Regularity of roots of polynomials

We show that smooth curves of monic complex polynomials $P_a (Z)=Z^n+\sum_{j=1}^n a_j Z^{n-j}$, $a_j : I \to \mathbb C$ with $I \subset \mathbb R$ a compact interval, have absolutely continuous roots in a uniform way. More precisely, there exists a positive integer $k$ and a rational number $p >1$, both depending only on the degree $n$, such that if $a_j \in C^{k}$ then any continuous choice of roots of $P_a$ is absolutely continuous with derivatives in $L^q$ for all $1 \le q < p$, in a uniform way with respect to $\max_j\|a_j\|_{C^k}$. The uniformity allows us to deduce also a multiparameter version of this result. The proof is based on formulas for the roots of the universal polynomial $P_a$ in terms of its coefficients $a_j$ which we derive using resolution of singularities. For cubic polynomials we compute the formulas as well as bounds for $k$ and $p$ explicitly.Comment: 32 pages, 2 figures; minor changes; accepted for publication in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5); some typos correcte

A general tool for consistency results related to I1

In this paper we provide a general tool to prove the consistency of $I1(\lambda)$ with various combinatorial properties at $\lambda$ typical at settings with $2^\lambda>\lambda^+$, that does not need a profound knowledge of the forcing notions involved. Examples of such properties are the first failure of GCH, a very good scale and the negation of the approachability property, or the tree property at $\lambda^+$ and $\lambda^{++}$

Arcs on Determinantal Varieties

We study arc spaces and jet schemes of generic determinantal varieties. Using the natural group action, we decompose the arc spaces into orbits, and analyze their structure. This allows us to compute the number of irreducible components of jet schemes, log canonical thresholds, and topological zeta functions.Comment: 27 pages. This is part of the author's PhD thesis at the University of Illinois at Chicago. v2: Minor changes. To appear in Transactions of the American Mathematical Societ

Cluster-based feedback control of turbulent post-stall separated flows

We propose a novel model-free self-learning cluster-based control strategy for general nonlinear feedback flow control technique, benchmarked for high-fidelity simulations of post-stall separated flows over an airfoil. The present approach partitions the flow trajectories (force measurements) into clusters, which correspond to characteristic coarse-grained phases in a low-dimensional feature space. A feedback control law is then sought for each cluster state through iterative evaluation and downhill simplex search to minimize power consumption in flight. Unsupervised clustering of the flow trajectories for in-situ learning and optimization of coarse-grained control laws are implemented in an automated manner as key enablers. Re-routing the flow trajectories, the optimized control laws shift the cluster populations to the aerodynamically favorable states. Utilizing limited number of sensor measurements for both clustering and optimization, these feedback laws were determined in only $O(10)$ iterations. The objective of the present work is not necessarily to suppress flow separation but to minimize the desired cost function to achieve enhanced aerodynamic performance. The present control approach is applied to the control of two and three-dimensional separated flows over a NACA 0012 airfoil with large-eddy simulations at an angle of attack of $9^\circ$, Reynolds number $Re = 23,000$ and free-stream Mach number $M_\infty = 0.3$. The optimized control laws effectively minimize the flight power consumption enabling the flows to reach a low-drag state. The present work aims to address the challenges associated with adaptive feedback control design for turbulent separated flows at moderate Reynolds number.Comment: 32 pages, 18 figure