The valence of a function f at a point w is the number of distinct, finite solutions to f(z) = w. Let f be a complex-valued harmonic function in an open set R ⊆ C. Let S denote the critical set of f and C(f) the global cluster set of f . We show that f(S)∪C(f) partitions the complex plane into regions of constant valence. We give some conditions such that f(S) ∪ C(f) has empty interior. We also...

All of these are of the form Au ˆ = f where  is self-adjoint and positive-definite (assuming zero Dirichlet boundary conditions and an appropriate inner product 〈u, v〉), as we’ve seen previously in class, so they should have unique solutions (excluding pathological c functions). However, the solutions may in general be quite different from those of −∇u = f . Can we relate them to G0, the Green...

I revisit the Great Inflation and the Great Moderation for nominal and real variables. I document a dichotomy in the evolution of the cyclical volatility of financial variables since the mid-1980s. While financial price variables are smoother during the Great Moderation, financial quantity measures experience an immoderation. A model with financial frictions and financial shocks allowing for st...

This paper studies how well a search and matching model can describe aggregate Japanese labor market dynamics in a full information setting. We develop a discretetime search and matching model with productivity and separation shocks and use it as a data-generating process for our empirical analysis. Using Bayesian methods, we estimate the model for data on unemployment and vacancy postings in J...

We show that if the graph of a bounded analytic function in the unit disk D is not complete pluripolar in C then the projection of the closure of its pluripolar hull contains a fine neighborhood of a point p ∈ ∂D. On the other hand we show that if an analytic function f in D extends to a function F which is defined on a fine neighborhood of a point p ∈ ∂D and is finely analytic at p then the pl...

We present a model in which a gauge symmetry of a field theory is intrinsic in the geometry of an extended space time itself. A consequence is that the dimension of our space time is restricted through the BRS cohomology. If the Hilbert space is a dense subspace of the space of all square integrable C functions, the BRS cohomology classes are nontrivial only when the dimension is two or four. p...

Observed choices in Social Dilemma Games usually take the form of bounded integers. We propose a doubly-truncated count data framework to process such data. We compare this framework to past approaches based on ordered outcomes and truncated continuous densities using Bayesian estimation and model selection techniques. We find that all three frameworks (i) support the presence of unobserved het...

For any C contact Anosov flow with r ≥ 3, we construct a scale of Hilbert spaces, which are embedded in the space of distributions on the phase space and contain all the C functions, such that the oneparameter family of the transfer operators for the flow extend to them boundedly and that the extensions are quasi-compact. Further we give explicit bounds on the essential spectral radii of those ...

In this article, we present a detailed study of the complex calculus of variations introduced in [4]. This calculus is analogous to the conventional calculus of variations, but is applied here to C functions in C. It is based on new concepts involving the minimum and convexity of a complex function. Such an approach allows us to propose explicit solutions to complex Hamilton-Jacobi equations, i...

We correct for endogeneity bias in fixed-effects estimates of workerand firmspecific earnings heterogeneity using longitudinal linked employer-employee data. Realized job assignments may be determined by unobservable components of earnings. We exploit the relational structure of the data to model the evolution of the matched data as an evolving bipartite graph in a Bayesian latent class framewo...