ZND-ZeaD Models and Theoretics Including Proofs for Takagi Factorization of Complex Time-Dependent Symmetric Matrix

Author(s):  
Zhuosong Fu ◽  
Min Yang ◽  
Jinjin Guo ◽  
Jianrong Chen ◽  
Yunong Zhang
1995 ◽  
Vol 15 (2) ◽  
pp. 26-36 ◽  
Author(s):  
J. Dorsey ◽  
J. Arvo ◽  
D. Greenberg

2019 ◽  
Vol 38 (20) ◽  
pp. 3747-3763 ◽  
Author(s):  
Tobias Bluhmki ◽  
Hein Putter ◽  
Arthur Allignol ◽  
Jan Beyersmann ◽  

2001 ◽  
Vol 918 (1-2) ◽  
pp. 60-66 ◽  
Author(s):  
Sridhar Sunderam ◽  
Ivan Osorio ◽  
James F. Watkins ◽  
Steven B. Wilkinson ◽  
Mark G. Frei ◽  
...  

2014 ◽  
Vol 44 (9) ◽  
pp. 2498-2523 ◽  
Author(s):  
Olivier Marchal

Abstract This study examines the observability of a stratified ocean in a square flat basin on a midlatitude beta plane. Here, “observability” means the ability to establish, in a finite interval of time, the time-dependent ocean state given density observations over the same interval and with no regard for errors. The dynamics is linearized and hydrostatic, so that the motion can be decomposed into normal modes and the observability analysis is simplified. An observability Gramian (a symmetric matrix) is determined for the flows in an inviscid interior, in frictional boundary layers, and in a closed basin. Its properties are used to establish the condition for complete observability and to identify optimal data locations for each of these flows. It is found that complete observability of an oceanic interior in time-dependent Sverdrup balance requires that the observations originate from the westernmost location at each considered latitude. The degree of observability increases westward due to westward propagation of long baroclinic Rossby waves: data collected in the west are more informative than data collected in the east. Likewise, the best locations for observing variability in the western (eastern) boundary layer are near (far from) the boundary. The observability of a closed basin is influenced by the westward propagation and the boundaries. Optimal data locations that are identified for different resolutions (0.01 to 1 yr) and lengths of data records (0.2 to 20 yr) show a variable influence of the planetary vorticity gradient. Data collected near the meridional boundaries appear always less informative, from the viewpoint of basin observability, than data collected away from these boundaries.


2020 ◽  
Vol 35 (01) ◽  
pp. 2075001
Author(s):  
Naima Mana ◽  
Mustapha Maamache

Pedrosa et al.1 have recently used a [Formula: see text] symmetric linear invariant to study a unidimensional time-dependent [Formula: see text] symmetric harmonic oscillator with a complex time-dependent [Formula: see text] symmetric external force. We show in this comment that the normalization condition of the eigenfunctions of the invariant is not verified as claimed in Ref. 1. In order to obtain the normalization condition, we introduce a novel concept of the pseudoparity-time (pseudo-[Formula: see text]) symmetry.


2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Chia-Wei Hsu ◽  
An-Cheng Yang ◽  
Pei-Ching Kung ◽  
Nien-Ti Tsou ◽  
Nan-Yow Chen

AbstractEngineering simulation accelerates the development of reliable and repeatable design processes in various domains. However, the computing resource consumption is dramatically raised in the whole development processes. Making the most of these simulation data becomes more and more important in modern industrial product design. In the present study, we proposed a workflow comprised of a series of machine learning algorithms (mainly deep neuron networks) to be an alternative to the numerical simulation. We have applied the workflow to the field of dental implant design process. The process is based on a complex, time-dependent, multi-physical biomechanical theory, known as mechano-regulatory method. It has been used to evaluate the performance of dental implants and to assess the tissue recovery after the oral surgery procedures. We provided a deep learning network (DLN) with calibrated simulation data that came from different simulation conditions with experimental verification. The DLN achieves nearly exact result of simulated bone healing history around implants. The correlation of the predicted essential physical properties of surrounding bones (e.g. strain and fluid velocity) and performance indexes of implants (e.g. bone area and bone-implant contact) were greater than 0.980 and 0.947, respectively. The testing AUC values for the classification of each tissue phenotype were ranging from 0.90 to 0.99. The DLN reduced hours of simulation time to seconds. Moreover, our DLN is explainable via Deep Taylor decomposition, suggesting that the transverse fluid velocity, upper and lower parts of dental implants are the keys that influence bone healing and the distribution of tissue phenotypes the most. Many examples of commercial dental implants with designs which follow these design strategies can be found. This work demonstrates that DLN with proper network design is capable to replace complex, time-dependent, multi-physical models/theories, as well as to reveal the underlying features without prior professional knowledge.


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