Mathematical models for option pricing often result in partial differential equations.Recent enhancements are models driven by Lévy processes, which lead to a partialdifferential equation with an additional integral term. In the context of modelcalibration, these partial integro differential equations need to be solved quitefrequently. To reduce the computational cost the implementation of a reduced order modelhas shown to be very successful numerically. In this paper we give a priorierror estimates for the use of the proper orthogonal decomposition technique inthe context of option pricing models.

Nowadays, the Coupled Cluster (CC) method is the probably most widely used high precisionmethod for the solution of the main equation of electronic structure calculation, thestationary electronic Schrödinger equation. Traditionally, theequations of CC are formulated as a nonlinear approximation of a Galerkin solution of theelectronic Schrödinger equation, i.e. within a given discrete subspace.Unfortunately, this concept prohibits the direct application of concepts of nonlinearnumerical analysis to obtain e.g. existence and uniqueness results orestimates on the convergence of discrete solutions to the full solution. Here, thisshortcoming is approached by showing that based on the choice of anN-dimensional reference subspace R of H1(ℝ3 ×{± 1/2}), the original, continuous electronic Schrödingerequation can be reformulated equivalently as a root equation for an infinite-dimensionalnonlinear Coupled Cluster operator. The canonical projected CC equations may then beunderstood as discretizations of this operator. As the main step, continuity properties ofthe cluster operator S and its adjoint S† asmappings on the antisymmetric energy space H1 are established.

We present a reduced basis offline/online procedure for viscous Burgers initial boundaryvalue problem, enabling efficient approximate computation of the solutions of thisequation for parametrized viscosity and initial and boundary value data. This procedurecomes with a fast-evaluated rigorous error bound certifying the approximation procedure.Our numerical experiments show significant computational savings, as well as efficiency ofthe error bound.

Resistance to chemotherapies, particularly to anticancer treatments, is an increasingmedical concern. Among the many mechanisms at work in cancers, one of the most importantis the selection of tumor cells expressing resistance genes or phenotypes. Motivated bythe theory of mutation-selection in adaptive evolution, we propose a model based on acontinuous variable that represents the expression level of a resistance gene (or genes,yielding a phenotype) influencing in healthy and tumor cells birth/death rates, effects ofchemotherapies (both cytotoxic and cytostatic) and mutations. We extend previous work bydemonstrating how qualitatively different actions of chemotherapeutic and cytostatictreatments may induce different levels of resistance. The mathematical interest of ourstudy is in the formalism of constrained Hamilton–Jacobi equations in the framework ofviscosity solutions. We derive the long-term temporal dynamics of the fittest traits inthe regime of small mutations. In the context of adaptive cancer management, we alsoanalyse whether an optimal drug level is better than the maximal tolerated dose.

The least Steklov eigenvalue d1 for the biharmonic operatorin bounded domains gives a bound for the positivity preserving property for the hingedplate problem, appears as a norm of a suitable trace operator, and gives the optimalconstant to estimate the L2-norm of harmonic functions. Theseapplications suggest to address the problem of minimizing d1in suitable classes of domains. We survey the existing results and conjectures about thistopic; in particular, the existence of a convex domain of fixed measure minimizingd1 is known, although the optimal shape is still unknown. Weperform several numerical experiments which strongly suggest that the optimal planar shapeis the regular pentagon. We prove the existence of a domain minimizingd1 also among convex domains having fixed perimeter andpresent some numerical results supporting the conjecture that, among planar domains, thedisk is the minimizer.

In this paper, we study a class of Initial-Boundary Value Problems proposed by Colin andGhidaglia for the Korteweg-de Vries equation posed on a bounded domain(0,L). We show that this class of Initial-Boundary Value Problems islocally well-posed in the classical Sobolev spaceHs(0,L) for s > -3/4, which provides a positive answer to one of the openquestions of Colin and Ghidaglia [Adv. Differ. Equ. 6 (2001)1463–1492].

Dynamics of the nonlinear Schrödinger equation in the presence of a constant electricfield is studied. Both discrete and continuous limits of the model are considered. For thediscrete limit, a probabilistic description of subdiffusion is suggested and asubdiffusive spreading of a wave packet is explained in the framework of a continuous timerandom walk. In the continuous limit, the biased nonlinear Schrödinger equation is shownto be integrable, and solutions in the form of the Painlevé transcendents are obtained.

An overview of recently obtained authors’ results on traveling wave solutions of someclasses of PDEs is presented. The main aim is to describe all possible travelling wavesolutions of the equations. The analysis was conducted using the methods of qualitativeand bifurcation analysis in order to study the phase-parameter space of the correspondingwave systems of ODEs. In the first part we analyze the wave dynamic modes of populationsdescribed by the “growth - taxis - diffusion" polynomial models. It is shown that“suitable" nonlinear taxis can affect the wave front sets and generate non-monotone waves,such as trains and pulses, which represent the exact solutions of the model system.Parametric critical points whose neighborhood displays the full spectrum of possible modelwave regimes are identified; the wave mode systematization is given in the form ofbifurcation diagrams. In the second part we study a modified version of theFitzHugh-Nagumo equations, which model the spatial propagation of neuron firing. We assumethat this propagation is (at least, partially) caused by the cross-diffusion connectionbetween the potential and recovery variables. We show that the cross-diffusion version ofthe model, besides giving rise to the typical fast travelling wave solution exhibited inthe original “diffusion" FitzHugh-Nagumo equations, additionally gives rise to a slowtraveling wave solution. We analyze all possible traveling wave solutions of the model andshow that there exists a threshold of the cross-diffusion coefficient (for a given speedof propagation), which bounds the area where “normal" impulse propagation is possible. Inthe third part we describe all possible wave solutions for a class of PDEs withcross-diffusion, which fall in a general class of the classical Keller-Segel modelsdescribing chemotaxis. Conditions for existence of front-impulse, impulse-front, andfront-front traveling wave solutions are formulated. In particular, we show that anon-isolated singular point in the ODE wave system implies existence of free-boundaryfronts.

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.

One of the central results in Einstein’s theory of Brownian motion is that the meansquare displacement of a randomly moving Brownian particle scales linearly with time. Overthe past few decades sophisticated experiments and data collection in numerous biological,physical and financial systems have revealed anomalous sub-diffusion in which the meansquare displacement grows slower than linearly with time. A major theoretical challengehas been to derive the appropriate evolution equation for the probability density functionof sub-diffusion taking into account further complications from force fields andreactions. Here we present a derivation of the generalised master equation for an ensembleof particles undergoing reactions whilst being subject to an external force field. Fromthis general equation we show reductions to a range of well known special cases, includingthe fractional reaction diffusion equation and the fractional Fokker-Planck equation.

New conservation laws bifurcating from the classical form of conservation laws are constructed to the nonlinear Boussinesq model describing internal Kelvin waves propagating in a cylindrical wave field of an uniformly stratified water affected by the earth’s rotation. The obtained conservation laws are different from the well known energy conservation law for internal waves and they are associated with symmetries of the Boussinesq model. Particularly, it is shown that application of Lie group analysis provide three infinite sets of nontrivial integral conservation laws depending on two arbitrary functions, namely a(t, θ), b(t, r) and an arbitrary function c(t, θ, r) which is given implicitly as a nontrivial solution of a partial differential equation involving a(t, θ) and b(t, r).

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

The stability of a one-spike solution to a general class of reaction-diffusion (RD)system with both regular and anomalous diffusion is analyzed. The method of matchedasymptotic expansions is used to construct a one-spike equilibrium solution and to derivea nonlocal eigenvalue problem (NLEP) that determines the stability of this solution on an O(1) time-scale. For a particular sub-class of the reactionkinetics, it is shown that the discrete spectrum of this NLEP is determined in terms ofthe roots of certain simple transcendental equations that involve two key parametersrelated to the choice of the nonlinear kinetics. From a rigorous analysis of thesetranscendental equations by using a winding number approach and explicit calculations,sufficient conditions are given to predict the occurrence of Hopf bifurcations of theone-spike solution. Our analysis determines explicitly the number of possible Hopfbifurcation points as well as providing analytical formulae for them. The analysis isimplemented for the shadow limit of the RD system defined on a finite domain and for aone-spike solution of the RD system on the infinite line. The theory is illustrated fortwo specific RD systems. Finally, in parameter ranges for which the Hopf bifurcation isunique, it is shown that the effect of sub-diffusion is to delay the onset of the Hopfbifurcation.

We study origin, parameter optimization, and thermodynamic efficiency of isothermalrocking ratchets based on fractional subdiffusion within a generalized non-MarkovianLangevin equation approach. A corresponding multi-dimensional Markovian embedding dynamicsis realized using a set of auxiliary Brownian particles elastically coupled to the centralBrownian particle (see video on the journal web site). We show that anomalous subdiffusivetransport emerges due to an interplay of nonlinear response and viscoelastic effects forfractional Brownian motion in periodic potentials with broken space-inversion symmetry anddriven by a time-periodic field. The anomalous transport becomes optimal for asubthreshold driving when the driving period matches a characteristic time scale ofinterwell transitions. It can also be optimized by varying temperature, amplitude ofperiodic potential and driving strength. The useful work done against a load shows aparabolic dependence on the load strength. It grows sublinearly with time and thecorresponding thermodynamic efficiency decays algebraically in time because the energysupplied by the driving field scales with time linearly. However, it compares well withthe efficiency of normal diffusion rocking ratchets on an appreciably long time scale.

The release of chemicals following herbivore grazing on primary producers may providefeeding cues to carnivorous predators, thereby promoting multitrophic interactions. Inparticular, chemicals released following grazing on phytoplankton by microzooplanktonherbivores have been shown to elicit a behavioural foraging response in carnivorouscopepods, which may use this chemical information as a mechanism to locate and remainwithin biologically productive patches of the ocean. In this paper, we use a 1D spatialreaction-diffusion model to simulate a tri-trophic planktonic system in the water column,where predation at the top trophic level (copepods) is affected by infochemicals releasedby the primary producers forming the bottom trophic level. The effect of theinfochemical-mediated predation is investigated by comparing the case where copepodsforage randomly to the case where copepods adjust their vertical position to follow thedistribution of grazing-induced chemicals. Results indicate that utilization ofinfochemicals for foraging provides fitness benefits to copepods and stabilizes the systemat high nutrient load, whilst also forming a possible mechanism for phytoplankton bloomformation. We also investigate how the copepod efficiency to respond to infochemicalsaffects the results, and show that small increases (2%) in the ability of copepods tosense infochemicals can promote their persistence in the system. Finally we argue thateffectively employing infochemicals for foraging can be an evolutionarily stable strategyfor copepods.

We present an overview of recent results for the classic problem of the survivalprobability of an immobile target in the presence of a single mobile trap or of acollection of uncorrelated mobile traps. The diffusion exponent of the traps is taken tobe either γ = 1, associated with normal diffusive motion, or0 < γ < 1, corresponding to subdiffusive motion. We considertraps that can only die upon interaction with the target and, alternatively, traps thatmay die due to an additional evanescence process even before hitting the target. Theevanescence reaction is found to completely modify the survival probability of the target.Such evanescence processes are important in systems where the addition of scavengermolecules may result in the removal of the majority species, or ones where the mobiletraps have a finite intrinsic lifetime.

Water flow in plant tissues takes place in two different physical domains separated bysemipermeable membranes: cell insides and cell walls. The assembly of all cell insides andcell walls are termed symplast and apoplast,respectively. Water transport is pressure driven in both, where osmosis plays an essentialrole in membrane crossing. In this paper, a microscopic model of water flow and transportof an osmotically active solute in a plant tissue is considered. The model is posed on thescale of a single cell and the tissue is assumed to be composed of periodicallydistributed cells. The flow in the symplast can be regarded as a viscous Stokes flow,while Darcy’s law applies in the porous apoplast. Transmission conditions at the interface(semipermeable membrane) are obtained by balancing the mass fluxes through the interfaceand by describing the protein mediated transport as a surface reaction. Applyinghomogenization techniques, macroscopic equations for water and solute transport in a planttissue are derived. The macroscopic problem is given by a Darcy law with a force termproportional to the difference in concentrations of the osmotically active solute in thesymplast and apoplast; i.e. the flow is also driven by the local concentration differenceand its direction can be different than the one prescribed by the pressure gradient.

We prove the existence of a global bifurcation branch of 2π-periodic,smooth, traveling-wave solutions of the Whitham equation. It is shown that any subset ofsolutions in the global branch contains a sequence which converges uniformly to somesolution of Hölder class Cα, α < 1/2. Bifurcation formulas are given, as well as some properties along theglobal bifurcation branch. In addition, a spectral scheme for computing approximations tothose waves is put forward, and several numerical results along the global bifurcationbranch are presented, including the presence of a turning point and a ‘highest’, cuspedwave. Both analytic and numerical results are compared to traveling-wave solutions of theKdV equation.