A Generalized Two-Agent Location Problem: Asymmetric Dynamics and Coordination

We generalize a static two-agent location problem into dynamic, asymmetric settings. The dynamics is due to the ability of the agents to move at limited speeds. Since each agent has its own objective (demand) function and these functions are interdependent, decisions made by each agent may affect the performance of the other agent and thus affect the overall performance of the system. We show that, under a broad range of system’s parameters, centralized (system-wide optimal) and non-cooperative (Nash) behavior of the agents are characterized by a similar structure. The timing of these trajectories and the intermediate speeds are however different. Moreover, non-cooperative agents travel more and may never rest and thus the system performance deteriorates under decentralized decision-making. We show that a static linear reward approach, recently developed in Golany and Rothblum (Nav. Res. Logist. 53(1):1–15, 2006), can be generalized to provide coordination of the moving agents and suggest its dynamic modification. When the reward scheme is applied, the agents are induced to choose the system-wide optimal solution, even though they operate in a decentralized decision-making mode.     

Well-posedness for a mean field model of Ginzburg–Landau vortices with opposite degrees

In this paper we analyze the hydrodynamic equations for Ginzburg–Landau vortices as derived by E (Phys. Rev. B. 50(3):1126–1135, 1994). In particular, we are interested in the mean field model describing the evolution of two patches of vortices with equal and opposite degrees. Many results are already available for the case of a single density of vortices with uniform degree. This model does not take into account the vortex annihilation, hence it can also be seen as a particular instance of the signed measures system obtained in Ambrosio et al. (Ann. Inst. H. Poincaré Anal. Non Linéaire 28(2):217–246, 2011) and related to the Chapman et al. (Eur. J. Appl. Math. 7(2):97–111, 1996) formulation. We establish global existence of L ^p solutions, exploiting some optimal transport techniques introduced in this context in Ambrosio and Serfaty (Commun. Pure Appl. Math. LXI(11):1495–1539, 2008). We prove uniqueness for L ^∞ solutions, as expected by analogy with the incompressible Euler equations in fluidodynamics. We also consider the corresponding Dirichlet problem in a bounded domain. Moreover, we show some simple examples of 1-dimensional dynamic.     

Random Skew Plane Partitions with a Piecewise Periodic Back Wall

Random skew plane partitions of large size distributed according to an appropriately scaled Schur process develop limit shapes. In the present work, we consider the limit of large random skew plane partitions where the inner boundary approaches a piecewise linear curve with non-lattice slopes, describing the limit shape and the local fluctuations in various regions. This analysis is fairly similar to that in Okounkov and Reshetikhin (Commun Math Phys 269:571–609, 2007), but we do find some new behavior. For instance, the boundary of the limit shape is now a single smooth (not algebraic) curve, whereas the boundary in Okounkov and Reshetikhin (Commun Math Phys 269:571–609, 2007) is singular. We also observe the bead process introduced in Boutillier (Ann Probab 37(1):107–142, 2009) appearing in the asymptotics at the top of the limit shape.     

Pathwise Solutions of the 2-D Stochastic Primitive Equations

In this work we consider a stochastic version of the Primitive Equations (PEs) of the ocean and the atmosphere and establish the existence and uniqueness of pathwise, strong solutions. The analysis employs novel techniques in contrast to previous works (Ewald et al. in Anal. Appl. (Singap.) 5(2):183–198, 2007; Glatt-Holtz and Ziane in Discrete Contin. Dyn. Syst. Ser. B 10(4):801–822, 2008) in order to handle a general class of nonlinear noise structures and to allow for physically relevant boundary conditions. The proof relies on Cauchy estimates, stopping time arguments and anisotropic estimates.     

More Problems for MaxCon: Contingent Particularity and Stuff-Thing Coincidence

Ned Markosian argues (Australasian Journal of Philosophy 76:213-228, 1998a; Australasian Journal of Philosophy 82:332-340, 2004a, The Monist 87:405-428, 2004b) that simples are ‘maximally continuous’ entities. This leads him to conclude that there could be non-particular ‘stuff’ in addition to things. I first show how an ensuing debate on this issue McDaniel (Australasian Journal of Philosophy 81(2):265-275, 2003); Markosian (Australasian Journal of Philosophy 82:332-340, 2004a) ended in deadlock. I attempt to break the deadlock. Markosian’s view entails stuff-thing coincidence, which I show is just as problematic as the more oft-discussed thing-thing coincidence. Also, the view entails that every particular is only contingently so. If there is a world W like our own, but with ether, then there would be only one object in W. But, since merely adding ether to a world does not destroy the entities in it, then W contains counterparts of all the entities in the actual world—they just are not things. Hence, if simples are maximally continuous, then every actual particular is only contingently so. This in turn entails the following disjunction: (i) identity is contingent or intransitive, or (ii) there are no things at all in the actual world, or (iii) the distinction between stuff and things is one without a difference. I recommend that we reject this stuff-thing dualism.     

Statistical Causality,Extremal Measures and Weak Solutions of Stochastic Differential Equations with Driving Semimartingales

In this paper we consider different concepts of causality in filtered probability spaces. Especially, we consider a generalization of a causality relationship “G is a cause of J within H ” which was first given by Mykland (1986) and which is based on Granger’s definition of causality (Granger, Econometrica 37:424–438, 1969). Then we apply this concept on weak solutions of stochastic differential equations with driving semimartingales. We also show that the given causality concept is closely connected to the concept of extremality of measures and links Granger’s causality with the concept of adapted distribution. Finally, the concept of causality is applied on solution of martingale problem.     

Limiting Carleman weights and anisotropic inverse problems

In this article we consider the anisotropic Calderón problem and related inverse problems. The approach is based on limiting Carleman weights, introduced in Kenig et al. (Ann. Math. 165:567–591, 2007) in the Euclidean case. We characterize those Riemannian manifolds which admit limiting Carleman weights, and give a complex geometrical optics construction for a class of such manifolds. This is used to prove uniqueness results for anisotropic inverse problems, via the attenuated geodesic ray transform. Earlier results in dimension n≥3 were restricted to real-analytic metrics.     

Quasi exponential decay of a finite difference space discretization of the 1-d wave equation by pointwise interior stabilization

We consider the wave equation on an interval of length 1 with an interior damping at ξ. It is well-known that this system is well-posed in the energy space and that its natural energy is dissipative. Moreover, as it was proved in Ammari et al. (Asymptot Anal 28(3–4):215–240, 2001), the exponential decay property of its solution is equivalent to an observability estimate for the corresponding conservative system. In this case, the observability estimate holds if and only if ξ is a rational number with an irreducible fraction x = \fracpq,\xi=\frac{p}{q}, where p is odd, and therefore under this condition, this system is exponentially stable in the energy space. In this work, we are interested in the finite difference space semi-discretization of the above system. As for other problems (Zuazua, SIAM Rev 47(2):197–243, 2005; Tcheugoué Tébou and Zuazua, Adv Comput Math 26:337–365, 2007), we can expect that the exponential decay of this scheme does not hold in general due to high frequency spurious modes. We first show that this is indeed the case. Secondly we show that a filtering of high frequency modes allows to restore a quasi exponential decay of the discrete energy. This last result is based on a uniform interior observability estimate for filtered solutions of the corresponding conservative semi-discrete system.     

Constrained Optimization: Projected Gradient Flows

We consider a dynamical system approach to solve finite-dimensional smooth optimization problems with a compact and connected feasible set. In fact, by the well-known technique of equalizing inequality constraints using quadratic slack variables, we transform a general optimization problem into an associated problem without inequality constraints in a higher-dimensional space. We compute the projected gradient for the latter problem and consider its projection on the feasible set in the original, lower-dimensional space. In this way, we obtain an ordinary differential equation in the original variables, which is specially adapted to treat inequality constraints (for the idea, see Jongen and Stein, Frontiers in Global Optimization, pp. 223–236, Kluwer Academic, Dordrecht, 2003). The article shows that the derived ordinary differential equation possesses the basic properties which make it appropriate to solve the underlying optimization problem: the longtime behavior of its trajectories becomes stationary, all singularities are critical points, and the stable singularities are exactly the local minima. Finally, we sketch two numerical methods based on our approach.     

Rigorous computation of smooth branches of equilibria for the three dimensional Cahn–Hilliard equation

In this paper, we propose a new general method to compute rigorously global smooth branches of equilibria of higher-dimensional partial differential equations. The theoretical framework is based on a combination of the theory introduced in Global smooth solution curves using rigorous branch following (van den Berg et al., Math. Comput. 79(271):1565–1584, 2010) and in Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs (Gameiro and Lessard, J. Diff. Equ. 249(9):2237–2268, 2010). Using this method, one can obtain proofs of existence of global smooth solution curves of equilibria for large (continuous) parameter ranges and about local uniqueness of the solutions on the curve. As an application, we compute several smooth branches of equilibria for the three-dimensional Cahn–Hilliard equation.     

One-dimensional chemotaxis kinetic model

In this paper, we study a variation of the equations of a chemotaxis kinetic model and investigate it in one dimension. In fact, we use fractional diffusion for the chemoattractant in the Othmar–Dunbar–Alt system (Othmer in J Math Biol 26(3):263–298, 1988). This version was exhibited in Calvez in Amer Math Soc, pp 45–62, 2007 for the macroscopic well-known Keller–Segel model in all space dimensions. These two macroscopic and kinetic models are related as mentioned in Bournaveas, Ann Inst H Poincaré Anal Non Linéaire, 26(5):1871–1895, 2009, Chalub, Math Models Methods Appl Sci, 16(7 suppl):1173–1197, 2006, Chalub, Monatsh Math, 142(1–2):123–141, 2004, Chalub, Port Math (NS), 63(2):227–250, 2006. The model we study here behaves in a similar way to the original model in two dimensions with the spherical symmetry assumption on the initial data which is described in Bournaveas, Ann Inst H Poincaré Anal Non Linéaire, 26(5):1871–1895, 2009. We prove the existence and uniqueness of solutions for this model, as well as a convergence result for a family of numerical schemes. The advantage of this model is that numerical simulations can be easily done especially to track the blow-up phenomenon.     

Spikes for the Gierer–Meinhardt System with Discontinuous Diffusion Coefficients

We rigorously prove results on spiky patterns for the Gierer–Meinhardt system (Kybernetik (Berlin) 12:30–39, 1972) with a jump discontinuity in the diffusion coefficient of the inhibitor. Using numerical computations in combination with a Turing-type instability analysis, this system has been investigated by Benson, Maini, and Sherratt (Math. Comput. Model. 17:29–34, 1993a; Bull. Math. Biol. 55:365–384, 1993b; IMA J. Math. Appl. Med. Biol. 9:197–213, 1992). Firstly, we show the existence of an interior spike located away from the jump discontinuity, deriving a necessary condition for the position of the spike. In particular, we show that the spike is located in one-and-only-one of the two subintervals created by the jump discontinuity of the inhibitor diffusivity. This localization principle for a spike is a new effect which does not occur for homogeneous diffusion coefficients. Further, we show that this interior spike is stable. Secondly, we establish the existence of a spike whose distance from the jump discontinuity is of the same order as its spatial extent. The existence of such a spike near the jump discontinuity is the second new effect presented in this paper. To derive these new effects in a mathematically rigorous way, we use analytical tools like Liapunov–Schmidt reduction and nonlocal eigenvalue problems which have been developed in our previous work (J. Nonlinear Sci. 11:415–458, 2001). Finally, we confirm our results by numerical computations for the dynamical behavior of the system. We observe a moving spike which converges to a stationary spike located in the interior of one of the subintervals or near the jump discontinuity.      

Inventory models with ramp type demand rate,time dependent deterioration rate,unit production cost and shortages

Manna and Chaudhuri (Eur. J. Oper. Res. 171(2):557–566, 2006) presented a production-inventory system for deteriorating items with demand rate being a linearly ramp type function of time and production rate being proportional to the demand rate. The two models without shortages and with shortages were discussed. Both models were studied assuming that the time point at which the demand is stabilized occurs before the production stopping time. In this paper, we complete this model by considering that: (a) for the model with no shortages; the demand rate is stabilized after the production stopping time and (b) for the model with shortages; the demand rate is stabilized after the production stopping time or after the time when the inventory level reaches zero or after the production restarting time. In addition, we extend the work of Manna and Chaudhuri (Eur. J. Oper. Res. 171(2):557–566, 2006) assuming a general function of time for the variable part of the demand rate.     

The Order Completion Method for Systems of Nonlinear PDEs Revisited

In this paper we presents further developments regarding the enrichment of the basic Theory of Order Completion as presented in Oberguggenberger and Rosinger (Solution of continuous nonlinear PDEs through order completion, North-Holland, Amsterdam, 1994). In particular, spaces of generalized functions are constructed that contain generalized solutions to a large class of systems of continuous, nonlinear PDEs. In terms of the existence and uniqueness results previously obtained for such systems of equations (van der Walt, Acta Appl. Math. 103:1–17, 2008), one may interpret the existence of generalized solutions presented here as a regularity result. Furthermore, it is indicated how the methods developed in this paper may be adapted to solve initial and/or boundary value problems. In particular, we consider the Navier-Stokes equations in three spacial dimensions, subject to an initial condition on the velocity. In this regard, we obtain the existence of a generalized solution to a large class of such initial value problems.      

Further Results on the Robust Regulation of a One-Dimensional Dam-River System

In this paper, we consider a one-dimensional dam-river system studied by Chentouf and Wang (SIAM J. Control Optim. 47: 2275–2302, 2008). Then, using the frequency multiplier method, we provide a simple and alternative proof of stabilization and regulation results obtained in the work cited above. Moreover, we relax the conditions on the feedback gains involved in the feedback law and give a partial answer to the open problem left by the authors Chentouf and Wang (J. Optim. Theory Appl. 134: 223–239, 2007 and SIAM J. Control Optim. 47: 2275–2302, 2008) concerning the tuning of the gains.     

A note on meromorphic functions that share a set with their derivatives

In this paper, we study a problem of meromorphic functions that share an arbitrary set having three elements with their derivatives. A uniqueness result is derived which is an improvement of some related theorems given by Fang and Zalcman (J. Math. Anal. Appl. 280 (2003), 273–283) and Chang, Fang, and Zalcman (Arch. Math. 89 (2007), 561–569). As an application, we generalize the famous Brück conjecture with the idea of sharing a set.     

Multi-objective redundancy allocation optimization using a variable neighborhood search algorithm

A variable neighborhood search (VNS) algorithm has been developed to solve the multiple objective redundancy allocation problems (MORAP). The single objective RAP is to select the proper combination and redundancy levels of components to meet system level constraints, and to optimize the specified objective function. In practice, the need to consider two or more conflicting objectives simultaneously increases nowadays in order to assure managers or designers’ demand. Amongst all system level objectives, maximizing system reliability is the most studied and important one, while system weight or system cost minimization are two other popular objectives to consider. According to the authors’ experience, VNS has successfully solved the single objective RAP (Liang and Chen, Reliab. Eng. Syst. Saf. 92:323–331, 2007; Liang et al., IMA J. Manag. Math. 18:135–155, 2007). Therefore, this study aims at extending the single objective VNS algorithm to a multiple objective version for solving multiple objective redundancy allocation problems. A new selection strategy of base solutions that balances the intensity and diversity of the approximated Pareto front is introduced. The performance of the proposed multi-objective VNS algorithm (MOVNS) is verified by testing on three sets of complex instances with 5, 14 and 14 subsystems respectively. While comparing to the leading heuristics in the literature, the results show that MOVNS is able to generate more non-dominated solutions in a very efficient manner, and performs competitively in all performance measure categories. In other words, computational results reveal the advantages and benefits of VNS on solving multi-objective RAP.     

Uniqueness theorems of entire functions sharing a nonzero complex number with their difference operators

In this paper, we deal with a uniqueness question of entire functions sharing a nonzero complex number with their difference operators. The results in this paper improve Theorem 1.1 in Liu and Yang (Arch. Math. 92 (2009), 270–278) and deal with Question 1 in Liu and Yang (2009), where the entire functions are of finite orders.     

Equivalent Conditions for Jacobian Nonsingularity in Linear Symmetric Cone Programming

In this paper we consider the linear symmetric cone programming (SCP). At a Karush-Kuhn-Tucker (KKT) point of SCP, we present the important conditions equivalent to the nonsingularity of Clarke’s generalized Jacobian of the KKT nonsmooth system, such as primal and dual constraint nondegeneracy, the strong regularity, and the nonsingularity of the B-subdifferential of the KKT system. This affirmatively answers an open question by Chan and Sun (SIAM J. Optim. 19:370–396, 2008).     

Solving molecular distance geometry problems by global optimization algorithms

In this paper we consider global optimization algorithms based on multiple local searches for the Molecular Distance Geometry Problem (MDGP). Three distinct approaches (Multistart, Monotonic Basin Hopping, Population Basin Hopping) are presented and for each of them a computational analysis is performed. The results are also compared with those of two other approaches in the literature, the DGSOL approach (Moré, Wu in J. Glob. Optim. 15:219–234, 1999) and a SDP based approach (Biswas et al. in An SDP based approach for anchor-free 3D graph realization, Technical Report, Operations Research, Stanford University, 2005).