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Mean-field games (MFGs) and the best-reply strategy (BRS) are two methods of describing competitive optimisation of systems of interacting agents. The latter can be interpreted as an approximation of the respective MFG system. In this paper, we present an analysis and comparison of the two approaches in the stationary case. We provide novel existence and uniqueness results for the stationary boundary value problems related to the MFG and BRS formulations, and we present an analytical and numerical comparison of the two paradigms in some specific modelling situations. [ABSTRACT FROM AUTHOR]

In this paper we study the existence and the analytic dependence upon domain perturbation of the solutions of a nonlinear nonautonomous transmission problem for the Laplace equation. The problem is defined in a pair of sets consisting of a perforated domain and an inclusion whose shape is determined by a suitable diffeomorphism $\phi$. First we analyse the case in which the inclusion is a fixed domain. Then we will perturb the inclusion and study the arising boundary value problem and the dependence of a specific family of solutions upon the perturbation parameter $\phi$. [ABSTRACT FROM AUTHOR]

This paper examines thermoacoustic effects on the propagation of non-planar sound in a circular duct subjected to an axial temperature gradient. Of particular concern are thermoviscous diffusive effects, which are taken into account by the boundary-layer approximation in a framework of the linear theory. For disturbances expanded into Fourier and Fourier–Bessel series in the azimuthal and radial directions, respectively, the pressure in each mode is described by a one-dimensional, dispersive wave equation, if non-diffusive propagation is assumed. When the diffusive effects are included, each radial mode is coupled to the other radial modes through the boundary layer. Focusing on a single azimuthal and radial mode only, the dispersion relation for the propagation along an infinite duct of a uniform gas is first derived. Effects of the temperature gradient are then examined by solving boundary-value problems for a duct of finite length in four typical cases. Assuming that the wall temperature increases exponentially along the duct, eigenfrequencies and decay rates in the lowest axial mode are obtained as well as axial distributions of the sound pressure and the axial velocity in the duct. The frequency and the decay rate increase as the temperature ratio at both ends becomes higher. It is found from the acoustic energy equation that the dispersion combined with the diffusion acts to reduce the damping and that the temperature gradient makes little contribution to the production of the energy. However, it is unveiled that the non-uniformity in temperature yields thermoacoustic sound confinement in the vicinity of the cold end. [ABSTRACT FROM AUTHOR]

This paper is concerned with rapid-distortion theory on transversely sheared mean flows which (among other things) can be used to analyse the unsteady motion resulting from the interaction of a turbulent shear flow with a solid surface. It extends previous analyses of Goldstein et al. (J. Fluid Mech., vol. 736, 2013a, pp. 532-569; NASA/TM-2013-217862, 2013b) which showed that the unsteady motion is completely determined by specifying two arbitrary convected quantities. The present paper uses a pair of previously derived conservation laws to derive upstream boundary conditions that relate these quantities to experimentally measurable flow variables. The result is dependent on the imposition of causality on an intermediate variable that appears in the conservation laws. Goldstein et al. (2013a) related the convected quantities to the physical flow variables at the location of the interaction, but the results were not generic and hard to reconcile with experiment. That problem does not occur in the present formulation, which leads to a much simpler and more natural result than the one given in Goldstein et al. (2013a). We also show that the present formalism yields better predictions of the sound radiation produced by the interaction of a two-dimensional jet with the downstream edge of a flat plate than the Goldstein et al. (2013a) result. The role of causality is also discussed. [ABSTRACT FROM AUTHOR]

This paper is concerned with the problem of existence and uniqueness of weak and classical solutions for a fourth-order semilinear boundary value problem. The existence and uniqueness for weak solutions follows from standard variational methods, while similar uniqueness results for classical solutions are derived using maximum principles. [ABSTRACT FROM AUTHOR]

This article is the third one in a series of papers by the authors on vanishing-viscosity solutions to rate-independent damage systems. While in the first two papers (Knees, D. et al. 2013 Math. Models Methods Appl. Sci. 23 (4), 565–616; Knees, D. et al. 2015 Nonlinear Anal. Real World Appl. 24 , 126–162) the assumptions on the spatial domain Ω were kept as general as possible (i.e., non-smooth domain with mixed boundary conditions), we assume here that ∂Ω is smooth and that the type of boundary conditions does not change. This smoother setting allows us to derive enhanced regularity spatial properties both for the displacement and damage fields. Thus, we are in a position to work with a stronger solution notion at the level of the viscous approximating system. The vanishing-viscosity analysis then leads us to obtain the existence of a stronger solution concept for the rate-independent limit system. Furthermore, in comparison to [18, 19], in our vanishing-viscosity analysis we do not switch to an artificial arc-length parameterization of the trajectories but we stay with the true physical time. The resulting concept of Balanced Viscosity solution to the rate-independent damage system thus encodes a more explicit characterization of the system behaviour at time discontinuities of the solution. [ABSTRACT FROM AUTHOR]

The rapid development of new infrastructure programmes requires an accelerated deployment of new materials in new environments. Materials 4.0 is crucial to achieve these goals. The application of digital to the field of materials has been at the forefront of research for many years, but there does not exist a unified means to describe a framework for this area creating pockets of development. This is confounded by the broader expectations of a digital twin (DT) as the possible answer to all these problems. The issue being that there is no accepted definition of a component DT, and what information it should contain and how it can be implemented across the product lifecycle exist. Within this position paper, a clear distinction is made between the "manufacturing DT" and the "component DT"; the former being the starting boundary conditions of the latter. In order to achieve this, we also discuss the introduction of a digital thread as a key concept in passing data through manufacturing and into service. The stages of how to define a framework around the development of DTs from a materials perspective is given, which acknowledges the difference between creating new understanding within academia and the application of this knowledge on a per-component basis in industry. A number of challenges are identified to the broad application of a component DT; all lead to uncertainty in properties and locations, resolving these requires judgments to be made in the provision of safety-dependent materials property data. [ABSTRACT FROM AUTHOR]

This paper deals with the logistic Keller–Segel model \[ \begin{cases} u_t = \Delta u - \chi \nabla\cdot(u\nabla v) + \kappa u - \mu u^2, \\ v_t = \Delta v - v + u \end{cases} \] in bounded two-dimensional domains (with homogeneous Neumann boundary conditions and for parameters χ, κ ∈ ℝ and μ > 0), and shows that any nonnegative initial data (u0, v0) ∈ L1 × W1,2 lead to global solutions that are smooth in $\bar {\Omega }\times (0,\infty)$. [ABSTRACT FROM AUTHOR]

This paper is about the connection between certain Banach-algebraic properties of a commutative Banach algebra E with unit and the associated commutative Banach algebra C(X, E) of all continuous functions from a compact Hausdorff space X into E. The properties concern Ditkin's condition and bounded relative units. We show that these properties are shared by E and C(X, E). We also consider the relationship between these properties in the algebras E, B and $\~{B}$ that appear in the so-called admissible quadruples (X, E, B, $\~{B}$). [ABSTRACT FROM PUBLISHER]

This paper is mainly concerned with the global asymptotic behaviour of the unique solution to a class of singular Dirichlet problems − Δu = b(x)g(u), u > 0, x ∈ Ω, u|∂Ω = 0, where Ω is a bounded smooth domain in ℝn, g ∈ C1(0, ∞) is positive and decreasing in (0, ∞) with $\lim _{s\rightarrow 0^+}g(s)=\infty$ , b ∈ Cα(Ω) for some α ∈ (0, 1), which is positive in Ω, but may vanish or blow up on the boundary properly. Moreover, we reveal the asymptotic behaviour of such a solution when the parameters on b tend to the corresponding critical values. [ABSTRACT FROM AUTHOR]

This paper analyses a canonical class of one-dimensional superlinear indefinite boundary value problems of great interest in population dynamics under non-homogeneous boundary conditions; the main bifurcation parameter in our analysis is the amplitude of the superlinear term. Essentially, it continues the analysis of López-Gómez et al. (López-Gómez, J., Tellini, A. & Zanolin, F. (2014) High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems. Comm. Pure Appl. Anal. 13(1), 1–73) with empty overlapping, by computing the bifurcation diagrams of positive steady states of the model and by proving analytically a number of significant features, which have been observed from the numerical experiments carried out here. The numerics of this paper, besides being very challenging from the mathematical point of view, are imperative from the point of view of population dynamics, in order to ascertain the dimensions of the unstable manifolds of the multiple equilibria of the problem, which measure their degree of instability. From that point of view, our results establish that under facilitative effects in competitive media, the harsher the environmental conditions, the richer the dynamics of the species, in the sense discussed in Section 1. [ABSTRACT FROM PUBLISHER]

In the paper, Bifurcation analysis of the twist-Fréedericksz transition in a nematic liquid-crystal cell with pre-twist boundary conditions (2009 Eur. J. Appl. Math.20, 269–287) by da Costa et al. the twist-Fréedericksz transition in a nematic liquid-crystal one-dimensional cell of lenght L was studied, imposing an antisymmetric net twist Dirichlet condition at the cell boundaries. In the present paper, we extend that study to the more general case of net twist Dirichlet conditions without any kind of symmetry restrictions. We use phase-plane analysis tools and appropriately defined time maps to obtain the bifurcation diagrams of the model when L is the bifurcation parameter, and related these diagrams with the one in the antisymmetric situation. The stability of the bifurcating solutions is investigated by applying the method of Maginu (1978 J. Math. Anal. Appl.63, 224–243). [ABSTRACT FROM AUTHOR]

This paper investigates streaks and Görtler vortices in a boundary layer over a flat or concave wall in a contracting or expanding stream, which provides a favourable or adverse pressure gradient, respectively. We consider first the excitation of streaks and Görtler vortices by free stream vortical disturbances (FSVD), and their nonlinear evolution. The focus is on FSVD with sufficiently long wavelength, to which the boundary layer is most receptive. The formulation is directed at the general case where the Görtler number GΛ (based on the spanwise length scale Λ of FSVD) is of order one, and the FSVD is strong enough that the induced vortices acquire an O(1) streamwise velocity in the region where the boundary layer thickness becomes comparable with Λ, and the vortices are governed by the nonlinear boundary region equations (NBRE). An important effect of a pressure gradient is that the oncoming FSVD are distorted by the non-uniform inviscid flow outside the boundary layer through convection and stretching. This process is accounted for by using the rapid distortion theory. The impact of the distorting FSVD is analysed to provide the appropriate initial and boundary conditions, which form, along with the NBRE, the appropriate initial boundary value problem describing the excitation and nonlinear evolution of the vortices. Numerical results show that an adverse/favourable pressure gradient cause the Görtler vortices to saturate earlier/later, but at a lower/higher amplitude than that in the case of zero-pressure-gradient. On the other hand, for the same pressure gradient and at low levels of FSVD, the vortices saturate earlier and at a higher amplitude as GΛ increases. Raising FSVD intensity reduces the effects of the pressure gradient and curvature. At a high FSVD level of 14 %, the curvature has no impact on the vortices, while the pressure gradient only influences the saturation intensity. The unsteadiness of FSVD is found to reduce the boundary layer response significantly at FSVD levels, but that effect weakens as the turbulence level increases. A secondary instability analysis of the vortices is performed for moderate adverse and favourable pressure gradients. Three families of unstable modes have been identified, which may become dominant depending on the frequency and streamwise location. In the presence of an adverse pressure gradient, the secondary instability occurs earlier, but the unstable modes appear in a smaller band of frequency, and have smaller growth rates. The opposite is true for a favourable pressure gradient. The present theoretical framework, which accounts for the influence of the curvature, turbulence level and pressure gradient, allows for a detailed and integrated description of the key transition processes, and represents a useful step towards predicting the pretransitional flow and transition itself of the boundary layer over a blade in turbomachinery. [ABSTRACT FROM AUTHOR]

This paper presents a solution to the problem of generating constrained robust trajectory planning for nonlinear mechatronic systems. By using an indirect variational solution method, the necessary optimality conditions deriving from the Pontryagin's minimum principle are imposed, and lead to a differential Two-Point Boundary Value Problem (TPBVP); numerical solution of the latter is accomplished by means of collocation techniques. The robustness to parametric mismatches is obtained trough the use of sensitivity functions, while a hard constraint on actuator effort is obtained using a smoothing technique. Numerical results shows that the robustness can be greatly improved, and that the inclusion of constraints on actuator effort does not affect it. [ABSTRACT FROM AUTHOR]

The Polubarinova–Galin equation describes a parametric conformal map for the zero-surface-tension Hele-Shaw flow driven by injection or suction. In this paper, we classify degree three polynomial solutions to the Polubarinova–Galin equation into three categories: global solutions, solutions which can be continued after blow-up and solutions which cannot be continued after blow-up. The coefficient region of the initial functions in each category is obtained. [ABSTRACT FROM AUTHOR]

In this paper we survey recent developments in the classical theory of minimal surfaces in Euclidean spaces which have been obtained as applications of both classical and modern complex analytic methods; in particular, Oka theory, period dominating holomorphic sprays, gluing methods for holomorphic maps, and the Riemann–Hilbert boundary value problem. Emphasis is on results pertaining to the global theory of minimal surfaces, in particular, the Calabi–Yau problem, constructions of properly immersed and embedded minimal surfaces in $\mathbb{R}^{n}$ and in minimally convex domains of $\mathbb{R}^{n}$ , results on the complex Gauss map, isotopies of conformal minimal immersions, and the analysis of the homotopy type of the space of all conformal minimal immersions from a given open Riemann surface. [ABSTRACT FROM AUTHOR]

For a class of cell division processes in the Euclidean space Rd, spatial consistency is investigated. This addresses the problem whether the distribution of the generated structures, restricted to a bounded set V, depends on the choice of a larger region W D V where the construction of the cell division process is performed. This can also be understood as the problem of boundary effects in the cell division procedure. It is known that the STIT tessellations are spatially consistent. In the present paper it is shown that, within a reasonable wide class of cell division processes, the STIT tessellations are the only ones that are consistent. [ABSTRACT FROM AUTHOR]

When a droplet spreads on a solid substrate, it is unclear what the correct boundary conditions are to impose at the moving contact line. The classical no-slip condition is generally acknowledged to lead to a non-integrable singularity at the moving contact line, which a slip condition, associated with a small slip parameter, λ, serves to alleviate. In this paper, we discuss what occurs as the slip parameter, λ, tends to zero. In particular, we explain how the zero-slip limit should be discussed in consideration of two distinguished limits: one where time is held constant, t = O(1), and one where time tends to infinity at the rate t = O(|logλ|). The crucial result is that in the case where time is held constant, the λ → 0 limit converges to the slip-free equation, and contact line slippage occurs as a regular perturbative effect. However, if λ → 0 and t → ∞ then contact line slippage is a leading-order singular effect. [ABSTRACT FROM AUTHOR]

This paper addresses a free boundary problem for a steady, uniform patch of vorticity surrounding a single flat plate of zero thickness and finite length. Exact solutions to this problem have previously been found in terms of conformal maps represented by Cauchy-type integrals. Here, however, it is demonstrated how, by considering an associated circular-arc polygon and using ideas from automorphic function theory, these maps can be expressed in a simple non-integral form. [ABSTRACT FROM AUTHOR]

This paper is concerned with the small-amplitude unsteady motion of an inviscid non-heat-conducting compressible fluid on a transversely sheared mean flow. It extends previous analyses (Goldstein, J. Fluid Mech., vol. 84, 1978b, pp. 305–329; Goldstein, J. Fluid Mech., vol. 91, 1979a, pp. 601–632), which show that the hydrodynamic component of the motion is determined by two arbitrary convected quantities in the absence of solid surfaces and hydrodynamic instabilities. These results can be used to specify appropriate upstream boundary conditions for unsteady surface interaction problems on transversely sheared mean flows in the same way that the vortical component of the Kovasznay (J. Aero. Sci., vol. 20, 1953, pp. 657–674) decomposition is used to specify these conditions for surface interaction problems on uniform mean flows. But unlike Kovasznay’s result, the arbitrary convected quantities no longer bear a simple relation to the physical variables. A major purpose of this paper is to complete the formalism developed in Goldstein’s earlier two papers by obtaining the necessary relations between these quantities and the measurable flow variables. The results are important because they enable the complete extension of non-homogeneous rapid distortion theory to transversely sheared mean flows. Another purpose of the paper is to derive a generalization of the famous Ffowcs Williams and Hall (J. Fluid Mech., vol. 40, 1970, pp. 657–670) formula for the sound produced by the interaction of turbulence with an edge, which is frequently used as a starting point for predicting sound generation by turbulence–solid surface interactions. We illustrate the utility of this result by using it to calculate the sound radiation produced by the interaction of a two-dimensional jet with the downstream edge of a flat plate. [ABSTRACT FROM PUBLISHER]

This paper gives a modified Myers boundary condition in swirling inviscid flow, which differs from the standard Myers boundary condition by assuming a small but non-zero boundary layer thickness. The new boundary condition is derived and is shown to have the correct quadratic error behaviour with boundary layer thickness and also to agree with previous results when the swirl is set to zero. The boundary condition is initially derived for swirling flow with constant azimuthal velocity, but easily extends to radially varying swirling flow, with terms depending on the boundary layer model. The modified Myers boundary condition is then given in the time domain rather than in the frequency domain. The effect of the boundary layer profile is then considered, and shown to be small compared to the boundary layer thickness. The boundary condition is then used to analyse the eigenmodes and Green's function in a realistic flow. Modelling the thickness of the boundary layer properly is shown to be essential in order to get accurate results. [ABSTRACT FROM AUTHOR]

This paper investigates a family of dam-break problems over an erodible bed. The hydrodynamics is described by the shallow water equations, and the bed change by a sediment-conservation equation, coupled to the hydrodynamics by a sediment transport (bed-load) law. When the initial states Ul and Ur are sufficiently close to each other the resulting solutions are consistent with the theory proposed by Lax (Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, 1973, SIAM), that for a Riemann problem of n equations there are n waves associated with the n characteristic families. However, for wet-dry dam-break problems over a mobile bed, there are three governing equations, but only two waves. One wave vanishes because of the presence of the dry bed. When initial left and right bed levels (Bl and Br) are far apart, it is shown that a semi-characteristic shock may occur, which happens because, unlike in shallow water flow on a fixed bed, the flux function is non-convex. In these circumstances it is shown that it is necessary to reconsider the usual shock conditions. Instead, we propose an implied internal shock structure the concept of which originates from the fact that the stationary shock over a fixed-bed discontinuity can be regarded as a limiting case of flow over a sloping fixed bed. The Needham & Hey (Phil. Trans. R. Soc. Lond. A, vol. 334, 1991, pp. 25-53) approximation for the ambiguous integral term R h dB in the shock condition is improved based on this internal shock structure, such that mathematically valid solutions that incorporate a morphodynamic semi-characteristic shock are arrived at. [ABSTRACT FROM AUTHOR]

When decisions involve opting in or out of competition many decision makers will opt-in even when doing so leads to losses on average. In the current paper, we examine the generality of this effect in risky choices not involving competition. We found that re-framing a sure (certain) zero option as an option to observe the results of the other options without choosing would lead to increased consequential choice (i.e., increased selection of risky options rather than the zero option). Specifically, in two studies we compared the rate of consequential choice in a novel paradigm where decision makers decide to observe or to choose with consequence from a set of risky options (decisions-to-engage) to a full-feedback decisions-from-feedback paradigm where the choice set included a labeled sure zero option. Compared to decisions-from-feedback, participants were more likely to choose from mixed (risky) gambles with consequence (over a zero outcome) in decisions-to-engage. This occurred irrespective of whether doing so was advantageous (i.e., when choice led to monetary gains on average) or disadvantageous (i.e., when choice led to monetary losses on average), and when descriptions of the options outcomes and probabilities were provided (Study 2). These findings provide an important boundary condition for the positive effects of experience on the quality of choice, and suggest that decision makers' preference for agency can sometimes induce poorer choices. [ABSTRACT FROM AUTHOR]

We consider nonlocal equations of the general form \begin{equation} \left(a*u''\right)(\cdot)+\lambda f\big(\cdot,u(\cdot)\big)=0.\nonumber \end{equation} By developing a Green's function representation for the solution of the boundary value problem we study existence, uniqueness, and qualitative properties (e.g., positivity or monotonicity) of solutions to these problems. We apply our methods to fractional order differential equations. We also demonstrate an application of our methodology both to convolution equations with nonlocal boundary conditions as well as those with a nonlocal term in the convolution equation itself. [ABSTRACT FROM AUTHOR]

In one space dimension, we consider source-type (self-similar) solutions to the thin-film equation with vanishing slope at the edge of their support (zero contact-angle condition) in the range of mobility exponents $n\in\left(\frac 3 2,3\right)$. This range contains the physically relevant case n=2 (Navier slip). The existence and (up to a spatial scaling) uniqueness of these solutions has been established in [3] (Bernis, F., Peletier, L. A. & Williams, S. M. (1992) Source type solutions of a fourth-order nonlinear degenerate parabolic equation. Nonlinear Anal. 18, 217–234). It is also shown there that the leading-order expansion near the edge of the support coincides with that of a travelling-wave solution. In this paper we substantially sharpen this result, proving that the higher order correction is analytic with respect to two variables: the first one is just the spatial variable whereas the second one is a (generically irrational, in particular for n=2) power of it, which naturally emerges from a linearisation of the operator around the travelling-wave solution. This result shows that – as opposed to the case of n=1 (Darcy) or to the case of the porous medium equation (the second-order analogue of the thin-film equation) – in this range of mobility exponents, source-type solutions are not smooth at the edge of their support even when the behaviour of the travelling wave is factored off. We expect the same singular behaviour for a generic solution to the thin-film equation near its moving contact line. As a consequence, we expect a (short-time or small-data) well-posedness theory – of which this paper is a natural prerequisite – to be more involved than in the case n=1. [ABSTRACT FROM AUTHOR]

In this paper we study asymptotic consistency of law invariant convex risk measures and the corresponding risk averse stochastic programming problems for independent, identically distributed data. Under mild regularity conditions, we prove a law of large numbers and epiconvergence of the corresponding statistical estimators. This can be applied in a straightforward way to establish convergence with probability 1 of sample-based estimators of risk averse stochastic programming problems. [ABSTRACT FROM AUTHOR]

Bandit processes and the Gittins index have provided powerful and elegant theory and tools for the optimization of allocating limited resources to competitive demands. In this paper we extend the Gittins theory to more general branching bandit processes, also referred to as open bandit processes, that allow uncountable states and backward times. We establish the optimality of the Gittins index policy with uncountably many states, which is useful in such problems as dynamic scheduling with continuous random processing times. We also allow negative time durations for discounting a reward to account for the present value of the reward that was received before the present time, which we refer to as time-backward effects. This could model the situation of offering bonus rewards for completing jobs above expectation. Moreover, we discover that a common belief on the optimality of the Gittins index in the generalized bandit problem is not always true without additional conditions, and provide a counterexample. We further apply our theory of open bandit processes with time-backward effects to prove the optimality of the Gittins index in the generalized bandit problem under a sufficient condition. [ABSTRACT FROM AUTHOR]

This paper is concerned with the initial boundary value problem of a class of nonlinear wave equations and reaction–diffusion equations with several nonlinear source terms of different signs. For the initial boundary value problem of the nonlinear wave equations, we derive a blow up result for certain initial data with arbitrary positive initial energy. For the initial boundary value problem of the nonlinear reaction–diffusion equations, we discuss some probabilities of the existence and nonexistence of global solutions and give some sufficient conditions for the global and nonglobal existence of solutions at high initial energy level by employing the comparison principle and variational methods. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the existence of periodic solutions and the topological structure of the solution set of first-order differential equations involving the distributional Henstock–Kurzweil integral. The distributional Henstock–Kurzweil integral is a general integral, which includes the Lebesgue and Henstock–Kurzweil integrals. The main results extend some previously known results in the literature. [ABSTRACT FROM AUTHOR]

The multimodal method requires analytical (exact or approximate) natural sloshing modes that exactly satisfy the Laplace equation and boundary condition on the wetted tank surface. When dealing with the nonlinear sloshing problem, the modes should also allow for an analytical continuation throughout the mean free surface. Appropriate analytically approximate modes were constructed by Faltinsen & Timokha (J. Fluid Mech., vol. 695, 2012, pp. 467–477) for the two-dimensional circular tank. The present paper extends this result to the three-dimensional, spherical tank shape and, based on that, establishes specific properties of the linear liquid sloshing. [ABSTRACT FROM PUBLISHER]

When employing autonomous helicopters, it is desirable to use navigation approaches, which firmly ensure safety. In this paper, we propose and compare two approaches to navigation through environments containing obstacles. The first uses sliding mode boundary following to maintain a prespecified distance to obstacles, and the second uses a model predictive control approach to plan short horizon trajectories around detected objects, while ensuring that the helicopter is brought to a halt within the sensor visibility radius. The navigation approaches are subjected to analysis for robustness, and simulations are carried out with a realistic helicopter model for verification. Additional real-world experiments were performed with a wheeled robot to demonstrate potential for real-time application. [ABSTRACT FROM PUBLISHER]

This paper provides an algorithm for identifying the multipoles of a meromorphic function from the boundary measurements, which can be applied to inverse source problems for a Poisson equation to reconstruct its source such as linear combination of mono and dipolar point sources. After establishing an algorithm that allows to identify the number, the locations and the moments of poles, we provide a stability estimate of the algorithm under the measurement error. [ABSTRACT FROM PUBLISHER]

For the Keller–Segel model, it was conjectured by Childress and Percus (1984, Chemotactic collapse in two dimensions. In Lecture Notes in Biomath. Vol. 55, Springer, Berlin-Heidelberg-New York, 1984, pp. 61–66) that in a two-dimensional domain there exists a critical number C such that if the initial mass is strictly less than C, then the solution exists globally in time and if it is strictly larger than C blowup happens. For different versions of the Keller–Segel model, the conjecture has essentially been proved. The case of several chemotactic species introduces an additional question: What is the analogue for the critical mass obtained for the single species system? In this paper, we investigate for a two-species model for chemotaxis in 2 the conditions on the initial data, which determine blowup or global existence in time. Specifically, we find a curve in the plane of masses such that outside of it there is blowup and inside of it global existence in time is proved when the initial masses satisfy a threshold condition. Optimality of this condition is discussed through an analysis in the radial case. Finally, we show in the case of blowup for general data how it is possible to obtain a balance between entropies and prove what species should aggregates first. [ABSTRACT FROM AUTHOR]

This paper deals with some non-linear initial-boundary value problems under homogeneous Neumann boundary conditions, in which the solutions may blow up in finite time. Using a first-order differential inequality technique, lower bounds for blow-up time are determined. [ABSTRACT FROM AUTHOR]

In this paper, by using the fixed point index method first we obtain some existence and multiplicity results for sign-changing solutions of an (e1, B)-limit increasing operator equation. The main results can be applied to many non-linear boundary value problems to obtain the existence and multiplicity results for sign-changing solutions. We also give a clear description of locations of these sign-changing solutions through strict lower and upper solutions. As an example, in the last section we obtain some existence and multiplicity results for sign-changing solutions of some Sturm–Liouville differential boundary value problems. [ABSTRACT FROM AUTHOR]

The notion of completeness between two random elements has been considered recently to provide identification in nonparametric instrumental problems. This condition is quite abstract, however, and characterizations have been obtained only in special cases. This paper considers a nonparametric model between the two variables with an additive separability and a large support condition. In this framework, different versions of completeness are obtained, depending on which regularity conditions are imposed. This result allows one to establish identification in an instrumental nonparametric regression with limited endogenous regressor, a case where the control variate approach breaks down. [ABSTRACT FROM AUTHOR]

We consider in this paper the problem (1)\begin{equation}\label{eq:0.1} \left\{ \begin{array}{l@{\quad}l} -\-\Delta u=|x|^\alpha v^p, & x\in \Omega, \\ \displaystyle -\-\Delta v=|x|^\beta u^{q_\varepsilon}, & x\in \Omega,\\ \displaystyle u>0,\quad v>0, & x \in \Omega,\\ \displaystyle u=v=0, & x\in \partial \Omega,\\ \end{array}\right. \end{equation} where Ω is the unit ball in ℝN centred at the origin, 0 ≤ α < pN,β > 0, N ≥ 3. Suppose qϵ → q as ϵ → 0+ and qϵ, q satisfy, respectively, \begin{displaymath}\frac{N}{p+1}+\frac{N}{q_\varepsilon+1}>N-2,\quad\ \frac{N}{p+1}+\frac{N}{q+1}=N-2;\end{displaymath} we investigate the asymptotic estimates of the ground-state solutions (uϵ, vϵ) of (1) as β → + ∞ with p, qϵ fixed. We also show the symmetry-breaking phenomenon with α, β fixed and qϵ → q as ϵ → 0+. In addition, the ground-state solution is non-radial provided that ϵ > 0 is small or β is large enough. [ABSTRACT FROM PUBLISHER]

Type I domains are the domains of the self-adjoint operators determined by the weak formulation of formally self-adjoint differential expressions ℓ. This class of operators is defined by the requirement that the sesquilinear form q(u, v) obtained from ℓ by integration by parts agrees with the inner product <ℓu, v>. A complete characterisation of the boundary conditions assumed by functions in these domains for second-order differential expressions is given in this paper. In the singular case, the boundary conditions are stated in terms of certain 'boundary condition' functions and in the regular case they are given in terms of classical function values. [ABSTRACT FROM AUTHOR]

This paper presents a sharp boundary growth estimate for all positive superharmonic functions u in a smooth domain Ω in ℝ2 satisfying the nonlinear inequality -Δu(x) ≤ cδΩ(x)-αu(x)p for all x ϵ Ωwhere c > 0, α ϵ ℝ and p > 0, and δΩ(x) stands for the distance from a point x to the boundary of Ω. A result is applied to show the existence of nontangential limits of such superharmonic functions. [ABSTRACT FROM AUTHOR]

In this paper we show existence of positive solutions for a class of quasi-linear problems with Neumann boundary conditions defined in a half-space and involving the critical exponent. [ABSTRACT FROM AUTHOR]

AbstractIn this paper we study a non-linear elliptic equation involving p(x)-growth conditions and satisfying a Neumann boundary condition on a bounded domain. For that equation we establish the existence of two solutions using as a main tool an abstract linking argument due to Br?zis and Nirenberg. [ABSTRACT FROM AUTHOR]

Dense, dry granular avalanches are very efficient at sorting the larger particles towards the free surface of the flow, and finer grains towards the base, through the combined processes of kinetic sieving and squeeze expulsion. This generates an inversely graded particle-size distribution, which is fundamental to a variety of pattern formation mechanisms, as well as subtle size-mobility feedback effects, leading to the formation of coarse-grained lateral levees that create channels in geophysical flows, enhancing their run-out. In this paper we investigate some of the properties of a recent model [Gray, J. M. N. T. & Thornton, A. R. (2005) A theory for particle size segregation in shallow granular free-surface flows. Proc. R. Soc. 461, 1447?1473]; [Thornton, A. R., Gray, J. M. N. T. & Hogg, A. J. (2006) A three-phase mixture theory for particle size segregation in shallow granular free-surface flows. J. Fluid. Mech. 550, 1?25] for the segregation of particles of two sizes but the same density in a shear flow typical of shallow avalanches. The model is a scalar conservation law in space and time, for the volume fraction of smaller particles, with non-constant coefficients depending on depth within the avalanche. It is proved that for steady flow from an inlet, complete segregation occurs beyond a certain finite distance down the slope, no matter what the mixture at the inlet. In time-dependent flow, dynamic shock waves can develop; they are interfaces separating different mixes of particles. Shock waves are shown to be stable if and only if there is a greater concentration of large particles above the interface than below. Constructions with shocks and rarefaction waves are demonstrated on a pair of physically relevant initial boundary value problems, in which a region of all small particles is penetrated from the inlet by either a uniform mixture of particles or by a layer of small particles over a layer of large particles. In both cases, and under a linear shear flow, solutions are constructed for all time and shown to have similar structure for all choices of parameters. [ABSTRACT FROM AUTHOR]

In this paper, we study the multiplicity of solutions for the following problem: $$\begin{equation*} \begin{cases} -\Delta u-\Delta(|u|^{\alpha})|u|^{\alpha-2}u=g(x,u)+\theta h(x,u), \ \ x\in \Omega,\\ u=0, \ \ x\in \partial\Omega, \end{cases} \end{equation*}$$ where α ≥ 2, Ω is a smooth bounded domain in ${\mathbb{R}}$N, θ is a parameter and g, h ∈ C($\bar{\Omega}$ × ${\mathbb{R}}$). Under the assumptions that g(x, u) is odd and locally superlinear at infinity in u, we prove that for any j ∈ $\mathbb{N}$ there exists ϵj > 0 such that if |θ| ≤ ϵj, the above problem possesses at least j distinct solutions. Our results generalize some known results in the literature and are new even in the symmetric situation. [ABSTRACT FROM PUBLISHER]

This paper is concerned with the modified Wigner (respectively, Wigner–Fokker–Planck) Poisson equation. The quantum mechanical model describes the transport of charged particles under the influence of the modified Poisson potential field without (respectively, with) the collision operator. Existence and uniqueness of a global mild solution to the initial boundary value problem in one dimension are established on a weighted $L^{2}$-space. The main difficulties are to derive a priori estimates on the modified Poisson equation and prove the Lipschitz properties of the appropriate potential term. [ABSTRACT FROM PUBLISHER]

In this paper, we study Abelian and metabelian quotients of braid groups of oriented surfaces with boundary components. We provide group presentations and we prove rigidity results for these quotients arising from exact sequences related to (generalised) Fadell–Neuwirth fibrations. [ABSTRACT FROM AUTHOR]

In this paper, a method for the indirect solution of the optimal control problem (OCP) in the presence of pure state variable inequality constraints (SVICs) and mixed state-control inequality constraints (SCIC), without a need for a close initial guess is presented. In the proposed method, using the finite difference approximation (FDA), the pure SVICs are converted to SCIC. Here, the distance of the constraint function to the feasibility bounds of the constraint is computed in every situation and the control signal is chosen appropriately to facilitate the constraint stays safe. In this method, prior knowledge of the numbers and sequences of activation times is not required. So, it can be simply implemented in continuous boundary value problem (BVP) solvers. The proposed method simply applies the SVICs and since the constraint is directly applied on the control signal, it improves the convergence. On the other hand, because of the convergence problem in the indirect solution of OCP, the simple homotopy continuation method (HCM) is used to overcome the initial guess problem by deploying a secondary OCP for which the initial guess can be zero. The proposed approach is applied on a few comprehensive problems in the presence of different constraints. Simulations are compared with the direct solution of the OCP to confirm the accuracy and with the penalty function method and the sequential constraint-free OCP to confirm the convergence. The results indicate that the FDA method for handling the constraints along with the HCM is easy to apply with acceptable accuracy and convergence, even for highly nonlinear problems in robotic systems such as the constrained time optimal control of a two-link manipulator (TLM) and a three-link common industrial robot. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the influence of observed and unobserved initial values on the bias of the conditional maximum likelihood or conditional sum-of-squares (CSS, or least squares) estimator of the fractional parameter, d, in a nonstationary fractional time series model. The CSS estimator is popular in empirical work due, at least in part, to its simplicity and its feasibility, even in very complicated nonstationary models.We consider a process, Xt, for which data exist from some point in time, which we call –N0 + 1, but we only start observing it at a later time, t = 1. The parameter (d, μ, σ2) is estimated by CSS based on the model ${\rm{\Delta }}_0^d \left( {X_t - \mu } \right) = \varepsilon _t ,t = N + 1, \ldots ,N + T$, conditional on X1,..., XN. We derive an expression for the second-order bias of $\hat d$ as a function of the initial values, Xt, t = – N0 + 1,..., N, and we investigate the effect on the bias of setting aside the first N observations as initial values. We compare $\hat d$ with an estimator, $\hat d_c $, derived similarly but by choosing μ = C. We find, both theoretically and using a data set on voting behavior, that in many cases, the estimation of the parameter μ picks up the effect of the initial values even for the choice N = 0.If N0 = 0, we show that the second-order bias can be completely eliminated by a simple bias correction. If, on the other hand, N0 > 0, it can only be partly eliminated because the second-order bias term due to the initial values can only be diminished by increasing N. [ABSTRACT FROM PUBLISHER]

We derive a new equation for the optimal investment boundary of a general irreversible investment problem under exponential Lévy uncertainty. The problem is set as an infinite time-horizon, two-dimensional degenerate singular stochastic control problem. In line with the results recently obtained in a diffusive setting, we showthat the optimal boundary is intimately linked to the unique optional solution of an appropriate Bank-El Karoui representation problem. Such a relation and the Wiener-Hopf factorization allow us to derive an integral equation for the optimal investment boundary. In case the underlying Lévy process hits any point in R with positive probability we show that the integral equation for the investment boundary is uniquely satisfied by the unique solution of another equation which is easier to handle. As a remarkable by-product we prove the continuity of the optimal investment boundary. The paper is concluded with explicit results for profit functions of Cobb-Douglas type and CES type. In the former case the function is separable and in the latter case nonseparable. [ABSTRACT FROM AUTHOR]

Robots with micro- and nanoresolution of motion are becoming more practical and useful in many precision manufacturing processes and industries such as medical instruments and imaging tools. Apparently, the most important features of these devices are their precision and durability. As accuracy increases, more delicate tasks may be performed. Along this line, a spatial micromanipulator with three revolute–revolute–prismatic joints while equipped with nanometer motion resolution is considered here. At the end of the micromanipulator, a piezoresistive-based microcantilever operates as a force sensor to quantify the amount of the strain generated in the microcantilever and transduces it into a proper voltage for force sensing applications. In terms of the controller design, the value of the produced voltage can further be implemented as the feedback entering into the control loop and making the control unit to produce appropriate signals for manipulating the robot arm. A challenging and important problem is the need to control the applied boundary forces at the contact zone with external objects (specifically the biological samples). The ability to control the interaction force is of most interest today which has numerous applications in precision manufacturing and biomedical engineering. For this purpose, two types of controllers are presented here: a Lyapunov-based proportional-derivative (PD) controller and a robust adaptive (RA) controller. The performance and the stability of these two controllers are examined and discussed thoroughly in this paper so that it can be interpreted that the robust adaptive controller is robust under presence of uncertainties in force tracking control purposes. [ABSTRACT FROM AUTHOR]