Banach spaces with polynomial numerical index 1

We characterize Banach spaces with polynomial numerical index 1 when they have the Radon-Nikod\'ym property. The holomorphic numerical index is introduced and the characterization of the Banach space with holomorphic numerical index 1 is obtained when it has the Radon-Nikod\'ym property

Monotonicity and complex convexity in Banach lattices

AbstractThe goal of this article is to study the relations among monotonicity properties of real Banach lattices and the corresponding convexity properties in the complex Banach lattices. We introduce the moduli of monotonicity of Banach lattices. We show that a Banach lattice E is uniformly monotone if and only if its complexification EC is uniformly complex convex. We also prove that a uniformly monotone Banach lattice has finite cotype. In particular, we show that a Banach lattice is of cotype q for some 2⩽q<∞ if and only if there is an equivalent lattice norm under which it is uniformly monotone and its complexification is q-uniformly PL-convex. We also show that a real Köthe function space E is strictly (respectively uniformly) monotone and a complex Banach space X is strictly (respectively uniformly) complex convex if and only if Köthe–Bochner function space E(X) is strictly (respectively uniformly) complex convex

CFD Performance of Turbulence Models for Flow from Supersonic Nozzle Exhausts

The goal of this thesis is to compare the performance of several eddy-viscosity turbulence models for computing supersonic nozzle exhaust flows. These flows are of relevance in the development of future supersonic transport airplane. Flow simulations of exhaust flows from three supersonic nozzles are computed using ANSYS Fluent. Simulation results are compared to experimental data to assess the performance of various one- and two-equation turbulence models for accurately predicting the supersonic plume flow. One particular turbulence model of interest is the Wray-Agarwal (WA) turbulence model. This is a neat model which has demonstrated promising results mimicking the strength of two equation k-ω model while being a one equation model. Compressibility corrections are implemented for CFD simulations with SST k-ω, k-ε and low Reynolds versions of k-ε models which improved the results compared to the baseline models without compressibility correction. A compressibility correction for WA model is also developed to compare the performance of a compressibility correction to WA model with the compressibility correction to other models. Results show that the standard eddy-viscosity models can capture the shock structure and shear layer of the plume accurately when the thickness of the shear layer is small compared to plume diameter. However, when thickness of the shear layer is relatively large, a compressibility correction should be implemented to predict the supersonic jet flow. However, the use of compressibility correction consistently overestimates the length of potential core on the centerline of the plume although it improves the prediction of the velocity profile in other regions of the flow field such as the mixing region. Also, it is speculated that an accurate prediction of boundary layer profile at the nozzle exit has an influence in the model’s ability to predict the length of potential core as well as the shear layer growth rate. No single model appears to capture all features of the plumes’ flow fields without or with compressibility correction. In particular, WA model shows an excellent potential for computation of supersonic nozzles’ exhaust flows; however further improvements and investigations in WA model are warranted

Simultaneously continuous retraction and Bishop-Phelps-Bollob\'as type theorem

We study the existence of a retraction from the dual space $X^*$ of a (real or complex) Banach space $X$ onto its unit ball $B_{X^*}$ which is uniformly continuous in norm topology and continuous in weak-$*$ topology. Such a retraction is called a uniformly simultaneously continuous retraction. It is shown that if $X$ has a normalized unconditional Schauder basis with unconditional basis constant 1 and $X^*$ is uniformly monotone, then a uniformly simultaneously continuous retraction from $X^*$ onto $B_{X^*}$ exists. It is also shown that if $\{X_i\}$ is a family of separable Banach spaces whose duals are uniformly convex with moduli of convexity $\delta_i(\varepsilon)$ such that $\inf_i \delta_i(\varepsilon)>0$ and $X= \left[\bigoplus X_i\right]_{c_0}$ or $X=\left[\bigoplus X_i\right]_{\ell_p}$ for $1\le p<\infty$, then a uniformly simultaneously continuous retraction exists from $X^*$ onto $B_{X^*}$. The relation between the existence of a uniformly simultaneously continuous retraction and the Bishsop-Phelps-Bollob\'as property for operators is investigated and it is proved that the existence of a uniformly simultaneously continuous retraction from $X^*$ onto its unit ball implies that a pair $(X, C_0(K))$ has the Bishop-Phelps-Bollob\'as property for every locally compact Hausdorff spaces $K$. As a corollary, we prove that $(C_0(S), C_0(K))$ has the Bishop-Phelps-Bollob\'as property if $C_0(S)$ and $C_0(K)$ are the spaces of all real-valued continuous functions vanishing at infinity on locally compact metric space $S$ and locally compact Hausdorff space $K$ respectively.Comment: 15 page