On prime ends and local connectivity

Let U be a simply connected domain whose complement K = \ U contains more than one point. We show that the impressionof a prime end of U contains at most two points at which K islocally connected. This is achieved by establishing a characterizationof local connectivity of K at a point z₀ U in terms of theprime ends of U whose impressions contain z₀, and then invokinga result of Ursell and Young.     

The structure and distribution of prime ends

Summary In this paper, we study the prime ends of a simply connected, plane domain. We use the metric concept of the separation of subsets of a domain to define the lateral points of a prims end (Section 3). The idea of right and left side-chains of rankn then leads to the concept of right and left lateral points of rankn (n = 1, 2, ..., ) (Section 5). Thus we achieve an analysis which goes beyond Carathéodory's classification (sea Section 2) of the points of a prime end into principal and subsidiary points (Hauptpunkte andNebenpunkte). Each subsidiary point of a prime endP is a lateral point ofP; but even a principal point ofP can be a right and left lateral point of infinite rank ofP.The need for the new concepts and for the extended analysis arises from the following fact: There exist very simple domains with boundary points that are principal points of a prime end but which, at the same time, possess some of the properties of subsidiary points, relative to the same prime end. In other words, a boundary point may not only belong to two or more prime ends; but in each of the prime ends, it may play a multiple role.It is known [2, p. 349] that, in the space of the prime ends of a fixed domain, the set of prime ends having subsidiary points is a set of first category. In Section 3, we show that the analogue of this proposition for lateral points is false; all the prime ends of certain highly laminar domains have lateral points.A theorem of Lindelöf and others ([4, p. 28]; see [2, p. 347] for further references) asserts that if the function maps the unit disk ¦ ¦ < 1 conformally onto a simply connected domain, andL is a Stolz path approaching the point = 1, then the closure of the set (L) contains no subsidiary points of the prime end corresponding to the point = 1. In Section 4, we formulate and prove an analogous theorem for lateral points.In Section 6, we consider a special phenomenon which at first sight might be expected to re exceptional, but which we show can affect a residual set of prime ends. Section 7 deals with a theorem of Frankl.G. Piranian's contribution to this paper was made under Contract DA 20-018-ORD-13585, Office of Ordnance Research, U. S. Army.     

Bifurcations of basins of attraction from the view point of prime ends

In dynamical systems examples are common in which two or more attractors coexist, and in such cases the basin boundary is nonempty and the basins often have fractal basin boundaries. The purpose of this paper is to describe the structure and properties of unbounded basins and their boundaries for two-dimensional diffeomorphisms. Frequently, if not always, there is a periodic saddle on the boundary that is accessible from the basin. Carathéodory and many others developed an approach in which an open set (in our case a basin) is compactified using so-called prime end theory. Under the prime end compactification of the basin, boundary points of the basin (prime ends) can be characterized as either type 1, 2, 3, or 4. In all well-known examples, most points are of type 1. Many two-dimensional basins have a basin cell, that is, a trapping region whose boundary consists of pieces of the stable and unstable manifolds of a well chosen periodic orbit. Then the basin consists of a central body (the basin cell) and a finite number of channels attached to it, and the basin boundary is fractal. We present a result that says {a basin has a basin cell} if and only if {every prime end that is defined by a chain of unbounded regions (in the basin) is a prime end of type 3 and furthermore all other prime ends are of type 1}. We also prove as a parameter is varied, the basin cell for a basin B is created (or destroyed) if and only if either there is a saddle node bifurcation or the basin B has a prime end that is defined by a chain of unbounded regions and is a prime end of either type 2 or type 4.     

On rings whose prime ideals are completely prime

We investigate in this paper rings whose proper ideals are 2-primal. We concentrate on the connections between this condition and related concepts to this condition, and several kinds of π-regularities of rings which satisfy this condition. Moreover, we add counterexamples to the situations that occur naturally in the process of this note.     

Density functions for prime and relatively prime numbers

Letr ^*(x) denote the maximum number of pairwiserelatively prime integers which can exist in an interval (y,y+x] of lengthx, and let ^*(x) denote the maximum number ofprime integers in any interval (y,y+x] whereyx. Throughout this paper we assume the primek-tuples hypothesis. (This hypothesis could be avoided by using an alternative sievetheoretic definition of ^*(x); cf. the beginning of Section 1.) We investigate the differencer ^*(x)—^*(x): that is we ask how many more relatively prime integers can exist on an interval of lengthx than the maximum possible number of prime integers. As a lower bound we obtainr ^*(x)—^*(x)<x ^c for somec>0 (whenx). This improves the previous lower bound of logx. As an upper bound we getr ^*(x)—^*(x)=o[x/(logx)²]. It is known that ^*(x)—(x)>const.[x/(logx)²];.; thus the difference betweenr ^*(x) and ^*(x) is negligible compared to ^*(x)—(x). The results mentioned so far involve the upper bound or maximizing sieve. In Section 2, similar comparisons are made between two types of minimum sieves. One of these is the erasing sieve, which completely eliminates an interval of lengthx; and the other, introduced by Erdös and Selfridge [1], involves a kind of minimax for sets of pairwise relatively prime numbers. Again these two sieving methods produce functions which are found to be closely related.     

On prime,weakly prime ideals in ordered semigroups

Communicated by J. S. Ponizovsk?     

On the least prime primitive root modulo a prime

We derive a conditional formula for the natural density of prime numbers having its least prime primitive root equal to , and compare theoretical results with the numerical evidence.

     

Special prime numbers and discrete logs in finite prime fields

A set of primes involving numbers such as , where and , is defined. An algorithm for computing discrete logs in the finite field of order with is suggested. Its heuristic expected running time is for , where as , , and . At present, the most efficient algorithm for computing discrete logs in the finite field of order for general is Schirokauer's adaptation of the Number Field Sieve. Its heuristic expected running time is for . Using rather than general does not enhance the performance of Schirokauer's algorithm. The definition of the set and the algorithm suggested in this paper are based on a more general congruence than that of the Number Field Sieve. The congruence is related to the resultant of integer polynomials. We also give a number of useful identities for resultants that allow us to specify this congruence for some .

     

Fully prime rings

The structure of rings all of whose ideals are prime is studied and several examples of such rings are constructed.     

n-almost prime submodules

Let R be a commutative ring with identity. A proper submodule N of an R-module M will be called prime [resp. n-almost prime], if for r ∈ R and a ∈ M with ra ∈ N [resp. ra ∈ N \ (N: M) ^n?1 N], either a ∈ N or r ∈ (N: M). In this note we will study the relations between prime, primary and n-almost prime submodules. Among other results it is proved that:

1. If N is an n-almost prime submodule of an R-module M, then N is prime or N = (N: M)N, in case M is finitely generated semisimple, or M is torsion-free with dim R = 1.
2. Every n-almost prime submodule of a torsion-free Noetherian module is primary.
3. Every n-almost prime submodule of a finitely generated torsion-free module over a Dedekind domain is prime.
4. There exists a finitely generated faithful R-module M such that every proper submodule of M is n-almost prime, if and only if R is Von Neumann regular or R is a local ring with the maximal ideal m such that m ² = 0.
5. If I is an n-almost prime ideal of R and F is a flat R-module with IF ≠ F, then IF is an n-almost prime submodule of F.

     

Between ends and fibers

Let Γ be an infinite, locally finite, connected graph with distance function δ. Given a ray P in Γ and a constant C ≥ 1, a vertex‐sequence is said to be regulated by C if, for all n??, never precedes x_n on P, each vertex of P appears at most C times in the sequence, and . R. Halin (Math. Ann., 157, 3 , 125–137) defined two rays to be end‐equivalent if they are joined by infinitely many pairwise‐disjoint paths; the resulting equivalence classes are called ends. More recently H. A. Jung (Graph Structure Theory, Contemporary Mathematics, 147, 6 , 477–484) defined rays P and Q to be b‐equivalent if there exist sequences and VQ regulated by some constant C ≥ 1 such that for all n??; he named the resulting equivalence classes b‐fibers. Let denote the set of nondecreasing functions from into the set of positive real numbers. The relation (called f‐equivalence) generalizes Jung's condition to . As f runs through , uncountably many equivalence relations are produced on the set of rays that are no finer than b‐equivalence while, under specified conditions, are no coarser than end‐equivalence. Indeed, for every Γ there exists an “end‐defining function” that is unbounded and sublinear and such that implies that P and Q are end‐equivalent. Say if there exists a sublinear function such that . The equivalence classes with respect to are called bundles. We pursue the notion of “initially metric” rays in relation to bundles, and show that in any bundle either all or none of its rays are initially metric. Furthermore, initially metric rays in the same bundle are end‐equivalent. In the case that Γ contains translatable rays we give some sufficient conditions for every f‐equivalence class to contain uncountably many g‐equivalence classes (where ). We conclude with a variety of applications to infinite planar graphs. Among these, it is shown that two rays whose union is the boundary of an infinite face of an almost‐transitive planar map are never bundle‐ equivalent. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 125–153, 2007     

On sums of a prime and four prime squares in short intervals

In this paper, we are able to sharpen Hua's classical result by showing that each sufficiently large integer can be written as