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Home News & Events Pre-Election Statement on East Timor Elections, Aug. 16, 1999
Pre-Election Statement on East Timor Elections, Aug. 16, 1999
(Background: Former U.S. President Jimmy Carter and The Carter Center are closely monitoring the popular consultation process on autonomy in East Timor. Ten Carter Center observers have been traveling throughout East Timor since July 4 to assess preparations for the vote, the security environment, and the fairness of the campaign, while remaining strictly neutral and nonpartisan. The following is the fifth in a series of weekly reports to be issued by The Carter Center observer mission before and after the consultation.)
The popular consultation moved into a new phase this week, with the end of the registration process and the beginning of the political campaign period, which is scheduled to run through August 27. As the campaign period begins, The Carter Center notes no significant improvements in the security situation in East Timor or fulfillment by the Government of Indonesia (GOI) of its security obligations under the May 5 New York agreements. Indonesian Army, police, and civil officials have failed to intervene against or have actively participated in attacks on pro-independence supporters' activities.
The Carter Center does note the recently announced change in military command in Dili. Further, the U.N. Secretary-General's Special Envoy Jamsheed Marker has issued a statement concerning Indonesian military support for pro-integration militias, assuring that measures are being taken to correct the situation.
Nevertheless, of particular concern to The Carter Center observers during the past week were the following developments:
Intimidation, including explicit death threats, of UNAMET personnel, which has continued in several districts. In one location, UNAMET CIVPOLs officials were assaulted while attempting to protect students who were being attacked by pro-integration militia.
Harassment of pro-independence elements, particularly students and National Council of Timorese Resistance (CNRT) political party workers, which has increased sharply in some locations.
A substantial increase in the number of Internally Displaced Persons (IDPs), with thousands of new IDPs in Viqueque, where an Aug. 11 attack resulted in two deaths, and a worsening of conditions of IDPs in Suai and Same.
Carter Center observers also collected further information that reinforces our previous reports. This includes continued examples of the Indonesian military (TNI) supporting and directing armed pro-integration militias and the police watching passively as militia attacked pro-independence groups, although in at least one case police protected the opening of a CNRT office.
New evidence also shows a strengthening of existing militias and a spread across East Timor of more aggressive pro-integration militias. GOI officials continue to actively campaign for the integration option, and to provide food, other resources, and support to pro-integration groups. Carter Center observers also have seen an unevenly balanced distribution of campaign opportunities in some districts.
Meanwhile, there has been no perceptible movement toward the concentration of TNI troops into district centers, nor has there been any reduction in TNI troops in East Timor. In fact, according to one authoritative report, troop strength is being reinforced.
To date, we have seen little demonstration of GOI behavior designed to create the necessary conditions for a free, open, and democratic consultation in East Timor. There remains widespread concern that continued insecurity could jeopardize the consultation process.
This report is based on interviews with the Commission for Peace and Stability, police, military, local government and church officials, local residents, internally displaced persons, students, and militia members. It also includes interviews with Indonesian Government civilian, military, and police officials; Timorese non-governmental organizations; UNAMET officials; diplomats; and representatives of international organizations. Additional interviews were conducted with diplomats, members of pro-integration groups (which favor the autonomy option for East Timor), members of pro-independence groups, and others. All interviews were held in Bobonaro, Dili, Baucau, Lautem, Viqueque, Ainaro, Manufahi and Covalima districts during the past week.
Physical Attacks and Intimidation: The Aug. 11 attack on the newly-opened student office in Viqueque was the most serious act of intimidation since April. Two persons were killed and at least three wounded in the attack. Two international journalists had to take shelter for 45 minutes while firing continued. Some residents fled 10 miles to Ossu.
Meanwhile, security forces and pro-integration militias continued to threaten and intimidate others across East Timor. Center observers in Baucau noted the unusual presence of Aitarak militia in that district. Two Indonesian air force fighter aircraft swooped low Aug. 12 over the church in the town of Suai (simulating a bombing attack). On Aug. 13, in Dili's old market area, TNI soldiers and Aitarak militia reportedly fired scores of rounds into the air. Residents considered the firing as intimidation on the eve of the campaign period.
Internally Displaced Persons (IDPs): Numbers of IDPs increased significantly this week. A reported 5,000 persons fled the Viqueque attack. Their flight is the largest such movement since April. Some of them who had taken refuge in Ossu sub-district told Carter Center observers that militia members threatened with death all young men over age 15. In Suai, some 700 new IDPs joined others at the church. They told Carter Center observers that pro-integration militia threats had caused them to leave villages elsewhere in Covalima district. In Ritabou, Bobonoro, 62 new IDPs fled in fear.
In Covalima, church officials reported that fear of militia on the roads prevented food convoys from reaching IDPs. In Covalima and Manufahi, church officials said supplies of food, water, and medicine were exhausted.
Political Campaign: Preparations for the campaign period proceeded through the week.On Aug. 8, the two sides agreed on a code of conduct and the two symbols (pro-independence and pro-integration) to be used. While the symbols contain some similar elements, no one with whom Carter Center observers have spoken has expressed concern that this would lead to confusion. Plans for campaigning in the districts vary widely. In Bobonaro, three pro-integration groups have been allocated three days each for exclusive campaigning, while the pro-independence CNRT will have only three days in total. In contrast, in Baucau, the pro-integration party and the pro-independence party have agreedto share campaign days but in separate locations.
The Role of the Police: Failure of the Indonesian police to live up to their obligations under the May 5 New York agreements continues. Police units have not assumed responsibility for protecting the population, nor do they appear seriously to be investigating politically-related incidents reported to them. Police units failed to take action to protect pro-independence groups and foreign journalists during the Aug. 11 attack in Viqueque. The police should be commended, however, for successfully providing security at the Aug. 13 opening of the CNRT office in Suai.
The Role of the Military: Deployment of TNI units across East Timor down to small hamlet-based units has not changed. Repeated promises from government sources that such units will be "cantoned" at the district level have not been met. Carter Center personnel have observed TNI trucks filled with armed soldiers on full alert, apparently on patrol. Furthermore, a senior international observer reported that TNI personnel from West Timor have reinforced locally-based troops. Meanwhile, intimidation by the Indonesian military also continued, including TNI soldiers participating in an Aug. 11 attack on pro-independence activists in Viqueque.
The Role of UNAMET: UNAMET's successful coordination of the approval of a campaign code of conduct and polling symbols is a mark of the respect which most parties accord it. In the two districts for which we have evidence, UNAMET and The Commission for Peace and Stability (KPS) have played supportive roles in getting the parties together to plan for the campaign. Still, UNAMET's ability to function during the rest of the consultation will remain dependent on the perceived security of its local staff, many of whom are being put at risk by their employment.
Thus it remains essential that UN and Indonesian authorities allocate increased attention to the matter.(Background: Former U.S. President Jimmy Carter and The Carter Center are closely monitoring the popular consultation process on autonomy in East Timor. Ten Carter Center observers have been traveling throughout East Timor since July 4 to assess preparations for the vote, the security environment, and the fairness of the campaign, while remaining strictly neutral and nonpartisan. The following is the fifth in a series of weekly reports to be issued by The Carter Center observer mission before and after the consultation.)
Political Campaign: Preparations for the campaign period proceeded through the week.On Aug. 8, the two sides agreed on a code of conduct and the two symbols (pro-independence and pro-integration) to be used. While the symbols contain some similar elements, no one with whom Carter Center observers have spoken has expressed concern that this would lead to confusion.
Plans for campaigning in the districts vary widely. In Bobonaro, three pro-integration groups have been allocated three days each for exclusive campaigning, while the pro-independence CNRT will have only three days in total. In contrast, in Baucau, the pro-integration party and the pro-independence party have agreedto share campaign days but in separate locations.
The Role of UNAMET: UNAMET's successful coordination of the approval of a campaign code of conduct and polling symbols is a mark of the respect which most parties accord it. In the two districts for which we have evidence, UNAMET and The Commission for Peace and Stability (KPS) have played supportive roles in getting the parties together to plan for the campaign. Still, UNAMET's ability to function during the rest of the consultation will remain dependent on the perceived security of its local staff, many of whom are being put at risk by their employment. Thus it remains essential that UN and Indonesian authorities allocate increased attention to the matter.


How does a small town girl become a big city jetsetter? Ask Elena Murzello, who grew up in Wetaskiwin, a prairie outpost outside of Calgary, and now travels between two countries and two coasts from Vancouver to New York City. Her sense of urban cool brings the best of fashion, lifestyle and travel to her site ELENAMURZELLO.COM, where she shares what she absolutely loves about products and places.
Murzello, a young 30-something, always had big dreams. The daughter of a Filipino-Chinese mother and an Italian-East Indian father, she comes by her exotic global influences naturally. After pursuing degrees at the University of Victoria (Nursing), and the University of British Columbia where she got her MBA, she packed her passport and headed out to the airport. Stops in Buenos Aires, London, Paris, Barcelona, Rome, Dubai and Tokyo gave this world traveler a true appreciation of the international lifestyle. But she felt she needed a partner, so she scribbled down a "love list" and immediately met her boyfriend of ten years. When that relationship ended, she felt other women could benefit from her advice, so she authored the book The Love List: A Guide To Getting Who You Want, which was published in 2013.
Her sense of energy, fun and adventure makes her an ideal event planner. For her 30th birthday, she even rented out the Vancouver Aquarium!
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text:
A ThetaHealing egy, az Amerikai Egyesült Államokból származó, alternatív gyógyászati és terápiás módszer. A theta healinget a hivatalos eredettörténet szerint egy Vianna Stibal nevű hölgy találta fel, miután meditációval kigyógyította saját magát a rákból, eközben théta-állapotban kapcsolatba lépett a Teremtővel, és rájött, hogyan lehet a gondolat erejével a DNS-t közvetlenül manipulálni. Kritikusai áltudománynak tekintik.

Alkalmazása
A módszer alkalmazása során egy vezetett meditáción keresztül mind a kezelő (ún. konzulens), mind a kezelt személy (ún. kliens) théta agyhullámra hangolódik. A ThetaHealing szerint a théta állapot segítségével könnyen és gyorsan beazonosíthatóak a kezelt személy tudatalattijának rejtett hiedelmei (pl. fóbia, pszichoszomatikus tünetek). A módszert használó így képessé válik azokat tudatosan megváltoztatni. Alkalmazásával gyors javulás tapasztalható meg a kliens közérzetében. Alkalmazói szerint a megváltozott, nyugodt lelkiállapot jótékony hatással bír az egyén egészségi állapotára, ám erre tudományos bizonyíték nincs, Az MTA Orvosi Tudományok Osztálya állásfoglalása szerint "nem található adat arra vonatkozólag, hogy e kritériumok alapján kedvező hatása lenne a beavatkozásnak", emiatt az áltudományok közé sorolják.

Alapító

A módszert az amerikai Vianna Stibal alkotta meg 1995-ben, aki állítása szerint önmagán alkalmazva gyógyult ki súlyos betegségeiből. Eredeti könyve részletesen leírja gyógyulásának útját és az általa kidolgozott meditációs technika használatát.

Irodalom magyar nyelven
 Stibal, Vianna: ThetaHealing. BIOENERGETIC Kiadó Kft, , Megjelenés éve: 2014
 Stibal, Vianna: ThetaHealing haladóknak - A mindenség erejének kiaknázása. BIOENERGETIC Kiadó Kft, , Megjelenés éve: 2015
 Stibal, Vianna: ThetaHealing - Betegségek és rendellenességek. BIOENERGETIC Kiadó Kft, , Megjelenés éve: 2015
 Stibal, Vianna: ThetaHealing Ritmus – Álomsúly könnyedén. BIOENERGETIC Kiadó Kft, , Megjelenés éve: 2016
 Stibal, Vianna: A ThetaHealing filozófiája - A létezés hét síkja. BIOENERGETIC Kiadó Kft, , Megjelenés éve: 2016
 Stibal, Vianna: Találd meg a lélektársadat a ThetaHealing® segítségével. BIOENERGETIC Kiadó Kft, , Megjelenés éve: 2017

Jegyzetek

Források
 Stibal, Vianna: ThetaHealing
Ön elhiszi, hogy pár millióért visszanő a levágott láb?  index.hu

További információk
 About Vianna Stibal, Founder of ThetaHealing. Hungarian.thetahealing.com
 A ThetaHealing a gyakorlatban - Nagy Szilvia Jakab István vendége. Youtube.com

Alternatív gyógyászat

redpajama_set_name: RedPajamaWikipedia


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1 Articles tagged "papahanaumokuake-expansion"
OHA applauds PMNM expansion
HONOLULU (Aug. 25, 2016) -- In response to the release made by the White House Press Office to elevate the Office of Hawaiian Affairs (OHA) to a monument Co-Trustee and expand the boundaries of the Papahānaumokuākea Marine National Monument, OHA Ka Pouhana/Chief Executive Officer Kamanaʻopono Crabbe issued the following statement: "OHA applauds President Obama's decision to elevate the voice of Native Hawaiians in the management of the lands and waters in the northwestern Hawaiian Islands. Papahānaumokuākea is critical to Native Hawaiian spiritual wellbeing, and this action by the President helps revive our connection to our kūpuna islands and reinforce our…
FAQS on Proposed Papahānaumokuākea Expansion
Q: Where is Papahānaumokuākea Marine National Monument? A: Papahānaumokuākea Marine National Monument (Papahānaumokuākea) was established in June 2006 by Proclamation 8031 issued by President George W. Bush, using his powers under the Antiquities Act. It begins approximately 165 miles northwest of Kauaʻi and continues 1,200 miles to Kure Atoll, and currently encompasses 139,797 square miles. The current boundaries extend 50 nautical miles from the shore of each of the Northwest Hawaiian Islands (NWHI). Q: What is the proposal for additional protection? A: In March 2016, the Papahānaumokuākea - Native Hawaiian Cultural Working Group (NHCWG) specified that additional protection was necessary…

redpajama_set_name: RedPajamaCommonCrawl


Research areas related to Physics, Chemistry and Earth Sciences. This programmes aims to produce graduates who are better equipped to face life's challenges. The programmes are designed to develop responsible citizens and foster national unity, to develop students' physical, spiritual, emotional health and soft skills.


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\section{Introduction}
\label{sec:intro}

We study the existence of ``simple'' variables in boolean conjunctive normal forms, considered as clause-sets. ``Simple'' here means a variable occurring not very often. A major use of the existence of such variables is in inductive proofs of properties of minimally unsatisfiable clause-sets, using splitting on a variable to reduce $n$, the number of variables, to $n-1$: here it is vital that we have control over the changes imposed by the substitution, and so we want to split on a variable occurring as few times as possible. The background for these considerations is the enterprise of classifying minimal unsatisfiable clause-sets $F$ in dependency on the deficiency $\delta(F) := c(F) - n(F)$, the difference between the number $c(F) := \abs{F}$ of clauses of $F$ and the number $n(F) := \abs{\var(F)}$ of variables of $F$. The most basic fact is $\delta(F) \ge 1$, as first shown in \cite{AhLi86}. For $\delta(F) = 1$ the structure is completely known (\cite{AhLi86,DDK98,Ku99dKo}, for $\delta(F) = 2$ the structure after reduction of singular variables (occurring in one sign only once) is known (\cite{KleineBuening2000SubclassesMU}), while for $\delta(F) \in \set{3,4}$ only basic cases have been classified (\cite{XD99}).

The starting point of our investigation is Lemma C.2 in \cite{Ku99dKo}, where it is shown that a minimally unsatisfiable clause-set $F$ must have a variable $v$ with at most $\delta(F)$ positive and at most $\delta(F)$ negative occurrences; we write this as $\ldeg_F(v) \le \delta(F)$ and $\ldeg_F(\overline{v}) \le \delta(F)$, using the notion of \emph{literal degrees} (the number of occurrences of the literal). Thus we have $\vdeg_F(v) \le 2 \delta(F)$, using the \emph{variable degree} $\vdeg_F(v) := \ldeg_F(v) + \ldeg_F(\overline{v})$. Using the \emph{minimum variable degree} (min-var-degree) $\minvdeg(F) := \min_{v \in \var(F)} \vdeg_F(v)$ of $F$, this becomes $\minvdeg(F) \le 2 \delta(F)$. In this article we show a sharper bound on $\minvdeg(F)$ for a larger class of clause-sets $F$. More precisely, we show that the worst-cases $\ldeg_F(v), \ldeg_F(\overline{v}) \le \delta(F)$ can not occur at the same time (for a suitable variable), but actually $\ldeg_F(v) + \ldeg_F(\overline{v}) - \delta(F)$ only grows logarithmically in $\delta(F)$, and this for a larger class of formulas.

The larger class of clause-sets considered is the class $\mc{LEAN}$ of \emph{lean clause-sets}, which are clause-sets having no non-trivial autarky. For an overview on the theory of minimally unsatisfiable clause-sets and on the theory of autarkies see \cite{Kullmann2007HandbuchMU}. The deficiency $\delta(F) \in \mathbb{Z}$ of clause-sets is replaced by the \emph{surplus} $\surp(F) \in \mathbb{Z}$, which is the minimal deficiency over all clause-sets $F[V]$ for non-empty variable sets $V \subseteq \var(F)$, where $F[V]$ is obtained from $F$ by removing clauses which have no variables in $V$, and restricting the remaining clauses to $V$; see \cite{Kullmann2007ClausalFormZI} for more information on the surplus of (generalised) clause-sets. We need to count multiple occurrences of clauses here (which might arise during the process of removing literals with variables not in $V$), and thus actually multi-clause-sets $F$ are used here. Note that by considering $V = \var(F)$ we have $\surp(F) \le \delta(F)$, and by considering $V = \set{v}$ for $v \in \var(F)$ we get $\surp(F) \le \minvdeg(F) - 1$. Now the main result of this article (Theorem \ref{thm:leanminvardeg}) is
\begin{displaymath}
  \minvdeg(F) \le \nonmer(\surp(F))
\end{displaymath}
for lean $F$, where $\nonmer: \mathbb{N} \rightarrow \mathbb{N}$ (see Definition \ref{def:minvdegdef}) is a super-linear function with $\nonmer(k) \le k + 1 + \log_2(k)$. As an application we obtain (Corollary \ref{cor:leanminvardegsat}), that if a (multi-)clause-set $F$ has no variable occurring with degree at most $\delta(F) + 1 + \log_2(\delta(F))$, then $F$ has a non-trivial autarky. It is an open problem whether such an autarky can be found in polynomial time (for arbitrary clause-sets $F$); we conjecture (Conjecture \ref{con:findauthard}) that this is possible.

\paragraph{Related work}

This article appears to be the first systematic study of the problem of minimum variable occurrences in minimally unsatisfiable clause-sets and generalisations, in dependency on the deficiency, asking for the existence of a variable occurring ``infrequently'' in general, or for extremal examples where all variables occur not infrequently. The problem of maximum variable occurrences (asking for the existence of a variable occurring frequently in general, or for extremal examples where all variables occur not frequently) in uniform (minimally) unsatisfiable clause-sets, in dependency on the (constant) clause-length, has been studied in the literature, starting with \cite{Tovey1984NPcomplete}; for a recent article see \cite{GebauerSzaboTardos2010LocalLemma}.

\paragraph{Overview}

In Section \ref{sec:prelim} basic notions and concepts regarding clause-sets, autarkies and minimal unsatisfiability are reviewed. Section \ref{sec:nonmer} introduces the numbers $\nonmer(k)$ and proves exact formulas and sharp lower and upper bounds. Section \ref{sec:leansurp} contains the main results. First in Subsection \ref{sec:specialcaseMU} the bound is shown for minimally unsatisfiable clause-sets (Theorem \ref{thm:MUminvdegdef}). In Subsection \ref{sec:proofgencase} the bound then is lifted to lean clause-sets, proving Theorem \ref{thm:leanminvardeg}. The immediate corollary of Theorem \ref{thm:leanminvardeg} is, that if the asserted upper bound on the minimal variable degree is not fulfilled, then a non-trivial autarky must exist (Corollary \ref{cor:leanminvardegsat}). In Subsection \ref{sec:findaut} the problem of finding such autarky is discussed, with Conjecture \ref{con:findauthard} making precise our believe that one can find such autarkies efficiently. In Section \ref{sec:strengthbound} we discuss the sharpness of the bound, and the possibilities to generalise it further. Finally, in Section \ref{sec:open} open problems are stated, culminating in the central Conjecture \ref{con:finhit} about the classification of unsatisfiable hitting clause-sets (or ``disjoint tautologies'' in the terminology of DNFs).

\section{Preliminaries}
\label{sec:prelim}

We follow the general notations and definitions as outlined in \cite{Kullmann2007HandbuchMU}, where also further background on autarkies and minimal unsatisfiability can be found. We use $\mathbb{N} = \set{1,2,\dots}$ and $\NN_0 = \mathbb{N} \cup \set{0}$.

\subsection{Clause-sets}
\label{sec:prelimcls}

Complementation of literals $x$ is denoted by $\overline{x}$, while for a set $L$ of literals we define $\overline{L} := \set{\overline{x} : x \in L}$. A \textbf{clause} $C$ is a finite and clash-free set of literals (i.e., $C \cap \overline{C} = \emptyset$), while a \textbf{clause-set} is a finite set of clauses. We use $\var(F) := \bigcup_{C \in F} \var(C)$ for the set of variables of $F$, where $\var(C) := \set{\var(x) : x \in C}$ is the set of variables of clause $C$, while $\var(x)$ is the underlying variable for a literal $x$. For a clause-set $F$ we denote by $n(F) := \abs{\var(F)} \in \NN_0$ the number of variables and by $c(F) := \abs{F} \in \NN_0$ the number of clauses. The \textbf{deficiency} of a clause-set is denoted by $\delta(F) := c(F) - n(F) \in \mathbb{Z}$. We call a clause $C$ \textbf{full} for a clause-set $F$ if $\var(C) = \var(F)$, while a clause-set $F$ is called full if every clause is full. For a finite set $V$ of variables let $A(V)$ be the set of all $2^{\abs{V}}$ full clauses over $V$. Thus full clause-sets are exactly the sub-clause-sets of some $A(V)$. A \textbf{partial assignment} is a map $\varphi: V \rightarrow \set{0,1}$ for some (possibly empty) set $V$ of variables. The application of a partial assignment $\varphi$ to a clause-set $F$ is denoted by $\varphi * F$, which yields the clause-set obtained from $F$ by removing all satisfied clauses (which have at least one literal set to $1$), and removing all falsified literals from the remaining clauses. A clause-set $F$ is satisfiable iff there is a partial assignment $\varphi$ with $\varphi * F = \top := \emptyset$, otherwise $F$ is unsatisfiable. All $A(V)$ are unsatisfiable.

These notions are generalised to \textbf{multi-clause-sets}, which are pairs $(F, m)$, where $F$ is a clause-set and $m: F \rightarrow \mathbb{N}$ determines the multiplicity of the clauses. Now $c((F,m)) := \sum_{C \in F} m(C)$, while the application of partial assignments $\varphi$ to a multi-clause-set $F$ yields a \emph{multi-}clause-set $\varphi * F$, where the multiplicity of a clause $C$ in $\varphi * F$ is the sum of all multiplicities of clauses in $F$ which are shortened to $C$ by $\varphi$. For example if $\varphi$ is a total assignment for $F$ (assigns all variables of $F$) which does not satisfying $F$ (i.e., $\varphi * F \not= \top$), then $\varphi * F$ is $(\set{\bot},(f)_{C \in \set{\bot}})$, where $\bot := \emptyset$ is the empty clause, while $f \in \mathbb{N}$ is the number of clauses (with their multiplicities) of $F$ falsified by $\varphi$.

For the number of occurrences of a literal $x$ in a (multi-)clause-set $(F,m)$ we write $\ldeg_F(x) := \sum_{C \in F, x \in C} m(C)$, called the \textbf{literal-degree}, while the \textbf{variable-degree} of a variable $v$ is defined as $\vdeg_F(v) := \ldeg_F(v) + \ldeg_F(\overline{v})$. A \textbf{singular variable} in a (multi-)clause-set $F$ is a variable occurring in one sign only once (i.e., $1 \in \set{\ldeg_F(v), \ldeg_F(\overline{v})}$). A (multi-)clause-set is called \textbf{non-singular} if it does not have singular variables.

For a set $V$ of variables and a multi-clause-set $F$ by $F[V]$ the \textbf{restriction} of $F$ to $V$ is denoted, which is obtained by removing clauses from $F$ which have no variables in common with $V$, and removing from the remaining clauses all literals where the underlying variable is not in $V$ (note that this can increase multiplicities of clauses).

\subsection{Autarkies}
\label{sec:prelimAut}

An \textbf{autarky} for a clause-set $F$ is a partial assignment $\varphi$ which satisfies every clause $C \in F$ it touches, i.e., with $\var(\varphi) \cap \var(C) \not= \emptyset$. The empty partial assignment is always an autarky for every $F$, the \textbf{trivial autarky}. If $\varphi$ is an autarky for $F$, then $\varphi * F \subseteq F$ holds, and thus $\varphi * F$ is satisfiability-equivalent to $F$. A clause-set $F$ is \textbf{lean} if there is no non-trivial autarky for $F$. A weakening is the notion of a \textbf{matching-lean} clause-set $F$, which has no non-trivial \textbf{matching autarky}, which are special autarkies given by a matching condition (for every clause touched, a unique variable underlying a satisfied literal must be selectable). The process of applying autarkies as long as possible to a clause-set is confluent, yielding the \textbf{lean kernel} of a clause-set. Computation of the lean kernel is NP-hard, but the \textbf{matching-lean kernel}, obtained by applying matching autarkies as long as possible, which is also a confluent process, is computable in polynomial time. Note that a clause-set $F$ is lean resp.\ matching lean iff the lean resp.\ matching-lean kernel is $F$ itself. For every matching-lean multi-clause-set $F \not= \top$ we have $\delta(F) \ge 1$, while in general a multi-clause-set $F \not= \top$ is matching lean iff $\surp(F) \ge 1$, where the \textbf{surplus} $\sigma(F) \in \mathbb{Z}$ is defined as the minimum of $\delta(F[V])$ for all $\emptyset \not= V \subseteq \var(F)$. Note that while w.r.t.\ general autarkies there is no difference between a multi-clause-set and the underlying clause-set, for matching autarkies there is a difference, due to the matching condition. For more information on autarkies see \cite{Kullmann2007HandbuchMU,Kullmann2007ClausalFormZI}.

\subsection{Minimally unsatisfiable clause-sets}
\label{sec:MUprelim}

The set of minimally unsatisfiable clause-sets is $\Musat$, the set of all clause-sets which are unsatisfiable, while removal of any clause makes them satisfiable. Furthermore the set of saturated minimally unsatisfiable clause-sets is $\Smusat \subset \Musat$, which is the set of minimally unsatisfiable clause-sets such that addition of any literal to any clause renders them satisfiable. We recall the fact that every minimally unsatisfiable clause-set $F \in \Musat$ can be \textbf{saturated}, i.e., by adding literal occurrences to $F$ we obtain $F' \in \Smusat$ with $\var(F') = \var(F)$ such that there is a bijection $\alpha: F \rightarrow F'$ with $C \subseteq \alpha(C)$ for all $C \in F$. Some basic properties of $\Musat$ and $\Smusat$ w.r.t.\ the application of partial assignments are given in the following lemma.

\begin{lem}\label{lem:auxSMUSAT}
  For all clause-sets $F$ we have:
  \begin{enumerate}
  \item\label{lem:auxSMUSAT1} $F \in \Smusat$ iff for all $v \in \var(F)$ and $\varepsilon \in \set{0,1}$ we have $\pao{v}{\varepsilon} * F \in \Musat$.
  \item\label{lem:auxSMUSAT2} If for some variable $v$ holds $\pao{v}{0} * F \in \Smusat$ and $\pao{v}{1} * F \in \Smusat$, then $F \in \Smusat$.
  \item\label{lem:auxSMUSAT3} If for some variable $v$ holds $\pao{v}{0} * F \in \Musat$ and $\pao{v}{1} * F \in \Musat$, then $F \in \Musat$.
  \end{enumerate}
\end{lem}
For more information on minimal unsatisfiability see \cite{Kullmann2007HandbuchMU,Kullmann2007ClausalFormZII}.

\section{Non-Mersenne numbers}
\label{sec:nonmer}

Splitting on variables with minimum occurrence in minimally unsatisfiable clause-sets leads by Theorem \ref{thm:MUminvdegdef} to the following recursion. The understanding of this recursion is the topic of this section. On a first reading, only Definition \ref{def:minvdegdef} and the main results, Lemma \ref{lem:solveN} and Corollary \ref{cor:upperboundnonmer}, need to be considered.

\begin{defi}\label{def:minvdegdef}
  For $k \in \mathbb{N}$ let $\nonmer(k) := 2$ if $k = 1$, while else
  \begin{displaymath}
    \nonmer(k) := \max_{i \in \tb{2}{k}} \min(2 \cdot i, \nonmer(k-i+1) + i).
  \end{displaymath}
\end{defi}
Remarks:
\begin{enumerate}
\item This is sequence \url{http://oeis.org/A062289} in the ``On-Line Encyclopedia of Integer Sequences''. It can be defined as the enumeration of those natural numbers containing the string ``10'' (at consecutive positions). The sequence leaves out exactly the number of the form $2^n-1$ for $n \in \mathbb{N}$, and thus the name. The sequence consists of arithmetic progressions of slope $1$ and length $2^m - 1$, $m = 1, 2, \dots$, each such progression separated by an additional step of $+1$. The recursion in Definition \ref{def:minvdegdef} is new, and so we can not use these characterisations, but must directly prove the basic properties.
\item The value of $\nonmer(k)$ for $k = (1), (2,3,4), (5,\dots,11)$, $(12,\dots,26)$ is $(2), (4,5,6),\\ (8,\dots,14)$, $(16,\dots,30)$.
\item For $k \ge 2$ we have $\nonmer(k) \ge 4$. This holds since $\nonmer(2) = 4$, while the induction step for $k \ge 3$ is $\nonmer(k) = \max_{i \in \tb 2k} \min(2 i, \nonmer(k-i+1)+i) \ge \min(4, \min(4+2,1+3)) = 4$.
\item By induction and by definition we have $k+1 \le \nonmer(k) \le 2 \cdot k$ for $k \in \mathbb{N}$.
\end{enumerate}

For a sequence $a: \mathbb{N} \rightarrow \mathbb{R}$ and $k \in \mathbb{N}$ let $\bmm{\Delta a(k)} := a(k+1) - a(k)$ be the step in the value of the sequence from $k$ to $k+1$. The next number in the sequence of non-Mersenne numbers is obtained by adding $1$ or $2$ to the previous number:
\begin{lem}\label{lem:stepNM}
  For $k \in \mathbb{N}$ holds $\Delta \nonmer(k) \in \set{1,2}$.
\end{lem}
\begin{proof} {\sloppy
  For $k=1$ we get $\Delta \nonmer(1) = 2$. Now consider $k \ge 2$. We have $\nonmer(k+1) = \max(\min(4,\nonmer(k)+2), \max_{i \in \tb 3{k+1}} \min(2 i, \nonmer(k-i+2)+i)) =
  \max_{i \in \tb 3{k+1}} \min(2 i, \nonmer(k-i+2)+i) = \max_{i \in \tb 2k} \min(2 (i+1), \nonmer(k-(i+1)+2)+(i+1)) =
  \max_{i \in \tb 2k} \min(2 i + 2, \nonmer(k-i+1)+i+1) = 1 + \max_{i \in \tb 2k} \min(2 i + 1, \nonmer(k-i+1)+i)$.

Thus on the one hand we have $\nonmer(k+1) \ge 1 + \max_{i \in \tb 2k} \min(2 i, \nonmer(k-i+1)+i) = 1 + \nonmer(k)$, and on the other hand $\nonmer(k+1) \le 1 + \max_{i \in \tb 2k} \min(2 i + 1, \nonmer(k-i+1)+i+1) = 2 + \nonmer(k)$.
}
\end{proof}

\begin{corol}\label{cor:NMmon}
  $\nonmer: \mathbb{N} \rightarrow \mathbb{N}$ is strictly increasing.
\end{corol}

\begin{corol}\label{cor:NMdiff}
  We have $\nonmer(a+b) \ge \nonmer(a) + b$ for $a \in \mathbb{N}$ and $b \in \NN_0$, and thus $\nonmer(a - b) \le \nonmer(a) - b$ for $b \le a$.
\end{corol}

Instead of considering the maximum over $k-1$ cases $i \in \tb 2k$ to compute $\nonmer(k)$, we can now simplify the recursion to only one case $i(k) \in \tb 2k$, and for that case also consideration of the minimum is dispensable:
\begin{lem}\label{lem:simprecnm}
  For $k \in \mathbb{N}$, $k \ge 2$, let $i(k) \in \mathbb{N}$ be the smallest $i \in \tb 2k$ with $i \ge \nonmer(k-i+1)$ (note that $k \ge \nonmer(k-k+1) = 2$, and thus $i(k)$ is well-defined). For example we have $i(2) = 2$, $i(3) = 3$, $i(4) = 4$ and $i(5) = 4$. Then we have:
  \begin{enumerate}
  \item\label{lem:simprecnm1} $i(k) - \nonmer(k-i(k)+1) \le 2$.
  \item\label{lem:simprecnm2} $\nonmer(k) = \nonmer(k-i(k)+1)+i(k)$.
  \item\label{lem:simprecnm3} $\Delta i(k) \in \set{0,1}$.
  \end{enumerate}
\end{lem}
\begin{proof} We have $i(k) = 2$ iff $k=2$, while for $k=2$ the assertions hold trivially; so assume $k \ge 3$ and $i(k) \ge 3$. Part \ref{lem:simprecnm1} follows by Lemma \ref{lem:stepNM} from the facts that the sequence $i \in \tb 2k \mapsto i$ moves up in steps of $+1$, while the sequence $i \in \tb 2k \mapsto \nonmer(k-i+1)$ moves down in steps of $-1$ or $-2$. It remains to show Part \ref{lem:simprecnm2}. By Lemma \ref{lem:stepNM} the sequence $i \in \tb 2k \mapsto \nonmer(k-i+1)+i$ is monotonically decreasing, and thus by definition we obtain $\nonmer(k) = \max(2 \cdot (i(k)-1), \nonmer(k-i(k)+1)+i(k))$. That the maximum here is actually always attained in the second component follows by Part \ref{lem:simprecnm1}. Finally Part \ref{lem:simprecnm3} follows again from Lemma \ref{lem:stepNM}.
\end{proof}

After these preparations we are able to characterise the ``jump positions'', the set $J \subset \mathbb{N}$ of $k \in \mathbb{N}$ with $\Delta \nonmer(k) = 2$. Thus $\Delta \nonmer(k) = 1$ iff $k \notin J$, and $J = \set{1,4,11,26,\dots}$. Note $\nonmer(k) = 1 + k + \abs{\set{k' \in J : k' < k}}$.
\begin{lem}\label{lem:characjump}
  Let $i'(k) := k-i(k)+1$ and $h(k) := \nonmer(i'(k))$ for $k \in \mathbb{N}$, $k \ge 2$. Thus $\Delta i'(k) \in \set{0,1}$ and $\Delta i(k) = 1 - \Delta i'(k)$. Furthermore we have $\nonmer(k) = h(k) + i(k)$, thus $\Delta \nonmer(k) = \Delta h(k) + \Delta i(k)$, and $i(k) - h(k) \in \set{0,1,2}$. Consider $k \ge 2$.
  \begin{enumerate}
  \item\label{lem:characjump1} If $\Delta i(k) = 0$, then:
    \begin{enumerate}
    \item\label{lem:characjump1a} $\Delta i(k+1) = 1$
    \item\label{lem:characjump1b} $i(k) \not= h(k)$.
    \item\label{lem:characjump1c} $i(k+1) = h(k+1)$.
    \end{enumerate}
  \item\label{lem:characjump2} If $\Delta i(k) = 1$, then:
    \begin{enumerate}
    \item\label{lem:characjump2a} $\Delta h(k) = 0$, and so $k \notin J$
    \item\label{lem:characjump2b} $i(k) \not= h(k)+2$.
    \end{enumerate}
  \item\label{lem:characjump3} The following conditions are equivalent:
    \begin{enumerate}
    \item\label{lem:characjump3a} $k \in J$
    \item\label{lem:characjump3c} $\Delta h(k) = 2$
    \item\label{lem:characjump3d} $i(k) = h(k) + 2$
    \item\label{lem:characjump3g} $\Delta i(k-1) = 1$ and $i(k-1) = h(k-1) + 1$
    \item\label{lem:characjump3e} $\Delta i(k-2) = \Delta i(k-1) = 1$
    \item\label{lem:characjump3b} $i'(k) = i'(k-1) = i'(k-2)$ and $i'(k) \in J$.
    \end{enumerate}
  \item\label{lem:characjump4} If $k \in J$, then $i'(k) = \max({k' \in J : k' < k})$.
  \end{enumerate}
\end{lem}
\begin{proof} Part \ref{lem:characjump1a} follows by definition. For Part \ref{lem:characjump1b} note $i(k+1) = i(k)$ while $h(k+1) \ge h(k) + 1$. For Part \ref{lem:characjump1c} assume $i(k+1) > h(k+1)$. Then we have $i(k) = h(k)+2$ and $h(k+1) = h(k)+1$. However then $i(k)-1 = h(k)+1 = h(k+1) = \nonmer(k - (i(k)-1) + 1)$ contradicting the definition of $i(k)$. For Part \ref{lem:characjump2a} assume $i(k) = i(k+1) = i(k+2)$. We have $i(k) \ge h(k+2) = \nonmer(k-i(k)+3)$, while $i(k) - 1 < \nonmer(k-(i(k)-1)+1) = \nonmer(k-i(k)+2)$, i.e., $i(k) \le \nonmer(k-i(k)+2)$, contradicting the strict monotonicity of $\nonmer$. Part \ref{lem:characjump2b} follows by $i(k+1) \le h(k+1) + 2$ and $i(k+1) = i(k)+1$, $h(k+1) = h(k)$. Now consider Part \ref{lem:characjump3}.

Condition \ref{lem:characjump3a} implies condition \ref{lem:characjump3c} due to $\Delta i(k) = 0$ in case of $k \in J$ by Part \ref{lem:characjump2a}. Condition \ref{lem:characjump3c} implies condition \ref{lem:characjump3d}, since $\Delta h(k) = 2$ implies $\Delta i(k) = 0$ (otherwise we had $\Delta \nonmer(k) = 3$), and so by Part \ref{lem:characjump1c} we have $i(k) = i(k+1) = h(k+1)$, while the assumption says $h(k+1) = h(k) + 2$. In turn condition \ref{lem:characjump3d} implies condition \ref{lem:characjump3a}, since by Part \ref{lem:characjump2b} we get $\Delta i(k) = 0$, and thus $\Delta \nonmer(k) = \Delta h(k)$, while in case of $\Delta h(k) \le 1$ we would have $i(k)-1 \ge \nonmer(k-(i(k)-1)+1)$ contradicting the definition of $i(k)$, due to $\nonmer(k-(i(k)-1) + 1) = \nonmer((k+1) - i(k+1) + 1) = h(k+1) \le h(k) + 1 = i(k) - 1$. So now we can freely use the equivalence of these three conditions.

Condition \ref{lem:characjump3d} implies condition \ref{lem:characjump3g}, since we have $\Delta i(k) = 0$, and thus $\Delta i(k-1) = 1$ with Part \ref{lem:characjump1a}, from which we furthermore get $i(k) = i(k-1) + 1$ and $h(k-1) = h(k)$, and so $i(k-1) = i(k) - 1 = h(k) + 1 = h(k-1) + 1$. Condition \ref{lem:characjump3g} implies condition \ref{lem:characjump3e}, since in case of $\Delta i(k-2) = 0$ we had $i(k-1) = h(k-1)$ with Part \ref{lem:characjump1c}. In turn condition \ref{lem:characjump3e} implies condition \ref{lem:characjump3d}, since $i(k) = i(k-1) + 1 = i(k-2) + 2$, while $h(k) = h(k-1) = h(k-2)$, where by definition $i(k-2) \ge h(k-2)$ holds, whence $i(k) \ge h(k) + 2$, which implies $i(k) = h(k) + 2$. So now the first five conditions have been shown to be equivalent.

Now condition \ref{lem:characjump3e} implies condition \ref{lem:characjump3b}, since it only remains to show $i'(k) \in J$, which follows with condition \ref{lem:characjump3c} (using $\Delta i(k) = 0$). In turn condition \ref{lem:characjump3b} implies immediately condition \ref{lem:characjump3e}.

Finally, we prove Part \ref{lem:characjump4} by induction on $k$ (regarding the enumeration of $J$). We have $i'(4) = 1$, and so the induction holds for $k=4$, the smallest jump position $k \ge 2$. Now assume that the assertion holds for all elements of $J \cap \tb 1{k-1}$, where $k > 4$, and we have to show the assertion for $k$. By Part \ref{lem:characjump3b} we know $i'(k) \in J$, where $2 \le i'(k) < k$. Assume there is $k' \in J$ with $i'(k) < k' < k$. Now by induction hypothesis we get $i'(k) \le i'(k') < k'$. However by Part \ref{lem:characjump1} we get $\Delta i'(k') = 1$, and thus $i'(k) > i'(k')$ (since $k > k'$).
\end{proof}

\begin{corol}\label{cor:CharacJ}
  We have $J = \set{2^{m+1} - m - 2 : m \in \mathbb{N}}$.
\end{corol}
\begin{proof} Let $k_m$ for $m \in \mathbb{N}$ be the $m$th element of $J$; so the assertion is $k_m = 2^{m+1} - m - 2$. We have $k_1 = 4 - 1 - 2 = 1 = \min J$; in the remainder assume $m \ge 2$. We prove the assertion by induction, in parallel with $i(k_m) = 2^{m+1} - 2^m$. For $m=2$ we have $k_2 = 8 - 2 - 2 = 4 = \min J \setminus \set{1}$, while $i(4)$ is the smallest $i \in \set{2,3,4}$ with $i \ge \nonmer(5-i)$, which yields $i(4) = 4 = 2^3 - 2^2$. Now we consider the induction step, from $m-1$ to $m$. The induction hypothesis yields $k_{m-1} = 2^m - m - 1$ and $i(k_{m-1}) = 2^m - 2^{m-1}$. Lemma \ref{lem:characjump}, Part \ref{lem:characjump4} yields $i'(k_m) = k_{m-1}$, from which by $i'(k_m) = k_m - i(k_m) + 1$ follows $k_m = 2^m - m - 2 + i(k_m)$. By definition we get $i(k_m) = \Delta i(k_m-1) + \dots + \Delta i(k_{m-1}) + i(k_{m-1})$. By Lemma \ref{lem:characjump}, Parts \ref{lem:characjump1} - \ref{lem:characjump3} the sequence of $\Delta$-values has the form (starting with the lowest index) $0,1,\: 0,1,\: \dots, 0,1,1$, and thus their sum has the value $\frac 12 (k_m - k_{m-1} - 1) + 1$. So we get $i(k_m) = \frac 12 (k_m - k_{m-1} - 1) + 1 + i(k_{m-1}) = \frac 12 (2^m - m - 2 + i(k_m) - 2^m + m + 1 - 1) + 1 + 2^m - 2^{m-1} = \frac 12 i(k_m) - 1 + 1 + 2^m - 2^{m-1}$, from which $i(k_m) = 2^{m+1} - 2^m$ follows. Finally $k_m = 2^m - m - 2 + 2^{m+1} - 2^m = 2^{m+1} - m - 2$.
\end{proof}

Now the closed formula for $\nonmer(k)$ can be proven (using $\ld(x) := \log_2(x)$):
\begin{lem}\label{lem:solveN}
  For $k \in \mathbb{N}$ let $\fld(k) := \floor{\ld(k)}$ (``floor of logarithm dualis''). Then we have for $k \in \mathbb{N}$ the equality $\nonmer(k) = k + \fld(k+1 + \fld(k+1))$.
\end{lem}
\begin{proof} Let $g(k) := \fld(k+1 + \fld(k+1))$ and $f(k) := k + g(k)$ (so $\nonmer(k) = f(k)$ is to be shown, for $k \ge 1$). We have $f(1) = 1 + \fld(2 + \fld(2)) = 1 + \fld(3) = 2 = \nonmer(1)$. We will now prove that the function $g(k)$ changes values exactly at the transitions $k \mapsto k+1$ for $k \in J$, that is, for indices $k = k_m := 2^{m+1} - m - 2$ (using Corollary \ref{cor:CharacJ}) with $m \in \mathbb{N}$ we have $\Delta g(k_m) = 1$, while otherwise we have $\Delta g(k_m) = 0$, from which the assertion follows (by the definition of $J$).

We have $g(1) = 1$ and $g(2) = 2$. Now consider $m \in \mathbb{N}$ and $k_m + 1 \le k \le k_{m+1}$. We show $g(k) = m+1$, which proves the claim. Note that $g(k)$ is monotonically increasing. Now $g(k) \ge g(k_m+1) = \floor{\ld(2^{m+1}-m + \floor{\ld(2^{m+1}-m)})} = \floor{\ld(2^{m+1}-m + m)} = m+1$ and $g(k) \le g(k_{m+1}) = \floor{\ld(2^{m+2}-m-2 + \floor{\ld(2^{m+2}-m-2)})} \le \floor{\ld(2^{m+2}-m-2 + m+1)} = \floor{\ld(2^{m+2}-1)} = m+1$.
\end{proof}

As a result, we obtain very precise bounds:
\begin{corol}\label{cor:upperboundnonmer}
  $k + \fld(k+1) \le \nonmer(k) \le k+1+\fld(k)$ holds for $k \in \mathbb{N}$.
\end{corol}
\begin{proof} The lower bound follows trivially. The upper bound holds (with equality) for $k \le 2$, so assume $k \ge 3$. We have to show $g(k) = \fld(k+1+\fld(k+1)) \le 1 + \fld(k)$, which follows from $\ld(k+1+ \fld(k+1)) \le 1 + \ld(k)$. Now $\ld(k+1+ \fld(k+1)) \le \ld(k+1+ \ld(k+1)) \le \ld(k+k) = 1 + \ld(k)$.
\end{proof}

\section{Lean clause-sets and the surplus}
\label{sec:leansurp}

In this section we prove the main result of this paper, Theorem \ref{thm:leanminvardeg}. The proof consists in first handling a special case, minimally unsatisfiable clause-sets instead of lean clause-sets, in Subsection \ref{sec:specialcaseMU}, and then lifting the result to the general case in Subsection \ref{sec:proofgencase}. In Subsection \ref{sec:findaut} we consider the algorithmic implications of this result.

\begin{thm}\label{thm:leanminvardeg}
  We have $\minvdeg(F) \le \nonmer(\surp(F))$ for a lean multi-clause-set $F$ with $n(F) > 0$. More precisely, there exists a variable $v \in \var(F)$ with $\vdeg_F(v) \le \nonmer(\surp(F))$ and $\ldeg_F(v), \ldeg_F(\overline{v}) \le \surp(F)$.
\end{thm}

We obtain a sufficient criterion for the existence of a non-trivial autarky.
\begin{corol}\label{cor:leanminvardegsat}
  Consider a multi-clause-set $F$ with $n(F) > 0$. If $\surp(F) \le 0$, then $F$ has a non-trivial matching autarky. So assume $\surp(F) \ge 1$. If we have $\minvdeg(F) > \nonmer(\surp(F))$, then for every $\emptyset \not= V \subseteq \var(F)$ with $\delta(F[V]) = \surp(F)$ we have an autarky $\varphi$ for $F$ with $\var(\varphi) = V$ (and thus $F$ has a non-trivial autarky).
\end{corol}

The quantities $\minvdeg(F)$ and $\nonmer(\surp(F))$ (resp.\ $\nonmer(\delta(F))$) are computable in polynomial time, and so the applicability of Corollary \ref{sec:proofgencase} can be checked in polynomial time. We conjecture that also ``constructivisation'' of Corollary \ref{cor:leanminvardegsat} can be done in polynomial time:
\begin{conj}\label{con:findauthard}
   There is a poly-time algorithm for computing a non-trivial autarky in case of $\minvdeg(F) > \nonmer(\surp(F))$ (or $\minvdeg(F) > \nonmer(\delta(F))$) for matching-lean clause-sets $F$.
\end{conj}
See Subsection \ref{sec:findaut} for more discussion on Conjecture \ref{con:findauthard} (there also the remaining details of Corollary \ref{cor:leanminvardegsat} are proven).

\subsection{The special case of minimally unsatisfiable clause-sets}
\label{sec:specialcaseMU}

The main auxiliary lemma is the following statement, which receives its importance from the fact that every minimally unsatisfiable clause-set can be saturated (this method was first applied in \cite{Ku99dKo}).
\begin{lem}\label{lem:auxminvardeg}
  Consider $F \in \Smusati{\delta=k}$ for $k \in \mathbb{N}$ and a variable $v \in \var(F)$ realising the minimal var-degree (i.e., $\vdeg_F(v) = \minvdeg(F)$). Using $m_0 := \ldeg_F(\overline{v})$ and $m_1 := \ldeg_F(v)$ we have $\pao v{\varepsilon} * F \in \Musati{k-m_{\varepsilon}+1}$ for $\varepsilon \in \set{0,1}$, where $n(\pao v{\varepsilon} * F) = n(F) - 1$. Since minimally unsatisfiable clause-sets have deficiency at least one, we get $m_{\varepsilon} \le k$.
\end{lem}
\begin{proof} We have $n(\pao v{\varepsilon} * F) = n(F) - 1$ since $F$ contains no pure variable, while $v$ realises the minimum of var-degrees. Thus $\delta(\pao v{\varepsilon} * F) = \delta(F) - m_{\varepsilon} + 1$, while $\pao v{\varepsilon} * F \in \Musat$ by Lemma \ref{lem:auxSMUSAT}, Part \ref{lem:auxSMUSAT1}.
\end{proof}

\begin{thm}\label{thm:MUminvdegdef}
  For all $k \in \mathbb{N}$ and $F \in \Musati{\delta \le k}$ we have $\minvdeg(F) \le \nonmer(k)$. More precisely, for $n(F) > 0$ there exists a variable $v \in \var(F)$ with $\vdeg_F(v) \le \nonmer(k)$ and $\ldeg_F(v), \ldeg_F(\overline{v}) \le k$.
\end{thm}
\begin{proof} The assertion is known for $k=1$, so assume $k > 1$, and we apply induction on $k$. Assume $\delta(F) = k$ (due to $k > 1$ we have $n(F) > 1$). Saturate $F$ and obtain $F'$. Consider a variable $v \in \var(F')$ realising the min-var-degree of $F'$. If $\vdeg_{F'}(v) = 2$ then we are done, so assume $\vdeg_{F'}(v) \ge 3$. Let $i := \max(\ldeg_{F'}(v),\ldeg_{F'}(\overline{v}))$; so $\vdeg_{F'}(v) \le 2 i$. W.l.o.g.\ assume that $i = \ldeg_{F'}(v)$. By Lemma \ref{lem:auxminvardeg} we get $2 \le i \le k$. Applying the induction hypothesis and Lemma \ref{lem:auxminvardeg} we obtain a variable $w \in \var(G)$ for $G := \pao v1 * F$ with $\vdeg_G(w) \le \nonmer(k-i+1)$. By definition we have $\vdeg_{F'}(w) \le \vdeg_G(w) + \ldeg_{F'}(v)$. Altogether we get $\minvdeg(F) \le \min(2 i, \nonmer(k-i+1) + i) \le \nonmer(k)$.
\end{proof}

It is interesting to generalise Theorem \ref{thm:MUminvdegdef} for generalised clause-sets (see \cite{Kullmann2007ClausalFormZI,Kullmann2007ClausalFormZII} for a systematic study, and \cite{Kullmann2011ClausalForm} for the underlying report). Generalised clause-sets have literals ``$v \not= \varepsilon$'' for variables $v$ with domains $D_v$ and values $\varepsilon \in D_v$, and the deficiency is generalised by giving every variable a weight $\abs{D_v} - 1$ (which is $1$ in the boolean case). The base case of deficiency $k=1$ is handled in Lemma 5.4 in \cite{Kullmann2007ClausalFormZII}, showing that for generalised clause-sets we have here $\minvdeg(F) \le \max_{v \in \var(F)} \abs{D_v}$. But $k \ge 2$ requires more work:
\begin{enumerate}
\item The basic method of saturation is not available for generalised clause-sets, as discussed in Subsection 5.1 in \cite{Kullmann2007ClausalFormZII}. Thus the proofs for the boolean case seem not to be generalisable.
\item Stipulating the effects of saturation via the ``substitution stability parameter regarding irredundancy'', in Corollary 5.10 in \cite{Kullmann2007ClausalFormZII} one finds a first approach towards generalising the basic bound $\minvdeg(F) \le 2 \delta(F)$ (for the boolean case) by $\minvdeg(F) \le \max_{v \in \var(F)} \abs{D_v} \cdot \delta(F)$.
\item Another approach uses translations to boolean clause-sets. The ``generic translation scheme'' (see \cite{Kullmann2010GreenTao,Kullmann2007ClausalFormZII}) allows (for certain instances) to preserve the deficiency and the other structures relevant here. So we get general upper bounds for the minimum number of occurrences of variables in generalised clause-sets from the boolean case. But further investigations are needed in these bounds.
\end{enumerate}

\subsection{Proof of the general case}
\label{sec:proofgencase}

Now consider an arbitrary (multi-)clause-set $F$. Consider a set of variables $\emptyset \not= V \subseteq \var(F)$ realising the surplus of $F$, i.e., such that $\delta(F[V])$ is minimal. If $F[V]$ would be satisfiable, then a satisfying assignment would give a non-trivial autarky for $F$. Assuming that $F$ is lean thus yields that $F[V]$ must be unsatisfiable. So there exists a minimally unsatisfiable $F' \subseteq F[V]$. If now $\var(F') \not= \var(F[V]) = V$ would be the case, then we would loose control over the deficiency of $F'$. Fortunately this can not happen, as the following lemma shows.

\begin{lem}\label{lem:auxminvardegsigma}
  Consider a multi-clause-set $F$ with $\surp(F) = \delta(F)$. Then for every unsatisfiable sub-multi-clause-set $F' \le F$ we have $\var(F') = \var(F)$.
\end{lem}
\begin{proof} Assume $\var(F') \subset \var(F)$, and consider a minimally unsatisfiable sub-clause-set $F'' \subseteq F'$. By definition we have $\delta(F'') + \delta(F[\var(F) \setminus \var(F'')]) \le \delta(F)$, where $\delta(F[\var(F) \setminus \var(F'')]) \ge \surp(F) = \delta(F)$, from which we conclude $\delta(F'') \le 0$, but $\delta(F'') \ge 1$ must hold since $F''$ is minimally unsatisfiable.
\end{proof}

Finally we are able to prove Theorem \ref{thm:leanminvardeg}. Recall that $F$ is a lean multi-clause-set with $n(F) > 0$, and we have to show the existence of a variable $v$ with $\vdeg_F(v) \le \nonmer(\surp(F))$ and $\ldeg_F(v), \ldeg_F(\overline{v}) \le \surp(F)$.

Consider $\emptyset \not= V \subseteq \var(F)$ with $\delta(F[V]) = \surp(F)$, and let $F' := F[V]$. $F'$ is unsatisfiable, since $F$ is lean. Because of $\delta(F') = \surp(F)$ we have $\delta(F') = \surp(F')$. Consider some minimally unsatisfiable $F'' \subseteq F'$. By Lemma \ref{lem:auxminvardegsigma} we have $\var(F'') = \var(F')$. So we get $\delta(F'') = \delta(F') - (c(F') - c(F''))$. By Theorem \ref{thm:MUminvdegdef} there is $v \in \var(F'')$ with $\vdeg_{F''}(v) \le \nonmer(\delta(F'')) = \nonmer(\delta(F') - (c(F') - c(F''))) \le \nonmer(\delta(F')) - (c(F') - c(F''))$ and $\ldeg_{F''}(v), \ldeg_{F''}(\overline{v}) \le \delta(F'') = \delta(F') - (c(F') - c(F''))$. Finally we have $\vdeg_F(v) \le \vdeg_{F''}(v) + (c(F') - c(F''))$ (note that all occurrences of $v$ in $F$ are also in $F'$), and similarly for the literal degrees. QED

\begin{corol}\label{cor:leanminvardegdef}
  For a lean multi-clause-set $F$ with $n(F) > 0$ we have $\minvdeg(F) \le \nonmer(\delta(F))$.
\end{corol}

\begin{corol}\label{cor:charsurp1}
  Consider a lean multi-clause-set $F$.
  \begin{enumerate}
  \item\label{cor:charsurp1a} $\surp(F) = 1$ holds if and only if $\minvdeg(F) = 2$ holds.
  \item\label{cor:charsurp1b} $\minvdeg(F) = 3$ implies $\surp(F) = 2$.
  \end{enumerate}
\end{corol}
\begin{proof} First consider Part \ref{cor:charsurp1a}. If $\surp(F) = 1$ (so $n(F) > 0$), then by Theorem \ref{thm:leanminvardeg} we have $\minvdeg(F) \le \nonmer(1) = 2$, while in case of $\minvdeg(F) = 1$ there would be a matching autarky for $F$. If on the other hand $\minvdeg(F) = 2$ holds, then by definition $\surp(F) \le 2 - 1 = 1$, while $\surp(F) \ge 1$ holds since $F$ is matching lean. For Part \ref{cor:charsurp1b} note that due to $\surp(F)+1 \le \minvdeg(F)$ we have $\surp(F) \le 2$, and then the assertion follows by Part \ref{cor:charsurp1a}.
\end{proof}

Remarks:
\begin{enumerate}
\item If $F$ is lean, then $\surp(F) = 2$ implies $\minvdeg(F) \in \set{3,4}$. An example for $\minvdeg(F) = 4$ is given by the full unsatisfiable clause-set with $2$ variables.
\item Is there a minimally unsatisfiable $F$ with $\minvdeg(F) = 4$ and $\surp(F)=3$?
\item More generally, is there for every $k \in \mathbb{N}$ a minimally unsatisfiable $F$ with $\sigma(F) = k$ and $\minvdeg(F) = k+1$?
\end{enumerate}

\subsection{On finding the autarky}
\label{sec:findaut}

The following lemma (with Theorem \ref{thm:leanminvardeg}) yields the proof of Corollary \ref{cor:leanminvardegsat}:
\begin{lem}\label{lem:charakappcor}
  Consider a matching-lean multi-clause-set $F$ with $n(F) > 0$. If we have $\minvdeg(F) > \nonmer(\surp(F))$, then all $F[V]$ for $\emptyset \subset V \subseteq \var(F)$ with $\delta(F[V]) = \surp(F)$ are satisfiable.
\end{lem}
\begin{proof} If some $F[V]$ would be unsatisfiable, then by the proof of Theorem \ref{thm:leanminvardeg} in Subsection \ref{sec:proofgencase} there would be a variable $v$ with $\vdeg_F(v) \le \nonmer(\surp(F))$.
\end{proof}

Now consider a matching-lean multi-clause-set $F$ with $n(F) > 0$, where Corollary \ref{cor:leanminvardegsat} is applicable (recall that we have $\surp(F) \ge 1$), that is, we have $\minvdeg(F) > \nonmer(\surp(F))$. So we know that $F$ has a non-trivial autarky. Conjecture \ref{con:findauthard} states that finding such a non-trivial autarky in this case can be done in polynomial time (recall that finding a non-trivial autarky in general is NP-complete, which was shown in \cite{Ku00f}).

The task of actually finding the autarky can be considered as finding a satisfying assignment for the following class $\Mlcr \subset \mc{SAT} \cap \mc{M}\Lean$ of satisfiable(!) clause-sets $F$, obtained by considering all $F[V]$ for minimal sets of variables $V$ with $\delta(F[V]) = \surp(F)$ (where ``CR'' stands for ``critical''):
\begin{defi}\label{def:Mlcr}
  Let \bmm{\Mlcr} be the class of clause-sets $F$ fulfilling the following three conditions:
  \begin{enumerate}
  \item\label{def:Mlcr1} $F$ is matching-lean, has at least one variable, and does not contain the empty clause.
  \item\label{def:Mlcr2} The only $\emptyset \not= V \subseteq \var(F)$ with $\delta(F[V]) = \surp(F)$ is $V = \var(F)$ (and thus we have $\delta(F) = \surp(F)$).
  \item\label{def:Mlcr3} $\minvdeg(F) > \nonmer(\surp(F))$.
  \end{enumerate}
\end{defi}
It is sufficient to find a non-trivial autarky for this class of satisfiable clause-sets.
\begin{lem}\label{lem:HardMlcr}
  Conjecture \ref{con:findauthard} is equivalent to the statement, that finding a non-trivial autarky for clause-sets in $\Mlcr$ can be achieved in polynomial time.
\end{lem}
At the time of writing this article, we are not aware of elements of $\Mlcr$ with a deficiency at least $2$.

\section{On strengthening the bound}
\label{sec:strengthbound}

For a class $\mathcal{C}$ of clause-sets let $\minvdeg(\mathcal{C})$ be the supremum of $\minvdeg(F)$ for $F \in \mathcal{C}$ with $n(F) > 0$. So by Theorem \ref{thm:MUminvdegdef} we have $\minvdeg(\Musati{\delta=k}) \le \nonmer(k)$ for all $k \in \mathbb{N}$. The task of precisely determining $\minvdeg(\Musati{\delta=k})$ for all $k$ will be pursued in the forthcoming \cite{KullmannZhao2010Extremal}; we need more theory for minimally unsatisfiable clause-sets (especially for unsatisfiable hitting clause-sets), and so here we can only mention some results connected with this article.
\begin{itemize}
\item We can show for infinitely many $k$ that $\minvdeg(\Musati{\delta=k}) = \nonmer(k)$.
\item We can also show that the smallest $k$ where we don't have equality is $k=6$, namely $\minvdeg(\Musati{\delta=6}) = 8 = \nonmer(6) - 1$.
\item Let $\nonmer_1: \mathbb{N} \rightarrow \mathbb{N}$ be defined by the recursion as in Definition \ref{def:minvdegdef}, however with different start values, namely $\nonmer_1(k) := \nonmer(k)$ for $1 \le k \le 5$, while $\nonmer_1(6) := \nonmer(6) - 1 = 8$. We have $\nonmer_1(k) = \nonmer(k)$ for $k \notin \set{2^m-m+1 : m \in \mathbb{N}, m \ge 3}$, while for $k = 2^m-m+1$ we have $\nonmer_1(k) = \nonmer(k) - 1 = 2^m$.
\item With the same proof as for Theorem \ref{thm:MUminvdegdef} we can show $\minvdeg(\Musati{\delta=k}) \le \nonmer_1(k)$ for all $k \in \mathbb{N}$.
\item It seems that this bound can not be generalised to lean clause-sets (as in Theorem \ref{thm:leanminvardeg}).
\end{itemize}

\begin{conj}\label{con:sharpness}
  For all $k \in \mathbb{N}$ we have $\minvdeg(\Musati{\delta=k}) \ge \nonmer(k)-1$.
\end{conj}

Now we consider the question whether the bound holds for a larger class of clause-sets, that is, whether Theorem \ref{thm:leanminvardeg} can be generalised further, incorporating non-lean clause-sets. We consider the large class $\mc{M}\Lean$ of matching lean clause-sets, as introduced in \cite{Ku00f}, which is natural, since a basic property of $F \in \Musat$ used in the proof of Theorem \ref{thm:leanminvardeg} is $\delta(F) \ge 1$ for $F \not= \top$, and this actually holds for all $F \in \mc{M}\Lean$. We will construct for arbitrary deficiency $k \in \mathbb{N}$ and $K \in \mathbb{N}$ clause-sets $F \in \mc{M}\Lean$ of deficiency $k$ where every variable occurs positively at least $K$ times. Thus neither the upper bound $\max(\ldeg_F(v), \ldeg_F(\overline{v})) \le f(\delta(F))$ nor $\ldeg_F(v) + \ldeg_F(\overline{v}) = \vdeg_F(v) \le f(\delta(F))$ for some chosen variable $v$ and for any function $f$ does hold for $\mc{M}\Lean$.

An example for $F \in \Mleani{\delta=1}$ with $\minldeg(F) \ge 2$ (and thus $\minvdeg(F) \ge 4$) is given in Section 5 in \cite{Ku2003b}, displaying a ``star-free'' (thus satisfiable) clause-set $F$ with deficiency $1$. In Subsection 9.3 in \cite{Kullmann2007ClausalFormZI} it is shown that this clause-set is matching lean. ``Star-freeness'' in our context means, that there are no singular variables (occurring in one sign only once). Our simpler construction pushes the number of positive occurrences arbitrary high, but there are variables with only one negative occurrence (i.e., there are singular variables).

For a finite set $V$ of variables let $M(V) \subseteq A(V)$ be the full clause-set over $V$ containing all full clauses with at most one complementation. Obviously $\delta(F) = 1$ holds, and it is easy to see that $M(V) \in \mc{M}\Lean$ (for every $\emptyset \not= F' \subset F \subseteq A(V)$ we have $\delta(F') < \delta(F)$, and thus a full clause-set $F$ is matching lean iff $\delta(F) \ge 1$). Furthermore by definition we have $\ldeg_{M(V)}(v) = \abs{V}$ and $\ldeg_{M(V)}(\overline{v}) = 1$ for $v \in V$.

\begin{lem}\label{lem:exmpmlean}
  For $k \in \mathbb{N}$ and $K \in \mathbb{N}$ there are clause-sets $F \in \Mleani{\delta=k}$ such that for all variables $v \in \var(F)$ we have $\ldeg_F(v) \ge K$.
\end{lem}
\begin{proof} For $k = 1$ we can set $F := M(\tb {v_1}{v_K})$; so assume $k \ge 2$. Consider any clause-set $G \in \Mleani{\delta=k-1}$ with $n := n(G) \ge K$ (for example we could use $F \in \Musati{\delta=k-1}$), and let $V := \var(G)$. Consider a disjoint copy of $V$, that is a set $V'$ of variables with $V' \cap V = \emptyset$ and $\abs{V'} = \abs{V}$, and consider two enumerations of the clauses $M(V) = \set{C_1, \dots, C_{n+1}}$, $M(V') = \set{C_1', \dots, C_{n+1}'}$. Now
\begin{displaymath}
  F := G \cup \setb{ C_i \cup C_i' : i \in \tb{1}{n+1}}
\end{displaymath}
has no matching autarky: If $\varphi$ is a matching autarky for $F$, then $\var(\varphi) \cap V = \emptyset$ since $G$ is matching lean, whence $\var(\varphi) \cap V' = \emptyset$ since $M(V')$ is matching lean, and thus $\varphi$ must be trivial. Furthermore we have $n(F) = 2 n$ and $c(F) = c(G) + n + 1$, and thus $\delta(F) = c(G) + n + 1 - 2 n = \delta(G)+1 = k$. By definition for all variables $v \in \var(F)$ we have $\ldeg_F(v) \ge n$.
\end{proof}

Remarks:
\begin{enumerate}
\item It remains open whether for deficiency $k \in \mathbb{N}$ we find examples $F \in \Mleani{\delta=k}$ with $\minldeg(F) \ge k+1$ (the above mentioned star-free clause-sets shows that this is the case for $k=1$), or stronger, $\minldeg(F) \ge K$ for arbitrary $K \in \mathbb{N}$.
\item The clause-sets $F$ constructed in  Lemma \ref{lem:exmpmlean} are not elements of $\Mlcr_{\delta=k}$ for $k \ge 2$, since $\delta(F[V']) = n+1 - n = 1$, thus $\surp(F) = 1$, and so Condition \ref{def:Mlcr2} of Definition \ref{def:Mlcr} is not fulfilled. The corresponding autarky is a satisfying assignment of $M(V')$, which is easy to find.
\end{enumerate}

\section{Conclusion and open problems}
\label{sec:open}

We have shown the upper bound $\minvdeg(F) \le \nonmer(\surp(F))$ for lean clause-sets (Theorem \ref{thm:leanminvardeg}). The function $\nonmer(k)$ has been characterised in Lemma \ref{lem:solveN} and Corollary \ref{cor:upperboundnonmer}. We presented first initial results regarding the sharpness of the bound and regarding the constructive aspects of the bound (i.e., what happens if the bound is violated). There remain several open problems:
\begin{enumerate}
\item Prove Conjecture \ref{con:findauthard}, which says that such an autarky, which must exist if a clause-set does not fulfil the upper bound on the minimum variable degree of Theorem \ref{thm:leanminvardeg}, can be found in polynomial time. See Subsection \ref{sec:findaut} for more information on this topic.
\item Generalise Theorem \ref{thm:MUminvdegdef} to clause-sets with non-bool\-ean variables; see the discussion after Theorem \ref{thm:MUminvdegdef}.
\item See the remarks to Corollary \ref{cor:charsurp1} (an underlying question is to understand better the quantity ``surplus'').
\item Strengthen the bound on the minimum variable degree for minimally unsatisfiable clause-sets (see the forthcoming \cite{KullmannZhao2010Extremal}).
\item Strengthen the construction of Lemma \ref{lem:exmpmlean} (perhaps completely different constructions are needed).
\end{enumerate}

As mentioned in the introduction, a major motivation for us is the project of the classification of minimally unsatisfiable clause-sets for deficiencies $\delta = 1, 2, \dots$. Especially the classification of unsatisfiable hitting clause-sets in dependency on the deficiency seems very interesting (recall that a hitting clause-set $F$ is defined by the condition that every two clauses $C, C' \in F$, $C \not= C'$, clash in at least one variable, that is $\abs{C \cap \overline{C'}} \ge 1$). The main conjecture is:
\begin{conj}\label{con:finhit}
  For every deficiency $k \in \mathbb{N}$ there are only finitely many isomorphism types of non-singular unsatisfiable hitting clause-sets.
\end{conj}
For $k \le 2$ this conjecture follows from known results, while recently we were able to prove it for $k=3$.

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


text:
Home > Glossary
Aerosol droplet transmission
Transmission of infectious agents in droplets from respiratory secretions.
'Active' heating and cooling systems
'Active' means a heating and cooling system that requires additional energy to make the house warmer or cooler; includes gas, fire and electricity systems.
Accessibility/Remoteness Index of Australia, an agreed numeric measure of how remote any town is within Australia
'As-built' drawings
A set of drawings that include all changes made during the construction process.
(AP) Lands
Anangu Pitjantjatjara Lands, an Aboriginal controlled area of land in the north west of South Australia
AS/NZS
Joint Australian and New Zealand Standard
Building Code of Australia. The Building Code of Australia (BCA) is Volumes One and Two of the National Construction Code (NCC). The BCA is produced and maintained by the Australian Building Codes Board (ABCB) on behalf of the Australian Government and State and Territory Governments. The BCA has been given the status of building regulation by all States and Territories.
Electric element that can be switched on to supplement (boost) the temperature of hot water (usually on solar hot water systems).
Blackwater is a term used to describe wastewater containing faecal matter and urine. It is also known as foul water, or sewage. It is distinct from greywater or sullage, the residues of washing processes.
Level tap handles with a single outlet (spout) that can be used by people with limited mobility
Copper Chrome Arsenate treatment for preserving timber
Common effluent disposal, waste water is treated in local septic tanks and combined into a 'common' effluent line and piped to a point for final disposal
Documentation indicating a building is safe to inhabit.
Certificate of compliance/certification
Document confirming correct construction and/or installation by a licensed tradesperson, for example electrical or water and drainage works.
Visual inspection to confirm the location of an item or to determine whether there are safety risks–for example, check there are hot and cold water taps specifically for the washing machine.
A tube or trough to protect electrical wiring.
Convective cooling
Venting of high level hot air to draw in cooler air from lower levels.
When the population of the house regularly exceeds the capacity of functioning health hardware facilities of the house.
Cyclical maintenance
Regular assessment and fixing of houses as opposed to emergency maintenance needed when things fail. For example, a cycle of maintenance that ensures a pre-summer fix and a pre-winter fix or an annual maintenance check of all houses.
Detailed drawing
Large-scale drawing showing exactly how components of a house are to be made. For example, a drawing may show a bathroom is three metres by four metres and the location of the bath, shower, window and door; a detailed drawing would show how the waterproofing of the floor meets the walls, the exact fall of tiles to the floor waste, and the location and heights of towel rails, hooks and soap holder.
Earth leakage circuit breaker, may be similar to an RCD – residual current device. Both aim to disconnect the power supply before injury to people occurs.
A covering piece usually associated with protecting water from entering the junction between a tap and a wall.
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General Power Outlet, more commonly known as a power point.
Waste water from the shower, laundry, basins and kitchen.
Health hardware
Originally used by Dr Fred Hollows to describe the physical equipment necessary for healthy, hygienic living. The equipment must have design and installation characteristics that allow it to function and to maintain or improve health status. In a water supply system, health hardware includes both the bore and the basin plug, as well as the shower rose, taps and drain.
Hot water system/supply
A bacterial skin infection characterized by microscopic, pus filled blisters.
A light bulb that contains a filament which glows white (incandesces) when powered by an electrical current.
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An electrical load applied to the electrical system of the house, to ensure all wiring and electrical fittings have been installed correctly and have not been damaged during the building process.
Nib wall
A small wall at right angles to a main wall
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An electrical device that prevents the automatic re-starting of electrical appliances after power failure to reduce combined loads on generating systems, often used with air conditioners
Package treatment system
A self contained system of tanks and pumps, usually for an individual house (but can apply to a small community), for treating both grey and black wastewater.
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As used in this guide, usually refers to a pre-existing solution for a building component that can be purchased directly from a supplier and not needing to be designed and constructed by each project. For example a proprietary shelving system may contain the shelves, support brackets and wall fixing system that can be purchased as an item.
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Means safe to drink (usually associated with water).
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An additional wastewater disposal trench able to accept excess waste water at peak load times, which is designed to be empty at low load times.
A type of electrical switch.
Sacrificial anode
A replaceable component of a material specifically selected for the prevailing water quality that corrodes before other parts of a hot water system, such as the element or cylinder.
An electrical device to disconnect the power supply before injury to any person. May also be known as an RCD or ELCB
A device that detects the presence of smoke and activates an alarm, also known as 'smoke detector'.
Written description of work usually accompanying a detailed drawing, particularly information that may not be easily shown on the drawing, for example type of kitchen bench material, fittings and method of waterproofing the kitchen sink.
Structure, structural component
Structure of a building includes the parts needed for the structure to be stable, or to stand up. For example, floor beams and joists are structural components that support the floor, and if some were removed the floor would collapse, sag or move noticeably.
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Activity that can be performed by housing managers or residents, for example test power points using a power point tester available from electrical and major hardware stores.
Mass within a building such as brick, concrete, stone or earth that stores both heat and cold.
How efficiently a building can provide a comfortable living environment for residents, particularly when external conditions are extreme.
Trade test
Requires a licensed tradesperson to conduct more complex or dangerous assessment of parts of a house or infrastructure.The licensed tradesperson should be able to certify, in writing, that the test has been conducted and that the item passed the test.
Trench doming
Plastic hoop segments that have been designed to create underground effluent disposal soakage trenches
Tundish
A way to collect usually small quantities or wastewater produced from hot water systems.
Uwankara Palyanyku Kanyintjaku, the report title for a public and environmental health review produced in 1987 for the Anangu Pitjantjatjara Lands. Work in this report established the nine healthy living practices and the connections between the living environment and health.


On Thursday's Andrea Mitchell Reports on MSNBC, host Mitchell not only fretted that First Daughter Ivanka Trump was not doing enough to "lobby" against aggressive enforcement of immigration laws, but she also seemed taken aback that Attorney General Jeff Sessions pushed for such enforcement as a "deterrent" as if such a motivation were a scandal, even though the Barack Obama administration also pursued what it called an "aggressive deterrence policy" to ward off asylum seekers from Central America.
After playing clips of an administration official recalling that the issue of how family separations would impact children had been discussed within the administration, MItchell turned to KPBS immigration reporter Jean Guerrero and fretted: "To hear from a health official that they argued against separation only validates suspicions that what the Attorney General did in announcing this was to propose a deterrent policy. They just wanted to stop people from coming across and seeking refuge."
It would be more accurate to state that what the administration was trying to deter was people traveling across the desert with children to cross the border illegally with an expectation that they would not face aggressive prosecution if they had children with them. Legitimate asylum seekers can travel to ports of entry and avoid such prosecution and family separation.
But her suggestion that it is somehow outrageous or unprecedented to deter such behavior is also odd considering that the Obama administration openly spoke of enacting an "aggressive deterrence policy," in the words of then-Homeland Security Secretary Jeh Johnson, after thousands of Central American immigrants began flooding across the border and seeking asylum.
There were complaints from the left about President Barack Obama trying to deter asylum seekers after there was a dramatic increase in detentions after migrants started flooding into the country from Central America.
Homeland Security Secretary Jeh Johnson, who oversees ICE, heard from border patrollers that the emergency was brewing momentum. People kept coming because word was out that the United States was granting permisos to new arrivals, allowing them to walk right into the country.
According to lawmakers and administration officials, Johnson determined that the United States could cut down the surge only by demonstrating that asylum seekers wouldn't receive leniency. Johnson won approval from the White House to explore ramping up family detention for asylum seekers on a scale never before seen in America, part of what he called an "aggressive deterrence strategy."
The left-wing The Nation also complained about President Obama deporting asylum seekers as a deterrent.
Additionally, the denial rate of asylum claims has been increasing for several years since during the Obama administration, and, for Central American immigrants, has tended to be around 75 to 80 percent.
Back to Andrea Mitchell Reports, a bit later in the segment, after playing a clip of Ivanka Trump speaking negatively of the family separations, the MSNBC host followed up by complaining: "It's very interesting that she would say that because the next followup question would be: What have you done to argue against it inside the White House?"


You are advertising a 20% off sale but when you click on it, everything is still full price. Liar, liar. iTunes doesn't look fondly on that.
We have addressed this a bit in other threads as well in support. It's being worked on and last time I checked when things like this happen they do get honored on our side regardless. I know it can be confusing and very frustrating but we will get it sorted out.
We will be sorting this out via charge back on all the gems I purchased for war. Gree has failed to deliver on a satisfactory gaming experience. Something I paid good money for.
3 days later and the gem sale isn't fixed yet. Pretty unbelievable that Gree is impacting their revenue for so long.
Maybe those programmers that got laid off did a number on Gree before they left. Too bad they couldn't have given us a bug free game. Then maybe they would still have jobs.
This is either Gree's complete incapability to do anything right or an illegal trick to make you buy the limited time supply crate (which is better value anyway but is beside the point).
This is false advertising and I'm sure it's illegal.
I have heard rumors that some people are buying many mountains and then getting itunes to refund them 100%. For those of us who think that is too sneaky and want to legitimately buy gems at the sale price, will gree extend the sale once they get things fixed?
No word from Gree yet? I have purchased gems twice at full price this weekend for the war effort. I would like to get credit for that with additional gems or something. Hope they resolve it soon.
This will probably sound more cheeky than I mean it... but as sad as it is to say this, I believe we need to move past considering Gree an above-board firm. If we give them money at this point, it should be assumed that the chance of shenanigans is at least +/- 50%--whether that is in the form of glitches, mischarges, etc. If that makes anyone uncomfortable, then I would really recommend not spending any money. I spend an amount I am comfortable watching them set on fire, which is to say not very much.
Pretty unbelievable that Gree is impacting their revenue for so long.
Pretty unbelievable you guys are still buying gems after being screwed for so long.


text:
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HOW LEGAL/FINANCIAL PROBLEMS ARE SOLVED BY NYC CO-OPS AND CONDOS
> LEGAL/FINANCIAL > LOFTLANDIA: THE SOUTH BRONX IS THE NEW SOHO
Loftlandia: The South Bronx Is the New Soho
The Loft Law was passed in the 1980s to protect pioneering artists in rundown neighborhoods like Soho.
Gov. Andrew Cuomo has signed legislation that extends the state's Loft Law, Crain's reports, yet another boost to tenants and yet another blow to landlords dished out by the state Legislature, which is under full Democratic control for the first time in a decade.
The Loft Law allows an owner of a manufacturing property to formally convert a building to residential use when occupants are already living at the premises illegally. The law was conceived in the 1980s to prevent pioneering artists who had moved into rundown properties in lower Manhattan, primarily in Soho and Tribeca, from being booted out of the dwellings after the neighborhoods became more attractive and affluent.
Residents in buildings with loft status can live rent-free until their landlord brings the building into compliance with residential health and safety codes – a process that often takes years and can be prolonged by the tenants themselves. After the units are certified for residential occupation, residents qualify for rent stabilization. The new legislation signed by Gov. Cuomo allows tenants or landlords to apply for loft status in buildings where the tenants were in residence in 2015 and 2016. The new applicants likely will be in Brooklyn, the South Bronx and Queens – where artists have tended to migrate in recent years.
Proponents say the law makes buildings safer without displacing residents. But landlords say the law has in the past allowed tenants to drag out renovations by blocking access, which they say allows them to continue living rent-free. "It can be a years-long process to compel a tenant legally to provide access to their apartment so the conversion can be planned and then completed," says Jason Frosch, an attorney with Borah Goldstein who has represented several landlords stuck in such circumstances. "It's a huge loophole that is being exploited."


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Home » Brain & Nerves » Scientists program cells to carry out gene-guided construction projects
Scientists program cells to carry out gene-guided construction projects
Researchers have developed a method for building nanoscale structures with genetically reprogrammed cells.
Stanford researchers have developed a technique that reprograms cells to use synthetic materials, provided by the scientists, to build artificial structures able to carry out functions inside the body.
"We turned cells into chemical engineers of a sort, that use materials we provide to construct functional polymers that change their behaviors in specific ways," said Karl Deisseroth, MD, PhD, the D. H. Chen Professor and professor of bioengineering and of psychiatry and behavioral sciences.
A paper describing the research was published March 20 in Science. Deisseroth shares senior authorship with Zhenan Bao, PhD, the K.K. Lee Professor and professor of chemical engineering. The lead authors are former postdoctoral scholars Jia Liu, PhD, and Ariane Tom, PhD; graduate student Yoon Seok Kim; and research scientist Claire Richardson, PhD.
In the paper, researchers explained how they developed genetically targeted chemical assembly and used the new method to build artificial structures on mammalian brain cells and on neurons in the tiny worm called C. elegans. The structures were made using two different biocompatible materials, each with a different electronic property. One material was an insulator, the other a conductor.
Changing neuronal properties
Bao said that while the current experiments focused mainly on brain cells or neurons, genetically targeted chemical assembly should also work with other cell types. "We've developed a technology platform that can tap into the biochemical processes of cells throughout the body," Bao said.
The researchers began by genetically reprogramming cells they wanted to affect. They did this by using standard bioengineering techniques to deliver instructions for adding an enzyme, called APEX2, into specific neurons.
Next, the scientists immersed the worms and other experimental tissues in a solution with two active ingredients — an extremely low, nonlethal dose of hydrogen peroxide, and billions of molecules of the raw material they wanted the cells to use for their building projects.
Contact between the hydrogen peroxide and the neurons with the APEX2 enzyme triggered a series of chemical reactions that fused the raw-material molecules together into a chain known as a polymer to form a mesh-like material. In this way, the researchers were able to weave artificial nets, with either insulative or conductive properties, around only the neurons they wanted.
The polymers changed the properties of the neurons. Depending on which polymer was formed, the neurons fired faster or slower, and when these polymers were created in cells of C. elegans, the worms' crawling movements were altered in opposite ways.
In the mammalian cell experiments, the researchers ran similar polymer-forming experiments on living slices from mouse brains and on cultured neurons from rat brains, and verified the conducting or insulating properties of the synthesized polymers. Finally, they injected a low-concentration hydrogen peroxide solution along with millions of the raw-material molecules into the brains of mice to verify that these elements were not toxic together.
Tools for exploration
Rather than a medical application, Deisseroth said, "what we have are tools for exploration." But these tools could be used to study how multiple sclerosis, caused by the fraying of myelin insulation around nerves, might respond if diseased cells could be induced to form insulating polymers as replacements. Researchers might also explore whether forming conductive polymers atop misfiring neurons in autism or epilepsy might modify those conditions.
Going forward, the researchers would like to explore variants of their cell-targeted technology. Genetically targeted chemical assembly could be used to produce a wide range of functional materials, implemented by diverse chemical signals. "We're imagining a whole world of possibilities at this new interface of chemistry and biology," Deisseroth said.
Deisseroth is a Howard Hughes Medical Institute investigator and a member of Stanford Bio-X and the Wu Tsai Neurosciences Institute at Stanford. Bao is a senior fellow at the Precourt Institute for Energy, a member of Stanford Bio-X and Stanford ChEM-H, and an affiliate of the Stanford Woods Institute for the Environment.
Other Stanford co-authors are Kang Shen, MD, PhD, professor of pathology and biology; Sergiu Pasca, MD, PhD, assistant professor of psychiatry and behavioral sciences; laboratory director Jeffrey Tok, PhD; Lief Fenno, MD, PhD, a resident in psychiatry; Lydia-Marie Joubert, PhD, life science researcher; postdoctoral or visiting scholars Xiao Wang, PhD, Cheng Wang, PhD, Toru Katsumata, PhD, Huiliang Wang, PhD, Yuanwen Jiang, PhD, and Fikri Birey, PhD; former graduate student Shucheng Chen, PhD; and life science research assistant Charu Ramakrishnan.
A researcher at Lawrence Berkeley National Laboratory's Advanced Light Source also contributed to the work.
The research was supported in part by the National Institutes of Health, the National Science Foundation, Stanford Bio-X, Life Sciences Research Foundation, the Gordon and Betty Moore Foundation, the Wu-Tsai Neurosciences Institute and the Kwanjeong International Foundation.
By TOM ABATE
Tom Abate is the associate director of communications for the Stanford School of Engineering. Email him at [email protected].
Stanford Medicine integrates research, medical education and health care at its three institutions – Stanford University School of Medicine, Stanford Health Care (formerly Stanford Hospital & Clinics), and Lucile Packard Children's Hospital Stanford. For more information, please visit the Office of Communication & Public Affairs site at http://mednews.stanford.edu.
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On tantrik's advice, jobless graduate sets house on fire, killed in police firing
The father of the man admitted to a local hospital says that it was the tantrik who is to be blamed for this bizarre ritual.
New Delhi, News Nation Bureau | Updated : 09 July 2019, 10:35 AM
Black magic or suspicion of black magic has been reason behind many crimes, especially in rural India.
In a shocking case, a young man in Assam set his house and vehicles on fire to complete a bizarre ritual that was advised by a family 'guru'. According to a report by The Indian Express, the man has been identified as one Pulakesh Saharia. The 28-year-old was a graduate in computer application. On the fateful day, the jobless man forced his family to take part in the black magic ritual. Stripped naked, the Saharias set their home and vehicles on fire. When the neighbours informed the police about the blaze, the family attacked the cops with swords. An altercation broke out between the cops and the family and later Pulakesh Saharia was killed in the police firing, the Indian Express report said.
The father of the man admitted to a local hospital says that it was the tantrik who is to be blamed for this bizarre ritual. "He and his associate Bhuyan had told us that if this puja was held, a hidden treasure would be found, and Pulakesh would get a job. They tried to convince us that our daughter had committed suicide because she became a victim of sexual assault and this puja would ensure that the assaulters get punished," The Indian Express quoted Jadab Saharia as saying.
Black magic or suspicion of black magic has been reason behind many crimes, especially in rural India. Recently, a 30-year-old man had murdered his mother because he suspected that she was 'doing black magic on him.' According to news agency ANI, the incident took place on December 23 in a village in Telangana. Boinpalle Police of Rajanna Sircilla District said that, "On December 23, a person by the name of Srinivas, killed his mother by strangulating her to death. He suspected that his mother practiced black magic on him, from which he was falling ill often. So the accused decided to murder his 52-year-old mother at her residence." The man has been booked under Section 302 of the Indian Penal Code.
But it was the Burari case in the National Capital that rocked the nation. In this case, 11 members of a family allegedly committed suicide. Many reports suggested that the Chandawat family of Delhi's Burari area were under some influence and they were preforming some black magic ritual. However, the probe by the Delhi Police has pointed out the ritual gone wrong theory. According to the police, the family members never intended to end their lives. They were performing a ritual also known as 'badh pooja', in which, members tied themselves with the iron mesh in the house. But it was not a mass suicide. The probe has also ruled out any foul play or the theory that Lalit, one of the sons , killed everyone and then committed suicide.
First Published: Tuesday, July 09, 2019 10:13 AM
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