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Man'ko S.S.,NASU Pidstryhach Institute for Applied Problems in Mechanics and Mathematics
Journal of Mathematical Physics | Year: 2012

We study Schrödinger operators on star metric graphs with potentials of the form αε-2Q(ε-1x). In dimension 1 such potentials, with additional assumptions on Q, approximate in the sense of distributions as ε → 0 the first derivative of the Dirac delta-function. We establish the convergence of the Schrödinger operators in the uniform resolvent topology and show that the limit operator depends on α and Q in a very nontrivial way. © 2012 American Institute of Physics.


Novosyadlyj B.,Ivan Franko National University of Lviv | Sergijenko O.,Ivan Franko National University of Lviv | Durrer R.,University of Geneva | Pelykh V.,NASU Pidstryhach Institute for Applied Problems in Mechanics and Mathematics
Physical Review D - Particles, Fields, Gravitation and Cosmology | Year: 2012

The dynamics of expansion and large-scale structure formation of the Universe are analyzed for models with dark energy in the form of a phantom scalar field which initially mimics a Λ-term and evolves slowly to the Big Rip singularity. The discussed model of dark energy has three parameters-the density and the equation of state parameter at the current epoch, Ω de and w 0, and the asymptotic value of the equation of state parameter at a→, ca2. Their best-fit values are determined jointly with all other cosmological parameters by the Markov chain Monte Carlo method using observational data on cosmic microwave background anisotropies and polarization, supernovae type Ia luminosity distances, baryon acoustic oscillations measurements, and more. Similar computations are carried out for ΛCDM and a quintessence scalar field model of dark energy. It is shown that the current data slightly prefer the phantom model, but the differences in the maximum likelihoods are not statistically significant. It is also shown that the phantom dark energy with monotonically increasing density in the future will cause the decay of large-scale linear matter density perturbations due to the gravitational domination of dark energy perturbations long before the Big Rip singularity. © 2012 American Physical Society.


Malanchuk N.I.,NASU Pidstryhach Institute for Applied Problems in Mechanics and Mathematics
Materials Science | Year: 2011

We study the problem of contact interaction of two elastic isotropic bodies under the conditions of plane deformation with regard for slip caused by the local inhomogeneity of the friction coefficient under consecutive loading by normal and shear forces. By the method of complex potentials, this contact problem is reduced to a singular integral equation for the relative shift of the boundaries of the bodies in the region of slip whose solution is found in the analytic form. The influence of external loads on the relative shift of the boundaries of the bodies in this region, their length, and contact stresses is analyzed. © 2011 Springer Science+Business Media, Inc.


Malamud M.M.,NASU Pidstryhach Institute for Applied Problems in Mechanics and Mathematics
Russian Journal of Mathematical Physics | Year: 2010

Diverse closed (and selfadjoint) realizations of elliptic differential expressions, on smooth (bounded or unbounded) domains Ω in ℝn with compact boundary ∂Ω are considered. Trace-ideal properties of powers of resolvent differences for these closed realizations of A are proved by using the concept of boundary triples and operator-valued Weyl-Titchmarsh functions, and estimates for negative eigenvalues of certain selfadjoint extensions of the nonnegative minimal operator are derived. Our results extend classical theorems due to Vishik, Povzner, Birman, and Grubb. © 2010 Pleiades Publishing, Ltd.


Bulatsyk O.O.,NASU Pidstryhach Institute for Applied Problems in Mechanics and Mathematics
Proceedings of International Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory, DIPED | Year: 2013

The nonlinear integral equation of Hammerstein type arising in the antenna synthesis problem according to the given power pattern is investigated. This equation contains the nonlinearity of the third degree in the integrant. Its solutions are presented by a real positive function (amplitude pattern) and a polynomial with complex zeros (whose phase coincides with the phase pattern). The both amplitude pattern and zeros of the polynomials are found from a set of equations, one of which is the nonlinear integral equation, and the rest (as many as twice the polynomial degree) are the transcendental ones. Numerical results for a concrete problem are described. © 2013 Pidstryhach Institute for Applied Problems in Mechanics and Mathematics, NASU.

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