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## Travelling-wave solution in the Rapoport–Leas model

### Analysis and Mathematical Physics (2017-02-28): 1-7 , February 28, 2017

Rapoport–Leas model of motion of a two-phase flow on a plane is considered. Travelling-wave solutions for these equations are found. Wavefronts of these solutions may be circles, lines and parabolae. Ordinary differential equations for pressure and saturation on the wavefronts are established.

## Inverse problems for differential operators of variable orders on star-type graphs: general case

### Analysis and Mathematical Physics (2014-09-01) 4: 247-262 , September 01, 2014

We study inverse spectral problems for ordinary differential equations on compact star-type graphs when differential equations have different orders on different edges. As the main spectral characteristics we introduce and study the so-called Weyl-type matrices which are generalizations of the Weyl function (m-function) for the classical Sturm–Liouville operator. We provide a procedure for constructing the solution of the inverse problem and prove its uniqueness.

## Zak transform for semidirect product of locally compact groups

### Analysis and Mathematical Physics (2013-09-01) 3: 263-276 , September 01, 2013

Let $$H$$ be a locally compact group and $$K$$ be an LCA group also let $$\tau :H\rightarrow Aut(K)$$ be a continuous homomorphism and $$G_\tau =H\ltimes _\tau K$$ be the semidirect product of $$H$$ and $$K$$ with respect to $$\tau $$ . In this article we define the Zak transform $$\mathcal{Z }_L$$ on $$L^2(G_\tau )$$ with respect to a $$\tau $$ -invariant uniform lattice $$L$$ of $$K$$ and we also show that the Zak transform satisfies the Plancherel formula. As an application we analyze how these technique apply for the semidirect product group $$\mathrm SL (2,\mathbb{Z })\ltimes _\tau \mathbb{R }^2$$ and also the Weyl-Heisenberg groups.

## Preface

### Analysis and Mathematical Physics (2012-12-01) 2: 319-324 , December 01, 2012

## Subharmonic functions that are harmonic when they are large

### Analysis and Mathematical Physics (2014-06-01) 4: 115-130 , June 01, 2014

Suppose that $$u$$ is subharmonic in the plane and such that, for some $$c>1$$ and sufficiently large $$K_0=K_0(c)$$ , $$u$$ is harmonic in the disc $$\Delta (z,\tau (z)^{-c})$$ whenever $$u(z)>B(|z|,u)-K_0\log \tau (z)$$ , where $$\tau (z)=\max \{|z|,B(|z|,u)\}$$ and $$B(r,u)=\max _{|z|=r}u(z)$$ . It is shown that if in addition $$u$$ satisfies a certain lower growth condition, then there are ‘Wiman–Valiron discs’ in each of which $$u$$ is the logarithm of the modulus of an analytic function, and that the derivatives of the analytic functions have regular asymptotic growth.

## Effects of zonal flows on correlation between energy balance and energy conservation associated with nonlinear nonviscous atmospheric dynamics in a thin rotating spherical shell

### Analysis and Mathematical Physics (2016-12-28): 1-14 , December 28, 2016

The nonlinear Euler equations are used to model two-dimensional atmosphere dynamics in a thin rotating spherical shell. The energy balance is deduced on the basis of two classes of functorially independent invariant solutions associated with the model. It it shown that the energy balance is exactly the conservation law for one class of the solutions whereas the second class of invariant solutions provides and asymptotic convergence of the energy balance to the conservation law.

## Inverse problem for dirac system with singularities in interior points

### Analysis and Mathematical Physics (2016-03-01) 6: 1-29 , March 01, 2016

We study the non-selfadjoint Dirac system on a finite interval having non-integrable regular singularities in interior points with additional matching conditions at these points. Properties of spectral characteristics are established, and the inverse spectral problem is investigated. We provide a constructive procedure for the solution of the inverse problem, and prove its uniqueness. Moreover, necessary and sufficient conditions for the global solvability of this nonlinear inverse problem are obtained.

## Second order ODEs under area-preserving maps

### Analysis and Mathematical Physics (2015-03-01) 5: 87-111 , March 01, 2015

We study the geometry of real analytic second order ODEs under the local real analytic diffeomorphism of $$\mathbb {R}^2$$ which are area preserving, through the method of Cartan. We obtain a subdivision into three “parts”. The first one is the most symmetric case. It is characterized by the vanishing of an area-preserving relative invariant namely $$f_y+\dfrac{2}{9}f_{y^{\prime }}^{2}-\dfrac{1}{3}\mathfrak {D}(f_{y^{\prime }})$$ . In this situation we associate a local affine normal Cartan connection on the first jet $$J^{1}(\mathbb {R},\mathbb {R})$$ space whose curvature contains all the area-preserving relative differential invariants, to any second order ODE under study. The second case which includes all the Painlevé transcendents is given by the ODEs for which $$f_y+\dfrac{2}{9}f_{y^{\prime }}^{2}-\dfrac{1}{3}\mathfrak {D}(f_{y^{\prime }})\not \equiv 0$$ . In the latter case we give all necessary steps in order to obtain an $$e$$ -structure on $$J^{1}(\mathbb {R},\mathbb {R})$$ for a generic second order ODE equation of that type. Finally we give the method to reduce to an $$e$$ -structure on $$J^{1}$$ when $$f_{y^\prime y^\prime y^\prime y^\prime }\not \equiv 0$$ .

## On a Schauder basis related to the eigenvectors of a family of non-selfadjoint analytic operators and applications

### Analysis and Mathematical Physics (2013-12-01) 3: 311-331 , December 01, 2013

In the present paper, we deal with the perturbed operator $$\begin{aligned} T(\varepsilon ):=T_0+\varepsilon T _1+\varepsilon ^2T_2+\cdots +\varepsilon ^kT_k+\cdots , \end{aligned}$$ where $$\varepsilon \in \mathbb C ,\,T_0$$ is a closed densely defined linear operator on a separable Banach space $$X$$ with domain $$\mathcal D (T_0),$$ while $$T_1, T_2, \ldots $$ are linear operators on $$X$$ with the same domain $$\mathcal D \supset \mathcal D (T_0)$$ and satisfying a specific growing inequality. The basic idea here is to investigate under sufficient conditions assuring the invariance of the closure of the perturbed operator $$T(\varepsilon )$$ which enables us to study the changed spectrum. Moreover, we prove that the system formed by some eigenvectors of $$T(\varepsilon )$$ which are analytic on $$\varepsilon ,$$ forms a Schauder basis in $$X.$$ After that, we apply the obtained results to a nonself-adjoint problem describing the radiation of a vibrating structure in a light fluid and to a nonself-adjoint Gribov operator in Bargmann space.

## Finite Blaschke products with prescribed critical points, Stieltjes polynomials, and moment problems

### Analysis and Mathematical Physics (2017-09-20): 1-29 , September 20, 2017

The determination of a finite Blaschke product from its critical points is a well-known problem with interrelations to several other topics. Though existence and uniqueness of solutions are established for long, we present new aspects which have not yet been explored to their full extent. In particular, we show that the following three problems are equivalent: (i) determining a finite Blaschke product from its critical points, (ii) finding the equilibrium position of moveable point charges interacting with a special configuration of fixed charges, and (iii) solving a moment problem for the canonical representation of power moments on the real axis. These equivalences are not only of theoretical interest, but also open up new perspectives for the design of algorithms. For instance, the second problem is closely linked to the determination of certain Stieltjes and Van Vleck polynomials for a second order ODE and characterizes solutions as global minimizers of an energy functional.