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.

The notion of inside dynamics of traveling waves has been introduced in the recent paper[14]. Assuming that a traveling waveu(t,x) = U(x − c   t)is made of several components υi ≥ 0(i ∈ I ⊂ N), the inside dynamics of the wave is thengiven by the spatio-temporal evolution of the densities of the componentsυi. For reaction-diffusion equations of theform∂tu(t,x) = ∂xxu(t,x) + f(u(t,x)),where f is of monostable or bistable type, the results in [14] show that traveling waves can be classified intotwo main classes: pulled waves and pushed waves. Using the same framework, we study thepulled/pushed nature of the traveling wave solutions of delay equations

            ∂tu(t,x) = ∂xxu(t,x) + F(u(t −τ,x),u(t,x))

We begin with areview of the latest results on the existence of traveling wave solutions of suchequations, for several classical reaction terms. Then, we give analytical and numericalresults which describe the inside dynamics of these waves. From a point of view ofpopulation ecology, our study shows that the existence of a non-reproductive andmotionless juvenile stage can slightly enhance the genetic diversity of a speciescolonizing an empty environment.

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.

During the last two decades, molecular genetic studies and the completion of thesequencing of the Arabidopsis thaliana genome have increased knowledge ofhormonal regulation in plants. These signal transduction pathways act in concert throughgene regulatory and signalling networks whose main components have begun to be elucidated.Our understanding of the resulting cellular processes is hindered by the complex, andsometimes counter-intuitive, dynamics of the networks, which may be interconnected throughfeedback controls and cross-regulation. Mathematical modelling provides a valuable tool toinvestigate such dynamics and to perform in silico experiments that may not be easilycarried out in a laboratory. In this article, we firstly review general methods formodelling gene and signalling networks and their application in plants. We then describespecific models of hormonal perception and cross-talk in plants. This mathematicalanalysis of sub-cellular molecular mechanisms paves the way for more comprehensivemodelling studies of hormonal transport and signalling in a multi-scale setting.

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.

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 bifurcation to one-dimensional weakly subcritical periodic patterns is described bythe cubic-quintic Ginzburg-Landau equation

       At = µA + Axx + i(a1|A|2Ax + a2A2Ax*) + b|A|2A - |A|4A.

These periodic patterns may in turn become unstable through one of two differentmechanisms, an Eckhaus instability or an oscillatory instability. We study the dynamicsnear the instability threshold in each of these cases using the corresponding modulationequations and compare the results with those obtained from direct numerical simulation ofthe equation. We also study the stability properties and dynamical evolution of differenttypes of fronts present in the protosnaking region of this equation. The results providenew predictions for the dynamical properties of generic systems in the weakly subcriticalregime.

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.

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.

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.

This paper is concerned with the existence and stability of travelling front solutionsfor more general autocatalytic chemical reaction systems ut = duxx − uf(v), vt = vxx + uf(v)with d > 0 and d ≠ 1, wheref(v) has super-linear or linear degeneracy atv = 0. By applying Lyapunov-Schmidt decomposition method in someappropriate exponentially weighted spaces, we obtain the existence and continuousdependence of wave fronts with some critical speeds and with exponential spatial decay ford near 1. By applying special phase plane analysis and approximatecenter manifold theorem, the existence of traveling waves with algebraic spatial decay orwith some lower exponential decay is also obtained for d > 0. Further,by spectral estimates and Evans function method, the wave fronts with exponential spatialdecay are proved to be spectrally or linearly stable in some suitable exponentiallyweighted spaces. Finally, by adopting the main idea of proof in [12] and some similar arguments as in [21], the waves with critical speeds or with non-critical speeds are proved to belocally exponentially stable in some exponentially weighted spaces and Lyapunov stable inCunif(ℝ) space, if the initial perturbation of the waves issmall in both the weighted and unweighted norms; the perturbation of the waves also stayssmall in L1(ℝ) norm and decays algebraically inCunif(ℝ) norm, if the initial perturbation is in additionsmall in L1 norm.

Plants, algae, and fungi are essential for nearly all life on earth. Throughphotosynthesis, plants and algae convert solar energy to chemical energy in the form oforganic compounds that sustains essentially all life on earth. In addition, plants andalgae convert the carbon dioxide produced by respiring organisms to oxygen that is neededfor respiration. Fungi decompose complex organic compounds produced by respiring organismsso that molecules can be recycled in photosynthesis and respiration. Plants, algae, andfungi have one important feature in common, their cells have walls. Expansive growth andits regulation are central to the life and development of plant, algal, and fungal cells,i.e. cells with walls. In recent decades there has been an explosion of informationrelevant to expansive growth of cells with walls. Mathematical models have beenconstructed in an attempt to organize and evaluate this information, to gain insight, toevaluate hypotheses, and to assist in the selection and development of new experimentalstudies. In this article some of the mathematical models constructed to study expansivegrowth of cells with walls are reviewed. It is nearly impossible to review all relevantresearch conducted in this area. Instead, the review focuses on the development ofmathematical equations that have been used to model expansive growth, morphogenesis, andgrowth rate regulation of cells with walls. Also, relevant experimental findings arereviewed, conceptual models are presented, and suggestions for future research areproposed. The authors have attempted to provide an overview that is accessible toresearchers that are not working in this field.

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.

The introduction of the ragweed leaf beetle in the South of Russia in 1978–1989 wasaccompanied by a number of spectacular phenomena that determined the general success ofthe ragweed control and further dispersal and acclimatization of the beetles:(i) formation of solitary population waves (SPW), characterized by anextremely high density of the phytophage population at the narrow band of the front of amoving wave defoliating nearly all ragweed plants, and (ii) rapid, within5-6 generations, development of flight in the leaf beetle species that in its homelandlost the ability to fly. We present here a demogenetic model capable of reproducing boththese phenomena, assuming that the flight ability of a phytophage population is governedby a single diallelic locus with flight and flightless alleles that determine threegenotypes of the ragweed leaf beetle. Simulation results agree well with the practicalrecommendation of retaining a high density of common ragweed in the release area in orderto provide the necessary conditions for the initial increase of the leaf beetle populationand the formation of the wave. The model confirms the earlier hypothesis that the SPW isthe key factor that determines efficiency of weed biocontrol program. We demonstrate alsothat the formation of the wave has crucially accelerated the development of the beetles’ability to fly.

We focus on a subdiffusion–reaction system in which substances are separated at theinitial moment. This system is described by nonlinear differential subdiffusion–reactionequations with a fractional time derivative. These equations are very difficult to solvebut there exist methods which allow us to solve them approximately. We discuss how usefulsuch methods are, in particular, the quasistatic approximation method and the perturbationmethod in analytical solving subdiffusion–reaction equations.

This work deals with the consequences on structural stability of Gause type predator-preymodels, when are considered three standard functional responses and the prey growth rateis subject to an Allee effect.

An important consequence of this ecological phenomenon is the existence of a separatrixcurve dividing the behavior of trajectories in the phase plane. The origin is an attractorfor any set of parameters and the existence of heteroclinic curves can be also shown.

Conditions on the parameter values are established to ensure the existence of a uniquepositive equilibrium, which can be either an attractor or a repellor surrounded by one ormore limit cycles.

The influence of the Allee effect on the number of limit cycles is analyzed and theresults are compared with analogous models without this phenomenon, and which mainfeatures have been given in various above works. Ecological interpretations of theseresults are also given.

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.

Pollen tubes are tip growing plant cells that display oscillatory growth behavior. It hasbeen demonstrated experimentally that the reduction of the average pollen tube growth ratethrough elevated extracellular calcium or borate concentrations coincides with a greateramplitude of the growth rate oscillation and a lower oscillation frequency. We present asimple numerical model of pollen tube growth that reproduces these results, as well asanalytical calculations that suggest an underlying mechanism. These data show that thepollen tube oscillator is non-isochronous, and is different from harmonic oscillation.

Habitat fragmentation is an important area of concern in species conservation. Habitatfragmentation can affect population distributions through reductions in suitable habitat,and through organism responses to different habitat types and the transitions betweenthem. In earlier work, the effect of habitat fragmentation on cyclic populations wasinvestigated in the context of populations that show no behavioural response to theinterface between habitat types. In this paper, we extend the earlier work by addingedge-mediated behaviour to the models. That is, we investigate the dynamics that resultwhen oscillatory predator and prey species also exhibit behavioural responses to habitatinterfaces. Our results show generally that habitat loss decreases the amplitude and theaverage density of the prey and predator populations, but that most of the reponsesobserved in the two models exhibit marked differences. This work highlights the complexityof the interplay between population cycles, habitat fragmentation, and edge-mediatedbehaviour, and the need to study such systems in greater detail.