Limitations to Frechet's Metric Embedding Method

Frechet's classical isometric embedding argument has evolved to become a major tool in the study of metric spaces. An important example of a Frechet embedding is Bourgain's embedding. The authors have recently shown that for every e>0 any n-point metric space contains a subset of size at least n^(1-e) which embeds into l_2 with distortion O(\log(2/e) /e). The embedding we used is non-Frechet, and the purpose of this note is to show that this is not coincidental. Specifically, for every e>0, we construct arbitrarily large n-point metric spaces, such that the distortion of any Frechet embedding into l_p on subsets of size at least n^{1/2 + e} is \Omega((\log n)^{1/p}).Comment: 10 pages, 1 figur

Konvencija o zabrani kemijskoj oružja i uloga inženjera i znanstvenika

Chemical weapons, like all military technology, are associated with activities of scientists and engineers. However, chemical weapons differ from any other military technology because they were invented, and their first mass use directly developed by famous chemists. The active contribution of engineers and scientists and their organisations in the negotiations on chemical disarmament, including drafting the Chemical Weapons Convention, is described. Their present and future role in implementing the Convention is analysed, taking into consideration the threats and benefits of advances in science and technology, and stressing the independent expertise of the OPCW Scientific Advisory Board.Kemijsko oružje, kao i cijela vojna tehnologija, povezana je s aktivnostima inženjera i znanstvenika. Međutim, kemijsko oružje se razlikuje u odnosu na druge vojne tehnologije po tome, što je ono nastalo i prvi puta masovno upotrijebljeno izravnim sudjelovanjem znamenitih kemičara. Opisan je znatan doprinos inženjera i znanstvenika te njihovih organizacija pregovorima o kemijskom razoružanju, uključujući i stvaranje (pisanje) Konvencije o zabrani kemijskog oružja. Razmatra se njihova sadašnja i buduća uloga u provedbi Konvencije uzimajući u obzir ugroze i dobrobit koje donose napredak znanosti i tehnologije, naglašavajući neovisnu ekspertizu Znanstvenog savjeta Organizacije za zabranu kemijskog oružja

Kemijsko razoružanje: Aktualni problemi u provedbi Konvencije o kemijskom oružju

The Chemical Weapons Convention (CWC) is briefly characterised by stressing its main pillars, such as verified destruction ofCWstockpiles and destruction/conversion ofCWproduction facilities (CWPFs), verified non-production of CW by the chemical industries, assistance and protection), and international cooperation. The CWC´s leading principle in defining theCW(protecting it generally against scientific and technological development, i. e. so called General Purpose Criterion) is thoroughly elucidated showing its relation to the CWC´s sophisticated verification system. Status of implementation (as of August 2005) shows main data obligatory declared by the States Parties (SP), among them 6 possessors of CW stockpiles (Russia, USA, India, South Korea, Albania and Libya). From the declared 71 373 agent-tons, 12 889 have been destroyed, from the declared 8 679 M items of munitions (containers), 2 420 have been destroyed, which means that the anticipated 10 years deadline for CW destruction (after entry into force EIF) will be not managed. For Russia and USA the allowed extension by another 5 years has been already agreed. From the 64 CWPFs (operational after 1946), declared by 12 SPs, 53 have been certified as destroyed/converted. The Organisation for the Prohibition of Chemical Weapons (OPCW) is briefly presented and main results of the First Review Conference (2003) analysed on the base of the adopted principal documents. Future problems of implementing the CWC are connected in the first line with its universality, because among 16 non-SPs, several countries (located mainly in the Near East and on the Korean peninsula) are presumed to be CW-possessors. Special emphasis is laid on both, threats and benefits of the scientific and technological development for current implementing the CWC as well as of its implementation in future after all CW stockpiles have been destroyed.Kratko je opisana Konvencija o kemijskom oružju (Chemical Weapons Convention, CWC), naglašavajući glavne stavke, kao što je verificirano uništavanje zaliha kemijskog oružja i uništavanje ili prenamjena postrojenja za proizvodnju, verifikacija prestanka proizvodnje kemijskog oružja u kemijskim industrijama, pomoć i zaštita, te međunarodna suradnja. Vodeće načelo Konvencije o kemijskom oružju u definiranju kemijskog oružja (štiteći ga općenito od znanstvenog i tehnološkog napretka) je tzv. Kriterij opće namjene koji je potpuno razjašnjen te se pokazuje njegov odnos prema sofisticiranom sustavu verifikacije CWC-a. Stanje primjene Konvencije (od kolovoza 2005.) pokazuje osnovne glavne podatke koje su kao obvezu dostavile države članice, a među njima je šest država koje imaju zalihe kemijskog oružja (Rusija, SAD, Indija, Južna Koreja, Albanija i Libija). Od deklariranih 71 373 tona agensa, 12 889 je uništeno; od deklariranih 8,679 milijuna komada streljiva (kontejnera) 2420 je uništeno, što znači da se predviđenih deset godina kao krajnji rok za uništavanje kemijskog oružja (od početka primjene Konvencije), neće moći održati. Za Rusiju i SAD je već dogovoreno produljenje od daljnjih pet godina. Od 64 postrojenja za proizvodnju bojnih otrova (operativnih poslije 1946.) koje je deklariralo 12 država članica, 53 je verificirano kao uništeno odnosno prenamijenjeno. Kratko je predstavljena Organizacija za zabranu kemijskog oružja (Organisation for the Prohibition of Chemical Weapons, OPCW), a glavni rezultati Prve pregledne konferencije (održane 2003.) su analizirani na temelju prihvaćenih osnovnih dokumenata. Budući problemi primjene CWC povezani su u prvom redu s njezinom univerzalnošću, s obzirom na to da je među 18 nečlanica i nekoliko zemalja (lociranih uglavnom na Bliskom istoku i Korejskom poluotoku) za koje se pretpostavlja da posjeduju kemijsko oružje. Posebni naglasak stavljen je na prijetnje i koristi od znanstvenog i tehnološkog razvoja na tekuće primjene Konvencije kao i njezine primjene u budućnosti nakon što sve zalihe kemijskog oružja budu uništene

Kemijsko razoružanje: Aktualni problemi u provedbi Konvencije o kemijskom oružju

The Chemical Weapons Convention (CWC) is briefly characterised by stressing its main pillars, such as verified destruction ofCWstockpiles and destruction/conversion ofCWproduction facilities (CWPFs), verified non-production of CW by the chemical industries, assistance and protection), and international cooperation. The CWC´s leading principle in defining theCW(protecting it generally against scientific and technological development, i. e. so called General Purpose Criterion) is thoroughly elucidated showing its relation to the CWC´s sophisticated verification system. Status of implementation (as of August 2005) shows main data obligatory declared by the States Parties (SP), among them 6 possessors of CW stockpiles (Russia, USA, India, South Korea, Albania and Libya). From the declared 71 373 agent-tons, 12 889 have been destroyed, from the declared 8 679 M items of munitions (containers), 2 420 have been destroyed, which means that the anticipated 10 years deadline for CW destruction (after entry into force EIF) will be not managed. For Russia and USA the allowed extension by another 5 years has been already agreed. From the 64 CWPFs (operational after 1946), declared by 12 SPs, 53 have been certified as destroyed/converted. The Organisation for the Prohibition of Chemical Weapons (OPCW) is briefly presented and main results of the First Review Conference (2003) analysed on the base of the adopted principal documents. Future problems of implementing the CWC are connected in the first line with its universality, because among 16 non-SPs, several countries (located mainly in the Near East and on the Korean peninsula) are presumed to be CW-possessors. Special emphasis is laid on both, threats and benefits of the scientific and technological development for current implementing the CWC as well as of its implementation in future after all CW stockpiles have been destroyed.Kratko je opisana Konvencija o kemijskom oružju (Chemical Weapons Convention, CWC), naglašavajući glavne stavke, kao što je verificirano uništavanje zaliha kemijskog oružja i uništavanje ili prenamjena postrojenja za proizvodnju, verifikacija prestanka proizvodnje kemijskog oružja u kemijskim industrijama, pomoć i zaštita, te međunarodna suradnja. Vodeće načelo Konvencije o kemijskom oružju u definiranju kemijskog oružja (štiteći ga općenito od znanstvenog i tehnološkog napretka) je tzv. Kriterij opće namjene koji je potpuno razjašnjen te se pokazuje njegov odnos prema sofisticiranom sustavu verifikacije CWC-a. Stanje primjene Konvencije (od kolovoza 2005.) pokazuje osnovne glavne podatke koje su kao obvezu dostavile države članice, a među njima je šest država koje imaju zalihe kemijskog oružja (Rusija, SAD, Indija, Južna Koreja, Albanija i Libija). Od deklariranih 71 373 tona agensa, 12 889 je uništeno; od deklariranih 8,679 milijuna komada streljiva (kontejnera) 2420 je uništeno, što znači da se predviđenih deset godina kao krajnji rok za uništavanje kemijskog oružja (od početka primjene Konvencije), neće moći održati. Za Rusiju i SAD je već dogovoreno produljenje od daljnjih pet godina. Od 64 postrojenja za proizvodnju bojnih otrova (operativnih poslije 1946.) koje je deklariralo 12 država članica, 53 je verificirano kao uništeno odnosno prenamijenjeno. Kratko je predstavljena Organizacija za zabranu kemijskog oružja (Organisation for the Prohibition of Chemical Weapons, OPCW), a glavni rezultati Prve pregledne konferencije (održane 2003.) su analizirani na temelju prihvaćenih osnovnih dokumenata. Budući problemi primjene CWC povezani su u prvom redu s njezinom univerzalnošću, s obzirom na to da je među 18 nečlanica i nekoliko zemalja (lociranih uglavnom na Bliskom istoku i Korejskom poluotoku) za koje se pretpostavlja da posjeduju kemijsko oružje. Posebni naglasak stavljen je na prijetnje i koristi od znanstvenog i tehnološkog razvoja na tekuće primjene Konvencije kao i njezine primjene u budućnosti nakon što sve zalihe kemijskog oružja budu uništene

A Tverberg type theorem for matroids

Let b(M) denote the maximal number of disjoint bases in a matroid M. It is shown that if M is a matroid of rank d+1, then for any continuous map f from the matroidal complex M into the d-dimensional Euclidean space there exist t \geq \sqrt{b(M)}/4 disjoint independent sets \sigma_1,\ldots,\sigma_t \in M such that \bigcap_{i=1}^t f(\sigma_i) \neq \emptyset.Comment: This article is due to be published in the collection of papers "A Journey through Discrete Mathematics. A Tribute to Jiri Matousek" edited by Martin Loebl, Jaroslav Nesetril and Robin Thomas, due to be published by Springe

On the discrepancy of random low degree set systems

Motivated by the celebrated Beck-Fiala conjecture, we consider the random setting where there are $n$ elements and $m$ sets and each element lies in $t$ randomly chosen sets. In this setting, Ezra and Lovett showed an $O((t \log t)^{1/2})$ discrepancy bound in the regime when $n \leq m$ and an $O(1)$ bound when $n \gg m^t$. In this paper, we give a tight $O(\sqrt{t})$ bound for the entire range of $n$ and $m$, under a mild assumption that $t = \Omega (\log \log m)^2$. The result is based on two steps. First, applying the partial coloring method to the case when $n = m \log^{O(1)} m$ and using the properties of the random set system we show that the overall discrepancy incurred is at most $O(\sqrt{t})$. Second, we reduce the general case to that of $n \leq m \log^{O(1)}m$ using LP duality and a careful counting argument

On the optimality of gluing over scales

We show that for every $\alpha > 0$, there exist $n$-point metric spaces (X,d) where every "scale" admits a Euclidean embedding with distortion at most $\alpha$, but the whole space requires distortion at least $\Omega(\sqrt{\alpha \log n})$. This shows that the scale-gluing lemma [Lee, SODA 2005] is tight, and disproves a conjecture stated there. This matching upper bound was known to be tight at both endpoints, i.e. when $\alpha = \Theta(1)$ and $\alpha = \Theta(\log n)$, but nowhere in between. More specifically, we exhibit $n$-point spaces with doubling constant $\lambda$ requiring Euclidean distortion $\Omega(\sqrt{\log \lambda \log n})$, which also shows that the technique of "measured descent" [Krauthgamer, et. al., Geometric and Functional Analysis] is optimal. We extend this to obtain a similar tight result for $L_p$ spaces with $p > 1$.Comment: minor revision

Bed roughness at high bed shear in open channels and pressurized pipes

River hydrodynamicsBed roughness and flow resistanc

Deterministic Sampling and Range Counting in Geometric Data Streams

We present memory-efficient deterministic algorithms for constructing epsilon-nets and epsilon-approximations of streams of geometric data. Unlike probabilistic approaches, these deterministic samples provide guaranteed bounds on their approximation factors. We show how our deterministic samples can be used to answer approximate online iceberg geometric queries on data streams. We use these techniques to approximate several robust statistics of geometric data streams, including Tukey depth, simplicial depth, regression depth, the Thiel-Sen estimator, and the least median of squares. Our algorithms use only a polylogarithmic amount of memory, provided the desired approximation factors are inverse-polylogarithmic. We also include a lower bound for non-iceberg geometric queries.Comment: 12 pages, 1 figur

Blocking Coloured Point Sets

This paper studies problems related to visibility among points in the plane. A point $x$ \emph{blocks} two points $v$ and $w$ if $x$ is in the interior of the line segment $\bar{vw}$. A set of points $P$ is \emph{$k$-blocked} if each point in $P$ is assigned one of $k$ colours, such that distinct points $v,w\in P$ are assigned the same colour if and only if some other point in $P$ blocks $v$ and $w$. The focus of this paper is the conjecture that each $k$-blocked set has bounded size (as a function of $k$). Results in the literature imply that every 2-blocked set has at most 3 points, and every 3-blocked set has at most 6 points. We prove that every 4-blocked set has at most 12 points, and that this bound is tight. In fact, we characterise all sets $\{n_1,n_2,n_3,n_4\}$ such that some 4-blocked set has exactly $n_i$ points in the $i$-th colour class. Amongst other results, for infinitely many values of $k$, we construct $k$-blocked sets with $k^{1.79...}$ points