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We exploit changes in wind speeds at surfing locations in Switzerland as a source of variation in users' propensity to post content about their surfing activity on an online social network. We exploit this variation to test whether users' online content-generation activity is codetermined with their social ties. Economically significant effects of this type can produce positive feedback that generates local network effects in content generation. When quantitatively significant, the increased content and tie density arising from the network effect induces more visitation and browsing on the site, which fuels growth by generating advertising revenue. We find evidence consistent with such network effects. This paper was accepted by J. Miguel Villas-Boas, marketing.

We propose a prototypical Split Inverse Problem (SIP) and a new variational problem, called the Split Variational Inequality Problem (SVIP), which is a SIP. It entails finding a solution of one inverse problem (e.g., a Variational Inequality Problem (VIP)), the image of which under a given bounded linear transformation is a solution of another inverse problem such as a VIP. We construct iterative algorithms that solve such problems, under reasonable conditions, in Hilbert space and then discuss special cases, some of which are new even in Euclidean space.

We propose new primal-dual decomposition algorithms for solving systems of inclusions involving sums of linearly composed maximally monotone operators. The principal innovation in these algorithms is that they are block-iterative in the sense that, at each iteration, only a subset of the monotone operators needs to be processed, as opposed to all operators as in established methods. Flexible strategies are used to select the blocks of operators activated at each iteration. In addition, we allow lags in operator processing, permitting asynchronous implementation. The decomposition phase of each iteration of our methods is to generate points in the graphs of the selected monotone operators, in order to construct a half-space containing the Kuhn–Tucker set associated with the system. The coordination phase of each iteration involves a projection onto this half-space. We present two related methods: the first method provides weakly convergent primal and dual sequences under general conditions, while the second is a variant in which strong convergence is guaranteed without additional assumptions. Neither algorithm requires prior knowledge of bounds on the linear operators involved or the inversion of linear operators. Our algorithmic framework unifies and significantly extends the approaches taken in earlier work on primal-dual projective splitting methods.

This paper introduces time-continuous numerical schemes to simulate stochastic differential equations (SDEs) arising in mathematical finance, population dynamics, chemical kinetics, epidemiology, biophysics, and polymeric fluids. These schemes are obtained by spatially discretizing the Kolmogorov equation associated with the SDE in such a way that the resulting semi-discrete equation generates a Markov jump process that can be realized exactly using a Monte Carlo method. In this construction the jump size of the approximation can be bounded uniformly in space, which often guarantees that the schemes are numerically stable for both finite and long time simulation of SDEs. By directly analyzing the infinitesimal generator of the approximation, we prove that the approximation has a sharp stochastic Lyapunov function when applied to an SDE with a drift field that is locally Lipschitz continuous and weakly dissipative. We use this stochastic Lyapunov function to extend a local semimartingale representation of the approximation. This extension makes it possible to quantify the computational cost of the approximation. Using a stochastic representation of the global error, we show that the approximation is (weakly) accurate in representing finite and infinite-time expected values, with an order of accuracy identical to the order of accuracy of the infinitesimal generator of the approximation. The proofs are carried out in the context of both fixed and variable spatial step sizes. Theoretical and numerical studies confirm these statements, and provide evidence that these schemes have several advantages over standard methods based on time-discretization. In particular, they are accurate, eliminate nonphysical moves in simulating SDEs with boundaries (or confined domains), prevent exploding trajectories from occurring when simulating stiff SDEs, and solve first exit problems without time-interpolation errors.

The problem whether a given pair of functions can be used as the kernels of a generalized fractional derivative and the associated generalized fractional integral is reduced to the problem of existence of a solution to the Sonine equation. It is shown for some selected classes of functions that a necessary condition for a function to be the kernel of a fractional derivative is an integrable singularity at 0. It is shown that locally integrable completely monotone functions satisfy the Sonine equation if and only if they are singular at 0.

We give an overview of the different concepts and methods that are commonly used when studying the asymptotic properties of isotonic estimators. After introducing the inverse process, we illustrate its use in establishing weak convergence of the estimators at a fixed point and also weak convergence of global distances, such as the 𝕃𝑝-distance and supremum distance. Furthermore, we discuss the developments on smooth isotonic estimation.

The purpose of this paper is to introduce and investigate θ-monotone operators and θ-monotone bifunctions in the context of Banach space. Local boundedness of θ-monotone bifunctions in the interior of their domains is proved. Also, the difference of two θ-monotone operators is studied. Moreover, some relations between θ-monotonicity and θ-convexity are investigated.

We develop a perturbation theory for stochastic differential equations (SDEs) by which we mean both stochastic ordinary differential equations (SODEs) and stochastic partial differential equations (SPDEs). In particular, we estimate the Lp -distance between the solution process of an SDE and an arbitrary Itô process, which we view as a perturbation of the solution process of the SDE, by the Lq -distances of the differences of the local characteristics for suitable p, q > 0. As one application of the developed perturbation theory, we establish strong convergence rates for numerical approximations of a class of SODEs with nonglobally monotone coefficients. As another application of the developed perturbation theory, we prove strong convergence rates for spatial spectral Galerkin approximations of solutions of semilinear SPDEs with nonglobally monotone nonlinearities including Cahn–Hilliard–Cook-type equations and stochastic Burgers equations. Further applications of the developed perturbation theory include regularity analyses of solutions of SDEs with respect to their initial values as well as small-noise analyses for ordinary and partial differential equations.

In this work, we propose and analyse forward-backward-type algorithms for finding a zero of the sum of finitely many monotone operators, which are not based on reduction to a two operator inclusion in the product space. Each iteration of the studied algorithms requires one resolvent evaluation per set-valued operator, one forward evaluation per cocoercive operator, and two forward evaluations per monotone operator. Unlike existing methods, the structure of the proposed algorithms are suitable for distributed, decentralised implementation in ring networks without needing global summation to enforce consensus between nodes.

Endogenous demand composition across sectors due to income elasticity differences, or Engel's Law for brevity, affects (i) sectoral compositions in employment and in value-added, (ii) variations in innovation rates and in productivity change across sectors, (iii) intersectoral patterns of trade across countries, and (iv) product cycles from rich to poor countries. Using a two-country model of directed technical change with a continuum of sectors under nonhomothetic preferences, which is rich enough to capture all these effects as well as their interactions, this paper offers a unifying perspective on how economic growth and globalization affect the patterns of structural change, innovation, and trade across countries and across sectors in the presence of Engel's Law. Among the main messages is that globalization amplifies, instead of reducing, the power of endogenous domestic demand composition differences as a driver of structural change.