Higgledy-piggledy subspaces and uniform subspace designs

In this article, we investigate collections of `well-spread-out' projective (and linear) subspaces. Projective $k$-subspaces in $\mathsf{PG}(d,\mathbb{F})$ are in `higgledy-piggledy arrangement' if they meet each projective subspace of co-dimension $k$ in a generator set of points. We prove that the set $\mathcal{H}$ of higgledy-piggledy $k$-subspaces has to contain more than $\min{|\mathbb{F}|,\sum_{i=0}^k\lfloor\frac{d-k+i}{i+1}\rfloor}$ elements. We also prove that $\mathcal{H}$ has to contain more than $(k+1)\cdot(d-k)$ elements if the field $\mathbb{F}$ is algebraically closed. An $r$-uniform weak $(s,A)$ subspace design is a set of linear subspaces $H_1,..,H_N\le\mathbb{F}^m$ each of rank $r$ such that each linear subspace $W\le\mathbb{F}^m$ of rank $s$ meets at most $A$ among them. This subspace design is an $r$-uniform strong $(s,A)$ subspace design if $\sum_{i=1}^N\mathrm{rank}(H_i\cap W)\le A$ for $\forall W\le\mathbb{F}^m$ of rank $s$. We prove that if $m=r+s$ then the dual ($\{H_1^\bot,...,H_N^\bot\}$) of an $r$-uniform weak (strong) subspace design of parameter $(s,A)$ is an $s$-uniform weak (strong) subspace design of parameter $(r,A)$. We show the connection between uniform weak subspace designs and higgledy-piggledy subspaces proving that $A\ge\min{|\mathbb{F}|,\sum_{i=0}^{r-1}\lfloor\frac{s+i}{i+1}\rfloor}$ for $r$-uniform weak or strong $(s,A)$ subspace designs in $\mathbb{F}^{r+s}$. We show that the $r$-uniform strong $(s,r\cdot s+\binom{r}{2})$ subspace design constructed by Guruswami and Kopprty (based on multiplicity codes) has parameter $A=r\cdot s$ if we consider it as a weak subspace design. We give some similar constructions of weak and strong subspace designs (and higgledy-piggledy subspaces) and prove that the lower bound $(k+1)\cdot(d-k)+1$ over algebraically closed field is tight.Comment: 27 pages. Submitted to Designs Codes and Cryptograph

Lines in higgledy-piggledy position

We examine sets of lines in PG(d,F) meeting each hyperplane in a generator set of points. We prove that such a set has to contain at least 1.5d lines if the field F has more than 1.5d elements, and at least 2d-1 lines if the field F is algebraically closed. We show that suitable 2d-1 lines constitute such a set (if |F| > or = 2d-1), proving that the lower bound is tight over algebraically closed fields. At last, we will see that the strong (s,A) subspace designs constructed by Guruswami and Kopparty have better (smaller) parameter A than one would think at first sight.Comment: 17 page

A finite word poset : In honor of Aviezri Fraenkel on the occasion of his 70th birthday

Our word posets have �nite words of bounded length as their elements, with the words composed from a �nite alphabet. Their partial ordering follows from the inclusion of a word as a subsequence of another word. The elemental combinatorial properties of such posets are established. Their automorphism groups are determined (along with similar result for the word poset studied by Burosch, Frank and R¨ohl [4]) and a BLYM inequality is veri�ed (via the normalized matching property)

A characterization of multiple (n-k)-blocking sets in projective spaces of square order

In [10], it was shown that small t-fold (n - k)-blocking sets in PG(n, q), q = p(h), p prime, h >= 1, intersect every k-dimensional space in t (mod p) points. We characterize in this article all t-fold (n k)-blocking sets in PG(n, q), q square, q >= 661, t < c(p)q(1/6)/2, vertical bar B vertical bar < tq(n-k) + 2tq(n-k-1) root q, intersecting every k-dimensional space in t (mod root q) points

Proton-proton elastic scattering at the LHC energy of {\surd} = 7 TeV

Proton-proton elastic scattering has been measured by the TOTEM experiment at the CERN Large Hadron Collider at {\surd}s = 7 TeV in dedicated runs with the Roman Pot detectors placed as close as seven times the transverse beam size (sbeam) from the outgoing beams. After careful study of the accelerator optics and the detector alignment, |t|, the square of four-momentum transferred in the elastic scattering process, has been determined with an uncertainty of d t = 0.1GeV p|t|. In this letter, first results of the differential cross section are presented covering a |t|-range from 0.36 to 2.5GeV2. The differential cross-section in the range 0.36 < |t| < 0.47 GeV2 is described by an exponential with a slope parameter B = (23.6{\pm}0.5stat {\pm}0.4syst)GeV-2, followed by a significant diffractive minimum at |t| = (0.53{\pm}0.01stat{\pm}0.01syst)GeV2. For |t|-values larger than ~ 1.5GeV2, the cross-section exhibits a power law behaviour with an exponent of -7.8_\pm} 0.3stat{\pm}0.1syst. When compared to predictions based on the different available models, the data show a strong discriminative power despite the small t-range covered.Comment: 12pages, 5 figures, CERN preprin

First Results from the TOTEM Experiment

The first physics results from the TOTEM experiment are here reported, concerning the measurements of the total, differential elastic, elastic and inelastic pp cross-section at the LHC energy of $\sqrt{s}$ = 7 TeV, obtained using the luminosity measurement from CMS. A preliminary measurement of the forward charged particle $\eta$ distribution is also shown.Comment: Conference Proceeding. MPI@LHC 2010: 2nd International Workshop on Multiple Partonic Interactions at the LHC. Glasgow (UK), 29th of November to the 3rd of December 201

LHC Optics Measurement with Proton Tracks Detected by the Roman Pots of the TOTEM Experiment

Precise knowledge of the beam optics at the LHC is crucial to fulfil the physics goals of the TOTEM experiment, where the kinematics of the scattered protons is reconstructed with the near-beam telescopes -- so-called Roman Pots (RP). Before being detected, the protons' trajectories are influenced by the magnetic fields of the accelerator lattice. Thus precise understanding of the proton transport is of key importance for the experiment. A novel method of optics evaluation is proposed which exploits kinematical distributions of elastically scattered protons observed in the RPs. Theoretical predictions, as well as Monte Carlo studies, show that the residual uncertainty of this optics estimation method is smaller than 0.25 percent.Comment: 20 pages, 11 figures, 5 figures, to be submitted to New J. Phy

Elastic Scattering and Total Cross-Section in p+p reactions measured by the LHC Experiment TOTEM at sqrt(s) = 7 TeV

Proton-proton elastic scattering has been measured by the TOTEM experiment at the CERN Large Hadron Collider at $\sqrt{s} = 7$ TeV in special runs with the Roman Pot detectors placed as close to the outgoing beam as seven times the transverse beam size. The differential cross-section measurements are reported in the |t|-range of 0.36 to 2.5 GeV^2. Extending the range of data to low t values from 0.02 to 0.33 GeV^2,and utilizing the luminosity measurements of CMS, the total proton-proton cross section at sqrt(s) = 7 TeV is measured to be (98.3 +- 0.2(stat) +- 2.8(syst)) mb.Comment: Proceedings of the XLI International Symposium on Multiparticle Dynamics. Accepted for publication in Prog. Theor. Phy