And if we take a closer look, we realize that the courses and textbooks pose exactly the same problems as printed encyclopedias — problems solved by wikis and Wikipedia’s openness:

Restrained audience (limited to students of a certain grade)

Potentially less practical to consult than a digital format, which can be used at any time and in any circumstances more easily than a binder or a course book

An improvable editorial model: each teacher prepares and writes his or her own course, or a few associates in the case of textbooks

A content that can’t be improved once printed

So why not take the idea behind Wikipedia and apply it to school lessons? It is the same type of media and the same problems are faced, so a similar solution would certainly bring similar advantages.

The Wikipedia of school lessons

I think that an incredibly positive impact could result from the creation of a “Wikipedia of school lessons”: a repository of courses, available for free on the Web, which can be improved by everyone and used without restrictions (under free license). That’s why I undertook to create the draft of such a website.

I would see it as a platform intended for teachers and students alike. On the one hand, students could use it to study, learn and revise their lessons. A bit like revision books with courses, revision sheets and exercises, but with all the advantages of the Web and of the wikis. On the other hand, teachers could use it to prepare their courses, thus avoiding that each teacher reinvents the wheel and prepares, all by himself, from scratch, the same course as thousands of others. Currently, each teacher tries, with varying degrees of success, to create his or her own course, with his or her own illustrations, formulations, exercises and controls. Why not share this work? The quality of the content produced would be better and a considerable amount of time would be saved.

There are currently very few lessons or educational resources in general on the web, and those available are scattered, of varying quality, in a repulsive form, paid or locked by copyrights. The goal of this website is to be a unique resource for high-quality, well presented and well-explained courses.

Ideally, I would imagine it as a kind of mix between the Khan Academy, with their interactive tests and videos (which make it as easy to revise our lessons as it is to lose time on YouTube), Kartable, which centralizes well-presented courses for all subjects in one place, and MOOCs, with their community of students helping out each other on forums.

And publishing courses on the Web is not an end in itself for me, but I rather hope it will be a step towards a renewal of teaching and a better educational system.

In class, a huge amount of time is spent copying the lesson. If high-quality courses are available freely on the Web, teachers will waste less time reinventing the wheel, students will no longer waste time copying the lesson, and they will spend more time learning and understanding it. Lessons could also be read and assimilated by the students before going to class, and classes could then cease to be a dictation session and become an exchange between the students and the teacher, much more useful and effective to learn.

Using most of the time when teachers and students are together to copy the lesson is misusing that time. Of course, copying the lesson can help learn it. But copying the lesson is intended to ensure that each student has a copy of it. So let’s use the most effective method to achieve this goal. And if copying the lesson helps students learn, great! But everyone uses the method that suits them best to learn the lesson: explaining it to someone else, making revision sheets, doing exercises, hand-copying the lesson, listening to it in audio, etc.

Students should not be forced to waste the precious and limited time that they have with their teachers to do a task that they could just as well do on their own and that not everyone needs to do. It would be much more profitable to use this privileged time for something else: so that students can ask questions and try to better understand the course, so that teachers can show them concrete applications of what they learn (this is something they are particularly fond of), or so that they can pass on to students the taste of what they learn, which is essential to good learning. Things for which the teacher is not only necessary, but also probably the ideal person, unlike for copying the course.

So, for those who might invoke this kind of argument, no, such a tool will never replace teachers, quite the opposite.

Where the teacher brings value is by better explaining the lesson, through concrete examples and by adapting to what the students understood or not, by answering their questions, by passing on their passion and developing the students’ curiosity and desire to learn. And this, no website or robot can do it in their place.

And I’m not the only one to think this way. Sal Khan, the founder of the Khan Academy, gave two very good TED Talks on the subject, which I highly recommend: Let’s use video to reinvent education, which address how to make better use of the time when teachers and students are together, and Let’s teach for mastery — not test scores, where he explains his vision to avoid having students accumulating gaps throughout their school careers until they prevent them from moving forward. He even wrote a book about his vision for the future of education, The One World Schoolhouse, and he knows his stuff, he is running an online school which has more than 40 million students from all over the world!

Wikicours

The website I plan to launch will be for school lessons in French, for quite a simple reason: I’m French and I live in France. I chose “Wikicours” as name, which translates to “Wikicourse(s)” in English, a name that I think is quite explicit and expresses both the content available (the courses) and its format (the wiki).

Initially, the site will be extremely rudimentary and the courses incomplete, but I prefer to set up only the bare minimum, and then improve it based on user feedback. My aim is to cover most of the subjects from middle school to high school, and eventually add other sections, such as pedagogical advice for teachers, guides to accompany students and their parents during their studies, or extracurricular content that would complement well the lessons.

Whether you are a teacher, a student, a web developer or just curious, you can help! Everyone can contribute and improve both the site itself and the content. Everything will be explained soon in another article!

The site will be available soon at wikicours.org, in a few months at most. (For the moment this address does not lead anywhere and it’s normal: the site does not exist yet.)

The source code of the site will be published on GitHub under MIT license, as well as the courses’ content, which will probably be under Creative Commons license.

And you, would you use such a site, or contribute to it, and if so how? Tell me in the comments, I’d love to hear what you think!


Madoka Kyouno is an energetic girl who is full of passion As the proud, and only, member of the Kamogawa Girls High School Jersey Club, she goes around helping people in need Madoka s li ... 2011-12-25 16:13:46

This is the PV from Lagrange The Flower of Rin-ne (Rinne no Lagrange) Game＆OVA Hybrid Disc what contains the OVA Rinne no Lagrange Kamogawa Days the stroy is after of events from the Seas ... 2012-06-10 19:53:34


A Ward-Round Non-Technical Skills for Surgery (WANTSS) Taxonomy.
Around half of surgical adverse events occur outside the operating room. However the majority of nontechnical skills (NTS) training programs have been developed for the intraoperative environment. Ward rounds are a crucial part of extraoperative care and to date no specific NTS training manual has been developed targeting emergency general surgical ward rounds. To develop a NTS taxonomy for emergency general surgical ward rounds that can be used to improve surgical team members' NTS and improve outcomes. A literature review of existing NTS taxonomies was conducted, followed by semistructured interviews and observational data collection, to determine good and poor surgical ward-round behaviors. These behaviors were reviewed by a panel of subject matter experts and categorized into a taxonomy, using the Non-Technical Skills for Surgeons taxonomy framework as a guide. The Ward-round Non-Technical Skills for Surgery taxonomy includes examples of good and poor ward-round-specific behaviors, grouped into elements and categories. The taxonomy can be used as both a training and teaching manual for the surgical team. Ward rounds are a crucial part of extraoperative surgical care. The Ward-round Non-Technical Skills for Surgery taxonomy provides surgical teams with a manual to help them improve their ward-round NTS.


1. Field of the Invention
An apparatus and method for protecting the fingerboard of a stringed musical instrument from the natural acidity and friction of the players fingers that can cause damage and wear to the fingerboard during normal playing of the instrument.
2. Description of the Prior Art
Stringed musical instruments that embody fingerboards, such as guitars, bass guitars, ukuleles, mandolins, banjos, violins, fiddles and cellos are all susceptible to damage and wear merely by being played by a musician. While playing these types of instruments, the musician must apply pressure and push the instruments strings into the frets and the fingerboard to create notes or chords. This physical contact in addition to the natural acidity of the players fingers can wear down the fingerboard creating indentations as well as tarnishing the fingerboard with friction and grime. In the case of a fretless instrument, the strings are pushed directly into the fingerboard causing the wear to occur at a much faster rate. This embodiment sets out to solve this problem by creating a protective barrier between the players fingers and the fingerboard, safeguarding it from undesirable wear during normal playing of the instrument as well as giving the instrument a unique new look.
U.S. Pat. No. 7,238,870, to Stewart, entitled “Enhanced Fret Saving Device and Process” (2007) describes a specialized at least bilayered device similiformic with the fretboard/fingerboard/keyboard of an instrument is designed to be removingly disposed between the strings and fretboard or frets of the instrument and prevents or mitigates fret damages (from, for example, dimpling or string grooving) owing to string pressures, such as during transport and storage of the instrument in a case or other. Fret integrity and active or functional life is extended in an industrially efficient, economical and elegant way heretofore uncontemplated and/or commercially realized.
This prior art is limited to only protecting the frets of an instrument only when the instrument is not being played, such as during transport and storage or while the instrument is in a case or otherwise and makes playing the instrument impossible while the prior art is installed on the instrument. This embodiment creates a protective barrier between the players fingers and the fingerboard, safeguarding it from undesirable wear while the instrument is being played by the musician.
U.S. Pat. No. 8,704,066, to Jacobsen, entitled “Removable Fretboard Stickers With Musical Indicia” (2014) describes a device and method for teaching note locations, fretboard patterns, scales and chords on the fingerboard of a fretted stringed instrument (“fretboard”), such as an electric or acoustic guitar. The invention comprises a set of independently affixable fretboard adhesive labels (“Stickers”) that adhere to the surface of the stringed instrument, using a mild adhesive, so as not to damage the instrument, and a number of markings, which indicated musical indicia. The device is applied to the instrument by slipping the various individual stickers between the corresponding frets and strings of the stringed instrument and affixing the stickers directly to the fretboard.
Even though the prior art embodies affixable fretboard adhesive labels (“stickers”) to be installed directly to the fretboard of the stringed instrument, the primary object of this prior art is to create a visual device to aid students in learning the location of various musical concepts and features. The prior art calls for the “stickers” to be installed and removed in sets according to what musical concept the student is learning. By either removing or not installing certain sets of “stickers”, the prior art allows areas of the fingerboard to remain uncovered, thus exposing those areas to the natural acidity and friction of the players fingers that can cause damage and wear to the fingerboard during normal playing of the instrument. A great disadvantage of the prior art that I have found, calls for an amount of the “sticker” to be folded over the edge of the fingerboard as to grip the sides of the neck of the instrument. I have found that any amount of a “sticker” that extends onto the sides of the neck of the instrument is prone to peeling and interfering with the players hand movements while it travels up and down the neck during normal playing of the instrument. This embodiment does not interfere with the musicians ability to play the instrument, whereas it is applied only to the face of the fingerboard and does not overlap onto the neck of the instrument. Because the prior art calls for the height of the “stickers” to be smaller than the distance between the frets, as shown on FIG. 3, large areas of the fingerboard are left exposed. Thus the prior art is not suitable as a protective barrier but merely a teaching tool. This embodiment is designed to be used during normal playing of the instrument thereby creating a protective barrier which protects the fingerboard from the natural acidity and friction of the players fingers that can cause damage and wear to the fingerboard. This embodiment will not affect the sound quality of the instrument.
U.S. Pat. No. 6,452,080, to Coonce, entitled “Note Locator for Stringed Instruments” (2002) describes an apparatus and method for a note locator for stringed instruments and method of use. Note locator (10) comprises a flexible planar top and bottom surface wherein the top surface has note position indicators (60) of color and the bottom surface has an adhesive for placement beneath the strings of the instrument along the fingerboard without interfering with the normal operation of the instrument. The user learns correct finger placement by visually identifying colors with notes of the equitempered chromatic scale. Diatonic scale note position indicators (32, 36, 38, 42, 46, 50 and 52) are indicated with various distinct colors while other half step note position indicators (30, 34, 40, 44 and 48) are indicated with a color different from that of the diatonic scale note position indicators. Segmentation (70,72) is provided for dividing note locator (10) so that segments can be added or removed from the fingerboard if desired.
Although the prior art embodies a thin vinyl, or similarly flexible top material with an adhesive bottom surface that is to be installed onto an instruments fingerboard, the primary objective of the prior art is to precisely indicate to the player the location on the strings where the notes are to be played. It is stated in the summary of the prior art, that segmentations means are provided for dividing the note locator so that segments can be added or removed from the fingerboard if desired. By removing segments from the fingerboard, the prior art exposes areas of the fingerboard to the natural acidity and friction of the players fingers that can cause damage and wear to the fingerboard during normal playing of the instrument. Although the prior art is not stating itself to be a fingerboard protector, I have found that it would have a great disadvantage as a fingerboard protector due to the suggested removal of segments of the prior art from the fingerboard, thus exposing said areas of the fingerboard to the natural acidity and friction of the players fingers that can cause damage and wear to the fingerboard during normal playing of the instrument. I have found another disadvantage of the prior art which calls for sufficient additional material for overhang, to wrap around the fingerboard to stabilize the device on the curved fingerboard of the instrument. I have found that any amount of material that extends onto the sides of the neck of the instrument is prone to peeling and interfering with the players hand movements while the players hand travels up and down the instruments neck during normal playing of the instrument. This embodiment does not interfere with the musicians ability to play the instrument, whereas it is applied only to the face of the fingerboard and does not overlap onto the neck of the instrument. This embodiment will not affect the sound quality of the instrument.


Prime Minister Narendra Modi visited Azamgarh in Uttar Pradesh to inaugurate the Poorvanchal Expressway project, which was apparently already inaugurated by Akhilesh Yadav in 2016. Never to lose an opportunity to make loud self-aggrandising speeches, the PM attacked Congress president Rahul Gandhi for allegedly stating that the "Congress is a Muslim party" and also raked up the issue of Muslim women and triple talaq claiming that while he is trying to "save" lakhs of Muslim women, Congress is creating obstacles by protesting against the Triple Talaq Bill.

At the outset, Rahul Gandhi has never made any such statement. It was reported by Urdu daily Inquilab on July 12 that in a meeting with Muslim intellectuals Rahul said, "Yes, Congress is a Muslim party," but the report was soon denied by Congress spokesperson Priyanka Chaturvedi on Twitter. Thereafter, several eminent personalities, such as historian Irfan Habib and Farah Naqvi, who were present at the same meeting, confirmed that no such statement was made.

This is absolute lies&shameful spin by the newspaper. The BJP using it to further its own agenda, yet again, exposes their desperate tactics to mislead the nation. However this will not deter us to fight for India's founding principles& its democratic values. https://t.co/EN15vL7oNa — Priyanka Chaturvedi (@priyankac19) July 12, 2018

Yet none of this deterred Modi from going ahead and creating a blatantly false narrative involving communal hate. Prior to this, during Gujarat elections, Modi had raked up similar completely baseless allegations against respected dignitaries like ex-prime minister of India, ex-vice president of India, Pakistan's high commissioner, Pakistan foreign minister and Congress leaders insinuating that there is a conspiracy being hatched by Congress and Pakistan to defeat the BJP in Gujarat elections.

It is bad enough that these allegations sound like tragic delusions of a schizophrenic patient. It gets worse when the PMO actually gives it in writing, in response to a RTI application, that Modi had no basis whatsoever to make these allegations.

Modi, however, continues to stoop lower and lower even as the entire country calls out his bluffs and condemns his actions which are unbecoming of the office he holds. Not only are the content of his speeches in bad taste and full of blatant lies and misinformation, even his body language and mannerism have become bizarrely threatening and provocative.

Modi's speech at Azamgarh was not just another normal campaign speech. It was a frantic display of sheer panic. He came across as a ruthless autocrat enraged and crippled with fear of losing power. He trembled with rage, his voice shrill, eyes popping out, wrinkles on the forehead, eyebrows twitched.

To give a movie reference, Adolf Hitler from the popular "Hitler reacts" meme featuring the pinnacle scene from German WW II drama film Der Untergang (Downfall) comes to mind.

It is rather intriguing why Modi is showing the signs of fear and desperation of an uncanny kind. In a democracy you win some elections you lose some elections, but Modi seems to have a fear larger than that of just losing elections. Typically, such fear is seen among autocratic leaders who might be sitting over human rights violation and are at the risk of facing criminal prosecution if they lose power.

BJP's 2014 election campaign was a refreshing change. They mainly fought on development agenda. The Gujarat model, acche din, corruption-free nation, women's safety, lower petrol prices — were some of the promises. Hindu Rashtra or Hindutva was nowhere in the scene. People of India thought BJP had left Babri and Godhra behind and were willing to make a fresh start based on real issues, not religion.

But the pretence was dropped soon after coming to power. Initially, it was the fringe elements, which started giving out the message that nothing has changed in the BJP camp, a theocratic homogenous Hindu Rashtra was still their main agenda. The same narrative has now been taken up by senior party leaders, ministers, and even PM Modi. The line between fringe and core began to blur.

The first killing by self-proclaimed cow vigilantism was heard in September 2015 in Dadri, UP. The incidents of such killings kept increasing through the last four years. Data collected in 2017 showed there has been 80 per cent rise in such killings after Modi came to power, and 90 per cent of those killed were Muslims. Amnesty International's Halt the Hate website currently documents over 200 such killings. Only one of these cases have seen a trial and conviction, when in March 2018 a Special Fast Track court in Ramgarh, Jharkhand, awarded life imprisonment to 11 of the 12 accused persons, including a local BJP leader.

By June 2018, the Jharkhand High Court granted bail to eight out of the 11 of them. As soon as these eight came out of jail BJP's Union minister Jayant Sinha publicly honoured them with garlands and sweets, making a global spectacle of how firmly India's largest political party puts its weight behind men who lynch Muslims.

The line between fringe and core has now been fully wiped out.

Meanwhile, several reports and statistics stand proof to the fact that Modi government failed miserably on real issues — economy, jobs, women's safety, social harmony, agrarian crisis, foreign policy, defence, environment, internal and external stability. None of the poll promises have been fulfilled. On the contrary, all issue that were used as poll planks during 2014 elections, stopped being issues post 2014. And if any of us made noise about them, we were told, we either don't understand issues or are anti-nationals:

"Bahat hua beti par waar, ab ki baar Modi Sarkar" — NCRB data shows violence against women has increased in the last four years but it is no longer an issue because "we should not politicise rapes".

Rising prices of petrol was a major issue before 2014. Post 2014, prices kept rising and in June 2018 India saw the highest ever petrol diesel prices. We were told we don't understand how prices work, and that if these prices are lowered, there will be no money left for social schemes.

During the 2014 campaign, we were told there is so much black money in Swiss banks that if all are brought back to India, every citizen could potentially get Rs 15 lakh in their accounts. Post 2014, Amit Shah said "Rs 15 lakh in every account" was a figure of speech, wasn't to be taken seriously.

In June 2018, reports came that Indian money in Swiss banks has gone up by 50 per cent. That's right, nothing came to India, on the contrary the money stashed abroad saw a jump of 50 per cent. But we are now being told that all the money in Swiss banks is not black money.

In April 2018, the Association for Democratic Reforms (ADR) reported that in these four years BJP has emerged as India's richest political party. PM Modi recently inaugurated BJP party headquarters which is claimed to be world's largest political party office. Amit Shah's son Jay Shah's company saw a turnover that reportedly grew 16,000 times in these four years. Certain cooperative banks in Gujarat related to Amit Shah collected over notes worth Rs 3000 crore within first five days of demonetisation before it could be announced that cooperative banks will not deposit the demonetised notes.

I can go on but for brevity sake, take my word for it, on nine out of 10 markers, this government has failed and on rest there are question marks. With this track record Modi has to go to polls in 2019. What do you think his poll narrative would be? He cannot possibly resell the same acche din and Gujarat model. In fact, it is imperative that he diverts people's mind from those issues by raking up the tried and tested communal issues:

Dropping the last curtain on the show of Sabka Saath Sabka Vikas, Modi has therefore unabashedly taken up communal polarisation as the main poll plank for 2019. Congress party is anti-Hindu, vote for us; Rahul Gandhi is fake Hindu, vote for us; Rahul Gandhi is meeting Muslims, vote for us; Hindus are in danger, vote for us; AIMPLB are proposing Sharia courts, vote for us — BJP will seek vote on Hindutva plank alone. They might even orchestrate communal riots and blame it on the Congress.

What remains to be seen is whether people of India would fall for communal narrative or will they vote on issues.

Also read: Shashi Tharoor's 'Hindu Pakistan' analogy is an electoral gift to BJP


Contents

Synopsis

As Emerald takes on the Battle Palace challenge, he and Spenser are informed Guile had captured Jirachi. Spenser tells Emerald he can leave his Pokémon he trusts most in the Battle Palace to take on the challenge. Emerald, however, feels at doubt, since he never actually wanted to make particular bonds with his Pokémon.

Chapter Plot

The Frontier Brains are shocked they witnessed how Guile captured Jirachi. Guile notices Noland, claiming he did not see him two days ago. Noland confronts Guile, since he was the one that attacked Noland and stole all the rental Pokémon. Guile laughs, claiming Noland was weak in that moment. However, Guile thinks of thanking Noland for "borrowing" the rental Pokémon, since he was able to conduct his plans successfully. Brandon marches to attack Guile, but is blown away by his sword. Guile notices the Smeargle, who were captured at Artisan Cave, realizing they had used Sketch to copy the Wish move Jirachi used - like now, to guide the Frontier Brains to its location. Guile commends them for their intelligence, but claims they are late, as he has his Starmie attack the Frontier Brains.

The visitors coming out of the Battle Dome look at the rooftop, seeing the attack on the Frontier Brains. Guile sees the crowd and claims he does not need to lurk in the shadows, since he managed to capture Jirachi to fulfill his wish - to drown the Battle Frontier in the depths of the ocean. Guile releases his rental Pokémon to attack the crowd. Guile disappears, promising to drown the Frontier using Jirachi's wish, as the Frontier Brains move to calm the situation down. Noland realizes Guile must be heading to the Battle Tower, which is the tallest building in the Battle Frontier - closest to the night sky, where starts shine brightly. Noland remembers he did not decipher the last page of the Jirachi report. Noland explains Guile mentioned going to the place closest to the sky. However, Noland thinks Guile does not know about the part of the third eye to let Jirachi grant him the wish. Noland asks others to inform Spenser and Anabel about this.

At the Battle Palace, the phone rings. Emerald asks Spenser won't he answer the phone. Spenser replies once the doors of the Palace close, there won't be any disturbances. Spenser explains Emerald must win six sets of seven virtual trainers to face Spenser himself. However, Spenser thinks Emerald does not have to go through all that trouble. Emerald wants to know the rules to ask Crystal which most suitable Pokémon should he use. Spenser is intrigued, since this facility tests the trainer's "Spirit" - the three Pokémon Emerald will use have to be the ones Emerald trusts most. Emerald is shocked, while Spenser confirms Emerald has to trust his Pokémon, since here, the trainer does not give commands to his Pokémon; they do their battle on their own. Spenser sends Crobat, Slaking and Lapras, the Pokémon he'd use for the battle. Spenser wonders if Emerald has three Pokémon he can fully trust. Emerald looks at his Sceptile, Dusclops and Sudowoodo, displeasing Spenser, since Emerald did not trust the three during the tournament at the Battle Dome; instead, he swapped them for new Pokémon, causing his ultimate downfall.

Hearing the phone still ringing, Spenser answers it and is informed of dark news. Spenser reports to Emerald the Frontier Brains have encountered Jirachi, who had been captured by Guile himself. Emerald fears he has to go straight away to the Battle Tower and will postpone the battle. Spenser halts Emerald, for he can simply leave the Pokémon here for battle; it all depends on how Emerald is spirited enough to trust the Pokémon. Emerald looks at his Pokémon once more, recalling his past adventures. Emerald grins, seeing like this was planned and bound to happen that he has to trust his Pokémon - even if he hates such philosophies. Emerald considers this was likely necessary to happen. Without further ado, Emerald sends Sceptile, Dusclops and Sudowoodo and tells them he fully trusts them to conquer this facility and wishes them the best in this battle - one they are finished, they should catch up with Emerald at the Battle Tower.

Ruby and Sapphire hear something unusual and run off to investigate this threat. Emerald runs as well and sees the two had joined him, towards the Battle Tower.


#608c2c hex color

#608c2c Color Information

In a RGB color space, hex #608c2c is composed of 37.6% red, 54.9% green and 17.3% blue.
Whereas in a CMYK color space, it is composed of 31.4% cyan, 0% magenta, 68.6% yellow and 45.1% black.
It has a hue angle of 87.5 degrees, a saturation of 52.2% and a lightness of 36.1%.
#608c2c color hex could be obtained by blending #c0ff58 with #001900.
Closest websafe color is: #669933.