A New Bound for Nil U-Rings

1970 ◽  
Vol 22 (2) ◽  
pp. 403-407 ◽  
Author(s):  
R. G. Biggs
Keyword(s):  
Ordinal Number ◽  
Multiplicative Semigroup ◽  
Good Bound ◽  
Nilpotent Ring ◽  
Limit Ordinal ◽  
Locally Nilpotent ◽  
A Chain ◽  
Image Position

A U-ring is a ring in which every subring is a meta ideal. A meta ideal of a ring R is a subring I of R which lies in a chain of subrings,with the properties:(1) Iλ is an ideal of Iλ+1 for all λ < β;(2) If α is a limit ordinal number, then Iα = ∪λ<αIλ.Freidman [3] proved that every nil U-ring is a locally nilpotent ring. Since there are many locally nilpotent rings which are not U-rings, the class of locally nilpotent rings is not a very good bound for the class of nil U-rings. This paper establishes a new bound for nil U-rings based on a property of the multiplicative semigroup of the ring.

Some Simple Properties of Simple Nil Rings

1966 ◽  
Vol 9 (2) ◽  
pp. 197-200 ◽  
Author(s):  
W. A. McWorter
Keyword(s):  
Unsolved Problem ◽  
Finitely Generated ◽  
Nilpotent Ring ◽  
Locally Nilpotent ◽  
Nil Ring

An outstanding unsolved problem in the theory of rings is the existence or non-existence of a simple nil ring. Such a ring cannot be locally nilpotent as is well known [ 5 ]. Hence, if a simple nil ring were to exist, it would follow that there exists a finitely generated nil ring which is not nilpotent. This seemed an unlikely situation until the appearance of Golod's paper [ 1 ] where a finitely generated, non-nilpotent ring is constructed for any d ≥ 2 generators over any field.

The statistical distribution of the length of a rubber molecule

1948 ◽  
Vol 44 (3) ◽  
pp. 342-344 ◽  
Author(s):  
P. A. P. Moran
Keyword(s):  
Normal Distribution ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Statistical Distribution ◽  
Equal Length ◽  
Fixed Angle ◽  
A Chain ◽  
Image Position ◽  
Three Degrees Of Freedom

A rubber molecule containing n + 1 carbon atoms may be represented by a chain of n links of equal length such that successive links are at a fixed angle to each other but are otherwise at random. The statistical distribution of the length of the molecule, that is, the distance between the first and last carbon atoms, has been considered by various authors (Treloar (1) gives references). In particular, if the first atom is kept fixed at the origin of a system of coordinates and the chain is otherwise at random, it has been conjectured that the distribution of the (n + 1)th atom will tend, as n increases, towards a three-dimensional normal distribution of the formwhere σ depends on n. Thus r2 (= x2 + y2 + z2) will be approximately distributed as σ2χ2 with three degrees of freedom.

A ratio limit theorem for (sub) Markov chains on {1,2, …} with bounded jumps

1995 ◽  
Vol 27 (3) ◽  
pp. 652-691 ◽  
Author(s):  
Harry Kesten
Keyword(s):  
Harmonic Function ◽  
Invariant Measure ◽  
Probability Measure ◽  
Markov Chains ◽  
Positive Matrices ◽  
Ratio Limit Theorems ◽  
Ratio Limit ◽  
A Chain ◽  
Image Position ◽  
Quasi Stationary Distribution

We consider positive matrices Q, indexed by {1,2, …}. Assume that there exists a constant 1 L < ∞ and sequences u1< u2< · ·· and d1d2< · ·· such that Q(i, j) = 0 whenever i < ur < ur + L < j or i > dr + L > dr > j for some r. If Q satisfies some additional uniform irreducibility and aperiodicity assumptions, then for s > 0, Q has at most one positive s-harmonic function and at most one s-invariant measure µ. We use this result to show that if Q is also substochastic, then it has the strong ratio limit property, that is for a suitable R and some R–1-harmonic function f and R–1-invariant measure µ. Under additional conditions µ can be taken as a probability measure on {1,2, …} and exists. An example shows that this limit may fail to exist if Q does not satisfy the restrictions imposed above, even though Q may have a minimal normalized quasi-stationary distribution (i.e. a probability measure µ for which R–1µ = µQ).The results have an immediate interpretation for Markov chains on {0,1,2, …} with 0 as an absorbing state. They give ratio limit theorems for such a chain, conditioned on not yet being absorbed at 0 by time n.

Maximal branching processes and ‘long-range percolation’

1970 ◽  
Vol 7 (01) ◽  
pp. 89-98
Author(s):  
John Lamperti
Keyword(s):  
Probability Distribution Function ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
Transition Matrices ◽  
A Chain ◽  
Image Position ◽  
The Right

In the first part of this paper, we will consider a class of Markov chains on the non-negative integers which resemble the Galton-Watson branching process, but with one major difference. If there are k individuals in the nth “generation”, and are independent random variables representing their respective numbers of offspring, then the (n + 1)th generation will contain max individuals rather than as in the branching case. Equivalently, the transition matrices Pij of the chains we will study are to be of the form where F(.) is the probability distribution function of a non-negative, integervalued random variable. The right-hand side of (1) is thus the probability that the maximum of i independent random variables distributed by F has the value j. Such a chain will be called a “maximal branching process”.

Differential polynomial rings in several variables over locally nilpotent rings

2019 ◽  
Vol 30 (01) ◽  
pp. 117-123 ◽  
Author(s):  
Fei Yu Chen ◽  
Hannah Hagan ◽  
Allison Wang
Keyword(s):  
Polynomial Ring ◽  
Differential Polynomial ◽  
Polynomial Rings ◽  
Nilpotent Ring ◽  
Several Variables ◽  
Locally Nilpotent ◽  
Differential Polynomial Ring

We show that a differential polynomial ring over a locally nilpotent ring in several commuting variables is Behrens radical, extending a result by Chebotar.

On Cantor-Bendixson spectra containing (1,1). II

1985 ◽  
Vol 50 (3) ◽  
pp. 611-618 ◽  
Author(s):  
Annalisa Marcja ◽  
Carlo Toffalori
Keyword(s):  
Boolean Algebras ◽  
Stable Theory ◽  
Definable Subset ◽  
Limit Ordinal ◽  
Image Position ◽  
Countable Models

Let T be a (countable, complete, quantifier eliminable) ω-stable theory; an analysis of T, and consequently a classification of ω-stable theories, can be done by looking at the Boolean algebras B(M) of definable subsets of its countable models M (as usual, we often confuse a definable subset of M with the class of formulas defining it). If M ⊨ T, ∣M∣ = ℵ0, then, for every LM-formula ϕ(v) and for every ordinal α, we define a relation(CB = Cantor-Bendixson, of course) by induction on α:CB-rank ϕ(v) ≥ 0 if ϕ(M) ≠ ∅CB-rank ϕ(v) ≥ λ for λ a limit ordinal, if CB-rank ϕ(v) ≥ for all v < λ;CB-rank ϕ(v)≥ α + 1 if, for all n ∈ ω,(*) there are LM-formulas ϕ0(v), …, ϕn − 1(v) such thatIt is well known that the ω-stability of T implies that, for every consistent LM-formula ϕ(v), there is exactly one ordinal α < ω1 such that CB-rank ϕ(v) ≥ α and CB-rank ϕ(v)≱α + 1. Therefore we define:CB-rank ϕ(v) = αCB-degree ϕ(v) = d if d is the maximal n ∈ ω satisfying (*); andCB-type ϕ(v) = (α, d).

On a Theorem of Herstein

1969 ◽  
Vol 21 ◽  
pp. 1348-1353 ◽  
Author(s):  
M. Chacron
Keyword(s):  
Division Ring ◽  
Monic Polynomial ◽  
Constant Term ◽  
Multiplicative Semigroup ◽  
Simple Ring ◽  
Left Hand ◽  
Ring Of Integers ◽  
Image Position

Throughout this paper, Z is the ring of integers, ƒ*(t) (ƒ(t)) is an integer monic (co-monic) polynomial in the indeterminate t (i.e., each coefficient of ƒ* (ƒ) is in Z and its highest (lowest) coefficient is 1 (5, p. 121, Definition) and M* (M) is the multiplicative semigroup of all integer monic (co-monic) polynomials ƒ* (ƒ) having no constant term. In (3, Theorem 2), Herstein proved that if R is a division ring with centre C such that1then R = C. In this paper we seek a generalization of Herstein's result to semi-simple rings. We also study the following condition:(1)*Our results are quite complete for a semi-simple ring R in which there exists a bound for the codegree ofƒ (ƒ*) (i.e., the degree of the lowest monomial of ƒ(ƒ*)) appearing in the left-hand side of (1) ((1)*).

A Caeretan Hydria in Dunedin

1955 ◽  
Vol 75 ◽  
pp. 1-6
Author(s):  
J. K. Anderson
Keyword(s):  
Good Deal ◽  
The Body ◽  
A Chain ◽  
Image Position

The vase here described was recently presented to the Otago Museum in commemoration of the distinguished services of Dr. H. D. Skinner, for many years Director of the Museum. It was formerly on the Rome market. It is restored from fragments, and missing pieces of the neck, mouth, and shoulder have been replaced by plaster. The joints and plaster restorations have been carefully painted over, and there has been a good deal of repainting where the glaze was worn. On the mouth, neck, and shoulder the restorations, though extensive, merely fill gaps in a well-defined pattern, and can therefore be passed over without a detailed description. The repainting of the figures on the body of the vase will be described at greater length below. The clay is a fine, clear red, rather lighter than the usual colour of Attic. The principal dimensions of the vase are as follows (measurements in metres):The body is ovoid, with high, flat shoulders. It is separated from the wide flaring foot by a low, raised ridge. A similar ridge separates the shoulder from the neck, which is cylindrical with slightly concave sides. The lip flares widely. The side handles are small and slope slightly upward; they are attached just above the widest part of the vase and below the sharpest curve of the shoulder. The vertical handle is divided by three deep, vertical grooves. The inside of the mouth and the upper surface of the foot are ornamented with rounded tongues of black glaze. These were painted alternately red and white, but the paint, which was applied on top of the black glaze, is now much worn. On the lower part of the body are short black rays; above these is a rather wider zone with a chain of five-petalled lotuses linked to five-leaved palmettes.

AN IMAGE ANALYSIS OF MULTIPLE‐LAYER RESISTIVITY PROBLEMS

Geophysics ◽  
1959 ◽  
Vol 24 (3) ◽  
pp. 485-509 ◽  
Author(s):  
Irwin Roman
Keyword(s):  
Recursion Formula ◽  
Test Point ◽  
Reflection Factor ◽  
Primary Sequence ◽  
Method Of Images ◽  
Original Source ◽  
Exterior Regions ◽  
A Chain ◽  
Image Position ◽  
Upper Boundary

The Kelvin method of images is expressible by a transflection at a boundary. The original source is augmented by a supplement and a complement. The supplement contributes to the potential on the same side of the boundary as the source, but it lies at the optical image position of the source in the boundary. The complement lies at the position of the source but contributes to the potential on the opposite side of the boundary. For two or more boundaries, there are two exterior regions and one or more interior regions. For a source in the top layer, a primary sequence starts with a downward transflection and a secondary sequence with an upward transflection. To each primary sequence of transflections there corresponds a secondary sequence with an upward transflection at the upper boundary ahead of it. The exterior images are not transflected again. Successive transflections occur at adjacent boundaries, suggesting a link of two transflections. To a sequence of links, called a chain, there corresponds an associated sequence, obtained by dropping the last transflection. Exterior images follow from interior, associated from chain, and secondary from primary. Thus, only primary, interior, chain images need to be traced. Each potential is the sum of terms of the form m/r where m is the strength of a specific image, r is the distance of that image from the test point, and the sum includes all images contributing to that potential. The addition of each boundary introduces images and potentials that must be added to those existing prior to the introduction, but it does not otherwise alter them. For the three‐boundary problem, the separate image strengths are determined by simple multiplication after a kernel polynomial is calculated. The latter is a finite polynomial in the reflection‐factor at the middle boundary and can be tabulated. For the images of a specific potential and depth group, the strengths satisfy a recursion formula that serves as a check on direct evaluations.

Note on a property of matrices for Lewis and Langford's calculi of propositions

1940 ◽  
Vol 5 (4) ◽  
pp. 150-151 ◽  
Author(s):  
James Dugundji
Keyword(s):  
Characteristic Matrix ◽  
Finite Characteristic ◽  
Class 1 ◽  
A Chain ◽  
Image Position

The object of this note is to show that there is no finite characteristic matrix for any one of Lewis and Langford's systems.Theorem I. There is no finite characteristic matrix for any one of the systems S1–S5.Proof. Let M be a matrix with less than n elements for which all the provable formulas in S1, or S2, or S3, or S4, or S5, are satisfied. Let Fn represent the formulawhere ∑ stands for a ∨-chain, and the pi, are variables in any one of the calculi. Using M, there is always at least one summand in Fn where pi and pk have the same value. Therefore, Fn can always be written in the form (a = a)∨ B, and thus will give, for any B, a “designated” value, since the formula (p = p) ∨ q is provable in any one of the systems S1–S5.Give to any one of the systems S1–S5, the following matrix, due to Henle:1. Elements: all possible classes formed from the integers 1, 2, 3, …, n.2. “Designated” element: the class {1, 2, 3, …,n}.3. Boole-Schröder algebra on the elements.4. ◊N = N (N the null class)◊A = {1, 2, …, n} (A any non-null class).5

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