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In this paper we introduce a bulk-surface reaction-diffusion (BSRD) model in three space dimensions that extends the DIB morphochemical model to account for the electrolyte contribution in the application, in order to study structure formation during discharge-charge processes in batteries. Here we propose to approximate the model by the Bulk-Surface Virtual Element Method on a tailor-made mesh that proves to be competitive with fast bespoke methods for PDEs on Cartesian grids. We present a selection of numerical simulations that accurately match the classical morphologies found in experiments. Finally, we compare the Turing patterns obtained by the coupled 3D BS-DIB model with those obtained with the original 2D version., Comment: 25 pages, 11 figures, 1 table

We introduce the notions of Collective Arbitrage and of Collective Super-replication in a discrete-time setting where agents are investing in their markets and are allowed to cooperate through exchanges. We accordingly establish versions of the fundamental theorem of asset pricing and of the pricing-hedging duality. A reduction of the price interval of the contingent claims can be obtained by applying the collective super-replication price.

Shortfall systemic (multivariate) risk measures $\rho$ defined through an $N$-dimensional multivariate utility function $U$ and random allocations can be represented as classical (one dimensional) shortfall risk measures associated to an explicitly determined $1$-dimensional function constructed from $U$. This finding allows for simplifying the study of several properties of $\rho$, such as dual representations, law invariance and stability.

In this work we propose deep learning-based algorithms for the computation of systemic shortfall risk measures defined via multivariate utility functions. We discuss the key related theoretical aspects, with a particular focus on the fairness properties of primal optima and associated risk allocations. The algorithms we provide allow for learning primal optimizers, optima for the dual representation and corresponding fair risk allocations. We test our algorithms by comparison to a benchmark model, based on a paired exponential utility function, for which we can provide explicit formulas. We also show evidence of convergence in a case for which explicit formulas are not available., Comment: 4 figures, 5 tables

In this work we present a novel bulk-surface virtual element method (BSVEM) for the numerical approximation of elliptic bulk-surface partial differential equations (BSPDEs) in three space dimensions. The BSVEM is based on the discretisation of the bulk domain into polyhedral elements with arbitrarily many faces. The polyhedral approximation of the bulk induces a polygonal approximation of the surface. Firstly, we present a geometric error analysis of bulk-surface polyhedral meshes independent of the numerical method. Then, we show that BSVEM has optimal second-order convergence in space, provided the exact solution is $H^{2+3/4}$ in the bulk and $H^2$ on the surface, where the additional $\frac{3}{4}$ is due to the combined effect of surface curvature and polyhedral elements close to the boundary. We show that general polyhedra can be exploited to reduce the computational time of the matrix assembly. To demonstrate optimal convergence results, a numerical example is presented on the unit sphere., Comment: 25 pages, 4 figures, 1 table. This replacement improves figures, updates references, and avoids redundancies. arXiv admin note: substantial text overlap with arXiv:2002.11748

When numerical solution of elliptic and parabolic partial differential equations is required to be highly accurate in space, the discrete problem usually takes the form of large-scale and sparse linear systems. In this work, as an alternative, for spatial discretization we provide a Matrix-Oriented formulation of the classical Finite Element Method, called MO-FEM, of arbitrary order $k\in\mathbb{N}$. On structured 2D domains (e.g. squares or rectangles) the discrete problem is then reformulated as a Sylvester matrix equation, that we solve by the reduced approach in the associated spectral space. On a quite general class of domains, namely normal domains, and even on special surfaces, the MO-FEM yields a multiterm Sylvester matrix equation where the additional terms account for the geometric contribution of the domain shape. In particular, we obtain a sequence of these equations after time discretization of parabolic problems by the IMEX Euler method. We apply the matrix-oriented form of the Preconditioned Conjugate Gradient (MO-PCG) method to solve each multiterm Sylvester equation for MO-FEM of degree $k=1,\dots,4$ and for the lumped $\mathbb{P}_1$ case. We choose a matrix-oriented preconditioner with a single-term form that captures the spectral properties of the whole multiterm Sylvester operator. For several numerical examples, we show a gain in computational time and memory occupation wrt the classical vector approach solving large sparse linear systems by a direct method or by the vector PCG with same preconditioning. As an application, we show the advantages of the MO-FEM-PCG to approximate Turing patterns with high spatial resolution in a reaction-diffusion PDE system for battery modeling., Comment: 33 pages, 8 figures, 5 tables

In this note we consider a system of financial institutions and study systemic risk measures in the presence of a financial market and in a robust setting, namely, where no reference probability is assigned. We obtain a dual representation for convex robust systemic risk measures adjusted to the financial market and show its relation to some appropriate no-arbitrage conditions.

We investigate to which extent the relevant features of (static) Systemic Risk Measures can be extended to a conditional setting. After providing a general dual representation result, we analyze in greater detail Conditional Shortfall Systemic Risk Measures. In the particular case of exponential preferences, we provide explicit formulas that also allow us to show a time consistency property. Finally, we provide an interpretation of the allocations associated to Conditional Shortfall Systemic Risk Measures as suitably defined equilibria. Conceptually, the generalization from static to conditional Systemic Risk Measures can be achieved in a natural way, even though the proofs become more technical than in the unconditional framework.

The objective of this paper is to develop a duality between a novel Entropy Martingale Optimal Transport problem (A) and an associated optimization problem (B). In (A) we follow the approach taken in the Entropy Optimal Transport (EOT) primal problem by Liero et al. "Optimal entropy-transport problems and a new Hellinger-Kantorovic distance between positive measures", Invent. math. 2018, but we add the constraint, typical of Martingale Optimal Transport (MOT) theory, that the infimum of the cost functional is taken over martingale probability measures, instead of finite positive measures, as in Liero et al.. The Problem (A) differs from the corresponding problem in Liero et al. not only by the martingale constraint, but also because we admit less restrictive penalization terms $\mathcal{D}_{U}$, which may not have a divergence formulation. In Problem (B) the objective functional, associated via Fenchel conjugacy to the terms $\mathcal{D}_{U}$, is not any more linear, as in OT or in MOT. This leads to a novel optimization problem which also has a clear financial interpretation as a nonlinear subhedging value. Our theory allows us to establish a nonlinear robust pricing-hedging duality, which covers a wide range of known robust results. We also focus on Wasserstein-induced penalizations and we study how the duality is affected by variations in the penalty terms, with a special focus on the convergence of EMOT to the extreme case of MOT.

In addressing the problem of commodity production out of feedstock imports, an eco-environmentally rational agent aims at minimizing the cost of feedstock imports and their transportation, but also the water footprint of the feedstock production process and the water scarcity in the exporting countries. The problem is formulated as a nonlinear program. This study proves the existence of solutions and quantitatively demonstrates that transportation costs and non-uniform feedstock characteristics inhibit feedstock interchangeability. Moreover, it is shown that the interplay between water footprint and water scarcity across countries can inhibit or foster feedstock interchangeability., Comment: 14 pages

In this paper we consider a coupled bulk-surface PDE in two space dimensions. The model consists of a PDE in the bulk that is coupled to another PDE on the surface through general nonlinear boundary conditions. For such a system we propose a novel method, based on coupling a virtual element method [Beir\~ao da Veiga et al., 2013] in the bulk domain to a surface finite element method [Dziuk & Elliott, 2013] on the surface. The proposed method, which we coin the Bulk-Surface Virtual Element Method (BSVEM) includes, as a special case, the bulk-surface finite element method (BSFEM) on triangular meshes [Madzvamuse & Chung, 2016]. The method exhibits second-order convergence in space, provided the exact solution is $H^{2+1/4}$ in the bulk and $H^2$ on the surface, where the additional $\frac{1}{4}$ is required only in the simultaneous presence of surface curvature and non-triangular elements. Two novel techniques introduced in our analysis are (i) an $L^2$-preserving inverse trace operator for the analysis of boundary conditions and (ii) the Sobolev extension as a replacement of the lifting operator [Elliott & Ranner, 2013] for sufficiently smooth exact solutions. The generality of the polygonal mesh can be exploited to optimize the computational time of matrix assembly. The method takes an optimised matrix-vector form that also simplifies the known special case of BSFEM on triangular meshes [Madzvamuse & Chung, 2016]. Three numerical examples illustrate our findings., Comment: 39 pages, 9 figures, 5 tables

A Systemic Optimal Risk Transfer Equilibrium (SORTE) was introduced in: "Systemic optimal risk transfer equilibrium", Mathematics and Financial Economics (2021), for the analysis of the equilibrium among financial institutions or in insurance-reinsurance markets. A SORTE conjugates the classical B\"{u}hlmann's notion of a risk exchange equilibrium with a capital allocation principle based on systemic expected utility optimization. In this paper we extend such a notion to the case when the value function to be optimized is multivariate in a general sense, and it is not simply given by the sum of univariate utility functions. This takes into account the fact that preferences of single agents might depend on the actions of other participants in the game. Technically, the extension of SORTE to the new setup requires developing a theory for multivariate utility functions and selecting at the same time a suitable framework for the duality theory. Conceptually, this more general framework allows us to introduce and study a Nash Equilibrium property of the optimizer. We prove existence, uniqueness, and the Nash Equilibrium property of the newly defined Multivariate Systemic Optimal Risk Transfer Equilibrium.

We propose a novel concept of a Systemic Optimal Risk Transfer Equilibrium (SORTE), which is inspired by the B\"uhlmann's classical notion of an Equilibrium Risk Exchange. We provide sufficient general assumptions that guarantee existence, uniqueness, and Pareto optimality of such a SORTE. In both the B\"uhlmann and the SORTE definition, each agent is behaving rationally by maximizing his/her expected utility given a budget constraint. The two approaches differ by the budget constraints. In B\"uhlmann's definition the vector that assigns the budget constraint is given a priori. On the contrary, in the SORTE approach, the vector that assigns the budget constraint is endogenously determined by solving a systemic utility maximization. SORTE gives priority to the systemic aspects of the problem, in order to optimize the overall systemic performance, rather than to individual rationality.

In our previous paper, "A Unified Approach to Systemic Risk Measures via Acceptance Set" (\textit{Mathematical Finance, 2018}), we have introduced a general class of systemic risk measures that allow for random allocations to individual banks before aggregation of their risks. In the present paper, we prove the dual representation of a particular subclass of such systemic risk measures and the existence and uniqueness of the optimal allocation related to them. We also introduce an associated utility maximization problem which has the same optimal solution as the systemic risk measure. In addition, the optimizer in the dual formulation provides a \textit{risk allocation} which is fair from the point of view of the individual financial institutions. The case with exponential utilities which allows for explicit computation is treated in details., Comment: Keywords}: Systemic risk measures, random allocations, risk allocation, fairness

We propose a method to assess the intrinsic risk carried by a financial position $X$ when the agent faces uncertainty about the pricing rule assigning its present value. Our approach is inspired by a new interpretation of the quasiconvex duality in a Knightian setting, where a family of probability measures replaces the single reference probability and is then applied to value financial positions. Diametrically, our construction of Value\&Risk measures is based on the selection of a basket of claims to test the reliability of models. We compare a random payoff $X$ with a given class of derivatives written on $X$ , and use these derivatives to \textquotedblleft test\textquotedblright\ the pricing measures. We further introduce and study a general class of Value\&Risk measures $% R(p,X,\mathbb{P})$ that describes the additional capital that is required to make $X$ acceptable under a probability $\mathbb{P}$ and given the initial price $p$ paid to acquire $X$.

We develop a robust framework for pricing and hedging of derivative securities in discrete-time financial markets. We consider markets with both dynamically and statically traded assets and make minimal measurability assumptions. We obtain an abstract (pointwise) Fundamental Theorem of Asset Pricing and Pricing--Hedging Duality. Our results are general and in particular include so-called model independent results of Acciao et al. (2016), Burzoni et al. (2016) as well as seminal results of Dalang et al. (1990) in a classical probabilistic approach. Our analysis is scenario--based: a model specification is equivalent to a choice of scenarios to be considered. The choice can vary between all scenarios and the set of scenarios charged by a given probability measure. In this way, our framework interpolates between a model with universally acceptable broad assumptions and a model based on a specific probabilistic view of future asset dynamics.

We present and analyze a Virtual Element Method (VEM) of arbitrary polynomial order $k\in\mathbb{N}$ for the Laplace-Beltrami equation on a surface in $\mathbb{R}^3$. The method combines the Surface Finite Element Method (SFEM) [Dziuk, Elliott, \emph{Finite element methods for surface PDEs}, 2013] and the recent VEM [Beirao da Veiga et al, \emph{Basic principles of Virtual Element Methods}, 2013] in order to handle arbitrary polygonal and/or nonconforming meshes. We account for the error arising from the geometry approximation and extend to surfaces the error estimates for the interpolation and projection in the virtual element function space. In the case $k=1$ of linear Virtual Elements, we prove an optimal $H^1$ error estimate for the numerical method. The presented method has the capability of handling the typically nonconforming meshes that arise when two ore more meshes are pasted along a straight line. Numerical experiments are provided to confirm the convergence result and to show an application of mesh pasting., Comment: 25 pages, 6 figures

We demonstrate a novel way of synthesizing spin-orbit interactions in ultracold quantum gases, based on a single-photon optical clock transition coupling two long-lived electronic states of two-electron $^{173}$Yb atoms. By mapping the electronic states onto effective sites along a synthetic "electronic" dimension, we have engineered synthetic fermionic ladders with tunable magnetic fluxes. We have detected the spin-orbit coupling with fiber-link-enhanced clock spectroscopy and directly measured the emergence of chiral edge currents, probing them as a function of the magnetic field flux. These results open new directions for the investigation of topological states of matter with ultracold atomic gases., Comment: Minor changes with respect to v1 (we have corrected some typos, fixed the use of some mathematical symbols, added one reference)

We propose and analyse a novel surface finite element method that preserves the invariant regions of systems of semilinear parabolic equations on closed compact surfaces in $\mathbb{R}^3$ under discretisation. We also provide a fully-discrete scheme by applying the implicit-explicit (IMEX) Euler method in time. We prove the preservation of the invariant rectangles of the continuous problem under spatial and full discretizations. For scalar equations, these results reduce to the well-known discrete maximum principle. Furthermore, we prove optimal error bounds for the semi- and fully-discrete methods, that is the convergence rates are quadratic in the meshsize and linear in the timestep. Numerical experiments are provided to support the theoretical findings. In particular we provide examples in which, in the absence of lumping, the numerical solution violates the invariant region leading to blow-up due to the nature of the kinetics., Comment: 34 pages, 5 figures, 4 tables

Global Positioning System (GPS) dissemination of frequency standards is ubiquitous at present, providing the most widespread time and frequency reference for the majority of industrial and research applications worldwide. On the other hand, the ultimate limits of the GPS presently curb further advances in high-precision, scientific and industrial applications relying on this dissemination scheme. Here, we demonstrate that these limits can be reliably overcome even in laboratories without a local atomic clock by replacing the GPS with a 642-km-long optical fiber link to a remote primary caesium frequency standard. Through this configuration we stably address the $^1$S$_0$---$^3$P$_0$ clock transition in an ultracold gas of $^{173}$Yb, with a precision that exceeds the possibilities of a GPS-based measurement, dismissing the need for a local clock infrastructure to perform high-precision tasks beyond GPS limit. We also report an improvement of two orders of magnitude in the accuracy on the transition frequency reported in literature.

In a model free discrete time financial market, we prove the superhedging duality theorem, where trading is allowed with dynamic and semi-static strategies. We also show that the initial cost of the cheapest portfolio that dominates a contingent claim on every possible path $\omega \in \Omega$, might be strictly greater than the upper bound of the no-arbitrage prices. We therefore characterize the subset of trajectories on which this duality gap disappears and prove that it is an analytic set.

The financial crisis has dramatically demonstrated that the traditional approach to apply univariate monetary risk measures to single institutions does not capture sufficiently the perilous systemic risk that is generated by the interconnectedness of the system entities and the corresponding contagion effects. This has brought awareness of the urgent need for novel approaches that capture systemic riskiness. The purpose of this paper is to specify a general methodological framework that is flexible enough to cover a wide range of possibilities to design systemic risk measures via multi-dimensional acceptance sets and aggregation functions, and to study corresponding examples. Existing systemic risk measures can usually be interpreted as the minimal capital needed to secure the system after aggregating individual risks. In contrast, our approach also includes systemic risk measures that can be interpreted as the minimal capital funds that secure the aggregated system by allocating capital to the single institutions before aggregating the individual risks. This allows for a possible ranking of the institutions in terms of systemic riskiness measured by the optimal allocations. Moreover, we also allow for the possibility of allocating the funds according to the future state of the system (random allocation). We provide conditions which ensure monotonicity, convexity, or quasi-convexity properties of our systemic risk measures.

In a model independent discrete time financial market, we discuss the richness of the family of martingale measures in relation to different notions of Arbitrage, generated by a class $\mathcal{S}$ of significant sets, which we call Arbitrage de la classe $\mathcal{S}$. The choice of $\mathcal{S}$ reflects into the intrinsic properties of the class of polar sets of martingale measures. In particular: for S=${\Omega}$ absence of Model Independent Arbitrage is equivalent to the existence of a martingale measure; for $\mathcal{S}$ being the open sets, absence of Open Arbitrage is equivalent to the existence of full support martingale measures. These results are obtained by adopting a technical filtration enlargement and by constructing a universal aggregator of all arbitrage opportunities. We further introduce the notion of market feasibility and provide its characterization via arbitrage conditions. We conclude providing a dual representation of Open Arbitrage in terms of weakly open sets of probability measures, which highlights the robust nature of this concept.

It is well known that temperature variations and acoustic noise affect ultrastable frequency dissemination along optical fiber. Active stabilization techniques are in general adopted to compensate for the fiber-induced phase noise. However, despite this compensation, the ultimate link performances remain limited by the so called delay-unsuppressed fiber noise that is related to the propagation delay of the light in the fiber. In this paper, we demonstrate a data post-processing approach which enables us to overcome this limit. We implement a subtraction algorithm between the optical signal delivered at the remote link end and the round-trip signal. In this way, a 6 dB improvement beyond the fundamental limit imposed by delay-unsuppressed noise is obtained. This result enhances the resolution of possible comparisons between remote optical clocks by a factor of 2. We confirm the theoretical prediction with experimental data obtained on a 47 km metropolitan fiber link, and propose how to extend this method for frequency dissemination purposes as well.

To significantly improve the frequency references used in radio-astronomy and precision measurements in atomic physics, we provide frequency dissemination through a 642 km coherent optical fiber link, that will be also part of a forthcoming European network of optical links. We obtained a resolution of 3e-19 at 1000 s on the frequency transfer, and an accuracy of 5e-19. The ultimate link performance has been evaluated by doubling the link to 1284 km, demonstrating a new characterization technique based on the double round-trip on a single fiber. The arming of a second fiber is avoided: this is beneficial to long hauls realizations in view of a continental fiber network for frequency and time metrology. The data analysis is based on the Allan deviation; its expression is theoretically derived for the observed noise power spectrum, which is seldom found in the literature.

We performed a two-way remote optical phase comparison on optical fiber. Two optical frequency signals were launched in opposite directions in an optical fiber and their phases were simultaneously measured at the other end. In this technique, the fiber noise was passively cancelled, and we compared two optical frequencies at the ultimate 1E-21 stability level. The experiment was performed on a 47 km fiber that is part of the metropolitan network for Internet traffic. The technique relies on the synchronous measurement of the optical phases at the two ends of the link, that is made possible by the use of digital electronics. This scheme offers several advantages with respect to active noise cancellation, and can be upgraded to perform more complex tasks.

Evenly convex sets in a topological vector space are defined as the intersection of a family of open half spaces. We introduce a generalization of this concept in the conditional framework and provide a generalized version of the bipolar theorem. This notion is then applied to obtain the dual representation of conditionally evenly quasi-convex maps.

In order to evaluate the quality of the scientific research, we introduce a new family of scientific performance measures, called Scientific Research Measures (SRM). Our proposal originates from the more recent developments in the theory of risk measures and is an attempt to resolve the many problems of the existing bibliometric indices. The SRM that we introduce are based on the whole scientist's citation record and are: coherent, as they share the same structural properties; flexible to fit peculiarities of different areas and seniorities; granular, as they allow a more precise comparison between scientists, and inclusive, as they comprehend several popular indices. Another key feature of our SRM is that they are planned to be calibrated to the particular scientific community. We also propose a dual formulation of this problem and explain its relevance in this context.

We propose a generalization of the classical notion of the $V@R_{\lambda}$ that takes into account not only the probability of the losses, but the balance between such probability and the amount of the loss. This is obtained by defining a new class of law invariant risk measures based on an appropriate family of acceptance sets. The $V@R_{\lambda}$ and other known law invariant risk measures turn out to be special cases of our proposal. We further prove the dual representation of Risk Measures on $\mathcal{P}(% \mathbb{R}).$

In the conditional setting we provide a complete duality between quasiconvex risk measures defined on $L^{0}$ modules of the $L^{p}$ type and the appropriate class of dual functions. This is based on a general result which extends the usual Penot-Volle representation for quasiconvex real valued maps.

The accuracy of mass estimates by gravitational lensing using the thin-lens approximation applied to Navarro-Frenk-White mass models with a soft truncation mechanism recently proposed by Baltz, Marshall and Oguri is studied. The gravitational lens scenario considered is the case of the inference of lens mass from the observation of Einstein rings (strong lensing). It is found that the mass error incurred by the simplifying assumption of thin lenses is below 0.5%. As a byproduct, the optimal tidal radius of the soft truncation mechanism is found to be at most 10 times the virial radius of the mass model., Comment: accepted to MNRAS

We provide a dual representation of quasiconvex maps between two lattices of random variables in terms of conditional expectations. This generalizes the dual representation of quasiconvex real valued functions and the dual representation of conditional convex maps., Comment: Date changed Added one remark on assumption (c), page 6

For utility functions $u$ finite valued on $\mathbb{R}$, we prove a duality formula for utility maximization with random endowment in general semimartingale incomplete markets. The main novelty of the paper is that possibly non locally bounded semimartingale price processes are allowed. Following Biagini and Frittelli \cite{BiaFri06}, the analysis is based on the duality between the Orlicz spaces $(L^{\widehat{u}}, (L^{\widehat{u}})^*)$ naturally associated to the utility function. This formulation enables several key properties of the indifference price $\pi(B)$ of a claim $B$ satisfying conditions weaker than those assumed in literature. In particular, the indifference price functional $\pi$ turns out to be, apart from a sign, a convex risk measure on the Orlicz space $L^{\widehat{u}}$., Comment: Submitted to Mathematical Finance on April 18, 2008

We consider a stochastic financial incomplete market where the price processes are described by a vector-valued semimartingale that is possibly nonlocally bounded. We face the classical problem of utility maximization from terminal wealth, with utility functions that are finite-valued over $(a,\infty)$, $a\in\lbrack-\infty,\infty)$, and satisfy weak regularity assumptions. We adopt a class of trading strategies that allows for stochastic integrals that are not necessarily bounded from below. The embedding of the utility maximization problem in Orlicz spaces permits us to formulate the problem in a unified way for both the cases $a\in\mathbb{R}$ and $a=-\infty$. By duality methods, we prove the existence of solutions to the primal and dual problems and show that a singular component in the pricing functionals may also occur with utility functions finite on the entire real line., Comment: Published in at http://dx.doi.org/10.1214/07-AAP469 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We examine the accuracy of strong gravitational lensing determinations of the mass of galaxy clusters by comparing the conventional approach with the numerical integration of the fully relativistic null geodesic equations in the case of weak gravitational perturbations on Robertson-Walker metrics. In particular, we study spherically-symmetric, three-dimensional singular isothermal sphere models and the three-dimensional matter distribution of Navarro et al. (1997), which are both commonly used in gravitational lensing studies. In both cases we study two different methods for mass-density truncation along the line of sight: hard truncation and conventional (no truncation). We find that the relative error introduced in the total mass by the thin lens approximation alone is less than 0.3% in the singular isothermal sphere model, and less than 2% in the model of Navarro et al. (1997). The removal of hard truncation introduces an additional error of the same order of magnitude in the best case, and up to an order of magnitude larger in the worst case studied. Our results ensure that the future generation of precision cosmology experiments based on lensing studies will not require the removal of the thin-lens assumption, but they may require a careful handling of truncation., Comment: accepted to ApJ

We present a new computational framework (LEO), that enables us to carry out the very first large-scale, high-resolution computations in the context of the characteristic approach in numerical relativity. At the analytic level, our approach is based on a new implementation of the ``eth'' formalism, using a non-standard representation of the spin-raising and lowering angular operators in terms of non-conformal coordinates on the sphere; we couple this formalism to a partially first-order reduction (in the angular variables) of the Einstein equations. The numerical implementation of our approach supplies the basic building blocks for a highly parallel, easily extensible numerical code. We demonstrate the adaptability and excellent scaling of our numerical code by solving, within our numerical framework, for a scalar field minimally coupled to gravity (the Einstein-Klein-Gordon problem) in 3-dimensions. The nonlinear code is globally second-order convergent, and has been extensively tested using as reference a calibrated code with the same boundary-initial data and radial marching algorithm. In this context, we show how accurately we can follow quasi-normal mode ringing. In the linear regime, we show energy conservation for a number of initial data sets with varying angular structure. A striking result that arises in this context is the saturation of the flow of energy through the Schwarzschild radius. As a final calibration check we perform a large simulation with resolution never achieved before., Comment: RevTeX4, 22 pages, 21 figures, to appear in Phys. Rev. D

In an incomplete market the price of a claim f in general cannot be uniquely identified by no arbitrage arguments. However, the ``classical'' super replication price is a sensible indicator of the (maximum selling) value of the claim. When f satisfies certain pointwise conditions (e.g., f is bounded from below), the super replication price is equal to sup_QE_Q[f], where Q varies on the whole set of pricing measures. Unfortunately, this price is often too high: a typical situation is here discussed in the examples. We thus define the less expensive weak super replication price and we relax the requirements on f by asking just for ``enough'' integrability conditions. By building up a proper duality theory, we show its economic meaning and its relation with the investor's preferences. Indeed, it turns out that the weak super replication price of f coincides with sup_{Q\in M_{\Phi}}E_Q[f], where M_{\Phi} is the class of pricing measures with finite generalized entropy (i.e., E[\Phi (\frac{dQ}{dP})]<\infty) and where \Phi is the convex conjugate of the utility function of the investor., Comment: Published at http://dx.doi.org/10.1214/105051604000000459 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

A choice of first-order variables for the characteristic problem of the linearized Einstein equations is found which casts the system into manifestly well-posed form. The concept of well-posedness for characteristic problems invoked is that there exists an \textit{a priori} estimate of the solution of the characteristic problem in terms of the data. The notion of manifest well-posedness consists of an algebraic criterion sufficient for the existence of the estimates, and is to characteristic problems as symmetric hyperbolicity is to Cauchy problems. Both notions have been made precise elsewere., Comment: 6 pages

An algebraic criterion that is sufficient to establish the existence of certain a priori estimates for the solution of first-order homogeneous linear characteristic problems is derived. Estimates of such kind ensure the stability of the solutions under small variations of the data. Characteristic problems that satisfy this criterion are, in a sense, manifestly well posed., Comment: 6 pages

We calculate certain estimates for the solution of the characteristic problem of the wave equation reduced to first order, in terms of the free data prescribed on two transverse surfaces, one of which is characteristic. Estimates of such kind ensure the stability of the solutions under small variations of the data. Similar estimates exist for the derivatives of the solution as well., Comment: 10 pages, 2 figures, to appear in J. Phys. A

Second-order formulations of the 3+1 Einstein equations obtained by eliminating the extrinsic curvature in terms of the time derivative of the metric are examined with the aim of establishing whether they are well posed, in cases of somewhat wide interest, such as ADM, BSSN and generalized Einstein-Christoffel. The criterion for well-posedness of second-order systems employed is due to Kreiss and Ortiz. By this criterion, none of the three cases are strongly hyperbolic, but some of them are weakly hyperbolic, which means that they may yet be well posed but only under very restrictive conditions for the terms of order lower than second in the equations (which are not studied here). As a result, intuitive transferences of the property of well-posedness from first-order reductions of the Einstein equations to their originating second-order versions are unwarranted if not false., Comment: v1:6 pages; v2:7 pages, discussion extended, to appear in Phys. Rev. D; v3: typos corrected, published version

In relation to the BSSN formulation of the Einstein equations, we write down the boundary conditions that result from the vanishing of the projection of the Einstein tensor normally to a timelike hypersurface. Furthermore, by setting up a well-posed system of propagation equations for the constraints, we show explicitly that there are three constraints that are incoming at the boundary surface and that the boundary equations are linearly related to them. This indicates that such boundary conditions play a role in enforcing the propagation of the constraints in the region interior to the boundary. Additionally, we examine the related problem for a strongly hyperbolic first-order reduction of the BSSN equations and determine the characteristic fields that are prescribed by the three boundary conditions, as well as those that are left arbitrary., Comment: 11 pages

It is known that a fully relativistic integration of the null geodesics of a weak perturbation of flat spacetime leads to a correction of order $v/c$ to the bending angle and time delay due to a gravitational lens in slow motion with small acceleration. The existence of the $v/c$ correction was verified by the VLBI experiment of the bending of light by Jupiter on September 8, 2002. Here the $v/c$ correction is interpreted by means of standard aberration of light in an optically active medium with an effective index of refraction induced by the gravitational field of a lens in motion., Comment: 3 pages