Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThermodynamic and vortic fine structures of real Schur flows
52
0
0.0
(
0
)
Authors
Jian-Zhou Zhu
Ask ChatGPT about the research
No Arabic abstract
A two-component-two-dimensional coupled with one-component-three-dimensional (2C2Dcw1C3D) flow may also be called a real Schur flow (RSF), as its velocity gradient is uniformly of real Schur form, the latter being the intrinsic local property of any general flows. The thermodynamic and `vortic fine structures of RSF are exposed and, in particular, the complete set of equations governing a (viscous and/or driven) 2C2Dcw1C3D flow are derived. The Lie invariances of the decomposed vorticity 2-forms of RSFs in $d$-dimensional Euclidean space $mathbb{E}^d$ for any interger $dge 3$ are also proven, and many Lie-invariant fine results, such as those of the combinations of the entropic and vortic quantities, including the invariances of the decomposed Ertel potential vorticity (and their multiplications by any interger powers of entropy) 3-forms, then follow.
rate research
Read More
The problem of a flow with its velocity gradient being of textit{real Schur form} uniformly in a cyclic box is formulated for numerical simulation, and a semi-analytic algorithm is developed from the precise structures. Computations starting from two-component-two-dimensional-coupled-with-one-component-three-dimensional initial velocity fields of the Taylor-Green and Arnold-Beltrami-Childress fashions are carried out, and some discussions related to turbulence are offered for the multi-scale eddies which, though, present precise order and symmetry. Plenty of color pictures of patterns of these completely new flows are presented for general and specific conceptions.
Solar filaments, also called solar prominences when appearing above the solar limb, are cold, dense materials suspended in the hot tenuous solar corona, consisting of numerous long, fibril-like threads. These threads are the key to disclosing the physics of solar filaments. Similar structures also exist in galaxy clusters. Besides their mysterious formation, filament threads are observed to move with alternating directions, which are called counterstreaming flows. However, the origin of these flows has not been clarified yet. Here we report that turbulent heating at the solar surface is the key, which randomly evaporates materials from the solar surface to the corona, naturally reproducing the formation and counterstreamings of the sparse threads in the solar corona. We further suggest that while the cold H$alpha$ counterstreamings are mainly due to longitudinal oscillations of the filament threads, there are million-kelvin counterstreamings in the corona between threads, which are alternating unidirectional flows.
In these notes, we consider the problem of finding the logarithm or the square root of a real matrix. It is known that for every real n x n matrix, A, if no real eigenvalue of A is negative or zero, then A has a real logarithm, that is, there is a real matrix, X, such that e^X = A. Furthermore, if the eigenvalues, xi, of X satisfy the property -pi < Im(xi) < pi, then X is unique. It is also known that under the same condition every real n x n matrix, A, has a real square root, that is, there is a real matrix, X, such that X^2 = A. Moreover, if the eigenvalues, rho e^{i theta}, of X satisfy the condition -pi/2 < theta < pi/2, then X is unique. These theorems are the theoretical basis for various numerical methods for exponentiating a matrix or for computing its logarithm using a method known as scaling and squaring (resp. inverse scaling and squaring). Such methods play an important role in the log-Euclidean framework due to Arsigny, Fillard, Pennec and Ayache and its applications to medical imaging. Actually, there is a necessary and sufficient condition for a real matrix to have a real logarithm (or a real square root) but it is fairly subtle as it involves the parity of the number of Jordan blocks associated with negative eigenvalues. As far as I know, with the exception of Highams recent book, proofs of these results are scattered in the literature and it is not easy to locate them. Moreover, Highams excellent book assumes a certain level of background in linear algebra that readers interested in the topics of this paper may not possess so we feel that a more elementary presentation might be a valuable supplement to Higham. In these notes, I present a unified exposition of these results and give more direct proofs of some of them using the Real Jordan Form.
The proofs that the real numbers are denumerable will be shown, i.e., that there exists one-to-one correspondence between the natural numbers $N$ and the real numbers $Re$. The general element of the sequence that contains all real numbers will be explicitly specified, and the first few elements of the sequence will be written. Remarks on the Cantors nondenumerability proofs of 1873 and 1891 that the real numbers are noncountable will be given.
In this paper we discuss various philosophical aspects of the hyperstructure concept extending networks and higher categories. By this discussion we hope to pave the way for applications and further developments of the mathematical theory of hyperstructures.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
