We study in detail main properties of two families of the basic hypergeometric ${}_2\phi_1$-polynomials, which are natural $q$-extensions of the classical Chebyshev polynomials $T_n(x)$ and $U_n(x)$. In particular, we show that they are expressible as special cases of the big $q$-Jacobi polynomials $P_n(x;a,b,c;q)$ with some chosen parameters $a$, $b$ and $c$. We derive quadratic transformations that relate these polynomials to the little $q$-Jacobi polynomials $p_n(x;a,b\,|q)$. Explicit forms of discrete orthogonality relations on a finite interval, $q$-difference equations and Rodrigues-type difference formulas for these $q$-Chebyshev polynomials are also given.

We establish a long time decay for linearized dynamics of nonlinear Hamilton system which describes interaction of a charged particle with the Klein-Gordon field. The main contribution is deriving the decay for the frozen linearized system for initial data which are symplectic orthogonal to the root space of the linear differential operator involved.

The paper is devoted to the applications of the time-frequency integrals and the two-dimensional stationary phase method for the problems of waves propagation from moving sources in acoustic dispersive media. Applying the stationary phase method we obtain the effective formula for the acoustic fields in the dispersive media generated by non-uniformly moving modulated source.

We consider standard European as well as double-barrier European options for underlyings that are given by the superposition of a Guassian and a compound Poisson (jump) process with discrete values. We derive a formula for calculating such options and furthermore show that as the barriers tend to $\pm\infty$, the value of the double-barrier option tends asymptotically to that of the standard option. Numerical examples are provided.

We studied numerically the frequency spectrum of photons in a multilayered microsphere coated by a quasiperiodic (Pascal) dielectric stack. It is found that the transmittancy spectrum of such a stack consists of quasiband gaps and narrow resonances caused by re-reflection of optical waves. When the number (Pascal order) of layers increases, the band gaps and resonances split, and the structure of the frequency spectrum acquires a fractal form. Some parts of the spectrum show the self-similarity in the different frequency scales.

In the paper we deal with Toeplitz operators acting on the Bergman space $\mathcal{A}^2(\mathbb{B}^n)$ of square integrable analytic functions on the unit ball $\mathbb{B}^n$ in $\mathbb{C}^n$. A bounded linear operator acting on the space $\mathcal{A}^2(\mathbb{B}^n)$ is called radial if it commutes with unitary changes of variables. Zhou, Chen, and Dong [9] showed that every radial operator $S$ is diagonal with respect to the standard orthonormal monomial basis $(e_\alpha)_{\alpha\in\mathbb{N}^n}$. Extending their result we prove that the corresponding eigenvalues depend only on the length of multi-index $\alpha$, i.e. there exists a bounded sequence $(\lambda_k)_{k=0}^\infty$ of complex numbers such that $Se_\alpha=\lambda_{|\alpha|}e_\alpha$. Toeplitz operator is known to be radial if and only if its generating symbol $g$ is a radial function, i.e., there exists a function $a$, defined on $[0,1]$, such that $g(z)=a(|z|)$ for almost all $z\in\mathbb{B}^n$. In this case $T_g e_\alpha = \gamma_{n,a}(|\alpha|)e_\alpha$, where the eigenvalue sequence $\bigl(\gamma_{n,a}(k)\bigr)_{k=0}^\infty$ is given by \[ \gamma_{n,a}(k) =2(k+n)\int_0^1 a(r)\,r^{2k+2n-1}\,dr =(k+n)\int_0^1 a(\sqrt{r})\,r^{k+n-1}\,dr. \] Denote by $\Gamma_n$ the set $\{\gamma_{n,a}\colon a\in L^\infty([0,1])\}$. By a result of Suárez [8], the $C^\ast$-algebra generated by $\Gamma_1$ coincides with the closure of $\Gamma_1$ in $\ell^\infty$ and is equal to the closure of $d_1$ in $\ell^\infty$, where $d_1$ consists of all bounded sequences $x=(x_k)_{k=0}^\infty$ such that \[ \sup_{k\ge0}\,\Bigl((k+1)\,|x_{k+1}-x_k|\Bigr) \lt +\infty. \] We show that the $C^\ast$-algebra generated by $\Gamma_n$ does not actually depend on $n$, and coincides with the set of all bounded sequences $(x_k)_{k=0}^\infty$ that are slowly oscillating in the following sense: $|x_j-x_k|$ tends to $0$ uniformly as $\frac{j+1}{k+1}\to1$ or, in other words, the function $x\colon\{0,1,2,\ldots\}\to\mathbb{C}$ is uniformly continuous with respect to the distance $\rho(j,k)=|\ln(j+1)-\ln(k+1)|$. At the same time we give an example of a complex-valued function $a\in L^1([0,1],r\,dr)$ such that its eigenvalue sequence $\gamma_{n,a}$ is bounded but is not slowly oscillating in the indicated sense. This, in particular, implies that a bounded Toeplitz operator having unbounded defining symbol does not necessarily belong to the $C^*$-algebra generated by Toeplitz operators with bounded defining symbols.

We consider nonlinear relativistic wave equations in one space dimension and prove the spreading rate estimates for a general class of potentials. Such estimates play an important role in studying the asymptotic stability of solitons.

Regularization of the Navier-Stokes equations by adding hyperviscosity term $\mu(-\Delta^{2})$, $\mu \gt 0$ is considered. We proved convergence of Galerkin's approximations to the strong generalized solution of the regularized Navier-Stokes equations; existence and uniqueness of the strong generalized solution.

We introduce poly-Bergman type spaces on the Siegel domain $D_n\subset \mathbb{C}^n$, and prove that they are isomorphic to tensor products of one-dimensional spaces generated by orthogonal polynomials of two kinds: Laguerre and Hermite polynomials. The linear span of all poly-Bergman type spaces is dense in the Hilbert space $L^2(D_n,d\mu_{\lambda})$, where $d\mu_{\lambda}=(\mathrm{Im}\, z_n |z_1|^2-\cdots -|z_{n-1}|^2)^{\lambda}dx_1dy_1\cdots dx_n dy_n$ and $\lambda>-1$.

We prove the existence of transmission eigenvalues in the case when the perturbation of the index of refraction may have singularity or degeneration on the boundary of its support. This singularity or degeneration is measured in terms of the distance to the boundary.

Orthogonal projectors on the subspace $\mathbf{H}_{n}$ of homogeneous spherical polynomials of degree $n$ and on the subspace $\mathbb{P}^{N}$ of spherical polynomials of degree $% n\leq N$ are defined for functions on the unit sphere $S$, and their derivatives $\Lambda ^{s}$ of real degree $s$ are introduced by using the multiplier operators. A family of Hilbert spaces $\mathbb{H}^{s}$ of generalized functions having fractional derivatives of real degree $s$ on $S$ is introduced, and some embedding theorems for functions from $\mathbb{H}^{s} $ and Banach spaces $\mathbb{L}^{p}(S)$ and $\mathbb{L} ^{p}(0,T;\mathbb{X})$ on $S$ are given. Non-stationary and stationary problems for barotropic vorticity equation (BVE) describing the vortex dynamics of viscous incompressible fluid on a rotating sphere $S$ are considered. A theorem on the unique weak solvability of nonstationary problem and theorem on the existence of weak solution to stationary problem are given, and a condition guaranteeing the uniqueness of such steady solution is also formulated. The asymptotic behaviour of solutions of nonstationary BVE as $t\rightarrow \infty $ is studied. Particular forms of the external vorticity source have been found which guarantee the existence of such bounded set $\mathbf{B}$ in a phase space $\mathbf{X}$ that eventually attracts all solutions to the BVE. It is shown that the asymptotic behaviour of the BVE solutions depends on both the structure and the smoothness of external vorticity source. Sufficient conditions for the global asymptotic stability of both smooth and weak solutions are also given.

In the paper we consider the class of nonlinear $n$-dimensional control systems that can be mapped to linear ones by change of variables and an additive change of control ($A$-linearizable systems). We show that for sufficiently small initial points the transferring to the origin is possible by means of bang-bang controls with no more than $n-1$ points of switching. Moreover in some cases such a transferring is extremal in the sense of time optimality. These results are based on technique of the power Markov min-problem. An algorithm of searching the mentioned above bang-bang controls is also given.