A finite basis theorem for product varieties of groups

1970 ◽  
Vol 2 (1) ◽  
pp. 39-44 ◽  
Author(s):  
M.S. Brooks ◽  
L.G. Kovács ◽  
M.F. Newman
Keyword(s):  
Product Variety ◽  
Finite Basis ◽  
Sufficient Condition ◽  
Nilpotent Variety ◽  
Basis Theorem ◽  
General Sufficient Condition ◽  
Special Cases

It is shown that, if U is a subvariety of the join of a nilpotent variety and a metabelian variety and if V is a variety with a finite basis for its laws, then UV also has a finite basis for its laws. The special cases U nilpotent and U metabelian have been established by Higman (1959) and Ivanjuta (1969) respectively. The proof here, which is independent of Ivanjuta's, depends on a rather general sufficient condition for a product variety to have a finite basis for its laws.

A sufficient condition for instability of BGK type waves in both unmagnetized and magnetized plasmas

1977 ◽  
Vol 17 (1) ◽  
pp. 57-68 ◽  
Author(s):  
E. Infeld ◽  
G. Rowlands
Keyword(s):  
Magnetic Field ◽  
Linear Theory ◽  
General Problem ◽  
Uniform Magnetic Field ◽  
Magnetized Plasmas ◽  
Sufficient Condition ◽  
General Sufficient Condition ◽  
Special Cases

This paper investigates the general problem of stability of Bernstein—Greene— Kruskal type waves. By investigating perturbations perpendicular to the wave, we obtain a general sufficient condition for instability. This is then extended to the case of magnetized plasmas with a uniform magnetic field in the direction of the BGK wave. New perturbed modes, having no counterpart in linear theory, are also found. Various special cases are considered and previous, more particular results confirmed.

A Combinatorial Distinction Between Unit Circles and Straight Lines: How Many Coincidences Can they Have?

2009 ◽  
Vol 18 (5) ◽  
pp. 691-705 ◽  
Author(s):  
GYÖRGY ELEKES ◽  
MIKLÓS SIMONOVITS ◽  
ENDRE SZABÓ
Keyword(s):  
Parameter Family ◽  
Sufficient Condition ◽  
Simple Application ◽  
Quadratic Number ◽  
General Sufficient Condition ◽  
Family Of Curves ◽  
Unit Circles ◽  
Triple Points ◽  
Straight Lines ◽  
Special Case

We give a very general sufficient condition for a one-parameter family of curves not to have n members with ‘too many’ (i.e., a near-quadratic number of) triple points of intersections. As a special case, a combinatorial distinction between straight lines and unit circles will be shown. (Actually, this is more than just a simple application; originally this motivated our results.)

scholarly journals A proof of Lyndon's finite basis theorem

1980 ◽  
Vol 29 (3) ◽  
pp. 229-233 ◽  
Author(s):  
Joel Berman
Keyword(s):  
Finite Basis ◽  
Basis Theorem

scholarly journals Convergence of the Chern–Moser–Beloshapka normal forms

2020 ◽  
Vol 2020 (765) ◽  
pp. 205-247
Author(s):  
Bernhard Lamel ◽  
Laurent Stolovitch
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Direct Proof ◽  
Sufficient Condition ◽  
Normalizing Transformation ◽  
General Sufficient Condition ◽  
Real Analytic ◽  
Moser Normal Form

AbstractIn this article, we give a normal form for real-analytic, Levi-nondegenerate submanifolds of{\mathbb{C}^{N}}of codimension{d\geq 1}under the action of formal biholomorphisms. We find a very general sufficient condition on the formal normal form that ensures that the normalizing transformation to this normal form is holomorphic. In the case{d=1}our methods in particular allow us to obtain a new and direct proof of the convergence of the Chern–Moser normal form.

FOURTH MOMENT STRUCTURE OF THE GARCH(p,q) PROCESS

1999 ◽  
Vol 15 (6) ◽  
pp. 824-846 ◽  
Author(s):  
Changli He ◽  
Timo Teräsvirta
Keyword(s):  
Statistical Theory ◽  
Autocorrelation Function ◽  
Fourth Moment ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Special Cases ◽  
The Moment ◽  
Necessary And Sufficient

In this paper, a necessary and sufficient condition for the existence of the unconditional fourth moment of the GARCH(p,q) process is given and also an expression for the moment itself. Furthermore, the autocorrelation function of the centered and squared observations of this process is derived. The statistical theory is further illustrated by a few special cases such as the GARCH(2,2) process and the ARCH(q) process.

scholarly journals A Basis Theorem for Perfect Sets

1998 ◽  
Vol 4 (2) ◽  
pp. 204-209 ◽  
Author(s):  
Marcia J. Groszek ◽  
Theodore A. Slaman
Keyword(s):  
Set Theory ◽  
General Theorem ◽  
Sufficient Condition ◽  
Specific Instance ◽  
Basis Theorem ◽  
Perfect Set ◽  
Models Of Set Theory

AbstractWe show that if there is a nonconstructible real, then every perfect set has a nonconstructible element, answering a question of K. Prikry. This is a specific instance of a more general theorem giving a sufficient condition on a pair M ⊂ N of models of set theory implying that every perfect set in N has an element in N which is not in M.

On the general bilinear time series model

1988 ◽  
Vol 25 (3) ◽  
pp. 553-564 ◽  
Author(s):  
Jian Liu ◽  
Peter J. Brockwell
Keyword(s):  
Time Series ◽  
Stationary Solution ◽  
Time Series Model ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Special Cases ◽  
Non Linear ◽  
Necessary And Sufficient ◽  
Special Case

A sufficient condition is derived for the existence of a strictly stationary solution of the general bilinear time series equations. The condition is shown to reduce to the conditions of Pham and Tran (1981) and Bhaskara Rao et al. (1983) in the special cases which they consider. Under the condition specified, a solution is constructed which is shown to be causal, stationary and ergodic. It is moreover the unique causal solution and the unique stationary solution of the defining equations. In the special case when the defining equations contain no non-linear terms, our condition reduces to the well-known necessary and sufficient condition for existence of a causal stationary solution.

Integer points on curves of genus 1

1970 ◽  
Vol 67 (3) ◽  
pp. 595-602 ◽  
Author(s):  
A. Baker ◽  
J. Coates
Keyword(s):  
Finite Number ◽  
Finite Basis ◽  
Rational Approximations ◽  
Algebraic Numbers ◽  
Integer Points ◽  
Basis Theorem

1. Introduction. A well-known theorem of Siegel(5) states that there exist only a finite number of integer points on any curve of genus ≥ 1. Siegel's proof, published in 1929, depended, inter alia, on his earlier work concerning rational approximations to algebraic numbers and on Weil's recently established generalization of Mordell's finite basis theorem. Both of these possess a certain non-effective character and thus it is clear that Siegel's argument cannot provide an algorithm for determining all the integer points on the curve. The purpose of the present paper is to establish such an algorithm in the case of curves of genus 1.

An extension of Willard?s Finite Basis Theorem: Congruence meet-semidistributive varieties of finite critical depth

2005 ◽  
Vol 52 (2-3) ◽  
pp. 289-302 ◽  
Author(s):  
Kirby A. Baker ◽  
George F. McNulty ◽  
Ju Wang
Keyword(s):  
Finite Basis ◽  
Critical Depth ◽  
Basis Theorem
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