We derive the conserved vectors for the nonlinear two-dimensional Euler equations describing nonviscous incompressible fluid flows on a three-dimensional rotating spherical surface superimposed by a particular stationary latitude dependent flow. Under the assumption of no friction and a distribution of temperature dependent only upon latitude, the equations in question can be used to model zonal west-to-east flows in the upper atmosphere between the Ferrel and Polar cells. As a particualr example, the conserved densities are analyzed by visualizing the exact invariant solutions associated with the given model for the particular form of finite disturbances for which the invariant solutions are also exact solutions of Navier-Stokes equations.

As one of the major directions in applied and computational harmonic analysis, the classical theory of wavelets and framelets has been extensively investigated in the function setting, in particular, in the function space L2(ℝd). A discrete wavelet transform is often regarded as a byproduct in wavelet analysis by decomposing and reconstructing functions in L2(ℝd) via nested subspaces of L2(ℝd) in a multiresolution analysis. However, since the input/output data and all filters in a discrete wavelet transform are of discrete nature, to understand better the performance of wavelets and framelets in applications, it is more natural and fundamental to directly study a discrete framelet/wavelet transform and its key properties. The main topic of this paper is to study various properties of a discrete framelet transform purely in the discrete/digital setting without involving the function space L2(ℝd). We shall develop a comprehensive theory of discrete framelets and wavelets using an algorithmic approach by directly studying a discrete framelet transform. The connections between our algorithmic approach and the classical theory of wavelets and framelets in the function setting will be addressed. Using tensor product of univariate complex-valued tight framelets, we shall also present an example of directional tight framelets in this paper.

Let {pn}∞n=0 be a sequence of orthogonal polynomials. We briefly review properties of pn that have been used to derive upper and lower bounds for the largest and smallest zero of pn. Bounds for the extreme zeros of Laguerre, Jacobi and Gegenbauer polynomials that have been obtained using different approaches are numerically compared and new bounds for extreme zeros of q-Laguerre and little q-Jacobi polynomials are proved.

Directional multiscale representations such as shearlets and curvelets have gained increasing recognition in recent years as superior methods for the sparse representation of data. Thanks to their ability to sparsely encode images and other multidimensional data, transform-domain denoising algorithms based on these representations are among the best performing methods currently available. As already observed in the literature, the performance of many sparsity-based data processing methods can be further improved by using appropriate combinations of dictionaries. In this paper, we consider the problem of 3D data denoising and introduce a denoising algorithm which uses combined sparse dictionaries. Our numerical demonstrations show that the realization of the algorithm which combines 3D shearlets and local Fourier bases provides highly competitive results as compared to other 3D sparsity-based denosing algorithms based on both single and combined dictionaries.

We discuss the hull of a multi-dimensional limit-periodic potential and show that such a hull is an inverse limit of product cyclic groups. We present the result in an explicit way, which will be useful for a future study of multi-dimensional limit-periodic Schrödinger operators.

The shearlet representation has gained increasing recognition in recent years as a framework for the efficient representation of multidimensional data. This representation consists of a countable collection of functions defined at various locations, scales and orientations, where the orientations are obtained through the use of shear matrices. While shear matrices offer the advantage of preserving the integer lattice and being more appropriate than rotations for digital implementations, the drawback is that the action of the shear matrices is restricted to cone-shaped regions in the frequency domain. Hence, in the standard construction, a Parseval frame of shearlets is obtained by combining different systems of cone-based shearlets which are projected onto certain subspaces of L2(ℝD) with the consequence that the elements of the shearlet system corresponding to the boundary of the cone regions lose their good spatial localization property. In this paper, we present a new construction yielding smooth Parseval frame of shearlets for L2(ℝD). Specifically, all elements of the shearlet systems obtained from this construction are compactly supported and C∞ in the frequency domain, hence ensuring that the system has also excellent spatial localization.

The paper presents theoretical and numerical results on the identifiability, i.e. the unique identification for the one-dimensional sine-Gordon equation. The identifiability for nonlinear sine-Gordon equation remains an open question. In this paper we establish the identifiability for a linearized sine-Gordon problem. Our method consists of a careful analysis of the Laplace and Fourier transforms of the observation of the system, conducted at a single point. Numerical results based on the best fit to data method confirm that the identification is unique for a wide choice of initial approximations for the sought test parameters. Numerical results compare the identification for the nonlinear and the linearized problems.

Propagation of nonlinear baroclinic Kelvin waves in a rotating column of uniformly stratified fluid under the Boussinesq approximation is investigated. The model is constrained by the Kelvin’s conjecture saying that the velocity component normal to the interface between rotating fluid and surrounding medium (e.g. a seashore) is possibly zero everywhere in the domain of fluid motion, not only at the boundary. Three classes of distinctly different exact solutions for the nonlinear model are obtained. The obtained solutions are associated with symmetries of the Boussinesq model. It is shown that one class of the obtained solutions can be visualized as rotating whirlpools along which the pressure deviation from the mean state is zero, is positive inside and negative outside of the whirlpools. The angular velocity is zero at the center of the whirlpools and it is monotonically increasing function of radius of the whirlpools.

We construct the heat kernels of the sub-Laplacian and the Laplacian on the h-Heisenberg group and compute the limits as h → 0 of the heat kernels.

The paper is devoted to optimization of resonances in a 1-D open optical cavity. The cavity’s structure is represented by its dielectric permittivity function ε(s). It is assumed that ε(s) takes values in the range 1 ≤ ε1 ≤ ε(s) ≤ ε2. The problem is to design, for a given (real) frequency α, a cavity having a resonance with the minimal possible decay rate. Restricting ourselves to resonances of a given frequency α, we define cavities and resonant modes with locally extremal decay rate, and then study their properties. We show that such locally extremal cavities are 1-D photonic crystals consisting of alternating layers of two materials with extreme allowed dielectric permittivities ε1 and ε2. To find thicknesses of these layers, a nonlinear eigenvalue problem for locally extremal resonant modes is derived. It occurs that coordinates of interface planes between the layers can be expressed via arg-function of corresponding modes. As a result, the question of minimization of the decay rate is reduced to a four-dimensional problem of finding the zeroes of a function of two variables.

We review recent stability and separation results in volume comparison problems and use them to prove several hyperplane inequalities for intersection and projection bodies.

Assuming the negative part of the potential is uniformly locally L1, we prove a pointwise Lp estimate on derivatives of eigenfunctions of one-dimensional Schrödinger operators. In particular, if an eigenfunction is in Lp, then so is its derivative, for 1 ≤ p ≤ ∞.

We introduce new classes of modulation spaces over phase space. By means of the Kohn-Nirenberg correspondence, these spaces induce norms on pseudo-differential operators that bound their operator norms on Lp–spaces, Sobolev spaces, and modulation spaces.

The heat equation is considered in the complex system consisting of many small bodies (particles) embedded in a given material. On the surfaces of the small bodies a Newton-type boundary condition is imposed. An equation for the limiting field is derived when the characteristic size a of the small bodies tends to zero, their total number \hbox{$\mathcal{N}(a)$}𝒩(a) tends to infinity at a suitable rate, and the distance d = d(a) between neighboring small bodies tends to zero a <  < d. No periodicity is assumed about the distribution of the small bodies.

Inverse scattering problem is discussed for the Maxwell’s equations. A reduction of the Maxwell’s system to a new Fredholm second-kind integral equation with a scalar weakly singular kernel is given for electromagnetic (EM) wave scattering. This equation allows one to derive a formula for the scattering amplitude in which only a scalar function is present. If this function is small (an assumption that validates a Born-type approximation), then formulas for the solution to the inverse problem are obtained from the scattering data: the complex permittivity ϵ′(x) in a bounded region D ⊂ R3 is found from the scattering amplitude A(β,α,k) known for a fixed k = ω √ϵ0μ0 >0 and all β,α ∈ S2, where S2 is the unit sphere in R3, ϵ0 and μ0 are constant permittivity and magnetic permeability in the exterior region D′ = R3\D. The novel points in this paper include: i) A reduction of the inverse problem for vector EM waves to a vector integral equation with scalar kernel without any symmetry assumptions on the scatterer, ii) A derivation of the scalar integral equation of the first kind for solving the inverse scattering problem, and iii) Presenting formulas for solving this scalar integral equation. The problem of solving this integral equation is an ill-posed one. A method for a stable solution of this problem is given.

Let L be a linear, closed, densely defined in a Hilbert space operator, not necessarily selfadjoint. Consider the corresponding wave equations

\begin{eqnarray} &(1) \quad \ddot{w}+ Lw=0, \quad w(0)=0,\quad \dot{w}(0)=f, \quad \dot{w}=\frac{dw}{dt}, \quad f \in H. \\ &(2) \quad \ddot{u}+Lu=f e^{-ikt}, \quad u(0)=0, \quad \dot{u}(0)=0, \end{eqnarray}$\begin{array}{ccc}& \mathrm{\left(}\mathrm{1}\mathrm{\right)}\hspace{1em}\begin{array}{}\mathrm{¨}\\ \mathit{w}\end{array}\mathrm{+}\mathit{Lw}\mathrm{=}\mathrm{0}\mathit{,}\hspace{1em}\mathit{w}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathit{,}\hspace{1em}\mathit{ẇ}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathit{f,}\hspace{1em}\mathit{ẇ}\mathrm{=}\frac{\mathit{dw}}{\mathit{dt}}\mathit{,}\hspace{1em}\mathit{f}\mathrm{\in }\mathit{H}\mathit{.}& \\ & \mathrm{\left(}\mathrm{2}\mathrm{\right)}\hspace{1em}\begin{array}{}\mathrm{¨}\\ \mathit{u}\end{array}\mathrm{+}\mathit{Lu}\mathrm{=}\mathit{f}{\mathit{e}}^{\mathrm{-}\mathit{ikt}}\mathit{,}\hspace{1em}\mathit{u}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathit{,}\hspace{1em}\mathit{u̇}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathit{,}& \end{array}$

where k > 0 is a constant. Necessary and sufficient conditions are given for the operator L not to have eigenvalues in the half-plane Rez < 0 and not to have a positive eigenvalue at a given point kd2 > 0. These conditions are given in terms of the large-time behavior of the solutions to problem (1) for generic f.

Sufficient conditions are given for the validity of a version of the limiting amplitude principle for the operator L.

A relation between the limiting amplitude principle and the limiting absorption principle is established.

Stockwell transforms as hybrids of Gabor transforms and wavelet transforms have been studied extensively. We introduce in this paper multi-dimensional Stockwell transforms that include multi-dimensional Gabor transforms as special cases. Continuous inversion formulas for multi-dimensional Stockwell transforms are proved.

In this paper, an open problem in the multidimensional complex analysis is presented that arises in the harmonic analysis related to the investigation of the regularity properties of Fourier integral operators and in the regularity theory for hyperbolic partial differential equations. The problem is discussed in a self-contained elementary way and some results towards its resolution are presented. A conjecture concerning the structure of appearing affine fibrations is formulated.

We establish sharp semiclassical upper bounds for the moments of some negative powers for the eigenvalues of the Dirichlet Laplacian. When a constant magnetic field is incorporated in the problem, we obtain sharp lower bounds for the moments of positive powers not exceeding one for such eigenvalues. When considering a Schrödinger operator with the relativistic kinetic energy and a smooth, nonnegative, unbounded potential, we prove the sharp Lieb-Thirring estimate for the moments of some negative powers of its eigenvalues.