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text:
Ürkmez bezeichnet:

 Ürkmez (Izmir), Ort im Distrikt Seferihisar der türkischen Provinz Izmir
 Ürkmez-Talsperre, Talsperre in der türkischen Provinz Izmir

Ürkmez ist der Familienname folgender Personen:

 Varol Ürkmez (1937–2021), türkischer Fußballspieler und Schauspieler

redpajama_set_name: RedPajamaWikipedia


The first obstacle facing the Federal Army in the September 1863 invasion of Walker County was getting over Lookout Mountain. Cooper's Gap, Stevens' Gap, and Dougherty's Gap proved to be serviceable. "Ascertaining that the rebels were just commencing to obstruct Cooper's Gap," General James Negley reported to 14th Corps headquarters on September 8, "I pushed forward a brigade and occupied the gap at 8 a.m. The rebels fled after cutting down about a dozen trees. I ordered one regiment to be sent to the foot of the gap and occupy and hold it. The road appears to be good at this point. I will report in the evening. One regiment was sent to seize and occupy Stevens' Gap, which is reported heavily obstructed with fallen trees; these will be cut out to-night if possible. The road to Frick's Gap branches off the Cooper's Gap road about 1 mile northeast of this point; the road is pronounced impassable for wagons. There is a great scarcity of water here; men and animals will suffer if we remain here. The batteries and brigade teams are all up the mountain. I hope to get the supply train up by dark. The only reliable news I have from the valley represents about 2,000 rebel cavalry on the road between Stevens's and LaFayette."
General Beatty commanded the Federal infantry brigade that captured Cooper's Gap. His report of this action provided additional information. "Started at four o'clock this morning," he stated, "and pushed for Cooper's Gap. Surprised a cavalry picket at the foot of the mountain, in McLemore's Cove, Chattanooga Valley. In this little affair we captured five sabers, one revolver, one carbine, one prisoner, and seriously wounded one man. While standing on a peak of Lookout, we saw far off the east long lines of dust trending slowly to the south, and inferred from this that Bragg had abandoned Chattanooga, and was either retiring before us or making preparations to check the center and right of our line."
General Negley elected to take his division over the mountain by way of the Stevens' Gap road. This movement was observed by Confederate cavalry under General William Martin who later informed General Braxton Bragg that the Federals were off the mountain and moving toward Davis' Crossroads, adding that they appeared to be vulnerable to a surprise Confederate attack. "During the 9th." General Bragg later wrote, "it was ascertained that a column, estimated at from 4,000 to 8,000, had crossed Lookout Mountain into the cove by way of Stevens' and Cooper's Gaps." Plans were made to block the Federals at Davis' Crossroads.
General Baird brought his division through Cooper's Gap on September 10th in time to support Negley when he was under attack at Davis' Crossroads. During the days that followed more and more Federal soldiers came through the gaps and began moving up the Cove in preparation for the Battle of Chickamauga. Cooper's Gap continued to be an important route for the rest of the campaign.


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text:
\section{Introduction}
It is known that there is a close relation between Nevanlinna theory and Diophantine Approximation due to the works of Osgood (see \cite{Os1, Os2}) and Vojta (see \cite{V}). Especially, Vojta has given a dictionary which provides the correspondences for the fundamental concepts of these two theories. With this dictionary, one can translate results from Nevanlinna theory into corresponding results from Diophantine Approximation and vice versa. In Nevanlinna theory, the main goal is to establish the second main theorem, which is an inequality where the characteristic functions of meromorphic functions or mappings are bounded above by a sum of certain counting functions. The result in Diophantine Approximation corresponding to the second main theorem is the so-called Schmidt's subspace theorem. Over the last decades much research on these theorems has been done. Many results have been given, for instance \cite{CRY,DT,LG,RS,RV,Shi}.To state some of them, we recall the following.

Let $k$ be a number field. Denote by $M_k$ the set of places (equivalence classes of absolute values) of $k$ and by $M^\infty_k$ the set of archimedean places of $k$. For each $v\in M_k$, we choose the normalized absolute value $|\cdot |_v$ such that $|\cdot |_v=|\cdot|$ on $\mathbf Q$ (the standard absolute value) if $v$ is archimedean, and $|p|_v=p^{-1}$ if $v$ is non-archimedean and lies above the rational prime $p$. For a valuation $v$ of $k$, denote by $k_v$ the completion of $k$ with respect to $v$ and set $$n_v := [k_v :\mathbf Q_v]/[k :\mathbf Q].$$
We put $||x||_v=|x|^{n_v}_v$. The product formula is stated as follows
$$\prod_{v\in M_k}||x||_v=1,\text{ for }  x\in k^*.$$

For $x = (x_0 ,\ldots , x_n)\in k^{n+1}$, define
$$||x||_v :=\max\{||x_0||_v,\ldots,||x_n||_v\},\ v\in M_k.$$
We define the absolute logarithmic height of a projective point $x=(x_0:\cdots :x_n)\in\P^n(k)$ by
$$h(x):=\sum_{v\in M_k}\log ||x||_v.$$
By the product formula, this does not depend on the choice of homogeneous coordinates $(x_0:\cdots :x_n)$ of $x$. If $x\in k^*$, we define the absolute logarithmic height of $x$ by
$$h(x):=\sum_{v\in M_k}\log^+ ||x||_v,$$
where $\log^+a=\log\max\{1,a\}.$

Let $\N_0=\{0,1,2,\ldots\}$ and for a positive integer $d$, we set
$$\mathcal T_d :=\{(i_0,\ldots, i_n)\in\N_0^{n+1}\ :\ i_0+\cdots +i_n=d\}.$$
Let $Q=\sum_{I\in\mathcal T_d}a_Ix^I$ be a homogeneous polynomial of degree $d$ in $k[x_0,\ldots, x_n]$, where $x^I = x^{i_0}\ldots x^{i_n}_n$ for $x=(x_0,\ldots, x_n)$  and $I = (i_0,\ldots,i_n)$. Define $||Q||_v =\max\{||a_I||_v; I\in\mathcal T_d\}$. The height of $Q$ is defined by
$$h(Q)=\sum_{v\in M_k}\log ||Q||_v.$$
For each $v\in M_k$, we define the Weil function $\lambda_{Q,v}$ by
$$\lambda_{Q,v}(x):=\log\frac{||x||_v^d\cdot ||Q||_v}{||Q(x)||_v},\ x\in\P^n(k)\setminus\{Q=0\}.$$

Let $\Lambda$ be an infinite index set. Let $A\subset\Lambda$ be an infinite index subset and $a$ a map $A\rightarrow k$.  We denote this map by $(A,a)$.

\begin{definition}\label{def1.1}{\rm
Let $A\subset\Lambda$ be an infinite index subset and $C_1,C_2\subset A$ subsets
of $A$ with finite complement. Two pairs $(C_1, a_1)$ and $(C_2, a_2)$ are said to be  equivalent if there is a subset $C\subset C_1\cap C_2$ such that $C$ has finite complement in $A$ and such that the restrictions of $a_1,a_2$ to $C$ coincide. Denote by $\mathcal R^0_A$ the set of all equivalence classes of pairs $(C, a)$. Then $\mathcal R^0_A$ has a ring structure. Moreover, we can embed $k$ into $\mathcal R^0_A$ as constant functions.}
\end{definition}

In this paper, we regard a collection of points $\{x(\alpha)\in\P^n(k)\ |\ \alpha\in\Lambda\}$ as a map $x:\Lambda\rightarrow\P^n(k)$. We call a moving hypersurface of degree $d$ in $\P^n(k)$ a homogeneous polynomial in $\mathcal R^0_\Lambda [x_0,\ldots, x_n]$ of the form
$$Q(\alpha)=\sum_{I\in\mathcal T_d}a_I(\alpha)x^I,\text{ for all }\alpha\in\Lambda,$$
where $a_I\ (I\in\mathcal T_d)$ are collections of points, not all zero.

Let $Q_1(\alpha),\ldots,Q_q(\alpha),\ (\alpha\in\Lambda)$ be $q$-moving hypersurfaces in $\P^n(k)$ of degrees $d_1,\ldots, d_q$ respectively. Suppose that $Q_i(\alpha)$ is given by
$$Q_i(\alpha)=\sum_{I^i\in\mathcal T_{d_i}}a_{i,I^i}(\alpha)x^{I^i}.$$

\begin{definition}\label{def1.2}{\rm
An infinite subset $A\subset\Lambda$  is said to be coherent with respect to
$\{Q_j\}^q_{j=1}$ if for every polynomial $P\in k[...,x_{1,I^1},...,x_{j,I^j},...,x_{q,I^q},...]$, $I^j\in\mathcal T_{d_j}$, which is homogeneous in $(...,x_{j,I^j},...)$ for each $j = 1,\ldots, q$, either $P(...,a_{1,I^1}(\alpha),\ldots,a_{j,I^j}(\alpha),...,a_{q,I^q},...)$ vanishes for all $\alpha\in A$, or it vanishes for only finitely many $\alpha\in A.$}
\end{definition}

\begin{lemma}[{see \cite[Lemma 1.4]{LG} and also \cite[Lemma 2.1]{CRY}}]\label{lem1.3}
There exists an infinite subset $A\in\Lambda$ which is coherent with respect
to $\{Q_j\}^q_{j=1}.$
\end{lemma}

Let $A\subset\Lambda$ be an infinite index subset which is coherent with respect to $\{Q_j\}^q_{j=1}$. If $j\in\{1,\ldots, q\}$ and $I\in\mathcal T_{d_j}$ satisfy $a_{j,I}(\alpha)\ne 0$ for some $\alpha\in  A$, then the set $\{\alpha\in A\ |\ a_{j,I}(\alpha)\ne 0\}$ has finite complement in $A$ by the coherence of $A$. Hence, the pair
\begin{align*}
\begin{array}{ccc}
\{\alpha\in  A\ |\ a_{j,I}(\alpha)\ne 0\}&\longrightarrow &k,\\
 \alpha&\mapsto &a_{j,J}(\alpha)/a_{j,I}(\alpha)
\end{array}
\end{align*}
belongs to $\mathcal R^0_A$. We denote by $\mathcal R^0_{A,\{Q_i\}_{i=1}^q}$ the subring of $\mathcal R^0_A$ generated over $k$ by all such pairs. Then $\mathcal R^0_{A,\{Q_i\}_{i=1}^q}$ is an entire ring (see \cite{LG}).

\begin{definition}\label{def1.4}{\rm
We define $\mathcal R_{A,\{Qj\}^q_{j=1}}$ to be the quotient field of $\mathcal R_{A,\{Qj\}^q_{j=1}}^0$.}
\end{definition}
\noindent
We would like to note that if $B\subset  A\subset\Lambda$ are two infinite index subsets and $A$ is coherent with respect to $\{Q_j\}_{j=1}^q$, then so is $B$, and $\mathcal R_{A,\{Qj\}^q_{j=1}}=\mathcal R_{B,\{Qj\}^q_{j=1}}$.

\begin{definition}\label{def1.5}{\rm
Let $x:\Lambda\rightarrow\P^n(k)$ be a collection of points. We say that $x$ is
algebraically non-degenerate (resp. linearly non-degenerate) over $\mathcal R_{\{Qj\}^q_{j=1}}$ if for all infinite subsets $A\subset\Lambda$ that are coherent with respect to $\{Q_j\}^q_{j=1}$, there is no non-zero homogeneous polynomial (resp. non-zero homogeneous polynomial of degree $1$) $Q\in\mathcal R_{A,\{Qj\}^q_{j=1}}[x_0,\ldots,x_n]$ such that $Q(x_0(\alpha),\ldots, x_n(\alpha))=0$, for all $\alpha\in A$ outside a finite subset of $A$.}
\end{definition}

\begin{definition}\label{def1.6}{\rm
We say that a set $\{Q_j\}^q_{j=1}\ (q\ge N+1)$ of homogeneous polynomials
in $\mathcal R^0_\Lambda [x_0,\ldots, x_n]$ is in weakly $N$-subgeneral position if there exists an infinite subset $A\subset\Lambda$ with finite complement such that for any $1\le j_0 <\cdots < j_{N+1}\le q$, and $\alpha\in  A$, the system of equations
\begin{align}\label{1.1}
Q_{j_i}(\alpha)(x_0,\ldots, x_n) = 0, 0\le i \le N,
\end{align}
has only the trivial solution $(x_0,\ldots,x_n)=(0,\ldots,0)$ in $\overline{k}^{n+1}$, where $\overline{k}$ is an algebraic closure of $k$. Furthermore, the set $\{Q_j\}^q_{j=1}\ (q\ge N+1)$ is said to be in $N$-subgeneral position if the equation (\ref{1.1}) has only the trivial solution $(x_0,\ldots,x_n)=(0,\ldots,0)$ in $\overline{k}^{n+1}$ for all $\alpha\in\Lambda$.}
\end{definition}
We will say that the set $\{Q_j\}^q_{j=1}\ (q\ge N+1)$ is in general position (resp. weakly general position) if it is in $n$-general position (resp. weakly $n-$general position).

We now state some pairs of corresponding second main theorem in Nevanlinna theory and Schmidt's subspace theorem for the case of moving targets. Firstly, M. Ru - W. Stoll \cite{RS} proved the following second main theorem for moving hyperplanes (this result was also reproved by M. Shirosaki \cite{Shi} with a simpler proof).

\vskip0.2cm
\noindent
\textbf{Theorem A} (cf. \cite{RS,Shi}). {\it Let $\{a_i\}_{i=1}^q$ be hyperplanes in general position in $\P^n(\C)$. Let $f:  {\C}^m \to \P^n(\C)$ be a meromorphic mapping such that $f$ is linearly nondegenerate over $\mathcal R_{\{a_i\}}$. Then for every $\epsilon >0$, we have}
$$||\ \ (q-n-1-\epsilon)T_f(r) \leq \sum_{i=1}^q N_{(f,a_i)}(r)+ o(T_f(r)).$$
Here, by the notation ``$\| P$'' we mean that the assertion $P$ holds for all $r\in [0,\infty)$ excluding a Borel subset $E$ of the interval $[0,\infty)$ with $\int_E dr<\infty$.

Corresponding to this theorem, M. Ru and P. Vojta \cite{RV} gave the following Schmidt's subspace theorem with moving hyperplane targets.

\vskip0.2cm
\noindent
\textbf{Theorem B} (cf. \cite{RV}). {\it Let $k$ be a number field, $S$ be a finite set of places of $k$, let $q\ge n+1$ be a positive integer and $\epsilon >0$. Let $\Lambda$ be an infinite index set, let $H_1,\ldots, H_q$ be moving hyperplanes in $\P^n(k)$ of degrees $d_1,\ldots, d_q$ respectively and let $x :\Lambda\rightarrow\P^n(k)$ be a collection of points such that:

$\mathrm{(1)}$ $H_1,\ldots, H_q$ is in general position;

$\mathrm{(2)}$ $x$ is linearly non-degenerate over $\mathcal R_{\{H_j\}^q_{j=1}}$;

$\mathrm{(3)}$ $ h(H_j(\alpha)) = o(h(x(\alpha)))\text{ for all } j=1,\ldots, q.$

\noindent
Then there exists an infinite index subset $A\subset\Lambda$ such that
$$\sum_{v\in S}\sum_{j=1}^q\lambda_{H_j(\alpha),v}(x(\alpha))\le ( n+1+\epsilon)h(x(\alpha))$$
for all $\alpha\in  A$.}

\vskip0.2cm
Later on, G. Dethloff and T. V. Tan extended the second main theorem of M. Ru and W. Stoll to the case of moving hypersurfaces. Following the method of  Dethloff and Tan, L. Giang \cite{LG} and Z. Chen, M. Ru and Q. Yan \cite{CRY} proved a corresponding Schmidt's subspace theorem for moving hypersurfaces in general position. Recently, in \cite{Q16-2}, we extended the result of Dethloff and Tan to the case of moving hypersurfaces in weakly subgeneral position as follows.

\vskip0.2cm
\noindent
\textbf{Theorem C} (cf. \cite{Q16-2}, see also \cite{Q16-1}). {\it Let $f$ be a nonconstant meromorphic map of $\mathbf{C}^m$ into $\P^n(\mathbf{C})$. Let $\{Q_i\}_{i=1}^q$ be a family of slowly (with respect to $f$) moving hypersurfaces in weakly $N$-subgeneral position with $\deg Q_j = d_j\ (1\le i\le q).$ Assume that $f$ is algebraically nondegenerate over $\mathcal R_{\{Q_i\}_{i=1}^q}$.  Then for any $\epsilon >0$, we have
$$  ||\ (q-(N-n+1)(n+1)-\epsilon)T_f(r)\le \sum_{i=1}^{q}\dfrac{1}{d_i}N^{[L_j]}_{Q_i(f)}(r)+o(T_f(r)),$$
where $L_j$ is a positive number (explicitly estimated).}

\vskip0.2cm
Our main result in this paper is motivated by the relationship between Nevanlinna theory and Diophantine approximation. We will prove a Schmidt's subspace fo moving hypersurfaces in weakly subgeneral position which corresponds to Theorem C. Namely, we will prove the following.

\noindent
\textbf{Main Theorem.}\ {\it Let $k$ be a number field, $S$ be a finite set of places of $k$, let $q\ge N+1$ be a positive integer and $\epsilon >0$. Let $\Lambda$ be an infinite index set, let $Q_1,\ldots, Q_q$ be moving hypersurfaces in $\P^n(k)$ of degrees $d_1,\ldots, d_q$ respectively and let $x :\Lambda\rightarrow\P^n(k)$ be a collection of points such that:

$\mathrm{(1)}$ the family of polynomials $Q_1,\ldots, Q_q$ is in weakly $N$-subgeneral position;

$\mathrm{(2)}$ $x$ is algebraically non-degenerate over $\mathcal R_{\{Qj\}^q_{j=1}}$;

$\mathrm{(3)}$ $ h(Q_j(\alpha)) = o(h(x(\alpha)))\text{ for all } j=1,\ldots, q.$

\noindent
Then there exists an infinite index subset $A\subset\Lambda$ such that
$$\sum_{v\in S}\sum_{j=1}^q\frac{1}{d_j}\lambda_{Q_j(\alpha),v}(x(\alpha))\le ( (N-n+1)(n+1)+\epsilon)h(x(\alpha))$$
for all $\alpha\in  A$.}

\vskip0.2cm
We would like to note that, when the family of moving hypersurfaces is in weakly general position, i.e., $N=n$, our above result will imply the Schmidt's subspace theorems of L. Giang and also of Z. Chen, M. Ru and Q. Yan.

\section{Some lemmas and theorems}

We recall some algebraic lemmas from \cite{CZ,DT}

\begin{lemma}[{see \cite[Lemma 2.2]{CZ}}]\label{2.5}
Let $A$ be a commutative ring and let $\{\phi_1,\ldots ,\phi_p\}$ be a regular sequence in $A$, i.e., for $i=1,\ldots ,p,\phi_i$ is not a zero divisor of $A/(\phi_1,\ldots ,\phi_{i-1})$. Denote by $I$ the ideal in $A$ generated by $\phi_1,\ldots ,\phi_p$. Suppose that for some $q,q_1,\ldots ,q_h\in A$, we have an equation
$$\phi_1^{i_1}\cdots\phi_p^{i_p}\cdot q=\sum_{r=1}^h\phi_1^{j_1(r)}\cdots\phi_p^{j_p(r)}\cdot q_r, $$
where $(j_1(r),\ldots ,j_p(r))>(i_1,\ldots ,i_p)$ for $r=1,\ldots ,h$. Then $q\in I$.
\end{lemma}
Here, as throughout this paper, we use the lexicographic order on $\mathbf N_0^p$. Namely,
$$(j_1,\ldots,j_p)>(i_1,\ldots,i_p)$$
iff for some $s\in\{1,\ldots ,p\}$ we have $j_l=i_l$ for $l<s$ and $j_s>i_s$.

\begin{lemma}[{see \cite[Lemma 3.2]{DT}}]\label{2.6}
Let $\{Q_i\}_{i=1}^q\ (q\ge n+1)$ be a set of homogeneous polynomials of common degree $d\ge 1$ in $\mathcal R^0_\Lambda[x_0,\ldots ,x_n]$ in weakly general position. Then for any pairwise different $1\le j_0,\ldots ,j_n\le q$, the sequence $\{Q_{j_0},\ldots,Q_{j_n}\}$ of elements in $\mathcal K_{\{Q_i\}}[x_0,\ldots,x_n]$ is a regular sequence, as well as all its subsequences.
\end{lemma}

Let $x$ be a map from an index set $\Lambda$ into $\P^n(k)$. A map $(C,a)\in\mathcal R^0_\Lambda$ is said to be small with respect to $x$ if
$$h(a(\alpha)) = o(h(x(\alpha))),$$
which means that, for every $\epsilon>0$, there exists a subset $C'\subset C$ with finite complement such that $h(a(\alpha))\le \epsilon h(x(\alpha))$ for all $\alpha\in C'$. Denote by $\mathcal K_x$ the set of all small maps. Then, $\mathcal K_x$ is a subring of $\mathcal R^0_\Lambda$.

We denote by $\mathcal C_x$ the set of all positive functions $f$ defined over $\Lambda$ outside a finite subset of $\Lambda$ such that
$$\log^+(f(\alpha)) = o(h(x(\alpha))).$$
Then, $\mathcal C_x$ is a ring. We see that if  $(C,a)\in\mathcal K_x$ then for every $v\in M_k$, the function $||a(\alpha)||_v\in\mathcal C_x$. Furthermore, if $(C,a)$ satisfies that $a(\alpha)\ne 0$ for all $\alpha\in C$ outside a finite subset then the function $\frac{1}{||a(\alpha)||_v}$ also belongs to $\mathcal C_x$

The following lemma is from \cite{LG}.
\begin{lemma}[{see \cite[Lemma 2.2]{LG}}]\label{lem2.3}
Let $\{Q_i\}_{i=0}^n$ be a set of homogeneous polynomials of common degree $d$ in $\mathcal R^0_\Lambda[x_0,...,x_n]$ in general position. Let $A$ be coherent with respect to $\{Q_i\}_{i=0}^n$. Assume that all coefficients of $Q_i,\ i\in\{0,...,n\}$ belong to the field $\mathcal R_{A,\{Q_i\}_{i=0}^n}$. Then, for every $v\in S$, there exist functions $l_{1,v}, l_{2,v}$ such that
$$l_{2,v}(\alpha)||x(\alpha)|^d_v\le\max_{0\le i\le n}||Q_i(\alpha)||_v\le l_{1,v}||x(\alpha)||^d_v,$$
for all $\alpha\in A$ outside a finite subset of $A$. Moreover, if the coefficients of $Q_i,\ i=0,...,n$ belong to $\mathcal K_x$ then $l_{1,v},l_{2,v}\in\mathcal C_x$.
\end{lemma}

To prove our main theorem, we need the following result of Ru and Vojta.

\begin{theorem}[{Schmidt's subspace theorem for moving hyperplane targets \cite[Theorem 1.1]{RV})}]\label{thm2.4}
Let $k$ be a number field, $S$ be a finite set of places of $k$, and let $\epsilon >0$. Let $\Lambda$ be an infinite index set. For every $v\in  S$, let $L^{(v)}_1,\ldots, L^{(v)}_q$ be moving hyperplanes $\Lambda\rightarrow \P^n(k)^*$ and let $x:\Lambda\rightarrow\P^n(k)$ be a collection of points such that we have the following:

$\mathrm{(1)}$ For every $v\in  S,\alpha\in\Lambda, L^{(v)}_1(\alpha),\ldots, L^{(v)}_q(\alpha)$ are in general position.

$\mathrm{(2)}$ For every $v\in  S$, $x$ is non-degenerate over $\mathcal R$ with respect to $L^{(v)}_1,\ldots, L^{(v)}_q$.

$\mathrm{(3)}$ For every $v\in  S, h(L^{(v)}_j(\alpha)) = o(h(x(\alpha))), j=1,\ldots, q.$

\noindent
Then, there exists an infinite index subset $A\subset\Lambda$ such that
$$\sum_{v\in S}\sum_{j=1}^q\lambda_{L_j^{(v)},v}(x(\alpha))\le (n+1+\epsilon)h(x(\alpha)),$$
for all $\alpha\in  A$.
\end{theorem}

We note that M. Ru and P. Vojta only stated their theorem in which the set of hyperplanes is fixed but their proof also valid for the case of moving hyperplanes as above.

\section{Schmidt's subspace theorem for moving hypersurfaces}

We first prove the following lemma.

\begin{lemma}\label{lem3.1}
Let $Q_1,...,Q_{N+1}$ homogeneous polynomials in $\mathcal R^0_\Lambda[x_0,...,x_n]$ of the same degree $d\ge 1$, in weakly $N$-subgeneral position. Then for an infinite subset $A\subset\Lambda$ which is coherent with respect to $\{Q_1,...,Q_{N+1}\}$, there exist $n$ homogeneous polynomials $P_{2},...,P_{n+1}$ in $\mathcal R^0_\Lambda[x_0,...,x_n]$ of the form
$$P_t=\sum_{j=2}^{N-n+t}c_{tj}Q_j,\ c_{tj}\in k,\ t=2,...,n+1,$$
such that $\{P_1,...,P_{n+1}\}$ are in weakly general position in $\mathcal R^0_A[x_0,...,x_n]$, where $P_1=Q_1$.
\end{lemma}
\begin{proof} We assume that $Q_i\ (1\le i\le N+1)$ has the following form
$$ Q_i=\sum_{I\in\mathcal T_d}a_{i,I}x^I.$$
By the definition of weakly subgeneral position, there exists an infinite subset $A'$ of $A$ with finite complement such that, for all $\alpha\in A'$, the system of equations
$$ Q_i(\alpha)(x_0,...,x_n)=0, 1\le i\le N+1, $$
has only the trivial solution $(0,...,0)$ in $\bar k$. We note that $A'$ is coherent with respect to $\{Q_i\}_{i=1}^{N+1}$. Replacing $A$ by $A'$ if necessary, we may assume that $A'=A$.

We fix $\alpha_0\in A$. For each homogeneous polynomial $Q\in\ k[x_0,...,x_n]$, we will denote by $Q^*$ the hypersurface in $\P^n(\bar k)$ defined by $Q$, i.e.,
$$ Q^*=\{(x_0:\cdots :x_n)\in\P^n(\bar k)\ |\ Q(x_0,...,x_n)=0\}.$$
Set $P_1=Q_1$. It is easy to see that
$$\dim\left(\bigcap_{i=1}^tQ^*_i(\alpha_0)\right)\le N-t\text{ for } t=N-n+2,...,N+1,$$
where $\dim\emptyset =-\infty$.

The homogeneous polynomial $P_2$ is constructed as follows. For every irreducible component $\Gamma$ of dimension $n-1$ of $Q^*_1(\alpha_0)$, we put
$$V_{1\Gamma}=\left\{c=(c_2,...,c_{N-n+2})\in k^{N-n+1}\ ;\ \Gamma\subset Q^*_c(\alpha_0),\text{ where }Q_c=\sum_{j=2}^{N-n+2}c_jQ_j\right\}.$$
Then $V_{1\Gamma}$ is a subspace of $k^{N-n+1}$. Since $\dim\left(\bigcap_{i=1}^{N-n+2}Q^*_i(\alpha_0)\right)\le n-2$, there exists $i\in\{2,...,N-n+2\}$ such that $\Gamma\not\subset Q^*_i(\alpha_0)$. This implies that $V_{1\Gamma}$ is a proper subspace of $k^{N-n+1}$. Since the set of irreducible components of dimension $n-1$ of $Q^*_1(\alpha_0)$ is finite,
$$k^{N-n+1}\setminus\bigcup_{\Gamma}V_{1\Gamma}\ne\emptyset. $$
Hence, there exists $(c_{12},...,c_{1(N-n+2)})\in k^{N-n+1}$ such that
$$\Gamma\not\subset P^*_2(\alpha_0)$$
for all irreducible components of dimension $n-1$ of $Q^*_1(\alpha_0)$, where
$P_2=\sum_{j=2}^{N-n+2}c_{1j}Q_j.$
This clearly implies that $\dim\left(P_1^*(\alpha_0)\cap P_2^*(\alpha_0)\right)\le n-2.$

Similarly, for every irreducible component $\Gamma'$ of dimension $n-2$ of $\left(P_1^*(\alpha_0)\cap P_2^*(\alpha_0)\right)$, put
$$V_{2\Gamma'}=\{c=(c_2,...,c_{N-n+3})\in k^{N-n+2}\ ;\ \Gamma\subset Q^*_c(\alpha_0),\text{ where }Q_c=\sum_{j=2}^{N-n+3}c_jQ_j\}.$$
Hence, $V_{2\Gamma'}$ is a subspace of $k^{N-n+2}$. Since $\dim\left(\bigcap_{i=1}^{N-n+3}Q_i^*(\alpha_0)\right)\le n-3$, there exists $i, (2\le i\le N-n+3)$ such that $\Gamma'\not\subset Q_i^*(z_0)$. This implies that $V_{2\Gamma'}$ is a proper subspace of $k^{N-n+2}$. Since the set of irreducible components of dimension $n-2$ of $\left(P_1^*(\alpha_0)\cap P_2^*(\alpha_0)\right)$ is finite,
$$k^{N-n+2}\setminus\bigcup_{\Gamma'}V_{2\Gamma'}\ne\emptyset. $$
Then, there exists $(c_{22},...,c_{2(N-n+3)})\in k^{N-n+2}$ such that
$$\Gamma'\not\subset P_3^*(\alpha_0) $$
for all irreducible components of dimension $n-2$ of $P_1^*(\alpha_0)\cap P_2^*(\alpha_0)$, where
$P_3=\sum_{j=2}^{N-n+3}c_{2j}Q_j.$
It is clear that $\dim\left(P_1^*(\alpha_0)\cap P_2^*(\alpha_0)\cap P_3^*(\alpha_0)\right)\le n-3.$

Repeating again the above step, after the $n$-th step we get the hypersurfaces $P_2,...,P_{n+1}$ satisfying
$$\dim\left(\bigcap_{j=1}^tP_j^*(\alpha_0)\right)\le n-t, \ 1\le t\le n+1. $$
In particular, $\bigcap_{j=1}^{n+1}P_j^*(\alpha_0)=\emptyset.$

Now we denote by $R$ the resultant of $P_1,...,P_{n+1}$. Then we see that
$$ R(\alpha_0)\ne 0. $$
This implies that $R(\alpha)\ne 0$ for all $\alpha\in A$ outside a finite set. Therefore, the system of equations
$$ P_i(\alpha)(x_0,...,x_n)=0,\ \ i=1,...,n+1, $$
has only the trivial solution $(0,...,0)$ in $\bar k^{n+1}$ for all $\alpha\in A$ outside a finite set. This means that $P_1,...,P_{n+1}$ are in weakly general position in $\mathcal R^0_A[x_0,...,x_n]$. We complete the proof of the lemma.
\end{proof}

We need the following lemma.
\begin{lemma}[{see \cite[Proposition 2.5]{LG} and also \cite{DT}}]\label{3.3}
Let $\{P_i\}_{i=1}^q\ (q\ge n+1)$ be a set of homogeneous polynomials of common degree $d\ge 1$ in $\mathcal R^0_\Lambda[x_0,\ldots ,x_n]$ in weakly general position. Then for any nonnegative integer $L$ and for any $J:=\{j_1,\ldots ,j_n\}\subset\{1,\ldots ,q\},$ the dimension of the vector space $\frac{V_L}{(P_{j_1},\ldots ,P_{j_n})\cap V_L}$ is equal to the number of $n$-tuples $(s_1,\ldots ,s_n)\in\mathbf N^n_0$ such that $s_1+\cdots +s_n\le L$ and $0\le s_1,\ldots,s_n\le d-1 $. In particular, for all $L\ge n(d-1)$, we have
$$\dim\frac{V_L}{(P_{j_1},\ldots ,P_{j_n})\cap V_L}=d^n. $$
\end{lemma}

In order to prove the main theorem, we first prove the following theorem, and then we will show that the main theorem is an implication of it.

\begin{theorem}\label{thm3.2}
Let $k$ be a number field, let $S$ be a finite set of places of $k$ and let $q\ge n+1$ be a positive integer and $\epsilon >0$. Further, let $\Lambda$ be an infinite index set, let $Q_1,...,Q_q$ be moving hypersurfaces in $\P^n(k)$ of the same degree $d$ and let $x: \Lambda\rightarrow\P^n(k)$ be a collection of points such that:

$\mathrm{(1)}$ the family of polynomials $Q_1,...,Q_q$ is in weakly $N$-subgeneral position, $N\ge n$,

$\mathrm{(2)}$ $x$ is algebraically nondegenerate over $\mathcal R_{\{Q_i\}_{i=1}^q}$,

$\mathrm{(3)}$ $h(Q_i(\alpha)) = o(h(x(\alpha)))$ for all $i=1,...,q$.

\noindent
Let $A\subset\Lambda$ be an infinite subset which is coherent with respect to $\{Q_i\}_{i=1}^q$. Suppose that:

$\mathrm{(4)}$ each $Q_i$ has coefficients in $\mathcal R_{A,\{Q_i\}_{i=1}^q}$ and has at least one coefficient equal to $1$,

$\mathrm{(5)}$ $Q_i(\alpha)(x(\alpha))\ne 0$ for all $\alpha\in A$, $i=1,...,q.$

\noindent
Then there exists an infinite index subset $B\subset A\subset\Lambda$ such that
$$\sum_{v\in S}\sum_{i=1}^q\frac{1}{d}\lambda_{Q_i(\alpha),v}(x(\alpha))\le ((N-n+1)(n+1)+\epsilon) h(x(\alpha)),$$
for all $\alpha\in B$.
\end{theorem}
\begin{proof}

We set
$$\mathcal I=\{(i_1,...,i_{N+1})\ ; 1\le i_j\le q\text{ for } j=1,\ldots,q, i_1,\ldots,i_{N+1}\text{ distinct}\}. $$
For each $I=(i_1,...,i_{N+1})\in\mathcal I$, we denote by $P_{I,1},...,P_{I,n+1}$ the moving hypersurfaces obtained in Lemma \ref{lem3.1} with respect to the family of moving hypersurfaces $\{Q_{i_1},...,Q_{i_{N+1}}\}$. It is easy to see that, for each $v\in S$ there exists a positive function $h_v\in\mathcal C_x$ such that
\begin{align}\label{3.1}
||P_{I,t}(\alpha)(x(\alpha))||_v\le h_v(\alpha)\max_{1\le j\le N+1-n+t}||Q_{i_j}(\alpha)(x(\alpha))||_v,
\end{align}
for all $I\in\mathcal I$ and $\alpha\in A$ outside a finite subset. Here the function $h_v$ may be chosen common for all $I\in\mathcal I$.

Replacing $A$ by an infinite subset, we may assume that: (\ref{3.1}) holds for all $\alpha\in A$, and for every $v\in S$ there is a permutation $(i_1(v),...,i_q(v))$ of $(1,...,q)$ such that
$$ ||Q_{i_1(v)}(\alpha)(x(\alpha))||_v\le ||Q_{i_2(v)}(\alpha)(x(\alpha))||_v\le\cdots\le ||Q_{i_q(v)}(\alpha)(x(\alpha))||_v. $$
Let $I(v)=(i_1(v),...,i_{N+1}(v))$. Since $P_{I(v),1},\ldots,P_{I(v),n+1}$ are in weakly general position, there exist functions $g_{0,v},g_v\in\mathcal C_x$, which may be chosen independent of $I$ and $\alpha$, such that
$$ ||x(\alpha)||_v^d\le g_{0,v}(\alpha)\max_{1\le j\le n+1}||P_{I(v),j}(\alpha)(x(\alpha))||_v\le g_v(\alpha)||Q_{i_{N+1}(v)}(\alpha)(x(\alpha))||_v.$$
Therefore, we have
\begin{align*}
\prod_{i=1}^q&\dfrac{||x(\alpha)||_v^d}{||Q_i(\alpha)(x(\alpha))||_v}\le g_v^{q-N}(\alpha)\prod_{j=1}^{N}\dfrac{||x(\alpha)||_v^d}{||Q_{i_j(v)}(\alpha)(x(\alpha))||_v}\\
&\le g_v^{q-N}(\alpha)h_v^{n-1}(\alpha)\dfrac{||x(\alpha)||_v^{Nd}}{\bigl (\prod_{j=2}^{N-n+1}||Q_{i_j(v)}(\alpha)(x(\alpha))||_v\bigl )\cdot\prod_{j=1}^{n}||P_{I(v),j}(\alpha)(x(\alpha))||_v}\\
&\le g_v^{q-N}(\alpha)h_v^{n-1}(\alpha)\dfrac{||x(\alpha)||_v^{Nd}}{||P_{I(v),1}(\alpha)(x(\alpha))||_v^{N-n+1}\cdot\prod_{j=2}^{n}||P_{I(v),j}(\alpha)(x(\alpha))||_v}\\
&\le g_v^{q-N}(\alpha)h_v^{n-1}(\alpha)\zeta_v^{(N-n)(n-1)}(\alpha)\dfrac{||x(\alpha)||_v^{Nd+(N-n)(n-1)d}}{\prod_{j=1}^{n}||P_{I(v),j}(\alpha)(x(\alpha))||_v^{N-n+1}},
\end{align*}
where $I(v)=(i_1(v),...,i_{N+1}(v))$ and $\zeta_v$ is a function in $\mathcal C_x$, which is chosen common for all $I(v)\in\mathcal I$, such that
$$||P_{I(v),j}(\alpha)(x(\alpha))||_v\le\zeta_v(\alpha) ||x(\alpha)||_v^d,\ \forall v\in S$$
and for all $\alpha\in A$ outside a finite subset (then by replacing $A$ by an infinite subset we may assume that this inequality holds for all $\alpha\in A$).

The above inequality implies that
\begin{align}\label{3.2}
\begin{split}
\log\prod_{i=1}^q\dfrac{||x(\alpha)||_v^d}{||Q_i(\alpha)(x(\alpha))||_v}\le&\log(g_v^{q-N}h_v^{n-1}\zeta_v^{(N-n)(n-1)})(\alpha)\\
&+(N-n+1)\log\dfrac{||x(\alpha)||_v^{nd}}{\prod_{j=1}^{n}||P_{I(v),j}(\alpha)(x(\alpha))||_v}.
\end{split}
\end{align}
Now, for each non-negative integer $L$, we denote by $V_L$ the vector space (over $\mathcal R^0_{A,\{Q_i\}}$) consisting of all homogeneous polynomials of degree $L$ in $\mathcal R^0_{A,\{Q_i\}}[x_0,\ldots ,x_n]$ and of the zero polynomial. Denote by $(P_{I,1},\ldots ,P_{I,n})$ the ideal in $\mathcal R^0_{A,\{Q_i\}}[x_0,\ldots ,x_n]$ generated by $P_{I,1},\ldots ,P_{I,n}$.

For each positive integer $L$ divisible by $d$ and for each $(i)=(i_1,\ldots,i_n)\in\mathbf N^n_0$ with $||(i)||=\sum_{s=1}^ni_s\le\frac{L}{d}$, we set
$$W^I_{(i)}=\sum_{(j)=(j_1,\ldots ,j_n)\ge (i)}P_{I,1}^{j_1}\cdots P_{I,n}^{j_n}\cdot V_{L-d||(j)||}. $$
It is clear that $W^I_{(0,\ldots,0)}=V_L$ and $W^I_{(i)}\supset W^I_{(j)}$ if $(i)<(j)$ in the lexicographic ordering. Hence, $W^I_{(i)}$ is a filtration of $V_L$.

Let $(i)=(i_1,\ldots ,i_n),(i')=(i_1',\ldots ,i_n')\in\mathbf N^n_0$. Suppose that $(i')$ follows $(i)$ in the lexicographic ordering and $d||(i)||<L$. We consider the following vector space homomorphism
$$\varphi:\gamma\in V_{L-d||(i)||}\mapsto [P_{I,1}^{i_1}\cdots P_{I,n}^{i_n}\gamma]\in\dfrac{W^I_{(i)}}{W^I_{(i')}}, $$
where $[P_{I,1}^{i_1}\cdots P_{I,n}^{i_n}\gamma]$ is the equivalent class in $\frac{W^I_{(i)}}{W^I_{(i')}}$ containing $P_{I,1}^{i_1}\cdots P_{I,n}^{i_n}\gamma$.
We see that $\varphi$ is surjective.
\begin{claim} $\ker\varphi = (P_{I,1},\ldots ,P_{I,n})\cap V_{L-d||(i)||}$.
\end{claim}
In fact, for any $\gamma\in\ker\varphi$, we have
\begin{align*}
P_{I,1}^{i_1}\cdots P_{I,n}^{i_n}\gamma& =\sum_{(j)=(j_1,\ldots,j_n)\ge (i')}P_{I,1}^{j_1}\cdots P_{I,n}^{j_n}g_{(j)}\ \
&=\sum_{(j)=(j_1,\ldots,j_n)> (i)}P_{I,1}^{j_1}\cdots P_{I,n}^{j_n}g_{(j)},
\end{align*}
where $g_{(j)}\in V_{L-d||(j)||}$. By Lemma \ref{2.5} and Lemma \ref{2.6}, we have $\gamma\in (P_{I,1},\ldots ,P_{I,n})$. Then
$$\ker\varphi\subset (P_{I,1},\ldots ,P_{I,n})\cap V_{L-d||(i)||}. $$
Conversely, for any $\gamma\in (P_{I,1},\ldots ,P_{I,n})\cap V_{L-d||(i)||},\ (\gamma\ne 0)$, we have
$$\gamma =\sum_{s=1}^nP_{I,s} h_s,\ \ h_s\in V_{L-d(||(i)||+1)}. $$
It implies that
$$\varphi (\gamma)=\sum_{s=1}^n[P_{I,1}^{i_1}\cdots P_{I,s-1}^{i_s-1}P_{I,s}^{i_s+1}P_{I,s+1}^{i_s+1}\cdots P_{I,n}^{i_n}h_s]. $$
It is clear that $P_{I,1}^{i_1}\cdots P_{I,s-1}^{i_s-1}P_{I,s}^{i_s+1}P_{I,s+1}^{i_s+1}\cdots P_{I,n}^{i_n}h_s\in W^I_{(i')}$, and hence $\varphi (\gamma)=0$, i.e., $\gamma\in\ker\varphi$. Therefore, we have
$$\ker\varphi = (P_{I,1},\ldots ,P_{I,n})\cap V_{L-d||(i)||}.$$
Hence the claim is proved.

This claim yields that
\begin{align}\label{3.4}
m^I_{(i)}:=\dim\dfrac{W^I_{(i)}}{W^I_{(i')}}=\dim\dfrac{V_{L-d||(i)||}}{(P_{I1},\ldots ,P_{In})\cap V_{L-d||(i)||}}.
\end{align}

Fix a number $L$ large enough (chosen later). Set $u=u_L:=\dim V_L=\binom{L+n}{n}$. We assume that
$$ V_L=W^I_{(i)_1}\supset W^I_{(i)_2}\supset\cdots\supset W^I_{(i)_K}, $$
where $W^I_{(i)_{s+1}}$ follows $W^I_{(i)_s}$ in the ordering and $(i)_K=(\frac{L}{d},0,\ldots ,0)$. It is easy to see that $K$ is the number of $n$-tuples $(i_1,\ldots,i_n)$ with $i_j\ge 0$ and $i_1+\cdots +i_n\le\frac{L}{d}$. Then we have
$$ K =\binom{\frac{L}{d}+n}{n}.$$
For each $k\in\{1,\ldots ,K-1\}$ we set $m^I_k=\dim\frac{W^I_{(i)_k}}{W^I_{(i)_{k+1}}}$, and set $m^I_K=1$. Then by Lemma \ref{3.3}, $m^I_k$ does not depends on $\{P_{I,1},\ldots ,P_{I,n}\}$ and $k$, but only on $||(i)_k||$. Hence, we set $m_k:=m^I_k$ and $m(l):=m^I_k$, where $l=||(i)_k||$. We also note that by Lemma \ref{3.3}
\begin{align}\label{3.5}
m_k=d^n
\end{align}
for all $k$ with $L-d||(i)_k||\ge n(d-1)$ (it is equivalent to $||(i)_k||\le\dfrac{L}{d}-n$).

From the above filtration, for each $v\in S$, we may choose a basis $\{\psi^{I(v)}_1,\cdots,\psi^{I(v)}_u\}$ of $V_L$ such that
$$\{\psi^{I(v)}_{u-(m_s+\cdots +m_K)+1},\ldots ,\psi^{I(v)}_u\}$$
 is a basis of $W^{I(v)}_{(i)_s}$. For each $k\in\{1,\ldots,K\}$ and $l\in\{u-(m_k+\cdots +m_k)+1,\ldots, u-(m_{k+1}+\cdots +m_k)\}$, we may write
$$\psi^{I(v)}_l=P_{{I(v)}1}^{i_{1k}}\cdots P_{{I(v)}n}^{i_{nk}}h_{v,l},\ \text{ where } (i_{1k},\ldots,i_{nk})=(i)_k, h_{v,l}\in W^{I(v)}_{L-d||(i)_k||}. $$
We may choose $h_{v,l}$ to be a monomial.

We have the following estimates:
Firstly, we see that
$$\sum_{k=1}^Km_ki_{sk}=\sum_{l=0}^{\frac{L}{d}}\sum_{k:||(i)_k||=l}m(l)i_{sk}=\sum_{l=0}^{\frac{L}{d}}m(l)\sum_{k:||(i)_k||=l}i_{sk}. $$
Note that, by the symmetry $(i_1,\ldots,i_n)\rightarrow (i_{\sigma (1)},\ldots ,i_{\sigma (n)})$ with $\sigma\in S(n)$,  $\sum_{k:||(i)_k||=l}i_{sk}$ does not depend on $s$. We set
$$ a:=\sum_{k=1}^Km_ki_{sk},\ \text{ which is independent of $s$ and $I$}.$$

Then we have
\begin{align*}
||\psi^{I(v)}_l(\alpha)(x(\alpha))||_v&\le ||P_{I(v),1} (\alpha)(x(\alpha))||_v^{i_{1k}}\cdots ||P_{I(v),n} (\alpha)(x(\alpha))||_v^{i_{nk}}||h_{v,l}(\alpha)(x(\alpha))||_v\\
&\le c_{v,l}(\alpha)||P_{I(v),1}(\alpha)(x(\alpha))||^{i_{1k}}_v\cdots ||P_{I(v),n}(\alpha)(x(\alpha))||_v^{i_{nk}}||x(\alpha)||_v^{L-d||(i)_k||}\\
&=c_{v,l}(\alpha)\left (\dfrac{||P_{I(v),1}(\alpha)(x(\alpha))||_v^{i_{1k}}}{||x(\alpha)||_v^d}\right)^{i_{1k}}\cdots\left (\dfrac{||P_{I(v),n} (\alpha)(x(\alpha))||_v}{||x(\alpha)||_v^d}\right)^{i_{nk}}||x(\alpha)||_v^L,
\end{align*}
where $c_{v,l}\in\mathcal C_x$. Taking the product on both sides of the above inequalities over all $l$ and then taking logarithms, we obtain
\begin{align}
\nonumber
\log\prod_{l=1}^u||\psi^{(I(v))}_l(\alpha)(x(\alpha))||_v&\le\sum_{k=1}^Km_k\biggl (i_{1k}\log\dfrac{||P_{I(v),1}(\alpha)(x(\alpha))||_v}{||x(\alpha)||_v^d}+\cdots\\
\label{3.6}
&+i_{nk}\log\dfrac{||P_{I(v),n}(\alpha)(x(\alpha))||_v}{||x(\alpha)||_v^d}\biggl)+uL\log ||x(\alpha)||_v+\log c_{v}(\alpha),
\end{align}
where $c_v=\prod_{l=1}^uc_{v,l}\in\mathcal C_x$.

Hence, (\ref{3.6}) gives
\begin{align*}
\log\prod_{l=1}^u||\psi^{I(v)}_l(\alpha)(x(\alpha))||_v&\le a\left(\log\prod_{i=1}^n\dfrac{||P_{I(v),i}(\alpha)(x(\alpha))||_v}{||x(\alpha)||_v^d}\right)+uL\log ||x(\alpha)||_v+\log c_{v}(\alpha),
\end{align*}
i.e.,
\begin{align*}
a\left(\log\prod_{i=1}^n\dfrac{||x(\alpha)||_v^d}{||P_{I(v),i}(\alpha)(x(\alpha))||_v}\right)\le\log\prod_{l=1}^u\frac{||x(\alpha)||_v^L}{||\psi^{I(v)}_l(\alpha)(x(\alpha))||_v}+\log c_v(\alpha),
\end{align*}

Set $c_{0,v}=g_v^{q-N}h_v^{n-1}\zeta_v^{(N-n)(n-1)}(1+c_v^{(N-n+1)/A})\in\mathcal C_x$. Combining the above inequality with (\ref{3.2}), we obtain that
\begin{align}\label{3.8}
\log\prod_{i=1}^q\dfrac{||x(\alpha)||_v^d}{||Q_i(\alpha)(x(\alpha))||_v}\le\frac{N-n+1}{a}\log\prod_{l=1}^u\dfrac{||x(\alpha)||_v^L}{||\psi^{I(v)}_l(\alpha)(x(\alpha))||_v}+\log c_{0,v}(\alpha).
\end{align}

We now fix a basic $\{\phi_1,...,\phi_u\}$ of $V_L$ such that $\phi_i$ is a monomial of the form $x_0^{i_0}...x_n^{i_n}$. Then $\{\psi^{I(v)}_j\}_{j=1}^u$ can be written as independent linear forms $\{L^{I(v)}_j\}$ in $\phi_1,...,\phi_u$, where $\psi^{I(v)}_j(x)=L^{I(v)}_j(F(x))$ with $F(x)=(\phi_1(x):\cdots :\phi_u(x))$.
Then we have
$$ ||\psi^{I(v)}_l(\alpha)(x(\alpha))||_v=||L^{I(v)}_l(\alpha)(x(\alpha))||_v,\ \ 1\le l\le u, \alpha\in A.$$
Hence, we see that
\begin{align*}
\log\prod_{l=1}^u\dfrac{||x(\alpha)||_v^N}{||\psi^{I(v)}_l(\alpha)(x(\alpha))||_v}&=\left (\log\prod_{l=1}^u\frac{||F(x(\alpha))||_v||L^{I(v)}_l(\alpha)||_v}{||L^{I(v)}_l(\alpha)(x(\alpha))||_v}-u\log ||F(x(\alpha))||_v\right )\\
&+uL\log ||x(\alpha)||_v-\log\prod_{l=1}^u||L^{I(v)}_l(\alpha)||_v.
\end{align*}
Combining this inequality with (\ref{3.8}), we get
\begin{align}\nonumber
\frac{a}{N-n+1}&\log\prod_{i=1}^q\dfrac{||x(\alpha)||_v^d||Q_i(\alpha)||_v}{||Q_i(\alpha)(x(\alpha))||_v}\le \biggl (\log\prod_{l=1}^u\frac{||F(x(\alpha))||_v||L^{I(v)}_l(\alpha)||_v}{||L^{I(v)}_l(\alpha)(x(\alpha))||_v}\\
\nonumber
&-u\log ||F(x(\alpha))||_v\biggl )+uL\log ||x(\alpha)||_v-\log\prod_{l=1}^u||L^{I(v)}_l(\alpha)||_v\\
\label{4.11}
&+\frac{a}{N-n+1}\biggl (\log c_{0,v}(\alpha)+\log\prod_{i=1}^q ||Q_i(\alpha)||_v\biggl ).
\end{align}
Since $\phi_i$ is chosen to be a monomial for every $1\le i\le u$, we now write
$$\psi^{I(v)}_l=\sum_{i=1}^uc^{I(v)}_{i,l}\phi_i\in V_L,\ \ c^{I(v)}_{i,l}\in\mathcal R_{A,\{Q_i\}}, $$
for all $l\in\{1,\ldots ,u\}$. This yields that
$$ L^{I(v)}_l(y_1,...,y_u)=\sum_{i=1}^uc^{I(v)}_{i,l}y_i,\ 1\le l\le u. $$

\begin{claim}\label{cl3.1}
For every $\epsilon'>0$, we have
\begin{align*}
\sum_{v\in S}\log\prod_{l=1}^u\frac{||F(x(\alpha))||_v||L^{I(v)}_l(\alpha)||_v}{||L^{I(v)}_l(\alpha)(x(\alpha))||_v}\le (Lu+\epsilon')h(x(\alpha)),
\end{align*}
for all $\alpha$ in an infinite subset of $A$.
\end{claim}
In deed, we will show that the family of linear forms $\{L^{I(v)l}_l\}_{1\le l\le u}$ will satisfy the assumptions (1), (2) and (3) of Theorem \ref{thm2.4} with respect to the collection of points $F(x): A\rightarrow\P^n(k)$ as follows:

\textbf{Step 1.} We verify the condition (1) of Theorem \ref{thm2.4}. Fix $v\in S$. Since $\{L^{I(v)}_l\}_{1\le l\le u}$ is linear independent over $\mathcal R_{A,\{Q_i\}}$, the determinant $\det (c^{I(v)}_{i,l})_{1\le i,l\le u}\ne 0$ in $\mathcal R_{A,\{Q_i\}}$. Then $\det (c^{I(v)}_{i,l}(\alpha))_{1\le i,l\le u}\ne 0$ for all $\alpha\in A$ outside a finite subset. By passing to an infinite subset of $A$, we may assume that  $\det (c^{I(v)}_{i,l}(\alpha))_{1\le i,l\le u}\ne 0$ for all $\alpha\in A$. This means that $\{L^{I(v)}_l\}_{1\le l\le u}$ is in general position.

\textbf{Step 2.} We verify the condition (2) of Theorem \ref{thm2.4}. Suppose to the contrary that $F(x)$ is linear degenerate over $\mathcal R_{\{L^{I(v)}_l\}_{1\le l\le u}}.$ Then there is an infinite subset $B$ of $A$ which is coherent with respect to $\{L^{I(v)}_l\}_{1\le l\le u}$ and a non-zero linear form
$$L=\sum_{i=1}^ua_iy_i,\ a_i\in\mathcal R_{B,\{L^{I(v)}_l\}_{l=1}^u}$$
 such that
$$ L(\alpha)(F(x(\alpha)))=0 $$
for all $\alpha\in B$ outside a finite subset.  This implies that
$$ Q(\alpha)(F(x(\alpha)))=0, $$
where $Q=\sum_{i=1}^ua_i\phi_i$ is a non-zero homogeneous polynomial in $\mathcal R_{B,\{L^{I(v)}_l\}_{l=1}^u}[x_0,...,x_n]\subset\mathcal R_{B,\{Q_i\}_{i=1}^q}$, for all $\alpha\in B$ outside a finite set. This contradicts the algebraically non-degeneracy of $x$ over $\mathcal R_{\{Q_i\}_{i=1}^q}$.

\textbf{Step 3.} We verify the condition (3) of Theorem \ref{thm2.4}. Since all coefficients of $L^{I(v)}_l\ (1\le l\le u)$ are coefficients of homogeneous polynomials $P^{j_1}_{I(v)1}...P^{j_n}_{I(v)n}$, hence are linear combinations of the coefficients of $Q^{j_1}_{i_1(v)}...Q^{j_{N+1}}_{i_{N+1}(v)}$, there is a positive constant $c_L$ (depending only on $L$) such that
$$ ||L^{I(v)}_l(\alpha)||_\omega=\max_{j}||c^{I(v)}_{j,l}(\alpha)||_\omega\le c_L\max_{1\le j\le n}||Q_{i_j(v)}(\alpha)||_\omega^{N/d}$$
for $\omega\in M_k$. Here, we would like to note that $Q_{i_j(v)}$ has at least one coefficient equal to $1$. Taking the product on both sides of this inequality over all $\omega\in M_k$, we obtain
\begin{align*}
h(L^{I(v)}_l(\alpha))\le \log c_L+\frac{L}{d}\sum_{i=1}^qh(Q_i(\alpha)).
\end{align*}
We also see that
$$ h(F(x(\alpha)))=Lh(x(\alpha))\text{ and }h(Q_i(\alpha))=o(h(x(\alpha))). $$
Therefore, we have
$$h(L^{I(v)}_l(\alpha))=o(h(x(\alpha)))=o(F(x(\alpha))).$$

Then, by Theorem \ref{thm2.4}, there exists an infinite subset $A'\subset A$ such that
\begin{align*}
\sum_{v\in S}\log\prod_{l=1}^u\frac{||F(x(\alpha))||_v||L^{I(v)}_l(\alpha)||_v}{||L^{I(v)}_l(\alpha)(x(\alpha))||_v}\le (u+\epsilon'/L)h(F(x(\alpha)))=(Lu+\epsilon')h(x(\alpha)),
\end{align*}
for all $\alpha\in A'$. Hence the claim is proved.

Without loss of generality we may assume that $A'=A$. Then we may assume that the claim holds for all $\alpha\in A$.

\begin{claim} For every $v\in S$, we have
\begin{align*}
\log ||L^{I(v)}_l(\alpha)||_v &=o(h(x(\alpha))),\\
\log ||Q_i(\alpha)||_v&=o(h(x(\alpha))),
\end{align*}
for $\alpha\in A$.
\end{claim}
Indeed, we have $\log ||L^{I(v)}_l(\alpha)||_v=\max_{j}\log ||c^{I(v)}_{j,l}(\alpha)||_v$ and for each $c^{I(v)}_{j,l}\not\equiv 0$,
\begin{align*}
\log ||c^{I(v)}_{j,l}(\alpha)||_v&\le\sum_{\omega\in M_k}\log^{+}||c^{I(v)}_{j,l}(\alpha)||_\omega =h(c^{I(v)}_{j,l}(\alpha))=o(h(x(\alpha))),\\
\log ||c^{I(v)}_{j,l}(\alpha)||_v&\ge \sum_{\omega\in M_k;||c^{I(v)}_{j,l}(\alpha)||_\omega <1}\log ||c^{I(v)}_{j,l}(\alpha)||_\omega =-h(c^{I(v)}_{j,l}(\alpha))=o(h(x(\alpha))),
\end{align*}
Therefore, we have
\begin{align*}
\log ||L^{I(v)}_l(\alpha)||_v&\le \sum_{\omega\in S; ||c^{I(v)}_{j,l}(\alpha)||_v\ge 1}\log ||c^{I(v)}_{j,l}(\alpha)||_v=o(h(x(\alpha))),\\
\log ||L^{I(v)}_l(\alpha)||_v&\ge \sum_{\omega\in S; ||c^{I(v)}_{j,l}(\alpha)||_v< 1}\log ||c^{I(v)}_{j,l}(\alpha)||_v=o(h(x(\alpha))).
\end{align*}
Thus
$$ \log ||L^{I(v)}_l(\alpha)||_v=o(h(x(\alpha))) $$
with $\alpha\in A$. Similarly, we have
$$ \log ||Q_i(\alpha)||_v=o(h(x(\alpha))) $$
for $\alpha\in A$. Hence, the claim is proved.

Summing both sides of (\ref{3.11}) over all $v\in S$ and using the above two claims with the note that $\log ||F(x(\alpha))||_v=L\log ||x(\alpha)||_v+o(h(x(\alpha)))$, we obtain
\begin{align*}
\frac{a}{N-n+1}\sum_{v\in S}\sum_{i=1}^q\lambda_{Q_i(\alpha),v}(x(\alpha))\le (Lu+\epsilon')h(x(\alpha))+o(h(x(\alpha)))
\end{align*}
with $\alpha\in A$. This is equivalent to
\begin{align}\label{3.9}
\sum_{v\in S}\sum_{i=1}^q\frac{1}{d}\lambda_{Q_i(\alpha),v}(x(\alpha))\le \frac{N-n+1}{da}(Lu+\epsilon')h(x(\alpha))+o(h(x(\alpha))).
\end{align}

We need some estimate for $a$ and $u$. Firstly, we have
$$ u=\binom{L+n}{n}=\frac{(L +1)\ldots (L+n)}{1\ldots n}=\frac{L^n}{n!}+O(L^{n-1}).$$
For each $I_k=(i_{1k},\ldots ,i_{nk})$ with $||(i)_k||\le \frac{L}{d}-n$, we set
$$i_{(n+1)k}=\dfrac{L}{d}-n-\sum_{s=1}^ni_s.$$
 Since the number of nonnegative integer $p$-tuples with summation $\le T$ is equal to the number of nonnegative integer $(p+1)$-tuples with summation exactly equal to $T\in\mathbf Z$, which is $\binom{T+n}{n}$, and since the sum below is independent of $s$, we have
\begin{align*}
a&=\sum_{||(i)_k||\le\frac{L}{d}}m^I_ki_{sk}\ge \sum_{||(i)_k||\le\frac{L}{d}-n}m^I_ki_{sk}=\dfrac{d^n}{n+1}\sum_{||(i)_k||\le\frac{L}{d}-n}\sum_{s=1}^{n+1}i_{sk}\\
&=\dfrac{d^n}{n+1}\cdot \binom{\frac{L}{d}}{n}\cdot (\dfrac{L}{d}-n)=d^n\binom{\frac{L}{d}}{n+1}\\
&=d^n\left (\frac{L^{n+1}}{d^{n+1}(n+1)!}+O(L^n)\right).
\end{align*}
Therefore,
$$ \dfrac{Lu+\epsilon'}{da}=\frac{\frac{L^{n+1}}{n!}+O(L^n)}{\frac{L^{n+1}}{(n+1)!}+O(L^n)}.$$
Then for $\epsilon >0$, we may choose $L$ big enough such that
$$ \dfrac{Lu+\epsilon'}{da}\le (n+1)+\frac{\epsilon}{2(N-n+1)}.$$
Hence (\ref{3.9}) implies that
$$ \sum_{v\in S}\sum_{i=1}^q\frac{1}{d}\lambda_{Q_i(\alpha),v}(x(\alpha))\le \left ((N-n+1)(n+1)+\frac{\epsilon}{2}\right )h(x(\alpha))+o(h(x(\alpha))).
 $$
Since the term $o(h(x(\alpha)))$ will be dominated by $\frac{\epsilon}{2}h(x(\alpha))$, the above inequality yields that
$$  \sum_{v\in S}\sum_{i=1}^q\frac{1}{d}\lambda_{Q_i(\alpha),v}(x(\alpha))\le \left ((N-n+1)(n+1)+\epsilon\right )h(x(\alpha)).$$
The theorem is proved.
\end{proof}

\begin{proof}[{\sc Proof of Main Theorem}]
Assume that
$$Q_i=\sum_{I\in\mathcal T_{d_i}}a_{i,I}x^I,\ a_{i,I}\in\mathcal R^0_\Lambda.$$
By changing the homogeneous coordinates of $\P^n(k)$ if necessary, we may assume that $a_{iI^i_0}\ne 0$ for all $i=1,...,q$, where $I^i_0=(d_i,0,...,0)$.

Let $A$ be an infinite subset of $\Lambda$ which is coherent with respect to $\{Q_i\}_{i=1}^q$. Replacing $A$ by an infinite subset with finite complement, we may assume that $a_{iI^i_0}(\alpha)\ne 0$ for all $\alpha\in A$. By the assumption that $x$ is algebraically nondegenerate over $\mathcal R_{\{Q_i\}_{i=1}^q}$, we have for each $Q_i$,
$$ Q_i(\alpha)(x(\alpha))\ne 0 $$
for all but finite many $\alpha\in A$. Then by passing to an infinite subset with finite complement of $A$, we may assume that
$$ Q_i(\alpha)(x(\alpha))\ne 0 $$
for all $i=1,...,q$ and $\alpha\in A.$

We set
$$ \tilde Q_i=\left (\dfrac{1}{a_{iI^i_0}}Q_i\right)^{d/d_i}, $$
where $d$ is the least common multiple of $d_1,...,d_q$. Then we see that $\tilde Q_i\ (1\le i\le q)$ have the same degree $d$ and if we write
$$ \tilde Q_i=\sum_{I\in\mathcal T_d}\tilde a_{i,I}x^I $$
then $a_{i,I_0}=1$, where $I_0=(d,0,...,0)$.

We now verify that $A$ is coherent with respect to $\{\tilde Q_i\}_{i=1}^q$. Indeed, suppose that we have a non-zero polynomial $P(...,x_{1,I},...,x_{i,J},...,x_{q,K},...)$, $I,J,K\in\mathcal T_d$, which is homogeneous in $(...,x_{i,I},...)$ for each $i\in\{1,...,q\}$ and satisfies
\begin{align}\label{3.10}
P(...,\tilde a_{1,I}(\alpha),...,\tilde a_{i,J}(\alpha),...,\tilde a_{q,K}(\alpha),...)\ne 0.
\end{align}
for some $\alpha\in A$. Then
\begin{align}\label{3.11}
P(...,a_{1I^1_0}^{d/d_1}\tilde a_{1,I}(\alpha),...,a_{i,I^i_0}^{d/d_i}\tilde a_{iJ}(\alpha),...,a_{qI^q_0}^{d/d_q}\tilde a_{qK}(\alpha),...)\ne 0
\end{align}
for some $\alpha\in A$. Since $A$ is coherent with respect to $\{Q_i\}_{i=1}^q$, (\ref{3.11}) holds for all $\alpha\in A$ outside a finite subset. This yields that $A$ is also coherent with respect to $\{\tilde Q_i\}_{i=1}^q$.

It is easy to see that $\{\tilde Q_i\}_{i=1}^q$ is in $N$-subgeneral position. And for an infinite subset $\mathcal B$ of $\Lambda$ we have
$$ \mathcal R_{\mathcal B,\{\tilde Q_i\}_{i=1}^q}=\mathcal R_{\mathcal B,\{Q_i\}_{i=1}^q},$$
and hence $x$ is algebraically non-degenerate over $\mathcal R_{\{\tilde Q_i\}_{i=1}^q}$. Moreover, we have
$$ h(\tilde Q_i(\alpha))=\dfrac{d}{d_i}h(Q_i(\alpha))=o(h(x(\alpha))),\ \ i=1,...,q.$$

Therefore, the family $\{\tilde Q_i\}_{i=1}^n$ satisfies the assumptions of Theorem \ref{thm3.2} with respect to the collection of points $x$. Applying Theorem \ref{thm3.2},  there exists an infinite index subset $B\subset A\subset\Lambda$ such that
$$\sum_{v\in S}\sum_{i=1}^q\frac{1}{d}\lambda_{\tilde Q_i(\alpha),v}(x(\alpha))\le ((N-n+1)(n+1)+\epsilon) h(x(\alpha)),$$
for all $\alpha\in B$. Since $\frac{1}{d}\lambda_{\tilde Q_i(\alpha),v}(x(\alpha))=\frac{1}{d_i}\lambda_{Q_i(\alpha),v}(x(\alpha))$, the above inequality implies that
$$\sum_{v\in S}\sum_{i=1}^q\frac{1}{d_i}\lambda_{Q_i(\alpha),v}(x(\alpha))\le ((N-n+1)(n+1)+\epsilon) h(x(\alpha)),$$
for all $\alpha\in B$. This proves the theorem.
\end{proof}

\noindent
{\bf Acknowledgements.} This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2015.03.

redpajama_set_name: RedPajamaArXiv


Insight into impermanence is central to Buddhist practice.
The Buddhas last words were: All conditioned things are impermanent. Strive on with diligence.
Many religions grapple with impermanence and suffering. Some spiritual traditions equate the world of impermanence with suffering. For these, the solution to suffering is to transcend the world of impermanence.
The Buddha approached suffering differently. He said that suffering is not inherent in the world of impermanence; suffering arises when we cling. When clinging disappears, impermanence no longer gives rise to suffering. The solution to suffering, then, is to end clinging, not to try to escape from the transient world.
One means of reducing clinging is to see the transient nature of what we cling to. This insight can either show us the futility of trying to find lasting happiness in what is impermanent, or it can encourage us to examine deeply why we cling.
Impermanence can be understood in three ways. First, is the obvious, ordinary understanding of impermanence. Second, is understanding from insight, from the intuitive, direct seeing of the nature of things. Finally, there is the way in which seeing impermanence can lead to liberation.
Beyond the ordinary experience of impermanence, Buddhist practice helps us open to the less immediately perceptible realm of impermanence, i.e., insight into the moment-to-moment arising and passing of every perceivable experience. With deep concentrated mindfulness, we see everything as constantly in flux, even experiences that ordinarily seem persistent.
Confronting impermanence profoundly, in this meditative way, can open us to liberation. The final, liberativelevel of impermanence is the movement towards letting go at the deepest level of our psyche. AjahnChahonce said, "If you let go a little, you'll have a little peace. If you let go a lot you'll have a lot of peace. If you let go completely, you'll have complete peace." This release is sometimes called Mahasukha, the Great Happiness, which is said to be the only happiness that is ultimately reliable.
Impermanence causes suffering.It is not simply that there are these two marks of existence, aniccaand dukkha, but rather that aniccabrings about dukkhawhen the world is regarded with ignorance, or to say it differently in the absence of wisdom.
So we see that for the Buddha, although there are three stated marks or characteristics of existence, anicca, dukkha, and anatta, when looked at carefully they appear to reduce more or less to a single, key characteristic: that of impermanence or anicca. The other two derive from that basic feature of reality, when wisdom is absent.
In a sense the entirety of Buddhist philosophy and metaphysics can be seen as a lengthy coming to terms with the truth of impermanence. This isn't surprising, since if we think about it impermanence is the only of the three characteristics that is mind-independent, that would be present in the universe were there no conscious beings. Without conscious beings there would be no dukkha, and there would be no locus around which anotion of self could coalesce. There would only be the impermanence of physical form.
This led us to a realization that anicca(impermanence), holds the key, since the other two characteristics of dukkha(unsatisfactoriness) and anatta(non-self) are brought about by impermanence, when in the presence of ignorance.
In the intellectual environment in which Buddhism evolved, the concept of something being stable and lasting was very important.
In the pre-Buddhist Indian world, the word nityawas often used to designate that foundation, that stability. The view put forward in the Upanishads, for example, suggests that within all the changes of the individual being there is a deep part of one's psyche, called the āmanor the self, that in some way either underlies or transcends…all of the changes that go on moment to moment. If we could only discover this subtle self in our experience and dwell in it moment to moment, we would manage to overcome the transience of the world and become established upon something eternal and everlasting.
This is the background against which Buddhism was working. And the Buddha, with his several excursions into the nature of human experience, basically came to the conclusion that this is an entirely constructed concept. The claim of stability articulated in these traditions is really just an idea that we project onto our world; it is not to be found in actual experience. So one of the principle insights of the whole Buddhist tradition is that the entire world of our experience… is fundamentally not permanent, not unchanging. Everything is in flux.
As far as classical Buddhism is concerned, impermanence is the number one inescapable, and essentially painful, fact of life. It is the singular existential problem that the whole edifice of Buddhist practice is meant to address. To understand impermanence at the deepest possible level (we all understand it at superficial levels), and to merge with it fully, is the whole of the Buddhist path. The Buddha's final words express this: Impermanence is inescapable. Everything vanishes. Therefore there is nothing more important than continuing the path with diligence. All other options either deny the problem or give it short shrift.
The happiness that spiritual practice promises is not endless bliss, endless joy, and soaring transcendence. Who would want that in a world in which there is so much injustice, so much tragedy, so much unhappiness, illness, and death? To feel the scourge of impermanence and loss and to appreciate it at the same time profoundly as the beautiful essence of what it means to be at all—this is the deep truth I hear reverberating in the Buddha's last words. Everything vanishes. Practice goes on.
Small Group Exercise: Choose a quote from below or one of your own & discuss how it relates to your thoughts on Impermance.
This entry was posted in buddhism, Home and tagged Anicca, discussion, Impermanece, Insight Meditation Center, Panna, Three Characteristics, Three Marks of Existence, Wisdom. Bookmark the permalink.


5.0 1 EILV Wed Mar 27 1985 Status on V2.2?
6.0 5 RAVEN1::HEFFELFINGER Mon Apr 01 1985 What IS DecMail?
8.0 3 PRSIS4::DTL Wed Apr 03 1985 rewritten?
16.0 22 RAYNAL::LUGRIN Wed May 15 1985 Where is the information?
21.0 SNICKR::MINER Mon Jul 01 1985 Decmail/Form length?
22.0 3 AUSSIE::BELL Tue Jul 09 1985 Which XXXmail should I use?
23.0 5 MARRHQ::RMURPHY Thu Jul 11 1985 DECmail Broadcast Bug?
26.0 2 USWAV3::EARLY Wed Aug 21 1985 All-in-1 V2.
27.0 8 KAFSV2::READ Thu Aug 22 1985 Converts and File Corruption ??
28.0 2 CHEAPR::WEINER Thu Aug 29 1985 What does the Asterisk Mean?
29.0 8 CHEAPR::WEINER Fri Aug 30 1985 How Does DECMAIL Editting Work?
33.0 3 ECCGY1::COURTS Tue Oct 01 1985 Memo recovery ?
34.0 8 MUDIS3::HOFBAUER Thu Oct 03 1985 DECmail in Cluster?
35.0 NY1MM::ARCHER Tue Oct 15 1985 EXTMAI going from 2.3 to 2.2?
36.0 3 HUMAN::CONKLIN Sat Oct 19 1985 Any way to AUTOFORWARD?
40.0 1 VIRGO::MARMAT Thu Jan 02 1986 Dumping of Folders?
47.0 1 MTV::MAILLET Mon Feb 03 1986 DECMAIL.VIR ???
56.0 2 PTOVAX::WILSONJ Thu Apr 17 1986 WHAT DOES THIS ERROR MEAN????
58.0 1 FURILO::BLESSLEY Tue May 06 1986 Don't DEFINE SYS, MFI, MEM etc.
59.0 2 GVAADG::ROUSSET Fri May 30 1986 LSEDIT Simulator?
60.0 3 CASV Mon Jun 02 1986 Is there a way?
62.0 1 NUWAVE::METZGER Tue Jul 08 1986 network kit???
66.0 BRUTUS::SCANLAND Mon Jul 28 1986 DecMail with Graphics?
70.0 CSSGY2::MCCREADY Tue Aug 19 1986 Quick delete of memos in multiple folders?
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73.0 5 BRAT::DAVISG Tue Sep 02 1986 Recovery from 'Quota Exceeded'????
76.0 1 1559 Tue Sep 30 1986 How long is dist. list name available?
83.0 2 HIGHFI::ZAPPIA Thu Oct 09 1986 Notification of new mail when logging in?
84.0 9 FRSBEE::ZAPPIA Fri Oct 10 1986 decmail system move - need to recreate accounts?
85.0 2 TIPPLE::LASSITER Wed Oct 15 1986 HP to Decmail?
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90.0 2 HIGHFI::ZAPPIA Sun Nov 02 1986 NODUNDEF error message/recovery?
91.0 1 HIGHFI::ZAPPIA Sun Nov 02 1986 What if SYSTEM account captive?
92.0 2 HIGHFI::ZAPPIA Thu Nov 06 1986 IMPORT/EXPORT filespec?
93.0 2 KBOV Fri Nov 07 1986 Search list for DECMAIL$DATA?
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99.0 6 HOTAIR::SHAW Wed Dec 24 1986 Volume set for DECMAIL$DATA ?
104.0 1 VLOV Tue Feb 03 1987 EVE/TPU with Decmail?
105.0 1 JEMEZ::DFREDERICK Thu Feb 05 1987 DECmail in a homogenous VAXcluster?
107.0 1 JEMEZ::DFREDERICK Wed Feb 18 1987 DECmail's replacement?
109.0 1 CSMSRE::VARGAS Thu Feb 26 1987 Can VAXMAIL send to DECMAIL without MTS & MRGATE ?
114.0 2 HARDY::CLARK Tue Mar 10 1987 Where to get documentation?
117.0 2 TROA Wed Mar 18 1987 Who has Done It?
118.0 SMAUG::GUNN Wed Mar 18 1987 Now Under New Management.
124.0 1 --UnknownUser-- Thu May 14 1987 DEFAULT RULER IN DECMAIL ???
126.0 1 LA78 Wed Jun 17 1987 DECmail Discontinued?
131.0 2 CSCMA::TERRY Wed Aug 05 1987 RSX 3.
134.0 2 KAOA Thu Sep 24 1987 Help on DECM and VTX !
137.0 1 MDREDU::NIEVES Mon Dec 14 1987 DECMAIL 2.2. URGENT NET KIT!!!
138.0 2 HPSTEK::SHERMAN Wed Dec 07 1988 How do you Use "@_MAIL"?
139.0 1 HPSTEK::SHERMAN Fri Jan 20 1989 SAVING_REPLY/FORWARD_MESSAGES?
145.0 1 BRYAN::ARCHER Mon Jul 10 1989 DECsystem to MAILbus?
147.0 1 HPSTEK::SHERMAN Fri Apr 13 1990 The WALK is QUICKER!
155.0 1 IOSG::MARSHALL Thu Apr 28 1994 Does DECMAIL still exist?


text:
This landed in our inbox on Sunday 11th December. Admirable speed, given the Durban meeting had only just finished.
A new briefing paper (http://climateactiontracker.org/news/116/Durban-Agreements-a-step-towards-a-global-agreement-but-risk-of-exceeding-3C-warming-remains-scientists.html) includes
– implications of Durban for meeting 2°C
– expected impacts in a 2°C vs. a 3-4°C world
As the climate talks in Durban concluded tonight with a groundbreaking establishment of the Durban Platform to negotiate a new global agreement by 2015, the world continues on a pathway of over 3°C warming with likely extremely severe impacts.
The agreement in Durban to establish a new body to negotiate a global agreement (Ad Hoc Working Group on the Durban Platform for Enhanced Action) by 2015 represents a major step forward. However, the agreement will not immediately affect the emissions outlook for 2020 and has postponed decisions on further emission reductions. Catching up on this postponed action will be increasingly costly.
The Climate Action Tracker estimates that global mean warming would reach about 3.5°C by 2100 with the current reduction proposals on the table. They are definitely insufficient to limit temperature increase to 2°C.
A warming over 3°C could bring the world close to several potential global-scale tipping points, such as:
· Possible dieback of the Amazon rainforest
· Corals reefs being irreversibly replaced by algae and sea grass
· Irreversible loss of the Greenland ice sheets of many centuries to thousands of years
· Risk of release of methane hydrates in ocean floor sediments further adding to the warming
· Permafrost thawing due to fast rising arctic temperatures
A depiction of the types of impacts likely from 1.5°C of warming, through 2°C and 3-4°C is attached at the bottom of the page.
The costs for adaptation and the residual damages from climate change will increase rapidly with warming. Approximate estimates indicate that the most extreme costs will be felt in West Africa and South Asia, with residual damage of 3.5% of regional GDP for 2°C warming and 5-6% for 3°C warming. With a 2°C warming, adaptation costs would be half those associated with a 3°C temperature rise.
The Climate Action Tracker is an independent science-based assessment, which tracks the emission commitments and actions of countries. The website provides an up-to-date assessment of individual national pledges to reduce their greenhouse gas emissions.
Niklas Höhne
for the Climate Action Tracker team from Ecofys, Climate Analytics and the Potsdam Institute for Climate Impact Research

redpajama_set_name: RedPajamaCommonCrawl


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