Trapping in the random conductance model

We consider random walks on $\Z^d$ among nearest-neighbor random conductances which are i.i.d., positive, bounded uniformly from above but whose support extends all the way to zero. Our focus is on the detailed properties of the paths of the random walk conditioned to return back to the starting point at time $2n$. We show that in the situations when the heat kernel exhibits subdiffusive decay --- which is known to occur in dimensions $d\ge4$ --- the walk gets trapped for a time of order $n$ in a small spatial region. This shows that the strategy used earlier to infer subdiffusive lower bounds on the heat kernel in specific examples is in fact dominant. In addition, we settle a conjecture concerning the worst possible subdiffusive decay in four dimensions.Comment: 21 pages, version to appear in J. Statist. Phy

The Two-Screen Measurement Setup to Indirectly Measure Proton Beam Self-Modulation in AWAKE

The goal of the first phase of the AWAKE \cite{AWAKE1,AWAKE2} experiment at CERN is to measure the self-modulation \cite{SMI} of the $\sigma_z = 12\,\rm{cm}$ long SPS proton bunch into microbunches after traversing $10\,\rm{m}$ of plasma with a plasma density of $n_{pe}=7\times10^{14}\,\rm{electrons/cm}^3$. The two screen measurement setup \cite{Turner2016} is a proton beam diagnostic that can indirectly prove the successful development of the self-modulation of the proton beam by imaging protons that got defocused by the transverse plasma wakefields after passing through the plasma, at two locations downstream the end of the plasma. This article describes the design and realization of the two screen measurement setup integrated in the AWAKE experiment. We discuss the performance and background response of the system based on measurements performed with an unmodulated Gaussian SPS proton bunch during the AWAKE beam commissioning in September and October 2016. We show that the system is fully commissioned and adapted to eventually image the full profile of a self-modulated SPS proton bunch in a single shot measurement during the first phase of the AWAKE experiment.Comment: 5 pages 8 figure

Mean-field driven first-order phase transitions in systems with long-range interactions

We consider a class of spin systems on $\Z^d$ with vector valued spins (\bS_x) that interact via the pair-potentials J_{x,y} \bS_x\cdot\bS_y. The interactions are generally spread-out in the sense that the $J_{x,y}$'s exhibit either exponential or power-law fall-off. Under the technical condition of reflection positivity and for sufficiently spread out interactions, we prove that the model exhibits a first-order phase transition whenever the associated mean-field theory signals such a transition. As a consequence, e.g., in dimensions $d\ge3$, we can finally provide examples of the 3-state Potts model with spread-out, exponentially decaying interactions, which undergoes a first-order phase transition as the temperature varies. Similar transitions are established in dimensions $d=1,2$ for power-law decaying interactions and in high dimensions for next-nearest neighbor couplings. In addition, we also investigate the limit of infinitely spread-out interactions. Specifically, we show that once the mean-field theory is in a unique ``state,'' then in any sequence of translation-invariant Gibbs states various observables converge to their mean-field values and the states themselves converge to a product measure.Comment: 57 pages; uses a (modified) jstatphys class fil

Indirect Self-Modulation Instability Measurement Concept for the AWAKE Proton Beam

AWAKE, the Advanced Proton-Driven Plasma Wakefield Acceleration Experiment, is a proof-of-principle R&D experiment at CERN using a 400 GeV/c proton beam from the CERN SPS (longitudinal beam size sigma_z = 12 cm) which will be sent into a 10 m long plasma section with a nominal density of approx. 7x10^14 atoms/cm3 (plasma wavelength lambda_p = 1.2mm). In this paper we show that by measuring the time integrated transverse profile of the proton bunch at two locations downstream of the AWAKE plasma, information about the occurrence of the self-modulation instability (SMI) can be inferred. In particular we show that measuring defocused protons with an angle of 1 mrad corresponds to having electric fields in the order of GV/m and fully developed self-modulation of the proton bunch. Additionally, by measuring the defocused beam edge of the self-modulated bunch, information about the growth rate of the instability can be extracted. If hosing instability occurs, it could be detected by measuring a non-uniform defocused beam shape with changing radius. Using a 1 mm thick Chromox scintillation screen for imaging of the self-modulated proton bunch, an edge resolution of 0.6 mm and hence a SMI saturation point resolution of 1.2 m can be achieved.Comment: 4 pages, 4 figures, EAAC conference proceeding

Usporedba kvalitete površine i životnog vijeka alata pri blanjanju lameliranih elemenata za prozore

The quality of the surface of wooden elements, that have been planed, has a crucial importance in the whole production process, since the obtained effects affect the quality of wooden surface after finishing (painting). The occurrence of defects is usually the reason for qualifying a workpiece as scrap or for requiring additional work. This paper presents the selected results of research of the effect of the cutting tool wear on the surface quality of elements after planing. Research experiments were conducted on the SCM Superset Class machine tool. Glulam elements of pine wood (Pinus sylvestris L.) were researched. The raw material samples (semi-finished products), 6 m long before planing, had been machined by suppliers also by planing. These workpieces were selected according to the plant requirements, e.g. their moisture content, straightness, and other defects. This paper presents the measuring results of surface roughness and some examples of surface profiles, as well as the dependence of total length of the planed elements on the type of blade material. From an economic point of view, the results showed that the use of solid carbide blades were more cost effective.Kvaliteta površine drvenih elemenata obrađenih blanjanjem ima presudnu važnost u cjelokupnome proizvodnom procesu jer kvaliteta blanjanja utječe na kvalitetu površine drva nakon završne obrade (nakon bojenja). Zbog nastalih grešaka elementi za prozore najčešće se kvalificiraju kao škart ili se moraju obaviti dodatni poslovi radi uklanjanja tih grešaka. U radu su prikazani odabrani rezultati istraživanja utjecaja zatupljenja alata na kvalitetu površine elemenata nakon blanjanja. Istraživanje je provedeno na stroju SCM Superset Class. Kao uzorci odabrani su lamelirani elementi od borovine (Pinus sylvestris L.). Uzorci lameliranih elemenata (poluproizvoda), duljine 6 m, već su prije eksperimentalnog blanjanja kod dobavljača također obrađeni blanjanjem. Ti su elementi selektirani prema zahtjevima tehnološkog procesa, npr. prema sadržaju vode, ravnosti i postojećim greškama drva. U radu su navedeni rezultati mjerenja kvalitete površine elemenata, kao i neki primjeri profila površine te ovisnost ukupne duljine oblanjanih elemenata o vrsti materijala oštrice alata. Rezultati su pokazali da je s ekonomskog stajališta isplativija uporaba alata s oštricama od tvrdog metala

Optimal designs for rational function regression

We consider optimal non-sequential designs for a large class of (linear and nonlinear) regression models involving polynomials and rational functions with heteroscedastic noise also given by a polynomial or rational weight function. The proposed method treats D-, E-, A-, and $\Phi_p$-optimal designs in a unified manner, and generates a polynomial whose zeros are the support points of the optimal approximate design, generalizing a number of previously known results of the same flavor. The method is based on a mathematical optimization model that can incorporate various criteria of optimality and can be solved efficiently by well established numerical optimization methods. In contrast to previous optimization-based methods proposed for similar design problems, it also has theoretical guarantee of its algorithmic efficiency; in fact, the running times of all numerical examples considered in the paper are negligible. The stability of the method is demonstrated in an example involving high degree polynomials. After discussing linear models, applications for finding locally optimal designs for nonlinear regression models involving rational functions are presented, then extensions to robust regression designs, and trigonometric regression are shown. As a corollary, an upper bound on the size of the support set of the minimally-supported optimal designs is also found. The method is of considerable practical importance, with the potential for instance to impact design software development. Further study of the optimality conditions of the main optimization model might also yield new theoretical insights.Comment: 25 pages. Previous version updated with more details in the theory and additional example

Superconductivity and charge carrier localization in ultrathin La1.85Sr0.15CuO4/La2CuO4\mathbf{{La_{1.85}Sr_{0.15}CuO_4}/{La_2CuO_4}} bilayers

$\mathrm{La_{1.85}Sr_{0.15}CuO_4}$/$\mathrm{La_2CuO_4}$ (LSCO15/LCO) bilayers with a precisely controlled thickness of N unit cells (UCs) of the former and M UCs of the latter ([LSCO15\_N/LCO\_M]) were grown on (001)-oriented {\slao} (SLAO) substrates with pulsed laser deposition (PLD). X-ray diffraction and reciprocal space map (RSM) studies confirmed the epitaxial growth of the bilayers and showed that a [LSCO15\_2/LCO\_2] bilayer is fully strained, whereas a [LSCO15\_2/LCO\_7] bilayer is already partially relaxed. The \textit{in situ} monitoring of the growth with reflection high energy electron diffraction (RHEED) revealed that the gas environment during deposition has a surprisingly strong effect on the growth mode and thus on the amount of disorder in the first UC of LSCO15 (or the first two monolayers of LSCO15 containing one $\mathrm{CuO_2}$ plane each). For samples grown in pure $\mathrm{N_2O}$ gas (growth type-B), the first LSCO15 UC next to the SLAO substrate is strongly disordered. This disorder is strongly reduced if the growth is performed in a mixture of $\mathrm{N_2O}$ and $\mathrm{O_2}$ gas (growth type-A). Electric transport measurements confirmed that the first UC of LSCO15 next to the SLAO substrate is highly resistive and shows no sign of superconductivity for growth type-B, whereas it is superconducting for growth type-A. Furthermore, we found, rather surprisingly, that the conductivity of the LSCO15 UC next to the LCO capping layer strongly depends on the thickness of the latter. A LCO capping layer with 7~UCs leads to a strong localization of the charge carriers in the adjacent LSCO15 UC and suppresses superconductivity. The magneto-transport data suggest a similarity with the case of weakly hole doped LSCO single crystals that are in a so-called {"{cluster-spin-glass state}"

Annealed lower tails for the energy of a polymer

We consider the energy of a randomly charged polymer. We assume that only charges on the same site interact pairwise. We study the lower tails of the energy, when averaged over both randomness, in dimension three or more. As a corollary, we obtain the correct temperature-scale for the Gibbs measure.Comment: 27 page

The topographical anatomy of the round window and related structures for the purpose of cochlear implant surgery

The treatment of total deafness using a cochlear implant has now become a routine medical procedure. The tendency to expand the audiological indications for cochlear stimulation and to preserve the remnants of hearing has brought new problems. The authors have studied the topographical anatomy of the internal structures of the ear in the area where cochleostomy is usually performed and an implant electrode inserted. Ten human temporal bones were obtained from cadavers and prepared in a formalin stain. After dissection of the bone in the area of round and oval windows, the following diameters were measured using a microscope with a scale: the transverse diameters of the cochlear and vestibular scalae at the level of the centre of the round window and 0.5 mm anteriorly to the round window, the distance between the windows and the distances from the end of the spiral lamina to the centre of the round window and to its anterior margin. The width of the cochlear scala at the level of the round window was 1.23 mm, and 0.5 mm anteriorly to the round window membrane it was 1.24 mm. The corresponding diameters for the vestibular scala are 1.34 and 1.27 mm. The distances from the end of the spiral lamina to the centre of the round window and to its anterior margin are 1.26 and 2.06 respectively. The authors noted that the two methods of electrode insertion show a difference of 2 mm in the length of the stimulated spiral lamina. The average total length of the unstimulated lamina is 2.06 and 4.06 in the two situations respectively