We derive the conserved vectors for the nonlinear two-dimensional Euler equationsdescribing nonviscous incompressible fluid flows on a three-dimensional rotating sphericalsurface superimposed by a particular stationary latitude dependent flow. Under theassumption 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 upperatmosphere between the Ferrel and Polar cells. As a particualr example, the conserveddensities are analyzed by visualizing the exact invariant solutions associated with thegiven model for the particular form of finite disturbances for which the invariantsolutions are also exact solutions of Navier-Stokes equations.

As one of the major directions in applied and computational harmonic analysis, theclassical theory of wavelets and framelets has been extensively investigated in thefunction setting, in particular, in the function spaceL2(ℝd). A discrete wavelettransform is often regarded as a byproduct in wavelet analysis by decomposing andreconstructing functions in L2(ℝd)via nested subspaces of L2(ℝd) ina multiresolution analysis. However, since the input/output data and all filters in adiscrete wavelet transform are of discrete nature, to understand better the performance ofwavelets and framelets in applications, it is more natural and fundamental to directlystudy a discrete framelet/wavelet transform and its key properties. The main topic of thispaper is to study various properties of a discrete framelet transform purely in thediscrete/digital setting without involving the function spaceL2(ℝd). We shall develop acomprehensive theory of discrete framelets and wavelets using an algorithmic approach bydirectly studying a discrete framelet transform. The connections between our algorithmicapproach and the classical theory of wavelets and framelets in the function setting willbe addressed. Using tensor product of univariate complex-valued tight framelets, we shallalso present an example of directional tight framelets in this paper.

Let {pn}∞n=0 be a sequence of orthogonal polynomials. We brieflyreview properties of pn that have been usedto derive upper and lower bounds for the largest and smallest zero ofpn. Bounds for theextreme zeros of Laguerre, Jacobi and Gegenbauer polynomials that have been obtained usingdifferent approaches are numerically compared and new bounds for extreme zeros ofq-Laguerre and little q-Jacobi polynomials are proved.

Directional multiscale representations such as shearlets and curvelets have gainedincreasing recognition in recent years as superior methods for the sparse representationof data. Thanks to their ability to sparsely encode images and other multidimensionaldata, transform-domain denoising algorithms based on these representations are among thebest performing methods currently available. As already observed in the literature, theperformance of many sparsity-based data processing methods can be further improved byusing appropriate combinations of dictionaries. In this paper, we consider the problem of3D data denoising and introduce a denoising algorithm which uses combined sparsedictionaries. Our numerical demonstrations show that the realization of the algorithmwhich combines 3D shearlets and local Fourier bases provides highly competitive results ascompared to other 3D sparsity-based denosing algorithms based on both single and combineddictionaries.

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

The shearlet representation has gained increasing recognition in recent years as aframework for the efficient representation of multidimensional data. This representationconsists of a countable collection of functions defined at various locations, scales andorientations, where the orientations are obtained through the use of shear matrices. Whileshear matrices offer the advantage of preserving the integer lattice and being moreappropriate than rotations for digital implementations, the drawback is that the action ofthe shear matrices is restricted to cone-shaped regions in the frequency domain. Hence, inthe standard construction, a Parseval frame of shearlets is obtained by combiningdifferent systems of cone-based shearlets which are projected onto certain subspaces ofL2(ℝD) with the consequence thatthe elements of the shearlet system corresponding to the boundary of the cone regions losetheir good spatial localization property. In this paper, we present a new constructionyielding smooth Parseval frame of shearlets forL2(ℝD). Specifically, allelements of the shearlet systems obtained from this construction are compactly supportedand C∞ in the frequency domain, hence ensuring that the systemhas also excellent spatial localization.

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

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

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

The paper is devoted to optimization of resonances in a 1-D open optical cavity. Thecavity’s structure is represented by its dielectric permittivity functionε(s). It is assumed thatε(s) takes values in the range1 ≤ ε1 ≤ ε(s) ≤ ε2.The problem is to design, for a given (real) frequency α, a cavity havinga resonance with the minimal possible decay rate. Restricting ourselves to resonances of agiven frequency α, we define cavities and resonant modes with locallyextremal decay rate, and then study their properties. We show that such locally extremalcavities are 1-D photonic crystals consisting of alternating layers of two materials withextreme allowed dielectric permittivities ε1 andε2. To find thicknesses of these layers, a nonlineareigenvalue problem for locally extremal resonant modes is derived. It occurs thatcoordinates of interface planes between the layers can be expressed via arg-function ofcorresponding modes. As a result, the question of minimization of the decay rate isreduced to a four-dimensional problem of finding the zeroes of a function of twovariables.

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

Assuming the negative part of the potential is uniformly locallyL1, we prove a pointwiseLp estimate on derivatives ofeigenfunctions of one-dimensional Schrödinger operators. In particular, if aneigenfunction is in Lp, then so is itsderivative, for 1 ≤ p ≤ ∞.

We introduce new classes of modulation spaces over phase space. By means of theKohn-Nirenberg correspondence, these spaces induce norms on pseudo-differential operatorsthat 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 aNewton-type boundary condition is imposed. An equation for the limiting field is derivedwhen the characteristic size a of the small bodies tends to zero, theirtotal number \hbox{$\mathcal{N}(a)$}𝒩(a) tends to infinity at a suitable rate, and the distanced = d(a) between neighboring smallbodies tends to zero a <  < d. No periodicity isassumed about the distribution of the small bodies.

Inverse scattering problem is discussed for the Maxwell’s equations. A reduction of theMaxwell’s system to a new Fredholm second-kind integral equation with a scalarweakly singular kernel is given for electromagnetic (EM) wave scattering. Thisequation allows one to derive a formula for the scattering amplitude in which only ascalar function is present. If this function is small (an assumption that validates aBorn-type approximation), then formulas for the solution to the inverse problem areobtained from the scattering data: the complex permittivityϵ′(x) in a bounded regionD ⊂ R3 is found from the scattering amplitudeA(β,α,k) known for a fixed k = ω √ϵ0μ0 >0 and allβ,α ∈ S2, whereS2 is the unit sphere in R3,ϵ0 and μ0 are constantpermittivity and magnetic permeability in the exterior regionD′ = R3\D. The novel pointsin this paper include: i) A reduction of the inverse problem for vectorEM waves to a vector integral equation with scalar kernelwithout any symmetry assumptions on the scatterer, ii) A derivation of thescalar integral equation of the first kind for solving the inversescattering problem, and iii) Presenting formulas for solving this scalar integralequation. The problem of solving this integral equation is an ill-posed one. A method fora 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, \quadw(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 aregiven for the operator L not to have eigenvalues in the half-planeRez < 0 and not to have a positive eigenvalue at a given point kd2 > 0. These conditions are given in terms of the large-timebehavior of the solutions to problem (1) for generic f.

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

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

Stockwell transforms as hybrids of Gabor transforms and wavelet transforms have beenstudied extensively. We introduce in this paper multi-dimensional Stockwell transformsthat include multi-dimensional Gabor transforms as special cases. Continuous inversionformulas for multi-dimensional Stockwell transforms are proved.

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

We establish sharp semiclassical upper bounds for the moments of some negative powers forthe eigenvalues of the Dirichlet Laplacian. When a constant magnetic field is incorporatedin the problem, we obtain sharp lower bounds for the moments of positive powers notexceeding one for such eigenvalues. When considering a Schrödinger operator with therelativistic kinetic energy and a smooth, nonnegative, unbounded potential, we prove thesharp Lieb-Thirring estimate for the moments of some negative powers of itseigenvalues.