Smooth counterexamples to strong unique continuation for a Beltrami system in C2\mathbb{C}^2

We construct an example of a smooth map $\mathbb{C}\to\mathbb{C}^2$ which vanishes to infinite order at the origin, and such that the ratio of the norm of the $\bar z$ derivative to the norm of the $z$ derivative also vanishes to infinite order. This gives a counterexample to strong unique continuation for a vector valued analogue of the Beltrami equation.Comment: 21 page

SOME NONLINEAR DIFFERENTIAL INEQUALITIES AND AN APPLICATION TO HÖLDER CONTINUOUS ALMOST COMPLEX STRUCTURES

Abstract. We consider some second order quasilinear partial differential inequalities for real valued functions on the unit ball and find conditions under which there is a lower bound for the supremum of nonnegative solutions that do not vanish at the origin. As a consequence, for complex valued functions f(z) satisfying ∂f/∂¯z = |f | α,0<α<1, and f(0) ̸ = 0, there is also a lower bound for sup |f | on the unit disk. For each α, we construct a manifold with an α-Hölder continuous almost complex structure where the Kobayashi-Royden pseudonorm is not upper semicontinuous. 1

Continuous solutions of nonlinear Cauchy-Riemann equations and pseudoholomorphic curves in normal coordinates

We establish elliptic regularity for nonlinear inhomogeneous Cauchy-Riemann equations under minimal assumptions, and give a counterexample in a borderline case. In some cases where the inhomogeneous term has a separable factorization, the solution set can be explicitly calculated. The methods also give local parametric formulas for pseudoholomorphic curves with respect to some continuous almost complex structures.Comment: Version 3 with new Section 4.2 based on suggestions of referee. Accepted to appear in Transactions AM

Development of a first generation perfusion process and medium for continuous processing based on existing fed-batch platform media

Process intensification leveraging perfusion offers tremendous potential for yield improvement over fed-batch processes for the production of monoclonal antibodies. In the context of continuous processing, the goal is to achieve highly intensified perfusion processes that allow substantial footprint reduction and enable flexible adaptation in new facilities. However developing a perfusion process and medium without prior technology requires leveraging the existing fed-batch platform knowledge. Evolving a medium for perfusion relies on designing suitable mixtures of basal and feed media that serve as adequate starting points for development. Focus on optimization of the medium to decrease byproduct waste, reduce unnecessary cell growth and enhance specific productivity is critical. Doing so would allow a more robust and controlled process, and allow steady-state to be more attainable which will aid in maintaining consistent product quality for continuous processing. Moreover, reducing medium utilization hence the ability to operate under lower cell specific perfusion rate was important in order to have a more economical and nimble process. In order to overcome the conventional perfusion medium bottlenecks of equipment capacity, liquid handling, transfer and storage, a different strategy to managing large bulk volume had to be undertaken in order to make fit for an existing small pilot plant. The approach to establishing a first generation perfusion process starting from a fed-batch platform will be shared. Examples demonstrating continuous perfusion and volumetric productivity of \u3e 1 g/L-day under low CSPR will be discussed