In this paper, we deal with the second order difference equation with asymmetric nonlinearities on the integer lattice and we investigate the distribution of zeros for continuous extensions of ...positive semi-waves. The distance between two consecutive zeros of two different positive semi-waves depends not only on the parameters of the problem but also on the position of one of these zeros with respect to the integer lattice. We provide an explicit formula for this distance, which allows us to obtain a new simple implicit description of all non-trivial Fučík curves for the discrete Dirichlet operator. Moreover, for fixed parameters of the problem, we show that this distance is bounded and attains its global extrema that are explicitly described in terms of Chebyshev polynomials of the second kind. Finally, for each non-trivial Fučík curve, we provide suitable bounds by two curves with a simple description similar to the description of the first non-trivial Fučík curve.
In this paper, we deal with the discrete Dirichlet operator of the second order and we investigate its Fučík spectrum, which consists of a finite number of algebraic curves. For each non-trivial ...Fučík curve, we are able to detect a finite number of its points, which are given explicitely. We provide the exact implicit description of all non-trivial Fučík curves in terms of Chebyshev polynomials of the second kind. Moreover, for each non-trivial Fučík curve, we give several different implicit descriptions, which differ in the level of depth of used nested functions. Our approach is based on the Möbius transformation and on the appropriate continuous extension of solutions of the discrete problem. Let us note that all presented descriptions of Fučík curves have the form of necessary and sufficient conditions. Finally, our approach can be also directly used in the case of difference operators of the second order with other local boundary conditions.
Indications of a discrepancy between simulations and data on the number of muons in cosmic ray (CR) showers exist over a large span of energies. We focus in particular on the excess of multi-muon ...bundles observed by the DELPHI detector at LEP and on the excess in the muon number in general reported by the Pierre Auger Observatory. Even though the primary CR energies relevant for these experiments differ by orders of magnitude, we can find a single mechanism which can simultaneously increase predicted muon counts for both, while not violating constraints from accelerators or from the longitudinal shower development as observed by the Pierre Auger Observatory. We present a brief motivation and describe a practical implementation of such a model, based on the addition of soft particles to interactions above a chosen energy threshold. Results of an extensive set of simulations show the behavior of this model in various parts of a simplified parameter space.
Inspired by recent astrophysical observations of leptonic excesses measured by satellite experiments, we consider the impact of some general models of the dark sector on the muon production in ...extensive air showers. We present a compact approximative expression for the bremsstrahlung of a massive photon from an electron and use it within Monte Carlo simulations to estimate the amount of weakly interacting photon-like massive particles that could be produced in an extensive air shower. We find that the resulting muon production is by many orders of magnitude below the average muon count in a shower and thus unobservable.
The Pierre Auger Observatory is a facility designed for the study of ultra-high energy cosmic rays. The Observatory combines two different types of detectors: a surface array of 1600 water Cherenkov ...stations placed on a 1.5 km triangular grid covering over 3000 km2; and a fluorescence detector of 24 telescopes located in 4 buildings at the perimeter of the surface array. The fluorescence telescopes, each consisting of 440 photomultipliers, collect the ultraviolet light produced when the charged secondary particles in an air shower excite nitrogen molecules in the atmosphere. Because the intensity of the nitrogen fluorescence is proportional to the energy deposited in the atmosphere during the air shower, the air fluorescence measurements can be used to make a calorimetric measurement of the cosmic ray primary energy. Showers observed independently by the surface array and fluorescence telescopes, called hybrid events, are critical to the function of the Observatory, as they allow for a model-independent calibration of the surface detector. In this paper I describe the detector and the most important measurements.
We study the structure of the Fucik spectra for the linear multi-point differential operators. We introduce a variational approach in order to obtain a robust and global algorithm which is suitable ...for the exploration of unknown Fucik spectrum structure. We apply our approach in the case of the four-point selfadjoint differential operator of the fourth order which is closely connected to the nonlinear model of a suspension bridge with two towers. Moreover, we reconstruct the Fucik spectra in the case of four-point non-selfadjoint ordinary differential operators of the second order in order to demonstrate their non-trivial and interesting structure.
Celotno besedilo
Dostopno za:
DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
The article deals with approximate solutions of a nonlinear ordinary differential equation with homogeneous Dirichlet boundary conditions. We provide a scheme of numerical-analytic method based upon ...successive approximations constructed in analytic form. We give sufficient conditions for the solvability of the problem and prove the uniform convergence of the approximations to the parameterized limit function. We provide a justification of the polynomial version of the method with several illustrating examples.
2000 Mathematics Subject Classification:
34B15; 65L10.
In this paper we present several nonlinear models of suspension bridges; most of them have been introduced by Lazer and McKenna. We discuss some results which were obtained by the authors and other ...mathematicians for the boundary value problems and initial boundary value problems. Our intention is to point out the character of these results and to show which mathematical methods were used to prove them instead of giving precise proofs and statements.PUBLICATION ABSTRACT
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