# From Sine kernel to Poisson statistics

*Authors*

- Allez, Romain
- Dumaz, Laure

*2010 Mathematics Subject Classification*

- 15B52 60B20 37A50

*Keywords*

- Random matrices, Dyson Brownian motion, Langevin diffusion, non-confining potentials

*DOI*

*Appeared in*

- Electron. J. Probab., 19 (2014), pp. 114/1-114/25.

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# Spectrahedral cones generated by rank 1 matrices

*Authors*

- Hildebrand, Roland

*2010 Mathematics Subject Classification*

- 15A48 90C22

*Keywords*

- Semi-definite relaxation, exactness, rank 1 extreme ray, quadratically constrained quadratic optimization problem

*DOI*

*Appeared in*

- J. Global Optim., 64 (2016) pp. 349--397.

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# Tracy--Widom at high temperature

*Authors*

- Allez, Romain
- Dumaz, Laure

*2010 Mathematics Subject Classification*

- 15B52 60B20 37A50

*Keywords*

- Random matrices, Tracy-Widom distribution, Random Schrödinger operator, Langevin diffusion

*DOI*

*Appeared in*

- J. Statist. Phys., 156 (2014) pp. 1146 -- 1183.

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# Random matrices in non-confining potentials

*Authors*

- Allez, Romain
- Dumaz, Laure

*2010 Mathematics Subject Classification*

- 15B52 60B20 37A50

*Keywords*

- Random matrices, Dyson Brownian motion, Langevin diffusion, non-confining potentials

*DOI*

*Appeared in*

- J. Statist. Phys., 160 (2015) pp. 681--714.

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# Femtosecond filamentation by intensity clamping at a Freeman resonance

*Authors*

- Hofmann, Michael
- Brée, Carsten

*2010 Mathematics Subject Classification*

- 78A60

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Tg 42.65.Jx 42.68.Ay 52.38.Hb

*Keywords*

- femtosecond filamentation, nonlinear optics, ultrafast optics, transient optical response, Freeman resonances

*DOI*

*Appeared in*

- Phys. Rev. A, 92 (2015) pp. 013813/1--013813/7.

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# Nonlocal isoperimetric problems

*Authors*

- di Castro, Agnese
- Novaga, Matteo
- Ruffini, Berardo
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35R11 49Q05 53A10

*Keywords*

- fractional perimeter, isoperimetric problem, existence, rigidity and regularity results

*DOI*

*Abstract*

We characterize the volume-constrained minimizers of a nonlocal free energy given by the difference of fractional perimeters. Exploiting the quantitative fractional isoperimetric inequality, we show that balls are the unique minimizers if the volume is sufficiently small, while the existence vs. nonexistence of minimizers for large volumes remains open. We also consider the corresponding isoperimetric problem and prove existence and regularity of minimizers.

*Appeared in*

- Calc. Var. Partial Differ. Equ., 54 (2015) pp. 2421--2464.

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# A nonlocal free boundary problem

*Authors*

- Dipierro, Serena
- Savin, Ovidiu
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35R11 35R35 49Q20 49Q05

*Keywords*

- Fractional perimeter, minimization problem, monotonicity formula, classification of cones

*DOI*

*Abstract*

We consider a nonlocal free boundary problem built by a fractional Dirichlet norm plus a fractional perimeter. Among other results, we prove a monotonicity formula for the minimizers, glueing lemmata, uniform energy bounds, convergence results, a regularity theory for the planar cones and a trivialization result for the flat case. Several classical free boundary problems are limit cases of the one that we consider in this paper.

*Appeared in*

- SIAM J. Math. Anal., 47 (2015) pp. 4559--4605.

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# Modeling of compressible electrolytes with phase transition

*Authors*

- Dreyer, Wolfgang
- Giesselmann, Jan
- Kraus, Christiane

*2010 Mathematics Subject Classification*

- 78A57 80A17 92E20 35C20 35R35 76T10 76T30 35Q30 35Q35 76D45 76N10 76T99 76A02 80A22 82B26 34B15

*Keywords*

- Multi-component flow, phase transition, electrochemical reactions, partial balances, entropy principle, asymptotic analysis, sharp interface limit, free boundary problems, Poisson-Boltzmann, Allen-Cahn equation, Navier-Stokes system, Euler system

*DOI*

*Abstract*

A novel thermodynamically consistent diffuse interface model is derived for compressible electrolytes with phase transitions. The fluid mixtures may consist of N constituents with the phases liquid and vapor, where both phases may coexist. In addition, all constituents may consist of polarizable and magnetizable matter. Our introduced thermodynamically consistent diffuse interface model may be regarded as a generalized model of Allen-Cahn/Navier-Stokes/Poisson type for multi-component flows with phase transitions and electrochemical reactions. For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques. We consider two scaling regimes, i.e. a non-coupled and a coupled regime, where the coupling takes place between the smallness parameter in the Poisson equation and the width of the interface. We recover in the sharp interface limit a generalized Allen-Cahn/Euler/Poisson system for mixtures with electrochemical reactions in the bulk phases equipped with admissible interfacial conditions. The interfacial conditions satisfy, for instance, a generalized Gibbs-Thomson law and a dynamic Young-Laplace law.

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# A boundary control problem for the viscous Cahn--Hilliard equation with dynamic boundary conditions

*Authors*

- Colli, Pierluigi
- Gilardi, Gianni
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35K55 35K50 82C26

*Keywords*

- Cahn-Hilliard equation, dynamic boundary conditions, phase separation, singular potentials, optimal control, optimality conditions, adjoint state system

*DOI*

*Abstract*

A boundary control problem for the viscous Cahn-Hilliard equations with possibly singular potentials and dynamic boundary conditions is studied and first order necessary conditions for optimality are proved.

*Appeared in*

- Appl. Math. Optim., 73 (2016), pp. 195--225, DOI 10.1007/s00245-015-9299-z .

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# A class of probabilistic models for the Schrödinger equation

*Authors*

- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 35Q41 60J25 81Q05

*Keywords*

- Schrödinger equation, probabilistic representation, stochastic particle model, piecewise deterministic Markov process

*DOI*

*Abstract*

A class of stochastic particle models for the spatially discretized time-dependent Schrödinger equation is constructed. Each particle is characterized by a complex-valued weight and a position. The particle weights change according to some deterministic rules between the jumps. The jumps are determined by the creation of offspring. The main result is that certain functionals of the particle systems satisfy the Schrödinger equation. The proofs are based on the theory of piecewise deterministic Markov processes.

*Appeared in*

- Monte Carlo Methods Appl., 21 (2015) pp. 121--137.

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# A comparative study of a direct discretization and an operator-splitting solver for population balance systems

*Authors*

- Anker, Felix
- Ganesan, Sashikumaar
- John, Volker

ORCID: 0000-0002-2711-4409 - Schmeyer, Ellen

*2010 Mathematics Subject Classification*

- 76T20

*Keywords*

- population balance systems, direct discretization, operator-splitting, urea synthesis, uni-variate population

*DOI*

*Abstract*

A direct discretization approach and an operator-splitting scheme are applied for the numerical simulation of a population balance system which models the synthesis of urea with a uni-variate population. The problem is formulated in axisymmetric form and the setup is chosen such that a steady state is reached. Both solvers are assessed with respect to the accuracy of the results, where experimental data are used for comparison, and the efficiency of the simulations. Depending on the goal of simulations, to track the evolution of the process accurately or to reach the steady state fast, recommendations for the choice of the solver are given.

*Appeared in*

- Comput. Chem. Engng., 75 (2015) pp. 95--104.

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# Functional a posteriori error estimation for stationary reaction-convection-diffusion problems

*Authors*

- Eigel, Martin
- Samrowski, Tatiana

*2010 Mathematics Subject Classification*

- 65N30 65N15 65J15 65N22 65J10

*DOI*

*Abstract*

A functional type a posteriori error estimator for the finite element discretisation of the stationary reaction-convection-diffusion equation is derived. In case of dominant convection, the solution for this class of problems typically exhibits boundary layers and shock-front like areas with steep gradients. This renders the accurate numerical solution very demanding and appropriate techniques for the adaptive resolution of regions with large approximation errors are crucial. Functional error estimators as derived here contain no mesh-dependent constants and provide guaranteed error bounds for any conforming approximation. To evaluate the error estimator, a minimisation problem is solved which does not require any Galerkin orthogonality or any specific properties of the employed approximation space. Based on a set of numerical examples, we assess the performance of the new estimator and compare it with some classic a posteriori error estimators often used in practice. It is observed that the new estimator exhibits a good efficiency also with convection-dominated problem settings.

*Appeared in*

- Comput. Methods Appl. Math., 14 (2014) pp. 135--150.

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# On the spectrum of the lattice spin-boson Hamiltonian for any coupling: 1D case

*Authors*

- Muminov, Mukhiddin
- Neidhardt, Hagen
- Rasulov, Tulkin

*2010 Mathematics Subject Classification*

- 81Q10 35P20 47N50

*Keywords*

- spin-boson Hamiltonian, block operator matrix, bosonic Fock space, annihilation and creation operators, Birman-Schwinger principle, essential spectrum, point and discrete spectrum

*DOI*

*Abstract*

A lattice model of radiative decay (so-called spin-boson model) of a two level atom and at most two photons is considered. The location of the essential spectrum is described. For any coupling constant the finiteness of the number of eigenvalues below the bottom of its essential spectrum is proved. The results are obtained by considering a more general model H for which the lower bound of its essential spectrum is estimated. Conditions which guarantee the finiteness of the number of eigenvalues of H below the bottom of its essential spectrum are found. It is shown that the discrete spectrum might be infinite if the parameter functions are chosen in a special form.

*Appeared in*

- J. Math. Phys., 56 (2015) pp. 053507/1--053507/24.

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# Bootstrap confidence sets under a model misspecification

*Authors*

- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427 - Zhilova, Mayya

*2010 Mathematics Subject Classification*

- 62F25 62F40 62E17

*2008 Physics and Astronomy Classification Scheme*

- 02.50.Tt 02.50.Ng

*Keywords*

- likelihood-based confidence set, misspecified model, finite sample size, multiplier bootstrap

*DOI*

*Abstract*

A multiplier bootstrap procedure for construction of likelihood-based confidence sets is considered for finite samples and possible model misspecification. Theoretical results justify the bootstrap consistency for small or moderate sample size and allow to control the impact of the parameter dimension: the bootstrap approximation works if the ratio of cube of the parameter dimension to the sample size is small. The main result about bootstrap consistency continues to apply even if the underlying parametric model is misspecified under the so called Small Modeling Bias condition. In the case when the true model deviates significantly from the considered parametric family, the bootstrap procedure is still applicable but it becomes a bit conservative: the size of the constructed confidence sets is increased by the modeling bias. We illustrate the results with numerical examples of misspecified constant and logistic regressions.

*Appeared in*

- Ann. Statist., 43 (2015), pp. 2653--2675.

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# Supercontinuum generation by multiple scatterings at a group velocity horizon

*Authors*

- Demircan, Ayhan
- Morgner, Uwe
- Amiranashvili, Shalva

ORCID: 0000-0002-8132-882X - Brée, Carsten
- Steinmeyer, Günter

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Re 42.65.Ky 42.65.Tg 42.81.Dp

*Keywords*

- Supercontinuum generation, Ultrashort pulses, Optical event horizons, Nonlinear Schrödinger equation, Hamiltonian methods, Pseudo-spectral scheme

*DOI*

*Abstract*

A new scheme for supercontinuum generation covering more than one octave and exhibiting extraordinary high coherence properties has recently been proposed in Phys. Rev. Lett. bf 110, 233901 (2013). The scheme is based on two-pulse collision at a group velocity horizon between a dispersive wave and a soliton. Here we demonstrate that the same scheme can be exploited for the generation of supercontinua encompassing the entire transparency region of fused silica, ranging from 300 to 2300nm. At this bandwidth extension, the Raman effect becomes detrimental, yet may be compensated by using a cascaded collision process. Consequently, the high degree of coherence does not degrade even in this extreme scenario.

*Appeared in*

- Opt. Express, 22 (2014) pp. 3866--3879.

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# Models for the two-phase flow of concentrated suspensions

*Authors*

- Ahnert, Tobias
- Münch, Andreas
- Wagner, Barbara

*2010 Mathematics Subject Classification*

- 76T20 35B25 35Q35

*Keywords*

- Suspensions, jamming, yield stress, averaging, multiphase model, phase-space methods, matched asymptotics, drift-flux

*DOI*

*Abstract*

A new two-phase model for concentrated suspensions is derived that incorporates a constitutive law combining the rheology for non-Brownian suspension and granular flow. The resulting model naturally exhibits a Bingham-type flow property. This property is investigated in detail for the simple geometry of plane Poiseuille flow, where an unyielded or jammed zone of finite width arises in the center of the channel. For the steady state of this problem, the governing equation are reduced to a boundary value problem for a system of ordinary differential equations and the dependence of its solutions are analyzed by using phase-space methods. For the general time-dependent case a new drift-flux model is derived for the first time using matched asymptotic expansions that take account of the boundary layers at the walls and the interface between the yielded and unyielded region. Using the drift-flux model, the behavior of the suspension flow, in particular the appearance and evolution of unyielded or jammed regions is then studied numerically for different choices of the parameters.

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# Numerical simulations and measurements of a droplet size distribution in a turbulent vortex street

*Authors*

- Schmeyer, Ellen
- Bordás, Robert
- Thévenin, Dominique
- John, Volker

ORCID: 0000-0002-2711-4409

*2010 Mathematics Subject Classification*

- 76F65 76T10

*Keywords*

- two-phase turbulent vortex street, disperse droplet population, non-intrusive measurements, population balance systmes, direct discretizations

*DOI*

*Abstract*

A turbulent vortex street in an air flow interacting with a disperse droplet population is investigated in a wind tunnel. Non-intrusive measurement techniques are used to obtain data for the air velocity and the droplet velocity. The process is modeled with a population balance system consisting of the incompressible Navier--Stokes equations and a population balance equation for the droplet size distribution. Numerical simulations are performed that rely on a variational multiscale method for turbulent flows, a direct discretization of the differential operator of the population balance equation, and a modern technique for the evaluation of the coalescence integrals. After having calibrated two unknown model parameters, a very good agreement of the experimental and numerical results can be observed.

Eine turbulente Wirbelstraße in einer Luftströmung mit einer dispergierten Tröpfchenpopulation wird in einem Windkanal untersucht. Nichtintrusive Messtechniken werden verwendet, um Daten bezüglich der Luft-- und Tröpfchengeschwindigkeiten zu gewinnen. Der zu Grunde liegende Prozess wird mit einem Populationsbilanzsystem modelliert, welches aus den inkompressiblen Navier--Stokes--Gleichungen und einer Populationsbilanzgleichung für die Tröpfchenverteilungsdichte besteht. Numerische Simulationen werden durchgeführt, welche ein variationelle Mehrskalenmethode für turbulente Strömungen, eine direkte Diskretisierung des Differentialoperators der Populationsbilanzgleichung und ein modernes Verfahren zur Berechnung der Koaleszensintegrale verwenden. Nachdem zwei unbekannte Modellparameter kalibriert worden sind, kann eine sehr gute Übereinstimmung der experimentellen und numerischen Ergebnisse beobachtet werden.

*Appeared in*

- Meteorol. Z., 23 (2014) pp. 387--396.

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# Some analytical results for an algebraic flux correction scheme for a steady convection-diffusion equation in 1D

*Authors*

- Barrenechea, Gabriel R.
- John, Volker

ORCID: 0000-0002-2711-4409 - Knobloch, Petr

*2010 Mathematics Subject Classification*

- 65N06 65N30

*Keywords*

- finite element method, convection-diffusion equation, algebraic flux correction, discrete maximum principle, fixed point iteration, solvability of linear subproblems, solvability of nonlinear problem

*DOI*

*Abstract*

Algebraic flux correction schemes are nonlinear discretizations of convection dominated problems. In this work, a scheme from this class is studied for a steady-state convection--diffusion equation in one dimension. It is proved that this scheme satisfies the discrete maximum principle. Also, as it is a nonlinear scheme, the solvability of the linear subproblems arising in a Picard iteration is studied, where positive and negative results are proved. Furthermore, the non-existence of solutions for the nonlinear scheme is proved by means of counterexamples. Therefore, a modification of the method, which ensures the existence of a solution, is proposed. A weak version of the discrete maximum principle is proved for this modified method.

*Appeared in*

- IMA J. Numer. Anal., 35:4 (2015), pp. 1729--1756, changed title: Some analytical results for an algebraic flux correction scheme for a steady convection-diffusion equation in one dimension.

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# Higher-quality tetrahedral mesh generation for domains with small angles by constrained Delaunay refinement

*Authors*

- Shewchuk, Jonathan Richard
- Si, Hang

*2010 Mathematics Subject Classification*

- 65M50 65N50 65D18

*Keywords*

- constrained Delaunay triangulation, Delaunay refinement algorithm, tetrahedral mesh generation, computational geometry

*DOI*

*Abstract*

Algorithms for generating Delaunay tetrahedral meshes have difficulty with domains whose boundary polygons meet at small angles. The requirement that all tetrahedra be Delaunay often forces mesh generators to overrefine near small domain angles---that is, to produce too many tetrahedra, making them too small. We describe a provably good algorithm that generates meshes that are constrained Delaunay triangulations, rather than purely Delaunay. Given a piecewise linear domain free of small angles, our algorithm is guaranteed to construct a mesh in which every tetrahedron has a radius-edge ratio of $2 sqrt2 / 3 doteq 1.63$ or better. This is a substantial improvement over the usual bound of $2$; it is obtained by relaxing the conditions in which boundary triangles are subdivided. Given a domain with small angles, our algorithm produces a mesh in which the quality guarantee is compromised only in specific places near small domain angles. We prove that most mesh edges have lengths proportional to the domain's minimum local feature size; the exceptions span small domain angles. Our algorithm tends to generate meshes with fewer tetrahedra than purely Delaunay methods because it uses the constrained Delaunay property, rather than vertex insertions, to enforce the conformity of the mesh to the domain boundaries. An implementation demonstrates that our algorithm does not overrefine near small domain angles.

*Appeared in*

- Proceedings of the Thirtieth Annual Symposium on Computational Geometry, Association for Computing Machinery, New York, NY, USA, 2014, pp. 290--299

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# On regularity, positivity and long-time behavior of solutions to an evolution system of nonlocally interacting particles

*Authors*

- Griepentrog, Jens André

*2010 Mathematics Subject Classification*

- 35K51 35R09 47J35 35B65 35B09 35B40

*Keywords*

- Nonlocal Cahn-Hilliard equations, nonconvex functionals, Sobolev-Morrey spaces, regularity theory, Łojasiewicz-Simon gradient inequality, asymptotic behavior

*DOI*

*Abstract*

An analytical model for multicomponent systems of nonlocally interacting particles is presented. Its derivation is based on the principle of minimization of free energy under the constraint of conservation of particle number and justified by methods established in statistical mechanics. In contrast to the classical Cahn-Hilliard theory with higher order terms, the nonlocal theory leads to an evolution system of second order parabolic equations for the particle densities, weakly coupled by nonlinear and nonlocal drift terms, and state equations which involve both chemical and interaction potential differences. Applying fixed-point arguments and comparison principles we prove the existence of variational solutions in suitable Hilbert spaces for evolution systems. Moreover, using maximal regularity for nonsmooth parabolic boundary value problems in Sobolev-Morrey spaces and comparison principles, we show uniqueness, global regularity and uniform positivity of solutions under minimal assumptions on the regularity of interaction. Applying a refined version of the Łojasiewicz-Simon gradient inequality, this paves the way to the convergence of solutions to equilibrium states. We conclude our considerations with the presentation of simulation results for a phase separation process in ternary systems.

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# Adjustable pulse compression scheme for generation of few-cycle pulses in the mid-infrared

*Authors*

- Demircan, Ayhan
- Amiranashvili, Shalva

ORCID: 0000-0002-8132-882X - Brée, Carsten
- Morgner, Uwe
- Steinmeyer, Günter

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Re 42.65.Ky 42.65.Tg 42.81.Dp

*Keywords*

- Pulse compression, Ultrashort pulses, Optical event horizons, Nonlinear Schrödinger equation

*DOI*

*Abstract*

An novel adjustable adiabatic soliton compression scheme is presented, enabling a coherent pulse source with pedestal-free few-cycle pulses in the infrared or mid-infrared regime. This scheme relies on interaction of a dispersive wave and a soliton copropagating at nearly identical group velocities in a fiber with enhanced infrared transmission. The compression is achieved directly in one stage, without necessity of an external compensation scheme. Numerical simulations are employed to demonstrate this scheme for silica and fluoride fibers, indicating ultimate limitations as well as the possibility of compression down to the single-cycle regime. Such output pulses appear ideally suited as seed sources for parametric amplification schemes in the mid-infrared.

*Appeared in*

- Opt. Lett., 39 (2014) pp. 2735--2738.

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# Optimal control of multiphase steel production

*Authors*

- Hömberg, Dietmar
- Krumbiegel, Klaus
- Togobytska, Nataliya

*2010 Mathematics Subject Classification*

- 35K05 49M37 49N90

*Keywords*

- hot rolling, dual phase steels, optimal contro

*DOI*

*Abstract*

An optimal control problem for the production of multiphase steel is investigated, where the state equations are a semilinear heat equation and an ordinary differential equation, which describes the evolution of the ferrite phase fraction. The optimal control problem is analyzed and the first-order necessary and second-order sufficient optimality conditions are derived. For the numerical solution of the control problem reduced sequential quadratic programming (rSQP) method with a primal-dual active set strategy (PDAS) was applied. The numerical results were presented for the optimal control of a cooling line for production of hot rolled Mo-Mn dual phase steel.

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# Anisotropy in wavelet based phase field models

*Authors*

- Korzec, Maciek
- Münch, Andreas
- Süli, Endre
- Wagner, Barbara

*2010 Mathematics Subject Classification*

- 34E13 74N20 74E10

*Keywords*

- Phase-field model, wavelets, sharp interface model, free boundaries

*DOI*

*Abstract*

Anisotropy is an essential feature of phase-field models, in particular when describing the evolution of microstructures in solids. The symmetries of the crystalline phases are reflected in the interfacial energy by introducing corresponding directional dependencies in the gradient energy coefficients, which multiply the highest order derivative in the phase-field model. This paper instead considers an alternative approach, where the anisotropic gradient energy terms are replaced by a wavelet analogue that is intrinsically anisotropic and linear. In our studies we focus on the classical coupled temperature - Ginzburg-Landau type phase-field model for dendritic growth. For the resulting derivative-free wavelet analogue existence, uniqueness and continuous dependence on initial data for weak solutions is proved. The ability to capture dendritic growth similar to the results obtained from classical models is investigated numerically.

*Appeared in*

- Discrete Contin. Dyn. Syst. Ser. B, 21 (2016) pp. 1167--1187.

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# Finite element method to fluid-solid interaction problems with unbounded periodic interfaces

*Authors*

- Hu, Guanghui
- Rathsfeld, Andreas
- Yin, Tao

*2010 Mathematics Subject Classification*

- 78A45 35Q74 74F10 35B27

*Keywords*

- fluid-solid interaction, periodic structure, variational approach, Helmholtz equation, Lamé system, convergence analysis, Rayleigh expansion

*DOI*

*Abstract*

Consider a time-harmonic acoustic plane wave incident onto a doubly periodic (biperiodic) surface from above. The medium above the surface is supposed to be filled with a homogeneous compressible inviscid fluid of constant mass density, whereas the region below is occupied by an isotropic and linearly elastic solid body characterized by its Lamé constants. This paper is concerned with a variational approach to the fluid-solid interaction problems with unbounded biperiodic Lipschitz interfaces between the domains of the acoustic and elastic waves. The existence of quasi-periodic solutions in Sobolev spaces is established at arbitrary frequency of incidence, while uniqueness is proved only for small frequencies or for all frequencies excluding a discrete set. A finite element scheme coupled with Dirichlet-to-Neumann mappings is proposed. The Dirichlet-to-Neumann mappings are approximated by truncated Rayleigh series expansions, and, finally, numerical tests in 2D are performed.

*Appeared in*

- Numer. Methods Partial Differential Equations, 32 (2016) pp. 5--35.

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# Uniqueness in inverse elastic scattering from unbounded rigid surfaces of rectangular type

*Authors*

- Elschner, Johannes
- Hu, Guanghui
- Yamamoto, Masahiro

*2010 Mathematics Subject Classification*

- 74J20 74J25 35Q74 35R30

*Keywords*

- inverse scattering, uniqueness, Navier equation, linear elasticity, Dirichlet boundary condition, rough surface, diffraction grating

*DOI*

*Abstract*

Consider the two-dimensional inverse elastic scattering problem of recovering a piecewise linear rigid rough or periodic surface of rectangular type for which the neighboring line segments are always perpendicular. We prove the global uniqueness with at most two incident elastic plane waves by using near-field data. If the Lamé constants satisfy a certain condition, then the data of a single plane wave is sufficient to imply the uniqueness. Our proof is based on a transcendental equation for the Navier equation, which is derived from the expansion of analytic solutions to the Helmholtz equation. The uniqueness results apply also to an inverse scattering problem for non-convex bounded rigid bodies of rectangular type.

*Appeared in*

- Inverse Probl. Imaging, 9 (2015) pp. 127--141.

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# Corners and edges always scatter

*Authors*

- Elschner, Johannes
- Hu, Guanghui

*2010 Mathematics Subject Classification*

- 35R30 78A46

*Keywords*

- Helmholtz equation, inverse medium scattering, uniqueness, shape identification, corner and wedge domains

*DOI*

*Abstract*

Consider time-harmonic acoustic scattering problems governed by the Helmholtz equation in two and three dimensions. We prove that bounded penetrable obstacles with corners or edges scatter every incident wave nontrivially, provided the function of refractive index is real-analytic. Moreover, if such a penetrable obstacle is a convex polyhedron or polygon, then its shape can be uniquely determined by the far-field pattern over all observation directions incited by a single incident wave. Our arguments are elementary and rely on the expansion of solutions to the Helmholtz equation.

*Appeared in*

- Inverse Problems, 015003/1--015003/17 (2015) pp. .

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# From heavy-tailed Boolean models to scale-free Gilbert graphs

*Authors*

- Hirsch, Christian

*2010 Mathematics Subject Classification*

- 60D05 60K35

*Keywords*

- scale-free network, Boolean model, random geometric graph, first-passage percolation, chemical distances

*DOI*

*Abstract*

Define the scale-free Gilbert graph based on a Boolean model with heavy-tailed radius distribution on the d-dimensional torus by connecting two centers of balls by an edge if at least one of the balls contains the center of the other. We investigate two asymptotic properties of this graph as the size of the torus tends to infinity. First, we determine the tail index associated with the asymptotic distribution of the sum of all power-weighted incoming and outgoing edge lengths at a randomly chosen vertex. Second, we study the behavior of chemical distances on scale-free Gilbert graphs and show the existence of different regimes depending on the tail index of the radius distribution. Despite some similarities to long-range percolation and ultra-small scale-free geometric networks, scale-free Gilbert graphs are actually more closely related to fractal percolation and this connection gives rise to different scaling limits. We also propose a modification of the graph, where the total number of edges can be reduced substantially at the cost of introducing a logarithmic factor in the chemical distances.

*Appeared in*

- Braz. J. Probab. Stat., 31 (2017) pp. 111-143.

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# On thermodynamic consistency of a Scharfetter--Gummel scheme based on a modified thermal voltage for drift-diffusion equations with diffusion enhancement

*Authors*

- Koprucki, Thomas

ORCID: 0000-0001-6235-9412 - Rotundo, Nella
- Farrell, Patricio

ORCID: 0000-0001-9969-6615 - Doan, Duy Hai
- Fuhrmann, Jürgen

ORCID: 0000-0003-4432-2434

*2010 Mathematics Subject Classification*

- 65N08 35K55

*Keywords*

- Scharfetter--Gummel scheme, thermodynamic consistency, Drift-diffusion equations, non-Boltzmann statistic distributions, diffusion enhancement

*DOI*

*Abstract*

Driven by applications like organic semiconductors there is an increased interest in numerical simulations based on drift-diffusion models with arbitrary statistical distribution functions. This requires numerical schemes that preserve qualitative properties of the solutions, such as positivity of densities, dissipativity and consistency with thermodynamic equilibrium. An extension of the Scharfetter-Gummel scheme guaranteeing consistency with thermodynamic equilibrium is studied. It is derived by replacing the thermal voltage with an averaged diffusion enhancement for which we provide a new explicit formula. This approach avoids solving the costly local nonlinear equations defining the current for generalized Scharfetter-Gummel schemes.

*Appeared in*

- Opt. Quantum Electron., 47 (2015) pp. 1327--1332.

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# Local equilibration error estimators for guaranteed error control in adaptive stochastic higher-order Galerkin FEM

*Authors*

- Eigel, Martin
- Merdon, Christian

*2010 Mathematics Subject Classification*

- 35R60 47B80 60H35 65C20 65N12 65N22 65J10

*Keywords*

- partial differential equations with random coefficients, equilibrated estimator, guaranteed bounds, uncertainty quantification, stochastic finite element methods, operator equations, adaptive methods

*DOI*

*Abstract*

Equilibration error estimators have been shown to commonly lead to very accurate guaranteed error bounds in the a posteriori error control of finite element methods for second order elliptic equations. Here, we extend previous results by the design of equilibrated fluxes for higher-order finite element methods with nonconstant coefficients and illustrate the favourable performance of different variants of the error estimator within two deterministic benchmark settings. After the introduction of the respective parametric problem with stochastic coefficients and the stochastic Galerkin FEM discretisation, a novel a posteriori error estimator for the stochastic error in the energy norm is devised. The error estimation is based on the stochastic residual and its decomposition into approximation residuals and a truncation error of the stochastic discretisation. Importantly, by using the derived deterministic equilibration techniques for the approximation residuals, the computable error bound is guaranteed for the considered class of problems. An adaptive algorithm allows the simultaneous refinement of the deterministic mesh and the stochastic discretisation in anisotropic Legendre polynomial chaos. Several stochastic benchmark problems illustrate the efficiency of the adaptive process.

*Appeared in*

- SIAM ASA J. Uncertain. Quantif., 4 (2016) pp. 1372--1397.

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# Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum

*Authors*

- Dávila, Juan
- del Pino, Manuel
- Dipierro, Serena
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35B44 35J10 35R11

*Keywords*

- nonlocal quantum mechanics, Green functions, concentration phenomena

*DOI*

*Abstract*

For a smooth, bounded Euclidean domain, we consider a nonlocal Schrödinger equation with zero Dirichlet datum. We construct a family of solutions that concentrate at an interior point of the domain in the form of a scaling of the ground state in entire space. Unlike the classical case, the leading order of the associated reduced energy functional in a variational reduction procedure is of polynomial instead of exponential order on the distance from the boundary, due to the nonlocal effect. Delicate analysis is needed to overcome the lack of localization, in particular establishing the rather unexpected asymptotics for the Green function in the expanding domain.

*Appeared in*

- Anal. PDE, 8 (2015) pp. 1165--1235.

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# Simultaneous Bayesian analysis of contingency tables in genetic association studies

*Authors*

- Dickhaus, Thorsten

*2010 Mathematics Subject Classification*

- 62J15 62C10

*DOI*

*Abstract*

Genetic association studies lead to simultaneous categorical data analysis. The sample for every genetic locus consists of a contingency table containing the numbers of observed genotype-phenotype combinations. Under case-control design, the row counts of every table are identical and fixed, while column counts are random. The aim of the statistical analysis is to test independence of the phenotype and the genotype at every locus. We present an objective Bayesian methodology for these association tests, utilizing the Bayes factor proposed by Good (1976) and Crook and Good (1980). It relies on the conjugacy of Dirichlet and multinomial distributions, where the hyperprior for the Dirichlet parameter is log-Cauchy. Being based on the likelihood principle, the Bayesian tests avoid looping over all tables with given marginals. Hence, their computational burden does not increase with the sample size, in contrast to frequentist exact tests. Making use of data generated by The Wellcome Trust Case Control Consortium (2007), we illustrate that the ordering of the Bayes factors shows a good agreement with that of frequentist p-values. Furthermore, we deal with specifying prior probabilities for the validity of the null hypotheses, by taking linkage disequilibrium structure into account and exploiting the concept of effective numbers of tests. Application of a Bayesian decision theoretic multiple test procedure to The Wellcome Trust Case Control Consortium (2007) data illustrates the proposed methodology. Finally, we discuss two methods for reconciling frequentist and Bayesian approaches to the multiple association test problem for contingency tables in genetic association studies.

*Appeared in*

- Stat. Appl. Genet. Mol. Biol., 14:4 (2015), pp. 347--360.

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# On a fractional harmonic replacement

*Authors*

- Dipierro, Serena
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 31A05 35R11 46E35

*Keywords*

- Harmonic replacement, fractional Sobolev spaces, energy estimates

*DOI*

*Abstract*

Given $s ∈(0,1)$, we consider the problem of minimizing the Gagliardo seminorm in $H^s$ with prescribed condition outside the ball and under the further constraint of attaining zero value in a given set $K$. We investigate how the energy changes in dependence of such set. In particular, under mild regularity conditions, we show that adding a set $A$ to $K$ increases the energy of at most the measure of $A$ (this may be seen as a perturbation result for small sets $A$). Also, we point out a monotonicity feature of the energy with respect to the prescribed sets and the boundary conditions.

*Appeared in*

- Discrete Contin. Dyn. Syst., 35 (2015) pp. 3377--3392.

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# Statistical Skorohod embedding problem and its generalizations

*Authors*

- Belomestny, Denis
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 62F10 62J12

*Keywords*

- Skorohod embedding problem, Levy process, Mellin transform, Laplace transform, variance mixture models, time-changed Levy processes

*DOI*

*Abstract*

Given a Levy process L, we consider the so-called statistical Skorohod embedding problem of recovering the distribution of an independent random time T based on i.i.d. sample from L(T). Our approach is based on the genuine use of the Mellin and Laplace transforms. We propose consistent estimators for the density of T, derive their convergence rates and prove their optimality. It turns out that the convergence rates heavily depend on the decay of the Mellin transform of T. We also consider the application of our results to the problem of statistical inference for variance-mean mixture models and for time-changed Levy processes.

*Appeared in*

- Stochastic Process. Appl., Vol. 126, 7, (2016) pp. 2092--2122 under the new title: Statistical inference for time-changed Lévy processes via Mellin transform approach.

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# Adaptive time step control for higher order variational time discretizations applied to convection-diffusion equations

*Authors*

- Ahmed, Naveed

ORCID: 0000-0002-9322-0373 - John, Volker

ORCID: 0000-0002-2711-4409

*2010 Mathematics Subject Classification*

- 65M60

*Keywords*

- Continuous Galerkin--Petrov method, discontinuous Galerkin method, post-processed solution, adaptive step length control, SUPG stabilization

*DOI*

*Abstract*

Higher order variational time stepping schemes allow an efficient post-processing for computing a higher order solution. This paper presents an adaptive algorithm whose time step control utilizes the post-processed solution. The algorithm is applied to convection-dominated convection-diffusion equations. It is shown that the length of the time step properly reflects the dynamics of the solution. The numerical costs of the adaptive algorithm are discussed.

*Appeared in*

- Comput. Methods Appl. Mech. Engrg., 285 (2014) pp. 83--101.

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# Existence of bounded discrete steady state solutions of the van Roosbroeck system with monotone Fermi--Dirac statistic functions

*Authors*

- Gärtner, Klaus

*2010 Mathematics Subject Classification*

- 65N08 65N12 35J55

*Keywords*

- generalized Scharfetter--Gummel scheme, Fermi--Dirac statistics, generalized Einstein relation, dissipativity, bounded discrete steady state solutions, unique thermodynamic equilibrium, degenerate semiconductors

*DOI*

*Abstract*

If the statistic function is modified, the equations can be derived by a variational formulation or just using a generalized Einstein relation. In both cases a dissipative generalization of the Scharfetter-Gummel scheme citeSch_Gu, understood as a one-dimensional constant current approximation, is derived for strictly monotone coefficient functions in the elliptic operator $nabla cdot bal ff(v) nabla $, $v$ chemical potential, while the hole density is defined by $p=cal F(v)le e^v.$ A closed form integration of the governing equation would simplify the practical use, but mean value theorem based results are sufficient to prove existence of bounded discrete steady state solutions on any boundary conforming Delaunay grid. These results hold for any piecewise, continuous, and monotone approximation of $bal ff(v)$ and $cal F(v)$.

*Appeared in*

- J. Comput. Electron., (2015) pp. DOI 10.1007/s10825-015-0712-2 .

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# Comparison and numerical treatment of generalized Nernst--Planck models

*Authors*

- Fuhrmann, Jürgen

ORCID: 0000-0003-4432-2434

*2010 Mathematics Subject Classification*

- 65N08 78A57

*Keywords*

- Finite Volumes, Electrolytes, Nernst-Planck-Equations

*DOI*

*Abstract*

In its most widespread, classical formulation, the Nernst-Planck-Poisson system for ion transport in electrolytes fails to take into account finite ion sizes. As a consequence, it predicts unphysically high ion concentrations near electrode surfaces. Historical and recent approaches to an approriate modification of the model are able to fix this problem. Several appropriate formulations are compared in this paper. The resulting equations are reformulated using absolute activities as basic variables describing the species amounts. This reformulation allows to introduce a straightforward generalisation of the Scharfetter-Gummel finite volume discretization scheme for drift-diffusion equations. It is shown that it is thermodynamically consistent in the sense that the solution of the corresponding discretized generalized Poisson-Boltzmann system describing the thermodynamic equilibrium is a stationary state of the discretized time-dependent generalized Nernst-Planck system. Numerical examples demonstrate the improved physical correctness of the generalised models and the feasibility of the numerical approach.

*Appeared in*

- Comput. Phys. Comm., 196 (2015) pp. 166--178.

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# A Stokes-consistent backflow stabilization for physiological flows

*Authors*

- Bertoglio, Cristobal
- Caiazzo, Alfonso

ORCID: 0000-0002-7125-8645

*2010 Mathematics Subject Classification*

- 62P10 76D05 76M10, 76Z05

*2008 Physics and Astronomy Classification Scheme*

- 47.63.-b, 47.11.-j

*Keywords*

- Navier-Stokes equations, backflow stabilization, blood flows, respiratory flows, finite element method

*DOI*

*Abstract*

In computational fluid dynamics incoming flow at open boundaries, or emphbackflow, often yields to unphysical instabilities for high Reynolds numbers. It is widely accepted that this is due to the incoming energy arising from the convection term, which cannot be empha priori controlled when the velocity field is unknown at the boundary. In order to improve the robustness of the numerical simulations, we propose a stabilized formulation based on a penalization of the residual of a weak Stokes problem on the open boundary, whose viscous part controls the incoming convective energy, while the inertial term contributes to the kinetic energy. We also present different strategies for the approximation of the boundary pressure gradient, which is needed for defining the stabilization term. The method has the advantage that it does not require neither artificial modifications or extensions of the computational domain. Moreover, it is consistent with the Womersley solution. We illustrate our approach on numerical examples - both academic and real-life - relevant to blood and respiratory flows. The results also show that the stabilization parameter can be reduced with the mesh size.

*Appeared in*

- J. Comput. Phys., 313 (2016) pp. 260--278.

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# Influence of cell shape, inhomogeneities and diffusion barriers in cell polarization models

*Authors*

- Giese, Wolfgang
- Eigel, Martin
- Westerheide, Sebastian
- Engwer, Christian
- Klipp, Edda

*2010 Mathematics Subject Classification*

- 35Q92 92C37 65M60 74S05 92-08 37N25 46N60 62P10

*Keywords*

- polarization models, surface FEM, bulk-surface PDE, computer simulation, spatial simulation, spatial inhomogenities, Cdc42

*DOI*

*Abstract*

In silico experiments bear the potential to further the understanding of biological transport processes by allowing a systematic modification of any spatial property and providing immediate simulation results for the chosen models. We consider cell polarization and spatial reorganization of membrane proteins which are fundamental for cell division, chemotaxis and morphogenesis. Our computational study is motivated by mating and budding processes of S. cerevisiae. In these processes a key player during the initial phase of polarization is the GTPase Cdc42 which occurs in an active membrane-bound form and an inactive cytosolic form. We use partial differential equations to describe the membrane-cytosol shuttling of Cdc42 during budding as well as mating of yeast. The membrane is modeled as a thin layer that only allows lateral diffusion and the cytosol is modeled as a volume. We investigate how cell shape and diffusion barriers like septin structures or bud scars influence Cdc42 cluster formation and subsequent polarization of the yeast cell. Since the details of the binding kinetics of cytosolic proteins to the membrane are still controversial, we employ two conceptual models which assume different binding kinetics. An extensive set of in silico experiments with different modeling hypotheses illustrate the qualitative dependence of cell polarization on local membrane curvature, cell size and inhomogeneities on the membrane and in the cytosol. We examine that spatial inhomogenities essentially determine the location of Cdc42 cluster formation and spatial properties are crucial for the realistic description of the polarization process in cells. In particular, our computer simulations suggest that diffusion barriers are essential for the yeast cell to grow a protrusion.

*Appeared in*

- Phys. Biol., 12 (2015) pp. 066014/1--18.

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# Coherent passive mode-locking in lasers: Qualitative analysis and numerical simulations

*Authors*

- Arkhipov, Rostislav M.
- Babushkin, Ihar
- Arkhipov, Mikhail V.

*2008 Physics and Astronomy Classification Scheme*

- 42.60.Fc 42.65.Re 42.50.Md 42.60.Da

*Keywords*

- mode-locking, ultrashort laser pulse, coherent pulse propagation, area theorem, self-induced transparency

*DOI*

*Abstract*

In the present work we report the possibility of passive mode-locking based on the coherent interaction of light with the amplifying and absorbing media in lasers with ring and linear cavities. We consider the realistic and practically interesting case when the absorbing and amplifying media are separated in the cavity space but not homogeneously mixed in the volume of the cavity, as was considered earlier in the literature. We perform qualitative consideration of coherent passive mode-locking based on the area theorem of McCall and Hahn and its graphical representation for the first time. We show that other, not soliton scenarios of passive mode-locking exist, and that coherent mode-locking is self-starting (lasing without an injection seeding pulse is possible). We point to the fact that the spectral width of the laser generation can be significantly larger than the spectral bandwidth of the gain medium. Numerical simulations were performed using the system of Maxwell-Bloch equations in the slowly varying envelope approximation.

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# Self-starting stable coherent mode-locking in a two-section laser

*Authors*

- Arkhipov, Rostislav M.
- Arkhipov, Mikhail V.
- Babushkin, Ihar

*2008 Physics and Astronomy Classification Scheme*

- 42.60.Fc 42.65.Re 42.50.Md 42.60.Da

*Keywords*

- mode-locking, ultrashort laser pulse, coherent pulse propagation, area theorem, self-induced transparency

*DOI*

*Abstract*

In the present work we report the possibility of passive mode-locking based on the coherent interaction of light with the amplifying and absorbing media in lasers with ring and linear cavities. We consider the realistic and practically interesting case when the absorbing and amplifying media are separated in the cavity space but not homogeneously mixed in the volume of the cavity, as was considered earlier in the literature. We perform qualitative consideration of coherent passive mode-locking based on the area theorem of McCall and Hahn and its graphical representation for the first time. We show that other, not soliton scenarios of passive mode-locking exist, and that coherent mode-locking is self-starting (lasing without an injection seeding pulse is possible). We point to the fact that the spectral width of the laser generation can be significantly larger than the spectral bandwidth of the gain medium. Numerical simulations were performed using the system of Maxwell-Bloch equations in the slowly varying envelope approximation.

*Appeared in*

- Opt. Commun., 361 (2016), pp. 73--78.

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# On evolutionary Gamma convergence for gradient systems

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 35B27 35K55 47H20 49S05 49J40 49J45

*Keywords*

- Variational evolution, energy functional, dissipation potential, dissipation distance, gradient flows, Gamma convergence, Mosco convergence, well-prepared initial conditions, rate-independent systems, abstract chain-rule, energy-dissipation balance, integrated evolutionary variational estimate, energetic solutions

*DOI*

*Abstract*

In these notes we discuss general approaches for rigorously deriving limits of generalized gradient flows. Our point of view is that a generalized gradient system is defined in terms of two functionals, namely the energy functional *E*_{ε} and the dissipation potential *R*_{ε} or the associated dissipation distance. We assume that the functionals depend on a small parameter and the associated gradients systems have solutions u_{ε}. We investigate the question under which conditions the limits u of (subsequences of) the solutions u_{ε} are solutions of the gradient system generated by the Γ-limits *E*_{0} and *R*_{0}. Here the choice of the right topology will be crucial as well as additional structural conditions.

We cover classical gradient systems, where *R*_{ε} is quadratic, and rate-independent systems as well as the passage from viscous to rate-independent systems. Various examples, such as periodic homogenization, are used to illustrate the abstract concepts and results.

*Appeared in*

- A. Mielke, Chapter 3: On Evolutionary $Gamma$-Convergence for Gradient Systems, in: Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, A. Muntean, J.D.M. Rademacher, A. Zagaris, eds., vol. 3 of Lecture Notes in Applied Mathematics and Mechanics, Springer International Publishing, Heidelberg et al., 2016, pp. 187--249

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# ``Entropic'' solutions to a thermodynamically consistent PDE system for phase transitions and damage

*Authors*

- Rocca, Elisabetta
- Rossi, Riccarda

*2010 Mathematics Subject Classification*

- 35D30 74G25 93C55 82B26 74A45

*Keywords*

- damage, phase transitions, thermoviscoelasticity, global-in-time weak solutions, time discretization

*DOI*

*Abstract*

In this paper we analyze a PDE system modelling (non-isothermal) phase transitions and dam- age phenomena in thermoviscoelastic materials. The model is thermodynamically consistent: in particular, no small perturbation assumption is adopted, which results in the presence of quadratic terms on the right-hand side of the temperature equation, only estimated in L^1. The whole system has a highly nonlinear character. We address the existence of a weak notion of solution, referred to as ``entropic'', where the temperature equation is formulated with the aid of an entropy inequality, and of a total energy inequality. This solvability concept reflects the basic principles of thermomechanics as well as the thermodynamical consistency of the model. It allows us to obtain global-in-time existence theorems without imposing any restriction on the size of the initial data. We prove our results by passing to the limit in a time discretization scheme, carefully tailored to the nonlinear features of the PDE system (with its ``entropic'' formulation), and of the a priori estimates performed on it. Our time-discrete analysis could be useful towards the numerical study of this model.

*Appeared in*

- SIAM J. Math. Anal., 74 (2015) pp. 2519--2586.

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# Near-field imaging of scattering obstacles with the factorization method

*Authors*

- Hu, Guanghui
- Yang, Jiaqing
- Zhang, Bo
- Zhang, Haiwen

*2010 Mathematics Subject Classification*

- 35R30 78A45 78A46

*Keywords*

- factorization method, inverse scattering, Helmholtz equation, point sources, near-field data

*DOI*

*Abstract*

In this paper we establish a factorization method for recovering the location and shape of an acoustic bounded obstacle with using the near-field data, corresponding to infinitely many incident point sources. The obstacle is allowed to be an impenetrable scatterer of sound-soft, sound-hard or impedance type or a penetrable scatterer. An outgoing-to-incoming operator is constructed for facilitating the factorization of the near-field operator, which can be easily implemented numerically. Numerical examples are presented to demonstrate the feasibility and effectiveness of our inversion algorithm, including the case where limited aperture near-field data are available only.

*Appeared in*

- Inverse Problems, 30 (2014) pp. 095005/1--095005/25.

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# Second-order analysis of a boundary control problem for the viscous Cahn--Hilliard equation with dynamic boundary condition

*Authors*

- Colli, Pierluigi
- Farshbaf Shaker, Mohammad Hassan
- Gilardi, Gianni
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35K55 35K50 82C26

*Keywords*

- Cahn--Hilliard equation, dynamic boundary conditions, phase separation, singular potentials, optimal control, first and second order optimality conditions, adjoint state system

*DOI*

*Abstract*

In this paper we establish second-order sufficient optimality conditions for a boundary control problem that has been introduced and studied by three of the authors in the preprint arXiv:1407.3916. This control problem regards the viscous Cahn--Hilliard equation with possibly singular potentials and dynamic boundary conditions.

*Appeared in*

- Ann. Acad. Rom. Sci. Math. Appl., 7 (2015) pp. 41--66 .

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# Hölder estimates for second-order operators with mixed boundary conditions

*Authors*

- ter Elst, A. F. M.
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 35J25 35B65 35D30 35A23

*Keywords*

- Mixed boundary problem, Hölder continuity, kernel estimates

*DOI*

*Abstract*

In this paper we investigate linear elliptic, second-order boundary value problems with mixed boundary conditions. Assuming only boundedness/ellipticity on the coefficient function and very mild conditions on the geometry of the domain -- including a very weak compatibility condition between the Dirichlet boundary part and its complement -- we prove first Hölder continuity of the solution. Secondly, Gaussian Hölder estimates for the corresponding heat kernel are derived. The essential instruments are De Giorgi and Morrey-Campanato estimates.

*Appeared in*

- Adv. Differential Equations, 20 (2015) pp. 299--360.

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# Damage processes in thermoviscoelastic materials with damage-dependent thermal expansion coefficients

*Authors*

- Heinemann, Christian
- Rocca, Elisabetta

*2010 Mathematics Subject Classification*

- 35D30 34B14 74A45

*Keywords*

- damage phenomena, thermoviscoelastic materials, global existence of weak solutions, nonlinear boundary value problems

*DOI*

*Abstract*

In this paper we prove existence of global in time weak solutions for a highly nonlinear PDE system arising in the context of damage phenomena in thermoviscoelastic materials. The main novelty of the present contribution with respect to the ones already present in the literature consists in the possibility of taking into account a damage-dependent thermal expansion coefficient. This term implies the presence of nonlinear couplings in the PDE system, which make the analysis more challenging.

*Appeared in*

- Math. Methods Appl. Sci., 38 (2015) pp. 4587--4612.

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# Optimal control for a phase field system with a possibly singular potential

*Authors*

- Colli, Pierluigi
- Gilardi, Gianni
- Marinoschi, Gabriela
- Rocca, Elisabetta

*2010 Mathematics Subject Classification*

- 80A22 35K55 49J20 49K20

*Keywords*

- Phase field system, phase transition, singular potentials, optimal control, optimality conditions, adjoint state system

*DOI*

*Abstract*

In this paper we study a distributed control problem for a phase-field system of Caginalp type with logarithmic potential. The main aim of this work would be to force the location of the diffuse interface to be as close as possible to a prescribed set. However, due to the discontinuous character of the cost functional, we have to approximate it by a regular one and, in this case, we solve the associated control problem and derive the related first order necessary optimality conditions.

*Appeared in*

- Math. Control Relat. Fields, 6 (2016) pp. 95--112.

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# Optimal distributed control of a nonlocal convective Cahn--Hilliard equation by the velocity in 3D

*Authors*

- Rocca, Elisabetta
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 49J20 49J50 35R09 45K05 74N99

*Keywords*

- Distributed optimal control, first-order necessary optimality conditions, nonlocal models, integrodifferential equations, convective Cahn-Hilliard equation, phase separation

*DOI*

*Abstract*

In this paper we study a distributed optimal control problem for a nonlocal convective Cahn-Hilliard equation with degenerate mobility and singular potential in three dimensions of space. While the cost functional is of standard tracking type, the control problem under investigation cannot easily be treated via standard techniques for two reasons: the state system is a highly nonlinear system of PDEs containing singular and degenerating terms, and the control variable, which is given by the velocity of the motion occurring in the convective term, is nonlinearly coupled to the state variable. The latter fact makes it necessary to state rather special regularity assumptions for the admissible controls, which, while looking a bit nonstandard, are however quite natural in the corresponding analytical framework. In fact, they are indispensable prerequisites to guarantee the well-posedness of the associated state system. In this contribution, we employ recently proved existence, uniqueness and regularity results for the solution to the associated state system in order to establish the existence of optimal controls and appropriate first-order necessary optimality conditions for the optimal control problem.

*Appeared in*

- SIAM J. Control Optim., 53 (2015) pp. 1654--1680.

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# Asymptotically linear problems driven by fractional Laplacian operators

*Authors*

- Fiscella, Alessio
- Servadei, Raffaella
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 49J35 35A15 35S15 47G20 45G05

*Keywords*

- Integrodifferential operators, fractional Laplacian, variational techniques, Saddle Point Theorem, Palais-Smale condition

*DOI*

*Abstract*

In this paper we study a non-local fractional Laplace equation, depending on a parameter, with asymptotically linear right-hand side. Our main result concerns the existence of weak solutions for this equation and it is obtained using variational and topological methods. We treat both the nonresonant case and the resonant one.

*Appeared in*

- Math. Methods Appl. Sci., 38 (2015) pp. 3551--3563.

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# Sharp interface control in a Penrose--Fife model

*Authors*

- Colli, Pierluigi
- Marinoschi, Gabriela
- Rocca, Elisabetta

*2010 Mathematics Subject Classification*

- 49J20 82B26 90C46

*Keywords*

- optimal control problems, Penrose-Fife model, sharp interface

*DOI*

*Abstract*

In this paper we study a singular control problem for a system of PDEs describing a phasefield model of Penrose-Fife type. The main novelty of this contribution consists in the idea of forcing a sharp interface separation between the states of the system by using heat sources distributed in the domain and at the boundary. We approximate the singular cost functional with a regular one, which is based on the Legendre-Fenchel relations. Then, we obtain a regularized control problem for which we compute the first order optimality conditions using an adapted penalization technique. The proof of some convergence results and the passage to the limit in these optimality conditions lead to the characterization of the desired optimal controller.

*Appeared in*

- ESAIM Control Optim. Calc. Var., 22 (2016) pp. 473--499.

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# Bifurcation results for a fractional elliptic equation with critical exponent in $R^n$

*Authors*

- Dipierro, Serena
- Medina, Maria
- Peral, Ireneo
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 5B40 35D30 35J20 35R11 49N60

*Keywords*

- Bifurcation, Lyapunov-Schmidt reduction, critical problem, fractional elliptic regularity

*DOI*

*Abstract*

In this paper we study some nonlinear elliptic equations obtained as a perturbation of the problem with the fractional critical Sobolev exponent. To construct solutions to this equation, we use the Lyapunov-Schmidt reduction, that takes advantage of the variational structure of the problem. Some cases of the parameter range are particularly difficult, due to the lack of regularity of the associated energy functional, and we need to introduce a new functional setting and develop an appropriate fractional elliptic regularity theory.

*Appeared in*

- Manuscripta Math., 153 (2017) pp. 183-230.

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# Cardiac contraction induces discordant alternans and localized block

*Authors*

- Radszuweit, Markus
- Alvarez-Lacalle, Enrique
- Bär, Markus
- Echebarria, Blas

*2010 Mathematics Subject Classification*

- 92B05 65M06

*2008 Physics and Astronomy Classification Scheme*

- 87.19.Hh, 87.19.R-, 05.45.-a, 89.75.-k

*Keywords*

- nonlinear dynamics, cardiac alternans, amplitude equations

*DOI*

*Abstract*

In this paper we use a simplified model of cardiac excitation-contraction coupling to study the effect of tissue deformation on the dynamics of alternans, i.e. alternations in the duration of the cardiac action potential, that occur at fast pacing rates and are known to be pro-arrhythmic. We show that small stretch-activated currents can produce large effects and cause a transition from in-phase to off-phase alternations (i.e. from concordant to discordant alternans) and to conduction blocks. We demonstrate numerically and analytically that this effect is the result of a generic change in the slope of the conduction velocity restitution curve due to electromechanical coupling. Thus, excitation-contraction coupling can potentially play a relevant role in the transition to reentry and fibrillation.

*Appeared in*

- Phys. Rev. E (3), 91 (4.2.2015) pp. 022703.

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# Existence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects

*Authors*

- Heinemann, Christian
- Kraus, Christiane

*2010 Mathematics Subject Classification*

- 35L20 35L51 35K85 35K55 49J40 49S05 74A45 74G25 34A12 82B26 82C26 35K92 35K35

*Keywords*

- Cahn-Hilliard system, phase separation, hyperbolic-parabolic systems, doubly nonlinear differential inclusions, existence results, energetic solutions, weak solutions, linear elasticity, rate-dependent systems

*DOI*

*Abstract*

In this paper, we consider a coupled PDE system describing phase separation and damage phenomena in elastically stressed alloys in the presence of inertial effects. The material is considered on a bounded Lipschitz domain with mixed boundary conditions for the displacement variable. The main aim of this work is to establish existence of weak solutions for the introduced hyperbolic-parabolic system. To this end, we first adopt the notion of weak solutions introduced in [C. Heinemann, C. Kraus: Existence results of weak solutions for Cahn-Hilliard systems coupled with elasticity and damage. Adv. Math. Sci. Appl. 21 (2011), 321-359]. Then we prove existence of weak solutions by means of regularization, time-discretization and different variational techniques.

*Appeared in*

- Discrete Contin. Dyn. Syst., 35 (2015) pp. 2565--2590.

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# Optimal boundary control of a viscous Cahn--Hilliard system with dynamic boundary condition and double obstacle potentials

*Authors*

- Colli, Pierluigi
- Farshbaf Shaker, Mohammad Hassan
- Gilardi, Gianni
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 74M15 49K20 35K61

*Keywords*

- Optimal control, parabolic obstacle problems, MPECs, dynamic boundary conditions, optimality conditions

*DOI*

*Abstract*

In this paper, we investigate optimal boundary control problems for Cahn--Hilliard variational inequalities with a dynamic boundary condition involving double obstacle potentials and the Laplace--Beltrami operator. The cost functional is of standard tracking type, and box constraints for the controls are prescribed. We prove existence of optimal controls and derive first-order necessary conditions of optimality. The general strategy, which follows the lines of the recent approach by Colli, Farshbaf-Shaker, Sprekels (see Appl. Math. Optim., 2014) to the (simpler) Allen--Cahn case, is the following: we use the results that were recently established by Colli, Gilardi, Sprekels in the preprint arXiv:1407.3916 [math.AP] for the case of (differentiable) logarithmic potentials and perform a so-called ``deep quench limit''. Using compactness and monotonicity arguments, it is shown that this strategy leads to the desired first-order necessary optimality conditions for the case of (non-differentiable) double obstacle potentials.

*Appeared in*

- SIAM J. Control Optim., 53 (2015) pp. 2696--2721.

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# Electromagnetic scattering by biperiodic multilayered gratings: A recursive integral equation approach

*Authors*

- Bugert, Beatrice
- Schmidt, Gunther

*2010 Mathematics Subject Classification*

- 31B10 35Q61 78A45 78M15

*Keywords*

- diffraction, biperiodic structure, multilayer grating, integral equation method, recursive algorithm

*DOI*

*Abstract*

In this paper, we propose a new recursive integral equation algorithm to solve the direct problem of electromagnetic scattering by biperiodic multilayered structures with polyhedral Lipschitz regular interfaces. We work with a combined potential approach that involves one unknown density on each of the grating profiles of the multilayered scatterer. Justified by the transmission conditions of the underlying electromagnetic scattering problem, we assume that densities in adjacent layers are linearly linked by a boundary integral operator and derive a recursion for these densities. It comprehends the inversion of one boundary integral equation on each scattering interface. Our algorithm is shown to be equivalent to the biperiodic multilayered electromagnetic scattering problem. Moreover, we obtain new existence and uniqueness results for our recursive integral equation algorithm, which promises to lead to an efficient numerical implementation of the considered scattering problem. These solvability results depend on the regularity of the grating interfaces and the values of the electromagnetic material parameters of the biperiodic multilayered structure at hand.

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# Unique determination of balls and polyhedral scatterers with a single point source wave

*Authors*

- Hu, Guanghui
- Liu, Xiaodong

*2010 Mathematics Subject Classification*

- 35R30 78A45 78A46

*Keywords*

- inverse acoustic scattering, uniqueness, polyhedral scatterers, balls, point source wave

*DOI*

*Abstract*

In this paper, we prove uniqueness in determining a sound-soft ball or polyhedral scatterer in the inverse acoustic scattering problem with a single incident point source wave in R^N (N=2,3). Our proofs rely on the reflection principle for the Helmholtz equation with respect to a Dirichlet hyperplane or sphere, which is essentially a 'point-to-point' extension formula. The method has been adapted to proving uniqueness in inverse scattering from sound-soft cavities with interior measurement data incited by a single point source. The corresponding uniqueness for sound-hard balls or polyhedral scatterers has also been discussed.

*Appeared in*

- Inverse Problems, 30 (2014) pp. 065010/1--065010/14.

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# Fluctuations near the limit shape of random permutations under a conservative measure

*Authors*

- Cipriani, Alessandra
- Zeindler, Dirk

*2010 Mathematics Subject Classification*

- 60F05 60F10 60F17

*Keywords*

- random permutation, multiplicative measure, algebraically growing cycle weights, limit shape, functional central limit theorem, saddle point method

*DOI*

*Abstract*

In this work we are considering the behavior of the limit shape of Young diagrams associated to random permutations on the set {1, ... n} under a particular class of multiplicative measures. Our method is based on generating functions and complex analysis (saddle point method). We show that fluctuations near a point behave like a normal random variable and that the joint fluctuations at different points of the limiting shape have an unexpected dependence structure. We will also compare our approach with the so-called randomization of the cycle counts of permutations and we will study the convergence of the limit shape to a continuous stochastic process.

*Appeared in*

- ALEA, Lat. Am. J. Probab. Math. Stat. 12:2 (2015), pp. 971--999, changed title: The limit shape of random permutations with polynomially growing cycle weights.

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# Theory and structure of the metal/electrolyte interface incorporating adsorption and solvation effects

*Authors*

- Dreyer, Wolfgang
- Guhlke, Clemens
- Landstorfer, Manuel

*2010 Mathematics Subject Classification*

- 78A57 35Q35 34B15

*2008 Physics and Astronomy Classification Scheme*

- 82.45.Gj 68.43.-h 68.35.Md

*Keywords*

- Double Layer, Adsorption, Solvation, Surface Mixture Theory, Gouy-Chapman-Stern Model, Electrode/Electrolyte Interface

*DOI*

*Abstract*

In this work we present a continuum theory for the metal/electrolyte interface which explicitly takes into account adsorption and partial solvation on the metal surface. It is based on a general theory of coupled thermo-electrodynamics for volumes and surfaces, utilized here in equilibrium and a 1D approximation. We provide explicit free energy models for the volumetric metal and electrolyte phases and derive a surface free energy for the species present on the metal surface. This surface mixture theory explicitly takes into account the very different amount of sites an adsorbate requires, originating from solvation effects on the surface. Additionally we account for electron transfer reactions on the surface and the associated stripping of the solvation shell. Based on our overall surface free energy we thus provide explicit expressions of the surface chemical potentials of all constituents. The equilibrium representations of the coverages and the overall charge are briefly summarized. Our model is then used to describe two examples: (i) a silver single crystal electrode with (100) face in contact to a (0.01M NaF + 0.01M KPF6) aqueous solution, and (ii) a general metal surface in contact to some electrolytic solution AC for which an electron transfer reaction occurs in the potential range of interest. We reflect the actual modeling procedure for these examples and discuss the respective model parameters. Due to the representations of the coverages in terms of the applied potential we provide an adsorption map and introduce adsorption potentials. Finally we investigate the structure of the space charge layer at the metal/surface/electrolyte interface by means of numerical solutions of the coupled Poisson-momentum equation system for various applied potentials. It turns out that various layers self-consistently form within the overall space charge region, which are compared to historic and recent pictures of the double layer. Based on this we present new interpretations of what is known as inner and outer Helmholtz-planes and finally provide a thermodynamic consistent picture of the metal/electrolyte interface structure.

*Appeared in*

- Electrochimica Acta, 201 (2016) pp. 187--219.

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# Nearly cloaking the elastic wave fields

*Authors*

- Hu, Guanghui
- Liu, Hongyu

*2010 Mathematics Subject Classification*

- 74B05 35R30 35J25 74J20

*Keywords*

- invisibility cloaking, elastic waves, transformation elastodynamics, Lamé system, regularization, asymptotic estimates

*DOI*

*Abstract*

In this work, we develop a general mathematical framework on regularized approximate cloaking of elastic waves governed by the Lamé system via the approach of transformation elastodynamics. Our study is rather comprehensive. We first provide a rigorous justification of the transformation elastodynamics. Based on the blow-up-a-point construction, elastic material tensors for a perfect cloak are derived and shown to possess singularities. In order to avoid the singular structure, we propose to regularize the blow-up-a-point construction to be the blow-up-a-small-region construction. However, it is shown that without incorporating a suitable lossy layer, the regularized construction would fail due to resonant inclusions. In order to defeat the failure of the lossless construction, a properly designed lossy layer is introduced into the regularized cloaking construction . We derive sharp asymptotic estimates in assessing the cloaking performance. The proposed cloaking scheme is capable of nearly cloaking an arbitrary content with a high accuracy.

*Appeared in*

- J. Math. Pures Appl., 104 (2015) pp. 1045--1074.

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# Simulation of multi-frequency-induction-hardening including phase transitions and mechanical effects

*Authors*

- Hömberg, Dietmar
- Liu, Qingzhe
- Montalvo-Urquizo, Jonathan
- Nadolski, Dawid
- Petzold, Thomas
- Schmidt, Alfred
- Schulz, Alwin

*2010 Mathematics Subject Classification*

- 35K55 35Q61 74F05 74F15

*Keywords*

- induction surface hardening, multi field problem, thermomechanics, TRIP, finite element simulation

*DOI*

*Abstract*

Induction hardening is a well known method for the heat treatment of steel components. With the concept of multi-frequency hardening, where currents with two different frequency components are provided on a single inductor coil, it is possible to optimize the hardening zone to follow a given contour, e.g. of a gear. In this article, we consider the simulation of multi-frequency induction hardening in 3D. The equations to solve are the vector potential formulation of Maxwell's equations describing the electromagnetic fields, the balance of momentum to determine internal stresses and deformations arising from thermoelasticity and transformation induced plasticity, a rate law to determine the distribution of different phases and the heat equation to determine the temperature distribution in the workpiece. The equations are solved using adaptive finite element methods. The simulation results are compared to experiments for discs and for gears. A very good agreement for the hardening profile and the temperature is observed. It is also possible to predict the distribution of residual stresses after the heat treatment.

*Appeared in*

- Finite Elem. Anal. Des., 121 (2016) pp. 86--100.

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# Spectral properties of limiting solitons in optical fibers

*Authors*

- Amiranashvili, Shalva

ORCID: 0000-0002-8132-882X - Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Akhmediev, Nail

*2008 Physics and Astronomy Classification Scheme*

- 42.81.Dp 42.65.Tg 05.45.Yv 42.65.Re

*Keywords*

- optical solitons, ultrashort pulses, dispersion

*DOI*

*Abstract*

It seems to be self-evident that stable optical pulses cannot be considerably shorter than a single oscillation of the carrier field. From the mathematical point of view the solitary solutions of pulse propagation equations should loose stability or demonstrate some kind of singular behavior. Typically, an unphysical cusp develops at the soliton top, preventing the soliton from being too short. Consequently, the power spectrum of the limiting solution has a special behavior: the standard exponential decay is replaced by an algebraic one. We derive the shortest soliton and explicitly calculate its spectrum for the so-called short pulse equation. The latter applies to ultra-short solitons in transparent materials like fused silica that are relevant for optical fibers.

*Appeared in*

- Opt. Express, 22 (2014) pp. 30251--30256.

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# Considering copositivity locally

*Authors*

- Dickinson, Peter J. C.
- Hildebrand, Roland

*2010 Mathematics Subject Classification*

- 15A48 52A20

*Keywords*

- Copositive matrix, face, irreducibility, extreme rays

*DOI*

*Abstract*

Let $A$ be an element of the copositive cone $coposn$. A zero $vu$ of $A$ is a nonnegative vector whose elements sum up to one and such that $vu^TAvu = 0$. The support of $vu$ is the index set $Suppvu subset 1,dots,n$ corresponding to the nonzero entries of $vu$. A zero $vu$ of $A$ is called minimal if there does not exist another zero $vv$ of $A$ such that its support $Suppvv$ is a strict subset of $Suppvu$. Our main result is a characterization of the cone of feasible directions at $A$, i.e., the convex cone $VarKA$ of real symmetric $n times n$ matrices $B$ such that there exists $delta > 0$ satisfying $A + delta B in coposn$. This cone is described by a set of linear inequalities on the elements of $B$ constructed from the set of zeros of $A$ and their supports. This characterization furnishes descriptions of the minimal face of $A$ in $coposn$, and of the minimal exposed face of $A$ in $coposn$, by sets of linear equalities and inequalities constructed from the set of minimal zeros of $A$ and their supports. In particular, we can check whether $A$ lies on an extreme ray of $coposn$ by examining the solution set of a system of linear equations. In addition, we deduce a simple necessary and sufficient condition on the irreducibility of $A$ with respect to a copositive matrix $C$. Here $A$ is called irreducible with respect to $C$ if for all $delta > 0$ we have $A - delta C notin coposn$.

*Appeared in*

- J. Math. Anal. Appl., 437 (2016) pp. 1184--1195

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# Minimal zeros of copositive matrices

*Authors*

- Hildebrand, Roland

*2010 Mathematics Subject Classification*

- 15A48 15A21

*Keywords*

- Copositive matrix, irreducibility, extreme ray

*DOI*

*Abstract*

Let $A$ be an element of the copositive cone $coposn$. A zero $u$ of $A$ is a nonzero nonnegative vector such that $u^TAu = 0$. The support of $u$ is the index set $Suppu subset 1,dots,n$ corresponding to the positive entries of $u$. A zero $u$ of $A$ is called minimal if there does not exist another zero $v$ of $A$ such that its support $Suppv$ is a strict subset of $Suppu$. We investigate the properties of minimal zeros of copositive matrices and their supports. Special attention is devoted to copositive matrices which are irreducible with respect to the cone $S_+(n)$ of positive semi-definite matrices, i.e., matrices which cannot be written as a sum of a copositive and a nonzero positive semi-definite matrix. We give a necessary and sufficient condition for irreducibility of a matrix $A$ with respect to $S_+(n)$ in terms of its minimal zeros. A similar condition is given for the irreducibility with respect to the cone $NNMn$ of entry-wise nonnegative matrices. For $n = 5$ matrices which are irreducible with respect to both $S_+(5)$ and $NNM5$ are extremal. For $n = 6$ a list of candidate combinations of supports of minimal zeros which an exceptional extremal matrix can have is provided.

*Appeared in*

- Linear Algebra and its Applications, 459 (2014) pp. 154--174.

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# Non-trivial linear bounds for a random walk driven by a simple symmetric exclusion process

*Authors*

- Soares dos Santos, Renato

*2010 Mathematics Subject Classification*

- 60K37 82C22

*Keywords*

- random walk, dynamic random environment, exclusion process, linear bounds, multiscale analysis, percolation

*DOI*

*Abstract*

Linear bounds are obtained for the displacement of a random walk in a dynamic random environment given by a one-dimensional simple symmetric exclusion process in equilibrium. The proof uses an adaptation of multiscale renormalisation methods of Kesten and Sidoravicius [11].

*Appeared in*

- Electron. J. Probab., 19 (2014) pp. 1--18.

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# Thick points for Gaussian free fields with different cut-offs

*Authors*

- Cipriani, Alessandra
- Hazra, Rajat Subhra

*2010 Mathematics Subject Classification*

- 60G60 60G15

*Keywords*

- Gaussian multiplicative chaos, cut-offs, Liouville quantum gravity, thick points, Hausdorff dimension

*DOI*

*Abstract*

Massive and massless Gaussian free fields can be described as generalized Gaussian processes indexed by an appropriate space of functions. In this article we study various approaches to approximate these fields and look at the fractal properties of the thick points of their cut-offs. Under some sufficient conditions for a centered Gaussian process with logarithmic variance we study the set of thick points and derive their Hausdorff dimension. We prove that various cut-offs for Gaussian free fields satisfy these assumptions. We also give sufficient conditions for comparing thick points of different cut-offs.

*Appeared in*

- Ann. Inst. H. Poincaré Probab. Statist., 53 (2017) pp. 79-97.

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# An active poroelastic model for mechanochemical patterns in protoplasmic droplets of Physarum polycephalum

*Authors*

- Radszuweit, Markus
- Engel, Harald
- Bär, Markus

*2010 Mathematics Subject Classification*

- 92C15

*2008 Physics and Astronomy Classification Scheme*

- 89.75.Kd

*Keywords*

- Physarum polycephalum, pattern formation, amoeboid movement, active gels, two-phase models, poroelasticity

*DOI*

*Abstract*

Motivated by recent experimental studies, we derive and analyze a two-dimensional model for the contraction patterns observed in protoplasmic droplets of Physarum polycephalum. The model couples a description of an active poroelastic two-phase medium with equations describing the spatiotemporal dynamics of the intracellular free calcium concentration. The poroelastic medium is assumed to consist of an active viscoelastic solid representing the cytoskeleton and a viscous fluid describing the cytosol. The equations for the poroelastic medium are obtained from continuum force balance and include the relevant mechanical fields and an incompressibility condition for the two-phase medium. The reaction-diffusion equations for the calcium dynamics in the protoplasm of Physarum are extended by advective transport due to the flow of the cytosol generated by mechanical stress. Moreover, we assume that the active tension in the solid cytoskeleton is regulated by the calcium concentration in the fluid phase at the same location, which introduces a mechanochemical coupling. A linear stability analysis of the homogeneous state without deformation and cytosolic flows exhibits an oscillatory Turing instability for a large enough mechanochemical coupling strength. Numerical simulations of the model equations reproduce a large variety of wave patterns, including traveling and standing waves, turbulent patterns, rotating spirals and antiphase oscillations in line with experimental observations of contraction patterns in the protoplasmic droplets.

*Appeared in*

- PLOS ONE, 9 (2014) e99220.

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# Optimal and pressure-independent $L^2$ velocity error estimates for a modified Crouzeix--Raviart Stokes element with BDM reconstructions

*Authors*

- Brennecke, Christian
- Linke, Alexander

ORCID: 0000-0002-0165-2698 - Merdon, Christian
- Schöberl, Joachim

*Keywords*

- variational crime, Crouzeix-Raviart finite element, divergence-free mixed method, incompressible Navier-Stokes equations, a priori error estimates

*DOI*

*Abstract*

Nearly all inf-sup stable mixed finite elements for the incompressible Stokes equations relax the divergence constraint. The price to pay is that a priori estimates for the velocity error become pressure-dependent, while divergence-free mixed finite elements deliver pressure-independent estimates. A recently introduced new variational crime using lowest-order Raviart-Thomas velocity reconstructions delivers a much more robust modified Crouzeix-Raviart element, obeying an optimal pressure-independent discrete H^{1} velocity estimate. Refining this approach, a more sophisticated variational crime employing the lowest-order BDM element is proposed, which also allows proving an optimal pressure independent L^{2 } velocity error. Numerical examples confirm the analysis and demonstrate the improved robustness in the Navier-Stokes case.

*Appeared in*

- J. Comput. Math., 33 (2015) pp. 191--208.

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# Optimal Sobolev regularity for linear second-order divergence elliptic operators occurring in real-world problems

*Authors*

- Disser, Karoline
- Kaiser, Hans-Christoph
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 35J25 35R05 35B65

*Keywords*

- Second-order divergence operators, elliptic regularity, mixed boundary conditions, discontinuous coefficients

*DOI*

*Abstract*

On bounded three-dimensional domains, we consider divergence-type operators including mixed homogeneous Dirichlet and Neumann boundary conditions and discontinuous coefficient functions. We develop a geometric framework in which it is possible to prove that the operator provides an isomorphism of suitable function spaces. In particular, in these spaces, the gradient of solutions turns out to be integrable with exponent larger than the space dimension three. Relevant examples from real-world applications are provided in great detail.

*Appeared in*

- SIAM J. Math. Anal., 47 (2015) pp. 1719--1746.

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# Multiscale modeling of palisade formation in gliobastoma multiforme

*Authors*

- Caiazzo, Alfonso

ORCID: 0000-0002-7125-8645 - Ramis-Conde, Ignacio

*2010 Mathematics Subject Classification*

- 65P99 92-08 92B05

*Keywords*

- Palisades, glioblastoma, multiscale hybrid modeling, force-based, finite elements

*DOI*

*Abstract*

Palisades are characteristic tissue aberrations that arise in glioblastomas. Observation of palisades is considered as a clinical indicator of the transition from a noninvasive to an invasive tumour. In this article we propose a computational model to study the influence of genotypic and phenotypic heterogeneity in palisade formation. For this we produced three dimensional realistic simulations, based on a multiscale hybrid model, coupling the evolution of tumour cells and the oxygen diffusion in tissue, that depict the shape of palisades during its formation. Our results can be summarized as the following: (1) we show that cell heterogeneity is a crucial factor in palisade formation and tumour growth; (2) we present results that can explain the observed fact that recursive tumours are more malignant than primary tumours; and (3) the presented simulations can provide to clinicians and biologists for a better understanding of palisades 3D structure as well as glioblastomas growth dynamics

*Appeared in*

- J. Theor. Biol., 383 (2015) pp. 145--156.

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# Hamiltonian framework for short optical pulses

*Authors*

- Amiranashvili, Shalva

ORCID: 0000-0002-8132-882X

*2008 Physics and Astronomy Classification Scheme*

- 45.20.Jj 5.45.-a 42.65.-k

*Keywords*

- Hamiltonian mechanics, Nonlinear waves, Nonlinear optics

*DOI*

*Abstract*

Physics of short optical pulses is an important and active research area in nonlinear optics. In what follows we theoretically consider the most extreme representatives of short pulses that contain only several oscillations of electromagnetic field. Description of such pulses is traditionally based on envelope equations and slowly varying envelope approximation, despite the fact that the envelope is not ?slow? and, moreover, there is no clear definition of such a ?fast? envelope. This happens due to another paradoxical feature: the standard (envelope) generalized nonlinear Schrödinger equation yields very good correspondence to numerical solutions of full Maxwell equations even for few-cycle pulses, a thing that should not be. In what follows we address ultrashort optical pulses using Hamiltonian framework for nonlinear waves. As it appears, the standard optical envelope equation is just a reformulation of general Hamiltonian equations. In a sense, no approximations are required, this is why the generalized nonlinear Schrödinger equation is so effective. Moreover, the Hamiltonian framework greatly contributes to our understanding of ''fast'' envelope, ultrashort solitons, stability and radiation of optical pulses. Even the inclusion of dissipative terms is possible making the Hamiltonian approach an universal theoretical tool also in extreme nonlinear optics.

*Appeared in*

- S. Amiranashvili, Chapter 6: Hamiltonian Framework for Short Optical Pulses, in: New Approaches to Nonlinear Waves, E. Tobisch, ed., vol. 908 of Lecture Notes in Physics, Springer International Publishing, Cham, 2016, pp. 153--196

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# On thermodynamical couplings of quantum mechanics and macroscopic systems

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 37N20 47N50 80A99 81V70 78A35

*Keywords*

- GENERIC, density matrix, Hamiltonian systems, Onsager systems, canonical correlation, heat reservoirs, non-equilibrium steady states, Maxwell-Bloch equation, thermo-opto-electronics

*DOI*

*Abstract*

Pure quantum mechanics can be formulated as a Hamiltonian system in terms of the Liouville equation for the density matrix. Dissipative effects are modeled via coupling to a macroscopic system, where the coupling operators act via commutators. Following Öttinger (2010) we use the GENERIC framework to construct thermodynamically consistent evolution equations as a sum of a Hamiltonian and a gradient-flow contribution, which satisfy a particular non-interaction condition:

We give three applications of the theory. First, we consider a finite-dimensional quantum system that is coupled to a finite number of simple heat baths, each of which is described by a scalar temperature variable. Second, we model quantum system given by a one-dimensional Schrödinger operator connected to a one-dimensional heat equation on the left and on the right. Finally, we consider thermo-opto-electronics, where the Maxwell-Bloch system of optics is coupled to the energy-drift-diffusion system for semiconductor electronics.*Appeared in*

- Mathematical Results in Quantum Mechanics. Proceedings of the QMath12 Conference, P. Exner, W. König, H. Neidhardt, eds., World Scientific Publishing, Singapore, 2015, pp. 331--348

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# Balanced-Viscosity solutions for multi-rate systems

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Rossi, Riccarda
- Savaré, Giuseppe

*2010 Mathematics Subject Classification*

- 34D15 47J30 49J40 49J45

*Keywords*

- Generalized gradient systems, vanishing-viscosity approach, energy-dissipation principle, jump curves

*DOI*

*Abstract*

Several mechanical systems are modeled by the static momentum balance for the displacement u coupled with a rate-independent flow rule for some internal variable z. We consider a class of abstract systems of ODEs which have the same structure, albeit in a finite-dimensional setting, and regularize both the static equation and the rate-independent flow rule by adding viscous dissipation terms with coefficients ε^{α} and ε, where 0<ε<1 and α>0 is a fixed parameter. Therefore for α different from 1 the variables u and z have different relaxation rates. We address the vanishing-viscosity analysis as ε tends to 0 in the viscous system. We prove that, up to a subsequence, (reparameterized) viscous solutions converge to a parameterized curve yielding a Balanced Viscosity solution to the original rate-independent system and providing an accurate description of the system behavior at jumps. We also give a reformulation of the notion of Balanced Viscosity solution in terms of a system of subdifferential inclusions, showing that the viscosity in u and the one in z are involved in the jump dynamics in different ways, according to whether α >1, α=1, or 0<α<1.

*Appeared in*

- MURPHYS-HSFS-2014: 7th MUlti-Rate Processes and HYSteresis (MURPHYS) & 2nd International Workshop on Hysteresis and Slow-Fast Systems (HSFS), O. Klein, M. Dimian, P. Gurevich, D. Knees, D. Rachinskii, S. Tikhomirov, eds., vol. 727 of Journal of Physics: Conference Series, IOP Publishing, 2016, pp. 012010/1--012010/26

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# Robust arbitrary order mixed finite element methods for the incompressible Stokes equations

*Authors*

- Linke, Alexander

ORCID: 0000-0002-0165-2698 - Matthies, Gunar
- Tobiska, Lutz

*2010 Mathematics Subject Classification*

- 76D07 65N30

*2008 Physics and Astronomy Classification Scheme*

- 47.11.Fg

*Keywords*

- mixed finite element methods, incompressible Stokes equations, divergence-free methods, conforming and nonconforming FEM, mass conservation

*DOI*

*Abstract*

Standard mixed finite element methods for the incompressible Navier-Stokes equations that relax the divergence constraint are not robust against large irrotational forces in the momentum balance and the velocity error depends on the continuous pressure. This robustness issue can be completely cured by using divergence-free mixed finite elements which deliver pressure-independent velocity error estimates. However, the construction of H1-conforming, divergence-free mixed finite element methods is rather difficult. Instead, we present a novel approach for the construction of arbitrary order mixed finite element methods which deliver pressure-independent velocity errors. The approach does not change the trial functions but replaces discretely divergence-free test functions in some operators of the weak formulation by divergence-free ones. This modification is applied to inf-sup stable conforming and nonconforming mixed finite element methods of arbitrary order in two and three dimensions. Optimal estimates for the incompressible Stokes equations are proved for the H1 and L2 errors of the velocity and the L2 error of the pressure. Moreover, both velocity errors are pressure-independent, demonstrating the improved robustness. Several numerical examples illustrate the results.

*Appeared in*

- ESAIM Math. Model. Numer. Anal., 50 (2016) pp. 289--309 .

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# Fluid and diffusion limits for the Poisson encounter-mating model

*Authors*

- Gün, Onur
- Yilmaz, Atilla

*2010 Mathematics Subject Classification*

- 92D25 60F37 60J28

*Keywords*

- population dynamics, fluid limit, diffusion limit, Lotka-Volterra equations, replicator equations, pair formation, encounter-mating, assortative mating, random mating, heterogamy, panmixia, homogamy, monogamy, mating preferences, mating pattern, contingency table, Poisson process

*DOI*

*Abstract*

Stochastic encounter-mating (SEM) models describe monogamous permanent pair formation in finite zoological populations of multitype females and males. In this article we study SEM with Poisson firing times. We prove that an infinite population corresponds to a fluid limit, i.e., the stochastic dynamics converges to a deterministic system governed by coupled ODEs. Moreover, we establish a functional central limit theorem and give a diffusion approximation for the model. Next, we convert the fluid limit ODEs to the well-known Lotka-Volterra and replicator equations from population dynamics. Under the so-called fine balance condition, which characterizes panmixia for finite populations, we solve the corresponding replicator equations and give an exact expression for the fluid limit. Finally, we consider the case with two types of females and males. Without the fine balance assumption, but under some symmetry conditions, we give an explicit formula for the limiting mating pattern, and then use it to fully characterize assortative mating.

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# Hausdorff metric BV discontinuity of sweeping processes

*Authors*

- Klein, Olaf

ORCID: 0000-0002-4142-3603 - Recupero, Vincenzo

*2010 Mathematics Subject Classification*

- 34A60 34C55 34G25, 47H30, 74C05

*Keywords*

- Sweeping process,, discontinuity,, bounded variation,, Hausdorff metric

*DOI*

*Abstract*

Sweeping processes are a class of evolution differential inclusions arising in elastoplasticity and were introduced by J.J. Moreau in the early seventies. The solution operator of the sweeping processes represents a relevant example of emphrate independent operator containing as a particular case the so called emphplay operator which is widely used in hysteresis. The continuity properties of these operators were studied in several works. In this note we address the continuity with respect to the strict metric in the space of functions of bounded variation with values in the metric space of closed convex subsets of a Hilbert space. We provide a counterexample showing that the solution operator of the sweeping process is not continuous when its domain is endowed with the strict topology of $BV$ and its codomain is endowed with the $L^1$-topology. This is at variance with the case of the play operator which instead is continuous in this sense.

*Appeared in*

- 727 of Journal of Physics: Conference Series, 2016, pp. 012006/1--012006/12.

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# Block preconditioners for linear systems arising from multiscale collocation with compactly supported RBFs

*Authors*

- Farrell, Patricio

ORCID: 0000-0001-9969-6615 - Pestana, Jennifer

*2010 Mathematics Subject Classification*

- 65F08 65N35 65N55

*Keywords*

- partial differential equation, multiscale collocation, compactly supported radial basis functions, Krylov subspace methods, preconditioning, additive Schwarz method

*DOI*

*Abstract*

Symmetric collocation methods with radial basis functions allow approximation of the solution of a partial differential equation, even if the right-hand side is only known at scattered data points, without needing to generate a grid. However, the benefit of a guaranteed symmetric positive definite block system comes at a high computational cost. This cost can be alleviated somewhat by considering compactly supported radial basis functions and a multiscale technique. But the condition number and sparsity will still deteriorate with the number of data points. Therefore, we study certain block diagonal and triangular preconditioners. We investigate ideal preconditioners and determine the spectra of the preconditioned matrices before proposing more practical preconditioners based on a restricted additive Schwarz method with coarse grid correction (ARASM). Numerical results verify the effectiveness of the preconditioners.

*Appeared in*

- Numer. Linear Algebra Appl., 22 (2015) pp. 731--747.

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# Efficient blood flow simulations for the design of stented valve reducer in enlarged ventricular outflow tracts

*Authors*

- Caiazzo, Alfonso

ORCID: 0000-0002-7125-8645 - Guibert, Romain
- Boudjemline, Younes
- Vignon-Clementel, Irene E.

*2010 Mathematics Subject Classification*

- 65Z05 74L15 76Z05 92C50

*Keywords*

- Device design, percutaneous pulmonary valve replacement, multi scale blood flow simulations, right ventricle outflow tract (RVOT), proper orthogonal decomposition (POD), repaired tetralogy of Fallot

*DOI*

*Abstract*

Tetralogy of Fallot is a congenital heart disease characterized over time, after the initial repair, by the absence of a functioning pulmonary valve, which causes regurgitation, and by progressive enlargement of the right ventricle and pulmonary arteries. Due to this pathological anatomy, available transcatheter valves are usually too small to be deployed in the enlarged right ventricular outflow tracts (RVOT). To avoid surgical valve replacement, an alternative consists in implanting a reducer prior to or in combination with a transcatheter valve. We describe a computational model to study the effect of a stented valve RVOT reducer on the hemodynamics in enlarged ventricular outflow tracts. To this aim, blood flow in the right ventricular outflow tract is modeled via the incompressible Navier--Stokes equations coupled to a simplified valve model, numerically solved with a standard finite element method and with a reduced order model based on Proper Orthogonal Decomposition (POD). Numerical simulations are based on a patient geometry obtained from medical imaging and boundary conditions tuned according to measurements of inlet flow rates and pressures. Different geometrical models of the reducer are built, varying its length and/or diameter, and compared with the initial device-free state. Simulations thus investigate multiple device configurations and describe the effect of geometry on hemodynamics. Forces exerted on the valve and on the reducer are monitored, varying with geometrical parameters. Results support the thesis that the reducer does not introduce significant pressure gradients, as was found in animal experiments. Finally, we demonstrate how computational complexity can be reduced with POD.

*Appeared in*

- Cardiovasc. Eng. Technol., 6 (2015) pp. 485--500.

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# On the Cahn--Hilliard equation with dynamic boundary conditions and a dominating boundary potential

*Authors*

- Colli, Pierluigi
- Gilardi, Gianni
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35K55 35K50 82C26

*Keywords*

- Cahn--Hilliard equation, dynamic boundary conditions, phase separation, irregular potentials, well-posedness

*DOI*

*Abstract*

The Cahn--Hilliard and viscous Cahn--Hilliard equations with singular and possibly nonsmooth potentials and dynamic boundary condition are considered and some well-posedness and regularity results are proved.

*Appeared in*

- J. Math. Anal. Appl., 419 (2014) pp. 972--994.

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# Probabilistic constraints via SQP solver: Application to a renewable energy management problem

*Authors*

- Bremer, Ingo
- Henrion, René
- Möller, Andris

*2010 Mathematics Subject Classification*

- 90C15 90B05

*Keywords*

- probabilistic constraints, renewable energies, multivariate Gaussian probability, SQP with low precision data

*DOI*

*Abstract*

The aim of this paper is to illustrate the efficient solution of nonlinear optimization problems with joint probabilistic constraints by means of an SQP method. Here, the random vector is assumed to obey some multivariate Gaussian distribution. The numerical solution approach is applied to a renewable energy management problem. We consider a coupled system of hydro and wind power production used in order to satisfy some local demand of energy and to sell/buy excessive or missing energy on a day-ahead and intraday market, respectively. A short term planning horizon of 2 days is considered and only wind power is assumed to be random. In the first part of the paper, we develop an appropriate optimization problem involving a probabilistic constraint reflecting demand satisfaction. Major attention will be payed to formulate this probabilistic constraint not directly in terms of random wind energy produced but rather in terms of random wind speed, in order to benefit from a large data base for identifying an appropriate distribution of the random parameter. The second part presents some details on integrating Genz' code for Gaussian probabilities of rectangles into the environment of the SQP solver SNOPT. The procedure is validated by means of a simplified optimization problem which by its convex structure allows to estimate the gap between the numerical and theoretical optimal values, respectively. In the last part, numerical results are presented and discussed for the original (nonconvex) optimization problem.

*Appeared in*

- Comput. Manag. Sci., 12 (2015) pp. 435--459.

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# Characterization of polynomials and higher-order Sobolev spaces in terms of nonlocal functionals involving difference quotients

*Authors*

- Ferreira, Rita
- Kreisbeck, Carolin
- Ribeiro, Ana Margarida

*2010 Mathematics Subject Classification*

- 46E35

*Keywords*

- Higher-order Sobolev spaces, nonlocal functionals

*DOI*

*Abstract*

The aim of this paper, which deals with a class of singular functionals involving difference quotients, is twofold: deriving suitable integral conditions under which a measurable function is polynomial and stating necessary and sufficient criteria for an integrable function to belong to a *k*th-order Sobolev space. One of the main theorems is a new characterization of W^{k,p}(Ω), *k*∈ ℕ and *p* ∈ (1, +∞), for arbitrary open sets Ω ⊂ ℝ^{n}. In particular, we provide natural generalizations of the results regarding Sobolev spaces summarized in Brézis' overview article [*Russ. Math. Surv.* **57** (2002), pp. 693-708] to the higher-order case, and extend the work by Borghol [*Asymptotic Anal.* **51** (2007), pp. 303-318] to a more general setting.

*Appeared in*

- Nonlinear Anal. Real World Appl., 112 (2015), pp. 199-214, DOI 10.1016/j.na.2014.09.007 .

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# Bistability and hysteresis in an optically injected two-section semiconductor laser

*Authors*

- Pimenov, Alexander
- Viktorov, Evgeniy A.
- Hegarty, Stephen P.
- Habruseva, Tatiana
- Huyet, Guillaume
- Rachinskii, Dmitrii
- Vladimirov, Andrei G.

*2008 Physics and Astronomy Classification Scheme*

- 05.45.-a 42.55.Px 42.60.Fc 42.65.Pc

*Keywords*

- Bistability and hysteresis, Numerical bifurcation analysis, semiconductor lasers, single-mode optical injection

*DOI*

*Abstract*

The effect of coherenct single frequency injection in two-section semiconductor lasers is studied numerically using a model based on a set of delay differential equations. The existence of bistability between different CW and non-stationary regimes of operation is demonstrated in the case of sufficiently large linewidth enhancement factors.

*Appeared in*

- Phys. Rev. E (3), 89 (2014) pp. 052903/1--052903/7.

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# An adaptive multi level Monte--Carlo method with stochastic bounds for quantities of interest in groundwater flow with uncertain data

*Authors*

- Eigel, Martin
- Merdon, Christian
- Neumann, Johannes

*2010 Mathematics Subject Classification*

- 35R60 47B80 60H35 65C20 65N12 65N22 65J10

*Keywords*

- partial differential equations with random coefficients, uncertainty quantification, multilevel Monte Carlo, adaptivity

*DOI*

*Abstract*

The focus of this work is the introduction of some computable a posteriori error control to the popular multilevel Monte Carlo sampling for PDE with stochastic data. We are especially interested in applications in the geosciences such as groundwater flow with rather rough stochastic fields for the conductive permeability. With a spatial discretisation based on finite elements, a goal functional is defined which encodes the quantity of interest. The devised goal-oriented error estimator enables to determine guaranteed a posteriori error bounds for this quantity. In particular, it allows for the adaptive refinement of the mesh hierarchy used in the multilevel Monte Carlo simulation. In addition to controlling the deterministic error, we also suggest how to treat the stochastic error in probability. Numerical experiments illustrate the performance of the presented adaptive algorithm for a posteriori error control in multilevel Monte Carlo methods. These include a localised goal with problem-adapted meshes and a slit domain example. The latter demonstrates the refinement of regions with low solution regularity based on an inexpensive explicit error estimator in the multilevel algorithm.

*Appeared in*

- SIAM ASA J. Uncertain. Quantif., 4 (2016) pp. 1219--1245.

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# An integral equation approach for electromagnetic scattering by biperiodic structures

*Authors*

- Bugert, Beatrice

*2010 Mathematics Subject Classification*

- 31B10 35Q60 35Q61 45A05 78A45

*Keywords*

- biperiodic scattering problems, Maxwell's equations, boundary integral equations, Lipschitz domains

*DOI*

*Abstract*

The objective of this paper is the analytical investigation of an integral equation formulation for electromagnetic scattering by 2π-biperiodic multilayered structures with polyhedral Lipschitz regular interfaces. Extending the combined potential ansatz from Preprint No. 1882 for the electric fields in the before mentioned electromagnetic scattering problem from single to N profile scattering yields an equivalent system of N integral equations. We present a uniqueness and two existence results for this system depending on the values of the electromagnetic material parameters of the considered biperiodic scatterer. This in particular includes the proof that the system of integral equations is of zero Fredholm index. The general case that the grating interfaces are of polyhedral Lipschitz regularity requires more strict assumptions than the special case of smooth grating interfaces. We exploit the solvability results of this work in a subsequent paper featuring a recursive integral equation algorithm for the 2π-biperiodic multilayered electromagnetic scattering problem.

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# Optimal stopping via pathwise dual empirical maximisation

*Authors*

- Belomestny, Denis
- Hildebrand, Roland
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 60G40 60G17

*Keywords*

- optimal stopping problem, dual martingale, convex optimization, variance reduction

*DOI*

*Abstract*

The optimal stopping problem arising in the pricing of American options can be tackled by the so called dual martingale approach. In this approach, a dual problem is formulated over the space of martingales. A feasible solution of the dual problem yields an upper bound for the solution of the original primal problem. In practice, the optimization is performed over a finite-dimensional subspace of martingales. A sample of paths of the underlying stochastic process is produced by a Monte-Carlo simulation, and the expectation is replaced by the empirical mean. As a rule the resulting optimization problem, which can be written as a linear program, yields a martingale such that the variance of the obtained estimator can be large. In order to decrease this variance, a penalizing term can be added to the objective function of the path-wise optimization problem. In this paper, we provide a rigorous analysis of the optimization problems obtained by adding different penalty functions. In particular, a convergence analysis implies that it is better to minimize the empirical maximum instead of the empirical mean. Numerical simulations confirm the variance reduction effect of the new approach.

*Appeared in*

- Appl. Math. Optim., published online on 8.11.2017, DOI 10.1007/s00245-017-9454-9 .

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# Adaptive behaviour in a predator-prey model leads to multiple equilibrium states

*Authors*

- Pimenov, Alexander
- Rachinskii, Dmitrii
- Korobeinikov, Andrei

*2010 Mathematics Subject Classification*

- 92D30 34D20

*Keywords*

- Lotka-Volterra model, predator-prey model, coexistence, stability of biosystems, multiple equilibria, predator pit, fold bifurcation

*DOI*

*Abstract*

There is evidence that multiple stable equilibrium states are possible in real-life ecological systems. In order to verify a hypothesis that such a multitude of equilibrium states can be caused by adapting of animal behaviour to changes of environmental conditions, we consider a simple predator-prey model where prey changes a mode of behaviour in response to the pressure of predation. This model exhibits two stable coexisting equilibrium states with basins of attraction separated by a separatrix of a saddle point.

*Appeared in*

- Theoret. Population Biol., 101 (2015) pp. 24--30.

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# Non-instantaneous polarization dynamics in dielectric media

*Authors*

- Hofmann, Michael
- Hyyti, Janne
- Birkholz, Simon
- Bock, Martin
- Das, Susanta K.
- Grunwald, Rüdiger
- Hoffmann, Mathias
- Nagy, Tamas
- Demircan, Ayhan
- Jupé, Marco
- Ristau, Detlev
- Morgner, Uwe
- Brée, Carsten
- Wörner, Michael
- Elsaesser, Thomas
- Steinmeyer, Günter

*2008 Physics and Astronomy Classification Scheme*

- 42.30.Rx 78.20.-e 78.20.Mg 78.20.Bh

*Keywords*

- nonlinear optics, ultrafast spectroscopy, interferometric FROG, pulse characterization, time-dependent Schrödinger equation

*DOI*

*Abstract*

Third-order optical nonlinearities play a vital role for generation and characterization of some of the shortest optical pulses to date, for optical switching applications, and for spectroscopy. In many cases, nonlinear optical effects are used far off resonance, and then an instantaneous temporal response is expected. Here, we show for the first time resonant frequency-resolved optical gating measurements that indicate substantial nonlinear polarization relaxation times up to 6.5,fs in dielectric media, i.e., significantly beyond the shortest pulses directly available from commercial lasers. These effects are among the fastest effects observed in ultrafast spectroscopy. Numerical solutions of the time-dependent Schrödinger equation are in excellent agreement with experimental observations. The simulations indicate that pulse generation and characterization in the ultraviolet may be severely affected by this previously unreported effect. Moreover, our approach opens an avenue for application of frequency-resolved optical gating as a highly selective spectroscopic probe in high-field physics.

*Appeared in*

- , 2 (2015) pp. 151--157.

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# Numerical study of SUPG and LPS methods combined with higher order variational time discretization schemes applied to time-dependent convection-diffusion-reaction equations

*Authors*

- Ahmed, Naveed

ORCID: 0000-0002-9322-0373 - Matthies, Gunar

*2010 Mathematics Subject Classification*

- 65M12 65M15 65M60

*Keywords*

- stabilized finite elements, discontinuous Galerkin, continuous Galerkin--Petrov, transient convection-diffusion-reaction equations

*DOI*

*Abstract*

This paper considers the numerical solution of time-dependent convection-diffusion-reaction equations. We shall employ combinations of streamline-upwind Petrov-Galerkin (SUPG) and local projection stabilization (LPS) methods in space with the higher order variational time discretization schemes. In particular, we consider time discretizations by discontinuous Galerkin (dG) methods and continuous Galerkin-Petrov (cGP) methods. Several numerical tests have been performed to assess the accuracy of combinations of spatial and temporal discretization schemes. Furthermore, the dependence of the results on the stabilization parameters of the spatial discretizations are discussed. Finally the long-time behavior of overshoots and undershoots is investigated.

*Appeared in*

- J. Sci. Comput., 67 (2016) pp. 988--1018.

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# Transient radiation from a circular string of dipoles excited at superluminal velocity

*Authors*

- Arkhipov, Rostislav M.
- Arkhipov, Mikhail V.
- Babushkin, Ihar
- Tolmachev, Yurii A.

*2008 Physics and Astronomy Classification Scheme*

- 41.60.Bq 42.25.Fx 42.25.Hz 42.65.Ky

*Keywords*

- Cherenkov radiation, ultrashort laser pulse, superliminal motions, optical Bloch equations

*DOI*

*Abstract*

This paper discusses the features of transient radiation from periodic one-dimensional resonant medium excited by ultrashort pulse. The case of circular geometry is considered for the harmonic distribution of the density of the particles along the circle. It is shown that a new frequency component arises in the spectrum of the scattered radiation in addition to the resonance frequency of medium. The new frequency appears both in the case of linear and nonlinear interaction, its value depends on the velocity of excitation pulse propagation and on the period of spatial modulation. In addition, the case when excitation moves at superluminal velocity and Cherenkov radiation arises is also studied.

*Appeared in*

- IEEE J. Quantum Electron., 45 (2015) pp. 590--596.

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# SUPG reduced order models for convection-dominated convection-diffusion-reaction equations

*Authors*

- Iliescu, Traian
- John, Volker

ORCID: 0000-0002-2711-4409 - Schyschlowa, Swetlana
- Wells, David

*2010 Mathematics Subject Classification*

- 65M60

*Keywords*

- Reduced order models (ROMs), convection-dominated equations, streamline-upwind Petrov--Galerkin (SUPG) method, Proper orthogonal decomposition (POD), stabilization parameter

*DOI*

*Abstract*

This paper presents a Streamline-Upwind Petrov--Galerkin (SUPG) reduced order model (ROM) based on Proper Orthogonal Decomposition (POD). This ROM is investigated theoretically and numerically for convection-dominated convection-diffusion-reaction equations. The SUPG finite element method was used on realistic meshes for computing the snapshots, leading to some noise in the POD data. Numerical analysis is used to propose the scaling of the stabilization parameter for the SUPG-ROM. Two approaches are used: One based on the underlying finite element discretization and the other one based on the POD truncation. The resulting SUPG-ROMs and the standard Galerkin ROM (G-ROM) are studied numerically. For many settings, the results obtained with the SUPG-ROMs are more accurate. Finally, one of the choices for the stabilization parameter is recommended.

*Appeared in*

- Comput. Methods Appl. Mech. Engrg., 289 (2015) pp. 454--474.

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# Guaranteed energy error estimators for a modified robust Crouzeix--Raviart Stokes element

*Authors*

- Linke, Alexander

ORCID: 0000-0002-0165-2698 - Merdon, Christian

*2010 Mathematics Subject Classification*

- 65N30 65N15 76D07

*2008 Physics and Astronomy Classification Scheme*

- 47.11.Fg

*Keywords*

- mixed finite elements, a posteriori error estimation, divergence-free method, incompressible Stokes equations, Crouzeix-Raviart element

*DOI*

*Abstract*

This paper provides guaranteed upper energy error bounds for a modified lowest-order nonconforming Crouzeix--Raviart finite element method for the Stokes equations. The modification from [A. Linke 2014, On the role of the Helmholtz-decomposition in mixed methods for incompressible flows and a new variational crime] is based on the observation that only the divergence-free part of the right-hand side should balance the vector Laplacian. The new method has optimal energy error estimates and can lead to errors that are smaller by several magnitudes, since the estimates are pressure-independent. An efficient a posteriori velocity error estimator for the modified method also should involve only the divergence-free part of the right-hand side. Some designs to approximate the Helmholtz projector are compared and verified by numerical benchmark examples. They show that guaranteed error control for the modified method is possible and almost as sharp as for the unmodified method.

*Appeared in*

- J. Sci. Comput., 64 (2015) pp. 541--558.

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# Dynamics of micro-integrated external-cavity diode lasers: Simulations, analysis and experiments

*Authors*

- Radziunas, Mindaugas
- Tronciu, Vasile Z.
- Luvsandamdin, Erdenetsetseg
- Kürbis, Christian
- Wicht, Andreas
- Wenzel, Hans

*2010 Mathematics Subject Classification*

- 78A60 35Q60 35B30 78-05

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px 42.65.bc 42.60.Pk 42.60.Mi

*Keywords*

- external cavity, diode laser, semiconductor, mode transitions, multi-stability, traveling wave model, cavity modes

*DOI*

*Abstract*

This paper reports the results of numerical and experimental investigations of the dynamics of an external cavity diode laser device composed of a semiconductor laser and a distant Bragg grating, which provides an optical feedback. Due to the influence of the feedback, this system can operate at different dynamic regimes. The traveling wave model is used for simulations and analysis of the nonlinear dynamics in the considered laser device. Based on this model, a detailed analysis of the optical modes is performed, and the stability of the stationary states is discussed. It is shown, that the results obtained from the simulation and analysis of the device are in good agreement with experimental findings.

*Appeared in*

- IEEE J. Quantum Electron., 51 (2015) pp. 2000408/1--2000408/8.

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# On existence and uniqueness of the equilibrium state for an improved Nernst--Planck--Poisson system

*Authors*

- Gajewski, Paul

*2010 Mathematics Subject Classification*

- 35J91 76T30 78A35

*Keywords*

- Equilibrium, Nernst-Planck-Poisson equations, Nonlinear Poisson equation

*DOI*

*Abstract*

This work deals with a model for a mixture of charged constituents introduced in [W. Dreyer et al. Overcoming the shortcomings of the Nernst-Planck model. emphPhys. Chem. Chem. Phys., 15:7075-7086, 2013]. The aim of this paper is to give a first existence and uniqueness result for the equilibrium situation. A main difference to earlier works is a momentum balance involving the gradient of pressure and the Lorenz force which persists in the stationary situation and gives rise to the dependence of the chemical potentials on the particle densities of every species.

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# Trace formulas for singular perturbations

*Authors*

- Malamud, Mark M.
- Neidhardt, Hagen

*2010 Mathematics Subject Classification*

- 47A55 47B25 34B44

*Keywords*

- Symmetric operators, perturbation determinants, trace formulas, spectral shift function

*DOI*

*Abstract*

Trace formulas for pairs of self-adjoint, maximal dissipative and other types of resolvent comparable operators are obtained. In particular, the existence of a complex-valued spectral shift function for a resolvent comparable pair H', H of maximal dissipative operators is proved. We also investigate the existence of a real-valued spectral shift function. Moreover, we treat in detail the case of additive trace class perturbations. Assuming that H and H'=H+V are maximal dissipative and V is of trace class, we prove the existence of a summable complex-valued spectral shift function. We also obtain trace formulas for a pair {A, A*} assuming only that A and A* are resolvent comparable. In this case the determinant of a characteristic function of A is involved in the trace formula.

In the case of singular perturbations we apply the technique of boundary triplets. It allows to express the spectral shift function of a pair of extensions in terms of abstract Weyl function and boundary operator.

We improve and generalize certain classical results of M.G. Krein for pairs of self-adjoint and dissipative operators, the results of A. Rybkin for such pairs, as well as the results of V. Adamyan, B. Pavlov, and M. Krein for pairs {A, A*} with a maximal dissipative operator A.

*Appeared in*

- Adv. Math., 274 (2015) pp. 736--832.

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# Pulse repetition-frequency multiplication in a passively mode-locked semiconductor laser coupled to an external passive cavity

*Authors*

- Arkhipov, Rostislav M.
- Amann, Andreas
- Vladimirov, Andrei G.

*2010 Mathematics Subject Classification*

- 78A60 78M35 78-05.

*2008 Physics and Astronomy Classification Scheme*

- 42.60.Fc 42.55.Px 42.65.Sf 85.30.De

*Keywords*

- mode-Locked lasers, mode selsction, delay differential equations model

*DOI*

*Abstract*

Using a delay differential equation model with two time delays, we investigate the dynamics of a semiconductor laser with an active cavity coupled to an external passive cavity. Our numerical simulations indicate that when the coupling between the two cavities is strong enough and the round-trip time of the active cavity is an integer multiple of the round-trip time of the external passive cavity, a harmonic mode-locking regime can develop in the laser with the pulse repetition period close to the passive cavity round trip time. We also demonstrate that the output field intensity sensitively depends on the relative phase between the electric fields in the two cavities giving rise to a resonant behavior. The period and width of the resonances depend on the ratio of the round-trip times and the coupling between the two cavities. We show that the coupled cavity system under consideration can demonstrate a bistability between different regimes of generation.

*Appeared in*

- Appl. Phys. B, 118 (2015) pp. 539--548.

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# On an extended interpretation of linkage disequilibrium in genetic case-control association studies

*Authors*

- Dickhaus, Thorsten
- Stange, Jens
- Demirhan, Haydar

*2010 Mathematics Subject Classification*

- 62J15 62P10 62E20

*Keywords*

- Asymptotic Gaussianity, chi-squared statistic, contingency table, correlation structure, Delta method, Fisher's exact test, odds ratio

*DOI*

*Abstract*

We are concerned with statistical inference for 2 x 2 x K contingency tables in the context of genetic case-control association studies. Multivariate methods based on asymptotic Gaussianity of vectors of test statistics require information about the asymptotic correlation structure among these test statistics under the global null hypothesis. We show that for a wide variety of test statistics this asymptotic correlation structure is given by the linkage disequilibrium matrix of the K loci under investigation. Three popular choices of test statistics are discussed for illustration.

*Appeared in*

- Appl. Genet. Mol. Biol., 14:5 (2015), pp. 497--505.

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# On multivariate chi-square distributions and their applications in testing multiple hypotheses

*Authors*

- Dickhaus, Thorsten
- Royen, Thomas

*2010 Mathematics Subject Classification*

- 62J15 62E15

*Keywords*

- Contingency tables, Kruskal-Wallis test, multiple Wald tests, multivariate analysis, multivariate gamma distributions, statistical genetics

*DOI*

*Abstract*

We are considered with three different types of multivariate chi-square distributions. Their members play important roles as limiting distributions of vectors of test statistics in several applications of multiple hypotheses testing. We explain these applications and provide formulas for computing multiplicity-adjusted $p$-values under the respective global hypothesis.

*Appeared in*

- Statistics, (2015) pp. 427--454 .

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# A diffuse interface model for two-phase incompressible flows with nonlocal interactions and nonconstant mobility

*Authors*

- Frigeri, Sergio Pietro
- Grasselli, Maurizio
- Rocca, Elisabetta

*2010 Mathematics Subject Classification*

- 35Q30 37L30 45K05 76D03 76T99

*Keywords*

- Navier-Stokes equations, nonlocal Cahn-Hilliard equations, degenerate mobility, incompressible, binary fluids, weak solutions, global attractors

*DOI*

*Abstract*

We consider a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids with matched constant densities. This model consists of the Navier-Stokes system coupled with a convective nonlocal Cahn-Hilliard equation with non-constant mobility. We first prove the existence of a global weak solution in the case of non-degenerate mobilities and regular potentials of polynomial growth. Then we extend the result to degenerate mobilities and singular (e.g. logarithmic) potentials. In the latter case we also establish the existence of the global attractor in dimension two. Using a similar technique, we show that there is a global attractor for the convective nonlocal Cahn-Hilliard equation with degenerate mobility and singular potential in dimension three.

*Appeared in*

- Nonlinearity, 28 (2015) pp. 1257--1293.

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# On a diffuse interface model of tumor growth

*Authors*

- Frigeri, Sergio Pietro
- Grasselli, Maurizio
- Rocca, Elisabetta

*2010 Mathematics Subject Classification*

- 35D30 35K57 35Q92 37L30 92C17

*Keywords*

- diffuse interface, tumor growth, Cahn-Hilliard equations, reaction-diffusion equations, weak solutions, well-posedness, global attractors

*DOI*

*Abstract*

We consider a diffuse interface model of tumor growth proposed by A. Hawkins-Daruud et al. This model consists of the Cahn-Hilliard equation for the tumor cell fraction *φ* nonlinearly coupled with a reaction-diffusion equation for *ψ* which represents the nutrient-rich extracellular water volume fraction. The coupling is expressed through a suitable proliferation function*p(φ)* multiplied by the differences of the chemical potentials for *φ* and *ψ*. The system is equipped with no-flux boundary conditions which entails the conservation of the total mass, that is, the spatial average of *φ*+*ψ*. Here we prove the existence of a weak solution to the associated Cauchy problem, provided that the potential *F* and *p* satisfy sufficiently general conditions. Then we show that the weak solution is unique and continuously depends on the initial data, provided that *p* satisfies slightly stronger growth restrictions. Also, we demonstrate the existence of a strong solution and that any weak solution regularizes in finite time. Finally, we prove the existence of the global attractor in a phase space characterized by an a priori bounded energy.

*Appeared in*

- European J. Appl. Math., 26 (2015) pp. 215--243.

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# On nonlocal Cahn--Hilliard--Navier--Stokes systems in two dimensions

*Authors*

- Frigeri, Sergio Pietro
- Gal, G. Ciprian
- Grasselli, Maurizio

*2010 Mathematics Subject Classification*

- 35Q30 37L30 45K05

*Keywords*

- Incompressible binary fluids, Navier-Stokes equations, nonlocal Cahn-Hilliard equations, weak solutions, uniqueness, strong solutions, global attractors, exponential attractors

*DOI*

*Abstract*

We consider a diffuse interface model which describes the motion of an incompressible isothermal mixture of two immiscible fluids. This model consists of the Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard equation. Several results were already proven by two of the present authors. However, in the two-dimensional case, the uniqueness of weak solutions was still open. Here we establish such a result even in the case of degenerate mobility and singular potential. Moreover, we show the strong-weak uniqueness in the case of viscosity depending on the order parameter, provided that the mobility is constant and the potential is regular. In the case of constant viscosity, on account of the uniqueness results we can deduce the connectedness of the global attractor whose existence was obtained in a previous paper. The uniqueness technique can be adapted to show the validity of a smoothing property for the difference of two trajectories which is crucial to establish the existence of an exponential attractor.

*Appeared in*

- J. Nonlinear Sci., 26 (2016), pp. 847--893.

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# Overdetermined problems for the fractional Laplacian in exterior and annular sets

*Authors*

- Soave, Nicola
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35N25 35R11 35A02

*Keywords*

- Rigidity and classification results, fractional Laplacian, unbounded domains, overdetermined problems

*DOI*

*Abstract*

We consider a fractional elliptic equation in an unbounded set with both Dirichlet and fractional normal derivative datum prescribed. We prove that the domain and the solution are necessarily radially symmetric. The extension of the result in bounded non-convex regions is also studied, as well as the radial symmetry of the solution when the set is a priori supposed to be rotationally symmetric.

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# A rigidity result for nonlocal semilinear equations

*Authors*

- Farina, Alberto
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35R11 35B53 35R09

*Keywords*

- Nonlocal integro-differential semilinear equations, Liouville-type theorems, nondecreasing nonlinearities

*DOI*

*Abstract*

We consider a possibly anisotropic integro-differential semilinear equation, run by a nondecreasing and nontrivial nonlinearity. We prove that if the solution grows at infinity less than the order of the operator, then it must be constant.

*Appeared in*

- Proc. Roy. Soc. Edinburgh Sec. A DOI: 10.1017/S0308210516000391, published online: 31 May 2017.

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# A functional limit theorem for limit order books

*Authors*

- Bayer, Christian

ORCID: 0000-0002-9116-0039 - Horst, Ulrich
- Qiu, Jinniao

*2010 Mathematics Subject Classification*

- 60B11 90B22 91B70

*Keywords*

- Limit order book, Scaling limit, Averaging principle, Queuing theory

*DOI*

*Abstract*

We consider a stochastic model for the dynamics of the two-sided limit order book (LOB). For the joint dynamics of best bid and ask prices and the standing buy and sell volume densities, we derive a functional limit theorem, which states that our LOB model converges to a continuous-time limit when the order arrival rates tend to infinity, the impact of an individual order arrival on the book as well as the tick size tend to zero. The limits of the standing buy and sell volume densities are described by two linear stochastic partial differential equations, which are coupled with a two-dimensional reflected Brownian motion that is the limit of the best bid and ask price processes.

*Appeared in*

- Ann. Appl. Probab., 27 (2017) pp. 2753-2806.

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# Existence of solutions to a two-dimensional model for nonisothermal two-phase flows of incompressible fluids

*Authors*

- Eleuteri, Michela
- Rocca, Elisabetta
- Schimperna, Giulio

*2010 Mathematics Subject Classification*

- 35Q35 35K25 76D05 35D30

*Keywords*

- Cahn-Hilliard, Navier-Stokes, incompressible non-isothermal binary fluid, global-in-time existence, a-priori estimates

*DOI*

*Abstract*

We consider a thermodynamically consistent diffuse interface model describing two-phase flows of incompressible fluids in a non-isothermal setting. The model was recently introduced in citeERS where existence of weak solutions was proved in three space dimensions. Here, we aim at studying the properties of solutions in the two-dimensional case. In particular, we can show existence of global in time solutions satisfying a stronger formulation of the model with respect to the one considered in citeERS. Moreover, we can admit slightly more general conditions on some material coefficients of the system.

*Appeared in*

- Ann. Inst. H. Poincare Anal. Non Lineaire, 33: 6 (2016), pp. 1431--1454

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# Homogenization and Orowan's law for anisotropic fractional operators of any order

*Authors*

- Patrizi, Stefania
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 74Q15 35B27 35R11 82D25

*Keywords*

- Crystal dislocation, homogenization, fractional operators

*DOI*

*Abstract*

We consider an anisotropic fractional operator and we consider the homogenization properties of an evolution equation. The scaling properties and the effective Hamiltonian that we obtain is different according to the fractional parameter. In the isotropic onedimensional case, we also prove a statement related to the so-called Orowan's law, that is an appropriate scaling of the effective Hamiltonian presents a linear behavior.

*Appeared in*

- Nonlinear Anal., 119 (2014) pp. 3--36.

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# Uniqueness for an inverse problem for a nonlinear parabolic system with an integral term by one-point Dirichlet data

*Authors*

- Hömberg, Dietmar
- Lu, Shuai
- Yamamoto, Masahiro

*2010 Mathematics Subject Classification*

- 80A22 49K20 49N45

*Keywords*

- inverse problem, bio-heat-equation, laser thermotherapy

*DOI*

*Abstract*

We consider an inverse problem arising in laser-induced thermotherapy, a minimally invasive method for cancer treatment, in which cancer tissues is destroyed by coagulation. For the dosage planning quantitatively reliable numerical simulation are indispensable. To this end the identification of the thermal growth kinetics of the coagulated zone is of crucial importance. Mathematically, this problem is a nonlinear and nonlocal parabolic heat source inverse problem. We show in this paper that the temperature dependent thermal growth parameter can be identified uniquely from a one-point measurement.

*Appeared in*

- J. Differential Equations, 266 (2019), pp. 7525--7544.

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# Computing and approximating multivariate chi-square probabilities

*Authors*

- Stange, Jens
- Loginova, Nina
- Dickhaus, Thorsten

*2010 Mathematics Subject Classification*

- 60E05 60E15 62E17 65D20

*Keywords*

- Bonferroni inequalities, chain factorization, correlation matrix, effective number of tests, linkage disequilibrium, m-factorial matrix, product-type probability approximations, sub-Markovian, Wishart matrix

*DOI*

*Abstract*

We consider computational methods for evaluating and approximating multivariate chi-square probabilities in cases where the pertaining correlation matrix or blocks thereof have a low-factorial representation. To this end, techniques from matrix factorization and probability theory are applied. We outline a variety of statistical applications of multivariate chi-square distributions and provide a system of MATLAB programs implementing the proposed algorithms. Computer simulations demonstrate the accuracy and the computational efficiency of our methods in comparison with Monte Carlo approximations, and a real data example from statistical genetics illustrates their usage in practice.

*Appeared in*

- J. Statist. Comput. Simul., 86:6 (2016), pp. 1233--1247.

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# Ground states and concentration phenomena for the fractional Schrödinger equation

*Authors*

- Fall, Mouhamed Moustapha
- Mahmoudi, Fethi
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35Q55 35R11 35B44 35B40 35J65 35J75

*Keywords*

- Fractional Laplacian, ground states, concentration phenomena, uniqueness

*DOI*

*Abstract*

We consider here solutions of the nonlinear fractional Schrödinger equation. We show that concentration points must be critical points for the potential. We also prove that, if the potential is coercive and has a unique global minimum, then ground states concentrate suitably at such minimal point. In addition, if the potential is radial, then the minimizer is unique.

*Appeared in*

- Nonlinearity, 28 (2015) pp. 1937--1961.

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# Monotonicity formulae and classification results for singular, degenerate, anisotropic PDEs

*Authors*

- Cozzi, Matteo
- Farina, Alberto
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35J92 35J93 35J20

*Keywords*

- Wulff shapes, energy monotonicity, rigidity and classification results

*DOI*

*Abstract*

We consider possibly degenerate and singular elliptic equations in a possibly anisotropic medium. We obtain monotonicity results for the energy density, rigidity results for the solutions and classi?cation results for the singularity/degeneracy/anisotropy allowed. As far as we know, these results are new even in the case of non-singular and non- degenerate anisotropic equations.

*Appeared in*

- Adv. Math., 293 (2016) pp. 343--381.

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# Reverse inequalities for slowly increasing sequences and functions

*Authors*

- Stephan, Holger

*2010 Mathematics Subject Classification*

- 26D15 35A23

*Keywords*

- reverse inequalities, slowly increasing sequences, slowly increasing functions

*DOI*

*Abstract*

We consider sharp inequalities involving slowly increasing sequences and functions, i.e., functions $f(t)$ with $f'(t) leq 1$ and sequences $(a_i)$ with $a_i+1-a_i leq 1$. The inequalities are reverse to mean inequalities, for example. In the continuous case, integrals of powers are estimated by powers of integrals, whereas in the discrete case powers of sums are estimated by sums of powers of sums. The problem is connected with interpolation theory in Banach spaces, one of them $W^1,infty$.

*Appeared in*

- Octogone Math. Mag., 22 (2015) pp. 621--633.

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# Recent trends and views on elliptic quasi-variational inequalities

*Authors*

- Alphonse, Amal
- Hintermüller, Michael

ORCID: 0000-0001-9471-2479 - Rautenberg, Carlos N.

ORCID: 0000-0001-9497-9296

*2010 Mathematics Subject Classification*

- 35J87 47J20 49K40 35J60

*Keywords*

- Quasi-variational inequalities, Mosco convergence, semismooth Newton methods

*DOI*

*Abstract*

We consider state-of-the-art methods, theoretical limitations, and open problems in elliptic Quasi-Variational Inequalities (QVIs). This involves the development of solution algorithms in function space, existence theory, and the study of optimization problems with QVI constraints. We address the range of applicability and theoretical limitations of fixed point and other popular solution algorithms, also based on the nature of the constraint, e.g., obstacle and gradient-type. For optimization problems with QVI constraints, we study novel formulations that capture the multivalued nature of the solution mapping to the QVI, and generalized differentiability concepts appropriate for such problems.

*Appeared in*

- Topics in Applied Analysis and Optimisation, M. Hintermüller, J.F. Rodrigues, eds., CIM Series in Mathematical Sciences, Springer Nature Switzerland AG, Cham, 2019, pp. 1--31.

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# Singular limit of Allen--Cahn equation with constraints and its Lagrange multiplier

*Authors*

- Farshbaf Shaker, Mohammad Hassan
- Fukao, Takeshi
- Yamazaki, Noriaki

*2010 Mathematics Subject Classification*

- 35K57 35R35 35B25

*Keywords*

- Allen--Cahn equation, constraint, double obstacle, singular limit, Lagrange multiplier, subdifferential

*DOI*

*Abstract*

We consider the Allen-Cahn equation with constraint. Our constraint is the subdifferential of the indicator function on the closed interval, which is the multivalued function. In this paper we give the characterization of the Lagrange multiplier to our equation. Moreover, we consider the singular limit of our system and clarify the limit of the solution and the Lagrange multiplier to our problem.

*Appeared in*

- Dynamical Systems, Differential Equations and Applications, AIMS Proceedings, 2015, pp. 418--427.

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# Modeling of electrochemical double layers in thermodynamic non-equilibrium

*Authors*

- Dreyer, Wolfgang
- Guhlke, Clemens
- Müller, Rüdiger

ORCID: 0000-0003-2643-722X

*2010 Mathematics Subject Classification*

- 35Q35 76T30 35C20

*2008 Physics and Astronomy Classification Scheme*

- 82.45.Gj 82.45.Fk

*Keywords*

- thermodynamics, electrode-electrolyte interface, double-layer, asymptotic analysis

*DOI*

*Abstract*

We consider the contact between an electrolyte and a solid electrode. At first we formulate a thermodynamic consistent model that resolves boundary layers at interfaces. The model includes charge transport, diffusion, chemical reactions, viscosity, elasticity and polarization under isothermal conditions. There is a coupling between these phenomena that particularly involves the local pressure in the electrolyte. Therefore the momentum balance is of major importance for the correct description of the layers. The width of the boundary layers is typically very small compared to the macroscopic dimensions of the system. In a second step we thus apply the method of asymptotic analysis to derive a simpler reduced model that does not resolve the boundary layers but instead incorporates the electrochemical properties of the layers into a set of new boundary conditions. For a metal-electrolyte interface, we derive a qualitative description of the double layer capacitance without the need to resolve space charge layers.

*Appeared in*

- Phys. Chem. Chem. Phys., 17 (2015), pp. 27176--27194, DOI 10.1039/C5CP03836G .

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# Moment bounds on the corrector of stochastic homogenization of non-symmetric elliptic finite difference equations

*Authors*

- Ben-Artzi, Jonathan
- Marahrens, Daniel
- Neukamm, Stefan

*2010 Mathematics Subject Classification*

- 35B27 35J08 60H25 60F17

*Keywords*

- stochastic homogenization, corrector equation, quantitative ergodicity

*DOI*

*Abstract*

We consider the corrector equation from the stochastic homogenization of uniformly elliptic finite-difference equations with random, possibly non-symmetric coefficients. Under the assumption that the coefficients are stationary and ergodic in the quantitative form of a Logarithmic Sobolev inequality (LSI), we obtain optimal bounds on the corrector and its gradient in dimensions d ≥ 2. Similar estimates have recently been obtained in the special case of diagonal coefficients making extensive use of the maximum principle and scalar techniques. Our new method only invokes arguments that are also available for elliptic systems and does not use the maximum principle. In particular, our proof relies on the LSI to quantify ergodicity and on regularity estimates on the derivative of the discrete Green's function in weighted spaces.

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# Recovering complex elastic scatterers by a single far-field pattern

*Authors*

- Hu, Guanghui
- Li, Jingzhi
- Liu, Hongyu

*2010 Mathematics Subject Classification*

- 74J20 74J25 35Q74 35R30

*Keywords*

- inverse elastic scattering, multiscale scatterers, asymptotic estimate, indicator functions, locating

*DOI*

*Abstract*

We consider the inverse scattering problem of reconstructing multiple impenetrable bodies embedded in an unbounded, homogeneous and isotropic elastic medium. The inverse problem is nonlinear and ill-posed. Our study is conducted in an extremely general and practical setting: the number of scatterers is unknown in advance; and each scatterer could be either a rigid body or a cavity which is not required to be known in advance; and moreover there might be components of multiscale sizes presented simultaneously. We develop several locating schemes by making use of only a single far-field pattern, which is widely known to be challenging in the literature. The inverse scattering schemes are of a totally ``direct" nature without any inversion involved. For the recovery of multiple small scatterers, the nonlinear inverse problem is linearized and to that end, we derive sharp asymptotic expansion of the elastic far-field pattern in terms of the relative size of the cavities. The asymptotic expansion is based on the boundary-layer-potential technique and the result obtained is of significant mathematical interest for its own sake. The recovery of regular-size/extended scatterers is based on projecting the measured far-field pattern into an admissible solution space. With a local tuning technique, we can further recover multiple multiscale elastic scatterers.

*Appeared in*

- J. Differential Equations, 257 (2014) pp. 469--489.

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# When do wireless network signals appear Poisson?

*Authors*

- Keeler, Paul

ORCID: 0000-0002-2063-1075 - Ross, Nathan
- Xia, Aihua

*2010 Mathematics Subject Classification*

- 60F05 60G55

*Keywords*

- Poisson point process, Cox point process, propagation process, vague topology, total variation distance

*DOI*

*Abstract*

We consider the point process of signal strengths from transmitters in a wireless network observed from a fixed position under models with general signal path loss and random propagation effects. We show via coupling arguments that under general conditions this point process of signal strengths can be well-approximated by an inhomogeneous Poisson or a Cox point processes on the positive real line. We also provide some bounds on the total variation distance between the laws of these point processes and both Poisson and Cox point processes. Under appropriate conditions, these results support the use of a spatial Poisson point process for the underlying positioning of transmitters in models of wireless networks, even if in reality the positioning does not appear Poisson. We apply the results to a number of models with popular choices for positioning of transmitters, path loss functions, and distributions of propagation effects.

*Appeared in*

- Bernoulli, 24:3 (2018) pp. 1973-1994.

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# Eigenvalue fluctuations for lattice Anderson Hamiltonians

*Authors*

- Biskup, Marek
- Fukushima, Ryoki
- König, Wolfgang

ORCID: 0000-0002-4212-0065

*2010 Mathematics Subject Classification*

- 60F05 82B44 60H25

*Keywords*

- Anderson model, spectra of random operators, central limit theorem

*DOI*

*Abstract*

We consider the random Schrödinger operator on a large box in the lattice with a large prefactor in front of the Laplacian part of the operator, which is proportional to the square of the diameter of the box. The random potential is assumed to be independent and bounded; its expectation function and variance function is given in terms of continuous bounded functions on the rescaled box. Our main result is a multivariate central limit theorem for all the simple eigenvalues of this operator, after centering and rescaling. The limiting covariances are expressed in terms of the limiting homogenized eigenvalue problem; more precisely, they are equal to the integral of the product of the squares of the eigenfunctions of that problem times the variance function.

*Appeared in*

- SIAM J. Math. Anal., 48 (2016), pp. 2674--2700.

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# Regular triangulation and power diagrams for Maxwell's equations

*Authors*

- Schlundt, Rainer

ORCID: 0000-0002-4424-4301

*2010 Mathematics Subject Classification*

- 35Q61 65F10 65F15 65N22 65N50

*Keywords*

- Maxwell's equations, finite integration technique, linear algebraic equations, eigenvalue problem, optimal triangulations, discrete Hodge star, microcell method

*DOI*

*Abstract*

We consider the solution of electromagnetic problems. A mainly orthogonal and locally barycentric dual mesh is used to discretize the Maxwell's equations using the Finite Integration Technique (FIT). The use of weighted duals allows greater flexibility in the location of dual vertices keeping the primal-dual orthogonality. The construction of the constitutive matrices is performed using either discrete Hodge stars or microcells. Hodge-optimized triangulations (HOT) can optimize the dual mesh alone to make it more self-centered while maintaining the primal-dual orthogonality, e.g., the weights are optimized in order to improve one or more of the discrete Hodge stars.

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# Asymptotic behaviour of a rigid body with a cavity filled by a viscous liquid

*Authors*

- Disser, Karoline

*2010 Mathematics Subject Classification*

- 35Q35 35Q30 74F10 76D03 35B40 37L15

*Keywords*

- Navier-Stokes equations, asymptotic behaviour of weak solutions, rigid body dynamics, conservation of angular momentum, strict Lyapunov functional

*DOI*

*Abstract*

We consider the system of equations modeling the free motion of a rigid body with a cavity filled by a viscous (Navier-Stokes) liquid. Zhukovskiy's Theorem states that in the limit of time going to infinity, the relative fluid velocity tends to 0 and the rigid velocity of the full structure tends to a steady rotation around one of the principle axes of inertia. We give a rigorous proof of this result. In particular, we prove that every global weak solution in a suitable class is subject to Zhukovskiy's Theorem, and note that existence of these solutions has been established. Independently of the geometry and of parameters, this shows that the presence of fluid prevents precession of the body in the limit. In general, we cannot predict which axis will be attained, but we can show stability of the largest axis and provide criteria on the initial data which are decisive in special cases.

*Appeared in*

- Archive for Rational Mechanics and Analysis, 221 (2016) pp. 487--526 under the title ``Inertial motions of a rigid body with a cavity filled with a viscous liquid''

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# Robust homoclinic orbits in planar systems with Preisach hysteresis operator

*Authors*

- Pimenov, Alexander
- Rachinskii, Dmitrii

*2010 Mathematics Subject Classification*

- 47J40 92D25 37L15

*Keywords*

- Robust homoclinic orbit, Operator-differential equations, predator-prey model

*DOI*

*Abstract*

We construct examples of robust homoclinic orbits for systems of ordinary differential equations coupled with the Preisach hysteresis operator. Existence of such orbits is demonstrated for the first time. We discuss a generic mechanism that creates robust homoclinic orbits and a method for finding them. An example of a homoclinic orbit in a population dynamics model with hysteretic response of the prey to variations of the predator is studied numerically

*Appeared in*

- J. Phys. Conf. Ser., 727 (2016) pp. 012012/1--012012/15, DOI 10.1088/1742-6596/727/1/012012 .

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# Impact of spatial inhomogeneities on on-axis pulse reconstruction in femtosecond filaments

*Authors*

- Brée, Carsten
- Kretschmar, Martin
- Nagy, Tamas
- Kurz, Heiko G.
- Morgner, Uwe
- Kovačev, Milutin

*2010 Mathematics Subject Classification*

- 78A60

*2008 Physics and Astronomy Classification Scheme*

- 42.65.-k 42.65.Jx 42.65.Re 52.38.Hb

*Keywords*

- pulse self-compression, femtosecond filamentation, nonlinear optics

*DOI*

*Abstract*

We demonstrate a strong influence of the spatial beam profile on the vacuum-propagated on-axis pulse shapes for a femtosecond filament in argon. The effects can be minimized by transmitting the filament into the far-field by a laser-drilled pinhole setup. Using this method, we can monitor the pulse compression dynamics along the entire longitudinal extension of the filament, including the ionization-induced plasma channel.

*Appeared in*

- J. Phys. B, 48 (2015) pp. 094002/1--094002/6.

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# Second-order subdifferential of 1- and maximum norm

*Authors*

- Emich, Konstantin

*2010 Mathematics Subject Classification*

- 49J52 49J53

*Keywords*

- second-order subdifferential, polyhedral norms

*DOI*

*Abstract*

We derive formulae for the second-order subdifferential of polyhedral norms. These formulae are fully explicit in terms of initial data. In a first step we rely on the explicit formula for the coderivative of normal cone mapping to polyhedra. Though being explicit, this formula is quite involved and difficult to apply. Therefore, we derive simple formulae for the 1-norm and -- making use of a recently obtained formula for the second-order subdifferential of the maximum function -- for the maximum norm.

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# Error estimates for nonlinear reaction-diffusion systems involving different diffusion length scales

*Authors*

- Reichelt, Sina

*2010 Mathematics Subject Classification*

- 35B25 35K57 35K65 35M10 41M25

*Keywords*

- Two-scale convergence, folding and unfolding, error estimates, nonlinear reaction, degenerating diffusion, Gronwall estimate

*DOI*

*Abstract*

We derive quantitative error estimates for coupled reaction-diffusion systems, whose coefficient functions are quasi-periodically oscillating modeling microstructure of the underlying macroscopic domain. The coupling arises via nonlinear reaction terms, and we allow for different diffusion length scales, i.e. whereas some species have characteristic diffusion length of order 1, other species may diffuse much slower, namely, with order of the characteristic microstructure-length scale. We consider an effective system, which is rigorously obtained via two-scale convergence, and we derive quantitative error estimates.

*Appeared in*

- MURPHYS-HSFS-2014: 7th MUlti-Rate Processes and HYSteresis (MURPHYS) & 2nd International Workshop on Hysteresis and Slow-Fast Systems (HSFS), O. Klein, M. Dimian, P. Gurevich, D. Knees, D. Rachinskii, S. Tikhomirov, eds., vol. 727 of Journal of Physics: Conference Series, IOP Publishing, 2016, pp. 012013/1--012013/15

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# Computations of quasiconvex hulls of isotropic sets

*Authors*

- Heinz, Sebastian
- Kružik, Martin

*2010 Mathematics Subject Classification*

- 49A50 26B25 52A30

*Keywords*

- Quasiconvexity, relaxation, L
^{∞}variational problems

*DOI*

*Abstract*

We design an algorithm for computations of quasiconvex hulls of isotropic compact sets in in the space of 2x2 real matrices. Our approach uses a recent result by the first author [Adv. Calc. Var. (2014), DOI: 10.1515acv-2012-0008] on quasiconvex hulls of isotropic compact sets in the space of 2x2 real matrices. We show that our algorithm has the time complexity of O(N log N ) where N is the number of orbits of the set. We show some applications of our results to relaxation of L^{∞} variational problems.

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# Hardy's inequality for functions vanishing on a part of the boundary

*Authors*

- Egert, Moritz
- Haller-Dintelmann, Robert
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 26D10 26D15 42B20 42B37

*Keywords*

- Hardy's inequality, uniform fatness, Poincaré's inequality, Sobolev extension operator

*DOI*

*Abstract*

We develop a geometric framework for Hardy's inequality on a bounded domain when the functions do vanish only on a closed portion of the boundary.

*Appeared in*

- Potential Anal., 43 (2015) pp. 49--78.

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# Deriving amplitude equations via evolutionary Gamma convergence

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 35Q56 76E30 35K55 35B35 47H20

*Keywords*

- Ginzburg-Landau equation, Swift-Hohenberg equation, gradient systems, Gamma convergence, evolutionary variational inequality

*DOI*

*Abstract*

We discuss the justification of the Ginzburg-Landau equation with real coefficients as an amplitude equation for the weakly unstable one-dimensional Swift-Hohenberg equation. In contrast to classical justification approaches we employ the method of evolutionary Gamma convergence by reformulating both equations as gradient systems. Using a suitable linear transformation we show Gamma convergence of the associated energies in suitable function spaces. The limit passage of the time-dependent problem relies on the recent theory of evolutionary variational inequalities for families of uniformly convex functionals as developed by Daneri and Savaré 2010. In the case of a cubic energy it suffices that the initial conditions converge strongly in L^{2}, while for the case of a quadratic nonlinearity we need to impose weak convergence in H^{1}. However, we do not need wellpreparedness of the initial conditions.

*Appeared in*

- Discrete Contin. Dyn. Syst., 35 (2015) pp. 2679--2700.

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# On a non-isothermal diffuse interface model for two-phase flows of incompressible fluids

*Authors*

- Eleuteri, Michela
- Rocca, Elisabetta
- Schimperna, Giulio

*2010 Mathematics Subject Classification*

- 35Q35 35K25 76D05 35D30

*Keywords*

- Cahn-Hilliard, Navier-Stokes, incompressible, non-isothermal binary fluid, global-in-time existence, weak solutions

*DOI*

*Abstract*

We introduce a diffuse interface model describing the evolution of a mixture of two different viscous incompressible fluids of equal density. The main novelty of the present contribution consists in the fact that the effects of temperature on the flow are taken into account. In the mathematical model, the evolution of the velocity **u** is ruled by the Navier-Stokes system with temperature-dependent viscosity, while the order parameter φ representing the concentration of one of the components of the fluid is assumed to satisfy a convective Cahn-Hilliard equation. The effects of the temperature are prescribed by a suitable form of the heat equation. However, due to quadratic forcing terms, this equation is replaced, in the weak formulation, by an equality representing energy conservation complemented with a differential inequality describing production of entropy. The main advantage of introducing this notion of solution is that, while the thermodynamical consistency is preserved, at the same time the energy-entropy formulation is more tractable mathematically. Indeed, global-in-time existence for the initial-boundary value problem associated to the weak formulation of the model is proved by deriving suitable a-priori estimates and showing weak sequential stability of families of approximating solutions.

*Appeared in*

- Discrete Contin. Dyn. Syst., 35 (2015) pp. 2497--2522.

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# Affine LIBOR models with multiple curves: Theory, examples and calibration

*Authors*

- Grbac, Zorana
- Papapantoleon, Antonis
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266 - Skovmand, David

*2010 Mathematics Subject Classification*

- 91G30 91G20 60G44

*Keywords*

- Multiple curve models, LIBOR, OIS, basis spread, affine LIBOR models caps, swaptions, basis swaptions, calibration

*DOI*

*Abstract*

We introduce a multiple curve LIBOR framework that combines tractable dynamics and semi-analytic pricing formulas with positive interest rates and basis spreads. The dynamics of OIS and LIBOR rates are specified following the methodology of the affine LIBOR models and are driven by the wide and flexible class of affine processes. The affine property is preserved under forward measures, which allows to derive Fourier pricing formulas for caps, swaptions and basis swaptions. A model specification with dependent LIBOR rates is developed, that allows for an efficient and accurate calibration to a system of caplet prices.

*Appeared in*

- SIAM Journal on Financial Mathematics, 6 (2015), pp. 984--1025

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# Nonlocal problems with Neumann boundary conditions

*Authors*

- Dipierro, Serena
- Ros-Oton, Xavier
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35R11 60G22

*Keywords*

- Nonlocal operators, fractional Laplacian, Neumann problem

*DOI*

*Abstract*

We introduce a new Neumann problem for the fractional Laplacian arising from a simple probabilistic consideration, and we discuss the basic properties of this model. We can consider both elliptic and parabolic equations in any domain. In addition,we formulate problems with nonhomogeneous Neumann conditions, and also with mixed Dirichlet and Neumann conditions, all of them having a clear probabilistic interpretation. We prove that solutions to the fractional heat equation with homogeneous Neumann conditions have the following natural properties: conservation of mass, decreasing energy, and convergence to a constant as time flows. Moreover, for the elliptic case we give the variational formulation of the problem, and establish existence of solutions. We also study the limit properties and the boundary behavior induced by this nonlocal Neumann condition.

*Appeared in*

- Rev. Mat. Iberoam., 33 (2017) pp. 377-416.

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# Delayed feedback control of the self-induced motion of localized structures of light

*Authors*

- Vladimirov, Andrei G.
- Pimenov, Alexander
- Gurevich, Svetlana V.
- Panajotov, Krassimir
- Averlant, Eugene
- Tlidi, Mustapha

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px 42.60.Mi 05.45.-a 02.30.Ks

*Keywords*

- semiconductor lasers, cavity solitons, delayed feedback, bifurcations, drift iinstability

*DOI*

*Abstract*

We investigate a control of the motion of localized structures of light by means of delay feedback in the transverse section of a broad area nonlinear optical system. The delayed feedback is found to induce a spontaneous motion of a solitary localized structure that is stationary and stable in the absence of feedback. We focus our analysis on an experimentally relevant system namely the Vertical-Cavity Surface-Emitting Laser (VCSEL). In the absence of the delay feedback we present experimental evidence of stationary localized structures in a 80 ?m aperture VCSEL. The spontaneous formation of localized structures takes place above the lasing threshold and under optical injection. Then, we consider the effect of the time-delayed optical feedback and investigate analytically the role of the phase of the feedback and the carrier lifetime on the self-mobility properties of the localized structures. We show that these two parameters affect strongly the space time dynamics of two-dimensional localized structures. We derive an analytical formula for the threshold associated with drift instability of localized structures and a normal form equation describing the slow time evolution of the speed of the moving structure.

*Appeared in*

- Phil. Trans. R. Soc. A, 372 (2014) pp. 20140013.

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# Local estimation of the noise level in MRI using structural adaptation

*Authors*

- Tabelow, Karsten

ORCID: 0000-0003-1274-9951 - Voss, Henning U.
- Polzehl, Jörg

ORCID: 0000-0001-7471-2658

*2010 Mathematics Subject Classification*

- 62G05 62P10

*Keywords*

- Magnetic Resonance Imaging, Noise estimation, Maximum Likelihood estimation

*DOI*

*Abstract*

We present a method for local estimation of the signal-dependent noise level in magnetic resonance images. The procedure uses a multi-scale approach to adaptively infer on local neighborhoods with similar data distribution. It exploits a maximum-likelihood estimator for the local noise level. The validity of the method was evaluated on repeated diffusion data of a phantom and simulated data using T1-data corrupted with artificial noise. Simulation results are compared with a recently proposed estimate. The method was applied to a high-resolution diffusion dataset to obtain improved diffusion model estimation results and to demonstrate its usefulness in methods for enhancing diffusion data.

*Appeared in*

- Med. Image Anal., 20 (2015) pp. 76--86.

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# Rate-independent damage in thermo-viscoelastic materials with inertia

*Authors*

- Lazzaroni, Giuliano
- Rossi, Riccarda
- Thomas, Marita
- Toader, Rodica

*2010 Mathematics Subject Classification*

- 35Q74 74H20 74R05 74C05 74F05

*Keywords*

- Partial damage, rate-independent systems, elastodynamics, phase-field models, heat equation, energetic solutions, local solutions

*DOI*

*Abstract*

We present a model for rate-independent, unidirectional, partial damage in visco-elastic materials with inertia and thermal effects. The damage process is modeled by means of an internal variable, governed by a rate-independent flow rule. The heat equation and the momentum balance for the displacements are coupled in a highly nonlinear way. Our assumptions on the corresponding energy functional also comprise the case of the Ambrosio-Tortorelli phase-field model (without passage to the brittle limit). We discuss a suitable weak formulation and prove an existence theorem obtained with the aid of a (partially) decoupled time-discrete scheme and variational convergence methods. We also carry out the asymptotic analysis for vanishing viscosity and inertia and obtain a fully rate-independent limit model for displacements and damage, which is independent of temperature.

*Appeared in*

- J. Dynam. Differential Equations, 30 (2018), pp. 1311--1364, DOI 10.1007/s10884-018-9666-y .

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# Numerics of thin-film free boundary problems for partial wetting

*Authors*

- Peschka, Dirk

ORCID: 0000-0002-3047-1140

*2010 Mathematics Subject Classification*

- 76A20 35R35 76M10

*Keywords*

- thin films, free boundary problem, numerics

*DOI*

*Abstract*

We present a novel framework to solve thin-film equations with an explicit non-zero contact angle, where the support of the solution is treated as an unknown. The algorithm uses a finite element method based on a gradient formulation of the thin-film equations coupled to an arbitrary Lagrangian-Eulerian method for the motion of the support. Features of this algorithm are its simplicity and robustness. We apply this algorithm in 1D and 2D to problems with surface tension, contact angles and with gravity.

*Appeared in*

- J. Comput. Phys., 295 (2015) pp. 770--778 under the title ``Thin-film free boundary problems for partial wetting''

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# A geometric discretization and a simple implementation for variational mesh generation and adaptation

*Authors*

- Huang, Weizhang
- Kamenski, Lennard

*2010 Mathematics Subject Classification*

- 65N50 65K10

*Keywords*

- variational mesh generation, mesh adaptation, moving mesh

*DOI*

*Abstract*

We present a simple direct discretization for functionals used in the variational mesh generation and adaptation. Meshing functionals are discretized on simplicial meshes and the Jacobian matrix of the continuous coordinate transformation is approximated by the Jacobian matrices of affine mappings between elements. The advantage of this direct geometric discretization is that it preserves the basic geometric structure of the continuous functional, which is useful in preventing strong decoupling or loss of integral constraints satisfied by the functional. Moreover, the discretized functional is a function of the coordinates of mesh vertices and its derivatives have a simple analytical form, which allows a simple implementation of variational mesh generation and adaptation on computer. Since the variational mesh adaptation is the base for a number of adaptive moving mesh and mesh smoothing methods, the result in this work can be used to develop simple implementations of those methods. Numerical examples are given.

*Appeared in*

- J. Comput. Phys., 301 (2015) pp. 322--337.

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# Self-concordant profile empirical likelihood ratio tests for the population correlation coefficient: A simulation study

*Authors*

- Dickhaus, Thorsten

*2010 Mathematics Subject Classification*

- 62F03 62F25 62G15 62G99

*Keywords*

- Confidence region, elliptical model, Lagrange multiplier, nonparametric tests, type I error rate, Wilks phenomenon

*DOI*

*Abstract*

We present results of a simulation study regarding the finite-sample type I error behavior of the self-concordant profile empirical likelihood ratio (ELR) test for the population correlation coefficient. Three different families of bivariate elliptical distributions are taken into account. Uniformly over all considered models and parameter configurations, the self-concordant profile ELR test does not keep the significance level for finite sample sizes, albeit the level exceedance monotonously decreases to zero as the sample size increases. We discuss some potential modifications to address this problem.

*Appeared in*

- Springer Proceedings in Mathematics & Statistics, Springer Berlin/Heidelberg, 2015, pp. 253--260

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# Higher order continuous Galerkin--Petrov time stepping schemes for transient convection-diffusion-reaction equations

*Authors*

- Ahmed, Naveed

ORCID: 0000-0002-9322-0373 - Matties, Gunar

*2010 Mathematics Subject Classification*

- 65M12 65M15 65M60

*Keywords*

- transient convection-diffusion-reaction problem, local projection stabilization, continuous Galerkin--Petrov method, discontinuous Galerkin method

*DOI*

*Abstract*

We present the analysis for the higher order continuous Galerkin--Petrov (cGP) time discretization schemes in combination with the one-level local projection stabilization in space applied to time-dependent convection-diffusion-reaction problems. Optimal a-priori error estimates will be proved. Numerical studies support the theoretical results. Furthermore, a numerical comparison between continuous Galerkin--Petrov and discontinuous Galerkin time discretization schemes will be given.

*Appeared in*

- ESAIM Math. Model. Numer. Anal., 49 (2015) pp. 1429--1450.

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# The stochastic encounter-mating model

*Authors*

- Gün, Onur
- Yilmaz, Atilla

*2010 Mathematics Subject Classification*

- 92D25 60J28 60G55

*Keywords*

- population dynamics, pair formation, encounter-mating, assortive mating, random mating, panmixia, homogamy, heterogamy, monogamy, mating preferences, mating pattern, contingency table, multiple hypergeometric distribution, simple point processes, Poisson process, Bernoulli process

*DOI*

*Abstract*

We propose a new model of permanent monogamous pair formation in zoological populations comprised of kge 2 types of females and males, which unifies and generalizes the encounter-mating models of Gimelfarb (1988). In our model, animals randomly encounter members of the opposite sex at their so-called firing times to form temporary pairs which then become permanent if mating happens. Given the distributions of the firing times and the mating preferences upon encounter, which depend on the sex and the type of the animals, we analyze the contingency table Q(t) of permanent pair types at any time tge 0. First, we consider definite mating upon encounter and provide a formula for the distribution of Q(t). In particular, at the terminal time T, the so-called mating pattern Q(T) has a multiple hypergeometric distribution. This implies panmixia which means that female and male types are uncorrelated in the expected mating pattern. Next, when the firing times come from Poisson and Bernoulli point processes, we formulate conditions that characterize panmixia. Moreover, when these conditions are satisfied, the underlying parameters of the model can be changed to yield definite mating upon encounter, and our results for the latter case carry over. Finally, when k=2, we fully characterize heterogamy/panmixia/homogamy, i.e., negative/zero/positive correlation of same type females and males in the expected mating pattern. We thereby rigorously prove, strengthen and generalize Gimelfarb's results.

*Appeared in*

- Acta Appl. Math., 148 (2017), pp. 71--102.

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# A phase-field model for solid-state dewetting and its sharp-interface limit

*Authors*

- Dziwnik, Marion
- Münch, Andreas
- Wagner, Barbara

*2010 Mathematics Subject Classification*

- 34E13 74N20 74E10

*Keywords*

- phase-field model, matched asymptotic expansions, sharp interface model, free boundaries, dewetting solid films

*DOI*

*Abstract*

We propose a phase field model for solid state dewetting in form of a Cahn-Hilliard equation with weakly anisotropic surface energy and a degenerate mobility together with a free boundary condition at the film-substrate contact line. We derive the corresponding sharp interface limit via matched asymptotic analysis involving multiple inner layers. The resulting sharp interface model is consistent with the pure surface diffusion model. In addition, we show that the natural boundary conditions, as indicated from the first variation of the total free energy, imply a contact angle condition for the dewetting front, which, in the isotropic case, is consistent with the well-known Young's equation.

*Appeared in*

- Nonlinearity 30 (2017) pp. 1465-1496, changed title: An anisotropic phase-field model...

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# A quenched functional central limit theorem for random walks in random environments under $(T)_gamma$

*Authors*

- Bouchet, Élodie
- Sabot, Christophe
- Soares dos Santos, Renato

*2010 Mathematics Subject Classification*

- 60F05 60G52

*Keywords*

- random walk in random environment, quenched central limit theorem, ballisticity condition

*DOI*

*Abstract*

We prove a quenched central limit theorem for random walks in i.i.d. weakly elliptic random environments in the ballistic regime. Such theorems have been proved recently under the assumption of large finite moments for the regeneration times. In this paper, we relax these moment assumptions under Sznitman's (T)_{γ} ballisticity condition, which allows the inclusion of new non-uniformly elliptic examples such as Dirichlet random environments.

*Appeared in*

- Stochastic Process. Appl., 126 (2016) pp. 1206--1225.

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# Uniform estimate of the relative free energy by the dissipation rate for finite volume discretized reaction-diffusion systems

*Authors*

- Fiebach, André
- Glitzky, Annegret

*2010 Mathematics Subject Classification*

- 65M08 35B40 35K57 35R05 46E39

*Keywords*

- Admissible finite volume mesh, reaction-diffusion system, free energy, discrete Poincare and, Sobolev-Poincare inequality

*DOI*

*Abstract*

We prove a uniform Poincare-like estimate of the relative free energy by the dissipation rate for implicit Euler, finite volume discretized reaction-diffusion systems. This result is proven indirectly and ensures the exponential decay of the relative free energy with a unified decay rate for admissible finite volume meshes.

*Appeared in*

- Finite Volumes for Complex Applications VII -- Methods and Theoretical Aspects -- FVCA 7, Berlin, June 2014, J. Fuhrmann, M. Ohlberger, Ch. Rohde, eds., vol. 77 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham et al., 2014, pp. 275--283

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# On the Simes inequality in elliptical models

*Authors*

- Bodnar, Taras
- Dickhaus, Thorsten

*2010 Mathematics Subject Classification*

- 60E15 62J15 62F03

*Keywords*

- Covariance matrix, distributional transform, multiple testing, multivariate normal distribution, p-value, Student's t, total positivity

*DOI*

*Abstract*

We provide necessary and sufficient conditions for the validity of the inequality of Simes (1986) in models with elliptical dependencies. Necessary conditions are presented in terms of sufficient conditions for the reverse Simes inequality. One application of our main results concerns the problem of model misspecification, in particular the case that the assumption of Gaussianity of test statistics is violated. Since our sufficient conditions require non-negativity of correlation coefficients between test statistics, we also develop exact tests for vectors of correlation coefficients.

*Appeared in*

- Ann. Inst. Statist. Math., (published online on Sept. 4, 2015) pp. 1--16, DOI 10.1007/s10463-015-0539-4 .

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# All functions are locally s-harmonic up to a small error

*Authors*

- Dipierro, Serena
- Savin, Ovidiu
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35R11 60G22 35A35 34A08

*Keywords*

- Density properties, approximation, s-harmonic functions

*DOI*

*Abstract*

We show that we can approximate locally every function with a fractional harmonic function in that vanishes outside a compact set. This result is clearly in contrast with the rigidity of harmonic functions in the classical case and can be viewed as a purely nonlocal feature.

*Appeared in*

- J. Eur. Math. Soc. (JEMS), 19 (2017) pp. 957-966.

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# Simulations and analysis of beam quality improvement in spatially modulated broad area edge-emitting devices

*Authors*

- Radziunas, Mindaugas
- Herrero, Ramon
- Botey, Muriel
- Staliunas, Kestutis

*2010 Mathematics Subject Classification*

- 35Q60 35B27 37M05 78A60 78A45

*2008 Physics and Astronomy Classification Scheme*

- 42.60.By 42.60.Da 42.60.Fc 42.60.Jf

*Keywords*

- semiconductor amplifier, edge emitting lasers, semiconductors, periodic structure, anisotrophy, beam shaping, spatial filtering, beam quality

*DOI*

*Abstract*

We simulate and analyze how beam quality improves while being amplified in edge emitting broad area semiconductor amplifiers with a periodic structuring of the electrical contacts, in both longitudinal and lateral directions. A spatio-temporal traveling wave model is used for simulations of the dynamics and nonlinear interactions of the optical fields, induced polarizations and carrier density. In the case of small beam amplification, the optical field can be expanded into few Bloch modes, so that the system is described by a set of ODEs for the evolution of the mode amplitudes. The analysis of such model provides a deep understanding of the impact of the different parameters on amplification and on spatial (angular) filtering of the beam. It is shown that under realistic parameters the twodimensional modulation of the current can lead not only to a significant reduction of the emission divergence, but also to an additional amplification of the emitted field.

*Appeared in*

- Proceedings SPIE Series, Semiconductor Lasers and Laser Dynamics VI, K. Panajatov, M. Sciamanna, A. Valle, R. Michalzik, eds., vol. 9134, Brussels, 2014

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# On asymptotic isotropy for a hydrodynamic model of liquid crystals

*Authors*

- Dai, Mimi
- Feireisl, Eduard
- Rocca, Elisabetta
- Schimperna, Giulio
- Schonbek, Maria E.

*2010 Mathematics Subject Classification*

- 76A15 74H40 35Q30

*Keywords*

- Liquid crystal, Q?tensor description, long-time behavior, Fourier splitting

*DOI*

*Abstract*

We study a PDE system describing the motion of liquid crystals by means of the Q?tensor description for the crystals coupled with the incompressible Navier-Stokes system. Using the method of Fourier splitting, we show that solutions of the system tend to the isotropic state at the rate (1 + t)?? as t ? ? 1 for a certain ? > 2 .

*Appeared in*

- Asymptot. Anal., 97 (2016) pp. 189--210.

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# Optimal distributed control of a nonlocal Cahn--Hilliard/Navier--Stokes system in 2D

*Authors*

- Frigeri, Sergio Pietro
- Rocca, Elisabetta
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 49J20 49J50 35R09 45K05 74N99

*Keywords*

- Distributed optimal control, first-order necessary optimality conditions, nonlocal models, integrodifferential equations, Navier-Stokes system, Cahn-Hilliard equation, phase separation

*DOI*

*Abstract*

We study a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids coupling the Navier-Stokes system with a convective nonlocal Cahn-Hilliard equation in two dimensions of space. We apply recently proved well-posedness and regularity results in order to establish existence of optimal controls as well as first-order necessary optimality conditions for an associated optimal control problem in which a distributed control is applied to the fluid flow.

*Appeared in*

- SIAM J. Control Optim., 54 (2016), pp. 221 -- 250.

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# Crystal dislocations with different orientations and collisions

*Authors*

- Patrizi, Stefania
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 82D25 35R09 74E15 35R11 47G20

*Keywords*

- Peierls-Nabarro model, nonlocal integro-differential equations, dislocation dynamics, attractive/repulsive potentials, collisions

*DOI*

*Abstract*

We study a parabolic differential equation whose solution represents the atom dislocation in a crystal for a general type of Peierls-Nabarro model with possibly long range interactions and an external stress. Differently from the previous literature, we treat here the case in which such dislocation is not the superpositions of transitions all occurring with the same orientations (i.e. opposite orientations are allowed as well). We show that, at a long time scale, and at a macroscopic space scale, the dislocations have the tendency to concentrate as pure jumps at points which evolve in time, driven by the external stress and by a singular potential. Due to differences in the dislocation orientations, these points may collide in finite time.

*Appeared in*

- Arch. Ration. Mech. Anal., 217 (2015) pp. 231--261.

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# 1D symmetry for semilinear PDEs from the limit interface of the solution

*Authors*

- Farina, Alberto
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 5J61 35J15

*Keywords*

- Phase transitions, symmetry results, limit interface

*DOI*

*Abstract*

We study bounded, entire, monotone solutions of the Allen-Cahn equation. We prove that under suitable assumptions on the limit interface and on the energy growth, the solution is 1D. In particular, differently from the previous literature, the solution is not assumed to have minimal properties. We think that this approach could be fruitful in concrete situations, where one can observe the phase separation at a large scale and whishes to deduce the values of the state parameter in the vicinity of the interface. As a simple example of the results obtained with this point of view, we mention that monotone solutions with energy bounds, whose limit interface does not contain a vertical line through the origin, are 1D, at least up to dimension 4.

*Appeared in*

- Comm. Partial Differential Equations, 41 (2016) pp. 665--682.

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# Timing jitter in passively mode-locked semiconductor lasers

*Authors*

- Pimenov, Alexander
- Rachinskii, Dmitrii
- Habruseva, Tatiana
- Hegarty, Stephen P.
- Guillaume, Huyet
- Vladimirov, Andrei G.

*2010 Mathematics Subject Classification*

- 37N20 65P30 78A60

*2008 Physics and Astronomy Classification Scheme*

- 05.45.-a 42.55.Px 42.60.Fc 42.60.Mi 42.65.Pc

*Keywords*

- Timing jitter, numerical bifurcation analysis, semiconductor lasers, Bistability

*DOI*

*Abstract*

We study the effect of noise on the dynamics of passively mode-locked semiconductor lasers both experimentally and theoretically. A method combining analytical and numeri- cal approaches for estimation of pulse timing jitter is proposed. We investigate how the presence of dynamical features such as wavelength bistability affects timing jitter.

*Appeared in*

- Opt. Lett., 39 (2014), pp. 6815--6818, new title "Effect of dynamical instability on timing jitter in passively mode-locked quantum-dot lasers''.

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# Point contacts and boundary triples

*Authors*

- Lotoreichik, Vladimir
- Neidhardt, Hagen
- Popov, Igor Yu.

*2010 Mathematics Subject Classification*

- 47A55 47B25 81Q10 81Q15

*Keywords*

- Boundary triples, Weyl function, point contacts, weak coupling, perturbation series

*DOI*

*Abstract*

We suggest an abstract approach for point contact problems in the framework of boundary triples. Using this approach we obtain the perturbation series for a simple eigenvalue in the discrete spectrum of the model self-adjoint extension with weak point coupling.

*Appeared in*

- Mathematical Results in Quantum Mechanics. Proceedings of the QMath12 Conference, P. Exner, W. König, H. Neidhardt, eds., World Scientific Publishing, Singapore, 2015, pp. 283--293

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# Longitudinal modes of multisection ring and edge-emitting semiconductor lasers

*Authors*

- Radziunas, Mindaugas

*2010 Mathematics Subject Classification*

- 78-04 78A60 35-04 78-05

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px 42.60.-v 02.30.Jr 02.60.Cb

*Keywords*

- traveling wave mode, semiconductor laser, optical mode, mode analysis

*DOI*

*Abstract*

We use the traveling wave model for simulating and analyzing nonlinear dynamics of multisection ring and edge-emitting semiconductor laser devices. We introduce the concept of instantaneous longitudinal optical modes and present an algorithm for their computation. A semiconductor ring laser was considered to illustrate the advantages of the mode analysis.

*Appeared in*

- Opt. Quantum Electron., 47 (2015) pp. 1319--1325.

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