Exponential bounds for the probability deviations of sums of random fields

Non-asymptotic exponential upper bounds for the deviation probability for a sum of independent random fields are obtained under Bernstein's condition and assumptions formulated in terms of Kolmogorov's metric entropy. These estimations are constructive in the sense that all the constants involved are given explicitly. In the case of moderately large deviations, the upper bounds have optimal log-asymptotices. The exponential estimations are extended to the local and global continuity modulus for sums of independent samples of a random field

The Hojman Construction and Hamiltonization of Nonholonomic Systems

In this paper, using the Hojman construction, we give examples of various Poisson brackets which differ from those which are usually analyzed in Hamiltonian mechanics. They possess a nonmaximal rank, and in the general case an invariant measure and Casimir functions can be globally absent for them

Finding of the optimal parameters of animated and stereographic rainbow diffractive images

We have considered the basic aspects of the technology of animated and stereographic rainbow images. These images can be included in Optical Security Devices (OSDs) in order to increase their structure complexity and to improve their protective properties. The cited technology provides, on the one hand, a simple identification on the visual level of verification and, on the other hand, the sufficient reliability against counterfeits. The last property is achieved at first by the division of the elemental unit on elemental regions of any adjusted shape with outline of the precision that is inaccessible for the recreation without Electron Beam Lithography Equipment (EBLE) which is used for the recording of OSDs. Second, the used encoding methods also assure the certain reliability. In the context of the paper, the theoretical discussion based a quantitative formulation of the Huygens--Fresnel law of the diffraction on an elemental diffractive grating is carried out. For other definite Conditions of Lighting and Observation of Diffracting Light (CLODL), the correlation between the slope angle of diffraction grating strokes and the corresponding horizontal parallax angle is got; and the parameter which defines the channel selection (the quality of splitting into separate channels) is introduced. The rule for the definition of the wavelength and the intensity of light that diffracts on a given grating under certain CLODL is derived as well. This fact allows one to create the software utility that models the behavior of anigrams or stereograms. An algorithm of the synthesis of anigrams and stereograms as parts of OSDs using the technology of a composite “figure” elemental unit (that is composed from parts of any shape) and applying the halftone encoding by the period, filling, and achromaticity or all these parameters at once is elaborated, and its software implementation is constructed. The criteria for the choice of such anigram’s parameters as the resolution, number of channels, and angle distances between them are elaborated. In order to check them, the results of tests are summarized. The investigation of the optimal parameters of halftone images’ encoding finding is implemented, and the results obtained for the tough and flexible linkings of a channel to the subregion of an elemental unit for hatching images are compared

Multilevel computer-generated holograms for reconstructing 3-D images in combined optical-digital security devices

Computer-generated holograms (CGHs) integrated within combined optical-digital security devices (CO/DSDs) are described in this work. They can restore the monochrome and color 3D images in white light. To record them, the Electron Beam Lithography (EBL) is used. Our investigations on optimization of synthesis and recording the CO/DSDs with the integrated in it multilevel CGHs of 3D images possessed horizontal parallax only (HPO) are presented here. The CGH fabrication process is mainly composed of two parts: calculation of the interferogram data (ID) and their recording. Calculation of the ID is done as follows: firstly, the geometrical and optical constants of recording scheme and the object surface represented by the elemental self-radiating areas, are determined, secondly, the basic parameters accounting for discretization of ID in hologram plane is defined. The ID values can be derived by calculation of the necessary elemental object areas bipolar intensities sum. Next, over suitable quantization of ID, recording the rectangle data appropriate for EBL onto glass coated with non-organic photoresist based on As₄₀S₄₀Se₂₀ is performed. We have also investigated reciprocal influence of an optical part of the CO/DSD and a digital one

The performance of the jet trigger for the ATLAS detector during 2011 data taking

The performance of the jet trigger for the ATLAS detector at the LHC during the 2011 data taking period is described. During 2011 the LHC provided proton–proton collisions with a centre-of-mass energy of 7 TeV and heavy ion collisions with a 2.76 TeV per nucleon–nucleon collision energy. The ATLAS trigger is a three level system designed to reduce the rate of events from the 40 MHz nominal maximum bunch crossing rate to the approximate 400 Hz which can be written to offline storage. The ATLAS jet trigger is the primary means for the online selection of events containing jets. Events are accepted by the trigger if they contain one or more jets above some transverse energy threshold. During 2011 data taking the jet trigger was fully efficient for jets with transverse energy above 25 GeV for triggers seeded randomly at Level 1. For triggers which require a jet to be identified at each of the three trigger levels, full efficiency is reached for offline jets with transverse energy above 60 GeV. Jets reconstructed in the final trigger level and corresponding to offline jets with transverse energy greater than 60 GeV, are reconstructed with a resolution in transverse energy with respect to offline jets, of better than 4 % in the central region and better than 2.5 % in the forward direction

Reduction and relative equilibria for the 2-body problem in spaces of constant curvature

We consider the two-body problem on surfaces of constant non-zero curvature and classify the relative equilibria and their stability. On the hyperbolic plane, for each $q>0$ we show there are two relative equilibria where the masses are separated by a distance $q$. One of these is geometrically of elliptic type and the other of hyperbolic type. The hyperbolic ones are always unstable, while the elliptic ones are stable when sufficiently close, but unstable when far apart. On the sphere of positive curvature, if the masses are different, there is a unique relative equilibrium (RE) for every angular separation except $\pi/2$. When the angle is acute, the RE is elliptic, and when it is obtuse the RE can be either elliptic or linearly unstable. We show using a KAM argument that the acute ones are almost always nonlinearly stable. If the masses are equal there are two families of relative equilibria: one where the masses are at equal angles with the axis of rotation (`isosceles RE') and the other when the two masses subtend a right angle at the centre of the sphere. The isosceles RE are elliptic if the angle subtended by the particles is acute and is unstable if it is obtuse. At $\pi/2$, the two families meet and a pitchfork bifurcation takes place. Right-angled RE are elliptic away from the bifurcation point. In each of the two geometric settings, we use a global reduction to eliminate the group of symmetries and analyse the resulting reduced equations which live on a 5-dimensional phase space and possess one Casimir function

Laser Mass Spectrometry Analysis of the Formation of Phosphazene-Containing Epoxy Oligomers

The main features of two methods for the synthesis of phosphazene-containing epoxy oligomers— namely, the methods based on oxidation of double bonds in organooxyphosphazenes and on the reaction of chlorocyclophosphazenes with diphenols and the subsequent interaction of the resulting hydroxy-aryloxy phosphazenes with epichlorohydrin—were examined. Using the example of hexa- and octa-eugenol derivatives of the corresponding cyclophosphazenes, optimal conditions were established for the oxidation of allyl groups of these compounds with peroxy acids and hexa- and octa-epoxide cyclophosphazenes were characterized. It was noted that the epoxidation of eugenol derivatives of a mixture of cyclophosphazenes with three to eight phosphazo groups is accompanied by side reactions leading to the formation of P–OH bonds and the partial opening of oxirane cycles. Bisphenol A phosphazene-containing oligoepoxides were synthesized both via the stage involving the formation of hydroxy-aryloxy cyclophosphazenes and their subsequent epoxidation with epichlorohydrin and via the direct interaction of chlorocyclophosphazenes with an excess of bisphenol A (BPA) in the presence of solid alkali. In the latter case, the resulting oligomers are mixtures of the conventional epoxide and phosphazene-containing epoxy oligomers. The content of the latter can be adjusted up to 50%. The synthesized oligomers contain 1–5% phosphorus. They can be cured by conventional hardeners to form flameproof or noncombustible compositions. © 2018, Pleiades Publishing, Ltd