Norm of Berezin transformon <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> space

It is shown that the Berezin transform B on L ^p (D), where D is the unit disc, has norm . Furthermore, the norms of a family of operators (on L ^p (D)) whose kernels are moduli of Bergman type kernels are also calculated. Partially supported by MNZZS, Grant ON144010     

Frequency modules and nonexistence of quasi-periodic solutions of nonlinear evolution equations

This paper gives lower estimates for the frequency modules of almost periodic solutions to equations of the form , where A generates a strongly continuous semigroup in a Banach space , F(t,x) is 2π-periodic in t and continuous in (t,x), and f is almost periodic. We show that the frequency module ℳ(u) of any almost periodic mild solution u of (*) and the frequency module ℳ(f) of f satisfy the estimate e ^{2π iℳ(f)}⊂e ^{2π iℳ(u)}. If F is independent of t, then the estimate can be improved: ℳ(f)⊂ℳ(u). Applications to the nonexistence of quasi-periodic solutions are also given.     

Uniqueness of weak solutions of the Navier-Stokes equations

Consider the Navier-Stokes equation with the initial data a ∈ L _σ ²(ℝ ^d ). Let u and v be two weak solutions with the same initial value a. If u satisfies the usual energy inequality and if ∇v ∈ L ²((0, T); (ℝ ^d ) ^d ) where (ℝ ^d ) is the multiplier space, then we have u = v.     

BMO from dyadic BMO via expectations on product spaces of homogeneous type

Using the random dyadic lattices developed by Hytönen and Kairema, we build up a bridge between BMO and dyadic BMO, and hence one between VMO and dyadic VMO, via expectations over dyadic lattices on spaces of homogeneous type, including both the one-parameter and product cases. We also obtain a similar relationship between _Ap$A_p$ and dyadic _Ap$A_p$, as well as one between the reverse Hölder class _RHp${\mathit{RH}}_p$ and dyadic _RHp${\mathit{RH}}_p$, via geometric–arithmetic expectations. These results extend the earlier theory along this line, developed by Garnett, Jones, Pipher, Ward, Xiao and Treil, to the more general setting of spaces of homogeneous type in the sense of Coifman and Weiss.     

Scheduling reconstruction operations with modes of execution

Since the decline of the Cold War, the risk of major conflict between powerful, industrialized nations has decreased. Insecurities in the 21st century are more apt to arise from the debris of imploding states. Such situations may require intervention—military or otherwise—by concerned states. To meet this new operational challenge, nations must adapt their planning procedures to account for Security, Stabilization, Transition, and Reconstruction Operations (SSTRO). This paper develops a project scheduling framework for post-conflict reconstruction that schedules reconstruction activities to maximize the positive impact to the local economy during the initial phase of SSTRO. Specifically, this paper builds on the Multimode Resource Constrained Project Scheduling Problem with Generalized Precedence Relations (MM-RCPSP-GPR) using goal programming to maximize the reconstruction operations’ positive impact to local population's economic welfare. This MM-RCPSP-GPR variant is applied to a notional example to illustrate its potential use in post-conflict SSTRO. The basic framework can be extended to other settings.     

Schauder estimates for higher-order parabolic systems with time irregular coefficients

We prove Schauder estimates for solutions to both divergence and non-divergence type higher-order parabolic systems in the whole space and a half space. We also provide an existence result for the divergence type systems in a cylindrical domain. All coefficients are assumed to be only measurable in the time variable and Hölder continuous in the spatial variables.     

Classical mean-variance model revisited: pseudo efficiency

Investigating the inverse problem of the classical Markowitz mean-variance formulation: Given a mean-variance pair, find initial investment levels and their corresponding portfolio policies such that the given mean-variance pair can be realized, we reveal that any mean-variance pair inside the reachable region can be achieved by multiple portfolio policies associated with different initial investment levels. Therefore, in the mean-variance world for a market of all risky assets, the common belief of monotonicity: ‘The larger you invest, the larger expected future wealth you can expect for a given risk (variance) level’ does not hold, which stimulates us to extend the classical two-objective mean-variance framework to an expanded three-objective framework: to maximize the mean and minimize the variance of the final wealth as well as to minimize the initial investment level. As a result, we eliminate from the policy candidate list the set of pseudo efficient policies that are efficient in the original mean-variance space, but inefficient in this newly introduced three-dimensional objective space.