An extremal problem on crossing vectors

For positive integers $w$ and $k$, two vectors $A$ and $B$ from $\mathbb{Z}^w$ are called $k$-crossing if there are two coordinates $i$ and $j$ such that $A[i]-B[i]\geq k$ and $B[j]-A[j]\geq k$. What is the maximum size of a family of pairwise $1$-crossing and pairwise non-$k$-crossing vectors in $\mathbb{Z}^w$? We state a conjecture that the answer is $k^{w-1}$. We prove the conjecture for $w\leq 3$ and provide weaker upper bounds for $w\geq 4$. Also, for all $k$ and $w$, we construct several quite different examples of families of desired size $k^{w-1}$. This research is motivated by a natural question concerning the width of the lattice of maximum antichains of a partially ordered set.Comment: Corrections and improvement

Anomalously old biotite ⁴⁰Ar/³⁹Ar ages in the NW Himalaya

Biotite 40Ar/39Ar ages older than corresponding muscovite 40Ar/39Ar ages, contrary to the diffusion properties of these minerals, are common in the Himalaya and other metamorphic regions. In these cases, biotite 40Ar/39Ar ages are commonly dismissed as “too old” on account of “excess Ar.” We present 32 step-heating 40Ar/39Ar ages from 17 samples from central Himachal Pradesh Himalaya, India. In almost all cases, the biotite ages are older than predicted from cooling histories. We document host-rock lithology and chemical composition, mica microstructures, biotite chemical composition, and chlorite and muscovite components of biotite separates to demonstrate that these factors do not offer an explanation for the anomalously old biotite 40Ar/39Ar ages. We discuss possible mechanisms that may account for extraneous Ar (inherited or excess Ar) in these samples. The most likely cause for “too-old” biotite is excess Ar, i.e., 40Ar that is separated from its parent K. We suggest that this contamination resulted from one or several of the following mechanisms: (1) 40Ar was released during Cenozoic prograde metamorphism; (2) 40Ar transport was restricted due to a temporarily dry intergranular medium; (3) 40Ar was released from melt into a hydrous fluid phase during melt crystallization. Samples from the Main Central Thrust shear zone may be affected by a different mechanism of excess-Ar accumulation, possibly linked to later-stage fluid circulation within the shear zone and chloritization. Different Ar diffusivities and/or solubilities in biotite and muscovite may explain why biotite is more commonly affected by excess Ar than muscovite

A fixed point theorem for the infinite-dimensional simplex

We define the infinite dimensional simplex to be the closure of the convex hull of the standard basis vectors in R^infinity, and prove that this space has the 'fixed point property': any continuous function from the space into itself has a fixed point. Our proof is constructive, in the sense that it can be used to find an approximate fixed point; the proof relies on elementary analysis and Sperner's lemma. The fixed point theorem is shown to imply Schauder's fixed point theorem on infinite-dimensional compact convex subsets of normed spaces.Comment: 8 pages; related work at http://www.math.hmc.edu/~su/papers.htm

On the average rank of LYM-sets

Let S be a finite set with some rank function r such that the Whitney numbers wi = |{x S|r(x) = i}| are log-concave. Given so that wk − 1 < wk wk + m, set W = wk + wk + 1 + … + wk + m. Generalizing a theorem of Kleitman and Milner, we prove that every F S with cardinality |F| W has average rank at least kwk + … + (k + m) wk + m/W, provided the normalized profile vector x1, …, xn of F satisfies the following LYM-type inequality: x0 + x1 + … + xn m + 1

Tur\'annical hypergraphs

This paper is motivated by the question of how global and dense restriction sets in results from extremal combinatorics can be replaced by less global and sparser ones. The result we consider here as an example is Turan's theorem, which deals with graphs G=([n],E) such that no member of the restriction set consisting of all r-tuples on [n] induces a copy of K_r. Firstly, we examine what happens when this restriction set is replaced just by all r-tuples touching a given m-element set. That is, we determine the maximal number of edges in an n-vertex such that no K_r hits a given vertex set. Secondly, we consider sparse random restriction sets. An r-uniform hypergraph R on vertex set [n] is called Turannical (respectively epsilon-Turannical), if for any graph G on [n] with more edges than the Turan number ex(n,K_r) (respectively (1+\eps)ex(n,K_r), no hyperedge of R induces a copy of K_r in G. We determine the thresholds for random r-uniform hypergraphs to be Turannical and to epsilon-Turannical. Thirdly, we transfer this result to sparse random graphs, using techniques recently developed by Schacht [Extremal results for random discrete structures] to prove the Kohayakawa-Luczak-Rodl Conjecture on Turan's theorem in random graphs.Comment: 33 pages, minor improvements thanks to two referee

Preparativne modifikacije askomicina. V. Dobivanje novih derivata pomoću zamjene cikloheksilvinilidenske podjedinice

Starting from the easily accessible 24-O-tert-butyldimethylsilyl-22(R)-dihydro-28-oxoascomycin, methodologies that allow replacement of the cyclohexylvinylidene moiety of ascomycin by various other substituents are described. In addition, a so far unknown reactivity of the masked tricarbonyl moiety of ascomycin towards a stabilized Wittig reagent is reported.Opisani su postupci koji počinju s lako pristupačnim 25-O-tert-butildimetilsilil-22(R)-dihidro-28-oksoaskomicinom i omogućuju zamjenu askomicinske cikloheksilvinilidenske podjedinice različitim substituentima

Sperner type theorems with excluded subposets

Let F be a family of subsets of an n-element set. Sperner's theorem says that if there is no inclusion among the members of F then the largest family under this condition is the one containing all ⌊ frac(n, 2) ⌋-element subsets. The present paper surveys certain generalizations of this theorem. The maximum size of F is to be found under the condition that a certain configuration is excluded. The configuration here is always described by inclusions. More formally, let P be a poset. The maximum size of a family F which does not contain P as a (not-necessarily induced) subposet is denoted by La (n, P). The paper is based on a lecture of the author at the Jubilee Conference on Discrete Mathematics [Banasthali University, January 11-13, 2009], but it was somewhat updated in December 2010. © 2011 Elsevier B.V. All rights reserved

Torn Between Two Plates: Exhumation of the Cer Massif (Internal Dinarides) as a Far‐Field Effect of Carpathian Slab Rollback Inferred From 40 Ar/ 39 Ar Dating and Cross Section Balancing

Abstract Extension across the southern Pannonian Basin and the internal Dinarides is characterized by Oligo‐Miocene metamorphic core complexes (MCCs) exhumed along mylonitic low‐angle extensional shear zones. Cer MCC at the transition between Dinarides and Pannonian Basin occupies a structural position within the distal‐most Adriatic thrust sheet and originates from two different tectonic processes: Late Cretaceous‐Paleogene nappe‐stacking during a continent‐continent collision with Adria in a lower plate position, and exhumation related to Miocene extension driven by the Carpathian slab‐rollback. Structural data and a balanced cross section across the Cer massif show linking of the exhuming shear zone to a breakaway fault, which reactivated the early Late Cretaceous most internal nappe contact. Paleozoic greenschist‐to amphibolite‐grade lithologies surround a polyphase intrusion composed of I‐ and S‐type granites and were exhumed along a shear zone characterized by top‐N transport. Thermobarometric analyses indicate an intrusion depth of 7–8 km of the Oligocene I‐type granite; cooling below ∼500°C occurred at 25.4 ± 0.6 Ma (1σ) yielded by 40 Ar/ 39 Ar dating of hornblende. Biotite and white mica from this intrusion as well as from the mylonitic shear zone yield 40 Ar/ 39 Ar cooling ages of 17–18 Ma independent of the used techniques (in situ laser ablation, single‐grain total fusion, single‐grain step heating, and multi‐grain step heating). White mica from the S‐type granite yield an 40 Ar/ 39 Ar cooling age of 16.7 ± 0.1 Ma (1σ). Associated dikes intruding the shear zone were also affected by N‐S extension resulting in the exhumation of the MCC, which was triggered by the opening of the Pannonian back‐arc basin in response to the Carpathian slab‐rollback.Plain Language Summary Horizontal stretching of continental plates induces thinning of the crustal upper part, melting of rocks, the sinking of the land surface, and formation of large basins. One of the world's best‐studied basins formed by such a process is the Central European Pannonian Basin. This basin is surrounded by the mountain belts of the Alps, Carpathians, and Dinarides. We have studied rocks between the Pannonian Basin and the southerly adjacent Dinaride Mountains, where rocks deposited in the basin are found right next to rocks that were initially about 7–8 km deep in the crust. These rocks are separated by a shear zone, along which they were brought to the surface. We have dated the activity of the shear zone by measuring concentrations of radioactive isotopes and their decay products contained in deformed minerals. The shear zone was active at a time when the Pannonian Basin started to open due to tectonic processes further NE underneath the Carpathian mountain chain. We also found evidence that the shear zone, which brought metamorphic rocks upwards was formerly one that brought rocks downwards into the crust during an earlier phase of mountain building, predating basin formation.Key Points Activity along the shear zone exhuming Cer metamorphic core complex in the internal Dinarides was dated by 40 Ar/ 39 Ar geochronology to ∼17 Ma Exhumation was facilitated by extensional reactivation of Late Cretaceous‐Paleogene nappe contacts resulting from Adria‐Europe collision Extensional reactivation of the thrusts is interpreted as a far‐field effect of Oligo‐Miocene Carpathian slab rollbac