Boolean Dimension, Components and Blocks

We investigate the behavior of Boolean dimension with respect to components and blocks. To put our results in context, we note that for Dushnik-Miller dimension, we have that if $\dim(C)\le d$ for every component $C$ of a poset $P$, then $\dim(P)\le \max\{2,d\}$; also if $\dim(B)\le d$ for every block $B$ of a poset $P$, then $\dim(P)\le d+2$. By way of constrast, local dimension is well behaved with respect to components, but not for blocks: if $\text{ldim}(C)\le d$ for every component $C$ of a poset $P$, then $\text{ldim}(P)\le d+2$; however, for every $d\ge 4$, there exists a poset $P$ with $\text{ldim}(P)=d$ and $\dim(B)\le 3$ for every block $B$ of $P$. In this paper we show that Boolean dimension behaves like Dushnik-Miller dimension with respect to both components and blocks: if $\text{bdim}(C)\le d$ for every component $C$ of $P$, then $\text{bdim}(P)\le 2+d+4\cdot2^d$; also if $\text{bdim}(B)\le d$ for every block of $P$, then $\text{bdim}(P)\le 19+d+18\cdot 2^d$.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1712.0609

Shattering-Extremal Set Systems of Small VC-Dimension

We say that a set system $\mathcal{F}\subseteq 2^{[n]}$ shatters a given set $S\subseteq [n]$ if $2^S={F \cap S : F \in \mathcal{F}}$. The Sauer inequality states that in general, a set system $\mathcal{F}$ shatters at least $|\mathcal{F}|$ sets. Here we concentrate on the case of equality. A set system is called shattering-extremal if it shatters exactly $|\mathcal{F}|$ sets. We characterize shattering extremal set systems of Vapnik-Chervonenkis dimension 1 in terms of their inclusion graphs. Also from the perspective of extremality, we relate set systems of bounded Vapnik-Chervonenkis dimension to their projections.Comment: 17 page

Exploring projective norm graphs

The projective norm graphs $\text{NG}(q,t)$ provide tight constructions for the Tur\'an number of complete bipartite graphs $K_{t,s}$ with $s>(t-1)!$. In this paper we determine their automorphism group and explore their small subgraphs. To this end we give quite precise estimates on the number of solutions of certain equation systems involving norms over finite fields. The determination of the largest integer $s_t$, such that the projective norm graph $\text{NG}(q,t)$ contains $K_{t,s_t}$ for all large enough prime powers $q$ is an important open question with far-reaching general consequences. The best known bounds, $t-1\leq s_t \leq (t-1)!$, are far apart for $t\geq 4$. Here we prove that $\text{NG}(q,4)$ does contain (many) $K_{4,6}$ for any prime power $q$ not divisble by $2$ or $3$. This greatly extends recent work of Grosu, using a completely different approach. Along the way we also count the copies of any fixed $3$-degenerate subgraph, and find that projective norm graphs are quasirandom with respect to this parameter. Some of these results also extend the work of Alon and Shikhelman on generalized Tur\'an numbers. Finally we also give a new, more elementary proof for the $K_{4,7}$-freeness of $\text{NG}(q,4)$.Comment: 41 pages + 6 pages of Appendi

On the toughness of thermoplastic polymer nanocomposites as assessed by the essential work of fracture (EWF) approach

The essential work of fracture (EWF) approach is widely used to determine the plane stress fracture toughness of highly ductile polymers and related systems. To shed light on how the toughness is affected by nanofillers EWF-suited model polymers, viz. amorphous copolyester and polypropylene block copolymer were modified by multiwall carbon nanotube (MWCNT), graphene (GR), boehmite alumina (BA), and organoclay (MMT) in 1 wt% each. EWF tests were performed on deeply double-edge notched tensile-loaded specimens under quasistatic loading conditions. Data reduction occurred by energy partitioning between yielding and necking/tearing. The EWF prerequisites were not met with the nanocomposites containing MWCNT and GR by contrast to those with MMT and BA. Accordingly, the toughness of nanocomposites with homogeneously dispersed and low aspect ratio fillers may be properly determined using the EWF. Results indicated that incorporation of nanofillers may result in an adverse effect between the specific essential and non-essential EWF parameters

A Possible Explanation of the Radio Afterglow of GRB980519: The Dense Medium Effect

GRB{980519} is characterized by its rapidly declining optical and X-ray afterglows. Explanations of this behavior include models invoking a dense medium environment which makes the shock wave evolve quickly into the sub-relativistic phase, a jet-like outflow, and a wind-shaped circumburst medium environment. Recently, Frail {et al}. (1999a) found that the latter two cases are consistent with the radio afterglow of this burst. Here, by considering the trans-relativistic shock hydrodynamics, we show that the dense medium model can also account for the radio light curve quite well. The potential virtue of the dense medium model for GRB{980519} is that it implies a smaller angular size of the afterglow, which is essential for interpreting the strong modulation of the radio light curve. Optical extinction due to the dense medium is not important if the prompt optical-UV flash accompanying the $\gamma$-ray emission can destroy dust by sublimation out to an appreciable distance. Comparisons with some other radio afterglows are also discussed.Comment: 5 pages, 1 figure, a few minor changes made and references up dated, MNRAS, in pres

TeV Neutrinos from Successful and Choked Gamma-Ray Bursts

Core collapse of massive stars resulting in a relativistic fireball jet which breaks through the stellar envelope is a widely discussed scenario for gamma-ray burst production. For very extended or slow rotating stars, the fireball may be unable to break through the envelope. Both penetrating and choked jets will produce, by photo-meson interactions of accelerated protons, a burst of neutrinos with energies in excess of 5 TeV while propagating in the envelope. The predicted flux, from both penetrating and chocked fireballs, should be easily detectable by planned cubic kilometer neutrino telescopes.Comment: Phys.Rev.Letters, in press, final version accepted 8/31/01 (orig. 3/17/01

The broad band spectral properties of galactic X-ray binary pulsars

BeppoSAX observed several galactic binary X-ray pulsars during the Science Verification Phase and in the first year of the regular program. The complex emission spectra of these sources are an ideal target for the BeppoSAX instrumentation, that can measure the emission spectra in an unprecedented broad energy band. Using this capability of BeppoSAX a detailed observational work can be done on the galactic X-ray pulsars. In particular the 0.1-200 keV energy band allows the shape of the continuum emission to be tightly constrained. A better determination of the underlying continuum allows an easier detection of features superimposed onto it, both at low energy (Fe K and L, Ne lines) and at high energies (cyclotron features). We report on the spectral properties of a sample of X-ray pulsars observed with BeppoSAX comparing the obtained results. Some ideas of common properties are also discussed and compared with our present understanding of the emission mechanisms and processes.Comment: 6 pages, 2 figures. Uses espcrc2.sty (included).To appear in Proceedings of "The Active X-ray Sky: Results from BeppoSAX and Rossi-XTE