Hilbert-Schmidt composition-differentiation operators on the unit ball

We use the notion of radial derivative to introduce composition-differentiation operators on the Hardy and Bergman spaces of the unit ball and the polydisk. We seek for necessary and sufficient conditions on the inducing functions to ensure that the composition-differentiation operator is Hilbert-Schmidt.Comment: 13 page

Frozen propagation of Reynolds force vector from high-fidelity data into Reynolds-averaged simulations of secondary flows

Successful propagation of information from high-fidelity sources (i.e., direct numerical simulations and large-eddy simulations) into Reynolds-averaged Navier-Stokes (RANS) equations plays an important role in the emerging field of data-driven RANS modeling. Small errors carried in high-fidelity data can propagate amplified errors into the mean flow field, and higher Reynolds numbers worsen the error propagation. In this study, we compare a series of propagation methods for two cases of Prandtl's secondary flows of the second kind: square-duct flow at a low Reynolds number and roughness-induced secondary flow at a very high Reynolds number. We show that frozen treatments result in less error propagation than the implicit treatment of Reynolds stress tensor (RST), and for cases with very high Reynolds numbers, explicit and implicit treatments are not recommended. Inspired by the obtained results, we introduce the frozen treatment to the propagation of Reynolds force vector (RFV), which leads to less error propagation. Specifically, for both cases at low and high Reynolds numbers, propagation of RFV results in one order of magnitude lower error compared to RST propagation. In the frozen treatment method, three different eddy-viscosity models are used to evaluate the effect of turbulent diffusion on error propagation. We show that, regardless of the baseline model, the frozen treatment of RFV results in less error propagation. We combined one extra correction term for turbulent kinetic energy with the frozen treatment of RFV, which makes our propagation technique capable of reproducing both velocity and turbulent kinetic energy fields similar to high-fidelity data

Weak proximal normal structure and coincidence quasi-best proximity points

[EN] We introduce the notion of pointwise cyclic-noncyclic relatively nonexpansive pairs involving orbits. We study the best proximity point problem for this class of mappings. We also study the same problem for the class of pointwise noncyclic-noncyclic relatively nonexpansive pairs involving orbits. Finally, under the assumption of weak proximal normal structure, we prove a coincidence quasi-best proximity point theorem for pointwise cyclic-noncyclic relatively nonexpansive pairs involving orbits. Examples are provided to illustrate the observed results.Fouladi, F.; Abkar, A.; Karapinar, E. (2020). Weak proximal normal structure and coincidence quasi-best proximity points. Applied General Topology. 21(2):331-347. https://doi.org/10.4995/agt.2020.13926OJS331347212A. Abkar and M. Gabeleh, Best proximity points for cyclic mappings in ordered metric spaces, J. Optim. Theorey. Appl. 150 (2011), 188-193. https://doi.org/10.1007/s10957-011-9810-xA.Abkar and M. Norouzian, Coincidence quasi-best proximity points for quasi-cyclic-noncyclic mappings in convex metric spaces, Iranian Journal of Mathematical Sciences and Informatics, to appear.M. A. Al-Thagafi and N. Shahzad, Convergence and existence results for best proximity points, Nonlinear Anal. 70 (2009), 3665-3671. https://doi.org/10.1016/j.na.2008.07.022M. S. Brodskii and D. P. Milman, On the center of a convex set, Dokl. Akad. Nauk USSR 59 (1948), 837-840 (in Russian).M. De la Sen, Some results on fixed and best proximity points of multivalued cyclic self mappings with a partial order, Abst. Appl. Anal. 2013 (2013), Article ID 968492, 11 pages. https://doi.org/10.1155/2013/968492M. De la Sen and R. P. Agarwal, Some fixed point-type results for a class of extended cyclic self mappings with a more general contractive condition, Fixed Point Theory Appl. 59 (2011), 14 pages. https://doi.org/10.1186/1687-1812-2011-59C. Di Bari, T. Suzuki and C. Verto, Best proximity points for cyclic Meir-Keeler contractions, Nonlinear Anal. 69 (2008), 3790-3794. https://doi.org/10.1016/j.na.2007.10.014A. A. Eldred, W. A. Kirk and P. Veeramani, Proximal normal structure and relatively nonexpansive mappings, Studia Math. 171 (2005), 283-293. https://doi.org/10.4064/sm171-3-5R. Espinola, M. Gabeleh and P. Veeramani, On the structure of minimal sets of relatively nonexpansive mappings, Numer. Funct. Anal. Optim. 34 (2013), 845-860. https://doi.org/10.1080/01630563.2013.763824A. F. Leon and M. Gabeleh, Best proximity pair theorems for noncyclic mappings in Banach and metric spaces, Fixed Point Theory 17 (2016), 63-84.M. Gabeleh, A characterization of proximal normal structure via proximal diametral sequences, J. Fixed Point Theory Appl. 19 (2017), 2909-2925. https://doi.org/10.1007/s11784-017-0460-yM. Gabeleh, O. Olela Otafudu and N. Shahzad, Coincidence best proximity points in convex metric spaces, Filomat 32 (2018), 1-12. https://doi.org/10.2298/FIL1801001DM. Gabeleh, H. Lakzian and N.Shahzad, Best proximity points for asymptotic pointwise contractions, J. Nonlinear Convex Anal. 16 (2015), 83-93.E. Karapinar, Best proximity points of Kannan type cyclic weak φ-contractions in ordered metric spaces, An. St. Univ. Ovidius Constanta. 20 (2012), 51-64. https://doi.org/10.2478/v10309-012-0055-yH. Aydi, E. Karapinar, I. M. Erhan and P. Salimi, Best proximity points of generalized almost -ψ Geraghty contractive non-self mappings, Fixed Point Theory Appl. 2014:32 (2014). https://doi.org/10.1186/1687-1812-2014-32N. Bilgili, E. Karapinar and K. Sadarangani, A generalization for the best proximity point of Geraghty-contractions, J. Ineqaul. Appl. 2013:286 (2013). https://doi.org/10.1186/1029-242X-2013-286E. Karapinar and I. M. Erhan, Best proximity point on different type contractions, Appl. Math. Inf. Sci. 3, no. 3 (2011), 342-353.E. Karapinar, Fixed point theory for cyclic weak $phi$-contraction, Appl. Math. Lett. 24, no. 6 (2011), 822-825. https://doi.org/10.1186/1687-1812-2011-69E. Karapinar, G. Petrusel and K. Tas, Best proximity point theorems for KT-types cyclic orbital contraction mappings, Fixed Point Theory 13, no. 2 (2012), 537-546. https://doi.org/10.1186/1687-1812-2012-42W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006. https://doi.org/10.2307/2313345W. A. Kirk, S. Reich and P. Veeramani, Proximinal retracts and best proximity pair theorems, Numer. Funct. Anal. Optim. 24 (2003), 851-862. https://doi.org/10.1081/NFA-120026380U. Kohlenbach, Some logical metatheorems with applications in functional analysis, Trans. Amer. Math. Soc. 357 (2005), 89-128. https://doi.org/10.1090/S0002-9947-04-03515-9V. Pragadeeswarar and M. Marudai, Best proximity points: approximation and optimization in partially ordered metric spaces, Optim. Lett. 7 (2013), 1883-1892. https://doi.org/10.1007/s11590-012-0529-xT. Shimizu and W. Takahashi, Fixed points of multivalued mappings in certain convex metric spaces, Topological Methods in Nonlin. Anal. 8 (1996), 197-203. https://doi.org/10.12775/TMNA.1996.028T. Suzuki, M. Kikkawa and C. Vetro, The existence of best proximity points in metric spaces with to property UC, Nonlinear Anal. 71 (2009), 2918-2926. https://doi.org/10.1016/j.na.2009.01.17