scholarly journals Free arrangements of hyperplanes over an arbitrary field

1983 ◽  
Vol 59 (7) ◽  
pp. 301-303 ◽  
Author(s):  
Hiroaki Terao
Keyword(s):  
Arbitrary Field ◽  
Arrangements Of Hyperplanes

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

scholarly journals Irreducible modules with highest weight vectors over modular Witt and special Lie superalgebras

2019 ◽  
Vol 17 (1) ◽  
pp. 1381-1391
Author(s):  
Keli Zheng ◽  
Yongzheng Zhang
Keyword(s):  
Lie Superalgebras ◽  
Arbitrary Field ◽  
Special Linear ◽  
Highest Weight ◽  
Irreducible Modules

Abstract Let 𝔽 be an arbitrary field of characteristic p > 2. In this paper we study irreducible modules with highest weight vectors over Witt and special Lie superalgebras of 𝔽. The same irreducible modules of general and special linear Lie superalgebras, which are the 0-th part of Witt and special Lie superalgebras in certain ℤ-grading, are also considered. Then we establish a certain connection called a P-expansion between these modules.

Derivation algebra of semi-direct sum of Lie algebras

2021 ◽  
pp. 2250063
Author(s):  
Mohammad Reza Alemi ◽  
Farshid Saeedi
Keyword(s):  
Lie Algebras ◽  
Arbitrary Field ◽  
Direct Sum ◽  
Derivation Algebra

Let [Formula: see text] and [Formula: see text] be two Lie algebras over an arbitrary field [Formula: see text], and let [Formula: see text] be the semidirect sum of [Formula: see text] by [Formula: see text]. In this paper, we give the structure of derivation algebra of [Formula: see text]; then as a consequence we illustrate the structure and dimension derivation algebra of Heisenberg Lie algebras.

Kolmogoroff's theory of locally isotropic turbulence

1947 ◽  
Vol 43 (4) ◽  
pp. 533-559 ◽  
Author(s):  
G. K. Batchelor
Keyword(s):  
Reynolds Number ◽  
Energy Dissipation ◽  
Isotropic Turbulence ◽  
Flow Characteristics ◽  
Turbulent Motion ◽  
Statistical Characteristics ◽  
Considerable Proportion ◽  
Arbitrary Field ◽  
Mean Flow ◽  
Local Mean

A new and fruitful theory of turbulent motion was published in 1941 by A. N. Kolmogoroff. It does not seem to be as widely known outside the U.S.S.R. as its importance warrants, and the present paper therefore describes the theory in some detail before presenting a number of extensions and making a comparison of experimental results with some of the theoretical predictions.Kolmogoroff's basic notion is that at high Reynolds number all kinds of turbulent motion, of arbitrary mean-flow characteristics, show a similar structure if attention is confined to the smallest eddies. The motion due to these eddies of limited size is conceived to be isotropic and statistically steady. Within this range of eddies we recognize two limiting processes. The influence of viscosity on the larger eddies of the range is negligible if the Reynolds number is large enough, so that their motion is determined entirely by the amount of energy which they are continually passing on to smaller eddies. This quantity of energy is the local mean energy dissipation due to turbulence. On the other hand, the smaller eddies of the range dissipate through the action of viscosity a considerable proportion of the energy which they receive, and the motion of the very smallest eddies is entirely laminar. The analytical expression of this physical picture is that the motion due to eddies less than a certain limiting size in an arbitrary field of turbulence is determined uniquely by two quantities, the viscosity and the local mean energy dissipation per unit mass of the fluid.The mathematical method of describing the motion due to eddies of a particular size is to construct correlations between the differences of parallel-velocity components at two points at an appropriate distance apart. Kinematical results analogous to those for turbulence which is isotropic in the ordinary sense are obtained, and then the scalar functions occurring in the expressions for the correlations are determined by dimensional analysis. The consequences of the theory in the case of turbulence which possesses ordinary isotropy are analysed and various predictions are made. One of these, namely that dimensionless ratios of moments of the probability distribution of the rate of extension of the fluid in any direction are universal constants, is confirmed by recent experiments, so far as the second and third moments are concerned. In several other cases it can be said that relations predicted by the theory have the correct form, but further experiments at Reynolds numbers higher than those hitherto used will be required before the theory can be regarded as fully confirmed. If valid, Kolmogoroff's theory of locally isotropic turbulence will provide a powerful tool for the analysis of problems of non-uniform turbulent flow, and for the determination of statistical characteristics of space and time derivatives of quantities influenced by the turbulence.

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