A geometric view of cryptographic equation solving

This paper considers the geometric properties of the Relinearisation algorithm and of the XL algorithm used in cryptology for equation solving. We give a formal description of each algorithm in terms of projective geometry, making particular use of the Veronese variety. We establish the fundamental geometrical connection between the two algorithms and show how both algorithms can be viewed as being equivalent to the problem of finding a matrix of low rank in the linear span of a collection of matrices, a problem sometimes known as the MinRank problem. Furthermore, we generalise the XL algorithm to a geometrically invariant algorithm, which we term the GeometricXL algorithm. The GeometricXL algorithm is a technique which can solve certain equation systems that are not easily soluble by the XL algorithm or by Groebner basis methods

Mechanical and Vacuum Stability Design Criteria for the LHC Experimental Vacuum Chambers

Four colliding beam experiments are planned for the Large Hadron Collider (LHC) requiring experimental vacuum chambers in the interaction region. The beam pipe should be as transparent as possible to scattered particles and detectors should be as close as possible to the interaction point, resulting in small diameter beam pipes. This, together with the bunched beam structure, makes ion induced pre ssure bump instability, well known from the Intersecting Storage Rings (ISR) at CERN, a potential problem. Adequate conductance, cleanliness of the beam pipes and efficient pumping are required to avo id this instability. Suppression of electron multipacting requires appropriate surface coatings and cleaning procedures. Small beam pipe diameters must provide the required beam stay clear and still a llow margin for alignment and stability inside detectors. Design criteria to ensure both local and global stability under static and dynamic mechanical loads are defined

Vacuum Stability for Ion Induced Gas Desorption

Ion induced vacuum instability was first observed in the Intersecting Proton Storage Rings (ISR) at CERN and in spite of substantial vacuum improvements, it remained a limitation of the maximum beam current throughout the operation of the machine. Extensive laboratory studies and dedicated machine experiments were made during this period to understand the details of this effect and to identify ways of increasing the limit to higher beam currents. Stimulated by the recent design work for the LHC vacuum system, the interest in this problem has been revived with a new critical review of the parameters which determine the pressure run-away in a given vacuum system with high intensity beams

The Secondary Electron Yield of Technical Materials and its Variation with Surface Treatments

Secondary electron emission of surfaces exposed to oscillating electromagnetic field is at the origin of the multipacting effect that could severely perturb the operation of particle accelerators. This contribution tries to illustrate by measurement results, the origin of the secondary electron emission as well as the main reasons for the discrepancies between technical materials and pure metals. The variation of the secondary electron yield with the incident electron energy will be discussed for various types of technical surfaces. The influence of a gas condensation on these surfaces will also be addressed in the context of the LHC accelerator. Various treatments aiming at a permanent reduction of the secondary electron yield will be presented. A special attention will be paid to the decrease of the secondary electron yield under electron or photon impact and to its possible beneficial consequences for the processing of devices prone to multipacting

The LEP Vacuum System: A Summary of 10 Years of Successful Operation

he LEP accelerator is now operating regularly above 100 GeV and its vacuum system is submitted to the impact of energetic photons with a critical energy approaching 1 MeV. The consequences of this high energy on the photon induced desorption will be reviewed in the light of the various photon absorption mechanisms for aluminium. A review will also be given of the ten years of operation of the LEP vacuum system concerning more especially the evolution of the dynamic pressure with the beam dose and energy, the main difficulties experienced and the actions taken to overcome them

Far-off-resonant wave interaction in one-dimensional photonic crystals with quadratic nonlinearity

We extend a recently developed Hamiltonian formalism for nonlinear wave interaction processes in spatially periodic dielectric structures to the far-off-resonant regime, and investigate numerically the three-wave resonance conditions in a one-dimensional optical medium with $\chi^{(2)}$ nonlinearity. In particular, we demonstrate that the cascading of nonresonant wave interaction processes generates an effective $\chi^{(3)}$ nonlinear response in these systems. We obtain the corresponding coupling coefficients through appropriate normal form transformations that formally lead to the Zakharov equation for spatially periodic optical media.Comment: 14 pages, 4 figure

Lateral diffusion in equimolar mixtures of natural sphingomyelins with dioleoylphosphatidylcholine

Cellular membranes of mammals are composed of a complex assembly of diverse phospholipids. Sphingomyelin (SM) and phosphatidylcholine (PC) are important lipids of eukaryotic cellular membranes and neuronal tissues, and presumably participate in the formation of membrane domains, known as "rafts," through intermolecular interaction and lateral microphase decomposition. In these two-dimensional membrane systems, lateral diffusion of lipids is an essential dynamic factor, which might even be indicative of lipid phase separation process. Here, we used pulsed field gradient nuclear magnetic resonance to study lateral diffusion of lipid components in macroscopically oriented bilayers composed of equimolar mixtures of natural SMs of egg yolk, bovine brain, bovine milk and dipalmitoylphosphatidylcholine (DPPC) with dioleoylphosphatidylcholine (DOPC). In addition, differential scanning calorimetry was used as a complementary technique to characterize the phase state of the lipid bilayers. In fully liquid bilayers, the lateral diffusion coefficients in both DOPC/DPPC and DOPC/SM systems exhibit mean values of the pure bilayers. For DOPC/SM bilayer system, this behavior can be explained by a model where most SM molecules form short-lived lateral domains with preferential SM-SM interactions occurring within them. However, for bilayers in the presence of their low-temperature gel phase, lateral diffusion becomes complicated and cannot simply be understood solely by a simple change in the liquid phase decomposition. © 2012 Elsevier Inc

Electron Cloud: Observations with LHC-Type Beams in the SPS

In August 1999, strong pressure increases were observed in the SPS in the presence of the new LHC-type beams. This paper reports on observations of the electron cloud phenomenon and the related pressure increase as a function of parameters such as the number of protons per bunch, the number of bunches per batch, the shape of the vacuum chamber and the electron current collected on pick-ups. Results of the observed clean-up, "beam scrubbing" will be presented as well as the consequences of the e-cloud phenomenon on the SPS operation with the LHC nominal beam intensity